Existence and κ-Mittag-Leffler-Ulam-Hyers stability results for implicit coupled (κ,ϑ)-fractional differential systems

Abstract

In this paper, we delve into the analysis of the existence and stability concerning the $\kappa$-Mittag-Leffler-Ulam-Hyers within a particular class of coupled systems related to boundary value problems. These problems entail implicit nonlinear fractional differential equations and $\kappa$-generalized ϑ-Hilfer fractional derivatives. To achieve our objectives, we employ the Mönch fixed point theorem, complemented by the application of the measure of noncompactness technique and an extension of the well-established Gronwall inequality. Furthermore, we include an illustrative example to showcase the practical utility of our results. The significance of our research lies in examining a comprehensive problem involving coupled systems, which serves as a generalization encompassing all the works mentioned in the introduction. It is viewed as a logical extension and continuation within the framework of this continually evolving theory.

Keywords:

• Banach spaces
• coupled systems
• existence
• generalized Gronwall inequality
• generalized Hilfer fractional derivative
• measure of noncompactness
• Mittag-Leffler function
• Ulam-Hyers stability

AMS (MOS) SUBJECT CLASSIFICATIONS:

• 26A33
• 34A12
• 34G20

1. Introduction

Fractional calculus, an intriguing approach that extends the concepts of differentiation and integration beyond integer orders, has attracted considerable interest in both theoretical investigations and real-world applications spanning a wide array of research fields. Its adaptability in tackling complex problems has firmly positioned it as an essential instrument in this domain. The past few years have witnessed a notable upswing in research related to fractional calculus, with scholars delving into a multitude of outcomes across different scenarios and formulations of fractional differential equations and inclusions. For a more comprehensive understanding of the practical applications of fractional calculus, readers are encouraged to consult the works by Herrmann (2011), Hilfer (2000), Kilbas et al. (2006) and Samko et al. (1993). Agrawal (2012) introduced certain generalizations of fractional integrals and derivatives, elucidating some of their fundamental properties. In the studies conducted by Benchohra et al. (2023a, 2023b), they explored existence, uniqueness, and stability outcomes for various problem classes under different conditions. Their approach involved an extension of the well-established Hilfer fractional derivative, unifying the Riemann-Liouville and Caputo fractional derivatives. Recent literature features numerous papers and books wherein authors delve into the existence, stability, and uniqueness of solutions for a diverse array of systems involving fractional differential equations and inclusions. These investigations encompass the utilization of various fractional derivatives and different types of conditions. Readers seeking further references and specific papers may refer to (Almalahi et al., 2021; Bedi et al., 2020, 2021; Derbazi & Baitiche, 2020; Dhaniya et al., 2023; Guida et al., 2020; Kaushik et al., 2023; Lin et al., 2021; Redhwan et al., 2022; Selvam et al., 2020; Wongcharoen et al., 2020), and the citations therein.

In a recent publication (Díaz & Teruel, 2005), Diaz introduced novel definitions for the special functions $\kappa$-gamma and $\kappa$-beta. To delve deeper into this topic, interested readers can refer to additional sources such as (Chu et al., 2020; Mubeen & Habibullah, 2012). Furthermore, in another work (Sousa & de Oliveira, 2018), Sousa et al. presented the ϑ-Hilfer fractional derivative with respect to another function and elucidated some crucial properties related to this type of fractional operators. For further insights and more results based on this operator, we suggest exploring the papers (Afshari et al., 2021; Almalahi & Panchal, 2020; Sousa & Capelas de Oliveira, 2019a, 2019b) and their respective reference lists. Drawing inspiration from the various papers cited earlier, we have introduced a novel extension of the renowned Hilfer fractional derivative (Salim et al., 2022). This new extension, referred to as the $\kappa$-generalized ϑ-Hilfer fractional derivative, has allowed us to establish a generalization of Grönwall’s lemma and explore several types of Ulam stability. Furthermore, we have conducted an in-depth investigation of qualitative and quantitative results for diverse classes of fractional differential problems (Krim et al., 2023; Salim et al., 2021, 2022; Salim, Benchohra, et al., 2023; Salim, Bouriah, et al., 2023), all made possible through the application of this new generalized fractional operator. See (Benchohra et al., 2023a, 2023b), for more details.

In many situations, solving differential equations with precision proves to be a challenging or even insurmountable task. In such cases, alongside nonlinear analysis and optimization, our focus turns to approximating solutions. It’s crucial to emphasize that we only accept solutions that demonstrate stability. Therefore, a variety of methodologies for stability analysis are employed, including well-known techniques like Lyapunov and exponential stability. The notion of stability in functional equations was initially brought to the forefront by mathematician S.M. Ulam during a 1940 lecture at the University of Wisconsin. Ulam posed a fundamental question (Ulam, 1964). The subsequent year, Hyers addressed Ulam’s inquiry for additive functions defined on Banach spaces (Hyers, 1941). Rassias, in 1978, generalized Hyers’ findings and demonstrated the existence of unique linear mappings in close proximity to approximately additive mappings (Rassias, 1978). Ulam-Hyers stability analysis, in contrast to Lyapunov and exponential stability analysis, shifts its focus to understanding how a function behaves when subjected to perturbations rather than examining the stability of a dynamical system or equilibrium point. A number of researchers, as seen in (Ahmed et al., 2021; Benchohra et al., 2023a; Khan et al., 2017; Luo et al. 2020; Shah & Tunç, 2017; Wang et al., 2019; Zada et al., 2020), have explored the Ulam stabilities of fractional differential problems under varying conditions.

In (Ahmad et al., 2020), Ahmed et al. the authors examined the subsequent coupled implicit ϑ-Hilfer fractional differential system: $$\{\begin{array}{c}{}_H{D}_{{\xi }_1{}^{+}}^{\lambda ,r;\vartheta }u(\delta )=\aleph \left(\delta ,u(\delta ){,}_H{D}_{{\xi }_1{}^{+}}^{\lambda ,r;\vartheta }v(\delta )\right),\hspace{1em}\delta \in \nabla =({\xi }_1,{\xi }_2],\hfill \\ {}_H{D}_{{\xi }_1{}^{+}}^{\lambda ,r;\vartheta }v(\delta )=g\left(\delta {,}_H{D}_{{\xi }_1{}^{+}}^{\lambda ,r;\vartheta }u(\delta ),v(\delta )\right),\hspace{1em}\gamma =\lambda +r-\lambda r,\hfill \\ {I_{{\xi }_1{}^{+}}^{1-\gamma ;\vartheta }u(\delta )|}_{\delta ={\xi }_1}=u_{{\xi }_1},{I_{{\xi }_1{}^{+}}^{1-\gamma ;\vartheta }v(\delta )|}_{\delta ={\xi }_1}=v_{{\xi }_1},\hspace{1em}{u}_{{\xi }_1},v_{{\xi }_1}\in \mathbb{R},\hfill \end{array}$$ where ${}_H{D}_{{\xi }_1{}^{+}}^{\lambda ,r;\vartheta }$ represent the ϑ-Hilfer fractional derivative of order λ and type r with $\lambda \in (0,1),r\in (0,1].$ $I_{{\xi }_1{}^{+}}^{1-\gamma ;\vartheta }$ is the ϑ-Riemann-Liouville fractional integral of order $1-\gamma .$ And, $\aleph ,g:\nabla \times \Psi \times \Psi \to \Psi$ are continuous and nonlinear functions on a Banach space $\Psi .$ The linear function $\vartheta :\nabla \to \mathbb{R}$ verifies ${\vartheta }^{\prime }(\delta )\ne 0,\delta \in \nabla .$

In (Abdo et al., 2020), Abdo et al. studied the following system: $$\{\begin{array}{c}{D}_{{\xi }_1{}^{+}}^{{\lambda }_1,{\zeta }_1;\vartheta }\mathfrak{q}(\delta )={\aleph }_1(\delta ,\mathfrak{p}(\delta )),{\xi }_1<\delta \le {\xi }_2,{\xi }_1>0,\hfill \\ D_{{\xi }_1{}^{+}}^{{\lambda }_2,{\zeta }_2;\vartheta }\mathfrak{p}(\delta )={\aleph }_2(\delta ,\mathfrak{q}(\delta )),{\xi }_1<\delta \le {\xi }_2,{\xi }_1>0,\hfill \\ \mathfrak{q}({\xi }_2)=w_1\in \mathbb{R},\hfill \\ \mathfrak{p}({\xi }_2)=w_2\in \mathbb{R},\hfill \end{array}$$ where $0<{\lambda }_{\mathit{ȷ}}<1,0\le {\zeta }_{\mathit{ȷ}}\le 1,D_{{\xi }_1{}^{+}}^{{\lambda }_{\mathit{ȷ}},{\zeta }_{\mathit{ȷ}}\vartheta }(\mathit{ȷ}=1,2)$ is the ϑ-Hilfer fractional derivative of order ${\lambda }_{\mathit{ȷ}}$ and type ${\zeta }_{\mathit{ȷ}}$ with respect to ϑ and $\aleph :({\xi }_1,{\xi }_2]\times \mathbb{R}\to \mathbb{R}$ is a given function.

Motivated by the previously mentioned publications and with the intention of building upon prior achievements, we consider: (1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) (2) $$\{\begin{array}{c}{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,\hfill \\ {\beta }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_1{}^{+})+{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_2)={\beta }_3,\hfill \end{array}$$(2) where for $\mathit{ȷ}=1,2,{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_{\mathit{ȷ}},{\zeta }_{\mathit{ȷ}};\vartheta }$ and ${\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_{\mathit{ȷ}}),\kappa ;\vartheta }$ are, respectively, the $\kappa$-generalized ϑ-Hilfer fractional derivative of order ${\varpi }_{\mathit{ȷ}}\in (0,\kappa )$ and type ${\zeta }_{\mathit{ȷ}}\in [0,1],$ and $\kappa$-generalized ϑ-fractional integral of order $\kappa (1-{\theta }_{\mathit{ȷ}}),$ where ${\theta }_{\mathit{ȷ}}=\frac{1}{\kappa }({\zeta }_{\mathit{ȷ}}(\kappa -{\varpi }_{\mathit{ȷ}})+{\varpi }_{\mathit{ȷ}}),\kappa >0,{\alpha }_3,{\beta }_3\in \mathbb{k},{\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in \mathbb{R}$ such that ${\alpha }_1+{\alpha }_2\ne 0$ and ${\beta }_1+{\beta }_2\ne 0,$ and ${\aleph }_{\mathit{ȷ}}:[{\xi }_1,{\xi }_2]\times {\mathbb{k}}^4\to \mathbb{k}$ are given functions, where $(\mathbb{k},\Vert ·\Vert )$ is a Banach space.

It is worth mentioning that our work is regarded as a logical extension and continuation on the results presented in the aforementioned works, specifically (Abdo et al., 2020; Ahmad et al., 2020). For more details on the particular cases of our problem, see (Benchohra et al., 2023a, 2023b).

The paper is organized as follows: In Section 2, we commence by introducing essential notations and providing an overview of the fundamentals associated with $\kappa$-generalized ϑ-Hilfer, as well as functions such as $\kappa$-Gamma, $\kappa$-Beta, $\kappa$-Mittag-Leffler, and various supporting results. Section 3 presents an existence result for the problem defined in Equations (1)–(2)(2) $$\{\begin{array}{c}{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,\hfill \\ {\beta }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_1{}^{+})+{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_2)={\beta }_3,\hfill \end{array}$$(2) , employing the Mönch fixed point theorem and the measure of noncompactness technique. Additionally, in Section 4, we give the proof of the stability result for the system denoted by Equations (1)–(2)(2) $$\{\begin{array}{c}{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,\hfill \\ {\beta }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_1{}^{+})+{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_2)={\beta }_3,\hfill \end{array}$$(2) . Finally, the last section is dedicated to offering a concrete illustrative example that effectively demonstrates the practical utility of our primary findings.

2. Preliminaries

We will commence by presenting the weighted spaces, notations, definitions, and fundamental concepts that will find application in this investigation. For more comprehensive information regarding the norms and definitions used, refer to the references such as (Salim et al., 2021, 2022), and (Kucche & Mali, 2021).

Let $0<{\xi }_1<{\xi }_2<\infty ,\nabla =[{\xi }_1,{\xi }_2],$ and let $\varpi \in (0,\kappa ),\zeta \in [0,1],\kappa >0$ and $\theta =\frac{1}{\kappa }(\zeta (\kappa -\varpi )+\varpi ).$

By $C(\nabla ,\mathbb{k})$ we denote the Banach space of all continuous functions from $\nabla$ into $\mathbb{k}$ with the norm $$\Vert \mathfrak{p}{\Vert }_{\infty }=\mathrm{\text{sup}}\{\Vert \mathfrak{p}(\delta )\Vert :\delta \in \nabla \}.$$

Let $A{C}^n(\nabla ,\mathbb{k})$ and $C^n(\nabla ,\mathbb{k})$ be the spaces of n-times absolutely continuous and n-times continuously differentiable functions on $\nabla ,$ respectively.

Consider the weighted Banach space $$C_{\theta ;\vartheta }(\nabla )=\left\{\mathfrak{p}:({\xi }_1,{\xi }_2]\to \mathbb{k}:\delta \to {\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)\mathfrak{p}(\delta )\in C(\nabla ,\mathbb{k})\right\},$$ where ${\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)={(\vartheta (\delta )-\vartheta ({\xi }_1))}^{1-\theta },$ with the norm $$\Vert \mathfrak{p}{\Vert }_{C_{\theta ;\vartheta }}=\underset{\delta \in \nabla }{\mathrm{\text{sup}}}\Vert {\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)\mathfrak{p}(\delta )\Vert ,$$ and $$C_{\theta ;\vartheta }^n(\nabla )=\left\{\mathfrak{p}\in C^{n-1}(\nabla ):{\mathfrak{p}}^{(n)}\in C_{\theta ;\vartheta }(\nabla )\right\},n\in \mathbb{N},C_{\theta ;\vartheta }^0(\nabla )=C_{\theta ;\vartheta }(\nabla ),$$ with the norm $$\Vert \mathfrak{p}{\Vert }_{C_{\theta ;\vartheta }^n}=\sum _{\mathit{ȷ}=0}^{n-1}\Vert {\mathfrak{p}}^{(\mathit{ȷ})}{\Vert }_{\infty }+\Vert {\mathfrak{p}}^{(n)}{\Vert }_{C_{\theta ;\vartheta }}.$$

Now, let us consider the Banach space $${\mathcal{F}}_{{\theta }_1,{\theta }_2}:=C_{{\theta }_1;\vartheta }(\nabla )\times C_{{\theta }_2;\vartheta }(\nabla ),$$ with the norm $$\Vert (\mathfrak{p},\mathfrak{q}){\Vert }_{{\mathcal{F}}_{{\theta }_1,{\theta }_2}}=\mathrm{\text{max}}\left\{\Vert \mathfrak{p}{\Vert }_{C_{{\theta }_1;\vartheta }},\Vert \mathfrak{q}{\Vert }_{C_{{\theta }_2;\vartheta }}\right\},$$

where $0<{\theta }_1,{\theta }_2\le 1.$

By $L^1(\nabla ),$ we denote the space of Bochner–integrable functions $\aleph :\nabla \to \mathbb{k}$ with the norm $$\Vert \aleph {\Vert }_1={\int }_{{\xi }_1}^{{\xi }_2}\Vert \aleph (\delta )\Vert d\delta .$$

Definition 2.1

(Díaz and Teruel, 2005). The $\kappa$-gamma function is defined by $${\Gamma }_{\kappa }(\xi )={\int }_0^{\infty }{\delta }^{\xi -1}{e}^{-\frac{{\delta }^{\kappa }}{\kappa }}d\delta ,\xi >0.$$

When $\kappa \to 1$ then $\Gamma (\xi )={\Gamma }_{\kappa }(\xi )$, and some other useful relations are ${\Gamma }_{\kappa }(\xi )={\kappa }^{\frac{\xi }{\kappa }-1}\Gamma \left(\frac{\xi }{\kappa }\right),{\Gamma }_{\kappa }(\xi +\kappa )=\xi {\Gamma }_{\kappa }(\xi )$ and ${\Gamma }_{\kappa }(\kappa )=1$. Moreover, the $\kappa$-beta function is given as $$B_{\kappa }(\xi ,\gamma )=\frac{1}{\kappa }{\int }_0^1{\delta }^{\frac{\xi }{\kappa }-1}{(1-\delta )}^{\frac{\gamma }{\kappa }-1}d\delta ,$$ so that $B_{\kappa }(\xi ,\gamma )=\frac{1}{\kappa }B\left(\frac{\xi }{\kappa },\frac{\gamma }{\kappa }\right)$ and $B_{\kappa }(\xi ,\gamma )=\frac{{\Gamma }_{\kappa }(\xi ){\Gamma }_{\kappa }(\gamma )}{{\Gamma }_{\kappa }(\xi +\gamma )}.$ The Mittag-Leffler function can also be refined into the $\kappa$-Mittag-Leffler function defined as follows $${\mathbb{k}}_{\kappa }^{\varpi ,\zeta }(\mathfrak{p})=\sum _{\mathit{ȷ}=0}^{\infty }\frac{{\mathfrak{p}}^{\mathit{ȷ}}}{{\Gamma }_{\kappa }(\varpi \mathit{ȷ}+\zeta )},\varpi ,\zeta >0,$$

Then, we can have $${\mathbb{k}}_{\kappa }^{\varpi }(\mathfrak{p})={\mathbb{k}}_{\kappa }^{\varpi ,\kappa }(\mathfrak{p})=\sum _{\mathit{ȷ}=0}^{\infty }\frac{{\mathfrak{p}}^{\mathit{ȷ}}}{{\Gamma }_{\kappa }(\varpi \mathit{ȷ}+\kappa )},\varpi >0.$$

Definition 2.2

(Rashid et al. 2020). Let $\vartheta (\delta )>0$ be an increasing function on $({\xi }_1,{\xi }_2]$ and ${\vartheta }^{\prime }(\delta )>0$ be continuous on $({\xi }_1,{\xi }_2)$, and $\varpi >0$. The generalized $\kappa$-fractional integral operators of a function $\varkappa$ of order $\varpi$ is defined by: $$\begin{array}{c}{\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\varkappa (\delta )={\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s){\vartheta }^{\prime }(s)\varkappa (s)\mathit{\text{ds}},\hfill \end{array}$$ with $\kappa >0$ and ${\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s)=\frac{{(\vartheta (\delta )-\vartheta (s))}^{\frac{\varpi }{\kappa }-1}}{\kappa {\Gamma }_{\kappa }(\varpi )}.$

Theorem 2.3

(Salim et al., 2021, 2022). Let $\varkappa \in L^1(\nabla )$ and take $\varpi >0$ and $\kappa >0$. Then ${\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\varkappa \in C([{\xi }_1,{\xi }_2],\mathbb{R}).$

Lemma 2.4

(Salim et al., 2021, 2022). Let $\varpi >0,\zeta >0$ and $\kappa >0$. Then, the semigroup properties that follow are met: $${\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }{\mathcal{J}}_{{\xi }_1+}^{\zeta ,\kappa ;\vartheta }\varkappa (\delta )={\mathcal{J}}_{{\xi }_1+}^{\varpi +\zeta ,\kappa ;\vartheta }\varkappa (\delta )={\mathcal{J}}_{{\xi }_1+}^{\zeta ,\kappa ;\vartheta }{\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\varkappa (\delta ).$$

Lemma 2.5

(Salim et al., 2021, 2022). Let $\varpi ,\zeta >0$ and $\kappa >0$. Then, we obtain $${\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }{\overline{\Phi }}_{\zeta }^{\kappa ,\vartheta }(\delta ,{\xi }_1)={\overline{\Phi }}_{\varpi +\zeta }^{\kappa ,\vartheta }(\delta ,{\xi }_1).$$

Theorem 2.6

(Salim et al., 2021, 2022). Let $0<{\xi }_1<{\xi }_2<\infty ,\varpi ,\zeta >0,0<\theta =\frac{1}{\kappa }(\zeta (\kappa -\varpi )+\varpi )\le 1,\kappa >0$ and $\mathfrak{p}\in C_{\theta ;\vartheta }(\nabla )$. If $\frac{\varpi }{\kappa }>1-\theta ,$ then $$\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1)=\underset{\delta \to {\xi }_1{}^{+}}{\mathrm{\text{lim}}}\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\mathfrak{p}\right)(\delta )=0.$$

Definition 2.7

($\kappa$-generalized ϑ-Hilfer derivative (Salim et al., 2021, 2022). Let $m-1<\frac{\varpi }{\kappa }\le m$ with $m\in \mathbb{N},-\infty \le {\xi }_1<{\xi }_2\le \infty$ and $\varkappa ,\vartheta \in C^m([{\xi }_1,{\xi }_2],\mathbb{R})$ two functions where ϑ is increasing and $\vartheta \prime (\delta )\ne 0$, for all $\delta \in \nabla$. The $\kappa$-generalized ϑ-Hilfer fractional derivatives ${}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }(·)$ of a function $\varkappa$ of order $\varpi$ and type $0\le \zeta \le 1$, with $\kappa >0$ is given by: $$\begin{array}{c}{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\varkappa \left(\delta \right)=\left({\mathcal{J}}_{{\xi }_1+}^{\zeta \left(\mathit{\text{km}}-\varpi \right),\kappa ;\vartheta }{\left(\frac{1}{{\vartheta }^{\prime }\left(\delta \right)}\frac{d}{d\delta }\right)}^m\left({\kappa }^m{\mathcal{J}}_{{\xi }_1+}^{\left(1-\zeta \right)\left(\mathit{\text{km}}-\varpi \right),\kappa ;\vartheta }\varkappa \right)\right)\left(\delta \right)\\ =\left({\mathcal{J}}_{{\xi }_1+}^{\zeta \left(\mathit{\text{km}}-\varpi \right),\kappa ;\vartheta }{\epsilon }_{\vartheta }^m\left({\kappa }^m{\mathcal{J}}_{{\xi }_1+}^{\left(1-\zeta \right)\left(\mathit{\text{km}}-\varpi \right),\kappa ;\vartheta }\varkappa \right)\right)\left(\delta \right),\end{array}$$

where ${\epsilon }_{\vartheta }^m={(\frac{1}{{\vartheta }^{\prime }(\delta )}\frac{d}{d\delta })}^m.$

Lemma 2.8

(Salim et al., 2021, 2022). Let $\delta >{\xi }_1,$ $0<\frac{\varpi }{\kappa }<1,0\le \zeta \le 1,\kappa >0.$ Then for $0<\theta <1;\theta =\frac{1}{\kappa }(\zeta (\kappa -\varpi )+\varpi )$, we have $$\left[{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }{\left({\Phi }_{\theta }^{\vartheta }(s,{\xi }_1)\right)}^{-1}\right](\delta )=0.$$

Theorem 2.9

(Salim et al., 2021, 2022). If $\varkappa \in C_{\theta ;\vartheta }^m[{\xi }_1,{\xi }_2],m-1<\frac{\varpi }{\kappa }<m,0\le \zeta \le 1$, where $m\in \mathbb{N}$ and $\kappa >0$, then $$\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\varkappa \right)(\delta )=\varkappa (\delta )-\sum _{\mathit{ȷ}=1}^m\frac{{\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\theta -\mathit{ȷ}}}{{\kappa }^{\mathit{ȷ}-m}{\Gamma }_{\kappa }(\kappa (\theta -\mathit{ȷ}+1))}\left\{{\epsilon }_{\vartheta }^{m-\mathit{ȷ}}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (m-\theta ),\kappa ;\vartheta }\varkappa ({\xi }_1)\right)\right\},$$ where $$\theta =\frac{1}{\kappa }\left(\zeta (\mathit{\text{km}}-\varpi )+\varpi \right).$$

If m = 1, we have $$\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\varkappa \right)(\delta )=\varkappa (\delta )-\frac{{(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\theta -1}}{{\Gamma }_{\kappa }(\zeta (\kappa -\varpi )+\varpi )}{\mathcal{J}}_{{\xi }_1+}^{(1-\zeta )(\kappa -\varpi ),\kappa ;\vartheta }\varkappa ({\xi }_1).$$

Lemma 2.10

(Salim et al., 2021, 2022). Let $\varpi >0,0\le \zeta \le 1,$ and $\mathfrak{p}\in C_{\theta ;\vartheta }^1(\nabla )$, where $\kappa >0$. Then for $\delta \in ({\xi }_1,{\xi }_2],$ we have $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }{\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\mathfrak{p}\right)(\delta )=\mathfrak{p}(\delta ).$$

Lemma 2.11

(Benchohra et al., 2023a). Let $\varpi >0$ and $\kappa >0$. Then, we have $${\mathcal{J}}_{a+}^{\varpi ,\kappa ;\vartheta }{\mathbb{k}}_{\kappa }^{\varpi }\left({\left(\vartheta (\delta )-\vartheta (a)\right)}^{\frac{\varpi }{\kappa }}\right)={\mathbb{k}}_{\kappa }^{\varpi }\left({\left(\vartheta (\delta )-\vartheta (a)\right)}^{\frac{\varpi }{\kappa }}\right)-1.$$

Theorem 2.12

(Benchohra et al., 2023a). Let $\mathfrak{p},$ $\mathfrak{q}$ be two integrable functions and g continuous, with domain $[{\xi }_1,{\xi }_2].$ Let $\vartheta \in C^1[{\xi }_1,{\xi }_2]$ an increasing function such that ${\vartheta }^{\prime }(\delta )\ne 0,\delta \in [{\xi }_1,{\xi }_2]$ and $\varpi >0$ with $\kappa >0$. Assume that

1. $\mathfrak{p}$ and $\mathfrak{q}$ are nonnegative;

2. w is nonnegative and nondecreasing.

If $$\mathfrak{p}\left(\delta \right)\le \mathfrak{q}\left(\delta \right)+\frac{w\left(\delta \right)}{\kappa }{\int }_{{\xi }_1}^{\delta }{\vartheta }^{\prime }\left(s\right){\left[\vartheta (\delta )-\vartheta (s)\right]}^{\frac{\varpi }{\kappa }-1}\mathfrak{p}\left(s\right)\mathit{\text{ds}},$$ then (3) $$\mathfrak{p}\left(\delta \right)\le \mathfrak{q}\left(\delta \right)+{\int }_{{\xi }_1}^{\delta }\stackrel{\infty }{\sum _{\mathit{ȷ}=1}}\frac{{\left[w\left(\delta \right){\Gamma }_{\kappa }\left(\varpi \right)\right]}^{\mathit{ȷ}}}{\kappa {\Gamma }_{\kappa }\left(\varpi \mathit{ȷ}\right)}{\vartheta }^{\prime }\left(s\right){\left[\vartheta (\delta )-\vartheta (s)\right]}^{\frac{\mathit{ȷ}\varpi }{\kappa }-1}\mathfrak{q}\left(s\right)\mathit{\text{ds}},$$(3) For all $\delta \in \left[{\xi }_1,{\xi }_2\right]$. And if $\mathfrak{q}$ is a nondecreasing function on $\left[{\xi }_1,{\xi }_2\right]$. Then, we have $$\mathfrak{p}\left(\delta \right)\le \mathfrak{q}\left(\delta \right){\mathbb{k}}_{\kappa }^{\varpi }\left(w\left(\delta \right){\Gamma }_{\kappa }\left(\varpi \right){\left(\vartheta \left(\delta \right)-\vartheta \left({\xi }_1\right)\right)}^{\frac{\varpi }{\kappa }}\right).$$

Definition 2.13

(Banas & Goebel, 1980). let $\Psi$ be a Banach space and let ${\Omega }_{\Psi }$ be the family of bounded subsets of $\Psi$. The Kuratowski measure of noncompactness is the map $\mu :{\Omega }_{\Psi }\to [0,\infty )$ defined by $$\mu (Ϝ)=\mathit{\text{inf}}\left\{ϵ>0:Ϝ\subset \underset{j=1}{\overset{m}{\cup }}{Ϝ}_j,\mathit{\text{diam}}({Ϝ}_j)\le ϵ\right\},$$ where $Ϝ\in {\Omega }_{\Psi }$ and μ verifies:

• $\mu (Ϝ)=0\iff \overline{Ϝ}$ is compact ($Ϝ$ is relatively compact).

• $\mu (Ϝ)=\mu (\overline{Ϝ}).$

• ${Ϝ}_1\subset {Ϝ}_2\Rightarrow \mu ({Ϝ}_1)\le \mu ({Ϝ}_2).$

• $\mu ({Ϝ}_1+{Ϝ}_2)\le \mu (B_1)+\mu (B_2).$

• $\mu (cϜ)=\left|c\right|\mu (Ϝ),c\in \mathbb{R}.$

• $\mu (\mathit{\text{conv}}Ϝ)=\mu (Ϝ).$

3. Existence of solutions

Firstly, we provide the following theorem in order to convert our system (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) –(2)(2) $$\{\begin{array}{c}{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,\hfill \\ {\beta }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_1{}^{+})+{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_2)={\beta }_3,\hfill \end{array}$$(2) into a coupled system of fractional integral equations.

Theorem 3.1.

Let $\theta =\frac{\zeta (\kappa -\varpi )+\varpi }{\kappa }$, where $\kappa >0,0<\varpi <\kappa ,0\le \zeta \le 1,$ and let $\phi (·)\in C(\nabla ,\mathbb{k})$. The function $\mathfrak{p}$ verifies: (4) $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )=\phi (\delta ),\delta \in ({\xi }_1,{\xi }_2],$$(4) (5) $${\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,$$(5)

If and only if it verifies the following integral equation: (6) $$\mathfrak{p}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(6) where ${\alpha }_1,{\alpha }_2\in \mathbb{R}$ such that ${\alpha }_1+{\alpha }_2\ne 0$ and ${\alpha }_3\in \mathbb{k}.$

Proof.

Assume that $\mathfrak{p}$ satisfies the equations (4)(4) $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )=\phi (\delta ),\delta \in ({\xi }_1,{\xi }_2],$$(4) -(5). We apply ${\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }(·)$ on both sides of (4)(4) $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )=\phi (\delta ),\delta \in ({\xi }_1,{\xi }_2],$$(4) to obtain $$\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )=\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ),$$ and using Theorem 2.9, we get (7) $$\mathfrak{p}(\delta )=\frac{{\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)}{{\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1){\Gamma }_{\kappa }(\kappa \theta )}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$(7)

Applying ${\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }(·)$ on both sides of (7)(7) $$\mathfrak{p}(\delta )=\frac{{\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)}{{\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1){\Gamma }_{\kappa }(\kappa \theta )}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$(7) , using Lemma 2.4, Lemma 2.5 and taking $\delta ={\xi }_2,$ we have (8) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(8)

Multiplying both sides of (8)(8) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(8) by α₂, we get $${\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_2{\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$

Using condition (5)(5) $${\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,$$(5) , we obtain $${\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3-{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+}).$$

Thus, $${\alpha }_3-{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})={\alpha }_2{\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$ Then, (9) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(9)

Substituting (9)(9) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(9) into (7)(7) $$\mathfrak{p}(\delta )=\frac{{\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}({\xi }_1)}{{\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1){\Gamma }_{\kappa }(\kappa \theta )}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$(7) , we obtain (6)(6) $$\mathfrak{p}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(6) .

Now, we show that if $\mathfrak{p}$ verifies Equation (6)(6) $$\mathfrak{p}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(6) , then it verifies (4)(4) $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )=\phi (\delta ),\delta \in ({\xi }_1,{\xi }_2],$$(4) -(5). We apply ${}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }(·)$ on (6)(6) $$\mathfrak{p}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(6) to get $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )={}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\left(\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}\right)+\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }{\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$

By Lemma 2.8 and Lemma 2.10, we get (4)(4) $$\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{\varpi ,\zeta ;\vartheta }\mathfrak{p}\right)(\delta )=\phi (\delta ),\delta \in ({\xi }_1,{\xi }_2],$$(4) . Now we apply the operator ${\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }(·)$ to Equation (6)(6) $$\mathfrak{p}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(6) , to obtain $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)(\delta )={\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\left(\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa \theta ){\Phi }_{\theta }^{\vartheta }(\delta ,{\xi }_1)}\right)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }{\mathcal{J}}_{{\xi }_1+}^{\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$

Now, using Lemma 2.4 and 2.5, we get (10) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)(\delta )=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$(10)

Using Theorem 2.6 with $\delta \to {\xi }_1,$ we obtain (11) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(11)

Next, taking $\delta ={\xi }_2$ in (10)(10) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)(\delta )=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)(\delta ).$$(10) , we have (12) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(12)

From (11)(11) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(11) and (12)(12) $$\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta ),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)=\frac{{\alpha }_3}{{\alpha }_1+{\alpha }_2}-\frac{{\alpha }_2}{{\alpha }_1+{\alpha }_2}\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2)+\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-\theta )+\varpi ,\kappa ;\vartheta }\phi \right)({\xi }_2).$$(12) , we obtain (5)(5) $${\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,$$(5) . This completes the proof. □

As a consequence of Theorem 3.1, we have the following result.

Lemma 3.2.

Let $\mathit{ȷ}=1,2,{\theta }_{\mathit{ȷ}}=\frac{{\zeta }_{\mathit{ȷ}}(\kappa -{\varpi }_{\mathit{ȷ}})+{\varpi }_{\mathit{ȷ}}}{\kappa }$ where $0<{\varpi }_{\mathit{ȷ}}<\kappa$ and $0\le {\zeta }_{\mathit{ȷ}}\le 1$, and let ${\aleph }_{\mathit{ȷ}}:\nabla \times {\mathbb{k}}^4\to \mathbb{k}$ be continuous functions. Then $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ satisfies the coupled system (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) if and only if $(\mathfrak{p},\mathfrak{q})$ is the fixed point of the operator $\mathfrak{R}:{\mathcal{F}}_{{\theta }_1,{\theta }_2}\to {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ defined by: (13) $$\mathfrak{R}(\mathfrak{p},\mathfrak{q})(\delta )=\left({\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta ),{\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\right),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(13) where ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$ are the operators defined for $\delta \in ({\xi }_1,{\xi }_2]$, as follows: (14) $${\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }{\phi }_1(s)\right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa {\theta }_1){\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }{\phi }_1(s)\right)(\delta ),$$(14) and (15) $${\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )=\frac{{\beta }_3-{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }{\phi }_2(s)\right)({\xi }_2)}{({\beta }_2+{\beta }_2){\Gamma }_{\kappa }(\kappa {\theta }_2){\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }{\phi }_2(s)\right)(\delta ),$$(15) where for $\mathit{ȷ}=1,2,$ ${\phi }_{\mathit{ȷ}}\in C(\nabla ,\mathbb{k})$ satisfy the following system of functional equations: $$\{\begin{array}{c}{\phi }_1(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta )\right),\hfill \\ {\phi }_2(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta )\right).\hfill \end{array}$$

We may employ Theorem 2.3 to easily demonstrate that for $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2},$ we have $\mathfrak{R}(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2},$ where $\mathfrak{R}$ is the operator defined in (13)(13) $$\mathfrak{R}(\mathfrak{p},\mathfrak{q})(\delta )=\left({\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta ),{\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\right),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(13) .

The hypotheses:

• $(C{d}_1)$ The functions ${\aleph }_{\mathit{ȷ}}:\nabla \times {\mathbb{k}}^4\to \mathbb{k};\mathit{ȷ}=1,2,$ are continuous.

• $(C{d}_2)$ There exist constants ${\mho }_{\mathit{ȷ}},{\mathfrak{I}}_{\mathit{ȷ}},{\overline{\mho }}_{\mathit{ȷ}},{\overline{\mathfrak{I}}}_{\mathit{ȷ}}>0$ such that $0<{\overline{\mho }}_1<1,0<{\overline{\mathfrak{I}}}_2<1$ and $$\begin{array}{c}\Vert {\aleph }_1(\delta ,{\mathfrak{p}}_1,{\mathfrak{q}}_1,w_1,z_1)-{\aleph }_1(\delta ,{\mathfrak{p}}_2,{\mathfrak{q}}_2,w_2,z_2)\Vert \\ \le {\mho }_1{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\Vert {\mathfrak{p}}_1-{\mathfrak{p}}_2\Vert +{\mathfrak{I}}_1{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\Vert {\mathfrak{q}}_1-{\mathfrak{q}}_2\Vert +{\overline{\mho }}_1\Vert w_1-w_2\Vert +{\overline{\mathfrak{I}}}_1\Vert z_1-z_2\Vert \end{array}$$ and $$\begin{array}{c}\Vert {\aleph }_2(\delta ,{\mathfrak{p}}_1,{\mathfrak{q}}_1,w_1,z_1)-{\aleph }_2(\delta ,{\mathfrak{p}}_2,{\mathfrak{q}}_2,w_2,z_2)\Vert \\ \le {\mho }_2{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\Vert {\mathfrak{p}}_1-{\mathfrak{p}}_2\Vert +{\mathfrak{I}}_2{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\Vert {\mathfrak{q}}_1-{\mathfrak{q}}_2\Vert +{\overline{\mho }}_2\Vert w_1-w_2\Vert +{\overline{\mathfrak{I}}}_2\Vert z_1-z_2\Vert \end{array}$$ For any ${\mathfrak{p}}_{\mathit{ȷ}},{\mathfrak{q}}_{\mathit{ȷ}},w_{\mathit{ȷ}},z_{\mathit{ȷ}}\in \mathbb{R}$ and $\delta \in ({\xi }_1,{\xi }_2],$ where $\mathit{ȷ}=1,2.$

• $(C{d}_3)$ For each bounded sets Ω and for each $\delta \in \nabla ,$ the following inequalities hold: $$\mu \left({\aleph }_1\left(\delta ,\Omega ,\Omega {,}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\Omega {,}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\Omega \right)\right)\le {\mho }_3{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\mu (\Omega )$$ and $$\mu \left({\aleph }_2\left(\delta ,\Omega ,\Omega {,}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\Omega {,}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\Omega \right)\right)\le {\mho }_4{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\mu (\Omega ),$$ where $${}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_{\mathit{ȷ}},{\zeta }_{\mathit{ȷ}};\vartheta }\Omega =\left\{{}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_{\mathit{ȷ}},{\zeta }_{\mathit{ȷ}};\vartheta }w:w\in \Omega \right\};\mathit{ȷ}=1,2.$$

Set the following: $${\epsilon }_1=\mathrm{\text{max}}\left\{{\mathcal{L}}_1,{\mathcal{L}}_2\right\},{\epsilon }_2=\mathrm{\text{max}}\left\{{\mathcal{L}}_3,{\mathcal{L}}_4\right\}\text{\hspace{0.5em}and\hspace{0.5em}}{\epsilon }_3={\mathcal{L}}_5+{\mathcal{L}}_6,$$ such that $${\mathcal{L}}_1=[\frac{\left|{\alpha }_2\right|{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1){\Gamma }_{\kappa }(2k-\kappa {\theta }_1+{\varpi }_1)}+\frac{{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )}]\times \left[\frac{{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\right],$$ $${\mathcal{L}}_2=[\frac{\left|{\beta }_2\right|{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2){\Gamma }_{\kappa }(2k-\kappa {\theta }_2+{\varpi }_2)}+\frac{{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )}]\hspace{1em}\times \left[\frac{{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\right],$$ $${\mathcal{L}}_3=[\frac{\left|{\alpha }_2\right|{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1){\Gamma }_{\kappa }(2k-\kappa {\theta }_1+{\varpi }_1)}+\frac{{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )}]\times \left[\frac{{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\right],$$ $${\mathcal{L}}_4=[\frac{\left|{\beta }_2\right|{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2){\Gamma }_{\kappa }(2k-\kappa {\theta }_2+{\varpi }_2)}+\frac{{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )}]\times \left[\frac{{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\right],$$ $${\mathcal{L}}_5=[\frac{\left|{\alpha }_2\right|{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1){\Gamma }_{\kappa }(2k-\kappa {\theta }_1+{\varpi }_1)}+\frac{{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )}]\times \left[\frac{{\aleph }^{*}{\overline{\mathfrak{I}}}_1+{\aleph }^{*}(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\right]+\frac{\Vert {\alpha }_3\Vert }{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1)},$$ $${\mathcal{L}}_6=[\frac{\left|{\beta }_2\right|{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2){\Gamma }_{\kappa }(2k-\kappa {\theta }_2+{\varpi }_2)}+\frac{{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )}]\times \left[\frac{{\aleph }^{*}(1-{\overline{\mho }}_1)+{\aleph }^{*}{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\right]+\frac{\Vert {\beta }_3\Vert }{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2)},$$ and $${\aleph }^{*}=\underset{\delta \in \nabla }{\mathrm{\text{sup}}}\Vert \aleph (\delta ,0,0,0,0)\Vert .$$

Theorem 3.3.

Suppose that the hypotheses $(C{d}_1)$-$(C{d}_3)$ hold. If (16) $$\begin{array}{c}\mathcal{l}=\mathrm{\text{max}}\{{\epsilon }_1+{\epsilon }_2,\frac{{\mho }_3{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )},\frac{{\mho }_4{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )},\frac{{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\}<1,\end{array}$$(16)

then problem (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) has at least one solution in ${\mathcal{F}}_{{\theta }_1,{\theta }_2}.$

Proof.

The proof will be given in several steps.

Claim 1: We show that the operator $\mathfrak{R}$ defined in (13)(13) $$\mathfrak{R}(\mathfrak{p},\mathfrak{q})(\delta )=\left({\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta ),{\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\right),\hspace{1em}\delta \in ({\xi }_1,{\xi }_2],$$(13) , transforms the ball $B_{\epsilon }=B(0,\epsilon )=\{(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}:\Vert (\mathfrak{p},\mathfrak{q}){\Vert }_{{\mathcal{F}}_{{\theta }_1,{\theta }_2}}\le \epsilon \}$ into itself. Let $\epsilon$a positive constant such that $$\epsilon \ge \frac{{\epsilon }_3}{1-{\epsilon }_1-{\epsilon }_2}.$$

For each $\delta \in ({\xi }_1,{\xi }_2],$ (14)(14) $${\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }{\phi }_1(s)\right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa {\theta }_1){\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }{\phi }_1(s)\right)(\delta ),$$(14) and (15)(15) $${\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )=\frac{{\beta }_3-{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }{\phi }_2(s)\right)({\xi }_2)}{({\beta }_2+{\beta }_2){\Gamma }_{\kappa }(\kappa {\theta }_2){\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }{\phi }_2(s)\right)(\delta ),$$(15) imply that we have (17) $$\Vert {\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )\Vert \le \frac{\Vert {\alpha }_3\Vert +\left|{\alpha }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)({\xi }_2)}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1){\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)(\delta ),$$(17) and (18) $$\Vert {\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\Vert \le \frac{\Vert {\beta }_3\Vert +\left|{\beta }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)({\xi }_2)}{({\beta }_2+{\beta }_2){\Gamma }_{\kappa }(\kappa {\theta }_2){\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)(\delta ).$$(18) By the hypothesis $(C{d}_2),$ for $\delta \in ({\xi }_1,{\xi }_2],$ we have $$\begin{array}{c}\Vert {\phi }_1(\delta )\Vert =\Vert {\aleph }_1(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta ))-\aleph (\delta ,0,0,0,0)\Vert +\Vert \aleph (\delta ,0,0,0,0)\Vert \\ \le {\mho }_1{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\Vert \mathfrak{p}(\delta )\Vert +{\mathfrak{I}}_1{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\Vert \mathfrak{q}(\delta )\Vert +{\overline{\mho }}_1\Vert {\phi }_1(\delta )\Vert +{\overline{\mathfrak{I}}}_1\Vert {\phi }_2(\delta )\Vert +{\aleph }^{*},\end{array}$$ Which implies that $$\Vert {\phi }_1(\delta )\Vert \le \frac{{\mho }_1}{1-{\overline{\mho }}_1}\Vert \mathfrak{p}{\Vert }_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_1}{1-{\overline{\mho }}_1}\Vert \mathfrak{q}{\Vert }_{C_{{\theta }_2;\vartheta }}+\frac{{\overline{\mathfrak{I}}}_1}{1-{\overline{\mho }}_1}\Vert {\phi }_2(\delta )\Vert +\frac{{\aleph }^{*}}{1-{\overline{\mho }}_1}.$$

Similarly, one can find that $$\Vert {\phi }_2(\delta )\Vert \le \frac{{\mho }_2}{1-{\overline{\mathfrak{I}}}_2}\Vert \mathfrak{p}{\Vert }_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2}{1-{\overline{\mathfrak{I}}}_2}\Vert \mathfrak{q}{\Vert }_{C_{{\theta }_2;\vartheta }}+\frac{{\overline{\mho }}_2}{1-{\overline{\mathfrak{I}}}_2}\Vert {\phi }_1(\delta )\Vert +\frac{{\aleph }^{*}}{1-{\overline{\mathfrak{I}}}_2}.$$

Therefore $$\begin{array}{c}\Vert {\phi }_1(\delta )\Vert \le \frac{{\mho }_1}{1-{\overline{\mho }}_1}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_1}{1-{\overline{\mho }}_1}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}+\frac{{\aleph }^{*}}{1-{\overline{\mho }}_1}+\frac{{\mho }_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}+\frac{{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\Vert {\phi }_1(\delta )\Vert +\frac{{\aleph }^{*}{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)},\end{array}$$

Then $$\Vert {\phi }_1(\delta )\Vert \le \frac{{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}+\frac{{\aleph }^{*}{\overline{\mathfrak{I}}}_1+{\aleph }^{*}(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}.$$

By following the same approach, we can also obtain the following: $$\Vert {\phi }_2(\delta )\Vert \le \frac{{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}+\frac{{\aleph }^{*}(1-{\overline{\mho }}_1)+{\aleph }^{*}{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}.$$

Thus, by the hypothesis $(C{d}_2),$ for $\delta \in ({\xi }_1,{\xi }_2],$ we have $$\Vert {\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )\Vert \le \frac{\Vert {\alpha }_3\Vert +\left|{\alpha }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)({\xi }_2)}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1)}+{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)(\delta ),$$ and $$\Vert {\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\Vert \le \frac{\Vert {\beta }_3\Vert +\left|{\beta }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)({\xi }_2)}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2)}\hspace{1em}+{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)(\delta ),$$ which implies $$\begin{array}{c}\Vert {\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )\Vert \\ \le [\frac{{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}\\ \hspace{1em}+\frac{{\aleph }^{*}{\overline{\mathfrak{I}}}_1+{\aleph }^{*}(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}]\\ \hspace{1em}\times (\frac{\left|{\alpha }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }(1)\right)({\xi }_2)}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1)}+{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }(1)\right)(\delta ))\\ \hspace{1em}+\frac{\Vert {\alpha }_3\Vert }{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1)},\end{array}$$ and $$\begin{array}{c}\Vert {\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\Vert \\ \le [\frac{{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}\\ \hspace{1em}+\frac{{\aleph }^{*}(1-{\overline{\mho }}_1)+{\aleph }^{*}{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}]\\ \hspace{1em}\times \left(\frac{\left|{\beta }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }(1)\right)({\xi }_2)}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2)}+{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }(1)\right)(\delta )\right)\\ \hspace{1em}+\frac{\Vert {\beta }_3\Vert }{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2)}.\end{array}$$

Lemma 2.5 implies $$\begin{array}{c}\Vert {\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )\Vert \\ \le [\frac{{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}\\ \hspace{1em}+\frac{{\aleph }^{*}{\overline{\mathfrak{I}}}_1+{\aleph }^{*}(1-{\overline{\mathfrak{I}}}_2)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}]{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}\\ \hspace{1em}\times [\frac{\left|{\alpha }_2\right|}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1){\Gamma }_{\kappa }(2k-\kappa {\theta }_1+{\varpi }_1)}+\frac{1}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )}]\\ \hspace{1em}+\frac{\Vert {\alpha }_3\Vert }{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1)},\end{array}$$ and $$\begin{array}{c}\Vert {\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\Vert \\ \le [\frac{{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}|{|}_{C_{{\theta }_1;\vartheta }}+\frac{{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}|{|}_{C_{{\theta }_2;\vartheta }}\\ \hspace{1em}+\frac{{\aleph }^{*}(1-{\overline{\mho }}_1)+{\aleph }^{*}{\overline{\mho }}_2}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}]{(\vartheta ({\xi }_2)-\vartheta ({\xi }_1))}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}\\ \hspace{1em}\times [\frac{\left|{\beta }_2\right|}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2){\Gamma }_{\kappa }(2k-\kappa {\theta }_2+{\varpi }_2)}+\frac{1}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )}]\\ \hspace{1em}+\frac{\Vert {\beta }_3\Vert }{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2)}.\end{array}$$

Thus, $$\Vert {\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta ){\Vert }_{C_{{\theta }_1;\vartheta }}\le {\epsilon }_1\Vert \mathfrak{p}{\Vert }_{C_{{\theta }_1;\vartheta }}+{\epsilon }_2\Vert \mathfrak{q}{\Vert }_{C_{{\theta }_2;\vartheta }}+{\mathcal{L}}_5,$$ and $$\Vert {\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta ){\Vert }_{C_{{\theta }_2;\vartheta }}\le {\epsilon }_1\Vert \mathfrak{p}{\Vert }_{C_{{\theta }_1;\vartheta }}+{\epsilon }_2\Vert \mathfrak{q}{\Vert }_{C_{{\theta }_2;\vartheta }}+{\mathcal{L}}_6.$$

Thus, for each $\delta \in ({\xi }_1,{\xi }_2]$ we get $$\Vert \mathfrak{R}(\mathfrak{p},\mathfrak{q}){\Vert }_{{\mathcal{F}}_{{\theta }_1,{\theta }_2}}\le ({\epsilon }_1+{\epsilon }_2)\epsilon +{\epsilon }_3\le \epsilon .$$

Claim 2: The operator $\mathfrak{R}:B_{\epsilon }\to B_{\epsilon }$ is continuous.

Let $\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\}$ be a sequence where $({\mathfrak{p}}_n,{\mathfrak{q}}_n)\to (\mathfrak{p},\mathfrak{q})$ in ${\mathcal{F}}_{{\theta }_1,{\theta }_2}.$ For each $\delta \in ({\xi }_1,{\xi }_2],$ we have $$\begin{array}{c}\Vert {\mathfrak{R}}_1({\mathfrak{p}}_n,{\mathfrak{q}}_n)(\delta )-{\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})(\delta )\Vert \\ \le \frac{\left|{\alpha }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_{1,n}(s)-{\phi }_1(s)\Vert \right)({\xi }_2)}{|{\alpha }_1+{\alpha }_2|{\Gamma }_{\kappa }(\kappa {\theta }_1){\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_{1,n}(s)-{\phi }_1(s)\Vert \right)(\delta ),\end{array}$$ and $$\begin{array}{c}\Vert {\mathfrak{R}}_2({\mathfrak{p}}_n,{\mathfrak{q}}_n)(\delta )-{\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})(\delta )\Vert \\ \le \frac{\left|{\beta }_2\right|\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_{2,n}(s)-{\phi }_2(s)\Vert \right)({\xi }_2)}{|{\beta }_2+{\beta }_2|{\Gamma }_{\kappa }(\kappa {\theta }_2){\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_{2,n}(s)-{\phi }_2(s)\Vert \right)(\delta ),\end{array}$$ where for $\mathit{ȷ}=1,2,$ ${\phi }_{\mathit{ȷ}}$ and ${\phi }_{\mathit{ȷ},n}$ verifies: $$\{\begin{array}{c}{\phi }_1(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta )\right),\hfill \\ {\phi }_2(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta )\right),\hfill \end{array}$$ and $$\{\begin{array}{c}{\phi }_{1,n}(\delta )={\aleph }_1\left(\delta ,{\mathfrak{p}}_n(\delta ),{\mathfrak{q}}_n(\delta ),{\phi }_{1,n}(\delta ),{\phi }_{2,n}(\delta )\right),\hfill \\ {\phi }_{2,n}(\delta )={\aleph }_2\left(\delta ,{\mathfrak{p}}_n(\delta ),{\mathfrak{q}}_n(\delta ),{\phi }_{1,n}(\delta ),{\phi }_{2,n}(\delta )\right).\hfill \end{array}$$

Since $({\mathfrak{p}}_n,{\mathfrak{q}}_n)\to (\mathfrak{p},\mathfrak{q}),$ then $${\phi }_{1,n}(\delta )\to {\phi }_1(\delta )$$ and $${\phi }_{2,n}(\delta )\to {\phi }_2(\delta ),$$ as $n\to \infty$ for each $\delta \in ({\xi }_1,{\xi }_2],$ and since ${\aleph }_1$ and ${\aleph }_2$ are continuous, then we have $$\Vert {\mathfrak{R}}_1({\mathfrak{p}}_n,{\mathfrak{q}}_n)-{\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q}){\Vert }_{C_{{\theta }_1;\vartheta }}\to 0\text{\hspace{0.5em}and\hspace{0.5em}}\Vert {\mathfrak{R}}_2({\mathfrak{p}}_n,{\mathfrak{q}}_n)-{\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q}){\Vert }_{C_{{\theta }_2;\vartheta }}\to 0\text{\hspace{0.5em}as\hspace{0.5em}}n\to \infty .$$

Thus, for each $\delta \in ({\xi }_1,{\xi }_2],$ we get $$\Vert \mathfrak{R}({\mathfrak{p}}_n,{\mathfrak{q}}_n)-\mathfrak{R}(\mathfrak{p},\mathfrak{q}){\Vert }_{{\mathcal{F}}_{{\theta }_1,{\theta }_2}}\to 0\text{\hspace{0.5em}as\hspace{0.5em}}n\to \infty .$$

Consequently, $\mathfrak{R}$ is continuous.

Claim 3: $\mathfrak{R}(B_{\epsilon })$ is bounded and equicontinuous.

Since $\mathfrak{R}(B_{\epsilon })\subset B_{\epsilon }$ and $B_{\epsilon }$ is bounded, then $\mathfrak{R}(B_{\epsilon })$ is bounded.

Let ${\varrho }_1,{\varrho }_2\in ({\xi }_1,{\xi }_2],{\varrho }_1<{\varrho }_2$ and let $(\mathfrak{p},\mathfrak{q})\in B_{\epsilon }.$ Thus $$\begin{array}{c}\Vert {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_1,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})({\varrho }_1)-{\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})({\varrho }_2)\Vert \\ \le \Vert {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_1,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1{}^{+}}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)({\varrho }_1)-{\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1{}^{+}}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)({\varrho }_2)\Vert \\ \le {\int }_{{\xi }_1}^{{\varrho }_1}\Vert {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_1,{\xi }_1){\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }({\varrho }_1,s)-{\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1){\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }({\varrho }_2,s)\Vert \Vert {\vartheta }^{\prime }(s){\phi }_1(s)\Vert \mathit{\text{ds}}\\ \hspace{1em}+\Vert {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1)\left({\mathcal{J}}_{{\varrho }_1^{+}}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)\Vert \right)({\varrho }_2)\Vert ,\end{array}$$ and $$\begin{array}{c}\Vert {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_1,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})({\varrho }_1)-{\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})({\varrho }_2)\Vert \\ \le \Vert {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_1,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1{}^{+}}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)({\varrho }_1)-{\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1{}^{+}}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)({\varrho }_2)\Vert \\ \le {\int }_{{\xi }_1}^{{\varrho }_1}\Vert {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_1,{\xi }_1){\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }({\varrho }_1,s)-{\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1){\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }({\varrho }_2,s)\Vert \Vert {\vartheta }^{\prime }(s){\phi }_2(s)\Vert \mathit{\text{ds}}\\ \hspace{1em}+\Vert {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1)\left({\mathcal{J}}_{{\varrho }_1^{+}}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)\Vert \right)({\varrho }_2)\Vert .\end{array}$$

Lemma 2.5 implies $$\begin{array}{c}\Vert {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_1,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})({\varrho }_1)-{\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1){\mathfrak{R}}_1(\mathfrak{p},\mathfrak{q})({\varrho }_2)\Vert \\ \le \epsilon {\int }_{{\xi }_1}^{{\varrho }_1}\Vert {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_1,{\xi }_1){\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }({\varrho }_1,s)-{\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1){\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }({\varrho }_2,s)\Vert \Vert {\vartheta }^{\prime }(s)\Vert \mathit{\text{ds}}\\ +\frac{\epsilon {\Phi }_{{\theta }_1}^{\vartheta }({\varrho }_2,{\xi }_1){\left(\vartheta ({\varrho }_2)-\vartheta ({\varrho }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )},\end{array}$$ and $$\begin{array}{c}\Vert {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_1,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})({\varrho }_1)-{\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1){\mathfrak{R}}_2(\mathfrak{p},\mathfrak{q})({\varrho }_2)\Vert \\ \le \epsilon {\int }_{{\xi }_1}^{{\varrho }_1}\Vert {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_1,{\xi }_1){\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }({\varrho }_1,s)-{\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1){\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }({\varrho }_2,s)\Vert \Vert {\vartheta }^{\prime }(s)\Vert \mathit{\text{ds}}\\ +\frac{\epsilon {\Phi }_{{\theta }_2}^{\vartheta }({\varrho }_2,{\xi }_1){\left(\vartheta ({\varrho }_2)-\vartheta ({\varrho }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )}.\end{array}$$

As ${\varrho }_1\to {\varrho }_2,$ the right-hand side of the above inequalities tends to zero. This shows the equicontinuity of $\mathfrak{R}(B_{\epsilon }).$

Step 4: The implication of Mönch’s fixed point Theorem (Mönch, 1980) holds.

Now let $\Lambda ={\Lambda }_1\cap {\Lambda }_2$ be an equicontinuous subset of $B_{\epsilon }$ such that ${\Lambda }_{\mathit{ȷ}}\subset \overline{{\mathcal{T}}_{\mathit{ȷ}}({\Lambda }_{\mathit{ȷ}})}\cup \left\{0\right\};\mathit{ȷ}=1,2,$ therefore the function $\delta \to d_{\mathit{ȷ}}(\delta )=\mu ({\Lambda }_{\mathit{ȷ}}(\delta ))$ are continuous on $({\xi }_1,{\xi }_2].$ By the hypotheses $(C{d}_3)$ and the properties of the measure μ, for each $\delta \in ({\xi }_1,{\xi }_2],$ we have $$\begin{array}{cc}{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)d_1(\delta )& \le \mu \left({\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)({\mathfrak{R}}_1{\Lambda }_1)(\delta )\cup \left\{0\right\}\right)\\ & \le \mu \left({\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)({\mathfrak{R}}_1{\Lambda }_1)(\delta )\right)\\ & \le {\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }{\mho }_3{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\mu ({\Lambda }_1(s))\right)(\delta )\\ & \le {\mho }_3\Vert d_1{\Vert }_{C_{{\theta }_1;\vartheta }}\left[\frac{{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )}\right],\end{array}$$ which implies that $$\begin{array}{c}\Vert d_1{\Vert }_{C_{{\theta }_1;\vartheta }}\le \mathcal{l}\Vert d_1{\Vert }_{C_{{\theta }_1;\vartheta }}.\end{array}$$

From (16)(16) $$\begin{array}{c}\mathcal{l}=\mathrm{\text{max}}\{{\epsilon }_1+{\epsilon }_2,\frac{{\mho }_3{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )},\frac{{\mho }_4{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )},\frac{{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\}<1,\end{array}$$(16) , we get $\Vert d_1{\Vert }_{C_{{\theta }_1;\vartheta }}=0,$ that is $d_1\left(\delta \right)=\mu \left({\Lambda }_1\left(\delta \right)\right)=0$, for each $\delta \in \nabla .$ Similarly, we have $$\begin{array}{cc}\Vert d_2{\Vert }_{C_{{\theta }_2;\vartheta }}& \le \frac{{\mho }_4{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )}\Vert d_2{\Vert }_{C_{{\theta }_2;\vartheta }}\\ & \le \mathcal{l}\Vert d_2{\Vert }_{C_{{\theta }_2;\vartheta }},\end{array}$$ that is $d_1(\delta )=\mu ({\Lambda }_1(\delta ))=0.$ Thus, $\mu (\Lambda (\delta ))\le \mu ({\Lambda }_1(\delta ))=0$ and $\mu (\Lambda (\delta ))\le \mu ({\Lambda }_2(\delta ))=0,$ which means that $\Lambda (\delta )$ is relatively compact in $\mathbb{k}\times \mathbb{k}.$ In view of the Ascoli-Arzela Theorem, $\Lambda$ is relatively compact in B_R. Applying now Mönch’s fixed point Theorem (Mönch, 1980), we conclude that $\mathfrak{R}$ has a fixed point, which is a solution to the system (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) –(2)(2) $$\{\begin{array}{c}{\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_2)={\alpha }_3,\hfill \\ {\beta }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_1{}^{+})+{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_2)={\beta }_3,\hfill \end{array}$$(2) .

4. $\kappa$-Mittag-Leffler-Ulam-Hyers stability

Now, we consider the $\kappa$-Mittag-Leffler-Ulam-Hyers stability for system (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2). In (Liu et al., 2019), Liu et al. introduced the concept of Ulam-Hyers-Mittag-Leffler which is suitable to describe the characteristic of fractional Ulam stability, by substituting the Mittag-Leffler function of their definitions with the more refined $\kappa$-Mittag-Leffler function, we give the following definitions. For this, we take inspiration from the following papers (Ali et al., 2017, 2019; Rassias, 1978; Rus, 2009) and the references therein. Let $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2},{ϵ}_1,{ϵ}_2>0.$ We consider the following inequalities: (19) $$\{\begin{array}{c}\Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right),\delta \in \nabla ,\hfill \\ \hfill \\ \Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .\hfill \end{array}$$(19)

Definition 4.1.

System (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) is $\kappa$-Mittag-Leffler-Ulam-Hyers stable with respect to $({\mathbb{k}}_{\kappa }^{{\varpi }_1}({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_1}{\kappa }});{\mathbb{k}}_{\kappa }^{{\varpi }_2}({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_2}{\kappa }}))$ if there exists a real number $a_{{\mathbb{k}}_{\kappa }^{\varpi }}>0$ such that for each $ϵ=\mathrm{\text{max}}\{{ϵ}_1,{ϵ}_2\}>0$ and for each solution $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ of inequality (19)(19) $$\{\begin{array}{c}\Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right),\delta \in \nabla ,\hfill \\ \hfill \\ \Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .\hfill \end{array}$$(19) there exists a solution $(\overline{\mathfrak{p}},\overline{\mathfrak{q}})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ of (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) with $$\Vert (\mathfrak{p},\mathfrak{q})(\delta )-(\overline{\mathfrak{p}},\overline{\mathfrak{q}})(\delta )\Vert \le ϵa_{{\mathbb{k}}_{\kappa }^{\varpi }}\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right],\delta \in \nabla .$$

Definition 4.2.

System (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) is generalized $\kappa$-Mittag-Leffler-Ulam-Hyers stable with respect to $({\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_1}{\kappa }}\right);{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_2}{\kappa }}\right))$ if there exists $v:C([0,\infty ),[0,\infty ))$ with $v(0)=0$ such that for each $ϵ=\mathrm{\text{max}}\{{ϵ}_1,{ϵ}_2\}>0$ and for each solution $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ of inequality (19)(19) $$\{\begin{array}{c}\Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right),\delta \in \nabla ,\hfill \\ \hfill \\ \Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .\hfill \end{array}$$(19) there exists a solution $(\overline{\mathfrak{p}},\overline{\mathfrak{q}})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ of (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) with $$\Vert (\mathfrak{p},\mathfrak{q})(\delta )-(\overline{\mathfrak{p}},\overline{\mathfrak{q}})(\delta )\Vert \le v(ϵ)\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right],\delta \in \nabla .$$

Remark 4.3.

It is clear that: Definition 4.1 $\Rightarrow$ Definition 4.2

Remark 4.4.

A function $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ is a solution of inequality (19)(19) $$\{\begin{array}{c}\Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right),\delta \in \nabla ,\hfill \\ \hfill \\ \Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .\hfill \end{array}$$(19) if and only if there exist ${\upsilon }_{{\aleph }_1},{\upsilon }_{{\aleph }_2}\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ such that

1. $\Vert {\upsilon }_{{\aleph }_1}(\delta )\Vert \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_1}{\kappa }}\right)$ and $\Vert {\upsilon }_{{\aleph }_2}(\delta )\Vert \le {ϵ}_2{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .$

2. $({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p})(\delta )={\aleph }_1(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p})(\delta ),({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q})(\delta ))+{\upsilon }_{{\aleph }_1}(\delta ),$ $\delta \in \nabla .$

3. $({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q})(\delta )={\aleph }_2(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p})(\delta ),({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q})(\delta ))+{\upsilon }_{{\aleph }_2}(\delta ),$ $\delta \in \nabla .$

Theorem 4.5.

Assume that the hypotheses $(C{d}_1)$-$(C{d}_3)$ and the condition (16)(16) $$\begin{array}{c}\mathcal{l}=\mathrm{\text{max}}\{{\epsilon }_1+{\epsilon }_2,\frac{{\mho }_3{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_1+\frac{{\varpi }_1}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_1+\kappa )},\frac{{\mho }_4{\left(\vartheta ({\xi }_2)-\vartheta ({\xi }_1)\right)}^{1-{\theta }_2+\frac{{\varpi }_2}{\kappa }}}{{\Gamma }_{\kappa }({\varpi }_2+\kappa )},\frac{{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\}<1,\end{array}$$(16) hold. Then, system (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) is $\kappa$-Mittag-Leffler-Ulam-Hyers stable with respect to $\chi (\delta )$ and consequently generalized $\kappa$-Mittag-Leffler-Ulam-Hyers stable.

Proof.

Let $(\mathfrak{p},\mathfrak{q})\in {\mathcal{F}}_{{\theta }_1,{\theta }_2}$ be a solution if inequality (19)(19) $$\{\begin{array}{c}\Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right),\delta \in \nabla ,\hfill \\ \hfill \\ \Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .\hfill \end{array}$$(19) , and let us assume that $(\overline{\mathfrak{p}},\overline{\mathfrak{q}})$ is the unique solution of the system $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\overline{\mathfrak{p}}\right)(\delta )={\aleph }_1\left(\delta ,\overline{\mathfrak{p}}(\delta ),\overline{\mathfrak{q}}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\overline{\mathfrak{p}}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\overline{\mathfrak{q}}\right)(\delta )\right),\hfill \\ \hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\overline{\mathfrak{q}}\right)(\delta )={\aleph }_2\left(\delta ,\overline{\mathfrak{p}}(\delta ),\overline{\mathfrak{q}}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\overline{\mathfrak{p}}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\overline{\mathfrak{q}}\right)(\delta )\right),\hfill \\ \hfill \\ {\alpha }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\overline{\mathfrak{p}}\right)({\xi }_1{}^{+})+{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\overline{\mathfrak{p}}\right)({\xi }_2)={\alpha }_3,\hfill \\ \hfill \\ {\beta }_1\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\overline{\mathfrak{q}}\right)({\xi }_1{}^{+})+{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\overline{\mathfrak{q}}\right)({\xi }_2)={\beta }_3,\hfill \\ \hfill \\ \left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\overline{\mathfrak{p}}\right)({\xi }_1{}^{+})=\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1),\kappa ;\vartheta }\mathfrak{p}\right)({\xi }_1{}^{+}),\hfill \\ \hfill \\ \left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\overline{\mathfrak{q}}\right)({\xi }_1{}^{+})=\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2),\kappa ;\vartheta }\mathfrak{q}\right)({\xi }_1{}^{+}).\hfill \end{array}$$

By Theorem 3.1, we obtain for each $\delta \in \nabla$ $$\overline{\mathfrak{p}}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }{\overline{\phi }}_1(s)\right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa {\theta }_1){\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }{\overline{\phi }}_1(s)\right)(\delta ),$$ and $$\overline{\mathfrak{q}}(\delta )=\frac{{\beta }_3-{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }{\overline{\phi }}_2(s)\right)({\xi }_2)}{({\beta }_2+{\beta }_2){\Gamma }_{\kappa }(\kappa {\theta }_2){\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }{\overline{\phi }}_2(s)\right)(\delta ),$$ where for $\mathit{ȷ}=1,2;$ ${\overline{\phi }}_{\mathit{ȷ}}\in C(\nabla ,\mathbb{R})$ satisfy the following system of functional equations: $$\{\begin{array}{c}{\overline{\phi }}_1(\delta )={\aleph }_1\left(\delta ,\overline{\mathfrak{p}}(\delta ),\overline{\mathfrak{q}}(\delta ),{\overline{\phi }}_1(\delta ),{\overline{\phi }}_2(\delta )\right),\hfill \\ {\overline{\phi }}_2(\delta )={\aleph }_2\left(\delta ,\overline{\mathfrak{p}}(\delta ),\overline{\mathfrak{q}}(\delta ),{\overline{\phi }}_1(\delta ),{\overline{\phi }}_2(\delta )\right).\hfill \end{array}$$

Since $(\mathfrak{p},\mathfrak{q})$ is a solution of the inequality (19)(19) $$\{\begin{array}{c}\Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right),\delta \in \nabla ,\hfill \\ \hfill \\ \Vert \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{p}\right)(\delta )-{\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)\Vert \hfill \\ \le {ϵ}_1{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right),\delta \in \nabla .\hfill \end{array}$$(19) , by Remark 4.4, we have (20) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)+{\upsilon }_{{\aleph }_1}(\delta ),\hfill \\ \hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)+{\upsilon }_{{\aleph }_2}(\delta ).\hfill \end{array}$$(20)

Clearly, from (20)(20) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)+{\upsilon }_{{\aleph }_1}(\delta ),\hfill \\ \hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right)+{\upsilon }_{{\aleph }_2}(\delta ).\hfill \end{array}$$(20) we can obtain $$\mathfrak{p}(\delta )=\frac{{\alpha }_3-{\alpha }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_1)+{\varpi }_1,\kappa ;\vartheta }({\phi }_1(s)+{\upsilon }_{{\aleph }_1}(s))\right)({\xi }_2)}{({\alpha }_1+{\alpha }_2){\Gamma }_{\kappa }(\kappa {\theta }_1){\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }({\phi }_1(s)+{\upsilon }_{{\aleph }_1}(s))\right)(\delta ),$$ and $$\mathfrak{q}(\delta )=\frac{{\beta }_3-{\beta }_2\left({\mathcal{J}}_{{\xi }_1+}^{\kappa (1-{\theta }_2)+{\varpi }_2,\kappa ;\vartheta }({\phi }_2(s)+{\upsilon }_{{\aleph }_2}(s))\right)({\xi }_2)}{({\beta }_2+{\beta }_2){\Gamma }_{\kappa }(\kappa {\theta }_2){\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }({\phi }_2(s)+{\upsilon }_{{\aleph }_2}(s))\right)(\delta ),$$ where for $\mathit{ȷ}=1,2;$ ${\phi }_{\mathit{ȷ}}\in C(\nabla ,\mathbb{R})$ satisfy the following system of functional equations: $$\{\begin{array}{c}{\phi }_1(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta )\right),\hfill \\ {\phi }_2(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta )\right).\hfill \end{array}$$

Hence, using Lemma 2.11, for each $\delta \in \nabla$ we have $$\begin{array}{c}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert \le \left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)-{\overline{\phi }}_1(s)\Vert \right)(\delta )+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }{\upsilon }_{{\aleph }_1}\right)(\delta )\\ \le ϵ{\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)-{\overline{\phi }}_1(s)\Vert \right)(\delta )\\ \le ϵ{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\Vert {\phi }_1(s)-{\overline{\phi }}_1(s)\Vert \right)(\delta ),\end{array}$$ and $$\begin{array}{c}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \le \left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)-{\overline{\phi }}_2(s)\Vert \right)(\delta )+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }{\upsilon }_{{\aleph }_2}\right)(\delta )\\ \le ϵ{\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)-{\overline{\phi }}_2(s)\Vert \right)(\delta )\\ \le ϵ{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\Vert {\phi }_2(s)-{\overline{\phi }}_2(s)\Vert \right)(\delta ).\end{array}$$

By the hypothesis $(C{d}_2),$ for $\delta \in ({\xi }_1,{\xi }_2],$ we have $$\begin{array}{c}\Vert {\phi }_1(\delta )-{\overline{\phi }}_1(\delta )\Vert =\Vert {\aleph }_1(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),{\phi }_1(\delta ),{\phi }_2(\delta ))-{\aleph }_1(\delta ,\overline{\mathfrak{p}}(\delta ),\overline{\mathfrak{q}}(\delta ),{\overline{\phi }}_1(\delta ),{\overline{\phi }}_2(\delta ))\Vert \\ \le {\mho }_1{\Phi }_{{\theta }_1}^{\vartheta }(\delta ,{\xi }_1)\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +{\mathfrak{I}}_1{\Phi }_{{\theta }_2}^{\vartheta }(\delta ,{\xi }_1)\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}+{\overline{\mho }}_1\Vert {\phi }_1(\delta )-{\overline{\phi }}_1(\delta )\Vert +{\overline{\mathfrak{I}}}_1\Vert {\phi }_2(\delta )-{\overline{\phi }}_2(\delta )\Vert ,\end{array}$$ which implies that $$\begin{array}{c}\Vert {\phi }_1(\delta )-{\overline{\phi }}_1(\delta )\Vert \le \frac{{\mho }_1{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{1-{\overline{\mho }}_1}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +\frac{{\mathfrak{I}}_1{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{1-{\overline{\mho }}_1}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}+\frac{{\overline{\mathfrak{I}}}_1}{1-{\overline{\mho }}_1}\Vert {\phi }_2(\delta )-{\overline{\phi }}_2(\delta )\Vert .\end{array}$$

Similarly, one can find that $$\begin{array}{c}\Vert {\phi }_2(\delta )-{\overline{\phi }}_2(\delta )\Vert \le \frac{{\mho }_2{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{1-{\overline{\mathfrak{I}}}_2}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +\frac{{\mathfrak{I}}_2{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{1-{\overline{\mathfrak{I}}}_2}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}+\frac{{\overline{\mho }}_2}{1-{\overline{\mathfrak{I}}}_2}\Vert {\phi }_1(\delta )-{\overline{\phi }}_1(\delta )\Vert .\end{array}$$

Therefore, $$\begin{array}{c}\Vert {\phi }_1(\delta )-{\overline{\phi }}_1(\delta )\Vert \le \frac{{\mho }_1{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{1-{\overline{\mho }}_1}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +\frac{{\mathfrak{I}}_1{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{1-{\overline{\mho }}_1}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}+\frac{{\mho }_2{\overline{\mathfrak{I}}}_1{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert \\ \hspace{1em}+\frac{{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert +\frac{{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)}\Vert {\phi }_1(\delta )\Vert .\end{array}$$

Then, $$\begin{array}{c}\Vert {\phi }_1(\delta )-{\overline{\phi }}_1(\delta )\Vert \le \frac{[{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert \\ \hspace{1em}+\frac{[{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert .\end{array}$$

Also, $$\begin{array}{c}\Vert {\phi }_2(\delta )-{\overline{\phi }}_2(\delta )\Vert \le \frac{[{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert \\ \hspace{1em}+\frac{[{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert .\end{array}$$

Thus, using Lemma 2.5, for each $\delta \in \nabla$ we have $$\begin{array}{c}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert \le ϵ{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)\\ \hspace{1em}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\frac{[{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}(s)-\overline{\mathfrak{p}}(s)\Vert \right)(\delta )\\ \hspace{1em}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_1,\kappa ;\vartheta }\frac{[{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}(s)-\overline{\mathfrak{q}}(s)\Vert \right)(\delta ),\end{array}$$ and $$\begin{array}{c}\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \le ϵ{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\\ \hspace{1em}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\frac{[{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}(s)-\overline{\mathfrak{p}}(s)\Vert \right)(\delta )\\ \hspace{1em}+\left({\mathcal{J}}_{{\xi }_1+}^{{\varpi }_2,\kappa ;\vartheta }\frac{[{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}(s)-\overline{\mathfrak{q}}(s)\Vert \right)(\delta ).\end{array}$$

Therefore, $$\begin{array}{c}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}\le ϵ\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right]\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }(\delta ,s){\vartheta }^{\prime }(s)\left(\frac{[{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}(s)-\overline{\mathfrak{p}}(s)\Vert \right)\mathit{\text{ds}}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }(\delta ,s){\vartheta }^{\prime }(s)\left(\frac{[{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}(s)-\overline{\mathfrak{q}}(s)\Vert \right)\mathit{\text{ds}}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }(\delta ,s){\vartheta }^{\prime }(s)\left(\frac{[{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}(s)-\overline{\mathfrak{p}}(s)\Vert \right)\mathit{\text{ds}}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }(\delta ,s){\vartheta }^{\prime }(s)\left(\frac{[{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}(s)-\overline{\mathfrak{q}}(s)\Vert \right)\mathit{\text{ds}}.\end{array}$$

For some constants $\varpi \in (0,\kappa ),\kappa >0$ and $\wp >0,$ we have $$\begin{array}{c}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}\le ϵ\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right]\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s)\frac{{\vartheta }^{\prime }(s)}{\wp }\left(\frac{[{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{p}(s)-\overline{\mathfrak{p}}(s)\Vert \right)\mathit{\text{ds}}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s)\frac{{\vartheta }^{\prime }(s)}{\wp }\left(\frac{[{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}\Vert \mathfrak{q}(s)-\overline{\mathfrak{q}}(s)\Vert \right)\mathit{\text{ds}}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s)\frac{{\vartheta }^{\prime }(s)}{\wp }\left(\frac{[{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{p}(s)-\overline{\mathfrak{p}}(s)\Vert \right)\mathit{\text{ds}}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s)\frac{{\vartheta }^{\prime }(s)}{\wp }\left(\frac{[{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}\Vert \mathfrak{q}(s)-\overline{\mathfrak{q}}(s)\Vert \right)\mathit{\text{ds}},\end{array}$$ where $${\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s)\ge \wp \mathrm{\text{max}}\{{\overline{\Phi }}_{{\varpi }_1}^{\kappa ,\vartheta }(\delta ,s),{\overline{\Phi }}_{{\varpi }_2}^{\kappa ,\vartheta }(\delta ,s)\},\hspace{1em}\text{for\hspace{0.5em}all\hspace{0.5em}}(\delta ,s)\in \nabla \times [{\xi }_1,\delta ].$$

Thus, $$\begin{array}{c}\Vert \mathfrak{p}(\delta )-\overline{\mathfrak{p}}(\delta )\Vert +\Vert \mathfrak{q}(\delta )-\overline{\mathfrak{q}}(\delta )\Vert \\ \hspace{1em}\le ϵ\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right]\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\mathcal{M}{\int }_{{\xi }_1}^{\delta }{\overline{\Phi }}_{\varpi }^{\kappa ,\vartheta }(\delta ,s){\vartheta }^{\prime }(s)\left(\Vert \mathfrak{p}(s)-\overline{s}(\delta )\Vert +\Vert \mathfrak{q}(s)-\overline{s}(\delta )\Vert \right)\mathit{\text{ds}},\end{array}$$ where $$\mathcal{M}=\mathrm{\text{max}}\{{\mathcal{M}}_1,{\mathcal{M}}_2\},$$ $${\mathcal{M}}_1=\frac{1}{\wp }[\frac{[{\mho }_2{\overline{\mathfrak{I}}}_1+{\mho }_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}+\frac{[{\mho }_2(1-{\overline{\mho }}_1)+{\mho }_1{\overline{\mho }}_2]{\Phi }_{{\theta }_1}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}],$$ and $${\mathcal{M}}_2=\frac{1}{\wp }[\frac{[{\mathfrak{I}}_2{\overline{\mathfrak{I}}}_1+{\mathfrak{I}}_1(1-{\overline{\mathfrak{I}}}_2)]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mathfrak{I}}}_2)(1-{\overline{\mho }}_1)-{\overline{\mho }}_2{\overline{\mathfrak{I}}}_1}+\frac{[{\mathfrak{I}}_2(1-{\overline{\mho }}_1)+{\mathfrak{I}}_1{\overline{\mho }}_2]{\Phi }_{{\theta }_2}^{\vartheta }({\xi }_2,{\xi }_1)}{(1-{\overline{\mho }}_1)(1-{\overline{\mathfrak{I}}}_2)-{\overline{\mathfrak{I}}}_1{\overline{\mho }}_2}].$$

By applying Theorem 2.12, we obtain $$\begin{array}{c}\Vert (\mathfrak{p},\mathfrak{q})(\delta )-(\overline{\mathfrak{p}},\overline{\mathfrak{q}})(\delta )\Vert \le ϵ\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right]\\ \hspace{1em}+{\int }_a^{\delta }\stackrel{\infty }{\sum _{\mathit{ȷ}=1}}\frac{{\mathcal{M}}^{\mathit{ȷ}}}{\kappa {\Gamma }_{\kappa }\left(\varpi \mathit{ȷ}\right)}{\vartheta }^{\prime }\left(s\right){\left[\vartheta (\delta )-\vartheta (s)\right]}^{\frac{\mathit{ȷ}\varpi }{\kappa }-1}\\ \hspace{1em}\times \left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (s)-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (s)-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right]\mathit{\text{ds}}\\ \le ϵ\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right]\\ \hspace{1em}\times {\mathbb{k}}_{\kappa }^{\varpi }\left(\mathcal{M}{\left(\vartheta \left(\delta \right)-\vartheta \left({\xi }_1\right)\right)}^{\frac{\varpi }{\kappa }}\right).\end{array}$$

Then, we conclude that for each $\delta \in \nabla ,$ we have $$\Vert (\mathfrak{p},\mathfrak{q})(\delta )-(\overline{\mathfrak{p}},\overline{\mathfrak{q}})(\delta )\Vert \le ϵa_{{\mathbb{k}}_{\kappa }^{\varpi }}\left[{\mathbb{k}}_{\kappa }^{{\varpi }_1}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_1}{\kappa }}\right)+{\mathbb{k}}_{\kappa }^{{\varpi }_2}\left({\left(\vartheta (\delta )-\vartheta ({\xi }_1)\right)}^{\frac{{\varpi }_2}{\kappa }}\right)\right],$$ where $$a_{{\mathbb{k}}_{\kappa }^{\varpi }}={\mathbb{k}}_{\kappa }^{\varpi }\left(\mathcal{M}{\left(\vartheta \left({\xi }_2\right)-\vartheta \left({\xi }_1\right)\right)}^{\frac{\varpi }{\kappa }}\right).$$

Hence, the system (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) is $\kappa$-Mittag-Leffler-Ulam-Hyers stable with respect to $({\mathbb{k}}_{\kappa }^{{\varpi }_1}({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_1}{\kappa }});{\mathbb{k}}_{\kappa }^{{\varpi }_2}({(\vartheta (\delta )-\vartheta ({\xi }_1))}^{\frac{{\varpi }_2}{\kappa }})).$ If we set $v(ϵ)=a_{{\mathbb{k}}_{\kappa }^{\varpi }}ϵ,$ then the problem (1)(1) $$\{\begin{array}{c}\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \\ \left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_1,{\zeta }_1;\vartheta }\mathfrak{p}\right)(\delta ),\left({}_{\kappa }^H{\mathcal{D}}_{{\xi }_1+}^{{\varpi }_2,{\zeta }_2;\vartheta }\mathfrak{q}\right)(\delta )\right),\delta \in ({\xi }_1,{\xi }_2],\hfill \end{array}$$(1) -(2) is also generalized $\kappa$-Mittag-Leffler-Ulam-Hyers stable. □

5. An example

Let $$\mathbb{k}=c_0=\left\{v=(v_1,v_2,\dots ,v_n,\dots ),v_n\to 0\text{\hspace{0.5em}as\hspace{0.5em}}n\to \infty \right\}$$ be the Banach space of real sequences converging to zero with $$\Vert v\Vert =\underset{n\ge 1}{\mathrm{\text{sup}}}\left|v_n\right|.$$

Suppose that $\nabla =[1,2],{\theta }_{\mathit{ȷ}}=\frac{1}{\kappa }({\zeta }_{\mathit{ȷ}}(\kappa -{\varpi }_{\mathit{ȷ}})+{\varpi }_{\mathit{ȷ}});\mathit{ȷ}=1,2.$

Example 5.1.

Taking ${\zeta }_1\to 1,{\zeta }_2\to 0,{\varpi }_1={\varpi }_2=\frac{1}{2},\kappa =1,\vartheta (\delta )=\delta ,{\alpha }_1={\beta }_2=1,{\alpha }_2={\beta }_1=0,{\alpha }_3={\beta }_3=(0,0,\dots ,0,\dots ),{\theta }_1=1$ and ${\theta }_2=\frac{1}{2},\delta \in (1,2]$, we obtain (21) $$\{\begin{array}{c}\left({}_1^H{\mathcal{D}}_{1+}^{\frac{1}{2},1;\vartheta }\mathfrak{p}\right)(\delta )=\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta ),\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )\right),\hfill \\ \left({}_1^H{\mathcal{D}}_{1+}^{\frac{1}{2},0;\vartheta }\mathfrak{q}\right)(\delta )=\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta ),\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )\right),\hfill \end{array}$$(21) (22) $$\{\begin{array}{c}\mathfrak{p}(1)=(0,0,\dots ,0,\dots ),\hfill \\ \left({\mathcal{J}}_{1+}^{\frac{1}{2},1;\vartheta }\mathfrak{q}\right)(e)=(0,0,\dots ,0,\dots ),\hfill \end{array}$$(22) where $$\mathfrak{p}=({\mathfrak{p}}_1,{\mathfrak{p}}_2,\dots ,{\mathfrak{p}}_n,\dots ),\mathfrak{q}=({\mathfrak{q}}_1,{\mathfrak{q}}_2,\dots ,{\mathfrak{q}}_n,\dots ),$$ $${\aleph }_1=({\aleph }_{1,1},{\aleph }_{1,2},\dots ,{\aleph }_{1,n},\dots ),{\aleph }_2=({\aleph }_{2,1},{\aleph }_{2,2},\dots ,{\aleph }_{2,n},\dots ),$$ $${}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}=\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{p}}_1{,}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{p}}_2,\dots {,}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{p}}_n,\dots \right),$$ $${}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}=\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{q}}_1{,}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{q}}_2,\dots {,}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{q}}_n,\dots \right),$$ $$\begin{array}{c}{\aleph }_{1,n}\left(\delta ,{\mathfrak{p}}_n(\delta ),{\mathfrak{q}}_n(\delta ),\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{p}}_n\right)(\delta ),\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{q}}_n\right)(\delta )\right)\\ =\frac{\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\delta )\left(1+|{\mathfrak{p}}_n(\delta )|+\sqrt{\delta -1}|{\mathfrak{q}}_n(\delta )|\right)}{373e^{\delta +5}\left(1+\Vert \mathfrak{p}(\delta )\Vert +\Vert \mathfrak{q}(\delta )\Vert +\Vert \left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta )\Vert +\Vert \left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )\Vert \right)},\delta \in (1,2],\end{array}$$ and $$\begin{array}{c}{\aleph }_{2,n}\left(\delta ,{\mathfrak{p}}_n(\delta ),{\mathfrak{q}}_n(\delta ),\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{p}}_n\right)(\delta ),\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}{\mathfrak{q}}_n\right)(\delta )\right)\\ =\frac{\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}(\delta )\sqrt{\delta -1}\left(1+|{\mathfrak{p}}_n(\delta )|+|{\mathfrak{q}}_n(\delta )|\right)}{73e^{\delta +2}\left(1+\Vert \mathfrak{p}(\delta )\Vert +\Vert \mathfrak{q}(\delta )\Vert +\Vert \left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta )\Vert +\Vert \left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )\Vert \right)},\delta \in (1,2],\end{array}$$ we have $$C_{{\theta }_1,\kappa ;\vartheta }(\nabla )=C_{1,1;\vartheta }(\nabla )=C(\nabla ,\mathbb{R}),$$ and $$C_{{\theta }_2,\kappa ;\vartheta }(\nabla )=C_{\frac{1}{2},1;\vartheta }(\nabla )=\left\{\mathfrak{p}:(1,2]\to \mathbb{R}:\mathfrak{p}\sqrt{\delta -1}\in C(\nabla ,\mathbb{R})\right\}.$$

Since it is clear to see that the functions ${\aleph }_1$ and ${\aleph }_2$ is continuous, then $(C{d}_1)$ is verified.

Further, for each ${\mathfrak{p}}_1,{\mathfrak{q}}_1,{\mathfrak{p}}_2,{\mathfrak{q}}_2,w_1,w_2,z_1,z_2\in \mathbb{k}$ and $\delta \in \nabla ,$ we have $$\begin{array}{c}\Vert {\aleph }_1(\delta ,{\mathfrak{p}}_1,{\mathfrak{q}}_1,w_1,z_1)-{\aleph }_1(\delta ,{\mathfrak{p}}_2,{\mathfrak{q}}_2,w_2,z_2)\Vert \\ \le \frac{\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\delta )}{373e^{\delta +5}}|{\mathfrak{p}}_1-{\mathfrak{p}}_2|+\frac{\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\delta )\sqrt{\delta -1}}{373e^{\delta +5}}|{\mathfrak{q}}_1-{\mathfrak{q}}_2|+\frac{\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\delta )}{373e^{\delta +5}}\left(|w_1-w_2|+|z_1-z_2|\right),\end{array}$$ and $$\begin{array}{c}|{\aleph }_2(\delta ,{\mathfrak{p}}_1,{\mathfrak{q}}_1,w_1,z_1)-{\aleph }_2(\delta ,{\mathfrak{p}}_2,{\mathfrak{q}}_2,w_2,z_2)|\\ \le \frac{\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}(\delta )\sqrt{\delta -1}}{73e^{\delta +2}}\left(|{\mathfrak{p}}_1-{\mathfrak{p}}_2|+|{\mathfrak{q}}_1-{\mathfrak{q}}_2|+|w_1-w_2|+|z_1-z_2|\right),\end{array}$$ thus, the condition $(C{d}_2)$ is satisfied with $${\mho }_1={\mathfrak{I}}_1={\overline{\mho }}_1={\overline{\mathfrak{I}}}_1=\frac{1}{373e^6},$$ and $${\mho }_2={\mathfrak{I}}_2={\overline{\mho }}_2={\overline{\mathfrak{I}}}_2=\frac{1}{73e^3}.$$

The hypothesis $(C{d}_3)$ is easily verified by taking $${\mho }_3=\frac{1}{373e^6},\mathit{\text{ and }}{\mho }_4=\frac{1}{73e^3}.$$

Also $$\mathcal{l}=\frac{1}{(373e^6-1)(73e^3-1)}<1.$$

As all the conditions of Theorem 3.3 are satisfied, then the problem (21)(21) $$\{\begin{array}{c}\left({}_1^H{\mathcal{D}}_{1+}^{\frac{1}{2},1;\vartheta }\mathfrak{p}\right)(\delta )=\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta )={\aleph }_1\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta ),\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )\right),\hfill \\ \left({}_1^H{\mathcal{D}}_{1+}^{\frac{1}{2},0;\vartheta }\mathfrak{q}\right)(\delta )=\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )={\aleph }_2\left(\delta ,\mathfrak{p}(\delta ),\mathfrak{q}(\delta ),\left({}^C{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{p}\right)(\delta ),\left({}^{\mathit{\text{RL}}}{\mathbb{D}}_{1^{+}}^{\frac{1}{2}}\mathfrak{q}\right)(\delta )\right),\hfill \end{array}$$(21) –(22)(22) $$\{\begin{array}{c}\mathfrak{p}(1)=(0,0,\dots ,0,\dots ),\hfill \\ \left({\mathcal{J}}_{1+}^{\frac{1}{2},1;\vartheta }\mathfrak{q}\right)(e)=(0,0,\dots ,0,\dots ),\hfill \end{array}$$(22) has at least one solution in $C_{1;\vartheta }([1,2])\times C_{\frac{1}{2};\vartheta }([1,2]),$ and is $\kappa$-Mittag-Leffler-Ulam-Hyers stable with respect to $({\mathbb{k}}_1^{\frac{1}{2}}\left(\sqrt{\delta -1}\right);{\mathbb{k}}_1^{\frac{1}{2}}\left(\sqrt{\delta -1}\right)).$

6. Conclusion

In this research endeavor, our aim is to establish the existence of solutions for coupled systems governed by the $(\kappa ,\vartheta )$-Hilfer fractional equations. These problems encompass nonlinear implicit fractional differential equations, complete with boundary conditions. Our method for demonstrating the existence of solutions relies on the application of Mönch’s fixed point theorem, complemented by the utilization of the measure of noncompactness technique. Furthermore, we explore the $\kappa$-Mittag-Leffler-Ulam-Hyers stability of our problem, facilitated by a generalized Gronwall inequality. To showcase the practical utility of our key findings and to illustrate that the conditions of our theorems can be met, we provide several illustrative examples. Significantly, the fractional operator under investigation serves as an extension, encompassing previously established fractional derivatives like the ϑ-Hilfer fractional derivative already well-documented in the literature. This broader framework enriches the ever-evolving field of fractional calculus, opening up promising avenues for future exploration and advancement in this dynamic and continually evolving domain.

Author’s contributions

The study was carried out in collaboration of all authors. All authors read and approved the final manuscript.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Acknowledgement

J. Alzabut would like to thank Prince Sultan University and OSTIM Technical University for their endless support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.