Analysis of Caputo-Hadamard fractional neutral delay differential equations involving Hadamard integral and unbounded delays: Existence and uniqueness

ABSTRACT

In this article, we considered the nonlinear Caputo-Hadamard fractional differential equations involving Hadamard integrals and unbounded delays. We employed some standard fixed-point theorems to establish the sufficient conditions for the existence and uniqueness of solutions of the problem. The uniqueness is guaranteed by the Boyd and Wong fixed-point theorems and the Banach fixed-point theorem (BFT), while the existence result is ensured by the Leray-Schauder nonlinear alternative fixed-point theorem by utilizing a generalized Gronwall inequality (GI), which is closely related to the Hadamard derivative, the Leray-Schauder nonlinear alternative fixed-point theorem (LSFT) establishes an apriori bounds. Moreover, a new kind of continuous nondecreasing function is employed by the Boyd and Wong fixed-point theorem to transform the operator of the problem into a nonlinear contraction and produce a unique solution. Continous dependency of solutions on initial conditions (ICs) is ensured via Grownwall inequality as well. We also provide examples to support the main findings we established.

KEYWORDS:

• Caputo-Hadamard fractional derivative
• Hadamard integral
• neutral delay differential equation
• existence and uniqueness
• fixed point theorems

1. Introduction

Fractional calculus is an extension of ordinary differential and integral calculus that investigates the possibility of computing derivatives and integrals of any real (or complex) order. This theory has proven to be a valuable tool in modeling many phenomena, such as an inextensible pendulum with fractional damping terms (Yin et al., 2007), cancer treatment by radiotherapy (Awadalla et al., 2019), continuum and statistical mechanics (Mainardi, 2012) and etc.

Fractional calculus is defined differently by different scholars. In those definitions, one fractional integral operator that is included is the Hadamard fractional integral operator, which was initially suggested by Hadamard in 1892 (Hadamard, 1892). It has various practical applications, such as fractional thermoelasticity (El-Karamany & Ezzat, 2011), physical phenomena in fluctuating environments (Tariboon et al., 2014) and probability (Garra et al., 2018). However, the Hadamard fractional derivative does not allow physically interpretable initial conditions with integer order derivative and the Hadamard derivative of a constant is not zero as well. In recent years, another derivative was proposed by modifying the Hadamard derivative with the Caputo one, known as Caputo-Hadamard derivative (Gambo et al., 2014). It is obtained from the Hadamard derivative by changing the order of its differentiation and integration so that the derivatives of a constant is 0 and these derivatives contain physically interpretable initial conditions similar to the ones in Caputo fractional derivatives. There were a few fractional models and problems generated when this operator was used, see (Awadalla et al., 2022; Dhaniya et al., 2023; Saeed, 2023) and the references therein (M. Cai et al., 2022; R. Cai et al., 2019) adopt the Caputo-Hadamard derivative to model the epidemic of COVID-19 caused by Omicron variant and the regional gradient controllability for ultra-slow diffusion processes.

On the other hand, differential equations with delays are effective tools in modeling real-world problems such as population dynamics, epidemiology, immunology, physiology, and neural networks, see (Baker et al., 1998; Bocharov & Rihan, 2000; Lakshmanan et al., 2014; Rakkiyappan et al., 2016; Rihan et al., 2021) and the references therein. In such equations, the rate of change at time t depends on the system’s history over a period of time $[\mathit{t}-\mathit{s},\mathit{s}],$ where s > 0 is a delay, in addition to the system’s current state. Mathematically, these models are characterized by functional differential equations with delay r. Equations with unbounded or infinite delays can be generated by the more general case $\mathit{r}=\mathrm{\infty }.$ Applications of systems with unbounded delays can be found in many fields such as networked control, biology, mechanics, and social science. For details, see (Atay, 2003; Chang et al., 2008; Culshaw et al., 2003; Dabas & Chauhan, 2013; Djema et al., 2018; Fridman, 2014; Gopalsamy & He, 1994; Jessop & Campbell, 2010; Josić et al., 2011; Kuang & Smith, 1993; Michiels et al., 2009; Roesch & Roth, 2005; Sipahi et al., 2008) and the references therein.

Moreover, delay terms could show up in the derivatives, also known as neutral delay differential equations, used in the mathematical modeling of a number of phenomena, such as electric networks with lossless transmission lines (those found in high-speed computers) (Kilbas, 2006), the force of sliding friction in connected pairs (Kyrychko & Hogan, 2010), Ociliating theory (Baculíková & Džurina, 2011; Moaaz et al., 2020; Tunc & Bazighifan, 2019).

Various fixed point theorems have been utilized in establishing sufficient conditions for the existence and uniqueness of solutions for different types of fractional differential problems, see for example (Abuasbeh et al., 2021; Arab et al., 2023; Awadalla et al., 2023; Derbazi & Hammouche, 2020) and their references. The BFT or Banach contraction principle is one of the most important tools in fixed point theory. This result establishes existence and uniqueness of a fixed point of a contraction defined on a Banach space. A number of generalizations have been provided for this theory, replacing the initial contraction condition with a weaker one. For example, Boyd and Wong established a condition weaker than the strict contraction and generated an operator with a unique fixed point in certain Banach space by substituting a continuous nondecreasing function F (4) for k (3), whose detail proof is found in (Boyd & Wong, 1969).

On the other hand, unlike BFT, LSFT only operates with continuous and compact non-self-mappings, not that it be a contraction; it only produces the existence of a fixed point without its uniqueness. But conversely, the Banach Theorem does not require compactness and self-mapping. A fixed point theorems such as LSFT can be used to establish existence of solutions to differential equations when it is proven that the equations have an apriori bound. One well-known method for determining an a priori bound is the Gronwall inequality which is defined in Theorm 2.5.

The Gronwall inequality is a widely used technique for analyzing the quantitative and qualitative properties of solution of fractional differential and integral equations. The inequality is used to establish a priori bounds that are utilized to demonstrate the stability, uniqueness, and global existence of differential equation results using a fixed point theorem (Mesloub & Gadain, 2020; Rezapour et al., 2021). Analyzing dependence solutions of fractional differential equations with initial conditions of any real order is another application for it.

The investigation of the existence of solutions for fractional differential equations with infinite delay frequently necessitates more advanced methodologies and approaches than the study of finite delay equations. For example, to avoid repetitions and better comprehend the intriguing features of the phase space, details of which can be found in (Yan, 2001). The phase space in the literature on equations with finite Delays usually refer to the space of all continuous functions on $[-\mathit{r},0],\mathit{r}>0$, endowed with the uniform norm topology; see the Hale and Lunel book (Hale & Lunel, 2013). The choice of the phase space when the delay is infinite plays a significant role in the investigation of both qualitative and quantitative theories. As introduced by Hale and Kato (Hale & Lunel, 2013), a semi-normed meeting of the appropriate axioms is a usual option.

The existence of solutions for differential equations with unbounded delay in the presence of various fractional derivatives has been researched by several scholars. Benchohra et al (Benchohra et al., 2008), investigated the existence of solutions for Riemman-Liouville fractional-order functional and neutral functional differential equations with infinite delay via the Banach fixed point theorem and the LSFT In 2013, Aissani and Benchohra (Aissani & Benchohra, 2013) studied the existence of mild solutions for Caputo fractional-order integro-differential equations with infinite delay. Their analysis is based on Monch’s fixed point theorem and the techniques of Kuratowski’s measure of noncompactness.

A class of Caputo fractional differential equations with infinite delay and over infinite interval in Banach space is discussed by Shanshan and Shuqin (Li & Zhang, 2020) in 2020. The existence of mild solutions to the fractional evolution equation is established in the study, applying the Schauder’s fixed-point theorem and the characteristics of analytic semigroups. Additionally, this work includes the Kuratowski measures of noncompactness and the Darbo-Sadovskii fixed point theorem’s application to the existence of mild solutions when the analytic semigroup is not compact. In 2021, Norouzi et al (Norouzi & N’guérékata, 2021), established the existence results of solution for a ϕ-Hilfer neutral fractional semilinear equation with infinite delay, and the semigroup operator, the Banach fixed-point theorem, and the nonlinear Leray-Schauder type alternative were used to arrive at the results

The Caputo-Hadamard fractional differential equations with an infinite delay scenario, however, are not studied in any articles. Hence, this study aims to close the gap that exists in this field of study. Therefore, the authors studied the establishment of the existence of solutions for the following initial value problem (IVP) of the Caputo-Hadamard-type fractional neutral delay differential equation with unbounded delays of the form:

(1) $\{\begin{array}{ll}{,}^{\mathit{CH}}{D}_{\mathit{a}+}^{{\mathit{\eta }}_1}{[}^{\mathit{CH}}{D}_{{\mathit{a}}^{+}}^{{\eta }_2}\mathit{y}(\mathit{t}){-}^{\mathit{H}}{I}_{{\mathit{a}}^{+}}^{{\eta }_3}\mathit{f}(\mathit{t},y_t)]=\mathit{g}(\mathit{t},y_{\mathit{t}}),& \text{if }\mathit{t}\in \mathit{J}=[\mathit{a},\mathit{b}],\mathit{b}>\mathit{a}>0\\ \mathit{y}(\mathit{a})=0,\mathit{y}{\mathit{ }}^{\prime }(\mathit{a})={\beta }_2\in \mathbb{R},\\ {\mathit{ }}^{\mathit{CH}}{D}^{\mathit{\eta }2}\mathit{y}(\mathit{a})={\beta }_1\in \mathbb{R},\\ \mathit{y}(\mathit{t})=\mathit{\psi }(\mathit{t}),& \mathit{t}\in (-\mathrm{\infty },\mathit{a}]\end{array}$(1)

where the operators ${\mathit{ }}^{\mathit{CH}}{D}_{{\mathit{a}}^{+}}^{{\eta }_1}$ and ${\mathit{ }}^{\mathit{CH}}{D}_{{\mathit{a}}^{+}}^{{\eta }_2}$ denote the Caputo-Hadamard fractional derivatives of order $0<{\eta }_1\le 1$ and $1<{\eta }_2\le 2$ respectively whereas ${\mathit{ }}^{\mathit{H}}{I}_{{\mathit{a}}^{+}}^{{\eta }_3}$ is the Hadamard integral of order ${\eta }_3>0.$ The functions $\mathit{f},\mathit{g}:\mathit{J}\times {\mathbb{B}}_2\to \mathbb{R}$ are given continuous functions and $\mathit{\psi }\in \mathit{C}((-\mathrm{\infty },\mathit{a}],\mathbb{R})$ is a given continuous initial function with $\mathit{\psi }(\mathit{a})=0$. When $\mathit{t}\in \mathit{J}$ and y is a function defined on $(-\mathrm{\infty },\mathit{b}]$; we denote y_t with $y_{\mathit{t}}(\mathit{\theta })=\mathit{y}(\mathit{t}+\mathit{\theta }),\mathit{\theta }\in (-\mathrm{\infty },0],$ corresponds to the admissible-phase space ${\mathbb{B}}_2$, which will be determined later.

The remaining portions of the paper are arranged as follows: Some important definitions and lemmas are discussed in Section 2. Part 3 provides proof for the main findings. Examples that illustrate how to apply our main findings are provided in Section 4. The conclusion is finally given in Section 5.

2. Preliminaries

Definitions, lemmas, and preliminary facts that are important for the sequel are introduced in this section. We give the following lemma to conveniently establish the abstract phase space ${\mathbb{B}}_2$ which is similar to (Chang, 2007).

Lemma 2.1.

(Yan, 2001) Assume that x is a measurable and bounded function on [c, d]. Then $\mathit{f}(\mathit{t})=\mathit{sup}\{|\mathit{x}(\mathit{r})|,\mathit{r}\in [\mathit{c},\mathit{t}]\}\mathit{is}$ a measurable function on [a, b].

Assume that $\mathit{h}=\mathit{C}((-\mathrm{\infty },0],(0,\mathrm{\infty }))$ with

(2) $\mathit{\kappa }={\int }_{-\mathrm{\infty }}^0\mathit{h}(\mathit{t})\mathit{dt}<\mathrm{\infty }$(2)

and $\mathit{h}(\mathit{t})>0$ for $\mathit{t}\in [0,\mathrm{\infty }).$Then, for any d > 0, we define

$\begin{array}{ll}{\mathbb{B}}_1=& \{\mathit{\varphi }(\mathit{t}):[-\mathit{d},0]\to \mathbb{R}\text{ such that}\mathit{\varphi }(\mathit{t})\text{ is measurable and}\\ \text{bounded on }[-\mathit{d},0]\}\end{array}$

with norm

$\Vert \mathit{\varphi }(\mathit{t}){\Vert }_{[-\mathit{d},0]}=\mathit{sup}\{|\mathit{\varphi }(\mathit{s})|,\mathit{s}\in [-\mathit{d},0]\}\mathrm{\forall \varphi }\in {\mathbb{B}}_1$

By Lemma 2.1 and the definition of ${\mathbb{B}}_1$, let us define

$\begin{array}{ll}{\mathbb{B}}_2=& \{\mathit{\varphi }:(-\mathrm{\infty },0]\to \mathbb{R}\text{ such that for any }\mathit{e}>0,\\ \mathit{\varphi }{|}_{[-\mathit{e},0]}\in {\mathbb{B}}_1\text{ and}{\int }_{-\mathrm{\infty }}^0\mathit{h}(\mathit{t})\Vert \mathit{\varphi }(\mathit{t}){\Vert }_{[\mathit{t},0]}\mathit{dt}<\mathrm{\infty }\}.\end{array}$

with norm

$\Vert \mathit{\varphi }(\mathit{t}){\Vert }_{{\mathbb{B}}_2}={\int }_{-\mathrm{\infty }}^0\mathit{h}(\mathit{t})\Vert \mathit{\varphi }(\mathit{t}){\Vert }_{[\mathit{t},0]}\mathit{dt},\mathrm{\forall \varphi }\in {\mathbb{B}}_2,$

then $({\mathbb{B}}_2,\Vert .{\Vert }_{{\mathbb{B}}_2})$ is a Banach space (Liu, 2003).

We now consider the space

${\mathbb{B}}_3=\{\mathit{\eta }:(-\mathrm{\infty },\mathit{b}]\to \mathbb{R}:\mathit{\eta }(\mathit{a})=\mathit{\psi }(\mathit{a})=0\}$

equipped with norm ${\Vert \mathit{\eta }\Vert }_{{\mathbb{B}}_3}=max\{\Vert \mathit{\psi }{\Vert }_{\mathrm{\infty }},\Vert \mathit{\eta }{\Vert }_{\mathit{b}}\}$ where

$\begin{array}{ll}\Vert \mathit{\psi }{\Vert }_{\mathrm{\infty }}=& \mathit{max}\{\Vert \mathit{\psi }{\Vert }_{{\mathbb{B}}_2},\mathit{sup}\{|\mathit{\psi }(\mathit{t})|:\mathit{t}\in [0,\mathit{a}]\}\},\\ |\mathit{\eta }{\Vert }_{\mathit{b}}=& \mathit{sup}\{|\mathit{\eta }(\mathit{s})|:\mathit{s}\in [\mathit{a},\mathit{b}]\}\}.\end{array}$

Thus, $({\mathbb{B}}_3,\Vert .{\Vert }_{{\mathbb{B}}_3})$ is a Banach space (Zhao & Ma, 2021).

Definition 2.1.

(Kilbas, 2006) The Hadamard fractional integral of order q > 0 for a function $\mathit{g}\in L^{\mathit{p}}[\mathit{a},\mathit{b}]$, $\mathit{a}\le \mathit{t}\le \mathit{b}\le \mathrm{\infty }$ , is defined as

${\mathit{ }}^{\mathit{H}}{I}_{{\mathit{a}}^{+}}^{\mathit{x}}\mathit{g}(\mathit{t})=\frac{1}{\mathrm{\Gamma }(\mathit{q})}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{\mathit{q}-1}\frac{\mathit{g}(\mathit{s})}{s}\mathit{ds},\mathit{q}>0,$

Definition 2.2.

(Kilbas, 2006) Let $[\mathit{c},\mathit{d}]\subset \mathbb{R},$ The Hadamard fractional derivative of order x > 0 for a function $\mathit{g}\in \mathit{A}{C}_{\delta }^m[\mathit{c},\mathit{d}]$ is defined as

$\mathit{align}{*}^H{D}_{c^{+}}^{\mathit{x}}\mathit{g}(\mathit{t})=\frac{1}{\mathit{\Gamma }(\mathit{m}-\mathit{x})}{(\mathit{\delta })}^n{\displaystyle {\int }_c^t(}\mathit{ln}\frac{t}{s}{)}^{\mathit{m}-\mathit{q}-1}\frac{\mathit{g}(\mathit{s})}{s}\mathit{ds},\mathit{m}-1<\mathit{x}\le \mathit{m},\mathit{m}=[\mathit{x}]+1.\mathit{align}*$

where $[\mathit{x}]$ denotes the integer part of the real number x, $\mathit{\delta }=\mathit{t}\frac{d}{\mathit{dt}}$ and AC[c, d] be the space of functions that are absolutely continuous on [c, d] and the space $\mathit{A}{C}_{\delta }^n[\mathit{c},\mathit{d}]$ which consists of functions g by,

$\mathit{A}{C}_{\delta }^m[\mathit{c},\mathit{d}]=\{\mathit{g}:[\mathit{c},\mathit{d}]\to \mathbb{R}]:{\delta }^{\mathit{m}-1}(\mathit{g}(\mathit{t})\in \mathit{AC}[\mathit{c},\mathit{d}]\}.$

Lemma 2.2.

(Jarad et al., 2012) Let $\mathrm{\Re }(\mathit{\beta })\ge 0$ and $\mathit{n}=[\mathit{Re}(\mathit{\beta })]+1$. If g(x) $\in \mathit{A}{C}_{\delta }^n[\mathit{c},\mathit{d}]$, where $0<\mathit{c}<\mathit{d}<\mathrm{\infty }$, then the Caputo-Hadamard fractional derivative of order

1. $\mathit{\beta }\notin {\mathbb{N}}_0$ is defined as

   ${\mathit{ }}^{\mathit{CH}}{D}_{{\mathit{c}}^{+}}^{\mathit{\beta }}\mathit{g}(\mathit{t})=\frac{1}{\mathrm{\Gamma }(\mathit{n}-\mathit{\beta })}{\int }_c^t(\mathit{ln}\frac{t}{s}{)}^{\mathit{n}-\mathit{\beta }-1}{\delta }^{\mathit{n}}\frac{\mathit{g}(\mathit{s})}{s}\mathit{ds},$

2. $\mathit{\beta }\in \mathbb{N}$ is defined as

   ${\mathit{ }}^{\mathit{CH}}{D}_{{\mathit{c}}^{+}}^{\mathit{\beta }}\mathit{g}(\mathit{t})={\delta }^{\mathit{n}}\mathit{g}(\mathit{t})$

where $\mathit{\delta }=\mathit{t}\frac{d}{\mathit{dt}}.$

In particular,

${\mathit{ }}^{\mathit{CH}}{D}_{{\mathit{c}}^{+}}^0\mathit{g}(\mathit{t})=\mathit{g}(\mathit{t})$

Lemma 2.3.

(Jarad et al., 2012) Let $\mathrm{\Re }(\mathit{\beta })\ge 0$ and $\mathit{n}=[\mathit{Re}(\mathit{\beta })]+1$. If y(x) $\in \mathit{A}{C}_{\delta }^n[\mathit{c},\mathit{d}]$, then

$\mathit{equation}{*}^H{I}_{c^{+}}^{\mathit{\beta }}{(}^{\mathit{CH}}{D}_{c^{+}}^{\mathit{\beta }}\mathit{y}(\mathit{t}))=\mathit{y}(\mathit{t})+{\displaystyle \sum _{\mathit{k}=0}^{\mathit{n}-1}{c}_k}{(\ln \frac{t}{a})}^{\mathit{k}}\mathit{equation}*$

where $c_{\mathit{k}}\in \mathbb{R}$, k = 1,2…, n-1.

Lemma 2.4.

(Kilbas, 2006) If $\mathit{\alpha }>0,\mathit{\beta }>0,$ then

${\mathit{ }}^{\mathit{H}}{I}_{{\mathit{a}}^{+}}^{\mathit{\alpha }}(\mathit{ln}\frac{t}{a}{)}^{\mathit{\beta }}=\frac{\mathrm{\Gamma }(\mathit{\beta }+1)}{\mathrm{\Gamma }(\mathit{\beta }+\alpha +1)}(\mathit{ln}\frac{t}{a}{)}^{\mathit{\beta }+\mathit{\alpha }}$

Lemma 2.5.

Contraction Mapping Principle (Deimling, 2010) Let E be a Banach space, $\mathit{Q}\subset \mathit{E}$ be closed and $\mathit{M}:\mathit{Q}\to \mathit{Q}$ a strict contraction, i.e.

(3) $|\mathit{Mx}-\mathit{My}|\le \mathit{r}|\mathit{x}-\mathit{z}|$(3)

for some $(0,1)$ and all $\mathit{x},\mathit{z}\in \mathit{Q}.$ Then M has a unique fixed point.

Definition 2.3.

(Aphithana et al., 2015) In a Banach space X, a mapping $\mathit{G}:\mathit{X}\to \mathit{X}$ is said to be a non-linear contraction if a continuous nondecreasing function $\mathit{F}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ exists, such that $\mathit{F}(0)=0$ and $\mathit{F}(\mathit{a})<\mathit{a},\mathrm{\forall a}>0$ and that

(4) $||\mathit{Gu}-\mathit{Gv}||\le \mathit{F}(||\mathit{u}-\mathit{v}||),\mathit{for}\mathit{all}\mathit{u},\mathit{v}\in \mathit{X}$(4)

Theorem 2.4.

(Boyd and Wong, 1969) Let X be a Banach space and $\mathit{G}:\mathit{X}\to \mathit{X}$ be a non-linear contraction. Then, F has a unique fixed point in X.

Lemma 2.6.

Nonlinear alternative for single valued maps (Granas & Dugundji, 2003) Let D be a closed, convex subset of a Banach space Y, W an open subset of D and $0\in \mathit{W}$. Assume that $\mathit{F}:\bar{W}\to \mathit{D}$ is a continuous, compact (that is, $\mathit{F}(\bar{W})$ is a relatively compact subset of D) map. Then either

1. F has a fixed point in $\bar{W}$, or

2. there is a $\mathit{w}\in \mathit{\partial W}$ and $0<\mathit{\beta }<1$ with $\mathit{W}=\mathit{\beta F}(\mathit{w})$..

Substituting the function $\mathit{\psi }(\mathit{t})$ of Theorem 1 of (Almeida et al., 2018) by the function $\mathit{ln}(\mathit{t})$, we get the following generalized Gronwall inequality for Hadamard fractional integrals

Theorem 2.5.

Suppose $\mathit{\alpha }>0,\mathit{a}(\mathit{t})$ and u(t) are integrable functions on $\mathit{a}\le \mathit{t}\le \mathit{b}$, and g(t) is a continuous function defined on $\mathit{a}\le \mathit{t}\le \mathit{b}$. Assume that

1. a and u are nonnegative

2. g is nonnegative and nondecreasing

If the following inequality

(5) $\mathit{u}(\mathit{t})\le \mathit{a}(\mathit{t})+\mathit{g}(\mathit{t}){\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{\mathit{q}-1}\frac{\mathit{u}(\mathit{s})}{s}\mathit{ds},\mathit{a}\le \mathit{t}\le \mathit{b},$(5)

holds. Then

$\begin{array}{ll}\mathit{u}(\mathit{t})\le & \mathit{a}(\mathit{t})+\sum _{\mathit{n}=1}^{\mathrm{\infty }}\frac{((g(t)\mathrm{\Gamma }(\alpha ){)}^{\mathit{n}}}{\mathrm{\Gamma }(\mathit{n\alpha })}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{\mathit{nq}-1}\frac{\mathit{a}(\mathit{s})}{s}\mathit{ds},\\ \mathrm{\forall t}\in [\mathit{a},\mathit{b}]\end{array}$

In particular,

$\mathit{u}(\mathit{t})\le \mathit{a}(\mathit{t})+\mathit{g}(\mathit{t}){\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{\mathit{q}-1}\frac{\mathit{a}(\mathit{s})}{s}\mathit{ds}$

For the existence and uniqueness of solutions for the problem (1) , we need the following auxiliary lemma.

Lemma 2.7.

A function $\mathit{y}(\mathit{t})\in {\mathbb{B}}_3$ is a solution of the Caputo-Hadamard-type fractional differential 1 if and only if it satisfies

(6) $\mathit{y}(\mathit{t})=\{\begin{array}{ll}\mathit{\psi }(\mathit{t}),& \text{if }\mathit{t}\in [-\mathrm{\infty },\mathit{a}],\\ \mathit{a}{\beta }_2(\mathit{ln}\frac{t}{a})+{\beta }_1\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds},& \text{if }\mathit{t}\in [\mathit{a},\mathit{b}].\end{array}$(6)

Proof.

Assume that $\mathit{y}(\mathit{t})\in {\mathbb{B}}_3$ is a solution of the problem (1). In view of Lemma 2.3 and applying ${\mathit{ }}^{\mathit{H}}{I}_{\mathit{a}+}^{{\mathit{\eta }}_1}$ to both sides of (1) for $\mathit{t}\in [\mathit{a},\mathit{b}]$, we obtain,

(7) $\begin{array}{ll}{ }^{\mathit{CH}}{D}_a^{{\eta }_2}\mathit{y}(\mathit{t})=\hfill & \frac{1}{\Gamma ({\eta }_3)}{\displaystyle {\int }_a^t(}\mathit{ln}\frac{t}{s}{)}^{{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds}\hfill \\ \hfill & +\frac{1}{\Gamma ({\eta }_1)}{\displaystyle {\int }_{a^{+}}^{\mathit{t}}(}\mathit{ln}\frac{t}{s}{)}^{-1+{\eta }_1}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}+c_0,\hfill \end{array}$(7)

Using the initial condition ${\mathit{ }}^{\mathit{CH}}{D}_a^{\eta 2}\mathit{y}(\mathit{a})={\beta }_1$, we find that $c_0={\beta }_1$ and 7 becomes

(8) $\begin{array}{ll}{ }^{\mathit{CH}}{D}_a^{{\eta }_2}\mathit{y}(\mathit{t})=\hfill & \frac{1}{\Gamma ({\eta }_3)}{\displaystyle {\int }_a^t(}\mathit{ln}\frac{t}{s}{)}^{{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds}\hfill \\ \hfill & ++\frac{1}{\Gamma ({\eta }_1)}{\displaystyle {\int }_a^t(}\mathit{log}\frac{t}{s}{)}^{-1+{\eta }_1}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}+{\beta }_1,\hfill \end{array}$(8)

Applying ${\mathit{ }}^{\mathit{H}}{I}_{\mathit{a}}^{{\mathit{\eta }}_2}$ to both sides of (8), in light of Lemma 2.3 and Lemma 2.4, we get,

(9) $\begin{array}{ll}\mathit{y}(\mathit{t})=& \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_1+{\eta }_2}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}\\ \mathit{ }+{\beta }_1\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}+c_0+c_1\mathit{ln}\frac{t}{a}\end{array}$(9)

Employing 9 and the initial conditions $\mathit{y}(\mathit{a})=0$ and $y^{{\mathit{ }}^{\prime }}(\mathit{a})={\beta }_2$ results $c_0=0$ and $c_1=\mathit{a}{\beta }_2.$ Substituting c₀ and c₁ into the above 9 , we obtain

(10) $\begin{array}{ll}\mathit{y}(\mathit{t})=& {\beta }_1\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}\\ \mathit{a}{\beta }_2\mathit{ln}\frac{t}{a}+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds}\end{array}$(10)

which is a solution of (6) for $\mathit{a}\le \mathit{t}\le \mathit{b}$. $\mathit{y}(\mathit{t})=\mathit{\psi }(\mathit{t})$ for $\mathit{t}\in (-\mathrm{\infty },\mathit{a}].$ Thus, we proved that

(11) $\mathit{y}(\mathit{t})=\{\begin{array}{ll}\mathit{\psi }(\mathit{t}),& \text{if }\mathit{t}\in [-\mathrm{\infty },\mathit{a}],\\ \mathit{a}{\beta }_2(\mathit{ln}\frac{t}{a})+{\beta }_1\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}+\\ \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}{\int }_a^t(\mathit{log}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds},& \text{if }\mathit{t}\in [\mathit{a},\mathit{b}].\end{array}$(11)

is a solution of (6) on $(-\mathrm{\infty },\mathit{b}]$. Direct computation yields the converse result. The proof is now complete.

3. Main result

We are now prepared to discuss our key findings. Using Lemma 2.7 , we define an operator $\mathit{R}:{\mathbb{B}}_{\mathit{\mu }}\to {\mathbb{B}}_{\mathit{\mu }}$ by

(12) $(\mathit{Ry})(\mathit{t}):=\{\begin{array}{ll}\mathit{\psi }(\mathit{t}),& \text{if }\mathit{t}\in [-\mathrm{\infty },\mathit{a}],\\ \mathit{a}{\beta }_2(\mathit{ln}\frac{t}{a})+{\beta }_1\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{g(s,y_s)}{\mathit{s}}\mathit{ds}+\\ \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}{\int }_a^t(\mathit{log}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,y_s)}{\mathit{s}}\mathit{ds},& \text{if }\mathit{t}\in [\mathit{a},\mathit{b}].\end{array}$(12)

where

(13) ${\mathbb{B}}_{\mathit{\mu }}=\{\mathit{y}\in {\mathbb{B}}_3:\Vert \mathit{y}{\Vert }_{{\mathbb{B}}_3}\le \mathit{\mu },\mathit{\mu }>0\}$(13)

which is a closed subset of Banach space ${\mathbb{B}}_3.$ Thus, ${\mathbb{B}}_{\mathit{\mu }}$ is a Banach space. It is easy to show that $\mathit{y}\in {\mathbb{B}}_{\mathit{\mu }}$ is a solution of the IVP (1) if and only if Ry = y. This implies that finding out the fixed point of the operator R, as stated by (12), is the same as solving problem (1).

For convenience’s sake, let’s set

$\begin{array}{ll}{\mathrm{\Delta }}_1=& \mathit{\kappa }[\frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}]\end{array}$

and

$\begin{array}{ll}{\mathrm{\Delta }}_2=& \mathit{a}|{\beta }_2|(\mathit{ln}\frac{t}{a})+|{\beta }_1|\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }+\frac{M_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{M_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\end{array}$

We provide the following uniqueness result based on Banach fixed point theorem2.5:

Theorem 3.1.

Suppose that the $\mathit{g},\mathit{f}:\mathit{J}\times {\mathbb{B}}_2\to \mathbb{R}$ are given continuous functions such that there exist two constants $k_1,k_2>0$ such that

1. $|\mathit{f}(\mathit{t},u_{\mathit{t}})-\mathit{f}(\mathit{t},v_{\mathit{t}})|\le k_1\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2},\text{for}\mathit{t}\in \mathit{J}\text{and every }{u}_{\mathit{t}},v_{\mathit{t}}\in {\mathbb{B}}_2$

2. $|\mathit{g}(\mathit{t},u_{\mathit{t}})-\mathit{g}(\mathit{t},v_{\mathit{t}})|\le k_2\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2}\text{for}\mathit{t}\in \mathit{J}\text{and every }{u}_{\mathit{t}},v_{\mathit{t}}\in {\mathbb{B}}_2$

If

(14) $\mathit{\kappa }[\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}+\frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}]<1,$(14)

then the initial value problem (1)has a unique solution on $(-\mathrm{\infty },\mathit{b}].$

Proof.

Consider the operator R defined by (12) and the set ${\mathbb{B}}_{\mathit{\mu }}$ by (13). Setting $\underset{\mathit{t}\in [\mathit{a},\mathit{b}]}{sup}|\mathit{f}(\mathit{t},0)|=M_1,\underset{\mathit{t}\in [\mathit{a},\mathit{b}]}{sup}|\mathit{g}(\mathit{t},0)|=M_2$ and fixing $\mathit{\mu }\ge \frac{{\mathrm{\Delta }}_2}{1-{\mathrm{\Delta }}_1}$,We show that the invariance of the set ${\mathbb{B}}_{\mathit{\mu }}$ with regard to the operator R; that is, $\mathit{R}({\mathbb{B}}_{\mathit{\mu }})\subseteq {\mathbb{B}}_{\mathit{\mu }}.$

For any $\mathit{y}\in {\mathbb{B}}_{\mathit{\mu }}$ and $\mathit{t}\in (-\mathrm{\infty },\mathit{a}],$ we have

$\Vert (\mathit{Ry})(\mathit{t}){\Vert }_{{\mathbb{B}}_3}=\Vert \mathit{\varphi }(\mathit{t}){\Vert }_{{\mathbb{B}}_3}=\mathit{max}\{\Vert \mathit{\varphi }{\Vert }_{\mathrm{\infty }}\}\le \Vert \mathit{y}{\Vert }_{{\mathbb{B}}_3}\le \mathit{\mu }$

For any $\mathit{y}\in {\mathbb{B}}_{\mathit{\mu }}$ and $\mathit{t}\in [\mathit{a},\mathit{b}]$, it follows from (1), (A1), and (A2) that,

$\begin{array}{ll}||\mathit{Ry}(\mathit{t})|{|}_{{\mathbb{B}}_3}& \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_1+{\eta }_2}\frac{|g(s,y_s)|}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_2+{\eta }_3}\frac{|\mathit{f}(\mathit{s},{\mathit{y}}_{\mathit{s}})|}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{t}{a})+|{\beta }_1|\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\\ \times \frac{|\mathit{g}(\mathit{s},{\mathit{y}}_{\mathit{s}})-\mathit{g}(\mathit{s},0)|+|\mathit{g}(\mathit{s},0)|}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\\ \times \frac{|\mathit{f}(\mathit{s},{\mathit{y}}_{\mathit{s}})-\mathit{f}(\mathit{s},0)|+|\mathit{f}(\mathit{s},0)|}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{t}{a})+|{\beta }_1|\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_1+{\eta }_2}\\ \frac{{\mathit{k}}_2\Vert y_{\mathit{s}}{\Vert }_{{\mathbb{B}}_2}+M_2}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\\ \frac{{\mathit{k}}_1\Vert y_{\mathit{s}}{\Vert }_{{\mathbb{B}}_2}+M_1}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{t}{a})+|{\beta }_1|\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{{\mathit{k}}_2\mathit{\kappa }\Vert \mathit{y}{\Vert }_{\mathit{T}}+M_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{{\mathit{k}}_1\mathit{\kappa }\Vert \mathit{y}{\Vert }_{\mathit{T}}+M_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3+1}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{{\mathit{k}}_2\mathit{\kappa }\Vert \mathit{y}{\Vert }_{{\mathit{B}}_3}+M_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{{\mathit{k}}_1\mathit{\kappa }\Vert \mathit{y}{\Vert }_{{\mathit{B}}_3}+M_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3+1}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \mathit{\kappa }\Vert \mathit{y}{\Vert }_{{\mathit{B}}_3}[\frac{{\mathit{k}}_1(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}+[\frac{{\mathit{k}}_2(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}]\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \mathit{\kappa \mu }[\frac{{\mathit{k}}_1(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}+[\frac{{\mathit{k}}_2(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}]\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }+\frac{M_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{M_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\\ \mathit{ }\le \mathit{\mu }\end{array}$

As a result, $\mathit{R}(\mathit{y})\in {\mathbb{B}}_{\mathit{\mu }}$ and thus $\mathit{R}({\mathbb{B}}_{\mathit{\mu }})\subseteq {\mathbb{B}}_{\mathit{\mu }}$.

The operator R is a strict contraction, as we will show next. If $\mathit{v},\mathit{w}\in {\mathbb{B}}_{\mathit{\mu }}$ and $\mathit{t}\in (-\mathrm{\infty },\mathit{a}]$ , and from the definition of R , we have:.

$\Vert \mathit{R}(\mathit{v})-\mathit{R}(\mathit{w}){\Vert }_{{\mathbb{B}}_3}=0=0\Vert \mathit{v}-\mathit{w}{\Vert }_{{\mathbb{B}}_3}\le \mathit{k}\Vert \mathit{v}-\mathit{w}{\Vert }_{{\mathbb{B}}_3}$

for all $\mathit{k}\in (0,1).$

For $\mathit{v},\mathit{w}\in {\mathbb{B}}_{\mathit{\mu }}$ and $\mathit{t}\in [\mathit{a},\mathit{b}],$ it follows from $(A_1),(A_2)$ and (14) that

$\begin{array}{l}\mathit{ }|\mathit{R}(\mathit{v})(\mathit{t})-\mathit{R}(\mathit{w})(\mathit{t})|\\ \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_1+{\eta }_2}\frac{|g(s,v_s)-\mathit{g}(\mathit{s},w_s)|}{\mathit{s}}\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{|f(s,v_s)-\mathit{f}(\mathit{s},w_s)|}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le \frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2)-1}\frac{\Vert v_s-w_s{\Vert }_{{\mathbb{B}}_2}}{\mathit{s}}\mathit{ds}\\ +\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert v_s-w_s{\Vert }_{{\mathbb{B}}_2}}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le \frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2)-1}\frac{\Vert \mathit{v}-\mathit{w}{\Vert }_{\mathit{b}}}{\mathit{s}}\mathit{ds}\\ +\frac{{\mathit{k}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert \mathit{v}-\mathit{w}{\Vert }_{\mathit{b}}}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le \frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\Vert \mathit{v}-\mathit{w}{\Vert }_{\mathit{b}}\\ \mathit{ }+\frac{{\mathit{k}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\Vert \mathit{v}-\mathit{w}{\Vert }_{\mathit{b}}\\ \mathit{ }\le \mathit{\kappa }[\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\\ \mathit{ }+\frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}]\Vert \mathit{v}-\mathit{w}{\Vert }_{{\mathit{B}}_3}.\end{array}$

Hence, we get

$\begin{array}{l}\Vert \mathit{R}(\mathit{v})(\mathit{t})-\mathit{R}(\mathit{w})(\mathit{t}){\Vert }_{{\mathit{B}}_3}\le \mathit{\kappa }[\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\\ +\frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}]\Vert \mathit{v}-\mathit{w}{\Vert }_{{\mathit{B}}_3}.\end{array}$

R is a contraction from the assumption (14). As a result, by BFT, R has a fixed point which is the unique solution of the problem (1) on $(-\mathrm{\infty },\mathit{b}]$. This completes the proof.

On the basis of nonlinear contractions, we then provide a second existence and uniqueness result.

Theorem 3.2.

Let $\mathit{g},\mathit{f}:\mathit{J}\times {\mathbb{B}}_2\to \mathbb{R}$ are given continuous functions satisfying the following assumptions:

1. $|\mathit{g}(\mathit{t},u_{\mathit{t}})-\mathit{g}(\mathit{t},v_{\mathit{t}})|\le \mathit{\rho }{k}_1(\mathit{t})(\mathit{ta}{n}^{-1}(\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2})),$

2. $|\mathit{f}(\mathit{t},u_{\mathit{t}})-\mathit{f}(\mathit{t},v_{\mathit{t}})|\le \mathit{\rho }{k}_2(\mathit{t})(\mathit{ta}{n}^{-1}(\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2}))$

for $\mathit{t}\in \mathit{J}$ , for every $u_{\mathit{t}},v_{\mathit{t}}\in {\mathbb{B}}_2$, $k_1,k_2:[\mathit{a},\mathit{b}]\to {\mathbb{R}}^{\mathbb{+}}$ are continous functions and ρ > 0 defined by

$\begin{array}{ll}{\rho }^{-1}& \mathit{ }=[\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^b(\mathit{ln}\frac{b}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{{\mathit{k}}_1(\mathit{s})}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^b(\mathit{ln}\frac{b}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3)-1}\frac{{\mathit{k}}_2(\mathit{s})}{\mathit{s}}\mathit{ds}\end{array}$

Then the IVP (1) has a unique solution on$(-\mathrm{\infty },\mathit{b}]$ if

(15) $0<\mathit{\kappa }\le 1$(15)

AS κ is defined by (2).

Proof.

Consider the operator R defined by (12) where $\mathit{R}:{\mathbb{B}}_3\to {\mathbb{B}}_3$ and the continuous non-decreasing function $\mathit{m}:{\mathbb{R}}^{\mathbb{+}}\to {\mathbb{R}}^{\mathbb{+}}$ defined by

$\mathit{m}(\mathit{x})={\tan }^{-1}\mathit{lx},\mathrm{\forall }\mathit{x}\ge 00<\mathit{l}\le 1$

Clearly, $\mathit{m}(0)=0$ and $\mathit{m}(\mathit{x})<\mathit{x}\mathrm{\forall x}>0.$

For $\mathit{u},\mathit{w}\in {\mathbb{B}}_3$ and $\mathit{t}\in [\mathit{a},\mathit{b}],$ it follows from $(A_3),(A_4)$ and (14) that

$\begin{array}{l}\mathit{ }\Vert \mathit{R}(\mathit{u})(\mathit{t})-\mathit{R}(\mathit{v})(\mathit{t}){\Vert }_{{\mathit{B}}_3}\\ \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2)-1}\frac{\mathit{\rho }{\mathit{k}}_1(\mathit{s})({\tan }^{-1}(\Vert u_{\mathit{s}}-v_{\mathit{s}}{\Vert }_{{\mathbb{B}}_2}))}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\mathit{\rho }{\mathit{k}}_2(\mathit{s})({\tan }^{-1}(\Vert u_{\mathit{s}}-v_{\mathit{s}}{\Vert }_{{\mathbb{B}}_2}))}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2)-1}\frac{\mathit{\rho }{\mathit{k}}_1(\mathit{s})({\tan }^{-1}(\mathit{\kappa }\Vert u_{\mathit{s}}-v_{\mathit{s}}{\Vert }_{{\mathbb{B}}_3}))}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\mathit{\rho }{\mathit{k}}_2(\mathit{s})({\tan }^{-1}(\mathit{\kappa }\Vert u_{\mathit{s}}-v_{\mathit{s}}{\Vert }_{{\mathbb{B}}_3}))}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le \mathit{\rho }\mathit{m}(\mathit{\kappa }\Vert \mathit{u}-\mathit{v}{\Vert }_{{\mathbb{B}}_3})[\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^b(\mathit{ln}\frac{b}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{{\mathit{k}}_1(\mathit{s})}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^b(\mathit{ln}\frac{b}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3)-1}\frac{{\mathit{k}}_2(\mathit{s})}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le \mathit{m}(\mathit{\kappa }\Vert \mathit{u}-\mathit{v}{\Vert }_{{\mathbb{B}}_3})\end{array}$

Since $\Vert \mathit{R}(\mathit{u})(\mathit{t})-\mathit{R}(\mathit{v})(\mathit{t}){\Vert }_{{\mathit{B}}_3}\le \mathit{m}(\Vert \mathit{u}-\mathit{v}{\Vert }_{{\mathbb{B}}_3})$ and R is a nonlinear contraction, by Lemma $2.4$ R has a fixed point in ${\mathbb{B}}_3,$ which is a unique solution of IVP (1).

Next, we use the Theorem 2.6to demonstrate the existence of solutions for the IVP problem (1).

Theorem 3.3.

Suppose that the functions $\mathit{G},\mathit{F}:\mathit{J}\times {\mathbb{B}}_2\to \mathbb{R}$ are given continuous functions. Assume that

1. $|\mathit{g}(\mathit{t},y_{\mathit{t}})|\le \mathit{\chi }(\mathit{t}),\mathrm{\forall }(\mathit{t},y_{\mathit{t}})\in \mathit{J}\times {\mathbb{B}}_2,\mathit{\chi }\in \mathit{C}(\mathit{J},{\mathbb{R}}^{\mathbb{+}}),\$\$\mathit{with}\underset{\mathit{t}\in [\mathit{a},\mathit{b}]}{sup}|\mathit{\chi }(\mathit{t})|=\Vert \mathit{\chi }{\Vert }_{\mathit{b}},$

2. $p_1>0$ and $p_2\ge 0$ such that

   $|\mathit{f}(\mathit{t},y_{\mathit{t}})|\le p_1\Vert y_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2}+p_2,\mathrm{\forall }(\mathit{t},y_{\mathit{t}})\in (\mathit{J},{\mathbb{B}}_2)$

3. there exists a positive constant $B^{\ast }>0$ such that all the solutions of the problem (1) satisfy the inequuality

   $\Vert \mathit{z}{\Vert }_{{\mathbb{B}}_3}\le B^{\ast }$

   where

   $\begin{array}{ll}{B}^{\ast }=& \frac{\Vert \mathit{\chi }{\Vert }_{\mathit{b}}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})\\ \mathit{ }+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}+p_2\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}\\ \mathit{ }+\frac{{\mathit{p}}_1\mathit{\kappa F}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\end{array}$

Then the IVP (1) has at least one solution in ${\mathbb{B}}_3.$

Proof.

The operator $\mathit{R}:{\mathbb{B}}_3\to {\mathbb{B}}_3$, as described by (12), will now be proved to satisfy Lemma 2.6’s assumptions. There will be four steps to the proof.

Step 1: Since g and f are continuous, the operator R is also continuous.

Step 2: R maps bounded sets into bounded sets, as we demonstrate. For a positive constant ${\ell }^{\ast }$

(16) ${\mathbb{B}}_{{\mathit{\ell }}^{\ast }}=\{\mathit{y}\in {\mathbb{B}}_3:\Vert \mathit{y}{\Vert }_{{\mathbb{B}}_3}\le {\ell }^{\ast }\}$(16)

is a bounded subset in ${\mathbb{B}}_3.$ For $\mathit{y}\in {\mathbb{B}}_3$, it follows from $(A_7)$ and $(A_8)$ that

$\begin{array}{ll}||\mathit{Ry}(\mathit{t})|{|}_{{\mathbb{B}}_3}& \mathit{ }\le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_1+{\eta }_2}\frac{|g(s,y_s)|}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-}1+{\eta }_2+{\eta }_3\frac{|\mathit{f}(\mathit{s},{\mathit{y}}_{\mathit{s}})|}{\mathit{s}}\mathit{ds}\\ \mathit{ }+|{\beta }_2|\mathit{a}(\mathit{ln}\frac{t}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{\Vert \mathit{\chi }{\Vert }_{\mathit{b}}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{\mathit{ds}}{s}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{{\mathit{p}}_1\Vert y_{\mathit{s}}{\Vert }_{{\mathbb{B}}_2}+p_2}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{\Vert \mathit{\chi }{\Vert }_{\mathit{b}}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{{\mathit{p}}_1\mathit{\kappa }\Vert \mathit{y}{\Vert }_{{\mathbb{B}}_3}+p_2}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }\le \frac{|\mathit{\chi }|}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\\ \mathit{ }+\frac{{\mathit{p}}_1\mathit{\kappa }{\ell }^{\ast }+p_2}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}\\ \mathit{ }+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}=p^{\ast }\end{array}$

for some positive constant $p^{\ast }.$

Step 3: We now show that R maps bounded sets into equicontinuous sets of ${\mathbb{B}}_3$, i.e $\mathit{R}({\mathbb{B}}_{{\mathit{\ell }}^{\ast }})$ is equicontinous. Set $\underset{(\mathit{t},{\mathit{u}}_{\mathit{t}})\times [\mathit{a},\mathit{b}]\times {\mathbb{B}}_{{\mathit{\ell }}^{\ast }}}{sup}|\mathit{f}(\mathit{t},y_{\mathit{t}})|=\bar{M}$

Let $t_1,t_2\in [\mathit{a},\mathit{b}]$ with $t_1<t_2$and $\mathit{u}\in {\mathbb{B}}_{{\mathit{\ell }}^{\ast }}.$ As $t_1\to t_2$, it follows from $(A_3)$ and $(A_4)$

$\begin{array}{l}\mathit{ }||\mathit{Ru}(t_2)-\mathit{Ru}(t_1)|{|}_{{\mathbb{B}}_3}\\ \le \frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}|{\int }_a^{t_1}(\mathit{ln}\frac{t_2}{\mathit{s}}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}-(\mathit{ln}\frac{t_1}{\mathit{s}}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\mathit{g}(\mathit{s},u_{\mathit{s}})\frac{\mathit{ds}}{s}|\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2}|{\int }_{t_1}^{t_2}(\mathit{ln}\frac{t_2}{\mathit{s}}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\mathit{g}(\mathit{s},u_{\mathit{s}})\frac{\mathit{ds}}{s}|\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}|{\int }_a^{t_1}(\mathit{ln}\frac{t_2}{\mathit{s}}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}-(\mathit{ln}\frac{t_1}{\mathit{s}}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\mathit{f}(\mathit{s},u_{\mathit{s}})\frac{\mathit{ds}}{s}|\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}|{\int }_{t_1}^{t_2}(\mathit{ln}\frac{t_2}{\mathit{s}}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\mathit{f}(\mathit{s},u_{\mathit{s}})\frac{\mathit{ds}}{s}|\\ +\mathit{a}|{\beta }_2|(\mathit{ln}\frac{t_2}{\mathit{a}})+|{\beta }_1|(|\frac{(\ln \frac{t_2}{\mathit{a}}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}-\frac{(\ln \frac{t_2}{t_1}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)})\\ \le \frac{\Vert \mathit{\chi }{\Vert }_{\mathit{b}}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}[|(\mathit{ln}\frac{t_2}{\mathit{a}}{)}^{{\mathit{\alpha }}_1+{\eta }_2}-(\mathit{ln}\frac{t_1}{\mathit{a}}{)}^{{\mathit{\eta }}_1+{\eta }_2}|\\ +2(\mathit{ln}\frac{t_2}{t_2}{)}^{{\mathit{\eta }}_1+{\eta }_2}]+\frac{\bar{M}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}[|(\mathit{ln}\frac{t_2}{\mathit{a}}{)}^{{\mathit{\eta }}_2+{\eta }_3}\\ \mathit{ }-(\mathit{ln}\frac{t_1}{\mathit{a}}{)}^{{\mathit{\eta }}_2+{\eta }_3}|+2(\mathit{ln}\frac{t_2}{t_2}{)}^{{\mathit{\alpha }}_2+{\eta }_3}]\\ +\mathit{a}|{\beta }_2|(\mathit{ln}\frac{t_2}{t_1})+|{\beta }_1|(|\frac{(\ln \frac{t_2}{\mathit{a}}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}-\frac{(\ln \frac{t_2}{t_1}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)})\to 0.\end{array}$

In a similar way, we can show for the cases where $-\mathrm{\infty }<t_1<\mathit{a}<t_2\le \mathit{b}$ and $-\mathrm{\infty }<t_1<t_2\le \mathit{a}.$

As a result of Step 1 through 3, R is continuous and completely continuous according to the Arzelá-Ascoli theorem.

We now determine an apriori bound for the problem’s solutions, from which we deduce the existence of the solutions using a fixed point theorem.

Step 4 : (A priori bounds) We now prove that an open set $\mathit{V}\subseteq {\mathbb{B}}_3$ exists with $\mathit{z}\ne \mathit{\lambda R}(\mathit{z})$ for $\mathit{\lambda }\in (0,1)$ and $\mathit{z}\in \mathit{\partial V}$.

Let $\mathit{t}\in {\mathbb{B}}_3\mathit{and}\mathit{z}=\mathit{\lambda R}(\mathit{z})$ for some $\mathit{\lambda }\in (0,1).$ Then for each $\mathit{t}\in [\mathit{a},\mathit{b}]$ we have

(17) $\begin{array}{l}\begin{array}{ll}\mathit{z}(\mathit{t})& =\mathit{\lambda }(\mathit{a}{\beta }_2(\mathit{ln}\frac{t}{a})+{\beta }_1\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{g(s,z_s)}{\mathit{s}}\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,z_s)}{\mathit{s}}\mathit{ds})\end{array}\end{array}$(17)

By $(A_7),(A_8)$ and the Gronwall inequality $2.5$ we obtain that

(18) $\begin{array}{l}\begin{array}{ll}||\mathit{z}(\mathit{t})|{|}_{{\mathbb{B}}_3}& \le \frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_1+{\alpha }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2-1}\frac{|g(s,z_s)|}{\mathit{s}}\mathit{ds}\\ +\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{|\mathit{f}(\mathit{s},{\mathit{z}}_{\mathit{s}})|}{\mathit{s}}\mathit{ds}\\ +\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \le \frac{\Vert \mathit{\chi }{\Vert }_{\mathit{b}}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})\\ +|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}+p_2\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}\\ +\frac{{\mathit{p}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{||z|{|}_{{\mathit{B}}_3}}{\mathit{s}}\mathit{ds}\\ \le \mathit{F}+\frac{{\mathit{p}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{F}{s}\\ \le \mathit{F}+\frac{{\mathit{p}}_1\mathit{\kappa F}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}=B^{\ast }\end{array}\end{array}$(18)

where

$\begin{array}{ll}\mathit{F}=& \frac{\Vert \mathit{\chi }{\Vert }_{\mathit{b}}}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}+\mathit{a}|{\beta }_2|(\mathit{ln}\frac{b}{a})\\ \mathit{ }+|{\beta }_1|\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}+p_2\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}\end{array}$

Set

$\mathit{V}=\{\mathit{z}\in {\mathbb{B}}_3:\Vert \mathit{z}{\Vert }_{{\mathbb{B}}_3}<B^{\ast }+1\}.$

Note that the operator $\mathit{R}:\bar{V}\to {\mathbb{B}}_3$ is continuous and compact. From the choice of V, there is no $\mathit{z}\in \mathit{\partial V}$ such that $\mathit{z}=\mathit{\lambda R}(\mathit{z})$ for some $\mathit{\lambda }\in (0,1)$. As a consequence of the nonlinear alternative of Leray-Schauder type (Lemma 2.6), we deduce that R has a fixed point $\mathit{z}\in \bar{V}$ which is the solution of the problem (1).

4. Continuous dependence of solutions with respect to initial value conditions via Gronwall inequality

This section addresses the solution’s continuity dependency in regard to the ICs

Theorem 4.1.

If the IVP (1) satisfying conditions A₁ and A₂, then the solution of IVP (1) depends continuously on its ICs.

Proof.

Consider the IVP (1) and its perturbation equation

(19) $[{,}^{\mathit{CH}}{D}_{a^{+}}^{{\eta }_1}{[}^{\mathit{CH}}{D}_{a^{+}}^{{\eta }_2}\mathit{y}(\mathit{t}){-}^H{I}_{a^{+}}^{{\eta }_3}\mathit{f}(\mathit{t},y_t)]=\mathit{g}(\mathit{t},y_t),\text{if }\mathit{t}\in \mathit{J}=[\mathit{a},\mathit{b}],\mathit{b}>\mathit{a}>0\mathit{y}(\mathit{a})={\delta }_1,\mathit{y}(\mathit{a})=({\beta }_2+{\delta }_2)\in \mathit{ℝ},\mathit{CH}{D}^{{\eta }_2}\mathit{y}(\mathit{a})=({\beta }_1+{\delta }_3)\in \mathit{ℝ},\mathit{y}(\mathit{t})=\mathit{\psi }(\mathit{t}),\mathit{t}\in (-\mathit{\infty },\mathit{a}]]$(19)

where the positive constants ${\delta }_2,{\delta }_2,{\delta }_3$ are sufficiently small.

Suppose that the solutions to the IVP (1) and (21), respectively, are y(t) and z(t) on [a, b]. It is easy to compute that

$\begin{array}{ll}\mathit{z}(\mathit{t})=& {\delta }_1+({\beta }_1+{\delta }_3)\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }(1+{\eta }_2)}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{-1+{\mathit{\eta }}_1+{\eta }_2}\frac{g(s,z_s)}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\frac{1}{\mathrm{\Gamma }({\mathit{\eta }}_2+{\eta }_3}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{f(s,z_s)}{\mathit{s}}\mathit{ds}\\ \mathit{ }+\mathit{a}({\beta }_2+{\delta }_2)\mathit{ln}\frac{t}{a}\mathrm{\forall t}\in [\mathit{a},\mathit{b}]\end{array}$

By Lemma 2.7 and applying A₁ , A₂, we have

$\begin{array}{ll}|\mathit{z}(\mathit{t})-\mathit{y}(\mathit{t})|& \mathit{ }\le {\delta }_1+\mathit{a}{\delta }_2\mathit{ln}\frac{t}{a}+{\delta }_3\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2)-1}\frac{\Vert z_s-y_s{\Vert }_{{\mathbb{B}}_2}}{\mathit{s}}\mathit{ds}\\ +\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert z_s-y_s{\Vert }_{{\mathbb{B}}_2}}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le {\delta }_1+\mathit{a}{\delta }_2\mathit{ln}\frac{t}{a}+{\delta }_3\frac{(\ln \frac{t}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_1+{\eta }_2)-1}\frac{\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}}{\mathit{s}}\mathit{ds}\\ +\frac{{\mathit{k}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le {\delta }_1+\mathit{a}{\delta }_2\mathit{ln}\frac{b}{a}+{\delta }_3\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ +\frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}\\ +\frac{{\mathit{k}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}}{\mathit{s}}\mathit{dS}\end{array}$

Rearranging terms, we obtain

(20) $\begin{array}{l}\mathit{ }|\mathit{z}(\mathit{t})-\mathit{y}(\mathit{t})|-\frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}\le {\delta }_1\\ \mathit{ }+\mathit{a}{\delta }_2\mathit{ln}\frac{b}{a}+{\delta }_3\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}\\ \mathit{ }+\frac{{\mathit{k}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}}{\mathit{s}}\mathit{ds}\end{array}$(20)

Thus

$\begin{array}{ll}\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}& \mathit{ }\le G_1+\le G_2{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{\Vert \mathit{z}-\mathit{y}{\Vert }_{{\mathbb{B}}_3}}{\mathit{s}}\mathit{ds}\end{array}$

where

$G_1=\frac{{\mathit{\delta }}_1+\mathit{a}{\delta }_2\mathit{ln}\frac{b}{a}+{\delta }_3\frac{(\ln \frac{b}{a}{)}^{{\mathit{\eta }}_2}}{\mathrm{\Gamma }({\eta }_2+1)}}{1-\frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}}$

and

$G_2=\frac{\frac{{\mathit{k}}_1\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_2+{\eta }_3)}}{1-\frac{{\mathit{k}}_2\mathit{\kappa }}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}}$

Applying Gronwall inequality Theorem 2.5 on (21), we get

$\begin{array}{ll}\Vert \mathit{z}(\mathit{t})-\mathit{y}(\mathit{t}){\Vert }_{{\mathbb{B}}_3}& \mathit{ }\le G_1+G_2{\int }_a^t(\mathit{ln}\frac{t}{s}{)}^{{\mathit{\eta }}_2+{\eta }_3-1}\frac{G_1}{\mathit{s}}\mathit{ds}\\ \mathit{ }\le G_1+\frac{{\mathit{G}}_1{G}_2(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}}{{\eta }_2+{\eta }_3}\end{array}$

It is simple to demonstrate that $\Vert \mathit{z}(\mathit{t})-\mathit{y}(\mathit{t}){\Vert }_{{\mathit{B}}_3}\to 0$ as ${\delta }_1,{\delta }_2,{\delta }_3\to 0$, then $\mathit{z}(\mathit{t})\to \mathit{y}(\mathit{t}).$, Hence, the proof is completed.

5. Examples

In order to illustrate the significance of our key findings, we provide examples in this section.

Example 5.1.

Consider the following neutral Caputo-Hadamard fractional differential equation with unbounded delays:

(21) ${,}^{\mathit{CH}}{D}_{1^{+}}^{\frac{1}{4}}{[}^{\mathit{CH}}{D}_{1^{+}}^{\frac{3}{2}}\mathit{y}(\mathit{t}){-}^H{I}_{1^{+}}^{\frac{1}{5}}\mathit{f}(\mathit{t},y_t)]=\mathit{g}(\mathit{t},y_t),\text{\hspace{1em}if }\mathit{t}\in [1,e^2],\mathit{y}(1)=0,{}^{\mathit{CH}}{D}_{1^{+}}^{\frac{3}{2}}\mathit{y}(1)=1,\mathit{y}(1)=0.1\mathit{y}(\mathit{t})=\mathit{\psi }(\mathit{t}),\mathit{t}\in (-\mathit{\infty },1]]$(21)

Here ${\eta }_1=\frac{1}{4},{\eta }_2=\frac{3}{2},{\eta }_3=\frac{1}{5},\mathit{a}=1,\mathit{b}=e^2\text{and}\mathit{\psi }(\mathit{t})=\mathit{sin}\mathit{\pi }\mathit{t}$

In order to demonstrate Theorem 3.1, we define continuous maps $\mathit{f},\mathit{g}:[1.e^2]\times {\mathbb{B}}_2$ by

$\begin{array}{l}\mathit{f}(\mathit{t},y_{\mathit{t}})=\frac{\sqrt{t}}{(t{e}^{\sqrt{\mathit{t}}}+3-\mathit{e}{)}^2}\frac{\Vert y_t{\Vert }_{{\mathbb{B}}_2}}{(1+\Vert y_t{\Vert }_{{\mathbb{B}}_2})}+\mathit{ln}\mathit{t},\\ \mathit{g}(\mathit{t},y_{\mathit{t}})=\frac{\mathit{ln}\mathit{t}}{((2+\ln \mathit{t}{)}^2-1)}\frac{\Vert y_t{\Vert }_{{\mathbb{B}}_2}}{(1+\Vert y_t{\Vert }_{{\mathbb{B}}_2})}+\mathit{sin}\mathit{t}\end{array}$

Let $\mathit{h}(\mathit{s})=e^{4\mathit{s}},\mathit{for}\mathit{s}<0$, then ${\int }_{-\mathrm{\infty }}^0\mathit{h}(\mathit{s})\mathit{ds}=\frac{1}{4}<\mathrm{\infty }$ Thus, from the definitions of ${\mathbb{B}}_1,{\mathbb{B}}_2,\mathit{and}{\mathbb{B}}_3,$ we can readily obtain ${\mathbb{B}}_1,{\mathbb{B}}_2,\mathit{and}{\mathbb{B}}_3,$ and their norms.

A simple calculation shows that

$|\mathit{f}(\mathit{t},u_{\mathit{t}})-\mathit{f}(\mathit{t},v_{\mathit{t}})|\le \frac{e}{9}\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2},$

and

$|\mathit{g}(\mathit{t},u_{\mathit{t}})-\mathit{g}(\mathit{t},v_{\mathit{t}})|\le \frac{2}{3}\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2},$

$\text{for }\mathit{t}\in [1,e^2]\text{ and every}{u}_{\mathit{t}},v_{\mathit{t}}\in {\mathbb{B}}_2$

Here $\mathit{\kappa }=\frac{1}{4},k_1=\frac{e}{9}$ and $k_2=\frac{2}{3}.$ Therefore,

$\begin{array}{l}\mathit{\kappa }[\frac{k_1}{\mathrm{\Gamma }({\eta }_2+{\eta }_3+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_2+{\eta }_3}+\frac{k_2}{\mathrm{\Gamma }({\eta }_1+{\eta }_2+1)}(\mathit{ln}\frac{b}{a}{)}^{{\mathit{\eta }}_1+{\eta }_2}]\\ \mathit{ }\approx 0.507371<1.\end{array}$

As a result, Theorem ${3.1}^{\prime }\mathit{s}$ conditions are all met, and as a consequence, the problem (21) has a unique solution on $(-\mathrm{\infty },e^2].$

Example 5.2.

Consider the above fractional differential equation problem (21) where $\mathit{f},\mathit{g}:[1.e^2]\times {\mathbb{B}}_2$ are continuous functions defined by

$\mathit{f}(\mathit{t},u_{\mathit{t}})=\frac{\sqrt{t}}{(t{e}^{\sqrt{\mathit{t}}}+3-\mathit{e}{)}^2}(\frac{{\mathit{tan}}^{-1}{u}_{\mathit{t}}}{1+{\tan }^{-1}{u}_{\mathit{t}}})+\mathit{ln}\mathit{t},$

$\mathit{g}(\mathit{t},u_{\mathit{t}})=\frac{\mathit{ln}\mathit{t}}{((2+\ln \mathit{t}{)}^2-1)}(\mathit{ta}{n}^{-1}{u}_{\mathit{t}})+\mathit{sin}\mathit{t}$

Let $\mathit{a}(\mathit{s})=e^{3\mathit{s}},\mathit{for}\mathit{s}<0$, then $\mathit{\kappa }={\int }_{-\mathrm{\infty }}^0\mathit{a}(\mathit{s})\mathit{ds}=\frac{1}{3}\le 1.$ Hence, we can easily determine ${\mathbb{B}}_1,{\mathbb{B}}_2,\mathit{and}{\mathbb{B}}_3,$ as well as their norms, from the definitions of ${\mathbb{B}}_1,{\mathbb{B}}_2,\mathit{and}{\mathbb{B}}_3,$ We choose

$k_1(\mathit{t})=\frac{\sqrt{t}}{(t{e}^{\sqrt{\mathit{t}}}+3-\mathit{e}{)}^2}\mathit{and}{k}_2(\mathit{t})=\frac{\mathit{ln}\mathit{t}}{(2+\ln \mathit{t}{)}^2-1}$

which are continuous functions on $[1,e^2]$ and we obtain

$\mathit{\rho }=3.7381>1$

An easy computation yields that

$\begin{array}{ll}|\mathit{f}(\mathit{t},u_{\mathit{t}})& \mathit{ }-\mathit{f}(\mathit{t},v_{\mathit{t}})|\\ \mathit{ }\le |k_2(\mathit{t})[\frac{{\mathit{tan}}^{-1}|u_{\mathit{t}}{|}_{{\mathbb{B}}_2}}{1+{\tan }^{-1}|u_{\mathit{t}}{|}_{{\mathbb{B}}_2}}-\frac{{\mathit{tan}}^{-1}|v_{\mathit{t}}{|}_{{\mathbb{B}}_2}}{1+{\tan }^{-1}|v_{\mathit{t}}{|}_{{\mathbb{B}}_2}}]|,\\ \mathit{ }\le k_2(\mathit{t}){\tan }^{-1}(\frac{1}{3}\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_3})\\ \mathit{ }\le \mathit{\rho }{k}_2(\mathit{t}){\tan }^{-1}(\frac{1}{3}\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_3})\end{array}$

Similarily, we can show that

$|\mathit{g}(\mathit{t},u_{\mathit{t}})-\mathit{g}(\mathit{t},v_{\mathit{t}})|\le \mathit{\rho }{k}_1(\mathit{t})({\tan }^{-1}\frac{1}{3}\Vert u_{\mathit{t}}-v_{\mathit{t}}{\Vert }_{{\mathbb{B}}_3}),$

$\text{for}\mathit{t}\in [1,e^2]\text{and every}{u}_{\mathit{t}},v_{\mathit{t}}\in {\mathbb{B}}_2.$

Therefore, the IVP (21) has a unique solution on $[1,e^2]$ according to Theorem 3.2.

Example 5.3.

Consider the above problem (21) such that $|\mathit{f}(\mathit{t},u_{\mathit{t}})|\le \frac{e}{9}\Vert u_{\mathit{t}}{\Vert }_{{\mathbb{B}}_2}+2$ and $|\mathit{g}(\mathit{t},u_{\mathit{t}})|\le \frac{\mathit{ln}\mathit{t}}{(2+\ln \mathit{t}{)}^2-1}+\mathit{sin}\mathit{t}$ from which $p_1=\frac{e}{9},p_2=2$ and $\Vert \mathit{\chi }{\Vert }_{\mathit{b}}=\frac{5}{3}$. Thus the assumptions A₇ and A₈ of Theorem 3.3 are satisfied. With these given assumptions and Gronwaall inequalities for Hadamard derivative, we get

$B^{\ast }=6.0235712$

With regard to Theorem 3.3’s conclusion, there is at least one solution for the problem (21) on $[1,e^2].$

6. Conclusions

This work investigated the IVP of Caputo-Hadamard neutral fractional differential equations with infinite delay, a topic that has not yet been studied by any scholars. We established an abstract phase space and, using the Leray-Schauder nonlinear alternative, Boyd and Wong and the Banach fixed point theorems, were able to derive some sufficient conditions for the existence and uniqueness of solutions for the given problem. This study is different from previous studies on such fractional differential equations in that in order to show uniqueness of solutions, it uses a new type of continuous nondecreasing function in the nonlinear contraction and a Gronwall inequality utilizing Hadamard fractional integral in establishing existence results. Compared to previous studies, the Gronwall inequality is used to prove the dependency of solutions with regard to initial conditions in a systematic way. Since there hasn’t been any real-world application of either finite or infinite delay yet, researchers need to work hard to seek out how to take these theoretical discoveries about delays and apply them to real-world problems.

Notes on contributor

Mesfin Teshome Beyene is a PhD candidate at Department of Mathematics, College of Natural and Computional Sciences, Bule Hora University, Ethiopia. An analysis of fractional differential equations is the focus of his research.

Acknowledgements

The authors would like to express their sincere gratitude to the reviewers for their helpful comments and suggestions that helped build up the paper.

Disclosure statement

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

Supplemental material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2024.2321669

Additional information

Funding

This research received no specific funding from public, private, or non-profit funding agencies.