Unique solutions, stability and travelling waves for some generalized fractional differential problems

ABSTRACT

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann–Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau–Ginzburg–Higgs equation and the Sine–Gordon one. Some numerical simulations are plotted and a conclusion is given at the end.

KEYWORDS:

• Caputo derivative
• fixed point
• Duffing equation
• conformable derivative
• tanh method
• travelling wave

AMS CLASSIFICATIONS:

• 30C45
• 39B72
• 34A08
• 35R11

1. Introduction

The Duffing phenomenon is named after Georg Duffing, and refers to the nonlinear behaviour of a mechanical system, such as a spring-mass system, that experiences both damping and a periodic forcing. The nonlinearity results in the occurrence of a variety of dynamic behaviours, including limit cycles and chaos, which are not present in linear systems. The Duffing equation is widely used as a model to study nonlinear vibrations and chaos in various fields, including mechanical engineering, physics, and control systems. The Duffing equation is a nonlinear, second-order ordinary differential equation that describes the dynamics of a system subjected to a periodic driving force and a nonlinear restoring force (see [1–5]).

Traveling waves can exhibit different types of symmetry, depending on the characteristics of the wave and the medium through which it travels. Here are some examples of symmetry in travelling waves:

• Reflection symmetry: A wave has reflection symmetry if it looks the same when reflected in a line or plane. For example, a sinusoidal wave that travels along a string has reflection symmetry because it looks the same when reflected in the string.

• Translational symmetry: A wave has translational symmetry if it looks the same after it has been shifted by a fixed distance. For example, a wave that travels along an infinitely long string has translational symmetry because it looks the same at any point along the string.

• Rotational symmetry: A wave has rotational symmetry if it looks the same after it has been rotated around a fixed point. For example, a circular wave that travels outward from a point source has rotational symmetry because it looks the same when viewed from any angle around the source. Time-reversal symmetry: A wave has time-reversal symmetry if it looks the same when time is reversed. For example, an electromagnetic wave that travels through free space has time-reversal symmetry because it looks the same whether it is moving forward or backward in time. In addition, the Duffing equation is symmetric under the transformation $x\to -x$, which is known as the parity transformation. This means that if the displacement x of the oscillator is replaced with its negative, the equation remains unchanged.

Fractional Differential Equations (FDEs) are mathematical equations that involve derivatives of non-integer order. Unlike traditional differential equations, which involve integer order derivatives, FDEs allow for the modelling of complex physical processes that exhibit memory and hereditary properties. FDEs are widely used in various fields including physics, engineering, finance, and biology to study problems such as anomalous diffusion, control theory, and viscoelasticity. The solution methods for FDEs include the Laplace transform, the Fourier transform, numerical techniques, and variational methods. The term ‘conformable derivative’ refers to a type of derivative that can be used in mathematical models to describe the rate of change of a function. It is similar to a traditional derivative, but is designed to handle functions with non-uniform scales. The concept is used in various branches of mathematics, including fractional calculus and mathematical finance [6].

The fractional Duffing phenomenon refers to the nonlinear behaviour of a mechanical system that experiences both damping and a periodic forcing, described by a fractional differential equation. Fractional differential equations are a generalization of the classical differential equations and can model complex dynamic behaviours, including memory and long-range dependence, which are not possible to capture using classical differential equations. The fractional Duffing equation is used to study nonlinear vibrations and chaos in systems that exhibit fractional dynamic behaviour, such as in viscoelastic materials, complex networks, and biological systems. The fractional Duffing equation has been found to exhibit a wider range of dynamic behaviours compared to the classical Duffing equation, including multistability, bifurcations, and chaotic attractors.

The oscillator phenomena have several applications in science and engineering. One of such oscillators is the Duffing phenomenon [7]. The equation of Duffing is a nonlinear problem [8,9]. It has been used successfully for modelling a variety of physical and chemical processes, reinforcing springs, beam buckling, nonlinear electronic circuits,…

The forced form of Duffing equation is: $\ddot{x}+p\dot{x}+qx+s{x}^3=r\cos \left(\omega t\right),$ where $x=x\left(t\right)$ is the displacement at time, $x^{\prime }\left(t\right)$ is the velocity, and $x^{″}\left(t\right)$ is the acceleration. The numbers p, q, s, r and ω are some given constants.

Up to now, many methods used for solving the above ‘forced’ equation and other nonlinear differential equations are developed, for example the first integral method [10], the exp-function method [11,12], the (G'/G) method [13], the Jacobi method [14], and the tanh method [15,16]. The tanh method seems to be one of the important algebraic methods serving to obtain solutions and explicit travelling waves for nonlinear differential problems, see for instance [15,17,18].

On the other hand, fractional differential problems have attracted attention of researchers for several times; they have been applied in physics, chemistry, biology to model many real word phenomena, see [19–23].

In this sense, in [24], by using Riemann–Liouville derivatives, V.A. Kim has considered the following problem of type Duffing: $\ddot{Z}\left(t\right)+\alpha D_{0t}^{q\left(t\right)}Z\left(\tau \right)-Z\left(t\right)+Z^3\left(t\right)=\text{β}\cos \left(\omega t\right),Z\left(0\right)=x_0,\dot{Z}\left(0\right)=y_0.$ Also in [25], Y. Gouari et al. have investigated the following problem with Duffing type that involves sequential derivatives: $\{\begin{array}{l}{D}^{\alpha }\left(D^{\text{β}}\right(D^{\delta }S\left(t\right)\left)\right)+f(t,S(t),D^{p}S(t\left)\right)+g(t,S(t),I^{q}S(t\left)\right)+h(t,S(t\left)\right)=b\left(t\right),\\ S\left(0\right)=\xi \in \mathbb{R},\\ {\displaystyle S\left(1\right)={\int }_0^{\eta }S\left(s\right)\mathrm{d}s,0<\eta <1,}\\ I^{e}S\left(\theta \right)=D^{\delta }S\left(1\right),0<u<1,\\ 0<\alpha ,\text{β},\delta ,p\le 1,q>0,t\in J,\end{array}$ Very recently in [12], M. Rakah et al. have taken into account the study of the following problem: $\{\begin{array}{l}{\displaystyle D^{\alpha }{D}^{\text{β}}{D}^{\gamma }Z\left(t\right)=\frac{a_{1}f(t,Z(t),D^{\gamma }Z(t\left)\right)+a_{2}g(t,D^{\gamma }Z(t),D^{\gamma }{D}^{\rho }Z(t\left)\right)+a_{3}h(t,Z(t\left)\right)}{K\left(Z\right(t\left)\right)},}\\ Z\left(0\right)+Z\left(1\right)={\int }_0^{\eta }bZ\left(s\right)\mathrm{d}s,0<\eta <1,\\ D^{\gamma }Z\left(0\right)+D^{\gamma }Z\left(1\right)=0,\\ D^{\mu }{D}^{\mu }Z\left(0\right)+D^{\mu }{D}^{\mu }Z\left(1\right)=0,\\ t\in [0,1],\\ 0<\alpha ,\text{β},\gamma ,\rho ,\mu \le 1,\\ a_1,a_2,a_3\in \mathbb{R}.\end{array}$ The existence of one solution has been treated by the authors.

In the work [26], the authors have looked for the investigation of the following sequential VdP Duffing equation: $\{\begin{array}{l}{D}^{\alpha }(D^{2-\text{β}}+\lambda D^{\alpha })Z\left(t\right)+k_1{f}_1(t,Z(t),D^{\alpha }Z(t\left)\right)+k_2{f}_2(t,Z(t),J^{p}Z(t\left)\right)=r\left(t\right),\\ Z\left(1\right)=0,D^{1-(\alpha -\text{β})}{D}^{\alpha -\text{β}}Z\left(1\right)=A^{\ast }\in \mathbb{R},Z\left(T\right)=0,\\ 0\le \text{β}<\alpha \le 1,0\le \alpha +\text{β}<1,p>0,t\in I.\end{array}$ They have studied the problem by using Caputo Hadamard approach. They have established new conditions for studying the solutions for the problem. The stability has also been discussed in their paper.

The aim of the first part of the present work is to study the following generalized fractional differential problem: (1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) For (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ), we take J the interval $[0,1],$ also, we consider that $0\le {\alpha }_i<1,i=1,2,\dots ,n$, and each derivative of the problem is of Caputo, $I^q$ is the Riemann–Liouville integral, the functions f and g are supposed to be defined from $J\times {\mathbb{R}}^2$ to $\mathbb{R}$, also $K:J\times {\mathbb{R}}^2\to {\mathbb{R}}_{+}^{\ast }$ are continuous, and $m_j$ are defined and continuous over $J\times \mathbb{R}$, $n_j,{\delta }_j$ are defined over J, ${\sigma }_j\in \mathbb{R},$ $j\in {\mathbb{N}}^{\ast }$, the constants ${\eta }_1,{\eta }_2$ are some reals.

We begin by proving an integral representation for our problem. Then, an existence and uniqueness result is established. Also, we study the stability of the unique ‘proved’ solution. An illustrative example is then discussed in details.

In the second part of the present work, we will use the method of tanh to obtain new travelling waves for three important problems of type: (2) $T_t^{2\alpha }u+a{T}_x^{2\text{β}}u+bu+c{u}^2+\mathrm{d}{u}^3=0,$(2) where $T_x^{\text{β}},T_t^{\alpha }$ are fractional derivatives of conformable type [27] (for more extended operator, one can see [28,29]), with $0<\alpha ,\text{β}\le 1$ and a, b, s, d are some real constants.

2. Background

In this Background second section, we have to present to the reader some definitions of fractional calculus theory, see [30,31].

Definition 2.1

We consider $v:(0,\mathrm{\infty })\to \mathbb{R}$ and $0<\alpha$. Then the derivative of Caputo with order α for v is: $D^{\alpha }v\left(x\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^x(x-s{)}^{n-\alpha -1}{v}^{\left(n\right)}(s)\mathrm{d}s,n-1<\alpha <n,n=[\alpha ]+1,$ such that $\left[\alpha \right]$ the integer part of the parameter α.

Definition 2.2

The Riemann–Liouville integral of order $0<\alpha$ for any function $v:(0,\mathrm{\infty })\to \mathbb{R}$ is given by $I^{\alpha }v\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^x(x-s{)}^{\alpha -1}v(s)\mathrm{d}s,t>0.$

Lemma 2.3

The solution of the generalized fractional equation $D^{\alpha }u\left(t\right)=k\left(t\right)$, $\mathrm{t}\in \mathrm{J}$, has the form: $I^{\alpha }{D}^{\alpha }u\left(\mathrm{t}\right)=I{}^{\alpha }k\left(\mathrm{t}\right)+c_0+c_1\mathrm{t}+\cdots +c_{n-1}{\mathrm{t}}^{n-1},$ taking into account that ${\mathrm{c}}_{\mathrm{i}}\in \mathbb{R},i=0,\dots \mathrm{n}-1$.

Lemma 2.4

If α and β are two positive parameters, then we have $D^{\alpha }{I}^{\alpha }v\left(t\right)=v\left(t\right),I^{\alpha }{I}^{\text{β}}v\left(t\right)=I^{\alpha +\text{β}}v\left(t\right).$

Lemma 2.5

Let G be a Banach space and $Z:G⟶G$ is an application that is contractible. Then, Z has exactly one fixed point in G.

Let us now present then prove another lemma.

Lemma 2.6

We consider K and $(H_j{)}_{j=1,\dots ,r},$ $r\in {\mathbb{N}}^{\ast }$ in $C\left(J\right)$. Then, the BVP (3) $\{\begin{array}{l}{D}^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=K\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{H}_j\left(t\right),\\ u\left(0\right)-\theta =0,\\ {\displaystyle u\left(1\right)-{\int }_0^{x}u\left(s\right)\mathrm{d}s=0,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j-1>0,\end{array}$(3) is equivalent to the following representation: (4) $\begin{array}{rl}u\left(t\right)& =I^{\sum _{i=1}^n{\alpha }_i}G\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(t\right)\\ & -{\zeta }_1\left[I^{{\alpha }_1}G\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}{H}_j(1\left)\right]t^{\sum _{i=2}^n{\alpha }_i}\\ & +{\zeta }_2\left[{\int }_0^x{I}^{\sum _{i=1}^n{\alpha }_i}G\right(s)\mathrm{d}s+{\int }_0^x\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j(s)\mathrm{d}s\\ & -I^{\sum _{i=1}^n{\alpha }_i}G\left(1\right)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(1\right)\\ & +{\zeta }_3\left(I^{{\alpha }_1}G\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}{H}_j(1\left)\right)+\theta (x-1)]\\ & \times \frac{t^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}+\theta ,\end{array}$(4) such that $\begin{array}{rl}{\zeta }_1& =\frac{1}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+1)},{\zeta }_2=\frac{\mathrm{\Gamma }({\alpha }_n+2)}{x^{{\alpha }_n+1}-{\alpha }_n-1},\\ {\zeta }_3& ={\left(\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+1)\right)}^{-1}-\frac{x^{1+\sum _{i=2}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+2)}.\end{array}$

Proof.

It is easy to see that the equation $D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=G\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{H}_j\left(t\right),$ permits us to write (5) $\begin{array}{rl}u\left(t\right)& =I^{\sum _{i=1}^n{\alpha }_i}G\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(t\right)\\ & +\frac{c_1{t}^{\sum _{i=2}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+1)}+\frac{c_2{t}^{\sum _{i=3}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=3}^n{\alpha }_i+1)}+\frac{c_3{t}^{\sum _{i=4}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=4}^n{\alpha }_i+1)}\\ & +\cdots +\frac{c_{n-1}{t}^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}+c_n.\end{array}$(5) Using some of the above initial conditions, we get: $\begin{array}{rl}u\left(0\right)& =\theta \Rightarrow c_n=\theta ,\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)& =0\Rightarrow c_2=0,\\ D^{{\alpha }_4}{D}^{{\alpha }_5}\cdots D^{{\alpha }_n}u\left(1\right)& =0\Rightarrow c_3=0,\\ ⋮& \\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)& =0\Rightarrow c_{n-2}=0.\end{array}$ By substitution, we get $\begin{array}{rl}u\left(t\right)& =I^{\sum _{i=1}^n{\alpha }_i}G\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(t\right)\\ & +\frac{c_1{t}^{\sum _{i=2}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+1)}+\frac{c_{n-1}{t}^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}+\theta .\end{array}$ On the other hand, we have $\begin{array}{rl}& D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ & \Rightarrow I^{{\alpha }_1}G\left(1\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}{H}_j\left(1\right)+c_1=0,\\ & \Rightarrow c_1=-I^{{\alpha }_1}G\left(1\right)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}{H}_j\left(1\right).\end{array}$ By using the fact that $u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s$, we observe that: $\begin{array}{rl}{c}_{n-1}& =\left[\frac{\mathrm{\Gamma }({\alpha }_n+2)}{x^{{\alpha }_n+1}-{\alpha }_n-1}\right]\left[{\int }_0^x{I}^{\sum _{i=1}^n{\alpha }_i}G\right(s)\mathrm{d}s-I^{\sum _{i=1}^n{\alpha }_i}G(1)\\ & +{\int }_0^x\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(s\right)\mathrm{d}s\\ & -\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(1\right)\\ & +\left(I^{{\alpha }_1}G\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}{H}_j(1\left)\right)\\ & \times (\frac{1}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+1)}-\frac{x^{1+\sum _{i=2}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+2)})+\theta (x-1)].\end{array}$ Inserting the values of $c_1,c_2,\dots ,c_n$ in (5(5) $\begin{array}{rl}u\left(t\right)& =I^{\sum _{i=1}^n{\alpha }_i}G\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}{H}_j\left(t\right)\\ & +\frac{c_1{t}^{\sum _{i=2}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i+1)}+\frac{c_2{t}^{\sum _{i=3}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=3}^n{\alpha }_i+1)}+\frac{c_3{t}^{\sum _{i=4}^n{\alpha }_i}}{\mathrm{\Gamma }(\sum _{i=4}^n{\alpha }_i+1)}\\ & +\cdots +\frac{c_{n-1}{t}^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}+c_n.\end{array}$(5) ), we end the proof.

3. Main results

3.1. Part 1: uniqueness of solutions and stability

Let us consider space $X:=\{v\in C(J,\mathbb{R}),D^{\gamma }v\in C(J,\mathbb{R}\left)\right\},$ with the norm is defined by $\Vert v{\Vert }_X=\Vert v{\Vert }_{\mathrm{\infty }}+\Vert D^{\gamma }v{\Vert }_{\mathrm{\infty }},$ where, $\Vert v{\Vert }_{\mathrm{\infty }}=\underset{t\in J}{sup}|v\left(t\right)|,\Vert D^{\gamma }v{\Vert }_{\mathrm{\infty }}=\underset{t\in J}{sup}|D^{\gamma }v\left(t\right)|.$ In addition, we define $W:X\to X$ by: (6) $\begin{array}{rl}Wu\left(t\right)& =I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}\left(H_j{)}_u^{\ast }\right(t)\\ & -{\zeta }_1\left[I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right)]t^{\sum _{i=2}^n{\alpha }_i}\\ & +{\zeta }_2\left[{\int }_0^x{I}^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\right(s)\mathrm{d}s+{\int }_0^x\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}(H_j{)}_u^{\ast }\left(s\right)\mathrm{d}s\\ & -I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\left(1\right)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}\left(H_j{)}_u^{\ast }\right(1)\\ & +{\zeta }_3\left(I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right))+\theta (x-1)]\\ & \times \frac{t^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}+\theta .\end{array}$(6) where $\begin{array}{rl}{G}_u^{\ast }\left(t\right)& =\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)},\\ \left(H_j{)}_u^{\ast }\right(t)& ={\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right].\end{array}$ Now, we have to consider the three hypotheses:

(F1):	The given functions of (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ) are continuous.
(F2):	There exist non-negative continuous functions ${\varphi }_1\left(t\right),{\varphi }_2\left(t\right),{\omega }_1\left(t\right),{\omega }_2\left(t\right),j\in \mathbb{N}$, such that for any $t\in J$, $u_i,v_i\in \mathbb{R},$ $\begin{array}{rl}\left|\frac{f}{K}\right(t,u_1,u_2)-\frac{f}{K}(t,v_1,v_2\left)\right|-\left[{\varphi }_1\right(t)|u_1-v_1|+{\varphi }_2(t)|u_2-v_2|]& \le 0,\\ \left|\frac{g}{K}\right(t,u_1,u_2)-\frac{f}{K}(t,v_1,v_2\left)\right|-\left[{\omega }_1\right(t)|u_1-v_1|+{\omega }_2(t)|u_2-v_2|)]& \le 0,\end{array}$ and there exist positive real numbers ${\lambda }_j$ such that, for any $t\in J$, $u,v\in \mathbb{R},$ $|m_j(t,u)-m_j(t,v|\le {\lambda }_j|u-v|,$ and $\sum _{j=1}^{\mathrm{\infty }}{\lambda }_j\le {\mathrm{\Lambda }}_m.$ We define also the quantities: $\begin{array}{rl}\varphi & =\mathrm{M}\mathrm{a}\mathrm{x}\left(\underset{t\in J}{sup}|{\varphi }_1\right(t)|,\underset{t\in J}{sup}|{\varphi }_2(t\left)|\right),\\ \omega & =\mathrm{M}\mathrm{a}\mathrm{x}\left(\underset{t\in J}{sup}|{\omega }_1\right(t),\underset{t\in J}{sup}|{\omega }_2(t\left)|\right).\end{array}$
(F3):	Suppose that ${\delta }_j$ is defined on J, and $M_{\delta }=\sum _{j=1}^{\mathrm{\infty }}\Vert {\sigma }_j{\delta }_j\left(t\right){\Vert }_{\mathrm{\infty }}<+\mathrm{\infty }.$

Now, we consider the quantities: $\begin{array}{rl}{\mathrm{\Psi }}_1& =M_{f,g,m}[{\left(\mathrm{\Gamma }(1+\sum _{i=1}^n{\alpha }_i)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+1+\sum _{i=1}^n{\alpha }_i)\right)}^{-1}\\ & \phantom{1+\sum _{i=1}^n}+|{\zeta }_1|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_1|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n+1)}[x^{1+\sum _{i=1}^n{\alpha }_i}{\left(\mathrm{\Gamma }(2+\sum _{i=1}^n{\alpha }_i)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{1+\sum _{i=1}^n}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}].\\ {\mathrm{\Psi }}_2& =M_{f,g,m}[{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i-\gamma +1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q-\gamma +\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i-\gamma +1)\right)}^{-1}\left)\right[\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}\left]\right]\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n-\gamma +1)}[x^{\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{1+\sum _{i=1}^n}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}].\end{array}$ and $M_{f,g,m}=|{\eta }_1|\varphi +|{\eta }_2|(\omega +\frac{\omega }{\mathrm{\Gamma }(q+1)})+{\mathrm{\Lambda }}_m{M}_{\delta }.$

3.1.1. A unique solution

We present the first result as follows:

Theorem 3.1

We consider that $\left(F_1\right),\left(F_2\right)$, and $\left(F_3\right)$ are verified. Also, we suppose $\mathrm{\Psi }<1;\mathrm{\Psi }={\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2.$ Thus, (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ) has a unique solution defined on J.

Proof.

We try to prove the contraction of W.

We take any $(u,v)\in X^2$, so we immediately get (7) $\begin{array}{rl}\Vert Wu-Wv{\Vert }_{\mathrm{\infty }}& \le M_{f,g,m}[{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\sum _{i=1}^n{\alpha }_i}+|{\zeta }_1|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_1|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-v{\Vert }_X\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n+1)}[x^{\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\sum _{i=1}^n{\alpha }_i}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-v{\Vert }_X.\end{array}$(7) We have also (8) $\begin{array}{rl}{D}^{\gamma }Wu\left(t\right)& =I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\left(t\right)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}\left(H_j{)}_u^{\ast }\right(t)\\ & -\left[I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right)]\frac{t^{\sum _{i=2}^n{\alpha }_i-\gamma }}{\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i-\gamma +1)}\\ & +{\zeta }_2\left[{\int }_0^x{I}^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\right(s)\mathrm{d}s+{\int }_0^x\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}(H_j{)}_u^{\ast }\left(s\right)\mathrm{d}s\\ & -I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\left(1\right)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}\left(H_j{)}_u^{\ast }\right(1)\\ & +{\zeta }_3\left(I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right))+\theta (x-1)]\\ & \times \frac{t^{{\alpha }_n-\gamma }}{\mathrm{\Gamma }({\alpha }_n-\gamma +1)}.\end{array}$(8) With the same arguments as before, we have (9) $\begin{array}{rl}& \Vert D^{\gamma }Wu-D^{\gamma }Wv{\Vert }_{\mathrm{\infty }}\\ & \le M_{f,g,m}[{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i-\gamma +1)\right)}^{-1}+(\mathrm{\Gamma }(q-\gamma +\sum _{i=1}^n{\alpha }_i+1){)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i-\gamma +1)\right)}^{-1}\left)\right[\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}\left]\right]\Vert u-v{\Vert }_X\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n-\gamma +1)}[x^{\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i-\gamma +1)\right)}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-v{\Vert }_X.\end{array}$(9) From (7(7) $\begin{array}{rl}\Vert Wu-Wv{\Vert }_{\mathrm{\infty }}& \le M_{f,g,m}[{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\sum _{i=1}^n{\alpha }_i}+|{\zeta }_1|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_1|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-v{\Vert }_X\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n+1)}[x^{\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\sum _{i=1}^n{\alpha }_i}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-v{\Vert }_X.\end{array}$(7) ) and (9(9) $\begin{array}{rl}& \Vert D^{\gamma }Wu-D^{\gamma }Wv{\Vert }_{\mathrm{\infty }}\\ & \le M_{f,g,m}[{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i-\gamma +1)\right)}^{-1}+(\mathrm{\Gamma }(q-\gamma +\sum _{i=1}^n{\alpha }_i+1){)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=2}^n{\alpha }_i-\gamma +1)\right)}^{-1}\left)\right[\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}\left]\right]\Vert u-v{\Vert }_X\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n-\gamma +1)}[x^{\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i-\gamma +1)\right)}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-v{\Vert }_X.\end{array}$(9) ), we get $\Vert W{u}_1-W{u}_2{\Vert }_X\le ({\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2)\Vert u_1-u_2{\Vert }_X.$ We see that W a contraction application. Therefore, we prove the uniqueness of fixed point for W, which is the solution of (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ).

Example:

As an example, we can take the special problem: (10) $\{\begin{array}{l}{\displaystyle D^{\frac{7}{8}}{D}^{\frac{2}{3}}{D}^{\frac{1}{2}}{D}^{\frac{6}{10}}{D}^{\frac{4}{5}}{D}^{\frac{9}{10}}{D}^{\frac{3}{5}}{D}^{\frac{7}{10}}u\left(s\right)}\\ {\displaystyle =\frac{2(1+|u\left(t\right)|+|D^{0.5}u\left(t\right)\left|\right)\sin \left(u\right(t)+D^{0.5}u(t\left)\right)+{\mathrm{e}}^{t^2+2}\left(\right|D^{0.5}u\left(t\right)|+|I^{\frac{1}{6}}u\left(t\right)\left|\right)}{2{\mathrm{e}}^{t^2+2}(1+|u\left(t\right)|+|D^{0.5}u\left(t\right)\left|\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}\frac{1}{j^2}{I}^{\frac{1}{6}}{\mathrm{e}}^{-j{t}^2-1}[\frac{1}{30{\mathrm{e}}^{jt}}\frac{\sin \left(2\right|u\left(t\right)\left|\right)}{j^2{\pi }^2}+\frac{1}{3j^2}\frac{t^2}{{\mathrm{e}}^{j{t}^2}}],0\le t\le 1,}\\ u\left(0\right)-e=0,\\ {\displaystyle u\left(1\right)-{\int }_0^{\frac{1}{4}}u\left(s\right)\mathrm{d}s=0,}\\ {\displaystyle D^{\frac{7}{10}}u\left(0\right)=0,}\\ {\displaystyle D^{\frac{3}{5}}{D}^{\frac{7}{10}}u\left(0\right)=0,}\\ {\displaystyle D^{\frac{9}{10}}{D}^{\frac{3}{5}}{D}^{\frac{7}{10}}u\left(0\right)=0,}\\ {\displaystyle D^{\frac{4}{5}}{D}^{\frac{9}{10}}{D}^{\frac{3}{5}}{D}^{\frac{7}{10}}u\left(0\right)=0}\\ {\displaystyle D^{\frac{1}{2}}{D}^{\frac{6}{10}}{D}^{\frac{4}{5}}{D}^{\frac{9}{10}}{D}^{\frac{3}{5}}{D}^{\frac{7}{10}}u\left(0\right)=0}\\ {\displaystyle D^{\frac{2}{3}}{D}^{\frac{1}{2}}{D}^{\frac{6}{10}}{D}^{\frac{4}{5}}{D}^{\frac{9}{10}}{D}^{\frac{3}{5}}{D}^{\frac{7}{10}}u\left(1\right)=0,}\end{array}$(10) where, $\begin{array}{rl}f(t,x_1,x_2)& =2(1+|x_1|+|x_2\left|\right)\sin (x_1+x_2),\\ g(t,x_1,x_2)& ={\mathrm{e}}^{t^2+2}\left(\right|x_1|+|x_2\left|\right),\\ K(t,x_1,x_2)& =2{\mathrm{e}}^{t^2+2}(1+|x_1|+|x_2\left|\right),\\ m_j(t,x)& =\frac{1}{30{\mathrm{e}}^{jt}}\frac{\sin \left(2\right|x\left|\right)}{j^2{\pi }^2},\\ n_j\left(t\right)& =\frac{1}{3j^2}\frac{t^2}{{\mathrm{e}}^{j{t}^2}},\\ {\mathrm{\Psi }}_1& =0.0742,{\mathrm{\Psi }}_2=0.1223,\\ \mathrm{\Psi }& =0.0742+0.1223=\mathrm{0.1965.}\end{array}$ Theorem 3.1 allows the reader to confirm for the above example, we have exactly one solution.

3.1.2. Stability of the unique solution

We introduce the following definitions

Definition 3.2

We say that (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ) is stable in the sense of Ulam Hyers if there is a $\mathrm{\Xi }>0$, such that for each positive ϵ and for an arbitrary solution $u\in X$ of (11) $\begin{array}{rl}& |D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u(t)-\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}\\ & -\sum _{j=1}^{\mathrm{\infty }}\phantom{\frac{2}{2}}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right]|\le ϵ,\end{array}$(11) we can obtain a solution $u^{\ast }\in X$ of (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ), with $\Vert u-u^{\ast }{\Vert }_X\le \mathrm{\Xi }ϵ.$

Definition 3.3

We say that (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ) is stabel in the generalised sens of Ulam Hyers if we can find a function $\chi \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$; $\chi \left(0\right)=0$; that verifies for any positive ϵ, and for each solution $u\in X$ of (11(11) $\begin{array}{rl}& |D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u(t)-\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}\\ & -\sum _{j=1}^{\mathrm{\infty }}\phantom{\frac{2}{2}}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right]|\le ϵ,\end{array}$(11) ), there is $u^{\ast }\in X$ solution of (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ); with $\Vert u-u^{\ast }{\Vert }_X<\chi \left(ϵ\right).$

We pass to consider the following main theorem.

Theorem 3.4

Suppose all the hypotheses of 3.1 are valid. Then, (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ) is stable in Ulam Hyers sense.

Proof.

We take a solution $u\in X$ of (11(11) $\begin{array}{rl}& |D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u(t)-\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}\\ & -\sum _{j=1}^{\mathrm{\infty }}\phantom{\frac{2}{2}}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right]|\le ϵ,\end{array}$(11) ). Therefore, Theorem 3.1 guarantees the existence of a unique solution $u^{\ast }\in X$ for (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ).

Using (11(11) $\begin{array}{rl}& |D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u(t)-\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}\\ & -\sum _{j=1}^{\mathrm{\infty }}\phantom{\frac{2}{2}}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right]|\le ϵ,\end{array}$(11) ), we observe that (12) $\begin{array}{rl}& \left|u\right(t)-I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }(t)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}(H_j{)}_u^{\ast }\left(t\right)\\ & +{\zeta }_1\left[I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right)]t^{\sum _{i=2}^n{\alpha }_i}\\ & -{\zeta }_2\left[{\int }_0^x{I}^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\right(s)\mathrm{d}s+{\int }_0^x\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}(H_j{)}_u^{\ast }\left(s\right)\mathrm{d}s\\ & -I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\left(1\right)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}\left(H_j{)}_u^{\ast }\right(1)\\ & +{\zeta }_3\left(I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right))+\theta (x-1)]\\ & \times \phantom{\sum _{i=1}^n}\frac{t^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}-\theta |\le \frac{ϵ}{\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)}.\end{array}$(12) Using (11(11) $\begin{array}{rl}& |D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u(t)-\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}\\ & -\sum _{j=1}^{\mathrm{\infty }}\phantom{\frac{2}{2}}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right]|\le ϵ,\end{array}$(11) ) and (12(12) $\begin{array}{rl}& \left|u\right(t)-I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }(t)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}(H_j{)}_u^{\ast }\left(t\right)\\ & +{\zeta }_1\left[I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right)]t^{\sum _{i=2}^n{\alpha }_i}\\ & -{\zeta }_2\left[{\int }_0^x{I}^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\right(s)\mathrm{d}s+{\int }_0^x\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}(H_j{)}_u^{\ast }\left(s\right)\mathrm{d}s\\ & -I^{\sum _{i=1}^n{\alpha }_i}{G}_u^{\ast }\left(1\right)-\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+\sum _{i=1}^n{\alpha }_i}\left(H_j{)}_u^{\ast }\right(1)\\ & +{\zeta }_3\left(I^{{\alpha }_1}{G}_u^{\ast }\right(1)+\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^{q+{\alpha }_1}(H_j{)}_u^{\ast }\left(1\right))+\theta (x-1)]\\ & \times \phantom{\sum _{i=1}^n}\frac{t^{{\alpha }_n}}{\mathrm{\Gamma }({\alpha }_n+1)}-\theta |\le \frac{ϵ}{\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)}.\end{array}$(12) ), we get $\begin{array}{rl}& \Vert u-u^{\ast }{\Vert }_{\mathrm{\infty }}\le \frac{ϵ}{\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)}\\ & +M_{f,g,m}[{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}+|{\zeta }_1|(\mathrm{\Gamma }({\alpha }_1+1){)}^{-1}\\ & +|{\zeta }_1|\left(\mathrm{\Gamma }\right(q+{\alpha }_1+1\left){)}^{-1}\right]\Vert u-u^{\ast }{\Vert }_{\mathrm{\infty }}\\ & +\frac{M_{f,g,m}|{\zeta }_2|}{\mathrm{\Gamma }({\alpha }_n+1)}[x^{\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}\\ & +x^{q+\sum _{i=1}^n{\alpha }_i+1}{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+2)\right)}^{-1}+{\left(\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & +{\left(\mathrm{\Gamma }(q+\sum _{i=1}^n{\alpha }_i+1)\right)}^{-1}\\ & \phantom{\left(\sum _{i=1}^n\right)}+|{\zeta }_3|\left(\mathrm{\Gamma }\right({\alpha }_1+1){)}^{-1}+|{\zeta }_3|(\mathrm{\Gamma }(q+{\alpha }_1+1){)}^{-1}]\Vert u-u^{\ast }{\Vert }_{\mathrm{\infty }}.\end{array}$ Therefore, $\Vert u-u^{\ast }{\Vert }_{\mathrm{\infty }}\le \frac{ϵ}{\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)}+{\mathrm{\Psi }}_1\Vert u-u^{\ast }{\Vert }_{\mathrm{\infty }}$ Thus, $\Vert u-u^{\ast }{\Vert }_{\mathrm{\infty }}\le \frac{ϵ}{\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)(1-{\mathrm{\Psi }}_1)}\le ϵ\mathrm{\Xi }.$ On the other hand, we have $\Vert D^{\gamma }(u-u^{\ast }){\Vert }_{\mathrm{\infty }}\le \frac{ϵ}{\mathrm{\Gamma }(\sum _{i=1}^n{\alpha }_i+1)(1-{\mathrm{\Psi }}_2)}\le ϵ{\mathrm{\Xi }}^{\ast }.$ Consequently, $\Vert u-u^{\ast }{\Vert }_X\le ϵ(\mathrm{\Xi }+{\mathrm{\Xi }}^{\ast }).$ In consequence, (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ) is stable in the sense of Ulam Hyers.

Remark 3.5

By considering the case where $\chi \left(ϵ\right)=ϵ(\mathrm{\Xi }+{\mathrm{\Xi }}^{\ast })$, we can obtain the stability in the generalised sense for (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ).

3.2. Part 2: new travelling wave solutions by tanh method

We are interested in applying the method of Tanh to solve some fractional conformable problems of type: $T_t^{2\alpha }u+a{T}_x^{2\text{β}}u+bu+c{u}^2+\mathrm{d}{u}^3=0,$ where $T_x^{\text{β}},T_t^{\alpha }$ are the conformable fractional derivative, with α and β are in $0,1\left]\right]$and $a,b,c,d\in \mathbb{R}$.

Note that when $\alpha =\text{β}=1,$ the above equation can be transformed into the following type of Duffing equation [32]: (13) $u_{tt}+u_{xx}+bu+c{u}^2+\mathrm{d}{u}^3=0.$(13)

3.2.1. Conformable derivatives

We need to introduce the following preliminaries (see [27,33,34])

Definition 3.6

Taking the function f defined over $(0,\mathrm{\infty })$. The conformable derivative with a fractional order α in $]0,1]$ is: $\left(T^{\alpha }f\right)\left(x\right)=\frac{{\mathrm{\partial }}^{\alpha }f(x,y)}{\mathrm{\partial }{x}^{\alpha }}=\underset{ϵ\to 0}{lim}\left(\frac{f(ϵx^{1-\alpha }+x)-f\left(x\right)}{ϵ}\right),x\in {\mathbb{R}}_{+}^{\ast }$

Definition 3.7

The conformable integral of any function defined over the infinite interval $(0,\mathrm{\infty })$ of order α is given by $\left(I^{\alpha }f\right)\left(t\right)=\stackrel{t}{\underset{0}{\int }}{\tau }^{\alpha -1}f\left(\tau \right)\mathrm{d}\tau ,0<\alpha \le 1.$

Based on the above definitions, we have: $I^{\alpha }{T}^{\alpha }f\left(t\right)=f\left(t\right)-f\left(0\right)$ and $\left(T^{\alpha }f\right)\left(t\right)=t^{1-\alpha }\frac{\mathrm{d}f\left(t\right)}{\mathrm{d}t}.$

3.2.2. Method of tanh

We consider the equation (14) $N(u,T_t^{\alpha }u,T_x^{\text{β}}u,T_t^{2\alpha }u,T_t^{\alpha }(T_x^{\text{β}}u),T_x^{2\text{β}}u,\dots )=0,$(14) where $T_t^{\alpha }u$ is the fractional conformable derivative of u of order $\alpha ,0<\alpha \le 1$. Introducing the new wave variable (15) $\xi =\frac{k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}},$(15) so, (15(15) $\xi =\frac{k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}},$(15) ) can be transformed to the equation: (16) $F(U,U^{{}^{\mathrm{\prime }}},U^{{}^{\mathrm{\prime }\mathrm{\prime }}},U^{{}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}},\dots )=0.$(16) We then put (17) $\mathrm{{\rm Y}}=\tanh \left(\xi \right).$(17) So, we have (18) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}\xi }& =(1-{\mathrm{{\rm Y}}}^2)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}},\\ \frac{{\mathrm{d}}^2}{\mathrm{d}{\xi }^2}& =2\mathrm{{\rm Y}}({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}}+{({\mathrm{{\rm Y}}}^2-1)}^2\frac{{\mathrm{d}}^2}{\mathrm{d}{\mathrm{{\rm Y}}}^2}.\end{array}$(18) Now, we consider (19) $u(x,t)=U\left(\xi \right)=F\left(\mathrm{{\rm Y}}\right)=\sum _{i=0}^m{a}_i{\mathrm{{\rm Y}}}^i,$(19) The parameter m is to be obtained.

3.2.3. Applications

Example 3.1

We take the Landau–Ginzburg–Higgs equation [35] (Figure 1): (20) $T_t^{2\alpha }u-T_x^{2\text{β}}u-g^{2}u+h^2{u}^3=0.$(20) Using (15(15) $\xi =\frac{k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}},$(15) ), we change (20(20) $T_t^{2\alpha }u-T_x^{2\text{β}}u-g^{2}u+h^2{u}^3=0.$(20) ) into the following nonlinear ODE (21) $(k^2-{\omega }^2)U_{\zeta \zeta }-g^{2}U+h^2{U}^3=0.$(21) Substituting (18(18) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}\xi }& =(1-{\mathrm{{\rm Y}}}^2)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}},\\ \frac{{\mathrm{d}}^2}{\mathrm{d}{\xi }^2}& =2\mathrm{{\rm Y}}({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}}+{({\mathrm{{\rm Y}}}^2-1)}^2\frac{{\mathrm{d}}^2}{\mathrm{d}{\mathrm{{\rm Y}}}^2}.\end{array}$(18) ) and (19(19) $u(x,t)=U\left(\xi \right)=F\left(\mathrm{{\rm Y}}\right)=\sum _{i=0}^m{a}_i{\mathrm{{\rm Y}}}^i,$(19) ) into (21(21) $(k^2-{\omega }^2)U_{\zeta \zeta }-g^{2}U+h^2{U}^3=0.$(21) ), we can get (22) $(k^2-{\omega }^2)\left[2\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}F}{\mathrm{d}\mathrm{{\rm Y}}}+({\mathrm{{\rm Y}}}^2-1{)}^2\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}]-g^{2}F+h^2{F}^3=0.$(22) We ‘balance’ ${\mathrm{{\rm Y}}}^4\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}$ with $F^3$ to obtain m = 1.

Figure 1. 3D-plot of case1-travelling wave solution of (20(20) $T_t^{2\alpha }u-T_x^{2\text{β}}u-g^{2}u+h^2{u}^3=0.$(20) ) with $0\le x\le 10$ and $0\le t\le 50$.

So, we have (23) $F\left(\mathrm{{\rm Y}}\right)=a_0+a_1\mathrm{{\rm Y}}.$(23) Substituting (23(23) $F\left(\mathrm{{\rm Y}}\right)=a_0+a_1\mathrm{{\rm Y}}.$(23) ) into (22(22) $(k^2-{\omega }^2)\left[2\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}F}{\mathrm{d}\mathrm{{\rm Y}}}+({\mathrm{{\rm Y}}}^2-1{)}^2\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}]-g^{2}F+h^2{F}^3=0.$(22) ), we can get (24) $(k^2-{\omega }^2)\left[2a_1\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1\left)\right]-g^2(a_0+a_1\mathrm{{\rm Y}})+h^2(a_0+a_1\mathrm{{\rm Y}}{)}^3=0$(24) from which we obtain $\{\begin{array}{l}3h^3{a}_0^2{a}_1-g^2{a}_1-2k^2{a}_1+2{\omega }^2{a}_1=0,\\ h^3{a}_1^3+2k^2{a}_1+2{\omega }^2{a}_1=0,\\ h^3{a}_0^3-g^2{a}_0=0,\\ 3h^3{a}_0{a}_1^2=0.\end{array}$ Solving the above system, we get the following new travelling waves:

Case 1: $a_0=0,a_1=\pm \frac{g}{h\sqrt{h}},\omega =\sqrt{\frac{g^2}{2}+k^2},$ (25) $u(x,t)=\pm a_1\tanh (\frac{k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}}).$(25) Case 2: $a_0=0,a_1=\pm \frac{g}{h\sqrt{h}},\omega =\sqrt{\frac{g^2}{2}+k^2},$ (26) $u(x,t)=\pm a_1\tanh (\frac{k}{\alpha }{t}^{\alpha }-\frac{\omega }{\text{β}}{x}^{\text{β}}).$(26)

Example 3.2

We take the example of Sine–Gordon equation [32,36] (Figure 2): (27) $T_t^{2\alpha }u-T_x^{2\text{β}}u+u-\frac{1}{6}{u}^3=0.$(27) We have (28) $(k^2-{\omega }^2)U_{\zeta \zeta }+U-\frac{1}{6}{U}^3=0.$(28) Substituting (18(18) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}\xi }& =(1-{\mathrm{{\rm Y}}}^2)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}},\\ \frac{{\mathrm{d}}^2}{\mathrm{d}{\xi }^2}& =2\mathrm{{\rm Y}}({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}}+{({\mathrm{{\rm Y}}}^2-1)}^2\frac{{\mathrm{d}}^2}{\mathrm{d}{\mathrm{{\rm Y}}}^2}.\end{array}$(18) ) and (19(19) $u(x,t)=U\left(\xi \right)=F\left(\mathrm{{\rm Y}}\right)=\sum _{i=0}^m{a}_i{\mathrm{{\rm Y}}}^i,$(19) ) into (28(28) $(k^2-{\omega }^2)U_{\zeta \zeta }+U-\frac{1}{6}{U}^3=0.$(28) ), we can get (29) $(k^2-{\omega }^2)\left[2\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}F}{\mathrm{d}\mathrm{{\rm Y}}}+({\mathrm{{\rm Y}}}^2-1{)}^2\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}]=\frac{1}{6}{F}^3-F.$(29) On the other hand, we have (30) $F\left(\mathrm{{\rm Y}}\right)=a_0+a_1\mathrm{{\rm Y}}.$(30) Substituting (30(30) $F\left(\mathrm{{\rm Y}}\right)=a_0+a_1\mathrm{{\rm Y}}.$(30) ) into (29(29) $(k^2-{\omega }^2)\left[2\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}F}{\mathrm{d}\mathrm{{\rm Y}}}+({\mathrm{{\rm Y}}}^2-1{)}^2\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}]=\frac{1}{6}{F}^3-F.$(29) ), yields (31) $(k^2+{\omega }^2)\left[2a_1\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1\left)\right]+(a_0+a_1\mathrm{{\rm Y}})-\frac{1}{6}(a_0+a_1\mathrm{{\rm Y}}{)}^3=0.$(31) Then, we have the algebraic system: $\{\begin{array}{l}{\displaystyle -2k^2{a}_1-2{\omega }^2{a}_1+a_1-\frac{1}{2}{a}_0^2{a}_1=0,}\\ {\displaystyle 2k^2{a}_1+2{\omega }^2{a}_1+-\frac{1}{6}{a}_1^3=0,}\\ {\displaystyle a_0-\frac{1}{6}{a}_0^3=0,}\\ a_0{a}_1^2=0.\end{array}$ We solve the above system and hence, we have the following cases giving travelling waves:

Figure 2. 3D plot of case1-travelling wave solution of (27(27) $T_t^{2\alpha }u-T_x^{2\text{β}}u+u-\frac{1}{6}{u}^3=0.$(27) ) with $0\le x\le 10$ and $0\le t\le 50$.

Case 1: $a_0=0,a_1=\pm \sqrt{6},\omega =\sqrt{-k^2+\frac{1}{2}},$ (32) $u(x,t)=\pm a_1\tanh (\frac{k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}}),k^2<\frac{1}{2}.$(32) Case 2: $a_0=0,a_1=\pm \sqrt{6},\omega =\sqrt{-k^2+\frac{1}{2}}$ (33) $u(x,t)=\pm a_1\tanh (\frac{k}{\alpha }{t}^{\alpha }-\frac{\omega }{\text{β}}{x}^{\text{β}}),k^2<\frac{1}{2}.$(33)

Example 3.3

Now, we would like to find the travelling waves for the problem [32,36] (Figure 3): (34) $T_t^{2\alpha }u+a{T}_x^{2\text{β}}u+bu-\mathrm{d}{u}^3=0,$(34) It can be transformed into the equation (35) $(k^2+a{\omega }^2)U_{\zeta \zeta }+bU-\mathrm{d}{U}^3=0.$(35) Substituting (18(18) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}\xi }& =(1-{\mathrm{{\rm Y}}}^2)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}},\\ \frac{{\mathrm{d}}^2}{\mathrm{d}{\xi }^2}& =2\mathrm{{\rm Y}}({\mathrm{{\rm Y}}}^2-1)\frac{\mathrm{d}}{\mathrm{d}\mathrm{{\rm Y}}}+{({\mathrm{{\rm Y}}}^2-1)}^2\frac{{\mathrm{d}}^2}{\mathrm{d}{\mathrm{{\rm Y}}}^2}.\end{array}$(18) ) and (19(19) $u(x,t)=U\left(\xi \right)=F\left(\mathrm{{\rm Y}}\right)=\sum _{i=0}^m{a}_i{\mathrm{{\rm Y}}}^i,$(19) ) into (35(35) $(k^2+a{\omega }^2)U_{\zeta \zeta }+bU-\mathrm{d}{U}^3=0.$(35) ), we can get (36) $(k^2+a{\omega }^2)\left[2\mathrm{{\rm Y}}\right(1-{\mathrm{{\rm Y}}}^2)\frac{\mathrm{d}F}{\mathrm{d}\mathrm{{\rm Y}}}+(1-{\mathrm{{\rm Y}}}^2{)}^2\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}]+bF-\mathrm{d}{F}^3=0.$(36) So, we have (37) $F\left(\mathrm{{\rm Y}}\right)=a_0+a_1\mathrm{{\rm Y}}.$(37) Substituting (37(37) $F\left(\mathrm{{\rm Y}}\right)=a_0+a_1\mathrm{{\rm Y}}.$(37) ) into (36(36) $(k^2+a{\omega }^2)\left[2\mathrm{{\rm Y}}\right(1-{\mathrm{{\rm Y}}}^2)\frac{\mathrm{d}F}{\mathrm{d}\mathrm{{\rm Y}}}+(1-{\mathrm{{\rm Y}}}^2{)}^2\frac{{\mathrm{d}}^{2}F}{\mathrm{d}{\mathrm{{\rm Y}}}^2}]+bF-\mathrm{d}{F}^3=0.$(36) ), we can get (38) $(k^2+a{\omega }^2)\left[2a_1\mathrm{{\rm Y}}\right({\mathrm{{\rm Y}}}^2-1\left)\right]+b(a_0+a_1\mathrm{{\rm Y}})-d(a_0+a_1\mathrm{{\rm Y}}{)}^3=0.$(38) The algebraic system allows us to obtain the following travelling wave solutions of (34(34) $T_t^{2\alpha }u+a{T}_x^{2\text{β}}u+bu-\mathrm{d}{u}^3=0,$(34) ):

Figure 3. 3D plot of travelling wave solution for case1 of (34(34) $T_t^{2\alpha }u+a{T}_x^{2\text{β}}u+bu-\mathrm{d}{u}^3=0,$(34) ) with: $0\le x\le 5$ and $0\le t\le 30$.

Case 1: $a_0=0,a_1=\pm \sqrt{\frac{b}{d}},k=\sqrt{-a{\omega }^2+\frac{b}{2}},$ (39) $u(x,t)=\pm a_1\tanh (\frac{k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}}).$(39) Case 2: $a_0=0,a_1=\pm \sqrt{\frac{b}{d}},k=\sqrt{-a{\omega }^2+\frac{b}{2}},$ (40) $u(x,t)=\pm a_1\tanh (\frac{-k}{\alpha }{t}^{\alpha }+\frac{\omega }{\text{β}}{x}^{\text{β}}).$(40)

4. Conclusion

We have first proposed a new sequential nonlinear differential problem with nonlocal integral conditions that involves convergent series in its right-hand sides by means of fractional derivatives (1(1) $\{\begin{array}{l}{\displaystyle D^{{\alpha }_1}{D}^{{\alpha }_2}\cdots D^{{\alpha }_n}u\left(t\right)=\frac{{\eta }_{1}f(t,u(t),D^{\gamma }u(t\left)\right)+{\eta }_{2}g(t,D^{\gamma }u(t),I^{q}u(t\left)\right)}{K(t,u(t),D^{\gamma }u(t\left)\right)}}\\ {\displaystyle +\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{I}^q{\delta }_j\left(t\right)\left[m_j\right(t,u\left(t\right))+n_j(t\left)\right],t\in [0,1],}\\ u\left(0\right)=\theta ,\theta \in \mathbb{R},\\ {\displaystyle u\left(1\right)={\int }_0^{x}u\left(s\right)\mathrm{d}s,0<x<1,}\\ D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_{n-1}}{D}^{{\alpha }_n}u\left(0\right)=0,\\ ⋮\\ D^{{\alpha }_3}{D}^{{\alpha }_4}\cdots D^{{\alpha }_n}u\left(0\right)=0,\\ D^{{\alpha }_2}{D}^{{\alpha }_3}\cdots D^{{\alpha }_n}u\left(1\right)=0,\\ {\alpha }_i+{\alpha }_j>1,\gamma <{\alpha }_n.\end{array}$(1) ). Our problem involves n sequential derivatives of Caputo type. Then, we have used Banach contraction principle to discuss a result on the existence of a unique solution. An example has been presented. A stability result has also been discussed. In the second part, we have used Khalil approach and applied the method of tanh to find new travelling waves for some interesting fractional differential equations that are similar to (2(2) $T_t^{2\alpha }u+a{T}_x^{2\text{β}}u+bu+c{u}^2+\mathrm{d}{u}^3=0,$(2) ); we cite Duffing, Landau–Ginzburg–Higgs and CTF Sine-Gordon fractional conformable equations. Some $3D-$graphs on the obtained travelling waves are plotted in different cases.

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