Nonlocal Neumann Boundary Value Problem for Fractional Symmetric Hahn Integrodifference Equations

Abstract

In this article, we present a nonlocal Neumann boundary value problems for separate sequential fractional symmetric Hahn integrodifference equation. The problem contains five fractional symmetric Hahn difference operators and one fractional symmetric Hahn integral of different orders. We employ Banach fixed point theorem and Schauder’s fixed point theorem to study the existence results of the problem.

1. Introduction

Quantum calculus is a study of calculus without limit that deals with a set of non-differentiable functions. It has been used in many studies such as approximation problems, particle physics problems, quantum mechanics, and calculus of variations. The q-calculus, one type of quantum calculus initiated by Jackson [1,2,3,4,5] has been employed in several fields of applied sciences and engineering such as physical problems, dynamical system, control theory, electrical networks, economics and so on [6,7,8,9,10,11,12,13,14].

Later, the motivation of quantum calculus based on two parameters $q,\omega$ was presented in 1949. W. Hahn [15] introduced the Hahn difference operator which is a combination of two well-known difference operators, the forward difference operator and the Jackson q-difference operator. In 2009, Aldwoah [16,17] defined the right inverse of $D_{q,\omega }$ in the terms of both the Jackson q-integral containing the right inverse of $D_q$ and Nörlund sum contaning the right inverse of ${\mathrm{\Delta }}_{\omega }$ [18]. Moreover, Fractional Hahn operators [19] was introduced in 2017. These calculus are also employed in many research works [20,21,22,23,24,25,26,27,28,29,30,31] including the studies of initial and boundary value problems [32,33,34,35,36,37,38,39,40].

For symmetry of Hahn calculus, Artur et al. [41] introduced symmetric Hahn difference operator in 2013. Recently, Patanarapeelert and Sitthiwirattham [42] introduced fractional symmetric Hahn difference operator. However, the study of the boundary value problems for fractional symmetric Hahn difference equation in the beginning, there exists only one paper on this subject [43].

The main motivation for this paper is to enrich the literature on the boundary value problems for fractional symmetric Hahn difference equations. We study the boundary value problem involving functions F and H which separate fractional symmetric Hahn integral and fractional symmetric Hahn difference, and the boundary condition is a Neumann boundary condition that is assigned values at two non-local points. In this paper, we aim to employ this recent work to study solutions to a boundary value problem for fractional symmetric Hahn integrodifference equations. Our problem is a nonlocal Neumann boundary value problems for sequential fractional symmetric Hahn integrodifference equation of the form

$$\begin{array}{ccc}\hfill {\tilde{D}}_{q,\omega }^{\alpha }{\tilde{D}}_{q,\omega }^{\beta }u\left(t\right)& =& \lambda F\left(t,u\left(t\right),\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)\left(t\right)\right)+\mu H\left(t,u\left(t\right),\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)\left(t\right)\right),t\in I_{q,\omega }^T,\hfill \\ \hfill {\tilde{D}}_{q,\omega }^{{\theta }_1}{g}_1\left({\eta }_1\right)u\left({\eta }_1\right)& =& {\varphi }_1\left(u\right),\hfill \\ \hfill {\tilde{D}}_{q,\omega }^{{\theta }_2}{g}_2\left({\eta }_2\right)u\left({\eta }_2\right)& =& {\varphi }_2\left(u\right),{\eta }_1,{\eta }_2\in I_{q,\omega }^T-\{{\omega }_0,T\}\hfill \end{array}$$

(1)

where $I_{q,\omega }^T:=\{q^{k}T+\omega {\left[k\right]}_q:k\in {\mathbb{N}}_0\}\cup \left\{{\omega }_0\right\}$, $\alpha ,\beta ,\gamma ,\nu ,{\theta }_1,{\theta }_2\in (0,1],$$\omega >0;q\in (0,1)$; $\lambda ,\mu \in {\mathbb{R}}^{+}$; $F,H\in C\left(I_{q,\omega }^T\times \mathbb{R}\times \mathbb{R},\mathbb{R}\right)$ and $g_1,g_2\in C\left(I_{q,\omega }^T,{\mathbb{R}}^{+}\right)$ are given functions, ${\varphi }_1,{\varphi }_2:C\left(I_{q,\omega }^T,\mathbb{R}\right)\to \mathbb{R}$ are given functionals; and for $\phi ,\psi \in \left(I_{q,\omega }^T\times I_{q,\omega }^T,[0,\infty )\right)$, we define the operators

$$\begin{array}{c}\hfill {\displaystyle \left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)\left(t\right):=\left({\tilde{\mathcal{I}}}_{q,\omega }^{\gamma }u\right)\left(t\right)=\frac{q^{\left(\genfrac{}{}{0pt}{}{\gamma }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\gamma \right)}{\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{\gamma -1}}\phi (t,{\sigma }_{q,\omega }^{\gamma -1}\left(s\right))u\left({\sigma }_{q,\omega }^{\gamma -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s,}\\ \hfill {\displaystyle \left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)\left(t\right):=\left({\tilde{D}}_{q,\omega }^{\nu }u\right)\left(t\right)=\frac{q^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(-\nu )}{\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{-\nu -1}}\psi (t,{\sigma }_{q,\omega }^{-\nu -1}\left(s\right))u\left({\sigma }_{q,\omega }^{-\nu -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s.}\end{array}$$

We first transform this nonlinear problem (1) into a fixed point problem, by in view of a linear variant of (1). When the fixed point operator is accessible, we use the classical fixed point theorems to find existence results. To study the solution of problem (1), we recall some definitions and basic knowledge, and we also study some properties of fractional symmetric Hahn integral that will be used in our main results in Section 2. In Section 3, we present the existence and uniqueness of a solution of problem (1) by using the Banach fixed point theorem. The existence of at least one solution of problem (1) is also investigated by using the Schuader’s fixed point theorem. In the last section, we give an example to illustrate our results.

2. Preliminaries

2.1. Basic Notions and Results

In this section, we introduce the definitions of fractional symmetric Hahn difference calculus and its properties [41,42,43,44,45] as follows.

For $0<q<1$, $\omega >0$, ${\omega }_0=\frac{\omega }{1-q}$ and ${\left[k\right]}_q=\frac{1-q^k}{1-q}$, we define

$$\begin{array}{ccc}\hfill {\widetilde{\left[k\right]}}_q& :=& \left\{\begin{array}{cc}{\displaystyle \frac{1-q^{2k}}{1-q^2}={\left[k\right]}_{q^2},}\hfill & k\in \mathbb{N}\hfill \\ 1,\hfill & k=0,\hfill \end{array}\right.\hfill \\ \hfill {\widetilde{\left[k\right]}}_q!& :=& \left\{\begin{array}{cc}{\displaystyle {\widetilde{\left[k\right]}}_q{\widetilde{[k-1]}}_q···{\widetilde{\left[1\right]}}_q=\prod _{i=1}^k\frac{1-q^{2i}}{1-q^2},}\hfill & k\in \mathbb{N}\hfill \\ 1,\hfill & k=0.\hfill \end{array}\right.\hfill \end{array}$$

The $q,\omega$-forward jump operator is defined by

$${\sigma }_{q,\omega }^k\left(t\right):=q^{k}t+\omega {\left[k\right]}_q,$$

and the $q,\omega$-backward jump operator is defined by

$${\displaystyle {\rho }_{q,\omega }^k\left(t\right):=\frac{t-\omega {\left[k\right]}_q}{q^k},}$$

where $k\in \mathbb{N}$.

For $n\in {\mathbb{N}}_0:=\{0,1,2,\cdots \}$, and $a,b\in \mathbb{R}$, the q-analogue of the power function is defined as

$${\displaystyle {(a-b)}_q^{\underline{0}}:=1,{(a-b)}_q^{\underline{n}}:=\prod _{i=0}^{n-1}(a-b{q}^i),}$$

the q-symmetric analogue of the power function is defined as

$${\displaystyle {\widetilde{(a-b)}}_q^{\underline{0}}:=1,{\widetilde{(a-b)}}_q^{\underline{n}}:=\prod _{i=0}^{n-1}(a-b{q}^{2i+1}),}$$

and, the $q,\omega$-symmetric analogue of the power function is defined as

$${\displaystyle {\widetilde{(a-b)}}_{q,\omega }^{\underline{0}}:=1,{\widetilde{(a-b)}}_{q,\omega }^{\underline{n}}:=\prod _{i=0}^{n-1}\left[a-{\sigma }_{q,\omega }^{2i+1}\left(b\right)\right].}$$

Generally, for $\alpha \in \mathbb{R}$, the power functions are defined as

$${\displaystyle {(a-b)}_q^{\underline{\alpha }}=a^{\alpha }\prod _{i=0}^{\infty }\frac{1-\left(\frac{b}{a}\right)q^i}{1-\left(\frac{b}{a}\right)q^{\alpha +i}},a\ne 0,}$$

$${\displaystyle {\widetilde{(a-b)}}_q^{\underline{\alpha }}=a^{\alpha }\prod _{i=0}^{\infty }\frac{1-\left(\frac{b}{a}\right)q^{2i+1}}{1-\left(\frac{b}{a}\right)q^{2(\alpha +i)+1}},a\ne 0,}$$

$${\displaystyle \widetilde{(a-b)}{)}_{q,\omega }^{\underline{\alpha }}={\widetilde{\left((a-{\omega }_0)-(b-{\omega }_0)\right)}}_q^{\underline{\alpha }}={(a-{\omega }_0)}^{\alpha }\prod _{i=0}^{\infty }\frac{1-\left(\frac{b-{\omega }_0}{a-{\omega }_0}\right)q^{2i+1}}{1-\left(\frac{b-{\omega }_0}{a-{\omega }_0}\right)q^{2(\alpha +i)+1}},a\ne {\omega }_0.}$$

Particularly, $a_q^{\underline{\alpha }}={\tilde{a}}_q^{\underline{\alpha }}=a^{\alpha }$ and ${\widetilde{(a-{\omega }_0)}}_{q,\omega }^{\underline{\alpha }}={(a-{\omega }_0)}^{\alpha }$ if $b=0$. If $a=b$, ${\left(0\right)}_q^{\underline{\alpha }}={\widetilde{\left(0\right)}}_q^{\underline{\alpha }}={\widetilde{\left({\omega }_0\right)}}_{q,\omega }^{\underline{\alpha }}=0$ for $\alpha >0$.

The q-symmetric gamma and q-symmetric beta functions are defined as

$$\begin{array}{ccc}\hfill {\tilde{\mathrm{\Gamma }}}_q\left(x\right)& :=& \left\{\begin{array}{cc}\frac{{(1-q^2)}_q^{\underline{x-1}}}{{(1-q^2)}^{x-1}}=\frac{{\widetilde{(1-q)}}_q^{\underline{x-1}}}{{(1-q^2)}^{x-1}},\hfill & x\in \mathbb{R}\backslash \{0,-1,-2,\cdots \}\hfill \\ {\widetilde{[x-1]}}_q!,\hfill & x\in \mathbb{N},\hfill \end{array}\right.\hfill \\ \hfill {\tilde{B}}_q(x,y)& :=& {\int }_0^1{\left(q^{-1}s\right)}^{x-1}{\widetilde{(1-s)}}_q^{\underline{y-1}}{\tilde{d}}_{q}s=\frac{{\tilde{\mathrm{\Gamma }}}_q\left(x\right){\tilde{\mathrm{\Gamma }}}_q\left(y\right)}{{\tilde{\mathrm{\Gamma }}}_q(x+y)},\hfill \end{array}$$

respectively.

Lemma 1

([42]). For $m,n\in {\mathbb{N}}_0$ and $\alpha \in \mathbb{R}$,

$\left(a\right)$

${\widetilde{(x-{\sigma }_{q,\omega }^n\left(x\right))}}_{q,\omega }^{\underline{\alpha }}={(x-{\omega }_0)}^k{\widetilde{(1-q^n)}}_q^{\underline{\alpha }}$,

$\left(b\right)$

${\widetilde{\left({\sigma }_{q,\omega }^m\left(x\right)\right)-{\sigma }_{q,\omega }^n\left(x\right))}}_{q,\omega }^{\underline{\alpha }}=q^{m\alpha }{(x-{\omega }_0)}^{\alpha }{\widetilde{(1-q^{n-m})}}_q^{\underline{\alpha }}$.

Definition 1

([41]). For $q\in (0,1)$, $\omega >0$, and f is a function defined on $I_{q,\omega }^T\subseteq \mathbb{R}$, the symmetric Hahn difference of f is defined by

$$\begin{array}{ccc}& & {\tilde{D}}_{q,\omega }f\left(t\right):=\frac{f\left({\sigma }_{q,\omega }\left(t\right)\right)-f\left({\rho }_{q,\omega }\left(t\right)\right)}{{\sigma }_{q,\omega }\left(t\right)-{\rho }_{q,\omega }\left(t\right)}t\in I_{q,\omega }^T-\left\{{\omega }_0\right\},\hfill \\ & & {\tilde{D}}_{q,\omega }f\left({\omega }_0\right)=f^{\prime }\left({\omega }_0\right)\mathrm{where}f\mathrm{is}\mathrm{differentiable}\mathrm{at}{\omega }_0.\hfill \end{array}$$

${\tilde{D}}_{q,\omega }f$ is called $q,\omega$-symmetric derivative of f, and f is $q,\omega$-symmetric differentiable on $I_{q,\omega }^T$. For $N\in \mathbb{N}$, ${\tilde{D}}_{q,\omega }^{N}f\left(x\right)={\tilde{D}}_{q,\omega }{\tilde{D}}_{q,\omega }^{N-1}f\left(x\right)$ where ${\tilde{D}}_{q,\omega }^{0}f\left(x\right)=f\left(x\right)$.

Remark 1.

If f and g are $q,\omega$-symmetric differentiable on $I_{q,\omega }^T$,

$\left(a\right)$

${\tilde{D}}_{q,\omega }[f\left(t\right)+g\left(t\right)]={\tilde{D}}_{q,\omega }f\left(t\right)+{\tilde{D}}_{q,\omega }g\left(t\right)$,

$\left(b\right)$

${\tilde{D}}_{q,\omega }\left[f\left(t\right)g\left(t\right)\right]=f\left({\rho }_{q,\omega }\left(t\right)\right){\tilde{D}}_{q,\omega }g\left(t\right)+g\left({\sigma }_{q,\omega }\left(t\right)\right){\tilde{D}}_{q,\omega }f\left(t\right)$,

$\left(c\right)$

${\tilde{D}}_{q,\omega }\left[{\displaystyle \frac{f\left(t\right)}{g\left(t\right)}}\right]={\displaystyle \frac{g\left({\rho }_{q,\omega }\left(t\right)\right){\tilde{D}}_{q,\omega }f\left(t\right)-f\left({\rho }_{q,\omega }\left(t\right)\right){\tilde{D}}_{q,\omega }g\left(t\right)}{g\left({\rho }_{q,\omega }\left(t\right)\right)g\left({\sigma }_{q,\omega }\left(t\right)\right)}},$$g\left({\rho }_{q,\omega }\left(t\right)\right)g\left({\sigma }_{q,\omega }\left(t\right)\right)\ne 0,$

$\left(d\right)$

${\tilde{D}}_{q,\omega }\left[C\right]=0$ where C is constant.

Definition 2

([41]). Let I be any closed interval of $\mathbb{R}$ containing $a,b$ and ${\omega }_0$ and $f:I\to \mathbb{R}$ be a given function. The symmetric Hahn integral of f from a to b is defind by

$${\displaystyle {\int }_a^{b}f\left(t\right){\tilde{d}}_{q,\omega }t:={\int }_{{\omega }_0}^{b}f\left(t\right){\tilde{d}}_{q,\omega }t-{\int }_{{\omega }_0}^{a}f\left(t\right){\tilde{d}}_{q,\omega }t,}$$

and

$$\begin{array}{c}\hfill {\displaystyle {\tilde{\mathcal{I}}}_{q,\omega }f\left(t\right)={\int }_{{\omega }_0}^{x}f\left(t\right){\tilde{d}}_{q,\omega }t:=(1-q^2)(x-{\omega }_0)\sum _{k=0}^{\infty }{q}^{2k}f\left({\sigma }_{q,\omega }^{2k+1}\left(x\right)\right),x\in I,}\end{array}$$

where the above series converges at $x=a$ and $x=b$. For $N\in \mathbb{N}$, ${\tilde{\mathcal{I}}}_{q,\omega }^{N}f\left(x\right)={\tilde{\mathcal{I}}}_{q,\omega }{\tilde{\mathcal{I}}}_{q,\omega }^{N-1}f\left(x\right)$ where ${\tilde{\mathcal{I}}}_{q,\omega }^{0}f\left(x\right)=f\left(x\right)$.

The following is the relation between the symmetric Hahn difference and integral.

$${\tilde{D}}_{q,\omega }{\tilde{\mathcal{I}}}_{q,\omega }f\left(x\right)=f\left(x\right)\mathrm{and}{\tilde{\mathcal{I}}}_{q,\omega }{\tilde{D}}_{q,\omega }f\left(x\right)=f\left(x\right)-f\left({\omega }_0\right).$$

Remark 2

([41]). Let $a,b\in I_{q,\omega }^T$ and $f,g$ be symmetric Hahn integrable on $I_{q,\omega }^T$. Then,

$\left(a\right)$

${\int }_a^{a}f\left(t\right){\tilde{d}}_{q,\omega }t=0,$

$\left(b\right)$

${\int }_a^{b}f\left(t\right){\tilde{d}}_{q,\omega }t=-{\int }_b^{a}f\left(t\right){\tilde{d}}_{q,\omega }t,$

$\left(c\right)$

${\int }_a^{b}f\left(t\right){\tilde{d}}_{q,\omega }t={\int }_c^{b}f\left(t\right){\tilde{d}}_{q,\omega }t+{\int }_a^{c}f\left(t\right){\tilde{d}}_{q,\omega }t,c\in I_{q,\omega }^T,a<c<b,$

$\left(d\right)$

${\int }_a^b\left[\alpha f\left(t\right)+\beta g\left(t\right)\right]{\tilde{d}}_{q,\omega }t=\alpha {\int }_a^{b}f\left(t\right){\tilde{d}}_{q,\omega }t+\beta {\int }_a^{b}g\left(t\right){\tilde{d}}_{q,\omega }t,\alpha ,\beta \in \mathbb{R},$

$\left(e\right)$

${\int }_a^b\left[f\left({\rho }_{q,\omega }\left(t\right)\right){\tilde{D}}_{q,\omega }g\left(t\right)\right]{\tilde{d}}_{q,\omega }t={\left[f\left(t\right)g\left(t\right)\right]}_a^b-{\int }_a^b\left[g\left({\sigma }_{q,\omega }\left(t\right)\right){\tilde{D}}_{q,\omega }f\left(t\right)\right]{\tilde{d}}_{q,\omega }t.$

Lemma 2

([41]). [Fundamental theorem of symmetric Hahn calculus]

Let $f:I\to \mathbb{R}$ be continuous at ${\omega }_0$. Then,

$$F\left(x\right):={\int }_{{\omega }_0}^{x}f\left(t\right){\tilde{d}}_{q,\omega }t,x\in I$$

is continuous at ${\omega }_0$ and ${\tilde{D}}_{q,\omega }F\left(x\right)$ exists for every $x\in {\sigma }_{q,\omega }\left(I\right):=\{qt+\omega :t\in I\}$ where

$${\tilde{D}}_{q,\omega }F\left(x\right)=f\left(x\right).$$

In addition,

$${\int }_a^b{\tilde{D}}_{q,\omega }f\left(t\right){\tilde{d}}_{q,\omega }t=f\left(b\right)-f\left(a\right)foralla,b\in I.$$

Lemma 3

([42]). Let $0<q<1$, $\omega >0$ and $f:I\to \mathbb{R}$ be continuous at ${\omega }_0$. Then,

$$\begin{array}{ccc}\hfill {\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{r}f\left(s\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }r& =& q{\int }_{{\omega }_0}^t{\int }_{qs+\omega }^{t}f(qs+\omega ){\tilde{d}}_{q,\omega }r{\tilde{d}}_{q,\omega }s.\hfill \end{array}$$

Definition 3

([42]). Let $\alpha ,\omega >0,0<q<1,$ and f be a function defined on $I_{q,\omega }^T$. The fractional symmetric Hahn integral is defined by

$$\begin{array}{ccc}\hfill {\tilde{\mathcal{I}}}_{q,\omega }^{\alpha }f\left(t\right)& :=& {\displaystyle \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{\alpha -1}}f\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s}\hfill \\ & =& {\displaystyle \frac{(1-q^2)q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}(t-{\omega }_0)}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}\sum _{k=0}^{\infty }{q}^{2k}{\left(t-{\sigma }_{q,\omega }^{2k+1}\left(t\right)\right)}_{q,\omega }^{\underline{\alpha -1}}f\left({\sigma }_{q,\omega }^{2k+\alpha }\left(t\right)\right)}\hfill \\ & =& {\displaystyle \frac{(1-q^2)q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{(t-{\omega }_0)}^{\alpha }}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}\sum _{k=0}^{\infty }{q}^{2k}{\widetilde{\left(1-q^{2k+1}\right)}}_q^{\underline{\alpha -1}}f\left({\sigma }_{q,\omega }^{2k+\alpha }\left(t\right)\right)}\hfill \end{array}$$

and ${\tilde{\mathcal{I}}}_{q,\omega }^{0}f)\left(t\right)=f\left(t\right)$.

Definition 4

([42]). For $\alpha ,\omega >0,0<q<1$ and f defined on $I_{q,\omega }^T$, the fractional symmetric Hahn difference operator of Riemann-Liouville type of order α is defined by

$$\begin{array}{ccc}\hfill {\tilde{D}}_{q,\omega }^{\alpha }f\left(t\right)& :=& {\tilde{D}}_{q,\omega }^N{\tilde{\mathcal{I}}}_{q,\omega }^{N-\alpha }f\left(t\right)\hfill \\ & =& {\displaystyle \frac{q^{\left(\genfrac{}{}{0pt}{}{-\alpha }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(-\alpha )}{\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{-\alpha -1}}f\left({\sigma }_{q,\omega }^{-\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s}\hfill \\ \hfill {\tilde{D}}_{q,\omega }^{0}f\left(t\right)& =& f\left(t\right)\hfill \end{array}$$

where $N-1<\alpha <N,N\in \mathbb{N}.$

Lemma 4

([42]). Let $\alpha ,\omega >0,0<q<1$ and $f:I_{q,\omega }^T\to \mathbb{R}$. Then,

$$\begin{array}{c}\hfill {\tilde{\mathcal{I}}}_{q,\omega }^{\alpha }{\tilde{D}}_{q,\omega }^{\alpha }f\left(t\right)=f\left(t\right)+C_1{(t-{\omega }_0)}^{\alpha -1}+C_2{(t-{\omega }_0)}^{\alpha -2}+\cdots +C_N{(t-{\omega }_0)}^{\alpha -N}\end{array}$$

for some $C_i\in \mathbb{R},i=1,2,\cdots ,N$ and $N-1<\alpha <N$ for $N\in \mathbb{N}$.

Lemma 5

([46] Arzelá-Ascoli theorem). A set of function in $C[a,b]$ with the sup norm, is relatively compact if and only if it is uniformly bounded and equicontinuous on $[a,b]$.

Lemma 6

([46]). If a set is closed and relatively compact then it is compact.

Lemma 7

([47] Schauder’s fixed point theorem). Let $(D,d)$ be a complete metric space, U be a closed convex subset of D, and $T:D\to D$ be the map such that the set $Tu:u\in U$ is relatively compact in D. Then the operator T has at least one fixed point $u^{*}\in U$: $T{u}^{*}=u^{*}.$

2.2. Auxiliary Lemmas

In this part, we establish some lemmas that are used to prove our main results.

Lemma 8.

Let $q\in (0,1),\omega >0$ and $n>0$. Then,

$$\begin{array}{c}\hfill {\int }_{{\omega }_0}^t{\tilde{d}}_{q,\omega }s=t-{\omega }_0\mathrm{and}{\int }_{{\omega }_0}^t{(s-{\omega }_0)}^n{\tilde{d}}_{q,\omega }s=\frac{q^n}{{\widetilde{[n+1]}}_q}{\left(t-{\omega }_0\right)}^{n+1}.\end{array}$$

Lemma 9.

Let $\alpha ,\beta >0,q\in (0,1)$ and $\omega >0$. Then,

(i) 

${\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{\alpha -1}}{\tilde{d}}_{q,\omega }s=\frac{{\left(t-{\omega }_0\right)}^{\alpha }}{{\widetilde{\left[\alpha \right]}}_q},$

(ii) 

${\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{\alpha -1}}{\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)-{\omega }_0\right)}^{\beta }{\tilde{d}}_{q,\omega }s=q^{\alpha \beta }{(t-{\omega }_0)}^{\alpha +\beta }{B}_q(\beta +1,\alpha ),$

(iii) 

${\displaystyle {\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\alpha -1}\left(s\right)}{\widetilde{(t-s)}}_{q,\omega }^{\underline{\alpha -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)-r\right)}}_{q,\omega }^{\underline{\beta -1}}{\tilde{d}}_{q,\omega }r{\tilde{d}}_{p,\omega }s=\frac{q^{\alpha \beta }}{{\widetilde{\left[\beta \right]}}_q}{(t-{\omega }_0)}^{\alpha +\beta }{B}_q(\beta +1,\alpha ).}$

Lemma 10.

Let $\alpha ,\beta ,\theta >0,q\in (0,1)$ and $\omega >0$. Then,

(a) 

${\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{-\theta -1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(s\right)-{\omega }_0\right)}^{\beta -1}{\tilde{d}}_{q,\omega }s=\frac{{\left(t-{\omega }_0\right)}^{\beta -\theta -1}}{q^{\theta (\beta -1)}}{\tilde{B}}_q(\beta ,-\theta ),$

(b) 

${\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\theta -1}\left(x\right)}{\widetilde{(t-x)}}_{q,\omega }^{\underline{-\theta -1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}_{q,\omega }^{\alpha -1}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x$$=\frac{{\left(t-{\omega }_0\right)}^{\alpha +\beta -\theta -1}}{q^{\theta (\alpha +\beta -1)-\beta (\alpha -1)}}{\tilde{B}}_q(\alpha ,\beta ){\tilde{B}}_q(\alpha +\beta ,-\theta ),$

(c) 

${\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\theta -1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{(t-y)}}_{q,\omega }^{\underline{-\theta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{-\theta -1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}\times$${\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y$$=\frac{{\left(t-{\omega }_0\right)}^{\alpha +\beta -\theta }}{q^{\theta (\alpha +\beta )-\alpha \beta }{\left[\tilde{\alpha }\right]}_q}{\tilde{B}}_q(\alpha +1,\beta ){\tilde{B}}_q(\alpha +\beta +1,-\theta ).$

Proof. 

We use the definition of $q,\omega$-symmetric analogue of the power function, Lemma 1 and Definition 2 to obtain

$$\begin{array}{cc}\hfill \left(a\right)& {\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{-\theta -1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(s\right)-{\omega }_0\right)}^{\beta -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill & =(1-q^2)\left(t-{\omega }_0\right)\sum _{j=0}^{\infty }{q}^{2j}{\widetilde{\left(t-{\sigma }_{q,\omega }^{2j+1}\left(t\right)\right)}}_{q,\omega }^{\underline{-\theta }}{\left(q^{-\theta -1}\left({\sigma }_{q,\omega }^{2j+1}\left(t\right)-{\omega }_0\right)\right)}^{\beta -1}\hfill \\ \hfill & =q^{-\theta (\beta -1)}(1-q^2){\left(t-{\omega }_0\right)}^{\beta -\theta -1}\sum _{j=0}^{\infty }{q}^{2j}{\widetilde{\left(1-q^{2j+1}\right)}}_q^{\underline{-\theta -1}}{\left(q^{-1}{q}^{2j+1}\right)}^{\beta -1}\hfill \\ \hfill & =q^{-\theta (\beta -1)}{(t-{\omega }_0)}^{\beta -\theta -1}{\int }_{{\omega }_0}^t{\widetilde{\left(1-s\right)}}_{q,\omega }^{-\theta -1}{\left(q^{-1}s\right)}^{\beta -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill & =\frac{{\left(t-{\omega }_0\right)}^{\beta -\theta -1}}{q^{\theta (\beta -1)}}{\tilde{B}}_q(\beta ,-\theta ),\hfill \\ \hfill \left(b\right)& {\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\theta -1}\left(x\right)}{\widetilde{(t-x)}}_{q,\omega }^{\underline{-\theta -1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}_{q,\omega }^{\alpha -1}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\hfill \\ \hfill & ={\int }_{{\omega }_0}^t{\widetilde{(t-x)}}_{q,\omega }^{\underline{-\theta -1}}\left[{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\theta -1}\left(x\right)}{\widetilde{\left({\sigma }_{q,\omega }^{-\theta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{-\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s\right]{\tilde{d}}_{q,\omega }x\hfill \\ \hfill & =q^{\beta (\alpha -1)}{B}_q(\alpha ,\beta ){\int }_{{\omega }_0}^t{\widetilde{(t-x)}}_{q,\omega }^{\underline{-\theta -1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(x\right)-{\omega }_0\right)}^{\alpha +\beta -1}{\tilde{d}}_{q,\omega }x\hfill \\ \hfill & =\frac{{\left(t-{\omega }_0\right)}^{\alpha +\beta -\theta -1}}{q^{\theta (\alpha +\beta -1)-\beta (\alpha -1)}}{\tilde{B}}_q(\alpha ,\beta ){\tilde{B}}_q(\alpha +\beta ,-\theta ),\hfill \\ \hfill \left(c\right)& {\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\theta -1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{(t-y)}}_{q,\omega }^{\underline{-\theta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{-\theta -1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y\hfill \\ \hfill & ={\int }_{{\omega }_0}^t{\widetilde{(t-y)}}_{q,\omega }^{\underline{-\theta -1}}\left[{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\theta -1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left({\sigma }_{q,\omega }^{-\theta -1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\right]{\tilde{d}}_{q,\omega }y\hfill \\ \hfill & =\frac{q^{\alpha \beta }}{{\left[\tilde{\alpha }\right]}_q}{B}_q(\alpha +1,\beta ){\int }_{{\omega }_0}^t{\widetilde{(t-y)}}_{q,\omega }^{\underline{-\theta -1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(x\right)-{\omega }_0\right)}^{\alpha +\beta }{\tilde{d}}_{q,\omega }y\hfill \\ \hfill & =\frac{{\left(t-{\omega }_0\right)}^{\alpha +\beta -\theta }}{q^{\theta (\alpha +\beta )-\alpha \beta }{\left[\tilde{\alpha }\right]}_q}{\tilde{B}}_q(\alpha +1,\beta ){\tilde{B}}_q(\alpha +\beta +1,-\theta ).\hfill \end{array}$$

□

2.3. Lemma for Linear Variant Form

In this part, a solution of a linear variant form of the problem (1) is investigated as shown in the following lemma.

Lemma 11.

Let $\mathrm{\Omega }\ne 0;\omega >0;q\in (0,1)$; $\alpha ,\beta ,{\theta }_1,{\theta }_2\in (0,1];$ $h\in C\left(I_{q,\omega }^T,\mathbb{R}\right)$ and $g_1,g_2\in C\left(I_{q,\omega }^T,{\mathbb{R}}^{+}\right)$ be given functions; ${\varphi }_1,{\varphi }_2\in C\left(I_{q,\omega }^T,\mathbb{R}\right)\to \mathbb{R}$ be given functionals. Then the problem

$$\begin{array}{ccc}\hfill {\tilde{D}}_{q,\omega }^{\alpha }{\tilde{D}}_{q,\omega }^{\beta }u\left(t\right)& =& h\left(t\right),t\in I_{q,\omega }^T,\hfill \\ \hfill {\tilde{D}}_{q,\omega }^{{\theta }_i}{g}_i\left({\eta }_i\right)u\left({\eta }_i\right)& =& {\varphi }_i\left(u\right),{\eta }_i\in I_{q,\omega }^T-\{{\omega }_0,T\},i=1,2,\hfill \end{array}$$

(2)

has the unique solution

$$\begin{array}{cc}\hfill u\left(t\right)=& \left[{\mathit{A}}_2{\mathit{O}}_1[{\varphi }_1,h]-{\mathit{A}}_1{\mathit{O}}_2[{\varphi }_2,h]\right]\frac{{(t-{\omega }_0)}^{\beta -1}}{\mathrm{\Omega }}\hfill \\ \hfill -& \left[{\mathit{B}}_2{\mathit{O}}_1[{\varphi }_1,h]-{\mathit{B}}_1{\mathit{O}}_2[{\varphi }_2,h]\right]\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{\mathrm{\Omega }{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left(t-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}h\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x.\hfill \end{array}$$

(3)

where the functionals ${\mathit{O}}_1[{\varphi }_1,h],{\mathit{O}}_2[{\varphi }_2,h]$ are defined by

$$\begin{array}{cc}\hfill {\mathit{O}}_1[{\varphi }_1,h]:=& {\varphi }_1\left(u\right)-\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_1}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-{\theta }_1)}{\int }_{{\omega }_0}^{{\eta }_1}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_1-1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left({\eta }_1-y\right)}}_{q,\omega }^{\underline{-{\theta }_1-1}}\times \hfill \\ \hfill & {\widetilde{\left({\sigma }_{q,\omega }^{-{\theta }_1-1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}{g}_1\left({\sigma }_{q,\omega }^{-{\theta }_1-1}\left(y\right)\right)\times \hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & h\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y,\hfill \\ \hfill {\mathit{O}}_2[{\varphi }_2,h]:=& {\varphi }_2\left(u\right)-\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_2}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-{\theta }_2)}{\int }_{{\omega }_0}^{{\eta }_2}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_2-1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left({\eta }_2-y\right)}}_{q,\omega }^{\underline{-{\theta }_2-1}}\times \hfill \\ \hfill & {\widetilde{\left({\sigma }_{q,\omega }^{-{\theta }_2-1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}{g}_2\left({\sigma }_{q,\omega }^{-{\theta }_2-1}\left(y\right)\right)\times \hfill \\ \hfill & h\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y,\hfill \end{array}$$

(5)

and the constants ${\mathit{A}}_1,{\mathit{A}}_2,{\mathit{B}}_1,{\mathit{B}}_2$ and Ω are defined by

$$\begin{array}{cc}\hfill {\mathit{A}}_1:=& \frac{1}{{\tilde{\mathrm{\Gamma }}}_q(-{\theta }_1)}{\int }_{{\omega }_0}^{{\eta }_1}{\widetilde{\left({\eta }_1-s\right)}}_{q,\omega }^{\underline{-{\theta }_1-1}}{\left({\sigma }_{q,\omega }^{\underline{-{\theta }_1-1}}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{g}_1\left({\sigma }_{q,\omega }^{-{\theta }_1-1}\left(s\right)\right){\tilde{d}}_{q,\omega }s,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathit{A}}_2:=& \frac{1}{{\tilde{\mathrm{\Gamma }}}_q(-{\theta }_2)}{\int }_{{\omega }_0}^{{\eta }_2}{\widetilde{\left({\eta }_2-s\right)}}_{q,\omega }^{\underline{-{\theta }_2-1}}{\left({\sigma }_{q,\omega }^{\underline{-{\theta }_2-1}}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{g}_2\left({\sigma }_{q,\omega }^{-{\theta }_2-1}\left(s\right)\right){\tilde{d}}_{q,\omega }s,\hfill \\ \hfill {\mathit{B}}_1:=& \frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_1}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(-{\theta }_1)}{\int }_{{\omega }_0}^{{\eta }_1}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_1-1}\left(x\right)}{\widetilde{\left({\eta }_1-x\right)}}_{q,\omega }^{\underline{-{\theta }_1-1}}{\left({\sigma }_{q,\omega }^{-{\theta }_1-1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\beta -1}}\times \hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\left({\sigma }_{q,\omega }^{-{\theta }_1-1}\left(s\right)-{\omega }_0\right)}_{q,\omega }^{\alpha -1}{g}_1\left({\sigma }_{q,\omega }^{-{\theta }_1-1}\left(x\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x,\hfill \\ \hfill {\mathit{B}}_2:=& \frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_2}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(-{\theta }_2)}{\int }_{{\omega }_0}^{{\eta }_2}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_2-1}\left(x\right)}{\widetilde{\left({\eta }_2-x\right)}}_{q,\omega }^{\underline{-{\theta }_2-1}}{\left({\sigma }_{q,\omega }^{-{\theta }_2-1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\beta -1}}\times \hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\left({\sigma }_{q,\omega }^{-{\theta }_2-1}\left(s\right)-{\omega }_0\right)}_{q,\omega }^{\alpha -1}{g}_2\left({\sigma }_{q,\omega }^{-{\theta }_2-1}\left(x\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \mathrm{\Omega }:=& {\mathit{A}}_2{\mathit{B}}_1-{\mathit{A}}_1{\mathit{B}}_2.\hfill \end{array}$$

(10)

Proof. 

From (2), we take a fractional symmetric Hahn integral of order $\alpha$ and find that

$$\begin{array}{cc}\hfill {\tilde{D}}_{q,\omega }^{\beta }u\left(t\right)=& C_0{(t-{\omega }_0)}^{\alpha -1}+\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{\alpha -1}}h\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s.\hfill \end{array}$$

(11)

Taking fractional symmetric Hahn integral of order (11), we obtain

$$\begin{array}{cc}\hfill u\left(t\right)=& C_1{(t-{\omega }_0)}^{\beta -1}+\frac{C_0{q}^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill & +\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{(t-x)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}h\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x.\hfill \end{array}$$

(12)

Next, we take fractional symmetric Hahn difference of order ${\theta }_i,i=1,2$ for (12). Thus, we have

$$\begin{array}{cc}\hfill {\tilde{D}}_{q,\omega }^{{\theta }_i}u\left(t\right)=& \frac{C_1{q}^{\left(\genfrac{}{}{0pt}{}{-{\theta }_i}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(-{\theta }_i)}{\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{-{\theta }_i-1}}{\left({\sigma }_{q,\omega }^{-\theta -1}\left(s\right)-{\omega }_0\right)}^{\beta -1}{\tilde{d}}_{q,\omega }s+\frac{C_0{q}^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_i}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-{\theta }_i)}\times \hfill \\ \hfill & {\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_i-1}\left(x\right)}{\widetilde{(t-x)}}_{q,\omega }^{\underline{-{\theta }_i-1}}{\widetilde{\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_i}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-{\theta }_i)}{\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{(t-y)}}_{q,\omega }^{\underline{-{\theta }_i-1}}{\widetilde{\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}\times \hfill \\ \hfill & {\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}^{\underline{\alpha -1}}h\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right){\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y.\hfill \end{array}$$

(13)

Substituting $i=1,2$ into (13) and employing the condition of (2), we have

$$\begin{array}{cc}\hfill {\mathit{A}}_1{C}_0+{\mathit{B}}_1{C}_1& ={\mathit{O}}_1[{\varphi }_1,h],\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\mathit{A}}_2{C}_0+{\mathit{B}}_2{C}_1& ={\mathit{O}}_2[{\varphi }_2,h].\hfill \end{array}$$

(15)

Solving the system of Equations (14) and (15), we get

$$\begin{array}{c}\hfill C_0=\frac{{\mathit{B}}_1{\mathit{O}}_2[{\varphi }_2,h]-{\mathit{B}}_2{\mathit{O}}_1[{\varphi }_1,h]}{\mathrm{\Omega }}\mathrm{and}{C}_2=\frac{{\mathit{A}}_2{\mathit{O}}_1[{\varphi }_1,h]-{\mathit{A}}_1{\mathit{O}}_2[{\varphi }_2,h]}{\mathrm{\Omega }}\end{array}$$

where ${\mathit{O}}_1[{\varphi }_1,h],{\mathit{O}}_2[{\varphi }_2,h],{\mathit{A}}_1,{\mathit{A}}_2,{\mathit{B}}_1,{\mathit{B}}_2$ and $\mathrm{\Omega }$ are defined as (4)–(10), respectively. Substituting $C_1,C_2$ into (12), we obtain the solution (3). The converse can be proved by direct computation. The proof is complete. □

3. Existence and Uniqueness of Solution of the Problem (1)

To consider the existence and uniqueness of solution to the problem (1), we use Banach fixed point theorem.

Let $\mathcal{C}=C\left(I_{q,\omega }^T,\mathbb{R}\right)$ be a Banach space of all function u with the norm defined by

$${\parallel u\parallel }_{\mathcal{C}}=\parallel u\parallel +\parallel {\tilde{D}}_{q,\omega }^{\nu }\parallel ,$$

where ${\displaystyle \parallel u\parallel =\underset{t\in I_{q,\omega }^T}{max}\left\{\left|u\right(t\left)\right|\right\}}$ and ${\displaystyle \parallel {\tilde{D}}_{q,\omega }^{\nu }\mu \parallel =\underset{t\in I_{q,\omega }^T}{max}\parallel \left({\tilde{D}}_{q,\omega }^{\nu }\mu \right)\left(t\right)\parallel }$. An operator $\mathcal{A}:\mathcal{C}\to \mathcal{C}$ is defined by

$$\begin{array}{cc}\hfill \left(\mathcal{A}u\right)& \left(t\right)\hfill \\ \hfill :=& \left[{\mathit{A}}_2{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]-{\mathit{A}}_1{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\right]\frac{{(t-{\omega }_0)}^{\beta -1}}{\mathrm{\Omega }}\hfill \\ \hfill -& \left[{\mathit{B}}_2{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]-{\mathit{B}}_1{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\right]\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{\mathrm{\Omega }{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left(t-x\right)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}^{\underline{\alpha -1}}\left[\lambda F\right({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right),\hfill \\ \hfill & {\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right))+\mu H\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right),{\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)\right)]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\hfill \end{array}$$

(16)

where the functionals ${\mathit{O}}_i^{*}[{\varphi }_i,F_u+H_u],i=1,2$ are given by

$$\begin{array}{cc}\hfill {\mathit{O}}_i^{*}& [{\varphi }_i,F_u+H_u]\hfill \\ \hfill :=& {\varphi }_i\left(u\right)-\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_i}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q\left({\theta }_i\right)}{\int }_{u_0}^{{\eta }_i}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left({\eta }_i-y\right)}}_{q,\omega }^{\underline{-{\theta }_i-1}}{\widetilde{\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}\hfill \\ \hfill & {\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}{g}_1\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)\right)[\lambda F\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right),{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)\right)\hfill \\ \hfill & +\mu H\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right),{\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)\right)]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y,\hfill \end{array}$$

(17)

and the constants ${\mathit{A}}_1,{\mathit{A}}_2,{\mathit{B}}_1,{\mathit{B}}_2,\mathrm{\Omega }$ are given in Lemma 11.

To prove the existence results to the problem (1), we first convert the given nonlinear problem (1) into a fixed point problem. If the operator $\mathcal{A}$ has fixed point, then the problem (1) has the solution.

Theorem 1.

Assume that $F,H:I_{q,\omega }^T\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, and $g_1,g_2:I_{q,\omega }^T\to {\mathbb{R}}^{+}$ are continuous, and $\phi ,\psi :I_{q,\omega }^T\times I_{q,\omega }^T\to [0,\infty )$ are continuous with ${\phi }_0=max\left\{\phi (t,s)\right\}:(t,s)and{\psi }_0=max\left\{\psi (t,s)\right\}$. Suppose that the following conditions hold:

$\left(H_1\right)$

There exist positive constants $M_i$ such that for each $t\in I_{q,\omega }^T$ and $u_i,v_i\in \mathbb{R},i=1,2$,

$$\begin{array}{ccc}& & |F\left[t,u_1,u_2\right]-F\left[t,v_1,v_2\right]|\le M_1|u_1-v_1|+M_2|u_2-v_2|.\hfill \end{array}$$

$\left(H_2\right)$

There exist positive constants $N_i$ such that for each $t\in I_{q,\omega }^T$ and $u_i,v_i\in \mathbb{R},i=1,2$,

$$\begin{array}{ccc}& & |H\left[t,u_1,u_2\right]-H\left[t,v_1,v_2\right]|\le N_1|u_1-v_1|+N_2|u_2-v_2|.\hfill \end{array}$$

$\left(H_3\right)$

There exist positive constants ${\omega }_i$ such that for each $u_i,v_i\in \mathcal{C},i=1,2$,

$$\begin{array}{ccc}& & |{\varphi }_1\left(u\right)-{\varphi }_1\left(v\right)|\le {\omega }_1{\parallel u-v\parallel }_{\mathcal{C}}and|{\varphi }_2\left(u\right)-{\varphi }_2\left(v\right)|\le {\omega }_2{\parallel u-v\parallel }_{\mathcal{C}}.\hfill \end{array}$$

$\left(H_4\right)$

For each $t\in I_{q,\omega }^T$, $g_i<g_i\left(t\right)<G_i,i=1,2$.

$\left(H_5\right)$

$\mathrm{\Xi }=\mathcal{L}\chi +{\omega }_1{\Theta }_2^{*}+{\omega }_2^{*}{\Theta }_1^{*}<1$,

where

$$\begin{array}{cc}\hfill \mathcal{L}:=& \lambda \left[M_1+M_2{\phi }_0{q}^{\left(\genfrac{}{}{0pt}{}{\gamma }{2}\right)}\frac{{(T-{\omega }_0)}^{\gamma }}{{\tilde{\mathrm{\Gamma }}}_q(\gamma +1)}\right]+\mu \left[N_1+N_2{\psi }_0{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{-\nu }}{{\tilde{\mathrm{\Gamma }}}_q(1-\nu )}\right]\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \chi :=& q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta +1)}+q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu +1)}+G_1{\mathrm{\Phi }}_1{\Theta }_2^{*}+G_2{\mathrm{\Phi }}_2{\Theta }_1^{*}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_i:=& q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_i}{2}\right)}\frac{{({\eta }_i-{\omega }_0)}^{\alpha +\beta -{\theta }_i}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_i+1)},i=1,2\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\Theta }_1^{*}:=& {\Theta }_1+{\overline{\Theta }}_1\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\Theta }_2^{*}:=& {\Theta }_2+{\overline{\Theta }}_2\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\Theta }_1:=& {(T-{\omega }_0)}^{\beta -1}max|A_1|+\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+(\alpha -1)\beta }{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){(T-{\omega }_0)}^{\alpha +\beta -1}}{\tilde{\mathrm{\Gamma }}(\alpha +\beta )}max\left|B_1\right|\hfill \\ \hfill {\overline{\Theta }}_1:=& {(T-{\omega }_0)}^{\beta -\nu -1}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{{\tilde{\mathrm{\Gamma }}}_q(\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)-\nu (\beta -1)}max\left|A_1\right|\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & +{(T-{\omega }_0)}^{\alpha +\beta -\nu -1}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu -1}{2}\right)-(\alpha -1)}max\left|B_1\right|\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\Theta }_2:=& {(T-{\omega }_0)}^{\beta -1}max|A_2|+\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+(\alpha -1)\beta }{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){(T-{\omega }_0)}^{\alpha +\beta -1}}{\tilde{\mathrm{\Gamma }}(\alpha +\beta )}max\left|B_2\right|\hfill \\ \hfill {\overline{\Theta }}_2:=& {(T-{\omega }_0)}^{\beta -\nu -1}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{{\tilde{\mathrm{\Gamma }}}_q(\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)-\nu (\beta -1)}max\left|A_2\right|\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & +{(T-{\omega }_0)}^{\alpha +\beta -\nu -1}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu -1}{2}\right)-(\alpha -1)}max\left|B_2\right|.\hfill \end{array}$$

(26)

Then, the problem (1) has a unique solution in $I_{q,\omega }^T$.

Proof. 

For each $t\in I_{q,\omega }^T$ and $u,v\in \mathcal{C}$, we find that

$$\begin{array}{cc}\hfill |{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\nu }u-{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\nu }v|\le & \frac{{\phi }_0{q}^{\left(\genfrac{}{}{0pt}{}{\gamma }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\gamma \right)}{\int }_{{\omega }_0}^t{\widetilde{(t-s)}}_{q,\omega }^{\underline{\gamma -1}}|u\left({\sigma }_{q,\omega }^{\gamma -1}\left(s\right)\right)-v\left({\sigma }_{q,\omega }^{\gamma -1}\left(s\right)\right)|{\tilde{d}}_{q,\omega }s\hfill \\ \hfill \le & \frac{{\phi }_0\parallel u-v\parallel q^{\left(\genfrac{}{}{0pt}{}{\gamma }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\gamma \right)}{\int }_{{\omega }_0}^T{\widetilde{(T-s)}}_{q,\omega }^{\underline{\gamma -1}}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill =& {\phi }_0{q}^{\left(\genfrac{}{}{0pt}{}{\gamma }{2}\right)}\frac{{(T-{\omega }_0)}^{\gamma }}{{\tilde{\mathrm{\Gamma }}}_q(\gamma +1)}.\hfill \end{array}$$

(27)

Similary,

$$\begin{array}{cc}\hfill |{\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u-{\widetilde{{\rm Y}}}_{q,\omega }^{\nu }v|=& {\phi }_0{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{-\nu }}{{\tilde{\mathrm{\Gamma }}}_q(1-\nu )}.\hfill \end{array}$$

(28)

Denote that

$$\begin{array}{cc}\hfill & \mathcal{F}|u-v|\left(t\right)=|F[t,u\left(t\right),\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)\left(t\right)]-F[t,v\left(t\right),\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }v\right)\left(t\right)]|\hfill \\ \hfill & \mathcal{H}|u-v|\left(t\right)=|H[t,u\left(t\right),\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)\left(t\right)]-H[t,v\left(t\right),\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }v\right)\left(t\right)]|,\hfill \end{array}$$

we obtain

$$\begin{array}{cc}\hfill & |{\mathit{O}}_i^{*}[{\varphi }_i,F_u+H_u]-{\mathit{O}}_i^{*}[{\varphi }_i,F_v+H_v]|\hfill \\ \hfill \le & |{\varphi }_i\left(u\right)-{\varphi }_i\left(v\right)|+\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_i}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-{\theta }_i)}{\int }_{{\omega }_0}^{{\eta }_i}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left({\eta }_i-y\right)}}_{q,\omega }^{\underline{-{\theta }_i-1}}\times \hfill \\ \hfill & {\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)-x\right)}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\alpha -1}}{g}_i\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)\right)\times \hfill \\ \hfill & \left[\lambda \mathcal{F}|u-v|\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)+\mu \mathcal{H}|u-v|\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)\right]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y\hfill \\ \hfill \le & {\omega }_i{\parallel u-v\parallel }_C\hfill \\ \hfill & +\left(\lambda \left[M_1|u-v|+M_2|{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u-{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }v|\right]+\mu \left[N_1|u-v|+N_2|{\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u-{\widetilde{{\rm Y}}}_{q,\omega }^{\nu }v|\right]\right)\times \hfill \\ \hfill & \frac{G_i{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_i}{2}\right)}{({\eta }_i-{\omega }_0)}^{\alpha +\beta -{\theta }_i}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_i+1)}\hfill \\ \hfill \le & \left({\omega }_i+\mathcal{L}\left(\lambda \left[M_1+M_2{\phi }_0{q}^{\left(\genfrac{}{}{0pt}{}{\gamma }{2}\right)}\frac{{(T-{\omega }_0)}^{\gamma }}{{\tilde{\mathrm{\Gamma }}}_q(\gamma +1)}\right]\mu \left[N_1+N_2{\mathrm{\Psi }}_0{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{-\nu }}{{\tilde{\mathrm{\Gamma }}}_q(1-\nu )}\right]\right)\right)\times \hfill \\ \hfill & \frac{G_i{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_i}{2}\right)}{({\eta }_i-{\omega }_0)}^{\alpha +\beta -{\theta }_i}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_i+1)}{\parallel u-v\parallel }_{\mathcal{C}}\hfill \\ \hfill =& \left({\omega }_i+\mathcal{L}\right)\frac{G_i{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_i}{2}\right)}{({\eta }_i-{\omega }_0)}^{\alpha +\beta -{\theta }_i}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_i+1)}{\parallel u-v\parallel }_{\mathcal{C}},\hfill \end{array}$$

(29)

for $i=1,2$. Therefore,

$$\begin{array}{cc}\hfill \left|\right(\mathcal{A}u\left)\right(t)& -\left(\mathcal{A}v\right)\left(t\right)|\hfill \\ \hfill \le & \frac{{(T-{\omega }_0)}^{\beta -1}}{\left|\mathrm{\Omega }\right|}\left\{\right|{\mathit{A}}_2\left|\right|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]-{\mathit{O}}_1^{*}[{\varphi }_1,F_v+H_v]+\left|{\mathit{A}}_1\right||{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\hfill \\ \hfill & -{\mathit{O}}_2^{*}[{\varphi }_2,F_v+H_v]\left|\right\}+\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^T{\widetilde{(T-s)}}_{q,\omega }^{\beta -1}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s\times \hfill \\ \hfill & \left\{|{\mathit{B}}_2\left|\right|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]-{\mathit{O}}_1^{*}[{\varphi }_1,F_v+H_v]+\left|{\mathit{B}}_1\right||{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]-{\mathit{O}}_2^{*}[{\varphi }_2,F_v+H_v]|\right\}\hfill \\ \hfill & +\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)+{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^T{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{(T-x)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\alpha -1}}\times \hfill \\ \hfill & \left[\lambda \mathcal{F}|u-v|\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)+\mu \mathcal{H}|u-v|\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)\right]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\hfill \\ \hfill \le & \{\mathcal{L}{q}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\alpha \beta }\frac{{(T-{\omega }_0)}^{\alpha +\beta }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -1)}+\frac{{(T-{\omega }_0)}^{\beta -1}}{\left|\mathrm{\Omega }\right|}[\left|{\mathit{A}}_2\right|\left({\omega }_1+\mathcal{L}{G}_1{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_1}{2}\right)}\frac{{({\eta }_1-{\omega }_0)}^{\alpha +\beta -{\theta }_1}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_1+1)}\right)\hfill \\ \hfill & +|{\mathit{A}}_1|\left({\omega }_2+\mathcal{L}{G}_2{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_2}{2}\right)}\frac{{({\eta }_2-{\omega }_0)}^{\alpha +\beta -{\theta }_2}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_2+1)}\right)]+\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+(\alpha -1)\beta }{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){(T-{\omega }_0)}^{\beta -1}}{\left|\mathrm{\Omega }\right|\tilde{\mathrm{\Gamma }}(\alpha +\beta )}\times \hfill \\ \hfill & \left[\right|{\mathit{B}}_2|\left({\omega }_1+\mathcal{L}{G}_1{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_1}{2}\right)}\frac{{({\eta }_1-{\omega }_0)}^{\alpha +\beta -{\theta }_1}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_1+1)}\right)\hfill \\ \hfill & +|{\mathit{B}}_1|\left({\omega }_2+\mathcal{L}{G}_2{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_2}{2}\right)}\frac{{({\eta }_2-{\omega }_0)}^{\alpha +\beta -{\theta }_2}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_2+1)}\right)\left]\right\}{\parallel u-v\parallel }_C\hfill \\ \hfill =& \left\{\mathcal{L}\right[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta +1)}+\frac{{(T-{\omega }_0)}^{\beta -1}}{\left|\mathrm{\Omega }\right|}(\left|{\mathit{A}}_2\right|G_1{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_1}{2}\right)}\frac{{({\eta }_1-{\omega }_0)}^{\alpha +\beta -{\theta }_1}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_1+1)}\hfill \\ \hfill & +|{\mathit{A}}_1|G_2{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_2}{2}\right)}\frac{{({\eta }_2-{\omega }_0)}^{\alpha +\beta -{\theta }_2}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_2+1)}\left)\right]+[\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+(\alpha -1)\beta }{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){(T-{\omega }_0)}^{\alpha +\beta -1}}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta )}\times \hfill \\ \hfill & \left(|{\mathit{B}}_2|G_1{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_1}{2}\right)}\frac{{({\eta }_1-{\omega }_0)}^{\alpha +\beta -{\theta }_1}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_1+1)}+\left|{\mathit{B}}_1\right|G_2{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_2}{2}\right)}\frac{{({\eta }_2-{\omega }_0)}^{\alpha +\beta -{\theta }_2}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_2+1)}\right)]\hfill \\ \hfill & +\frac{{(t-{\omega }_0)}^{\beta -1}}{\left|\mathrm{\Omega }\right|}({\omega }_1|{\mathit{A}}_2|+{\omega }_2|{\mathit{A}}_1\left|\right)+\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+(\alpha -1)\beta }{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){(T-{\omega }_0)}^{\alpha +\beta -1}}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta )}({\omega }_1|{\mathit{B}}_2|+{\omega }_2|{\mathit{B}}_1\left|\right)\}\times \hfill \\ \hfill & {\parallel u-v\parallel }_{\mathcal{C}}\hfill \\ \hfill =& \left\{\mathcal{L}\left[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta +1)}+G_1{\mathrm{\Phi }}_1{\Theta }_2+G_2{\mathrm{\Phi }}_2{\Theta }_1\right]+{\omega }_1{\Theta }_2+{\omega }_2{\theta }_1\right\}{\parallel u-v\parallel }_{\mathcal{C}}.\hfill \end{array}$$

(30)

Take fractional Hahn difference of order $\nu$ for (16). Then, we have

$$\begin{array}{cc}\hfill ({\tilde{D}}_{q,\omega }^{\nu }& \mathcal{A}u\left)\right(t)\hfill \\ \hfill =& \left[{\mathit{A}}_2{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]+{\mathit{A}}_1{\mathit{O}}_2^{*}[{\varphi }_2,F_v+H_v]\right]\frac{q^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{\mathrm{\Omega }{\tilde{\mathrm{\Gamma }}}_q(-\nu )}\times \hfill \\ \hfill & {\int }_{{\omega }_0}^t{\widetilde{\left(t-s\right)}}_{q,\omega }^{\underline{-\nu -1}}{\left({\sigma }_{q,\omega }^{-\nu -1}\left(s\right)-{\omega }_0\right)}^{\beta -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill -& \left[{\mathit{B}}_2{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]+{\mathit{B}}_1{\mathit{O}}_2^{*}[{\varphi }_2,F_v+H_v]\right]\frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{\mathrm{\Omega }{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-\nu )}\times \hfill \\ \hfill & {\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\nu -1}\left(x\right)}{\widetilde{\left(t-x\right)}}_{q,\omega }^{\underline{-\nu -1}}{\left({\sigma }_{q,\omega }^{-\nu -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{{\tilde{d}}_q\left(\alpha \right){\tilde{d}}_q\left(\beta \right){\tilde{d}}_q(-\nu )}{\int }_{{\omega }_0}^t{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\nu -1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left(t-y\right)}}_{q,\omega }^{\underline{-\nu -1}}{\left({\sigma }_{q,\omega }^{-\nu -1}\left(y\right)-x\right)}_{q,\omega }^{\underline{\beta -1}}\times \hfill \\ \hfill & {\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\alpha -1}}\{\lambda F[{\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u\left({\sigma }_{q,\omega }^{\alpha -1}\right),\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)\left({\sigma }_{q,\omega }^{\alpha -1}\right)]\hfill \\ \hfill & +\mu H[{\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u\left({\sigma }_{q,\omega }^{\alpha -1}\right),\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)\left({\sigma }_{q,\omega }^{\alpha -1}\right)]\}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y.\hfill \end{array}$$

(31)

Similary,

$$\begin{array}{cc}\hfill \left|\right(\tilde{D}& {}_{q,\omega }^{\nu }\mathcal{A}u)\left(t\right)-\left({\tilde{D}}_{q,\omega }^{\nu }\mathcal{A}v\right)\left(t\right)|\hfill \\ \hfill <& \left\{|{\mathit{A}}_2\left|\right|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]-{\mathit{O}}_1^{*}[{\varphi }_1,F_v+H_v]|+|{\mathit{A}}_1\left|\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]-{\mathit{O}}_2^{*}[{\varphi }_2,F_v+H_v]|\right\}\times \hfill \\ \hfill & \frac{q^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q(-\nu )}{\int }_{{\omega }_0}^T{\widetilde{\left(T-s\right)}}_{q,\omega }^{\underline{-\nu -1}}{\left({\sigma }_{q,\omega }^{-\nu -1}\left(s\right)-{\omega }_0\right)}^{\beta -1}{\tilde{d}}_{q,\omega }s\hfill \\ \hfill +& \left\{|{\mathit{B}}_2\left|\right|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]-{\mathit{O}}_1^{*}[{\varphi }_1,F_v+H_v]|+|{\mathit{B}}_1\left|\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]-{\mathit{O}}_2^{*}[{\varphi }_2,F_v+H_v]|\right\}\times \hfill \\ \hfill & \frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-\nu )}{\int }_{{\omega }_0}^T{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\nu -1}\left(x\right)}{\widetilde{\left(T-x\right)}}_{q,\omega }^{\underline{-\nu -1}}{\widetilde{\left({\sigma }_{q,\omega }^{-\nu -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-\nu )}{\int }_{{\omega }_0}^T{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-\nu -1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{\left(T-y\right)}}_{q,\omega }^{\underline{-\nu -1}}{\widetilde{\left({\sigma }_{q,\omega }^{-\nu -1}\left(y\right)-x\right)}}_{q,\omega }^{\beta -1}{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}_{q,\omega }^{\underline{\alpha -1}}\hfill \\ \hfill & \left[\lambda \mathcal{F}\right|u-v|\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)+\mu \mathcal{H}|u-v|\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right)\right)]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y\hfill \\ \hfill \le & \left\{\mathcal{L}\right[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu +1)}+\frac{{(T-{\omega }_0)}^{\beta -\nu -1}}{\left|\mathrm{\Omega }\right|}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{{\tilde{\mathrm{\Gamma }}}_q(\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)-\nu (\beta -1)}\times \hfill \\ \hfill & \left(|{\mathit{A}}_2|G_1{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_1}{2}\right)}\frac{{({\eta }_1-{\omega }_0)}^{\alpha +\beta -{\theta }_1}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_1+1)}+\left|{\mathit{A}}_1\right|G_2{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_2}{2}\right)}\frac{{({\eta }_2-{\omega }_0)}^{\alpha +\beta -{\theta }_2}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_2+1)}\right)\hfill \\ \hfill +& \frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu -1}}{\left|\mathrm{\Omega }\right|}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu -1}{2}\right)-(\alpha -1)}\times \hfill \\ \hfill & \left(|{\mathit{B}}_2|G_1{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_1}{2}\right)}\frac{{({\eta }_1-{\omega }_0)}^{\alpha +\beta -{\theta }_1}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_1+1)}+\left|{\mathit{B}}_1\right|G_2{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_2}{2}\right)}\frac{{({\eta }_2-{\omega }_0)}^{\alpha +\beta -{\theta }_2}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -{\theta }_2+1)}\right)]\hfill \\ \hfill +& \frac{{(T-{\omega }_0)}^{\beta -\nu -1}}{\left|\mathrm{\Omega }\right|}\frac{{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{{\tilde{\mathrm{\Gamma }}}_q(\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)-\nu (\beta -1)}({\omega }_1|{\mathit{A}}_2|+{\omega }_2|{\mathit{A}}_1\left|\right)+\frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu -1}}{\left|\mathrm{\Omega }\right|}\times \hfill \\ \hfill & \frac{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu )}{q}^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu -1}{2}\right)-(\alpha -1)}({\omega }_1|{\mathit{B}}_2|+{\omega }_2|{\mathit{B}}_1{\left|\right)\}\parallel u-v\parallel }_{\mathcal{C}}\hfill \\ \hfill =& \left\{\mathcal{L}\left[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu }}{\tilde{\mathrm{\Gamma }}(\alpha +\beta -\nu +1)}+G_1{\mathrm{\Phi }}_1{\overline{\Theta }}_2+G_2{\mathrm{\Phi }}_2{\overline{\Theta }}_1\right]+{\omega }_1{\overline{\Theta }}_2+{\omega }_2{\overline{\Theta }}_1\right\}{\parallel u-v\parallel }_{\mathcal{C}}.\hfill \end{array}$$

(32)

From (30) and (32), we find that

$$\begin{array}{cc}\hfill \parallel (& {\mathcal{A}u)\left(t\right)-\left(\mathcal{A}v\right)\left(t\right)\parallel }_{\mathcal{C}}\hfill \\ \hfill \le & \{\mathcal{L}\left[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta }}{\tilde{\mathrm{\Gamma }}(\alpha +\beta +1)}+q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu }}{\tilde{\mathrm{\Gamma }}\left(\right)\alpha +\beta -\nu +1}+G_1{\mathrm{\Phi }}_1({\Theta }_2+{\overline{\Theta }}_2)+G_2{\mathrm{\Phi }}_2({\Theta }_1+{\overline{\Theta }}_1)\right]\hfill \\ \hfill & +{\omega }_1({\Theta }_2+{\overline{\Theta }}_2)+{\omega }_2({\Theta }_1+{\overline{\Theta }}_1)\}{\parallel u-v\parallel }_{\mathcal{C}}\hfill \\ \hfill =& \left[\mathcal{L}\chi +{\omega }_1{\Theta }_2^{*}+{\omega }_2{\Theta }_1^{*}\right]{\parallel u-v\parallel }_{\mathcal{C}}\hfill \\ \hfill =& {\mathrm{\Xi }\parallel u-v\parallel }_{\mathcal{C}}.\hfill \end{array}$$

(33)

From $\left(H_3\right)$, we can conclude that $\mathcal{A}$ is a contraction. Hence, from Banach fixed point theorem, $\mathcal{A}$ has a fixed point which is a unique solution of problem (1) on $I_{q,\omega }^T$. □

4. Existence of at Least One Solution of Problem (1)

In this section, we further consider the existence of at least one solution of (1) by using the Schauder’s fixed point theorem as follows.

Theorem 2.

Suppose that $\left(H_1\right)$,$\left(H_2\right)$,$\left(H_4\right)$ and $\left(H_5\right)$ defined in Theorem 1 hold. Then, problem (1) has at least one solution on $I_{q,\omega }^T$.

Proof. 

We split the proof into several steps.

Step I. Verify $\mathcal{A}$ map bounded sets into bounded sets in $B_R$. Let $B_R:=\{u\in C\left(I_{q,\omega }^T\right){:\parallel u\parallel }_{\mathcal{C}}\le R\}$,${\displaystyle \underset{t\in I_{q,\omega }^T}{max}\left|F(t,0,0)\right|=F}$, ${\displaystyle \underset{t\in I_{q,\omega }^T}{max}\left|H(t,0,0)\right|=H}$, ${\displaystyle \underset{u\in C}{sup}\left|{\varphi }_i\left(u\right)\right|=P_i}$ and choose a constant

$$R\ge \frac{[\lambda F+\mu H]\chi +P_1{\Theta }_2^{*}+P_2{\Theta }_1^{*}}{1-\mathcal{L}\chi }.$$

(34)

Denote that

$$\left|\mathcal{F}(t,u,0)\right|=|F[t,u\left(t\right),\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)\left(t\right)]-F[t,0,0]|+\left|F[t,0,0]\right|,$$

$$\left|\mathcal{H}(t,u,0)\right|=|H[t,u\left(t\right),\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)\left(t\right)]-H[t,0,0]|+\left|H[t,0,0]\right|.$$

Consider that

$$\begin{array}{cc}\hfill & \left|{\mathit{O}}_i^{*}[{\varphi }_i,F_u+H_u]\right|\hfill \\ \hfill \le & P_i+\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)+\left(\genfrac{}{}{0pt}{}{-{\theta }_i}{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right){\tilde{\mathrm{\Gamma }}}_q(-{\theta }_i)}{\int }_{{\omega }_0}^{{\eta }_i}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)}{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{({\eta }_i-y)}}_{q,\omega }^{\underline{-{\theta }_i-1}}{\widetilde{\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)-x\right)}}_{q,\omega }^{\underline{\beta -1}}\times \hfill \\ \hfill & {\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}{g}_i\left({\sigma }_{q,\omega }^{-{\theta }_i-1}\left(y\right)\right)\left[\lambda |\mathcal{F}\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u,0\right)|+\mu |\mathcal{H}\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u,0\right)|\right]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y\hfill \\ \hfill \le & P_i+G_i\left[\lambda \left(M_1\left|u\right|+M_2\left|{\mathrm{\Psi }}_{q,\omega }^{\gamma }u\right|+F\right)+\mu \left(N_1\left|u\right|+N_2\left|{{\rm Y}}_{q,\omega }^{\nu }u\right|+H\right)\right]\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_i}{2}\right)}{({\eta }_i-{\omega }_0)}^{\alpha +\beta -{\theta }_i}}{\tilde{\mathrm{\Gamma }}(\alpha +\beta -{\theta }_i+1)}\hfill \\ \hfill \le & P_i+G_i\left[\lambda F+\mu H+{\mathcal{L}\parallel u\parallel }_C\right]\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -{\theta }_i}{2}\right)}{({\eta }_i-{\omega }_0)}^{\alpha +\beta -{\theta }_i}}{\tilde{\mathrm{\Gamma }}(\alpha +\beta -{\theta }_i+1)},\hfill \end{array}$$

(35)

where $i=i,2$, and

$$\begin{array}{cc}\hfill \left|\right(\mathcal{A}u\left)\right(t\left)\right|\le & \frac{{(T-{\omega }_0)}^{\beta -1}}{\left|\mathrm{\Omega }\right|}\left\{|{\mathit{A}}_2\left|\right|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]|+|{\mathit{A}}_1\left|\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]|\right\}\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^T{\widetilde{(T-s)}}_{q,\omega }^{\underline{\beta -1}}{\left({\sigma }_{q,\omega }^{\beta -1}\left(s\right)-{\omega }_0\right)}^{\alpha -1}{d}_{q,\omega }s\times \hfill \\ \hfill & \left\{|{\mathit{B}}_2\left|\right|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]|+\left|{\mathit{B}}_1\right||{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]|\right\}\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)+\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right){\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\int }_{{\omega }_0}^T{\int }_{{\omega }_0}^{{\sigma }_{q,\omega }^{\beta -1}\left(x\right)}{\widetilde{(T-x)}}_{q,\omega }^{\underline{\beta -1}}{\widetilde{\left({\sigma }_{q,\omega }^{\beta -1}\left(x\right)-s\right)}}_{q,\omega }^{\underline{\alpha -1}}\times \hfill \\ \hfill & \left[\lambda \left|\mathcal{F}\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u,0\right)\right|+\mu \left|\mathcal{H}\left({\sigma }_{q,\omega }^{\alpha -1}\left(s\right),u,0\right)\right|\right]{\tilde{d}}_{q,\omega }s{\tilde{d}}_{q,\omega }x{\tilde{d}}_{q,\omega }y\hfill \\ \hfill \le & {[\lambda F+\mu H+\mathcal{L}\parallel u\parallel }_C]\left[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta }}{\tilde{\mathrm{\Gamma }}(\alpha +\beta +1)}+G_1{\mathrm{\Phi }}_1{\Theta }_2+G_2{\mathrm{\Phi }}_2{\Theta }_1\right]+P_1{\Theta }_2+P_2{\Theta }_1.\hfill \end{array}$$

(36)

Next,

$$\begin{array}{cc}\hfill \left|\left({\tilde{D}}_{q,\omega }^{\nu }\mathcal{A}u\right)\left(t\right)\right|\le & {[\lambda F+\mu H+\mathcal{L}\parallel u\parallel }_C]\left[q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu }{2}\right)}\frac{{(T-{\omega }_0)}^{\alpha +\beta -\nu }}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu +1)}+G_1{\mathrm{\Phi }}_1{\overline{\Theta }}_2+G_2{\mathrm{\Phi }}_2{\overline{\Theta }}_1\right]\hfill \\ \hfill & +P_1{\overline{\Theta }}_2+P_2{\overline{\Theta }}_1.\hfill \end{array}$$

(37)

From (36) and (37), we have

$$\begin{array}{cc}\hfill \parallel \left(\mathcal{A}u\right)\left(t\right)\parallel \le & {[\lambda F+\mu H+\mathcal{L}\parallel u\parallel }_C]\chi +P_1{\Theta }_2^{*}+P_2{\Theta }_1^{*}\hfill \\ \hfill \le & R.\hfill \end{array}$$

(38)

Step II. Since $\mathcal{F}$ and $\mathcal{H}$ are continuous, the operator $\mathcal{A}$ is the continuous on $B_R$.

Step III. Examine that $\mathcal{A}$ is equicontinuous on $B_R$.

For any $t_1,t_2\in I_{q,\omega }^T$ with $t_2>t_1$, we obtain

$$\begin{array}{cc}\hfill \left|\right(\mathcal{A}u)& \left(t_2\right)-\left(\mathcal{A}u\right)\left(t_1\right)|\hfill \\ \hfill \le & |\lambda \parallel F\parallel +\mu \parallel H\parallel |\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta +1)}|{(t_2-{\omega }_0)}^{\alpha +\beta }-{(t_1-{\omega }_0)}^{\alpha +\beta }|\hfill \\ \hfill +& \frac{|{(t_2-{\omega }_0)}^{\beta -1}-{(t_1-{\omega }_0)}^{\beta -1}|}{\left|\mathrm{\Omega }\right|}\left\{|{\mathit{A}}_2|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]+\left|{\mathit{A}}_1\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\right\}\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\beta }{2}\right)-(\alpha -1)\beta }{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta )}|{(t_2-{\omega }_0)}^{\alpha +\beta -1}-{(t_1-{\omega }_0)}^{\alpha +\beta -1}|\times \hfill \\ \hfill & \left\{|{\mathit{B}}_2|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]+\left|{\mathit{B}}_1\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\right\}\hfill \end{array}$$

(39)

and

$$\begin{array}{cc}\hfill & |\left({\tilde{D}}_{q,\omega }^{\nu }u\right)\left(t_2\right)-\left({\tilde{D}}_{q,\omega }^{\nu }u\right)\left(t_1\right)|\hfill \\ \hfill \le & |\lambda \parallel F\parallel +\mu \parallel H\parallel |\frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu }{2}\right)}}{{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu +1)}|{(t_2-{\omega }_0)}^{\alpha +\beta -\nu }-{(t_1-{\omega }_0)}^{\alpha +\beta -\nu }|\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{-\nu }{2}\right)-\nu (\beta -1)}{\tilde{\mathrm{\Gamma }}}_q\left(\beta \right)}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q(\beta -\nu )}|{(t_2-{\omega }_0)}^{\beta -1}-{(t_1-{\omega }_0)}^{\beta -1}|\times \hfill \\ \hfill & \left\{|{\mathit{A}}_2|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]+\left|{\mathit{A}}_1\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\right\}\hfill \\ \hfill +& \frac{q^{\left(\genfrac{}{}{0pt}{}{\alpha +\beta -\nu -1}{2}\right)-(\alpha -1)}{\tilde{\mathrm{\Gamma }}}_q\left(\alpha \right)}{\left|\mathrm{\Omega }\right|{\tilde{\mathrm{\Gamma }}}_q(\alpha +\beta -\nu )}|{(t_2-{\omega }_0)}^{\alpha +\beta -\nu -1}-{(t_1-{\omega }_0)}^{\alpha +\beta -\nu -1}|\times \hfill \\ \hfill & \left\{|{\mathit{B}}_2|{\mathit{O}}_1^{*}[{\varphi }_1,F_u+H_u]+\left|{\mathit{B}}_1\right|{\mathit{O}}_2^{*}[{\varphi }_2,F_u+H_u]\right\}.\hfill \end{array}$$

(40)

We find that the right-hand side of (40) tends to be zero when $|t_2-t_1|\to 0$. Hence, $\mathcal{A}$ is relatively compact on $B_R$. Therefore, the set $\mathcal{A}\left(B_R\right)$ is an equicontinuous set. From Steps I to III and the Arzelá-Ascoli theorem, $\mathcal{A}:\mathcal{C}\to \mathcal{C}$ is completely continuous. By Schauder fixed point theorem, our problem (1) has at least one solution. □

5. Example

In this section, we provide an example to show our results. We let $\alpha =\frac{3}{4},\beta =\frac{1}{3},\gamma =\frac{1}{2},\nu =\frac{1}{4},{\theta }_1=\frac{1}{3},{\theta }_2=\frac{2}{3},q=\frac{1}{2},\omega =\frac{2}{3},{\omega }_0=\frac{4}{3},T=10,\lambda =e^{-4},\mu =e^{-\pi },{\eta }_1={\sigma }_{\frac{1}{2},\frac{2}{3}}^4\left(10\right)=\frac{15}{8},{\eta }_2={\sigma }_{\frac{1}{2},\frac{2}{3}}^5\left(10\right)=\frac{77}{48},g_1\left(t\right)={\left(\pi +sint\right)}^2,g_2\left(t\right)={\left(e+cost\right)}^2$, $F\left[t,u\left(t\right),\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)\left(t\right)\right]=\frac{e^{-\left[{sin}^2\left(2\pi t\right)\right]}}{100+e^{{cos}^2\left(2\pi t\right)}}·\frac{\left|u\left(t\right)\right|+e^{-\pi }\left|\left({\tilde{\mathrm{\Psi }}}_{\frac{1}{2},\frac{2}{3}}^{\frac{1}{2}}u\right)\left(t\right)\right|}{1+\left|u\left(t\right)\right|}$ and $H\left[t,u\left(t\right),\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)\left(t\right)\right]=\frac{e^{-\left[{cos}^2(2\pi t+\pi )\right]}}{{\left(t+\pi \right)}^3}·\frac{\left|u\left(t\right)\right|+e^{-\frac{\pi }{2}}\left|\left({\widetilde{{\rm Y}}}_{\frac{1}{2},\frac{2}{3}}^{\frac{1}{4}}u\right)\left(t\right)\right|}{1+\left|u\left(t\right)\right|}$ which are satisfied with the conditions of the problem (1). Therefore the problem (1) is represented by

$$\begin{array}{ccc}\hfill {\tilde{D}}_{\frac{1}{2},\frac{2}{3}}^{\frac{3}{4}}{\tilde{D}}_{\frac{1}{2},\frac{2}{3}}^{\frac{1}{3}}u\left(t\right)& =& \frac{e^{-\left[{sin}^2\left(2\pi t\right)+4\right]}}{100+e^{{cos}^2\left(2\pi t\right)}}·\frac{\left|u\left(t\right)\right|+e^{-\pi }\left|{\tilde{\mathrm{\Psi }}}_{\frac{1}{2},\frac{2}{3}}^{\frac{1}{2}}u\left(t\right)\right|}{1+\left|u\left(t\right)\right|}\hfill \\ & & +\frac{e^{-\left[{cos}^2(2\pi t+\pi )\right]}}{{\left(t+\pi \right)}^3}·\frac{\left|u\left(t\right)\right|+e^{-\frac{\pi }{2}}\left|{\widetilde{{\rm Y}}}_{\frac{1}{2},\frac{2}{3}}^{\frac{1}{4}}u\left(t\right)\right|}{1+\left|u\left(t\right)\right|},t\in I_{\frac{1}{2},\frac{2}{3}}^{10}\hfill \\ \hfill {\tilde{D}}_{\frac{1}{2},\frac{2}{3}}^{\frac{1}{3}}{\left(\pi +sin\left(\frac{15}{8}\right)\right)}^{2}u\left(\frac{15}{8}\right)& =& \frac{\left|u\left(t_i\right)\right|}{1000e^2}{sin}^2\left|\pi u\left(t_i\right)\right|,t_i\in {\sigma }_{\frac{1}{2},\frac{2}{3}}^i\left(10\right)\hfill \\ \hfill {\tilde{D}}_{\frac{1}{2},\frac{2}{3}}^{\frac{2}{3}}{\left(e+cos\left(\frac{77}{48}\right)\right)}^{2}u\left(\frac{77}{48}\right)& =& \frac{\left|u\left(t_i\right)\right|}{1000{\pi }^2}{cos}^2\left|\pi u\left(t_i\right)\right|,t_i\in {\sigma }_{\frac{1}{2},\frac{2}{3}}^i\left(10\right)\hfill \end{array}$$

(41)

where $\phi (t,s)=\frac{e^{-|t-s|}}{{(t+e)}^3}$ and $\psi (t,s)=\frac{e^{-|t-s|}}{{(t+\pi )}^3}$.

To investigate the values of $M_1,M_2,N_1,N_2,{\omega }_1,{\omega }_2,g_1,g_2,G_1,G_2,\mathcal{L},{\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,{\Theta }_1$, ${\Theta }_2,$${\Theta }_1^{*}$, ${\Theta }_2^{*}$, $\chi$ and $\mathrm{\Xi }$, we employ the assumptions $\left(H_1\right)$–$\left(H_5\right)$ to get the results as follows.

For all $t\in I_{\frac{1}{2},\frac{2}{3}}^{10}$ and $u,v\in \mathbb{R},$ we find that

$$\begin{array}{cc}\hfill \left|F[t,u,\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)]-F[t,v,\left({\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u\right)]\right|\le & \frac{1}{101}|u-v|+\frac{1}{101e\pi }\left|{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }u-{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\gamma }v\right|,\hfill \\ \hfill \left|H[t,u,\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }u\right)]-H[t,v,\left({\widetilde{{\rm Y}}}_{q,\omega }^{\nu }v\right)]\right|\le & \frac{1}{{\left(\frac{4}{3}+\pi \right)}^3}|u-v|+\frac{1}{{\left(\frac{4}{3}+\pi \right)}^3{e}^{\frac{\pi }{2}}}\left|{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\nu }u-{\tilde{\mathrm{\Psi }}}_{q,\omega }^{\nu }v\right|.\hfill \end{array}$$

Thus, $\left(H_1\right)$ and $\left(H_2\right)$ hold with $M_1=0.0099,M_2=0.000428,N_1=0.01116\mathrm{and}{N}_2=0.00232.$

For all $u,v\in \mathcal{C},$ we obtain

$\left|{\varphi }_1\left(u\right)-{\varphi }_1\left(v\right)\right||\le \frac{1}{1000e^2}\left||u-v\right|{|}_{\mathtt{C}},\left|{\varphi }_2\left(u\right)-{\varphi }_2\left(v\right)\right|{|\le \frac{1}{1000{\pi }^2}\left||u-v\right||}_{\mathtt{C}}.$

Then, $\left(H_3\right)$ holds with ${\omega }_1=0.0001353,{\omega }_2=0.0001013.$

The condition $\left(H_4\right)$ holds with $g_1=4.5864,g_2=2.9525,G_1=17.1528,G_2=13.8256.$

We next find that

$${\phi }_0=0.01504,{\psi }_0=0.01116,\left|A_1\right|\le 8.6795,\left|A_2\right|\le 1.83867,$$

$$\left|B_1\right|\le 41.5268,\left|B_2\right|\le 11.7567\mathrm{and}\left|\mathrm{\Omega }\right|\ge 1.4668,$$

and,

$$\mathcal{L}=0.000665,{\mathrm{\Phi }}_1=0.69789,{\mathrm{\Phi }}_2=4.{29227}^{-13},{\Theta }_1=45.7791,{\Theta }_2=12.8606,$$

$${\overline{\Theta }}_1=16.4749,{\overline{\Theta }}_2=4.65093,{\Theta }_1^{*}=62.2540,{\Theta }_2^{*}=17.5115\mathrm{and}\chi =253.8093.$$

So, $\left(H_5\right)$ hold with $\mathrm{\Xi }=0.17746<1.$

Therefore, by Theorem 1 problem (41) has a unique solution.

6. Conclusions

We present the new problem involving five fractional symmetric Hahn difference operators and one fractional symmetric Hahn integral of different orders where the new concepts of fractional symmetric Hanh calculus were used. By using the Schauder and Banach fixed point theorems we found conditions under which this problem, respectively, has a solution and has a unique solution. In addition, some properties of symmetric Hahn integral are also studied. The results of this article are new and enrich the field of boundary value problems for fractional symmetric Hahn integrodifference equations. In the future, we may expand this work by considering a new boundary value problem.