Stability in Nonlinear Neutral Caputo q-Fractional Difference Equations

Abstract

In this article, we consider a nonlinear neutral q-fractional difference equation. So, we apply the fixed point theorem of Krasnoselskii to obtain the existence of solutions under sufficient conditions. After that, we use the fixed point theorem of Banach to show the uniqueness, as well as the stability of solutions. Our main results extend and generalize previous results mentioned in the conclusion.

1. Introduction and Preliminaries

In the beginning of the twentieth century, the so-called q-calculus appeared, which was discussed in Jackson’s paper [1]; this concept deals with continuous functions that do not need to be smooth. Several investigators have continued to develop this theory and have given several applications in combinatorics and fluid mechanics [2,3,4,5].

In recent years, the q-fractional difference equations have received special attention. In fact, some mathematicians have recently pioneered the development of qualitative properties of q-fractional order difference equations, such as [6,7,8,9,10,11,12,13,14] who have studied transformation methods, properties of initial value problems, and their application. Furthermore, fixed point theorems are often used for existence and uniqueness purposes. In addition, they are used to study the attractiveness and stability and asymptotic stability of solutions, see [15,16,17,18,19,20,21,22,23,24] and the references contained therein.

For $0<q<1$, we define the time scale ${\mathbb{T}}_q=\left\{q^n,n\in \mathbb{Z}\right\}\cup \left\{0\right\}$, where $\mathbb{Z}$ is the set of integers.

For $a=q^{n_0}$ and $n_0\in \mathbb{Z}$, we denote ${\mathbb{T}}_a={\left[a,\infty \right)}_q=\left\{q^{-i}a,i=0,1,2,\cdots \right\}$.

Let ${\mathbb{R}}^m$ be the m-dimensional Euclidean space and define

$${\mathbb{I}}_{\tau }=\left\{\tau a,q^{-1}\tau a,q^{-2}\tau a,\cdots ,a\right\},$$

and

$${\mathbb{T}}_{\tau a}={\left[\tau a,\infty \right)}_q=\left\{q^{-i}\tau a,i=0,1,2,\cdots \right\},$$

where $\tau =q^d\in {\mathbb{T}}_q$, $d\in {\mathbb{N}}_0=\left\{0,1,2,\cdots \right\}$ and ${\mathbb{I}}_{\tau }=\left\{a\right\}$ with $d=0$ is the non-delay case.

Recently, Abdeljawad et al. [7] discussed only the existence of solutions for the q-fractional difference equation

$$\left\{\begin{array}{c}{}_q{C}_a^{\alpha }x\left(t\right)=f\left(t,x\left(t\right),x\left(\tau t\right)\right),t\in {\mathbb{T}}_a,\hfill \\ x\left(t\right)=\psi \left(t\right),t\in {\mathbb{I}}_{\tau },\hfill \end{array}\right.$$

(1)

where $f:{\mathbb{T}}_a\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and ${}_q{C}_a^{\alpha }$ denotes the Caputo’s q-fractional difference of order $\alpha \in \left(0,1\right)$. By employing the Krasnoselskii fixed point theorem, the authors obtained existence results.

In this paper, we are interested to study the existence, uniqueness, and stability of solutions for nonlinear neutral q-fractional difference equations

$$\left\{\begin{array}{c}{}_q{C}_a^{\alpha }\left(x\left(t\right)-g\left(t,x\left(\tau t\right)\right)\right)=f\left(t,x\left(t\right),x\left(\tau t\right)\right),t\in {\mathbb{T}}_a,\hfill \\ x\left(t\right)=\psi \left(t\right),t\in {\mathbb{I}}_{\tau },\hfill \end{array}\right.$$

(2)

where $f:{\mathbb{T}}_a\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},g:{\mathbb{T}}_a\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and ${}_q{C}_a^{\alpha }$ represents Caputo’s q-fractional difference of order $\alpha \in \left(0,1\right)$. To establish our results, we apply Krasnoselskii’s and Banach’s fixed point theorems, as well a Arzela–Ascoli’s theorem.

Now, we give some basic notations, definitions, and properties of q-calculus and fractional difference calculus, which are used throughout this paper; see [7]. For a function $f:{\mathbb{T}}_q\to \mathbb{R}$, its nabla q-derivative of f is written as

$${\nabla }_{q}f\left(t\right)=\frac{f\left(t\right)-f\left(qt\right)}{\left(1-q\right)t},t\in {\mathbb{T}}_q-\left\{0\right\}.$$

(3)

The nabla q-integral of f has the following form

$${\int }_0^{t}f\left(s\right){\nabla }_{q}s=\left(1-q\right)t\sum _{i=0}^{\infty }{q}^{i}f\left(q^{i}t\right).$$

(4)

For $a\in {\mathbb{T}}_q$, (4) becomes

$${\int }_a^{t}f\left(s\right){\nabla }_{q}s={\int }_0^{t}f\left(s\right){\nabla }_{q}s-{\int }_0^{a}f\left(s\right){\nabla }_{q}s.$$

(5)

Definition 1.

The q-factorial function for $n\in \mathbb{N}$ is given by

$${\left(t-s\right)}_q^n=\prod _{i=0}^{n-1}\left(t-q^{i}s\right).$$

(6)

In case is a non-positive integer, the q-factorial function is given by

$${\left(t-s\right)}_q^{\alpha }=t^{\alpha }\prod _{i=0}^{\infty }\frac{\left(1-\frac{s}{t}{q}^i\right)}{\left(1-\frac{s}{t}{q}^{i+\alpha }\right)}.$$

(7)

In the following Lemma, we present some properties of q-factorial functions.

Lemma 1

([8]). For $\alpha ,\beta ,a\in \mathbb{R}$, we have

(i) ${\left(t-s\right)}_q^{\alpha +\beta }={\left(t-s\right)}_q^{\alpha }{\left(t-q^{\alpha }s\right)}_q^{\beta }.$

(ii) ${\left(at-as\right)}_q^{\alpha }=a^{\alpha }{\left(t-s\right)}_q^{\alpha }.$

(iii) The nabla q-derivative of the q-factorial function with respect to t is

$${\nabla }_q{\left(t-s\right)}_q^{\alpha }=\frac{1-q^{\alpha }}{1-q}{\left(t-s\right)}_q^{\alpha -1}.$$

(8)

(iv) The nabla q-derivative of the q-factorial function with respect to s is

$${\nabla }_q{\left(t-s\right)}_q^{\alpha }=\frac{1-q^{\alpha }}{1-q}{\left(t-qs\right)}_q^{\alpha -1}.$$

(9)

Definition 2.

For a function $f:{\mathbb{T}}_q\to \mathbb{R}$, the left q-fractional integral ${}_q{\nabla }_a^{-\alpha }$ of order $\alpha \ne 0,-1,-2,\cdots$ and starting at $a=q^{n_0}\in {\mathbb{T}}_q,n_0\in \mathbb{Z}$, is defined by

$$\begin{array}{ccc}\hfill {}_q{\nabla }_a^{-\alpha }f\left(t\right)& =& \frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s\right){\nabla }_{q}s\hfill \\ & =& \frac{1-q}{{\Gamma }_q\left(\alpha \right)}\sum _{i=n}^{n_0-1}{q}^i{\left(q^n-q^{i+1}\right)}_q^{\alpha -1}f\left(q^i\right),\hfill \end{array}$$

(10)

where

$${\Gamma }_q\left(\alpha +1\right)=\frac{1-q^{\alpha }}{1-q}{\Gamma }_q\left(\alpha \right),{\Gamma }_q\left(1\right)=1,\alpha >0.$$

(11)

Remark 1.

The left q-fractional integral ${}_q{\nabla }_a^{-\alpha }$ maps functions defined on ${\mathbb{T}}_q$ to functions defined on ${\mathbb{T}}_q$.

Definition 3

([25]). Let $0<\alpha \notin \mathbb{N}$. Then,

(i) The left Caputo q-fractional derivative of order α of a function f defined on ${\mathbb{T}}_q$ is defined by

$${}_q{C}_a^{\alpha }f\left(t\right)\triangleq {\nabla }_a^{-\left(n-\alpha \right)}{\nabla }_q^{n}f\left(t\right)=\frac{1}{{\Gamma }_q\left(n-\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{n-\alpha -1}{\nabla }_q^{n}f\left(s\right){\nabla }_{q}s,$$

(12)

where $n=\left[\alpha \right]+1$. In case $\alpha \in \mathbb{N}$, then we may write ${}_q{C}_a^{\alpha }f\left(t\right)\triangleq {\nabla }_q^{n}f\left(t\right)$.

(ii) The left Riemann q-fractional derivative is defined by $\left({}_q{\nabla }_a^{\alpha }f\right)\left(t\right)=\left({\nabla }_q{\nabla }_a^{-\left(n-\alpha \right)}f\right)\left(t\right)$.

(iii) In virtue of [25], the Riemann and Caputo q-fractional derivatives are related by

$$\left({}_q{C}_a^{\alpha }f\right)\left(t\right)=\left({}_q{\nabla }_a^{\alpha }f\right)\left(t\right)-\frac{{\left(t-a\right)}_q^{-\alpha }}{{\Gamma }_q\left(1-\alpha \right)}f\left(a\right).$$

(13)

Remark 2.

Note that, for the right Caputo and Riemann q-fractional derivatives, we advise the reader to see reference [25].

Lemma 2.

Let $\alpha >0$ and f be defined in a suitable domain. Thus

$${}_q{\nabla }_a^{-\alpha }\left({}_q{C}_a^{\alpha }f\right)\left(t\right)=f\left(t\right)-\sum _{k=0}^{n-1}\frac{{\left(t-a\right)}_q^k}{{\Gamma }_q\left(k+1\right)}{\nabla }_q^{k}f\left(a\right),$$

(14)

and if $0<\alpha \le 1$ we have

$${}_q{\nabla }_a^{-\alpha }\left({}_q{C}_a^{\alpha }f\right)\left(t\right)=f\left(t\right)-f\left(a\right).$$

(15)

The following identity is crucial in solving the linear q-fractional equations

$${}_q{\nabla }_a^{-\alpha }{\left(x-a\right)}_q^{\mu }=\frac{{\Gamma }_q\left(\mu +1\right)}{{\Gamma }_q\left(\alpha +\mu +1\right)}{\left(x-a\right)}_q^{\mu +\alpha },0<a<x<b,$$

(16)

where $\alpha \in {\mathbb{R}}^{+}$ and $\mu \in \left(-1,\infty \right)$.

The space $l_{\infty }$ denotes the set of real bounded sequences with respect to the usual supremum norm. We recall that $l_{\infty }$ is a Banach space.

Definition 4.

A set D of sequences in $l_{\infty }$ is uniformly Cauchy if for every $ϵ>0$, there exists an integer $N^{*}$ such that $\left|x\left(t\right)-x\left(s\right)\right|<ϵ$ whenever $t,s>N^{*}$ for any $x=\left\{x\left(n\right)\right\}$ in D.

The following discrete version of Arzela–Ascoli’s theorem has a crucial role in the proof of our main theorem.

Lemma 3

(Arzela-Ascoli [26]). A bounded, uniformly Cauchy subset D of $l_{\infty }\left({\mathbb{T}}_a\right)$ (all bounded real-valued sequences with domain ${\mathbb{T}}_a$) is relatively compact.

The proof of the main theorem is achieved by employing the following fixed point theorems.

Lemma 4

(Banach fixed point [27]). Let $\left(X,∥·∥\right)$ be a Banach space and $A:X\to X$ be a contraction mapping, i.e.,

$$∥Ax-Ay∥\le \lambda ∥x-y∥,$$

for all $x,y\in X$, where $0<\lambda <1$. Then, A has a unique fixed point in X.

Lemma 5

(Krasnoselskii fixed point [27]). Let D be a nonempty, closed, convex, and bounded subset of a Banach space $\left(X,∥·∥\right)$. Suppose that $A_1:D\to X$ and $A_2:D\to X$ are two operators such that

(i) $A_1$ is a contraction,

(ii) $A_2$ is continuous and $A_2\left(D\right)$ resides in a compact subset of X,

(iii) for any $x,y\in D$, $A_{1}x+A_{2}y\in D$.

Then, the operator equation $A_{1}x+A_{2}x=x$ has a solution $x\in D$.

Lemma 6

([7]). We say that x is solution of Equation (1) if and only if it admits the following representation

$$x\left(t\right)=\psi \left(a\right)+\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s,$$

(17)

with $x\left(t\right)=\psi \left(t\right),t\in {\mathbb{I}}_{\tau }$.

We give the equivalence of problem for (2). So, the solvability of this equivalent equation implies the existence, uniqueness, and stability of solution to problem (2).

Lemma 7.

We say that x is solution of Equation (2) if and only if it admits the following representation

$$\begin{array}{cc}\hfill x\left(t\right)& =\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)+g\left(t,x\left(\tau t\right)\right)\hfill \\ \hfill & +\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s,\hfill \end{array}$$

(18)

where $x\left(t\right)=\psi \left(t\right),t\in {\mathbb{I}}_{\tau }$.

Proof. 

Let

$$z\left(t\right)=x\left(t\right)-g\left(t,x\left(\tau t\right)\right).$$

Then, we can write Equation (2) as

$${}_q{C}_a^{\alpha }z\left(t\right)=f\left(t,x\left(t\right),x\left(\tau t\right)\right).$$

By the same way used in [7], we obtain for $t\in {\mathbb{T}}_{a\tau }$, the initial value problem for (2) is equivalent to the following equation

$$z\left(t\right)=z\left(a\right)+\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s.$$

(19)

So

$$\begin{array}{cc}\hfill x\left(t\right)& =\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)+g\left(t,x\left(\tau t\right)\right)\hfill \\ \hfill & +\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s.\hfill \end{array}$$

□

Let $\mathbb{T}={\left[\tau a,T_1\right]}_q=\left\{q^{-i}\tau a,i=0,1,\cdots ,n_1+d\right\}$ where $T_1=q^{-n_1-d}\tau a$ with $n_1\in \left[d+3,\infty \right)\cap \mathbb{Z}$, and $B\left(\mathbb{T},\mathbb{R}\right)$ be the set of all real bounded sequences. $B\left(\mathbb{T},\mathbb{R}\right)$ is a Banach space endowed by the norm

$$∥x∥=\underset{t\in \mathbb{T}}{sup}\left|x\left(t\right)\right|,$$

and let

$$\mathsf{\Omega }=\left\{u\in B\left(\mathbb{T},\mathbb{R}\right),u\left(t\right)=\psi \left(t\right)\mathrm{for}t\in {\mathbb{I}}_{\tau }\mathrm{and}∥u∥\le R\right\},$$

(20)

a non-empty bounded closed and convex subset of $B\left(\mathbb{T},\mathbb{R}\right)$. According to Lemma 7, consider a mapping A on $\mathsf{\Omega }$ as given by

$$\begin{array}{cc}\hfill Ax\left(t\right)& =\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)+g\left(t,x\left(\tau t\right)\right)\hfill \\ \hfill & +\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s.\hfill \end{array}$$

(21)

We can write A as a sum of

$$A_{1}x\left(t\right)=\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)+g\left(t,x\left(\tau t\right)\right),$$

and

$$A_{2}x\left(t\right)=\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s.$$

So, x being fixed point of the operator $Ax=A_{1}x+A_{2}x$ is a solution of Equation (2).

We prove our main results under the following assumptions

$$\left|f\left(t,x,z\right)-f\left(t,y,w\right)\right|\le L_f\left(∥x-y∥+∥z-w∥\right),f\left(t,0,0\right)=0,$$

(22)

and

$$\left|g\left(t,x\right)-g\left(t,y\right)\right|\le L_g∥x-y∥,g\left(t,0\right)=0,$$

(23)

with $L_g<1$. In addition,

$$\left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+R{L}_g+\frac{2R{L}_{f}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha \right)}\le R,$$

(24)

and

$$L_g+\frac{2L_{f}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha \right)}:=\gamma <1,$$

(25)

where $C\left(\alpha \right)=\frac{\left(1-q\right){\left(T_1-a\right)}_q^{\alpha }}{1-q^{\alpha }}$ is a positive constant depending on $\alpha$ and $T_1$.

2. Main Results

We start with the following existence theorem.

Theorem 1.

Let Ω defined by (20). Assume the conditions (22)–(24) hold. Then, there exists at least one solution of Equation (2).

Proof. 

Following the three steps as mentioned in Lemma 5, we present our proof as follows.

Step 1. We prove that for any $x,y\in \mathsf{\Omega }$, $A_{1}x+A_{2}y\in \mathsf{\Omega }$

$$\begin{array}{cc}\hfill \left|A_{1}x\left(t\right)+A_{2}y\left(t\right)\right|& \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+\left|g\left(t,x\left(\tau t\right)\right)\right|\hfill \\ \hfill & +\left|\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s,y\left(s\right),y\left(\tau s\right)\right){\nabla }_{q}s\right|\hfill \\ \hfill & \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+L_g\left|x\left(\tau t\right)\right|\hfill \\ \hfill & +\frac{L_f}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}\left(\left|y\left(s\right)\right|+\left|y\left(\tau s\right)\right|\right){\nabla }_{q}s\hfill \\ \hfill & \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+R{L}_g\hfill \\ \hfill & +\frac{2R{L}_f}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}{\nabla }_{q}s.\hfill \end{array}$$

By the relations (11), (16), and since ${\left(t-a\right)}_q^0=1$, we have

$$\begin{array}{ccc}\hfill \frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}{\left(t-a\right)}_q^0{\nabla }_{q}s& =& {}_q{\nabla }_a^{\alpha }{\left(t-a\right)}_q^0=\frac{{\Gamma }_q\left(1\right){\left(t-a\right)}_q^{\alpha }}{{\Gamma }_q\left(\alpha +1\right)}\hfill \\ & \le & \frac{{\left(T_1-a\right)}_q^{\alpha }}{{\Gamma }_q\left(\alpha +1\right)}=\frac{\left(1-q\right){\left(T_1-a\right)}_q^{\alpha }}{\left(1-q^{\alpha }\right){\Gamma }_q\left(\alpha \right)},t<T_1.\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill \left|A_{1}x\left(t\right)+A_{2}y\left(t\right)\right|& \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+R{L}_g+\frac{2R{L}_{f}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha \right)}\hfill \\ \hfill & \le R.\hfill \end{array}$$

Step 2. Now we have to show that $A_2$ is relatively compact, for this purpose consider $0\le t_1\le t_2\le T_1$, so we obtain

$$\begin{array}{cc}\hfill & \left|A_{2}x\left(t_2\right)-A_{2}x\left(t_1\right)\right|\hfill \\ \hfill & \le \frac{1}{{\Gamma }_q\left(\alpha \right)}\left|{\int }_a^{t_2}{\left(t_2-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s\right.\hfill \\ \hfill & \left.-{\int }_a^{t_1}{\left(t_1-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s\right|\hfill \\ \hfill & \le \frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^{t_1}\left|{\left(t_2-qs\right)}_q^{\alpha -1}-{\left(t_1-qs\right)}_q^{\alpha -1}\right|\left|f\left(s,x\left(s\right),x\left(\tau s\right)\right)\right|{\nabla }_{q}s\hfill \\ \hfill & +\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_{t_1}^{t_2}{\left(t_2-qs\right)}_q^{\alpha -1}\left|f\left(s,x\left(s\right),x\left(\tau s\right)\right)\right|{\nabla }_{q}s.\hfill \end{array}$$

By the assumptions (22), (24), and Lemma 7, we obtain

$$\begin{array}{cc}\hfill \left|A_{2}x\left(t_2\right)-A_{2}x\left(t_1\right)\right|& \le R{L}_f\left[{\int }_a^{t_1}\left|{\left(t_2-qs\right)}_q^{\alpha -1}-{\left(t_1-qs\right)}_q^{\alpha -1}\right|{\nabla }_{q}s\right.\hfill \\ \hfill & \left.+\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_{t_1}^{t_2}{\left(t_2-qs\right)}_q^{\alpha -1}{\nabla }_{q}s\right].\hfill \end{array}$$

By using (10), we obtain

$$\left|A_{2}x\left(t_2\right)-A_{2}x\left(t_1\right)\right|\le 2R{L}_f\left({}_q{\nabla }_a^{-\alpha }\left({\left(t_2-a\right)}_q^0-{\left(t_1-a\right)}_q^0\right){+}_q{\nabla }_{t_1}^{-\alpha }{\left(t_2-t_1\right)}_q^0\right).$$

From (16), it follows that

$$\left|A_{2}x\left(t_2\right)-A_{2}x\left(t_1\right)\right|\le \frac{2R{L}_f}{{\Gamma }_q\left(\alpha +1\right)}\left({\left(t_2-a\right)}_q^{\alpha }-{\left(t_1-a\right)}_q^{\alpha }+{\left(t_2-t_1\right)}_q^{\alpha }\right).$$

Hence it follows that $\left|A_{2}x\left(t_2\right)-A_{2}x\left(t_1\right)\right|\to 0$ as $t_1\to t_2$.

Now in order to show the continuity of $A_2$, let us consider a sequence $x_n$ which converges to x, then we have

$$\begin{array}{cc}\hfill & \left|A_2{x}_n\left(t\right)-A_{2}x\left(t\right)\right|\hfill \\ \hfill & \le \frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}\left|f\left(s,x_n\left(s\right),x_n\left(\tau s\right)\right)-f\left(s,x\left(s\right),x\left(\tau s\right)\right)\right|{\nabla }_{q}s\hfill \\ \hfill & \le \frac{2L_{f_2}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha \right)}∥x_n-x∥\hfill \\ \hfill & =\gamma ∥x_n-x∥.\hfill \end{array}$$

So, $A_2{x}_n\to A_{2}x$ as $x_n\to x$. Hence using Lemma 3, $A_2\mathsf{\Omega }$, being a bounded and uniformly Cauchy subset of $B\left(\mathbb{T},\mathbb{R}\right)$, is relatively compact.

Step 3. We show that $A_1$ is contraction on $\mathsf{\Omega }$, then for any $x,y\in \mathsf{\Omega }$, we have

$$\begin{array}{cc}\hfill \left|A_{1}x\left(t\right)-A_{1}y\left(t\right)\right|& \le \left|g\left(t,x\left(\tau t\right)\right)-g\left(t,y\left(\tau t\right)\right)\right|\hfill \\ \hfill & \le L_g∥x_n-x∥,\hfill \end{array}$$

since $L_g<1$, then $A_1$ is contraction on $\mathsf{\Omega }$. Hence according to Lemma 5, A has a fixed point in $\mathsf{\Omega }$ which is a solution of Equation (2). □

The following theorem gives the uniqueness of solutions.

Theorem 2.

Assume the conditions (22)–(25) hold. Then, there exists a unique solution of Equation (2).

Proof. 

Let the set $\mathsf{\Omega }\subset B\left(\mathbb{T},\mathbb{R}\right)$ defined by (20). Consider a mapping A on $\mathsf{\Omega }$ as given in (21). So, the continuity of $Ax$ is followed by $x\in \mathsf{\Omega }$. To show A maps $\mathsf{\Omega }$ into itself, let $x\in \mathsf{\Omega }$, then

$$\begin{array}{cc}\hfill \left|Ax\left(t\right)\right|& \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+\left|g\left(t,x\left(\tau t\right)\right)\right|\hfill \\ \hfill & +\left|\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}f\left(s,x\left(s\right),x\left(\tau s\right)\right){\nabla }_{q}s\right|\hfill \\ \hfill & \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right|+L_g\left|x\left(\tau t\right)\right|\hfill \\ \hfill & +\frac{L_f}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}\left(\left|x\left(s\right)\right|+\left|x\left(\tau s\right)\right|\right){\nabla }_{q}s\hfill \\ \hfill & \le \left|\left(\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)\right)\right|+R{L}_g+\frac{2R{L}_{f}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha +1\right)}\hfill \\ \hfill & \le R.\hfill \end{array}$$

Hence A maps $\mathsf{\Omega }$ into itself.

Now we show that A is a contraction mapping. Let $x,y\in \mathsf{\Omega }$, then

$$\begin{array}{cc}\hfill & \left|Ax\left(t\right)-Ay\left(t\right)\right|\hfill \\ \hfill & \le \left|g\left(t,x\left(\tau t\right)\right)-g\left(t,y\left(\tau t\right)\right)\right|\hfill \\ \hfill & +\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}\left|f\left(s,x\left(s\right),x\left(\tau s\right)\right)-f\left(s,y\left(s\right),y\left(\tau s\right)\right)\right|{\nabla }_{q}s\hfill \\ \hfill & \le L_g∥x-y∥+\frac{2L_{f_2}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha \right)}∥x-y∥\hfill \\ \hfill & =\gamma ∥x-y∥.\hfill \end{array}$$

since (25) holds, then A is contraction on $\mathsf{\Omega }$. Hence by Lemma 4, Equation (2) has a unique solution. □

Now, we show that the solutions of Equation (2) is stable by giving sufficient conditions.

Theorem 3.

Let x be solution of Equation (2) and $\widehat{x}$ be a solution of Equation (2) satisfying the initial function $\widehat{x}\left(t\right)=\widehat{\psi }\left(t\right)$ on ${\mathbb{I}}_{\tau }$. Assume the conditions (22)–(25) hold. Moreover, for $ϵ>0$ small enough, there exists

$$\delta =\frac{1-\gamma }{1+L_g}ϵ.$$

Then, solutions of Equation (2) are stable.

Proof. 

We have

$$\begin{array}{cc}\hfill & \left|x\left(t\right)-\widehat{x}\left(t\right)\right|\hfill \\ \hfill & \le \left|\psi \left(a\right)-g\left(a,\psi \left(a\tau \right)\right)-\widehat{\psi }\left(a\right)+g\left(a,\widehat{\psi }\left(a\tau \right)\right)\right|\hfill \\ \hfill & +\left|g\left(t,x\left(\tau t\right)\right)-g\left(t,\widehat{y}\left(\tau t\right)\right)\right|\hfill \\ \hfill & +\frac{1}{{\Gamma }_q\left(\alpha \right)}{\int }_a^t{\left(t-qs\right)}_q^{\alpha -1}\left|f\left(s,x\left(s\right),x\left(\tau s\right)\right)-f\left(s,\widehat{x}\left(s\right),\widehat{x}\left(\tau s\right)\right)\right|{\nabla }_{q}s\hfill \\ \hfill & \le ∥\psi -\widehat{\psi }∥\left(1+L_g\right)+L_g∥x-\widehat{x}∥+\frac{2L_{f_2}C\left(\alpha \right)}{{\Gamma }_q\left(\alpha \right)}∥x-\widehat{x}∥\hfill \\ \hfill & \le ∥\psi -\widehat{\psi }∥\left(1+L_g\right)+\gamma ∥x-\widehat{x}∥.\hfill \end{array}$$

Hence

$$∥x-\widehat{x}∥\le \frac{1+L_g}{1-\gamma }∥\psi -\widehat{\psi }∥.$$

Then, for any $ϵ>0$, let $\delta =\frac{1-\gamma }{1+L_g}ϵ$, so for $∥\psi -\widehat{\psi }∥<\delta$ we have $∥x-\widehat{x}∥<ϵ$. Therefore, the solutions of Equation (2) are stable. This completes the proof. □

3. Conclusions and Remarks

In this paper, sufficient conditions guarantee the existence, uniqueness, and stability of solutions for a neutral type of q-fractional difference equation with a delay have been given. The main technique used here is the fixed point theorem. Further, Arzela–Ascoli’s theorem played an important and central role in the proof.

Note that, when $g\equiv 0$,

(i) Theorem 1 becomes the same Theorem 3 in [7], and this confirms the generality of the results.

(ii) Theorem 2 gives the uniqueness of solutions of Equation (1).

(iii) Theorem 3 gives the uniqueness of solutions of Equation (1).