Coupled implicit Caputo fractional q-difference systems

Abstract

This paper deals with some existence, uniqueness, and Ulam stability results for a coupled implicit Caputo fractional q-difference system in Banach and generalized Banach spaces. Some applications are made of some fixed point theorems for the existence and uniqueness of solutions. Next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Some illustrative examples are given in the last section.

1 Introduction

Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics, and other applied sciences [44]. For some fundamental results in the theory of fractional calculus and fractional differential equations, we refer the reader to the monographs [4,5,6, 33, 42, 47], the paper [46], and the references therein. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Caputo fractional derivative [5]. Implicit fractional differential equations were analyzed by many authors (see, for instance, [4, 5, 22, 23, 34, 43] and the references therein). Considerable attention has been given to the study of the Ulam stability of functional differential and integral equations; one can see the monograph [6], the papers [3, 17,18,19,20, 28, 29, 31, 32, 39,40,41], and the references therein.

Fractional q-difference equations initiated at the beginning of the nineteenth century [10, 24] and received significant attention in recent years [21, 26]. Some interesting details about initial and boundary value problems of q-difference and fractional q-difference equations can be found in [7,8,9, 12,13,14,15,16, 25, 27, 35] and the references therein.

In [1, 2], Abbas et al. considered some existence results for some coupled fractional differential systems in generalized Banach spaces.

In this paper we discuss the existence and Ulam–Hyers–Rassias stability of solutions for the following coupled implicit fractional q-difference system:

$$ \textstyle\begin{cases} ({}^{c}D_{q}^{\alpha_{1}}u_{1})(t)=f_{1}(t,u_{1}(t),u_{2}(t),({}^{c}D_{q}^{\alpha _{1}}u_{1})(t)),\\ ({}^{c}D_{q}^{\alpha_{2}}u_{2})(t)=f_{2}(t,u_{1}(t),u_{2}(t),({}^{c}D_{q}^{\alpha _{2}}u_{2})(t)), \end{cases}\displaystyle t\in I:=[0,T], $$

(1)

with the initial conditions

$$ \bigl(u_{1}(0),u_{2}(0) \bigr)=(u_{01},u_{02}), $$

(2)

where $q\in(0,1)$, $T>0$, $\alpha_{i}\in(0,1]$, $f_{i}:I\times{\Bbb {R}}\times {\Bbb {R}}\times{\Bbb {R}}\to{\Bbb {R}}$, $i=1,2$, are given continuous functions, and ${}^{c}D_{q}^{\alpha_{i}}$ is the Caputo fractional q-difference derivative of order $\alpha_{i}$, $i=1,2$.

Next, we discuss the existence and uniqueness of solutions for problem (1)–(2) in generalized Banach spaces, where $f_{i}:I\times{\Bbb {R}}^{3m}\to{\Bbb {R}}^{m}$, $i=1,2$, are given continuous functions, ${\Bbb {R}}^{m}$, $m\in{\Bbb {N}}^{\ast}$, is the Euclidian Banach space with a suitable norm $\|\cdot\|$. This paper initiates the study of implicit coupled Caputo fractional q-difference systems in Banach and generalized Banach spaces.

2 Preliminaries

Consider the Banach space $C(I):=C(I,{\Bbb {R}})$ of continuous functions from I into ${\Bbb {R}}$ equipped with the usual supremum (uniform) norm

$$\Vert u \Vert _{\infty}:=\sup _{t\in I} \bigl\vert u(t) \bigr\vert . $$

As usual, $L^{1}(I)$ denotes the space of measurable functions $v:I\rightarrow{\Bbb {R}}$ which are Lebesgue integrable with the norm

$$\Vert v \Vert _{1}= \int_{0}^{T} \bigl\vert v(t) \bigr\vert \, dt. $$

Let us recall some definitions and properties of fractional q-calculus. For $a\in\Bbb {R}$, we set

$$[a]_{q}=\frac{1-q^{a}}{1-q}. $$

The q-analogue of the power $(a-b)^{n}$ is

$$(a-b)^{(0)}=1,\qquad (a-b)^{(n)}=\prod _{k=0}^{n-1} \bigl(a-bq^{k} \bigr),\quad a,b \in{\Bbb {R}}, n\in{\Bbb {N}}. $$

In general,

$$(a-b)^{(\alpha)}=a^{\alpha}\prod_{k=0}^{\infty} \biggl(\frac {a-bq^{k}}{a-bq^{k+\alpha}} \biggr),\quad a,b,\alpha\in{\Bbb {R}}. $$

Definition 2.1

([30])

The q-gamma function is defined by

$$\varGamma_{q}(\xi)=\frac{(1-q)^{(\xi-1)}}{(1-q)^{\xi-1}},\quad \xi\in {\Bbb {R}}- \{0,-1,-2, \ldots\}. $$

Notice that the q-gamma function satisfies $\varGamma_{q}(1+\xi)=[\xi ]_{q}\varGamma_{q}(\xi)$.

Definition 2.2

([30])

The q-derivative of order $n\in{\Bbb {N}}$ of a function $u:I\to{\Bbb {R}}$ is defined by $(D_{q}^{0}u)(t)=u(t)$,

$$(D_{q}u) (t):= \bigl(D_{q}^{1}u \bigr) (t)= \frac{u(t)-u(qt) }{(1-q)t} , \quad t\neq0,\qquad (D_{q}u) (0)=\lim _{t\to0}(D_{q}u) (t), $$

and

$$\bigl(D_{q}^{n}u \bigr) (t)= \bigl(D_{q}D_{q}^{n-1}u \bigr) (t),\quad t\in I, n\in\{1,2,\ldots\}. $$

Set $I_{t}:=\{tq^{n}:n\in{\Bbb {N}}\}\cup\{0\}$.

Definition 2.3

([30])

The q-integral of a function $u:I_{t}\to{\Bbb {R}}$ is defined by

$$(I_{q}u) (t)= \int_{0}^{t}u(s)\,d_{q}s= \sum _{n=0}^{\infty}t(1-q)q^{n}f \bigl(tq^{n} \bigr), $$

provided that the series converges.

We note that $(D_{q}I_{q}u)(t)=u(t)$, while if u is continuous at 0, then

$$(I_{q}D_{q}u) (t)=u(t)-u(0). $$

Definition 2.4

([11])

The Riemann–Liouville fractional q-integral of order $\alpha\in{\Bbb {R}}_{+}:=[0,\infty)$ of a function $u:I\to{\Bbb {R}}$ is defined by $(I_{q}^{0}u)(t)=u(t)$, and

$$\bigl(I_{q}^{\alpha}u \bigr) (t)= \int_{0}^{t}\frac{(t-qs)^{(\alpha-1)}}{\varGamma _{q}(\alpha)}u(s) \,d_{q}s,\quad t\in I. $$

Lemma 2.5

([37])

For $\alpha\in{\Bbb {R}}_{+}:=[0,\infty)$and $\lambda\in(-1,\infty)$, we have

$$\bigl(I_{q}^{\alpha}(t-a)^{(\lambda)} \bigr) (t)= \frac{\varGamma_{q}(1+\lambda )}{\varGamma(1+\lambda+\alpha)}(t-a)^{(\lambda+\alpha)},\quad 0< a< t< T. $$

In particular,

$$\bigl(I_{q}^{\alpha}1 \bigr) (t)=\frac{1}{\varGamma_{q}(1+\alpha)}t^{(\alpha)}. $$

Definition 2.6

([38])

The Riemann–Liouville fractional q-derivative of order $\alpha\in {\Bbb {R}}_{+}$ of a function $u:I\to{\Bbb {R}}$ is defined by $(D_{q}^{0}u)(t)=u(t)$, and

$$\bigl(D_{q}^{\alpha}u \bigr) (t)= \bigl(D_{q}^{[\alpha]}I_{q}^{[\alpha]-\alpha}u \bigr) (t), \quad t\in I, $$

where $[\alpha]$ is the integer part of α.

Definition 2.7

([38])

The Caputo fractional q-derivative of order $\alpha\in{\Bbb {R}}_{+}$ of a function $u:I\to{\Bbb {R}}$ is defined by $({}^{C}D_{q}^{0}u)(t)=u(t)$, and

$$\bigl({}^{C}D_{q}^{\alpha}u \bigr) (t)= \bigl(I_{q}^{[\alpha]-\alpha}D_{q}^{[\alpha]}u \bigr) (t), \quad t \in I. $$

Lemma 2.8

([38])

Let $\alpha\in{\Bbb {R}}_{+}$. Then the following equality holds:

$$\bigl(I_{q}^{\alpha} {}^{C}D_{q}^{\alpha}u \bigr) (t)=u(t)-\sum _{k=0}^{[\alpha ]-1}\frac{t^{k}}{\varGamma_{q}(1+k)} \bigl(D_{q}^{k}u \bigr) (0). $$

In particular, if $\alpha\in(0,1)$, then

$$\bigl(I_{q}^{\alpha} {}^{C}D_{q}^{\alpha}u \bigr) (t)=u(t)-u(0). $$

From the above lemma, and in order to define the solution for problem (1)–(2), we conclude the following lemma.

Lemma 2.9

Let $f:I\times{\Bbb {R}}^{3m}\rightarrow {\Bbb {R}}^{m}$such that $f_{i}(\cdot,u_{i},v_{i},w_{i})\in C(I)$for each $u_{i},v_{i},w_{i}\in {\Bbb {R}}^{m}$. Then problem (1)–(2) is equivalent to the problem of obtaining the solutions of the integral equation

$$g_{i}(t)=f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha _{2}}g_{2} \bigr) (t),g_{i}(t) \bigr),\quad i=1,2, $$

and if $g_{i}(\cdot)\in C(I)$is the solution of this equation, then

$$u_{i}(t)=u_{0i}+ \bigl(I_{q}^{\alpha_{i}}g_{i} \bigr) (t). $$

3 Existence and Ulam stability results

In this section, we are concerned with the existence stability of solutions of system (1)–(2). We denote by $\mathcal {C}:=C(I)\times C(I)$ the Banach space with the norm

$$\bigl\Vert (u,v) \bigr\Vert _{\mathcal{C}}= \Vert u \Vert _{\infty}+ \Vert v \Vert _{\infty}. $$

Definition 3.1

By a solution of problem (1)–(2) we mean a coupled function $(u,v)\in\mathcal{C}$ that satisfies the system

$$\textstyle\begin{cases} ({}^{c}D_{q}^{\alpha_{1}}u)(t)=f_{1}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{1}}u)(t)),\\ ({}^{c}D_{q}^{\alpha_{2}}v)(t)=f_{2}(t,u(t),v(t),({}^{c}D_{q}^{\alpha_{2}}v)(t)) \end{cases} $$

on I and the initial condition $(u(0),v(0))=(u_{01},u_{02})$.

Now, we consider the Ulam stability for system (1)–(2). Let $\epsilon>0$ and $\varPhi:I\to{\Bbb {R}}_{+}$ be a continuous function. We consider the following inequalities:

$$\begin{aligned}& \textstyle\begin{cases} \vert ({}^{c}D_{q}^{\alpha_{1}}u)(t)-f_{1}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{1}}u)(t)) \vert \leq\epsilon,\\ \vert ({}^{c}D_{q}^{\alpha_{2}}v)(t)-f_{2}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{2}}v)(t)) \vert \leq\epsilon, \end{cases}\displaystyle t\in I; \end{aligned}$$

(3)

$$\begin{aligned}& \textstyle\begin{cases} \vert ({}^{c}D_{q}^{\alpha_{1}}u)(t)-f_{1}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{1}}u)(t)) \vert \leq\varPhi(t) \\ \vert ({}^{c}D_{q}^{\alpha_{2}}v)(t)-f_{2}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{2}}v)(t)) \vert \leq\varPhi(t), \end{cases}\displaystyle t\in I; \end{aligned}$$

(4)

$$\begin{aligned}& \textstyle\begin{cases} \vert ({}^{c}D_{q}^{\alpha_{1}}u)(t)-f_{1}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{1}}u)(t)) \vert \leq\epsilon\varPhi(t) \\ \vert ({}^{c}D_{q}^{\alpha_{2}}v)(t)-f_{2}(t,u(t),v(t),({}^{c}D_{q}^{\alpha _{2}}v)(t)) \vert \leq\epsilon\varPhi(t), \end{cases}\displaystyle t\in I. \end{aligned}$$

(5)

Set

$$\bigl\vert \bigl(u(t),v(t) \bigr) \bigr\vert := \bigl\vert u(t) \bigr\vert + \bigl\vert v(t) \bigr\vert . $$

Definition 3.2

([5, 40])

System (1)–(2) is Ulam–Hyers stable if there exists a real number $c_{f_{1},f_{2}}>0$ such that, for each $\epsilon>0$ and for each solution $(u_{1},v_{1})\in\mathcal {C}$ of inequalities (3), there exists a solution $(u,v)\in C(I)$ of (1)–(2) with

$$\bigl\vert \bigl(u_{1}(t)-u(t),v_{1}(t)-v(t) \bigr) \bigr\vert \leq\epsilon c_{f_{1},f_{2}},\quad t\in I. $$

Definition 3.3

([5, 40])

System (1)–(2) is generalized Ulam–Hyers stable if there exists $c_{f_{1},f_{2}}:C({\Bbb {R}}_{+},{\Bbb {R}}_{+})$ with $c_{f_{i}}(0)=0$, $i=1,2$, such that, for each $\epsilon>0$ and for each solution $(u_{1},v_{1})\in\mathcal{C}$ of inequalities (3), there exists a solution $(u,v)\in\mathcal{C}$ of (1)–(2) with

$$\bigl\vert \bigl(u_{1}(t)-u(t),v_{1}(t)-v(t) \bigr) \bigr\vert \leq c_{f_{1},f_{2}}(\epsilon),\quad t\in I. $$

Definition 3.4

([5, 40])

System (1)–(2) is Ulam–Hyers–Rassias stable with respect to Φ if there exists a real number $c_{f_{1},f_{2},\varPhi}>0$ such that, for each $\epsilon >0$ and for each solution $(u_{1},v_{1})\in\mathcal{C}$ of inequalities (5), there exists a solution $(u,v)\in\mathcal{C}$ of (1)–(2) with

$$\bigl\vert \bigl(u_{1}(t)-u(t),v_{1}(t)-v(t) \bigr) \bigr\vert \leq\epsilon c_{f_{1},f_{2},\varPhi}\varPhi (t),\quad t\in I. $$

Definition 3.5

([5, 40])

System (1)–(2) is generalized Ulam–Hyers–Rassias stable with respect to Φ if there exists a real number $c_{f_{1},f_{2},\varPhi}>0$ such that, for each solution $(u_{1},v_{1})\in\mathcal{C}$ of inequalities (4), there exists a solution $(u,v)\in\mathcal{C}$ of (1)–(2) with

$$\bigl\vert \bigl(u_{1}(t)-u(t),v_{1}(t)-v(t) \bigr) \bigr\vert \leq c_{f_{1},f_{2},\varPhi}\varPhi(t),\quad t\in I. $$

Remark 3.6

It is clear that

• (i) Definition 3.2 ⇒ Definition 3.3,

• (ii) Definition 3.4 ⇒ Definition 3.5,

• (iii) Definition 3.4 for $\varPhi(\cdot)=1$ ⇒ Definition 3.2.

One can have similar remarks for inequalities (3) and (5).

Theorem 3.7

(Schauder’s fixed point theorem)

LetXbe a Banach space, Dbe a bounded closed convex subset ofX, and $T:D\to D$be a compact and continuous map. ThenThas at least one fixed point inD.

The following hypotheses will be used in the sequel:

$(H_{1})$:

There exist functions $p_{i},d_{i},r_{i}\in C(I,[0,\infty))$, $i=1,2$, with $r_{i}(t)<1$ such that

$$\bigl\vert f_{i}(t,u,v,w) \bigr\vert \leq p_{i}(t)+d_{i}(t) \min \bigl( \vert u \vert , \vert v \vert \bigr)+r_{i}(t) \vert w \vert $$

for each $t\in I$ and $u,v,w\in{\Bbb {R}}$;

$(H_{2})$:

There exists $\lambda_{\varPhi}>0$ such that, for each $t\in I$, we have

$$\bigl(I_{q}^{\alpha_{i}}\varPhi \bigr) (t)\leq\lambda_{\varPhi}\varPhi(t),\quad i=1,2. $$

Set

$$\begin{gathered} L_{i}:=\frac{T^{\alpha_{i}}}{\varGamma_{q}(1+\alpha_{i})}, \\ p_{i}^{\ast}=\sup _{t\in I}p_{i}(t),\qquad d_{i}^{\ast}= \sup _{t\in I}d_{i}(t),\qquad r_{i}^{\ast}= \sup _{t\in I}r_{i}(t),\qquad \varPhi^{\ast}=\sup _{t\in I} \varPhi(t).\end{gathered} $$

Theorem 3.8

Assume that hypothesis $(H_{1})$holds. If

$$ r_{1}^{\ast}+r_{2}^{\ast}-r_{1}^{\ast}r_{2}^{\ast}+ \bigl(1-r_{2}^{\ast } \bigr)L_{1}d_{1}^{\ast}+ \bigl(1-r_{1}^{\ast} \bigr)L_{2}d_{2}^{\ast}< 1, $$

(6)

then system (1)–(2) has at least one solution defined onI. Moreover, if hypothesis $(H_{2})$holds, then system (1)–(2) is generalized Ulam–Hyers–Rassias stable.

Proof

Define the operators $N_{i}:C(I)\to C(I)$, $i=1,2$, by

$$ (N_{1}u) (t)=u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),\quad t\in I, $$

(7)

and

$$ (N_{2}v) (t)=u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),\quad t\in I, $$

(8)

where $g_{i}\in C(I)$ such that

$$g_{i}(t)=f \bigl(t,u(t),v(t),g_{i}(t) \bigr). $$

Consider the continuous operator $N:\mathcal{C}\to\mathcal{C}$ defined by

$$ \bigl(N(u,v) \bigr) (t)= \bigl((N_{1}u) (t),(N_{2}v) (t) \bigr). $$

(9)

Set

$$R\geq \frac{(1-r_{1}^{\ast})(1-r_{2}^{\ast})( \vert u_{01} \vert + \vert u_{02} \vert )+(1-r_{2}^{\ast })L_{1}p_{1}^{\ast}+(1-r_{1}^{\ast}) L_{2}p_{2}^{\ast}}{ 1-r_{1}^{\ast}-r_{2}^{\ast}+r_{1}^{\ast}r_{2}^{\ast}-(1-r_{2}^{\ast })L_{1}d_{1}^{\ast}-(1-r_{1}^{\ast})L_{2}d_{2}^{\ast}}, $$

and consider the closed and convex ball $B_{R}=\{u\in\mathcal{C}:\| (u,v)\|_{\mathcal{C}}\leq R\}$.

Let $u\in B_{R}$. Then, for each $t\in I$, we have

$$\bigl\vert (N_{1}u) (t) \bigr\vert \leq \vert u_{01} \vert + \int_{0}^{t}\frac{(t-qs)^{(\alpha _{1}-1)}}{\varGamma_{q}(\alpha_{1})} \bigl\vert g_{1}(s) \bigr\vert \,d_{q}s $$

and

$$\bigl\vert (N_{2}v) (t) \bigr\vert \leq \vert u_{02} \vert + \int_{0}^{t}\frac{(t-qs)^{(\alpha _{2}-1)}}{\varGamma_{q}(\alpha_{2})} \bigl\vert g_{2}(s) \bigr\vert \,d_{q}s. $$

By using $(H_{1})$, for each $t\in I$, we have

$$\begin{aligned} \bigl\vert g_{i}(t) \bigr\vert &\leq p_{i}(t)+d_{i}(t)\min \bigl( \bigl\vert u(t) \bigr\vert , \bigl\vert v(t) \bigr\vert \bigr) +r_{i}(t) \bigl\vert g_{i}(t) \bigr\vert \\ &\leq p_{i}^{\ast}+d_{i}^{\ast}R+r_{i}^{\ast} \bigl\vert g_{i}(t) \bigr\vert . \end{aligned} $$

Thus

$$\bigl\vert g_{i}(t) \bigr\vert \leq\frac{p_{i}^{\ast}+d_{i}^{\ast}R}{1-r_{i}^{\ast}}. $$

Hence

$$\bigl\Vert N_{1}(u) \bigr\Vert _{\infty}\leq \vert u_{01} \vert +\frac{L_{1}(p_{1}^{\ast}+d_{1}^{\ast }R)}{1-r_{1}^{\ast}} $$

and

$$\bigl\Vert N_{2}(v) \bigr\Vert _{\infty}\leq \vert u_{02} \vert +\frac{L_{2}(p_{2}^{\ast}+d_{2}^{\ast }R)}{1-r_{2}^{\ast}}. $$

This implies that

$$\begin{aligned} \bigl\Vert N(u,v) \bigr\Vert _{\mathcal{C}}&= \bigl\Vert N_{1}(u) \bigr\Vert _{\infty}+ \bigl\Vert N_{2}(v) \bigr\Vert _{\infty} \\ &\leq \vert u_{01} \vert + \vert u_{02} \vert + \sum _{i=1}^{2}\frac{L_{i}(p_{i}^{\ast }+d_{i}^{\ast}R)}{1-r_{i}^{\ast}} \\ &\leq R. \end{aligned} $$

This proves that N maps the ball $B_{R}$ into $B_{R}$. We shall show that the operator $N:B_{R}\to B_{R}$ is continuous and compact. The proof will be given in several steps.

Step 1: N is continuous.

Let $\{u_{n}\}_{n\in\mathbb{N}}$ and $\{v_{n}\}_{n\in\mathbb{N}}$ be two sequences such that $(u_{n},v_{n})\rightarrow(u,v)$ in $B_{R}$. Then, for each $t\in I$, we have

$$\bigl\vert (N_{1}u_{n}) (t)-(N_{1}u) (t) \bigr\vert \leq \int_{0}^{t}\frac{(t-qs)^{(\alpha _{1}-1)}}{\varGamma_{q}(\alpha_{1})} \bigl\vert \bigl(g_{1n}(s)-g_{1}(s) \bigr) \bigr\vert \,d_{q}s $$

and

$$\bigl\vert (N_{2}v_{n}) (t)-(N_{2}v) (t) \bigr\vert \leq \int_{0}^{t}\frac{(t-qs)^{(\alpha _{2}-1)}}{\varGamma_{q}(\alpha_{2})} \bigl\vert \bigl(g_{2n}(s)-g_{2}(s) \bigr) \bigr\vert \,d_{q}s, $$

where $g_{in},g_{i}\in C(I)$, $i=1,2$, such that

$$g_{in}(t)=f_{i} \bigl(t,u_{n}(t),v_{n}(t),g_{in}(t) \bigr) $$

and

$$g_{i}(t)=f_{i} \bigl(t,u(t),v(t),g_{i}(t) \bigr). $$

Since $(u_{n},v_{n})\rightarrow u$ as $n\rightarrow\infty$ and $f_{i}$ are continuous functions, we get

$$g_{in}(t)\rightarrow g_{i}(t)\quad \hbox{as } n\rightarrow \infty \hbox{ for each } t\in I. $$

Thus

$$\bigl\Vert N_{1}(u_{n})-N_{1}(u) \bigr\Vert _{\infty}\leq\frac{p_{1}^{\ast}+d_{1}^{\ast }R}{1-r_{1}^{\ast}} \Vert g_{1n}-g_{1} \Vert _{\infty} \rightarrow0 \quad\text{as }n\rightarrow\infty $$

and

$$\bigl\Vert N_{2}(v_{n})-N_{2}(v) \bigr\Vert _{\infty}\leq\frac{p_{2}^{\ast}+d_{2}^{\ast }R}{1-r_{2}^{\ast}} \Vert g_{2n}-g_{2} \Vert _{\infty} \rightarrow0 \quad\text{as }n\rightarrow\infty. $$

Hence

$$\bigl\Vert N(u_{n},v_{n})-N(u,v) \bigr\Vert _{\mathcal{C}}\rightarrow0 \quad\text{as }n\rightarrow\infty. $$

Step 2: $N(B_{R})$ is bounded. This is clear since $N(B_{R})\subset B_{R}$ and $B_{R}$ is bounded.

Step 3: N maps bounded sets into equicontinuous sets in $B_{R}$.

Let $t_{1},t_{2}\in I$ such that $t_{1}< t_{2}$, and let $(u,v)\in B_{R}$. Then we have

$$\begin{aligned} \bigl\vert (N_{i}u) (t_{1})-(N_{i}u) (t_{2}) \bigr\vert &\leq \int_{0}^{t_{1}}\frac { \vert (t_{2}-qs)^{(\alpha_{i}-1)}-(t_{1}-qs)^{(\alpha_{i}-1)} \vert }{\varGamma _{q}(\alpha_{i})} \bigl\vert g_{i}(s) \bigr\vert \,d_{q}s \\ &\quad{}+ \int_{t_{1}}^{t_{2}}\frac{ \vert (t_{2}-qs)^{(\alpha_{i}-1)} \vert }{\varGamma _{q}(\alpha_{i})} \bigl\vert g_{i}(s) \bigr\vert \,d_{q}s, \end{aligned} $$

where $g_{i}\in C(I)$ such that $g_{i}(t)=f(t,u(t),v(t),g_{i}(t))$. Hence

$$\begin{aligned} \bigl\vert (N_{1}u) (t_{1})-(N_{1}u) (t_{2}) \bigr\vert &\leq \frac{ p_{1}^{\ast}+d_{1}^{\ast}R}{1-r_{1}^{\ast}} \int_{0}^{t_{1}}\frac { \vert (t_{2}-qs)^{(\alpha_{1}-1)}-(t_{1}-qs)^{(\alpha_{1}-1)} \vert }{\varGamma _{q}(\alpha_{1})}\,d_{q}s \\ &\quad{}+\frac{p_{1}^{\ast}+d_{1}^{\ast}R}{1-r_{1}{\ast}} \int_{t_{1}}^{t_{2}}\frac { \vert (t_{2}-qs)^{(\alpha_{1}-1)} \vert }{\varGamma_{q}(\alpha_{1})}\,d_{q}s \\ &\rightarrow0 \quad\text{as }t_{1}\rightarrow t_{2}, \end{aligned} $$

and

$$\begin{aligned} \bigl\vert (N_{2}v) (t_{1})-(N_{2}v) (t_{2}) \bigr\vert &\leq \frac{ p_{2}^{\ast}+d_{2}^{\ast}R}{1-r_{2}^{\ast}} \int_{0}^{t_{1}}\frac { \vert (t_{2}-qs)^{(\alpha_{2}-1)}-(t_{1}-qs)^{(\alpha_{2}-1)} \vert }{\varGamma _{q}(\alpha_{2})}\,d_{q}s \\ &\quad{}+\frac{p_{2}^{\ast}+d_{2}^{\ast}R}{1-r_{2}{\ast}} \int_{t_{1}}^{t_{2}}\frac { \vert (t_{2}-qs)^{(\alpha_{2}-1)} \vert }{\varGamma_{q}(\alpha_{2})}\,d_{q}s \\ &\rightarrow0 \quad\text{as }t_{1}\rightarrow t_{2}. \end{aligned} $$

As a consequence of the above three steps with the Arzelá–Ascoli theorem, we can conclude that $N:B_{R}\to B_{R}$ is continuous and compact. From an application of Theorem 3.7, we deduce that N has at least a fixed point $(u,v)$ which is a solution of our system (1)–(2).

Step 4: Generalized Ulam–Hyers–Rassias stability.

Let $(u_{1},v_{1})$ be a solution of inequality (4), and let us assume that $(u,v)$ is a solution of system (1)–(2). Thus, we have

$$\bigl(u(t),v(t) \bigr)=(u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t), $$

where $g_{i}\in C(I)$, $i=1,2$, such that $g_{i}(t)=f(t,u(t),v(t),g_{i}(t))$.

From inequality (4) for each $t\in I$, we have

$$\bigl\vert u_{1}(t)-u_{01}- \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t) \bigr\vert \leq \bigl(I_{q}^{\alpha_{1}}\varPhi \bigr) (t) $$

and

$$\bigl\vert v_{1}(t)-u_{02}- \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t) \bigr\vert \leq \bigl(I_{q}^{\alpha_{2}}\varPhi \bigr) (t). $$

From hypotheses $(H_{1})$ and $(H_{2})$, for each $t\in I$, we have

$$\begin{aligned} \bigl\vert u(t)-u_{1}(t) \bigr\vert \leq& \bigl\vert u(t)-u_{01}- \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t)+ \bigl(I_{q}^{\alpha _{1}}(g_{1}-g_{2}) \bigr) (t) \bigr\vert \\ \leq& \bigl(I_{q}^{\alpha_{1}}\varPhi \bigr) (t) + \int_{0}^{t}\frac{(t-qs)^{(\alpha_{1}-1)}}{\varGamma_{q}(\alpha _{1})} \bigl( \bigl\vert g_{1}(s) \bigr\vert +\bigl|g_{2}(s) \bigr\vert \bigr) \,d_{q}s \\ \leq& \bigl(I_{q}^{\alpha_{1}}\varPhi \bigr) (t) + \frac{p_{1}^{\ast}+d_{1}^{\ast}}{1-r_{1}^{\ast}} \bigl(I_{q}^{\alpha _{1}}\varPhi \bigr) (t) \\ \leq&\lambda_{\phi}\varPhi(t)+2\lambda_{\phi}\frac{p_{1}^{\ast }+d_{1}^{\ast}}{1-r_{1}^{\ast}} \varPhi(t) \\ \leq& \biggl[1+2\frac{p_{1}^{\ast}+d_{1}^{\ast}}{1-r_{1}^{\ast}} \biggr]\lambda_{\phi}\varPhi(t) \\ :=&c_{f_{1},\varPhi}\varPhi(t). \end{aligned}$$

Also, we get

$$\begin{aligned} \bigl\vert v(t)-v_{1}(t) \bigr\vert \leq& \biggl[1+2 \frac{p_{2}^{\ast}+d_{2}^{\ast}}{1-r_{2}^{\ast}} \biggr]\lambda_{\phi}\varPhi(t) \\ :=&c_{f_{2},\varPhi}\varPhi(t). \end{aligned}$$

Thus

$$\begin{aligned} \bigl\vert \bigl(u(t),v(t) \bigr)- \bigl(u_{1}(t),v_{1}(t) \bigr) \bigr\vert =& \bigl\vert u(t)-u_{1}(t) \bigr\vert + \bigl\vert v(t)-v_{1}(t) \bigr\vert \\ \leq& \biggl[\lambda_{\phi} \sum _{i=1}^{2} \biggl(1+2\frac{p_{i}^{\ast }+d_{i}^{\ast}}{1-r_{i}^{\ast}} \biggr) \biggr]\varPhi(t) \\ :=&c_{f_{1},f_{2},\varPhi}\varPhi(t). \end{aligned}$$

Hence, problem (1)–(2) is generalized Ulam–Hyers–Rassias stable. □

4 Results in generalized Banach spaces

Now, we are concerned with the existence and uniqueness results of the coupled system (1)–(2) in generalized Banach spaces.

Let C be the Banach space of all continuous functions v from I into ${\Bbb {R}}^{m}$ with the supremum (uniform) norm

$$\Vert v \Vert _{C}:=\sup _{t\in I} \bigl\Vert v(t) \bigr\Vert . $$

By $L^{\infty}(I,{\Bbb {R}}_{+})$ we denote the Banach space of measurable functions from I into ${\Bbb {R}}_{+}$ which are essentially bounded.

Let $x,y\in{\Bbb {R}}^{m}$ with $x=(x_{1},x_{2},\dots,x_{m})$, $y=(y_{1},y_{2},\dots,y_{m})$.

By $x\leq y$ we mean $x_{i}\leq y_{i}$, $i=1,\dots,m$. Also,

$$\begin{gathered} \vert x \vert = \bigl( \vert x_{1} \vert , \vert x_{2} \vert ,\dots, \vert x_{m} \vert \bigr), \\ \max (x,y)= \bigl(\max (x_{1},y_{1}),\max (x_{2},y_{2}), \dots, \max (x_{m},y_{m}) \bigr),\end{gathered} $$

and

$${\Bbb {R}}^{m}_{+}= \bigl\{ x\in{\Bbb {R}}^{m}:x_{i}\in{ \Bbb {R}}_{+}, i=1,\dots,m \bigr\} . $$

If $c\in{\Bbb {R}}$, then $x\leq c$ means $x_{i}\leq c$, $i=1,\dots,m$.

Definition 4.1

Let X be a nonempty set. By a vector-valued metric on X we mean a map $d:X\times X\to{\Bbb {R}}^{m}$ with the following properties:

• (i) $d(x,y)\geq0$ for all $x,y\in X$, and if $d(x,y)=0$, then $x=y$;

• (ii) $d(x,y)=d(y,x)$ for all $x,y\in X$;

• (iii) $d(x,z)\leq d(x,y)+d(y,z)$ for all $x,y,z\in X$.

We call the pair $(X,d)$ a generalized metric space with

$$d(x,y):=\left ( \textstyle\begin{array}{c} d_{1}(x,y)\\ d_{2}(x,y)\\ \vdots\\ d_{m}(x,y) \end{array}\displaystyle \right ). $$

Notice that d is a generalized metric space on X if and only if $d_{i}$, $i=1,\dots,m$, are metrics on X.

Definition 4.2

([45])

A square matrix of real numbers is said to be convergent to zero if and only if its spectral radius $\rho(M)$ is strictly less than 1. In other words, this means that all the eigenvalues of M are in the open unit disc, i.e., $|\lambda|<1$ for every $\lambda\in{\Bbb {C}}$ with $\operatorname{det}(M-\lambda I)=0$, where I denotes the unit matrix of $M_{m\times m}({\Bbb {R}})$.

Example 4.3

The matrix $A\in M_{2\times2}({\Bbb {R}})$ defined by

$$A=\left ( \textstyle\begin{array}{c@{\quad}c} a & b \\ c & d \end{array}\displaystyle \right ) $$

converges to zero in the following cases:

• (1) $b=c=0$, $a,d>0$, and $\max\{a,d\}<1$.

• (2) $c=0$, $a,d>0$, $a+d<1$, and $-1< b<0$.

• (3) $a+b=c+d=0$, $a>1$, $c>0$, and $|a-c|<1$.

Definition 4.4

Let $(X,d)$ be a generalized metric space. An operator $N:X\to X$ is said to be contractive if there exists a matrix M convergent to zero such that

$$d \bigl(N(x),N(y) \bigr)\leq Md(x,y)\quad \text{for all }x,y\in X. $$

In the sequel we will make use of the following fixed point theorems in generalized Banach spaces.

Theorem 4.5

[36] Let $(X,d)$be a complete generalized metric space and $N:X\to X$be a contractive operator with Lipschitz matrixM. ThenNhas a unique fixed point $x_{0}$, and for each $x\in X$, we have

$$d \bigl(N^{k}(x),x_{0} \bigr)\leq M^{k}( M)^{-1}d \bigl(x,N(x) \bigr) \quad\textit{for all } k\in{\Bbb {N}}. $$

For $n = 1$, we recover the classical Banach contraction fixed point result.

Theorem 4.6

([36])

LetXbe a generalized Banach space, $D\subset E$be a nonempty closed convex subset ofE, and $N:D\to D$be a continuous operator with relatively compact range. ThenNhas at least a fixed point inD.

The following hypotheses will be used in the sequel.

$(H_{01})$:

There exist continuous functions $p_{i},d_{i},l_{i}:I\to {\Bbb {R}}_{+}$, $i=1,2$, such that $l_{i}<1$ and

$$\begin{gathered}\bigl\| f_{i}(t,u_{1},v_{1},w_{1})-f_{i}(t,u_{2},v_{2},w_{2})\bigr\| \\\quad\leq p_{i}(t) \Vert u_{1}-u_{2} \Vert +d_{i}(t) \Vert v_{1}-v_{2} \Vert +l_{i}(t) \Vert w_{1}-w_{2} \Vert \end{gathered} $$

for a.e. $t\in I$ and each $u_{i},v_{i},w_{i}\in{\Bbb {R}}^{m}$, $i=1,2$.

$(H_{02})$:

There exist continuous functions $K_{i},P_{i},D_{i},L_{i}:I\to{\Bbb {R}}_{+}$, $i=1,2$, such that

$$\bigl\Vert f_{i}(t,u,v,w) \bigr\Vert \leq K_{i}(t)+P_{i}(t) \Vert u \Vert +D_{i}(t) \Vert v \Vert +L_{i}(t) \Vert w \Vert $$

for a.e. $t\in I$ and each $u,v,w\in{\Bbb {R}}^{m}$, $i=1,2$.

Set

$$\begin{gathered} p_{i}^{\ast}:=\sup _{t\in I}p_{i}(t),\qquad d_{i}^{\ast}:=\sup _{t\in I}d_{i}(t),\qquad l_{i}^{\ast}:=\sup _{t\in I}l_{i}(t),\qquad K_{i}^{\ast}:=\sup _{t\in I}K_{i}(t),\qquad \\ P_{i}^{\ast}:=\sup _{t\in I}P_{i}(t),\qquad D_{i}^{\ast}:=\sup _{t\in I}D_{i}(t),\qquad L_{i}^{\ast}:=\sup _{t\in I}L_{i}(t),\end{gathered} $$

and

$$\ell_{i}:=\frac{T^{\alpha_{i}}}{\varGamma_{q}(1+\alpha_{i})},\quad i=1,2. $$

The space $C^{2}:=C\times C$ is a generalized Banach space with the norm

$$\bigl\Vert (u_{1},u_{2}) \bigr\Vert _{C^{2}} := \bigl( \Vert u_{1} \Vert _{C}, \Vert u_{2} \Vert _{C} \bigr) . $$

Definition 4.7

By a solution of problem (1)–(2) we mean a coupled continuous function $(u,v)\in C^{2}$ satisfying initial condition (2) and system (1) on I.

First, we prove an existence and uniqueness result for coupled system (1)–(2) by using Banach’s fixed point theorem type in generalized Banach spaces.

Theorem 4.8

Assume that hypothesis $(H_{01})$holds. If the matrix

$$M:=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{\ell_{1}p_{1}^{\ast}}{1-l_{1}^{\ast}} & \frac{\ell_{1}d_{1}^{\ast }}{1-l_{1}^{\ast}}\\ [3pt] \frac{\ell_{2}p_{2}^{\ast}}{1-l_{2}^{\ast}} & \frac{\ell_{2}d_{2}^{\ast }}{1-l_{2}^{\ast}} \end{array}\displaystyle \right ) $$

converges to 0, then coupled system (1)–(2) has a unique solution.

Proof

From Lemma 2.9, we can define the operators $N_{1},N_{2}:C^{2}\rightarrow C$ by

$$ \bigl(N_{i}(u_{1},u_{2}) \bigr) (t)=u_{0i}+ \bigl(I_{q}^{\alpha_{i}}g_{i} \bigr) (t),\quad i=1,2, t\in I, $$

(10)

where $g_{i}(\cdot)\in C(I)$, with

$$\begin{aligned} g_{i}(t)&=f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ &=f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha _{2}}g_{2} \bigr) (t),g_{i}(t) \bigr),\quad i=1,2. \end{aligned}$$

Consider the operator $N:C^{2}\to C^{2}$ defined by

$$ \bigl(N(u_{1},u_{2}) \bigr) (t)= \bigl( \bigl(N_{1}(u_{1},u_{2}) \bigr) (t), \bigl(N_{2}(u_{1},u_{2}) \bigr) (t) \bigr). $$

(11)

Clearly, the fixed points of the operator N are solutions of coupled system (1)–(2). We show that N satisfies all the conditions of Theorem 4.5.

For each $(u_{1},u_{2}), (v_{1},v_{2})\in C^{2}$ and $t\in I$, we have

$$ \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t)- \bigl(N_{i}(v_{1},v_{2}) \bigr) (t) \bigr\Vert \leq \int_{0}^{t}\frac {(t-qs)^{(\alpha-i)}}{\varGamma_{q}(\alpha_{i})} \bigl\Vert g_{i}(s)-h_{i}(s) \bigr\Vert \,d_{q}s, $$

(12)

where $g_{i}(\cdot),h_{i}(\cdot)\in C(I)$, $i=1,2$, with

$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr) \end{aligned}$$

and

$$\begin{aligned} h_{i}(t) =&f_{i} \bigl(t,v_{1}(t),v_{2}(t),h_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}h_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}h_{2} \bigr) (t),h_{i}(t) \bigr). \end{aligned}$$

From hypothesis $(H_{01})$, we have

$$\bigl\Vert g_{i}(t)-h_{i}(t) \bigr\Vert =p_{i}(t) \bigl\Vert u_{1}(t)-v_{1}(t) \bigr\Vert +d_{i}(t) \Vert u_{2}-v_{2} \Vert +l_{i}(t) \Vert g_{i}-h_{i} \Vert . $$

Then

$$\bigl\Vert g_{i}(t)-h_{i}(t) \bigr\Vert =p_{i}(t) \bigl\Vert u_{1}(t)-v_{1}(t) \bigr\Vert +d_{i}(t) \bigl\Vert u_{2}(t)-v_{2}(t) \bigr\Vert +l_{i}(t) \bigl\Vert g_{i}(t)-h_{i}(t) \bigr\Vert . $$

Thus

$$\Vert g_{i}-h_{i} \Vert _{C}=p_{i}^{\ast} \Vert u_{1}-v_{1} \Vert _{C}+i_{1}^{\ast} \Vert u_{2}-v_{2} \Vert _{C}+l_{i}^{\ast} \Vert g_{i}-h_{i} \Vert _{C}. $$

This implies that

$$\bigl(1-l_{i}^{\ast} \bigr) \Vert g_{i}-h_{i} \Vert _{C}=p_{i}^{\ast} \Vert u_{1}-v_{1} \Vert _{C}+d_{i}^{\ast} \Vert u_{2}-v_{2} \Vert _{C}. $$

Hence

$$\Vert g_{i}-h_{i} \Vert _{C}= \frac{p_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{1}-v_{1} \Vert _{C}+\frac {d_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{2}-v_{2} \Vert _{C}. $$

From (12), we get

$$\begin{aligned} \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr)- \bigl(N_{i}(v_{1},v_{2}) \bigr) \bigr\Vert _{C} \leq& \int_{0}^{t}\frac {(t-qs)^{(\alpha_{i}-1)}}{\varGamma_{q}(\alpha_{i}-1)} \bigl\Vert g_{i}(s)-h_{i}(s) \bigr\Vert \,d_{q}s \\ \leq&\frac{\ell_{i}p_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{1}-v_{1} \Vert _{C}+\frac {\ell_{i}d_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{2}-v_{2} \Vert _{C}. \end{aligned}$$

Consequently,

$$d \bigl( \bigl(N(u_{1},u_{2}) \bigr), \bigl(N(v_{1},v_{2}) \bigr) \bigr)\leq Md \bigl((u_{1},u_{2}),(v_{1},v_{2}) \bigr), $$

where

$$d \bigl((u_{1},u_{2}),(v_{1},v_{2}) \bigr)= \left ( \textstyle\begin{array}{c} \Vert u_{1}-v_{1} \Vert _{C} \\ \Vert u_{2}-v_{2} \Vert _{C} \end{array}\displaystyle \right ). $$

Since the matrix M converges to zero, then Theorem 4.5 implies that coupled system (1)–(2) has a unique solution. □

Now, we prove an existence result for coupled system (1)–(2) by using Schauder’s fixed point theorem type in a generalized Banach space.

Theorem 4.9

Assume that hypothesis $(H_{02})$holds. Then coupled system (1)–(2) has at least one solution.

Proof

Let $N:C^{2}\to C^{2}$ be the operator defined in (11). We show that N satisfies all the conditions of Theorem 4.6. The proof will be given in several steps.

Step 1. N is continuous.

Let $\{(u_{1n},u_{2n})\}_{n}$ be a sequence such that $(u_{1n},u_{2n})\to (u_{1},u_{2})\in C^{2}$ as $n\to\infty$. For any $i=1,2$ and each $t\in I$, we have

$$\bigl\Vert \bigl(N_{i}(u_{1n},u_{2n}) \bigr) (t)- \bigl(N_{i}(u_{1},u_{2}) \bigr) (t) \bigr\Vert \leq \int_{0}^{t}\frac {(t-qs)^{(\alpha_{i}-1)}}{\varGamma_{q}(\alpha_{i}-1)} \bigl\Vert g_{in}(s)-g_{i}(s) \bigr\Vert \,d_{q}s, $$

where $g_{i}(\cdot),g_{in}(\cdot)\in C(I)$, $i=1,2$, with

$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr) \end{aligned}$$

and

$$\begin{aligned} g_{in}(t) =&f_{i} \bigl(t,u_{1n}(t),u_{2n}(t),g_{in}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1n} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha _{2}}g_{2n} \bigr) (t),g_{in}(t) \bigr). \end{aligned}$$

From $(H_{02})$, we have

$$\Vert g_{in}-g_{i} \Vert _{C}\leq \frac{P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1n}-u_{1} \Vert _{C}+\frac{D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2n}-u_{2} \Vert _{C}. $$

Thus,

$$\begin{aligned} \bigl\Vert \bigl(N_{i}(u_{1n},u_{2n}) \bigr) (t)- \bigl(N_{i}(u_{1},u_{2}) \bigr) (t) \bigr\Vert \leq&\frac{\ell_{i}P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1n}-u_{1} \Vert _{C}+\frac{\ell_{i}D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2n}-u_{2} \Vert _{C} \\ \to&0\quad \text{as }n\to\infty. \end{aligned}$$

Hence, we get

$$\bigl\Vert N_{i}(u_{1n},u_{2n})-N_{i}(u_{1},u_{2}) \bigr\Vert _{C}\to0 \quad \text{as }n\to\infty. $$

Consequently,

$$\begin{aligned}& \bigl\Vert N(u_{1n},u_{2n})-N(u_{1},u_{2}) \bigr\Vert _{C^{2}} \\& \quad:= \bigl( \bigl\Vert N_{1}(u_{1n},u_{2n})-N_{1}(u_{1},u_{2}) \bigr\Vert _{C}, \bigl\Vert N_{2}(u_{1n},u_{2n})-N_{2}(u_{1},u_{2}) \bigr\Vert _{C} \bigr) \\& \quad\to(0,0) \quad \text{as }n\to\infty. \end{aligned}$$

Finally, N is continuous.

Step 2.Nmaps bounded sets into bounded sets in $C^{2}$.

Set

$$h_{i}^{\ast}:=\sup _{t\in I}h_{i}(t),\qquad k_{i}^{\ast}:=\sup _{t\in I}k_{i}(t),\qquad l_{i}^{\ast}:=\sup _{t\in I}l_{i}(t). $$

Let $R>0$ and set

$$B_{R}:= \bigl\{ (\mu,\nu)\in C^{2}: \Vert \mu \Vert _{C}\leq R, \Vert \nu \Vert _{C}\leq R \bigr\} . $$

For any $i=1,2$ and each $(u,v)\in B_{R}$ and $t\in I$, we have

$$\bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t) \bigr\Vert \leq \int_{0}^{t}\frac{(t-qs)^{(\alpha _{i}-1)}}{\varGamma_{q}(\alpha_{i}-1)} \bigl\Vert g_{i}(s) \bigr\Vert \,d_{q}s, $$

where $g_{i}(\cdot),\in C(I)$, $i=1,2$, with

$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr). \end{aligned}$$

Since

$$\Vert g_{i} \Vert _{C}\leq\frac{P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1} \Vert _{C}+\frac {D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2} \Vert _{C}, $$

we get

$$\bigl\Vert N_{i}(u_{1},u_{2}) \bigr\Vert _{C}\leq\frac{\ell_{i}P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1} \Vert _{C}+\frac{\ell_{i}D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2} \Vert _{C}. $$

Thus,

$$\bigl\Vert N_{i}(u_{1},u_{2}) \bigr\Vert _{C}\leq\frac{R\ell_{i}P_{i}^{\ast}}{1- L_{i}^{\ast }}+\frac{R\ell_{i}D_{i}^{\ast}}{1- L_{i}^{\ast}}:=M_{i}. $$

Hence,

$$\bigl\Vert \bigl(N(u,v) \bigr) \bigr\Vert _{C^{2}} \leq(M_{1},M_{2}):=M. $$

Step 3.Nmaps bounded sets into equicontinuous sets in $C^{2}$.

Let $B_{R}$ be the ball defined in Step 2. For each $t_{1},t_{2}\in I$ with $t_{1}\leq t_{2}$ and any $(u,v)\in B_{R}$ and $i=1,2$, we have

$$\begin{aligned}& \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \\& \quad\leq \int_{0}^{t_{1}}\frac{(t_{1}-qs)^{(\alpha_{i}-1)}}{\varGamma _{q}(\alpha_{i}-1)} \bigl\Vert g_{i}(s) \bigr\Vert \,d_{q}s - \int_{0}^{t_{2}}\frac{(t_{2}-qs)^{(\alpha_{i}-1)}}{\varGamma_{q}(\alpha _{i}-1)} \bigl\Vert g_{i}(s) \bigr\Vert \,d_{q}s, \end{aligned}$$

where $g_{i}(\cdot),\in C(I)$, $i=1,2$, with

$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr). \end{aligned}$$

Thus,

$$\begin{gathered} \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{1})-\bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \\ \quad\leq \biggl( \frac{R P_{i}^{\ast}}{1- L_{i}^{\ast}}+\frac{R D_{i}^{\ast}}{1- L_{i}^{\ast}} \biggr) \int_{0}^{t_{1}}\frac{ \vert (t_{2}-qs)^{(\alpha_{i}-1)}-(t_{1}-qs)^{(\alpha _{i}-1)} \vert }{\varGamma_{q}(\alpha_{1})}\,d_{q}s \\ \qquad{}+ \biggl(\frac{R P_{i}^{\ast}}{1- L_{i}^{\ast}}+\frac{R D_{i}^{\ast }}{1- L_{i}^{\ast}} \biggr) \int_{t_{1}}^{t_{2}}\frac{ \vert (t_{2}-qs)^{(\alpha_{i}-1)} \vert }{\varGamma_{q}(\alpha _{i})}\,d_{q}s \\ \quad\to0 \quad \text{as }t_{1}\to t_{2}. \end{gathered} $$

Hence,

$$\begin{aligned}& \bigl\Vert \bigl(N(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \\& \quad= \bigl( \bigl\Vert \bigl(N_{1}(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N_{1}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert , \bigl\Vert \bigl(N_{2}(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N_{2}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \bigr) \\& \quad\to(0,0) \quad \text{as }t_{1}\to t_{2}. \end{aligned}$$

As a consequence of steps 1 to 3 together with Theorem 4.6, we can conclude that N has at least one fixed point in $B_{R}$ which is a solution of our coupled system (1)–(2). □

5 Examples

Example 1

Consider the following coupled system of implicit fractional $\frac{1}{4}$-difference equation:

$$ \textstyle\begin{cases} ({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}u)(t)=f_{1}(t,u(t),v(t),({}^{c}D_{\frac {1}{4}}^{\frac{1}{2}}u)(t)),\\ ({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}v)(t)=f_{2}(t,u(t),v(t),({}^{c}D_{\frac {1}{4}}^{\frac{1}{2}}v)(t)),\\ (u(0),v(0))=(1,2), \end{cases}\displaystyle t\in[0,1], $$

(13)

where

$$\textstyle\begin{cases} f_{1}(t,x,y,z)=\frac{ct^{2}}{1+ \vert x \vert + \vert y \vert + \vert z \vert } (e^{-7}+\frac {1}{e^{t+5}} )(t^{2}+xt^{2}+z), \\ f_{2}(t,x,y,z)=\frac{ct^{2}}{e^{t+5}(1+ \vert x \vert + \vert y \vert + \vert z \vert )}(e^{t}+ty+z); \end{cases}\displaystyle t\in(0,1], $$

and $c>0$. Hypothesis $(H_{1})$ is satisfied with

$$\begin{gathered} p_{1}(t)=d_{1}(t)= \biggl(e^{-7}+ \frac{1}{e^{t+5}} \biggr)ct^{4}, \\ r_{1}(t)= \biggl(e^{-7}+\frac{1}{e^{t+5}} \biggr)ct^{2},\qquad p_{2}(t)=\frac {ct^{2}}{e^{t+4}},\qquad d_{2}=\frac{ct^{3}}{e^{t+5}},\qquad r_{2}(t)=\frac{ct^{2}}{e^{t+5}}.\end{gathered} $$

Also, condition (6) is satisfied. Indeed,

$$r_{1}^{\ast}+r_{2}^{\ast}-r_{1}^{\ast}r_{2}^{\ast}+ \bigl(1-r_{2}^{\ast } \bigr)L_{1}d_{1}^{\ast}+ \bigl(1-r_{1}^{\ast} \bigr)L_{2}d_{2}^{\ast}< 1 $$

implies the inequality

$$(2+3L)e^{-10}c^{2}-3(1+L)e^{-5}c+1>0, $$

with

$$L:=\frac{2}{\varGamma_{\frac{1}{4}}(\frac{1}{2})}, $$

which is satisfied for all $c\in{\Bbb {R}}_{+}$ because

$$\Delta=9(1+L)^{2}e^{-10}-4(2+3L)e^{-10} = \bigl(1+6L+9L^{2} \bigr)e^{-10}>0, $$

and $(2+3L)e^{-10}>0$. For example, if we take $c=1$, we can see that

$$r_{1}^{\ast}+r_{2}^{\ast}-r_{1}^{\ast}r_{2}^{\ast}+ \bigl(1-r_{2}^{\ast } \bigr)L_{1}d_{1}^{\ast}+ \bigl(1-r_{1}^{\ast} \bigr)L_{2}d_{2}^{\ast} =3e^{-5}(1+L)-e^{-10}(2+3L)< 1. $$

Hence, Theorem 3.8 implies that system (13) has at least a solution defined on $[0,1]$.

On the other hand, hypothesis $(H_{2})$ is satisfied with $\varPhi(t)=t^{2}$. Indeed, for each $t\in(0,1]$, there exists a real number $0<\epsilon <1$ such that $\epsilon< t\leq1$, and

$$\begin{aligned} \bigl(I_{q}^{\alpha}\varPhi \bigr) (t) \leq&\frac{t^{2}}{\epsilon ^{2}(1+q+q^{2})} \\ \leq&\frac{1}{\epsilon^{2}}\varPhi(t) \\ =&\lambda_{\varPhi}\varPhi(t). \end{aligned}$$

Consequently, problem (13) is generalized Ulam–Hyers–Rassias stable.

Example 2

Consider now the following coupled system of implicit fractional $\frac{1}{4}$-difference equations:

$$ \textstyle\begin{cases} ({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}u)(t)=f_{1}(t,u(t),v(t),({}^{c}D_{\frac {1}{4}}^{\frac{1}{2}}u)(t)),\\ ({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}v)(t)=f_{2}(t,u(t),v(t),({}^{c}D_{\frac {1}{4}}^{\frac{1}{2}}v)(t)),\\ (u(0),v(0))=(1,2), \end{cases}\displaystyle t\in[0,1], $$

(14)

where

$$\textstyle\begin{cases} f_{1}(t,x,y,z)=\frac{ct^{2}}{1+ \vert x \vert + \vert y \vert + \vert z \vert } (e^{-7}+\frac {1}{e^{t+5}} )(xt^{2}+z), \\ f_{2}(t,x,y,z)=\frac{ct^{2}}{e^{t+5}(1+ \vert x \vert + \vert y \vert + \vert z \vert )}(ty+z); \end{cases}\displaystyle t\in(0,1], $$

and $c>0$. Hypothesis $(H_{01})$ is satisfied with

$$\begin{gathered} p_{1}(t)= \biggl(e^{-7}+\frac{1}{e^{t+5}} \biggr)ct^{4},\qquad d_{1}(t)=0, \\ l_{1}(t)= \biggl(e^{-7}+\frac{1}{e^{t+5}} \biggr)ct^{2},\qquad p_{2}(t)=0,\qquad d_{2}= \frac{ct^{3}}{e^{t+5}},\qquad l_{2}(t)=\frac{ct^{2}}{e^{t+5}}.\end{gathered} $$

Also, with a good choice of the constant c, the matrix

$$M:=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{\ell_{1}p_{1}^{\ast}}{1-l_{1}^{\ast}} & 0\\ 0 & \frac{\ell_{2}d_{2}^{\ast}}{1-l_{2}^{\ast}} \end{array}\displaystyle \right ) $$

converges to 0. Hence, Theorem 4.8 implies that system (14) has a unique solution defined on $[0,1]$.

6 Conclusion

We have provided sufficient conditions for the existence, uniqueness, and Ulam stability of the solutions of two classes of coupled implicit Caputo fractional q-difference systems with initial conditions. Suitable fixed point theorems have been used. As illustration, we have presented two examples.