Existence Results for Block Matrix Operator of Fractional Orders in Banach Algebras

Abstract

This paper is concerned with proving the existence of solutions for a coupled system of quadratic integral equations of fractional order in Banach algebras. This result is a direct application of a fixed point theorem of Banach algebras. Some particular cases, examples and remarks are illustrated. Finally, the stability of solutions for that coupled system are studied.

1. Introduction and Preliminaries

Operators which have an operator matrix representation occur in various fields such as system theory, quantum mechanics, hydrodynamics and magnetohydro-dynamics (see [1,2,3]).

According to their origin, they may have rather different structure, and their study may require quite different approaches.

Let $\mathbb{A}$ be an operator which has the form

$$\mathbb{A}=\left(\begin{array}{cc}{T}_1& T_2·T_2^{\prime }\\ T_3& T_4\end{array}\right)$$

(1)

where $T_1,T_2,T_2^{\prime },T_3,T_4$ are nonlinear operators defined on Banach algebras. This kind of operators is studied by many researchers [4,5,6].

Amar and et al. [7] introduced and studied a coupled system of differential equations under boundary conditions of Rotenberg’s model type, the last one arising in growing cell populations. The entries of block operator matrix associated to this system are nonlinear and act on the Banach space.

Kaddachi and et al. [4] concentrated on answering the question: Under which conditions on its entries does the $2\times 2$ operator matrix (Equation (1)) acting on a product of Banach algebras has a fixed point? In [4], some fixed point theorems of a $2\times 2$ block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras are studied, where the entries are nonlinear operators. Furthermore, the obtained results are applied to a coupled system of nonlinear equations.

Let $X=\mathbb{C}(I,\mathbb{R}),I=[0,b]$ and $\alpha ,\beta >0.$ In this work, the following coupled system of fractional order

$$\begin{array}{ccc}\hfill v\left(t\right)& =& f_1(t,v\left(t\right))+g_1(t,w\left(t\right)){\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w\left(s\right))ds,t\in I,\hfill \\ \hfill w\left(t\right)& =& f_2(t,w\left(t\right))+g_2(t,v\left(t\right)){\int }_0^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}{u}_2(s,v\left(s\right))ds,t\in I.\hfill \end{array}$$

(2)

is studied in Banach algebras; some particular cases are given; and some examples and remarks are illustrated. Finally, the stability of solutions for the coupled system in Equation (2) is studied.

The solution of Equation (2) may be defined by a vector function $\left(\begin{array}{c}v\\ w\end{array}\right)\in X\times X$ that satisfies (2).

Now, we introduce the following definitions of fractional operators.

Definition 1

([8]). The Riemann–Liouville fractional integral of order $\beta >0$ of the function $f:[a,b]\to \mathbb{R}$ is given by

$$I_a^{\beta }f\left(t\right)={\int }_a^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}f\left(s\right)ds,t>a$$

(3)

and when $a=0,$ we have $I^{\beta }f\left(t\right)=I_0^{\beta }f\left(t\right),t>0$.

Definition 2

([8]). The Riemann–Liouville fractional derivative of order $\alpha \in (0,1)$ of a function f is defined as

$${}_R{D}_a^{\alpha }f\left(t\right)=\frac{d}{dt}{\int }_a^t\frac{{(t-s)}^{-\alpha }}{\Gamma (1-\alpha )}f\left(s\right)ds,t\in [a,b]$$

or

$${}_R{D}_a^{\alpha }f\left(t\right)=\frac{d}{dt}{I}_a^{1-\alpha }f\left(t\right),t\in [a,b].$$

2. Existence Theorem

Coupled systems of integral and differential equations are studied in many papers [9,10,11,12,13].

Especially, the investigation for coupled systems of fractional differential equations appears in many studies (e.g., [9,11,14,15,16,17]).

Assume that

(i)

$u_j:I\times \mathbb{R}\to \mathbb{R},j=1,2$ satisfies the Carathéodory condition and

$$|u_j(s,v)|\le m_j\left(s\right)\in L^1\left[I\right]\forall (s,v)\in I\times \mathbb{R},$$

$k_j=\underset{s\in I}{max}{I}^{{\gamma }_j}{m}_j\left(s\right)$ for any ${\gamma }_1\le \alpha$ and ${\gamma }_2\le \beta .$

(ii)

$f_j,g_j:I\times \mathbb{R}\to \mathbb{R}$ are continuous and

$M_j=\underset{(s,v)\in I\times \mathbb{R}}{max}|f_j(s,v)|,j=1,2$$N_j=\underset{(s,v)\in I\times \mathbb{R}}{max}|g_j(s,v)|,j=1,2$ respectively.

(iii)

There exist constants $l_j$ and $h_j$, which satisfy

$$|f_j(s,v)-f_j(s,w)|\le l_j|v-w|,j=1,2$$

and

$$|g_j(s,v)-g_j(s,w)|\le h_j|v-w|,j=1,2$$

$\forall s\in I$ and $v,w\in \mathbb{R}.$

Theorem 1.

Let Assumptions (i)–(iii) be satisfied. Moreover, if $h_1{k}_1{h}_2{k}_2<(1-l_1)(1-l_2)$ $\Gamma (\alpha -{\gamma }_1+1)$ $\Gamma (\beta -{\gamma }_2+1),$ then the exists at least one solution for Equation (2) in $X\times X.$

Proof. 

Consider the operators $T_1,T_2,T_3,T_4$ and $T_2^{\prime }$ on $X$ defined by:

$$\left\{\begin{array}{c}\left(T_{1}v\right)\left(t\right)=f_1(t,v\left(t\right))\hfill \\ \left(T_{2}w\right)\left(t\right)=g_1(t,w\left(t\right))\hfill \\ \left(T_{3}w\right)\left(t\right)=g_2(t,w\left(t\right)){\int }_0^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}{u}_2(s,w\left(s\right))ds\hfill \\ \left(T_{4}v\right)\left(t\right)=f_2(t,v\left(t\right))\hfill \\ \left(T_2^{\prime }w\right)\left(t\right)={\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w\left(s\right))ds.\hfill \end{array}\right.$$

(4)

The coupled system in Equation (2) may have the form:

$$\left\{\begin{array}{c}v\left(t\right)=T_{1}v\left(t\right)+T_{2}w\left(t\right)·T_2^{\prime }w\left(t\right)\hfill \\ w\left(t\right)=T_{4}w\left(t\right)+T_{3}v\left(t\right),\hfill \end{array}\right.$$

(5)

and

$$\left(\begin{array}{c}v\\ w\end{array}\right)=\left(\begin{array}{cc}{T}_1& T_2·T_2^{\prime }\\ T_3& T_4\end{array}\right)·\left(\begin{array}{c}v\\ w\end{array}\right).$$

(6)

Define

$$S=\{v\in X,||v||\le M_1+\frac{N_1{k}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}\}$$

$$S^{\prime }=\{w\in X,||w||\le M_2+\frac{N_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}\}.$$

For, let $v_1,v_2\in S.$ Thus,

$$||T_1{v}_1\left(t\right)-T_1{v}_2\left(t\right)||\le l_1||v_1-v_2||$$

(7)

and

$$||T_2{v}_1\left(t\right)-T_2{v}_2\left(t\right)||\le h_1||v_1-v_2||.$$

(8)

In addition, set

$$T_3{v}_1\left(t\right)=G{v}_1\left(t\right)·U{v}_1\left(t\right)=(G·U)v_1\left(t\right)$$

(9)

where $G{v}_1\left(t\right)=g_2(t,v_1\left(t\right))$ and $U{v}_1\left(t\right)=I^{\beta }{u}_2(t,v_1\left(t\right))$

$$\begin{array}{ccc}\hfill ||T_3{v}_1\left(t\right)-T_3{v}_2\left(t\right)||& =& ||(G·U)v_1\left(t\right)-(G·U)v_2\left(t\right)||\hfill \\ & \le & ||U{v}_1\left(t\right)||·||G{v}_1\left(t\right)-G{v}_2\left(t\right)||\hfill \\ & \le & \frac{k_2{h}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}·||v_1-v_2||.\hfill \end{array}$$

(10)

Furthermore,

$$\begin{array}{ccc}\hfill |T_{3}w\left(t\right)|& \le & |g_2(t,w\left(t\right))|{\int }_0^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}|u_2(s,w\left(s\right))|ds\hfill \\ & \le & \frac{N_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)},\hfill \end{array}$$

(11)

for each $t_1,t_2\in I$ and $t_1<t_2$, we get

$$\begin{array}{ccc}\hfill |\left(T_{3}w\right)\left(t_2\right)-\left(T_{3}w\right)\left(t_1\right)|& =& |g_2(t_2,w\left(t_2\right))I^{\beta }{u}_2(t_2,w\left(t_2\right))-g_2(t_1,w\left(t_1\right))I^{\beta }{u}_2(t_1,w\left(t_1\right))\hfill \\ & +& g_2(t_1,w\left(t_1\right))I^{\beta }{u}_2(t_2,w\left(t_2\right))-g_2(t_1,w\left(t_1\right))I^{\beta }{u}_2(t_2,w\left(\left(t_2\right)\right)|\hfill \\ & \le & |g_2(t_2,w\left(t_2\right))-g_2(t_1,w\left(t_1\right))|I^{\beta }|u_2(t_2,w\left(t_2\right))|\hfill \\ & +& |g(t_1,w\left(t_1\right))||I^{\beta }{u}_2(t_2,w\left(t_2\right))-I^{\beta }{u}_2(t_1,w\left(t_1\right))|,\hfill \end{array}$$

(12)

but

$$|I^{\beta }{u}_2(t_2,w\left(t_2\right))-I^{\beta }{u}_2(t_1,w\left(t_1\right))|\le {\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{\beta -1}}{\Gamma \left(\beta \right)}|u_2(s,w\left(s\right))|ds.$$

(13)

Then,

$$|I^{\beta }{u}_2(t_2,w\left(t_2\right))-I^{\beta }{u}_2(t_1,w\left(t_1\right))|\le k_2\frac{{(t_2-t_1)}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}.$$

(14)

Then, we get

$$|\left(T_{3}w\right)\left(t_2\right)-\left(T_{3}w\right)\left(t_1\right)|\le \frac{k_2{h}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}|w\left(t_2\right)-w\left(t_1\right)|+\frac{k_2{N}_2}{\Gamma (\beta -{\gamma }_2+1)}{(t_2-t_1)}^{\beta -{\gamma }_2}$$

(15)

Then, $\overline{U{S}^{\prime }}$ is relatively compact.

We prove that $T_3\left(S\right)\subseteq (I-T_4)\left(S^{\prime }\right),$ for $v_1\in S.$

Now, we can introduce a function ${\varphi }_{v_1}:X\to X$ by

$$w\to T_3{v}_1+T_4{v}_2,$$

(16)

then the function ${\varphi }_{v_1}$ is a contraction with a constant $l_2+\frac{h_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}.$ Then, there exists a unique point $w\in X$ where $T_{3}v+T_{4}w=w$ implies $T_{3}v=(I-T_4)w.$ Thus,

$$T_3\left(s\right)\subseteq (I-T_4)X.$$

(17)

For $w\in X,$, $\exists s^{*}\in I$ such that

$$\begin{array}{ccc}\hfill {||w||}_{\infty }=|w\left(s^{*}\right)|& =& |T_{3}v\left(s^{*}\right)+T_{4}w\left(s^{*}\right)|\hfill \\ & \le & |g_2(s^{*},v\left(s^{*}\right))I^{\beta }{u}_2(s^{*},v\left(s^{*}\right))+f_2(s^{*},w\left(s^{*}\right))|\hfill \\ & \le & \frac{N_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}+M_2.\hfill \end{array}$$

(18)

Then, $T_3\left(S\right)\subseteq (I-T_4)\left(S^{\prime }\right).$

For $v\in S^{\prime },$ then

$$\begin{array}{ccc}\hfill |T_2^{\prime }v\left(t_n\right)-T_2^{\prime }v\left(t\right)|& =& |I^{\alpha }{u}_1(t_n,v\left(t_n\right))-I^{\alpha }{u}_1(t,v\left(t\right))|\hfill \\ & \le & \frac{k_1}{\Gamma (\alpha -{\gamma }_1+1)}|t_n-t|,\hfill \end{array}$$

(19)

since $t_n\to t$ and $u_1$ is continuous in the second argument, then by Lebesgue Dominated Convergence Theorem, we have

$$u_1(t_n,v\left(t_n\right))\to u_1(t,v\left(t\right))\mathrm{in}\mathbb{R}\Rightarrow T_2^{\prime }v\left(t_n\right)\to T_2^{\prime }v\left(t\right)\mathrm{in}\mathbb{R},$$

thus $T_2^{\prime }v\in \mathbb{C}(I,\mathbb{R}).$

Defining $T^{\prime }=T_2^{\prime }{(1-T_4)}^{-1}{T}_3,$, Assumption (ii) implies that

$$\begin{array}{ccc}\hfill M& =& ||T^{\prime }\left(S\right)||=\underset{w\in S}{sup}|T^{\prime }\left(w\right)|\hfill \\ & \le & \underset{t\in I}{sup}|{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w\left(s\right))ds|\hfill \\ & \le & \frac{k_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)},\hfill \end{array}$$

(20)

and therefore $h_1{k}_1{h}_2{k}_2<(1-l_1)(1-l_2)\Gamma (\alpha -{\gamma }_1+1)\Gamma (\beta -{\gamma }_2+1).$

Let $v_1,v_2\in S$, then $\forall s\in I$. We obtain

$$\begin{array}{ccc}\hfill |T_1{v}_1\left(s\right)+T_2{(I-T_4)}^{-1}{T}_3{v}_1\left(s\right)T_2^{\prime }{(I-T_4)}^{-1}{T}_3{v}_2|& \le & M_1+\frac{N_1{k}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}.\hfill \end{array}$$

(21)

This implies that

$$\begin{array}{c}\hfill T_1{v}_1+T_2{(I-T_4)}^{-1}{T}_3{v}_1{T}_2^{\prime }{(I-T_4)}^{-1}{T}_3{v}_2\in S\mathrm{for} \mathrm{any}{v}_1,v_2\in S.\end{array}$$

(22)

Now, all conditions of Theorem 4.2 in [4] are verified and our results follows. □

Example 1.

Let $I=[0,1].$ Consider the fractional order coupled system

$$\begin{array}{ccc}\hfill v\left(t\right)& =& tsin(\frac{v\left(t\right)}{4})+\frac{w\left(t\right)}{3+t}{I}^{3/2}\frac{2tw\left(t\right)}{1+w\left(t\right)},t\in I\hfill \\ \hfill w\left(t\right)& =& \frac{t+sin(\frac{w\left(t\right)}{2})}{2+t^2}+\frac{v\left(t\right)}{4}{I}^{2/3}\frac{v\left(t\right)}{1+v\left(t\right)},t\in I.\hfill \end{array}$$

(23)

Set

$$\begin{array}{ccc}\hfill f_1(s,v)=ssin(\frac{v\left(s\right)}{4}),u_1(s,w)=\frac{2sw\left(s\right)}{1+w\left(s\right)},g_1(s,w)& =& \frac{w\left(s\right)}{3+s},\hfill \\ \hfill f_2(s,w)=\frac{s+sin(\frac{w\left(s\right)}{2})}{2+s^2},u_2(s,v)=\frac{v\left(s\right)}{1+v\left(s\right)},g_2(s,w)& =& \frac{w\left(s\right)}{4}.\hfill \end{array}$$

Then, we easily get

• $|u_1(s,v)|\le 2s=m_1\left(s\right)$ and $|u_2(s,v)|\le 1=m_2\left(s\right).$
  Choose ${\gamma }_1={\gamma }_2=1/2,$ then we can obtain $k_1=\frac{8}{3\sqrt{\pi }}$ and $k_2=\frac{2}{\sqrt{\pi }}$

  $$\begin{array}{ccc}\hfill |g_1(t,w_1)-g_1(t,w_2)|& \le & \frac{1}{3}|w_1-w_2|\hfill \\ \hfill |g_2(t,v_1)-g_2(t,v_2)|& \le & \frac{1}{4}|v_1-v_2|\hfill \end{array}$$

  and

  $$\begin{array}{ccc}\hfill |f_i(t,v_1)-f_i(t,v_2)|& \le & \frac{1}{2}|v_1-v_2|,i=1,2.\hfill \end{array}$$

Then, the inequality $h_1{k}_1{h}_2{k}_2<(1-l_1)(1-l_2)\Gamma (\alpha -\gamma +1)\Gamma (\beta -\gamma +1)$ is verified.

3. Stability of Solutions of the Coupled System

Here, asymptotic stability on ${\mathbb{R}}_{+}$ of the solution $z=(v,w)$ of the coupled system in Equation (2) is studied.

Definition 3.

A pair $z_1=(v_1,w_1)$ is said to be an asymptotically stable solution of Equation (2) if for any $\epsilon >0$ there exists $T^{\prime }=T^{\prime }\left(\epsilon \right)>0$ such that for very $t\ge T^{\prime }$ and for every other solution $z_2=(v_2,w_2)$ of (2),

$$|z_1\left(t\right)-z_2\left(t\right)|\le \epsilon .$$

Given two solutions $z_1$ and $z_2$ of Equation (2), then we have

$$\begin{array}{ccc}\hfill |v_1\left(t\right)-v_2\left(t\right)|& \le & |f_1(t,v_1\left(t\right))-f_1(t,v_2\left(t\right))|\hfill \\ & +& |g_1(t,w_1\left(t\right)){\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w_1\left(s\right))ds\hfill \\ & -& g_1(t,w_2\left(t\right)){\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w_2\left(s\right))ds|\hfill \\ & \le & l_1|v_1\left(t\right)-v_2\left(t\right)|\hfill \\ & +& g_1(t,w_1\left(t\right))|{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}|u_1(s,w_1\left(s\right))-u_1(s,w_2\left(s\right))|ds\hfill \\ & +& |g_1(t,w_2\left(t\right))-g_1(t,w_1\left(t\right))|{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}|u_1(s,w_2\left(s\right))|ds\hfill \\ & \le & l_1|v_1\left(t\right)-v_2\left(t\right)|+2|g_1(t,w_1\left(t\right))|{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{m}_1\left(s\right)ds\hfill \\ & +& |g_1(t,w_2\left(t\right))-g_1(t,w_1\left(t\right))|{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{m}_1\left(s\right)ds\hfill \\ & \le & l_1|v_1\left(t\right)-v_2\left(t\right)|+\frac{2k_1{N}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}+\frac{h_1{k}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}|w_1-w_2|,\hfill \end{array}$$

then

$$(1-l_1)||v_1\left(t\right)-v_2\left(t\right)||\le \frac{2k_1{N}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}+\frac{h_1{k}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}||w_1-w_2||.$$

(24)

In the same fashion, we obtain

$$\begin{array}{ccc}\hfill (1-l_2)||w_1\left(t\right)-w_2\left(t\right)||& \le & \frac{2k_2{N}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}+\frac{h_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}||v_1-v_2||,\hfill \end{array}$$

(25)

and

$$\begin{array}{c}[1-l_1-\frac{h_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}]||v_1-v_2||+[1-l_2-\frac{h_1{k}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}]||w_1-w_2||\le \frac{2k_1{N}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}+\frac{2k_2{N}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}\end{array}$$

(26)

Let $\Lambda =min\{1-l_1-\frac{h_2{k}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)},1-l_2-\frac{h_1{k}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}\},$ then

$$||v_1-v_2||+||w_1-w_2||\le {\Lambda }^{-1}[\frac{2k_1{N}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}+\frac{2k_2{N}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}].$$

(27)

Since

$$\begin{array}{ccc}\hfill z_1\left(t\right)-z_2\left(t\right)& =& (v_1\left(t\right)-v_2\left(t\right),w_1\left(t\right)-w_2\left(t\right)),\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill ||z_1\left(t\right)-z_2\left(t\right)||& \le & ||v_1\left(t\right)-v_2\left(t\right)||+||w_1\left(t\right)-w_2\left(t\right)||\hfill \\ & \le & {\Lambda }^{-1}[\frac{2k_1{N}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}+\frac{2k_2{N}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}]\hfill \\ & \le & \epsilon .\hfill \end{array}$$

(28)

Then, we obtain the following theorem.

Theorem 2.

Let assumptions of Theorem 1 be satisfied,

$$l_1\Gamma (\beta -{\gamma }_2+1)+h_2{k}_2{b}^{\beta -{\gamma }_2}<\Gamma (\beta -{\gamma }_2+1),l_2\Gamma (\alpha -{\gamma }_1+1)+h_1{k}_1{b}^{\alpha -{\gamma }_1}<\Gamma (\alpha -{\gamma }_1+1),$$

and

$${\Lambda }^{-1}[\frac{2k_1{N}_1{b}^{\alpha -{\gamma }_1}}{\Gamma (\alpha -{\gamma }_1+1)}+\frac{2k_2{N}_2{b}^{\beta -{\gamma }_2}}{\Gamma (\beta -{\gamma }_2+1)}]\le \epsilon .$$

Then, the solution of Equation (2) is asymptotically stable on ${\mathbb{R}}_{+}.$

4. Further Results

Consequently, we have the following results in $X\times X.$

(i)

Letting $\alpha ,\beta \to 1,$ then we have the coupled system of quadratic integral equations

$$\begin{array}{ccc}\hfill v\left(t\right)& =& f_1(t,v\left(t\right))+g_1(t,w\left(t\right)){\int }_0^t{u}_1(s,w\left(s\right))ds,\hfill \\ \hfill w\left(t\right)& =& f_2(t,w\left(t\right))+g_2(t,v\left(t\right)){\int }_0^t{u}_2(s,v\left(s\right))ds.\hfill \end{array}$$

(ii)

Letting $g_1=g_2=0,$ we get the coupled system of functional equations

$$\begin{array}{ccc}\hfill v\left(t\right)& =& f_1(t,v\left(t\right))\hfill \\ \hfill w\left(t\right)& =& f_2(t,w\left(t\right)).\hfill \end{array}$$

(iii)

Putting $f_1(t,w)=a_1\left(t\right),f_2(t,v)=a_2\left(t\right),$ then we have the coupled system of quadratic integral equations of fractional order

$$\begin{array}{ccc}\hfill v\left(t\right)& =& a_1\left(t\right)+g_1(t,w\left(t\right)){\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w\left(s\right))ds,\hfill \\ \hfill w\left(t\right)& =& a_2\left(t\right)+g_2(t,v\left(t\right)){\int }_0^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}{u}_2(s,v\left(s\right))ds.\hfill \end{array}$$

(v)

Letting $f_1(t,w)=a_1\left(t\right),f_2(t,v)=a_2\left(t\right),$ then we get the coupled system of fractional integral equations

$$\begin{array}{ccc}\hfill v\left(t\right)& =& a_1\left(t\right)+{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}_1(s,w\left(s\right))ds,\hfill \\ \hfill w\left(t\right)& =& a_2\left(t\right)+{\int }_0^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}{u}_2(s,v\left(s\right))ds.\hfill \end{array}$$

System of Fractional Differential Equations

Let

$$\begin{array}{ccc}\hfill {}_R{D}^{\alpha }v\left(t\right)& =& u_1(t,w\left(t\right)),t\in J\mathrm{and}v\left(0\right)=0,0<\alpha <1\hfill \\ \hfill {}_R{D}^{\beta }w\left(t\right)& =& u_2(t,v\left(t\right)),t\in J\mathrm{and}w\left(0\right)=0,0<\beta <1\hfill \end{array}$$

(29)

where ${}_R{D}^{\alpha }$ is a Riemann–Liouville fractional derivative of order $0<\alpha <1.$

Theorem 3.

Let assumptions of Theorem 1 be satisfied. Then, there exists at least one solution for Equation (29) in $X\times X.$

The proof is straight forward as in [11].

By direct calculations, we can prove an existence result for the following coupled systems

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& =& f_1(t,w\left(t\right)),v\left(0\right)=v_0,\hfill \\ \hfill w^{\prime }\left(t\right)& =& f_2(t,v\left(t\right)),w\left(0\right)=w_0.\hfill \end{array}$$

(30)

5. Conclusions

The theory of block operator matrices opens up a new line of attack of mathematical problems. During the past years, several papers are devoted to the investigation of linear operator matrices defined by $2\times 2$ block operator matrices (Equation (1)).

In this paper, we prove an existence theorem of solutions for a coupled system of quadratic integral equations of fractional order in Banach algebras, by a direct application of a block operator fixed point theorem [4]. This coupled system includes many key coupled systems of integral and differential equations that arise in nonlinear analysis and their applications. Some examples and remarks are illustrated. Finally, we study the stability of solutions for the coupled system in Equation (2).