On the solvability of a general class of a coupled system of stochastic functional integral equations

Abstract

The main objective in this work is to find some weaker conditions that guarantee the existence of continuous solutions of some stochastic coupled systems of the Urysohn–Volterra–Itô–Doob type. The uniqueness of the solution for these systems is investigated as well. It is worth mentioning that the results in this work include many previously published results as special cases.

Keywords:

• Coupled systems
• existence
• uniqueness
• Urysohn–Volterra–Itô–Doob stochastic integral equation

2010 MSC:

• 60H20
• 65C30
• 45R05
• 45Rxx

1. Introduction

Integral equations are considered significant tools in the analysis, and formulation of many phenomena arising in most applied sciences. In other words, integral equations can successfully describe the evolution of many real-world systems, such as systems in quantum mechanics, Newtonian mechanics, electrical engineering, electromagnetic theory, and many other fields, (see e.g. Poljak et al., 2018). Also, in ordinary and partial differential equations we can convert several initial and boundary value problems to an equivalent integral equations. At the same time examining coupled systems associated with integral equations is important as well, because such systems model many physical problems, (see e.g. El-Sayed & Al-Fadel, 2018; Hashem & El-Sayed, 2017; Zhang, 2018). Recently, A.M.A. El-Sayed et al. (El-Sayed & Kenawy, 2014b) used the fixed point principle, under sufficient conditions, to prove the existence of continuous solutions, (x, y), of the coupled system (1.1) $$\{\begin{array}{cc}x(t)=a_1(t)+{\int }_0^t{f}_1(t,s,y({\varphi }_1(s)))ds,\hfill & t\in [0,a].\hfill \\ \hfill & \hfill \\ y(t)=a_2(t)+{\int }_0^t{f}_2(t,s,x({\varphi }_2(s)))ds,\hfill & t\in [0,a].\hfill \end{array}$$(1.1) where x, and y are real-valued continuous functions defined on $[0,a], a<\infty ,$ and endowed with the Chebyshev’s norm. The functions ${\varphi }_j, j=1,2,$ are continuous selfmaps on $[0,a].$ Also, A.M.A. El-Sayed et al. utilized in (El-Sayed & Kenawy, 2014a) the same previous technique to study the existence of solution of the coupled system defined by (1.2) $$\{\begin{array}{cc}x(t)=a_1(t)+{\int }_0^1{f}_1(t,s,J^{{\beta }_1}y(s))ds,\hfill & t\in [0,1].\hfill \\ \hfill & \hfill \\ y(t)=a_2(t)+{\int }_0^1{f}_2(t,s,J^{{\beta }_2}x(s))ds,\hfill & t\in [0,1].\hfill \end{array}$$(1.2) where $0<{\beta }_j<1, j=1,2,$ and $J^{{\beta }_j}$ is the Riemann-Liouville fractional order integral operator. The uniqueness of the solution is investigated for Equations (1.1)(1.1) $$\{\begin{array}{cc}x(t)=a_1(t)+{\int }_0^t{f}_1(t,s,y({\varphi }_1(s)))ds,\hfill & t\in [0,a].\hfill \\ \hfill & \hfill \\ y(t)=a_2(t)+{\int }_0^t{f}_2(t,s,x({\varphi }_2(s)))ds,\hfill & t\in [0,a].\hfill \end{array}$$(1.1) , and (1.2)(1.2) $$\{\begin{array}{cc}x(t)=a_1(t)+{\int }_0^1{f}_1(t,s,J^{{\beta }_1}y(s))ds,\hfill & t\in [0,1].\hfill \\ \hfill & \hfill \\ y(t)=a_2(t)+{\int }_0^1{f}_2(t,s,J^{{\beta }_2}x(s))ds,\hfill & t\in [0,1].\hfill \end{array}$$(1.2) as well. But due to some unexpected disturbance which may effect the accuracy of the system, we believe that stochastic models give more accurate results than the proposed models in (El-Sayed & Kenawy, 2014a, 2014b), see for more details (Elborai, Abdou, & Youssef, 2013a, 2013b; Elborai & Youssef, 2019; Klyatskin, 2015; Umamaheswari, Balachandran, & Annapoorani, 2017, 2018). Therefore, our aim in this article is to generalize the deterministic models 1.1, and 1.2 to the stochastic form in a suitable sense. So, we shall study the existence of solutions of the coupled system of stochastic functional integral equations of the Urysohn–Volterra–Itô–Doob type defined by (1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) where $t\in I:=[0,a],a<\infty .$ The coupled system 1.3(1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) will be studied in the space $C(I,L_2(\Omega ,\mathcal{F},P))\times C(I,L_2(\Omega ,\mathcal{F},P)),$ which contains all ordered pairs, (x, y), of continuous stochastic processes in the mean square sense which are defined from I into $L_2(\Omega ,\mathcal{F},P)$ and adapted to the proposed filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\in I},$ see the next section for more explanations. The stochastic process W(t) is assumed to be ${\mathcal{F}}_t$-adapted real continuous martingale. The set $\{B(t):t\in I\}$ represents a collection of bounded linear operators defined on $C(I,L_2(\Omega ,\mathcal{F},P))$ into $C(I,L_2(\Omega ,\mathcal{F},P)),$ while the set $\{A(t):t\in I\}$ constitutes a family of closed linear operators defined on $C(I,L_2(\Omega ,\mathcal{F},P))$ taking values in $C(I,L_2(\Omega ,\mathcal{F},P)),$ see (Elborai & Youssef, 2019). The functions $h_j(t),f_j(t),$ and $g_j(t),$ j = 1, 2, are ${\mathcal{F}}_t$-measurable scalar functions satisfying some proposed conditions will be mentioned in the next section. It is clear that Equations (1.3)(1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) are more general than Equations (1.1)(1.1) $$\{\begin{array}{cc}x(t)=a_1(t)+{\int }_0^t{f}_1(t,s,y({\varphi }_1(s)))ds,\hfill & t\in [0,a].\hfill \\ \hfill & \hfill \\ y(t)=a_2(t)+{\int }_0^t{f}_2(t,s,x({\varphi }_2(s)))ds,\hfill & t\in [0,a].\hfill \end{array}$$(1.1) , and (1.2)(1.2) $$\{\begin{array}{cc}x(t)=a_1(t)+{\int }_0^1{f}_1(t,s,J^{{\beta }_1}y(s))ds,\hfill & t\in [0,1].\hfill \\ \hfill & \hfill \\ y(t)=a_2(t)+{\int }_0^1{f}_2(t,s,J^{{\beta }_2}x(s))ds,\hfill & t\in [0,1].\hfill \end{array}$$(1.2) , see Lemma 2.8 in (Kilbas, Srivastava, & Trujillo, 2006). The following sections are organized as follows. Section 2 provides the necessary definitions, and auxiliary theorems. New results are introduced in Section 3. Eventually, a conclusion and suggested future work are presented in Section 4.

2. Definitions and auxiliary results

This section includes some preliminaries which will be utilized to prove our main results. For more explanations, we refer to (Klyatskin, 2015; Pavliotis, 2014). Let $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\in I},P),$ where $\mathcal{F}:={\mathcal{F}}_a,$ be a filtered probability space satisfying the usual conditions, where $\Omega$ is known as the sample space and contains a collection abstract points ω, $\mathcal{F}$ is a sigma-algebra on $\Omega$ and includes all possible events which take place through the time interval I, P is a probability measure defined on $\mathcal{F}.$ Let $L_2:=L_2(\Omega ,\mathcal{F},P)$ be the space of all real stochastic processes $\{x(t):t\in I\}$ which have finite second moments $(i.e. E\{|x(t){|}^2\}<\infty ),$ for all $t\in I.$ Let the norm on the space L₂ be defined as $\Vert x(t){\Vert }_{L_2}={\left\{E(|x(t){|}^2)\right\}}^{\frac{1}{2}}={\left\{{\int }_{\Omega }|x(t){|}^{2}dP\right\}}^{\frac{1}{2}}$ for each $t\in I,$ where the expectation, E, is taken over x(t). Let $C:=C(I,L_2(\Omega ,\mathcal{F},P))$ be the space of all stochastic processes defined on I, continuous in the mean square sense and adapted to the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\in I}.$ The norm on C is defined as $\Vert x{\Vert }_C=\underset{t\in I}{\sup }\left\{\Vert x(t){\Vert }_{L_2}\right\}.$ Let $X:=C(I,L_2(\Omega ,\mathcal{F},P))\times C(I,L_2(\Omega ,\mathcal{F},P))=\{(x,y):x\in C, y\in C\}.$ That is, the space X is the Banach space of all ordered pairs (x, y) of stochastic processes which are defined on I, continuous in the mean square sense, and adapted to the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\in I}.$ Let the space X be equipped with the norm $\Vert (x,y){\Vert }_X:=\Vert x{\Vert }_C+\Vert y{\Vert }_C, \forall (x,y)\in X.$

Definition 2.1

(Pavliotis, 2014). Suppose $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\in I},P)$ is a filtered probability space. Let ${\{x(t)\}}_{t\in I}$ be a continuous real stochastic process adapted to the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\in I}.$ Then ${\{x(t)\}}_{t\in I}$ is called a continuous martingale with respect to the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\in I}$ if

1. $E(|x(t)|)<\infty ,$ $\forall t\in I.$

2. $E(x(t)|{\mathcal{F}}_s)=x(s)$ almost surely, $\forall s\le t.$

Remark 2.1.

In the above definition 2.1 if $E(x(t)|{\mathcal{F}}_s)=x(s)$ is replaced by $E(x(t)|{\mathcal{F}}_s)\ge x(s)$ $(E(x(t)|{\mathcal{F}}_s)\le x(s))$ we obtain the definition of submartingale (supermartingale), respectively.

Definition 2.2.

A stochastic process (x, y) is said to be a solution of the stochastic coupled system 1.3 if the process $(x,y)\in C(I,L_2(\Omega ,\mathcal{F},P))\times C(I,L_2(\Omega ,\mathcal{F},P)),$ and satisfies Equations (1.3)(1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) almost surely.

Theorem 2.1

(Kreyszig, 1978). Let X and Y be two normed spaces. Let T be a linear operator from $D(T)\subset X$ into Y. Then, the operator T is continuous if and only if it is bounded.

Theorem 2.2

(Kreyszig, 1978). A linear operator defined from a Banach space X into a Banach space Y is continuous if and only if it is closed.

Theorem 2.3

(Hochstadt, 1988). A completely continuous operator T defined on a closed bounded convex subset S in a Banach space X, such that $TS\subset S$ has at least one fixed point in the set S.

3. Main results

Suppose the following coupled system of stochastic functional integral equations. $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$ where, the first integral in each equation is the Riemann integral in the mean square sense while the second integral will be studied in the Itô-Doob sense. The stochastic process W(t) is assumed to be ${\mathcal{F}}_t$-adapted real martingale. The real random functions f_j, and g_j, j = 1, 2, are defined on $I\times I\times C\times C$ into the space L₂ and will be specified in the conditions below. The given random forcing functions h_j, j = 1, 2, are defined from I into the space C and called the stochastic perturbing terms. Now, let us propose some sufficient conditions.

H₁: Suppose there exists a non-decreasing continuous real function $F:I\to \mathbb{R}$ satisfying $$E\left\{\right|W(t_2)-W(t_1){|}^2\}=E\{|W(t_2)-W(t_1){|}^2\setminus {\mathcal{F}}_{t_1}\}=F(t_2)-F(t_1) a.s. \forall t_1<t_2.$$

H₂: The random functions $f_j(t,s,x,y),$ and $g_j(t,s,x,y),$ j = 1, 2, are measurable in $s\in I$ for each $(t,x,y)\in I\times C\times C,$ continuous in the mean square with respect to (t, x, y) for each $s\in I.$

H₃: There exist two real random functions $m_j, j=1,2,$ defined on I × I, and four positive constants $b_j,b_j^{*}, j=1,2,$ such that $$\begin{array}{c}|f_j(t,s,x,y){|}^2\le |m_j(t,s){|}^2+b_j|x{|}^2+b_j^{*}|y{|}^2, j=1,2.\\ {\int }_0^{a}E\left(|m_j(t,s){|}^2\right)ds<M_j, \forall 0\le s\le t\le a, M_j>0, j=1,2.\end{array}$$ where the symbol E represents the expectation with respect to $m_j, j=1,2.$

H₄: There exist two positive constants β_j, j = 1, 2, such that $$|g_j(t,s,x,y){|}^2\le {\beta }_j(1+|x{|}^2+|y{|}^2), \forall 0\le s\le t\le a.$$

Now, we define, for each $t\in I,$ the integral operator T by (3.1) $$T(x,y)(t):=(T_{1}y(t),T_{2}x(t)).$$(3.1)

Where, for each $t\in I,$ the integral operators T₁, and T₂ are defined by (3.2) $$\begin{array}{c}{T}_{1}y(t):=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ T_{2}x(t):=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ \hspace{1em}+{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(3.2)

Lemma 3.1.

The operator, T, defined by Equation (3.1)(3.1) $$T(x,y)(t):=(T_{1}y(t),T_{2}x(t)).$$(3.1) has the following properties.

1. $\forall (x,y)\in X$, we have $T(x,y)\in X$ ($i.e. T$ maps X into itself).

2. T is a mean square continuous operator on X.

Proof.

Clearly, the conditions H₁, H₃, and H₄ guarantee that the functions $T_{1}y,T_{2}x$ are ${\mathcal{F}}_t$-adapted for each $t\in I,$ and their second moments are finite, see (Elborai & Youssef, 2019), and hence the operator T makes sense. Now, to complete the proof of part (i) in Lemma 3.1, it remains to prove that T(x, y) is continuous in mean square $\forall (x,y)\in X.$ That is, we need to prove $T_{1}y\in C,$ and $T_{2}x\in C, \forall t\in I.$ Let $t_1\in I,t_2\in I,$ and assume, without loss of generality, that $t_2>t_1.$ $$\begin{array}{c}{\left|T_{1}y(t_2)-T_{1}y(t_1)\right|}^2\le 5|h_1(t_2)-h_1(t_1){|}^2\hfill \\ +5{\left|\underset{0}{\overset{t_1}{\int }}[f_1(t_2,s,y(s),B(s)y(s))-f_1(t_1,s,y(s),B(s)y(s))]ds\right|}^2\hfill \\ +5{\left|\underset{0}{\overset{t_1}{\int }}[g_1(t_2,s,y(s),A(s)y(s))-g_1(t_1,s,y(s),A(s)y(s))]dW(s)\right|}^2\hfill \\ +5{\left|\underset{t_1}{\overset{t_2}{\int }}{f}_1(t_2,s,y(s),B(s)y(s))ds\right|}^2+5{\left|\underset{t_1}{\overset{t_2}{\int }}{g}_1(t_2,s,y(s),A(s)y(s))dW(s)\right|}^2.\hfill \end{array}$$

From the Cauchy-Schwartz inequality, and condition H₁ we have $$\begin{array}{c}E\left\{|T_{1}y(t_2)-T_{1}y(t_1){|}^2\right\}\le 5E\left\{\right|h_1(t_2)-h_1(t_1){|}^2\}\hfill \\ +5a{\int }_0^{t_1}E\{|f_1(t_2,s,y(s),B(s)y(s))-f_1(t_1,s,y(s),B(s)y(s)){|}^2\}ds\hfill \\ +5{\int }_0^{t_1}E\{|g_1(t_2,s,y(s),A(s)y(s))-g_1(t_1,s,y(s),A(s)y(s)){|}^2\}dF(s)\hfill \\ +5(t_2-t_1){\int }_{t_1}^{t_2}E\left\{{\left|f_1(t_2,s,y(s),B(s)y(s))\right|}^2\right\}ds\hfill \\ +5{\int }_{t_1}^{t_2}E\left\{{\left|g_1(t_2,s,y(s),A(s)y(s))\right|}^2\right\}dF(s).\hfill \end{array}$$

Since $h\in C,$ so, $\Vert h_1(t_2)-h_1(t_1){\Vert }_C\to 0 as t_2\to t_1.$ Also, from condition H₂ the functions f, and g are mean square continuous in t. So, $t_2\to t_1$ implies $$\begin{array}{c}E\left\{{\left|f_1(t_2,s,y(s),B(s)y(s))-f_1(t_1,s,y(s),B(s)y(s))\right|}^2\right\}\to 0, \mathit{and}\hfill \\ E\left\{{\left|g_1(t_2,s,y(s),A(s)y(s))-g_1(t_1,s,y(s),A(s)y(s))\right|}^2\right\}\to 0.\hfill \end{array}$$

The operators A(t) are bounded on $C \forall t\in I$ from applying the closed graph theorem. So, there exists a non-negative real constant, γ₁, such that $\Vert A(t)y{\Vert }_C\le \gamma (t)\Vert y{\Vert }_C\le {\gamma }_1\Vert y{\Vert }_C,$ where ${\gamma }_1:=\underset{t\in I}{\max }\left\{\gamma (t)\right\}.$ Also, the operators B(t) are bounded on C. Therefore, there exists a non-negative real constant, ξ₁, such that $\Vert B(t)y{\Vert }_C\le \xi (t)\Vert y{\Vert }_C\le {\xi }_1\Vert y{\Vert }_C,$ where ${\xi }_1:=\underset{t\in I}{\max }\left\{\xi (t)\right\}.$ Let $\zeta :=\max \left\{{\xi }_1,{\gamma }_1\right\}.$ Applying condition H₃ yields $$\begin{array}{c}5(t_2-t_1){\int }_{t_1}^{t_2}E\left\{{\left|f_1(t_2,s,y(s),B(s)y(s))\right|}^2\right\}ds\hfill \\ \le 5(t_2-t_1)\left[M_1+(b_1+b_1^{*}{\zeta }^2)(t_2-t_1)\Vert y{\Vert }_C^2\right]\to 0 as t_2\to t_1.\hfill \end{array}$$

Also, applying condition H₄, and the continuity of F(t) give $$\begin{array}{c}5{\int }_{t_1}^{t_2}E\left\{{\left|g_1(t_2,s,y(s),A(s)y(s))\right|}^2\right\}dF(s)\hfill \\ \le 5{\beta }_1\left[1+(1+{\zeta }^2)\Vert y{\Vert }_C^2\right](F(t_2)-F(t_1))\to 0 as t_2\to t_1.\hfill \end{array}$$

Consequently, $\Vert T_{1}y(t_2)-T_{1}y(t_1){\Vert }_C\to 0$ as $t_2\to t_1,$ and hence the function $T_{1}y$ is continuous in the mean square sense on I. Using an arguments similar to the one used above, we can prove that $\Vert T_{2}x(t_2)-T_{2}x(t_1){\Vert }_C\to 0$ as $t_2\to t_1,$ that is $T_{2}x$ is also mean square continuous on I. Now we have $$\begin{array}{c}\Vert T(x,y)(t_2)-T(x,y)(t_1){\Vert }_X=\Vert (T_{1}y(t_2),T_{2}x(t_2))-(T_{1}y(t_1),T_{2}x(t_1)){\Vert }_X.\hfill \\ =\Vert (T_{1}y(t_2)-T_{1}y(t_1),T_{2}x(t_2)-T_{2}x(t_1)){\Vert }_X.\hfill \\ =\Vert T_{1}y(t_2)-T_{1}y(t_1){\Vert }_C+\Vert T_{2}x(t_2)-T_{2}x(t_1){\Vert }_C.\hfill \\ \to 0 as t_2\to t_1.\hfill \end{array}$$

So, T(x, y) is continuous in the mean square sense and thereby $T(x,y)\in X \forall (x,y)\in X.$ Therefore, the integral operator T maps the space X into itself. Let $(x_n,y_n)\to (x,y)$ in X almost surely, as $n\to \infty ,$ with ${\{(x_n,y_n)\}}_{n=1}^{\infty }\subseteq X.$ we need to prove $T(x_n,y_n)\to T(x,y)\in X,$ as $n\to \infty , \forall t\in I,$ where $T(x_n,y_n)(t)=(T_1{y}_n(t),T_2{x}_n(t)).$ That is, we will prove that $T_1{y}_n\to T_{1}y$ in C, and $T_2{x}_n\to T_{2}x$ in $C \forall t\in I.$ Using an arguments similar to the one used above yields $$\begin{array}{c}E\left\{|T_1{y}_n(t)-T_{1}y(t){|}^2\right\}\hfill \\ \le 2a{\int }_0^{t}E\{|f_1(t,s,y_n(s),B(s)y_n(s))-f_1(t,s,y(s),B(s)y(s)){|}^2\}ds\hfill \\ +2{\int }_0^{t}E\{|g_1(t,s,y_n(s),A(s)y_n(s))-g_1(t,s,y(s),A(s)y(s)){|}^2\}dF(s).\hfill \end{array}$$

From the continuity of the operators B(t), and A(t) it follows that $y_n\to y$ in C implies $B(t)y_n\to B(t)y,$ and $A(t)y_n\to A(t)y$ in $C \forall t\in I.$ Using conditions H₂, H₃, H₄, the Lebesgue dominated convergence theorem and let $n\to \infty$ yield $$\begin{array}{c}{\int }_0^{t}E\{|f_1(t,s,y_n(s),B(s)y_n(s))-f_1(t,s,y(s),B(s)y(s)){|}^2\}ds\to 0.\hfill \\ {\int }_0^{t}E\{|g_1(t,s,y_n(s),A(s)y_n(s))-g_1(t,s,y(s),A(s)y(s)){|}^2\}dF(s)\to 0.\hfill \end{array}$$

Therefore, $\Vert T_1{y}_n-T_{1}y{\Vert }_C\to 0$ as $\Vert y_n-y{\Vert }_C\to 0$ (that is, the operator T₁ is continuous on C). By the same way, we can prove that $\Vert T_2{x}_n-T_{2}x{\Vert }_C\to 0$ as $\Vert x_n-x{\Vert }_C\to 0,$ and thereby $\forall t\in I$ we have $$\begin{array}{c}\Vert T(x_n,y_n)-T(x,y){\Vert }_X=\Vert (T_1{y}_n,T_2{x}_n)-(T_{1}y,T_{2}x){\Vert }_X.\hfill \\ =\Vert (T_1{y}_n-T_{1}y,T_2{x}_n-T_{2}x){\Vert }_X.\hfill \\ =\Vert T_1{y}_n-T_{1}y{\Vert }_C+\Vert T_2{x}_n-T_{2}x{\Vert }_C.\hfill \\ \to 0, as n\to \infty .\hfill \end{array}$$

That is, the operator T is continuous on $X \forall t\in I.$ □

Remark 3.1.

In the sequel, let ${\sigma }_j:=3{\beta }_j[F(a)-F(0)],$ and ${\eta }_j:=3a^2[b_j+b_j^{*}{\zeta }^2], j=1, 2.$

Now, Let the sets $S_{r_1},S_{r_2},$ and S_r be defined as follows $$\begin{array}{c}{S}_{r_1}=\{y\in C:\Vert y{\Vert }_C\le r_1, r_1:=\sqrt{\frac{3\Vert h_1{\Vert }_C^2+3a{M}_1+{\sigma }_1}{1-[{\eta }_1+{\sigma }_1(1+{\zeta }^2)]}}, {\eta }_1+{\sigma }_1(1+{\zeta }^2)<1\}.\hfill \\ S_{r_2}=\{x\in C:\Vert x{\Vert }_C\le r_2, r_2:=\sqrt{\frac{3\Vert h_2{\Vert }_C^2+3a{M}_2+{\sigma }_2}{1-[{\eta }_2+{\sigma }_2(1+{\zeta }^2)]}}, {\eta }_2+{\sigma }_2(1+{\zeta }^2)<1\}.\hfill \\ S_r=\{u=(x,y)\in X:\Vert u{\Vert }_X\le r, r:=r_1+r_2\}.\hfill \end{array}$$

Theorem 3.1.

Suppose conditions $H_1-H_4$ are satisfied. Let the operator $T:S_r\to X$. Then the stochastic coupled system 1.3(1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) has at least one solution in S_r.

Proof.

it is clear that the set S_r is a closed bounded convex nonempty set contained in X. From Lemma 3.1, we deduce that $T:S_r\to X$ is a continuous operator. We suppose the sequence ${\{T(x_n,y_n)\}}_{n=1}^{\infty }$ whose elements are continuous functions in the set TS_r. Let $t_1\in I,t_2\in I,$ and assume, without loss of generality, that $t_2>t_1.$ Applying similar argument to those used in Lemma 3.1 yields $$\begin{array}{c}E\left\{|T_1{y}_n(t_2)-T_1{y}_n(t_1){|}^2\right\}\le 5E\left\{\right|h_1(t_2)-h_1(t_1){|}^2\}\hfill \\ +5a{\int }_0^{t_1}E\{|f_1(t_2,s,y_n(s),B(s)y_n(s))-f_1(t_1,s,y_n(s),B(s)y_n(s)){|}^2\}ds\hfill \\ +5{\int }_0^{t_1}E\{|g_1(t_2,s,y_n(s),A(s)y_n(s))-g_1(t_1,s,y_n(s),A(s)y_n(s)){|}^2\}dF(s)\hfill \\ +5(t_2-t_1){\int }_{t_1}^{t_2}E\left\{{|f_1(t_2,s,y_n(s),B(s)y_n(s))|}^2\right\}ds\hfill \\ +5{\int }_{t_1}^{t_2}E\left\{{|g_1(t_2,s,y_n(s),A(s)y_n(s))|}^2\right\}dF(s).\hfill \\ \to 0 as t_2\to t_1 \forall n\in \mathbb{N}.\hfill \end{array}$$

Therefore, the sequence ${\left\{T_1{y}_n\right\}}_{n=1}^{\infty }$ is a mean square equicontinuous. Applying similar arguments we can show that ${\left\{T_2{x}_n\right\}}_{n=1}^{\infty }$ is, also, a mean square equicontinuous, and thereby we have $$\begin{array}{c}E\left\{\right|T(x_n,y_n)(t_2)-T(x_n,y_n)(t_1){|}^2\}\hfill \\ =E\left\{\right|(T_1{y}_n(t_2),T_2{x}_n(t_2))-(T_1{y}_n(t_1),T_2{x}_n(t_1)){|}^2\}.\hfill \\ =E\left\{\right|(T_1{y}_n(t_2)-T_1{y}_n(t_1),T_2{x}_n(t_2)-T_2{x}_n(t_1)){|}^2\}.\hfill \\ =E\left\{\right|T_1{y}_n(t_2)-T_1{y}_n(t_1){|}^2\}+E\{|T_2{x}_n(t_2)-T_2{x}_n(t_1){|}^2\}.\hfill \\ \to 0 as t_2\to t_1 \forall n\in \mathbb{N}.\hfill \end{array}$$

So, the sequence ${\{T(x_n,y_n)\}}_{n=1}^{\infty }$ is a mean square equicontinuous in the set TS_r. It is easy to prove that the sequences ${\left\{T_1{y}_n\right\}}_{n=1}^{\infty },$ and ${\left\{T_2{x}_n\right\}}_{n=1}^{\infty }$ are uniformly bounded in the mean-square sense, see (Elborai & Youssef, 2019). So, the sequence ${\{T(x_n,y_n)\}}_{n=1}^{\infty }$ is also uniformly bounded in the mean-square sense. From the Arzela Ascoli theorem, we can find a convergent subsequence ${\{T(x_{n_k},y_{n_k})\}}_{k=1}^{\infty }$ in ${\{T(x_n,y_n)\}}_{n=1}^{\infty }$ whose convergence is uniformly in TS_r and thereby the set TS_r is compact. The previous discussion shows that the operator T is completely continuous. It is easy to show $T{S}_r\subset S_r$ because for every $y\in S_{r_1},$ we have $$\Vert T_{1}y{\Vert }_C^2\le 3\Vert h_1{\Vert }_C^2+3a{M}_1+{\sigma }_1+[{\eta }_1+{\sigma }_1(1+{\zeta }^2)]r_1^2.$$

Substituting $r_1:=\sqrt{\frac{3\Vert h_1{\Vert }_C^2+3a{M}_1+{\sigma }_1}{1-[{\eta }_1+{\sigma }_1(1+{\zeta }^2)]}}$ and simplifying yields $\Vert T_{1}y{\Vert }_C\le r_1$ for each $y\in S_{r_1}.$ Also, using similar arguments gives $\Vert T_{2}x{\Vert }_C\le r_2$ for each $x\in S_{r_2}.$ Now for every $(x,y)\in S_r,$ we have $$\Vert T(x,y){\Vert }_X=\Vert (T_{1}y,T_{2}x){\Vert }_X=\Vert T_{1}y{\Vert }_C+\Vert T_{2}x{\Vert }_C\le r_1+r_2=r.$$

So, for each $(x,y)\in S_r,$ we have $T(x,y)\in S_r,$ and therefore $T{S}_r\subset S_r.$ From the fixed point theorem due to Schauder there exists at least one fixed point for the operator T in the set S_r and hence the coupled system 1.3(1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) has at least one solution in S_r. □

Corollary 3.1.1.

Suppose conditions $H_1-H_4$ are satisfied. Consider the operator $T_1:S_{r_1}\to C$. Then the stochastic functional integral equation $$y(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds+{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).$$ has at least one solution in $S_{r_1}.$

Proof.

The proof follows directly from theorem 3.1 when $S_r=S_{r_1},$ X = C, T = T₁, h₁ = h₂, f₁ = f₂, and g₁ = g₂. □

Corollary 3.1.2.

Suppose conditions $H_1-H_3$ are satisfied. Let the operator $T:S_r\to X$. Then the following stochastic coupled system $$\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds.\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds.\hfill \end{array}$$ has at least one solution in S_r.

Proof.

The proof comes directly from theorem 3.1 when g_j = 0, and j = 1, 2. □

Corollary 3.1.3.

Suppose conditions H₁, H₂, and H₄ are satisfied. Let the operator $T:S_r\to X$. Then the following stochastic coupled system $$\begin{array}{c}x\left(t\right)=h_1\left(t\right)+{\int }_0^t{g}_1(t,s,y(s),A(s\left)y\right(s\left)\right)dW\left(s\right).\\ y\left(t\right)=h_2\left(t\right)+{\int }_0^t{g}_2(t,s,x(s),A(s\left)x\right(s\left)\right)dW\left(s\right).\end{array}$$ has at least one solution in S_r.

Proof.

The proof comes directly from theorem 3.1 when f_j = 0, and j = 1, 2. □

Theorem 3.2.

Suppose conditions H₁, H₃ and H₄ are satisfied. let the operator $T:S_r\to X$. Let the functions f_j and g_j,$j=1,2$, satisfy the following conditions:

H₅: There exists two real deterministic functions μ_j defined on I × I, such that $\forall 0\le s\le t\le a, x_j\in C,y_j\in C, j=1,2,$ we have $$\begin{array}{c}|f_j(t,s,x_2,y_2)-f_j(t,s,x_1,y_1)|\le {\mu }_j(t,s)\sqrt{(|x_2-x_1{|}^2+|y_2-y_1{|}^2)}.\hfill \\ {\int }_0^a|{\mu }_j(t,s){|}^{2}ds\le N_j, N_j>0, j=1,2.\hfill \end{array}$$

H₆: There exist two constants ${\rho }_j>0, j=1,2$ such that: $$|g_j(t,s,x_2,y_2)-g_j(t,s,x_1,y_1)|\le {\rho }_j\sqrt{(|x_2-x_1{|}^2+|y_2-y_1{|}^2)}.$$ $$\forall 0\le s\le t\le a, x_j\in C,y_j\in C, j=1,2.$$

Then the stochastic system 1.3 has a unique solution in S_r, provided that $$\begin{array}{c}0\le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)}<1.\hfill \\ \mathit{where}, \rho :=\max \{{\rho }_1,{\rho }_2\}, N:=\max \{N_1,N_2\}.\hfill \end{array}$$

Proof.

It is obvious the set S_r is a closed subspace of the Banach space X. So, it is also a Banach space. Using an argument similar to those in theorem 3.1 it is easy to show that, $T{S}_r\subset S_r.$ Therefore, it remains to show the contraction property of the operator T. Let (x, y) and $(x^{*},y^{*})$ belong to the set S_r. Using the Cauchy-Schwartz inequality, conditions H₅, H₆, and taking the supremum over $t\in I$ gives (3.3) $$\Vert T_1{y}^{*}-T_{1}y{\Vert }_C\le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)} \Vert y^{*}-y{\Vert }_C.$$(3.3) (3.4) $$\Vert T_2{x}^{*}-T_{2}x{\Vert }_C\le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)} \Vert x^{*}-x{\Vert }_C.$$(3.4)

Adding inequalities 3.3(3.3) $$\Vert T_1{y}^{*}-T_{1}y{\Vert }_C\le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)} \Vert y^{*}-y{\Vert }_C.$$(3.3) , 3.4(3.4) $$\Vert T_2{x}^{*}-T_{2}x{\Vert }_C\le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)} \Vert x^{*}-x{\Vert }_C.$$(3.4) and using the definition of the norm in X gives $$\begin{array}{c}\Vert (T_1{y}^{*}-T_{1}y,T_2{x}^{*}-T_{2}x){\Vert }_X\hfill \\ \le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)} \Vert (x^{*}-x,y^{*}-y){\Vert }_X.\hfill \end{array}$$ $$\begin{array}{c}\Vert T(x^{*},y^{*})-T(x,y){\Vert }_X\hfill \\ \le \sqrt{\left[2aN+2{\rho }^2(F(a)-F(0))\right](1+{\zeta }^2)} \Vert (x^{*},y^{*})-(x,y){\Vert }_X.\hfill \end{array}$$

So, T is a contraction operator, and hence has a unique fixed point in S_r from applying the Banach fixed point theorem. Therefore, the stochastic coupled system 1.3(1.3) $$\{\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds\hfill \\ +{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds\hfill \\ +{\int }_0^t{g}_2(t,s,x(s),A(s)x(s))dW(s).\hfill \end{array}$$(1.3) has a unique solution in S_r. □

Corollary 3.2.1.

Suppose conditions H₁, and $H_3-H_6$ are satisfied. Define an operator $T_1:S_{r_1}\to C$. Then the stochastic functional integral equation $$y(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds+{\int }_0^t{g}_1(t,s,y(s),A(s)y(s))dW(s).$$ has a unique solution in $S_{r_1}$. provided that $$\begin{array}{c}0\le \sqrt{\left[2a{N}_1+2{\rho }_1^2(F(a)-F(0))\right](1+{\zeta }^2)}<1.\hfill \end{array}$$

Proof.

The proof follows directly from theorem 3.2 when $S_r=S_{r_1},$ X = C, T = T₁, h₁ = h₂, f₁ = f₂, and g₁ = g₂. □

Corollary 3.2.2.

Suppose conditions H₁, H₃, and H₅ are satisfied. Assume the operator $T:S_r\to X$. Then the following stochastic coupled system $$\begin{array}{c}x(t)=h_1(t)+{\int }_0^t{f}_1(t,s,y(s),B(s)y(s))ds.\hfill \\ y(t)=h_2(t)+{\int }_0^t{f}_2(t,s,x(s),B(s)x(s))ds.\hfill \end{array}$$ has a unique solution in S_r. Provided that $$\begin{array}{c}0\le \sqrt{2aN(1+{\zeta }^2)}<1, \mathit{where}, N:=\max \{N_1,N_2\}.\hfill \end{array}$$

Proof.

The proof comes directly from theorem 3.2 when $g_j=0, j=1,2.$ □

Corollary 3.2.3.

Suppose conditions H₁, H₄, and H₆ are satisfied. Let the operator $T:S_r\to X$. Then the following stochastic coupled system $$\begin{array}{c}x\left(t\right)=h_1\left(t\right)+{\int }_0^t{g}_1(t,s,y(s),A(s\left)y\right(s\left)\right)dW\left(s\right).\\ y\left(t\right)=h_2\left(t\right)+{\int }_0^t{g}_2(t,s,x(s),A(s\left)x\right(s\left)\right)dW\left(s\right).\end{array}$$ has a unique solution in S_r. Provided that $$\begin{array}{c}0\le \sqrt{2{\rho }^2(F(a)-F(0))(1+{\zeta }^2)}<1, \mathit{where}, \rho :=\max \{{\rho }_1,{\rho }_2\}.\hfill \end{array}$$

Proof.

The proof comes directly from theorem 3.2 when $f_j=0, j=1,2.$ □

4. Conclusion

We have derived some new results on the existence, and uniqueness of continuous solution of a general class of stochastic coupled systems of the Urysohn–Volterra–Itô–Doob type. The main techniques which we used are the fixed point principle, the Carathèodory conditions, and the stochastic analysis due to Itô and Doob. We believe that our results are important in their own right. Our results offer a generalization to the results developed by A.M.A. El-Sayed et al. in (El-Sayed & Kenawy, 2014a, 2014b). In the future, many models can be investigated, such as fractal integral systems (Volterra or Fredholm) with more concentration on the singular types (He, 2020).

Author’s contributions

M. I. Youssef conceived this research, wrote the paper, and revised it.

https://orcid.org/0000-0002-6943-1197

twitter: @MIYoussef283

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Acknowledgments

We would like to thank the anonymous referees for their valuable comments which improved the current work so much.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Availability of data and material

All data generated or analyzed during this study are included in the article.