**1. Introduction**

Let (Ω, -, *P*) be a fixed probability space, where Ω is a sample space, is a *σ*-algebra and *P* is a probability measure.

The aim of this article is to extend the results of A.M.A. El-Sayed [1,2] on the stochastic fractional calculus operators defined on *C*([0, *T*], *L*2(Ω)) and the solution of stochastic differential equations subject to nonlocal integral conditions which have been considered in [3,4].

Moreover, we motivate the coupled system of integral equations in reflexive Banach space by A.M.A. El-Sayed and H.H.G.Hashem [5] to the coupled systems with random memory on the space of all second order stochastic process.

The continuous dependence of a unique solution has been studied on the random initial data and the random function which ensures the stability of the solution.

Nonlocal problem of differential equation have been studied by many authors (see for example [6–8]).

Let *Z*(*t*; *ω*) = *Z*(*t*), *t* ∈ [0, *T*], *ω* ∈ Ω be a second order stochastic process, i.e., *<sup>E</sup>*(*Z*2(*t*)) < <sup>∞</sup>, *<sup>t</sup>* ∈ [0, *<sup>T</sup>*].

Let *C* = *C*([0, *T*], *L*2(Ω)) be the space of all second order stochastic processes which is mean square (m.s) continuous on [0, *T*]. The norm of *Z* ∈ *C*([0, *T*], *L*2(Ω)) is given by

$$\|Z\|\_{\mathbb{C}} = \sup\_{t \in [0,T]} \|Z(t)\|\_{2\prime} \qquad\qquad \qquad \|Z(t)\|\_{2} = \sqrt{E(Z^2(t))}.$$

Let *T* ≥ 1. In this paper we study the existence of solutions (*x*, *y*) ∈ *C*([0, *T*], *L*2(Ω)) of the problem of the coupled system of random and stochastic differential equations

**Citation:** El-Sayed, A.M.A.; Fouad, H.A. On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions. *Mathematics* **2021**, *9*, 2111. https:// doi.org/10.3390/math9172111

Academic Editor: Alexandra Kashchenko

Received: 3 July 2021 Accepted: 27 August 2021 Published: 1 September 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

1

Faculty of Science, Alexandria University, Alexandria 21568, Egypt

$$\frac{dx(t)}{dt} \quad = \quad f\_1(t, y(\phi\_1(t))), \qquad t \in (0, T]. \tag{1}$$

$$dy(t) \quad = \quad f\_2(t, x(\phi\_2(t)))dW(t), \quad t \in (0, T] \tag{2}$$

subject to each one of the two nonlinear nonlocal stochastic integral conditions

$$\mathbf{x}(0) + \int\_0^\tau h\_1(s, \mathbf{y}(s)) \, dW(s) = \mathbf{x}\_{0\prime} \qquad \mathbf{y}(0) + \int\_0^\eta h\_2(s, \mathbf{x}(s)) \, ds = \mathbf{y}\_0 \tag{3}$$

and

$$\mathbf{x}(0) + \int\_0^\tau h\_1(\mathbf{s}, \mathbf{x}(\mathbf{s})) d\mathcal{W}(\mathbf{s}) = \mathbf{x}\_0, \qquad \mathbf{y}(0) + \int\_0^\eta h\_2(\mathbf{s}, \mathbf{y}(\mathbf{s})) d\mathbf{s} = \mathbf{y}\_0 \tag{4}$$

where *x*<sup>0</sup> and *y*<sup>0</sup> are two second order random variables.

Let *X* = *C*([0, *T*], *L*2(Ω)) × *C*([0, *T*], *L*2(Ω)) be the class of all ordered pairs (*x*, *y*), *x*, *y* ∈ *C* with the norm

$$\|\|(\mathbf{x}, y)\|\|\_{X} = \max\{\|\mathbf{x}\|\_{\mathbb{C}^\prime} \|y\|\_{\mathbb{C}}\} = \max\{\sup\_{t \in [0, T]} \|\mathbf{x}(t)\|\_{2\prime} \sup\_{t \in [0, T]} \|y(t)\|\_{2}\}.\tag{5}$$

Let *φ<sup>i</sup>* : [0, *T*] → [0, *T*] be continuous functions such that *φi*(*t*) ≤ *t* and consider the following assumptions

**Assumption 1.** *fi* : [0, *T*] × *L*2(Ω) → *L*2(Ω), *i* = 1, 2 *are measurable in t* ∈ [0, *T*] *for all x* ∈ *L*2(Ω) *and continuous in x* ∈ *L*2(Ω) *for all t* ∈ [0, *T*]. *There exist two bounded measurable functions mi* : [0, *T*] → *R and two positive constants bi such that*

$$\|f\_i(t, \mathbf{x})\|\_2 \le |m\_i(t)| + b\_i \|\mathbf{x}(t)\|\_2, \quad i = 1, 2. \tag{6}$$

**Assumption 2.** *hi* : [0, *T*] × *L*2(Ω) → *L*2(Ω), *i* = 1, 2 *are measurable in t* ∈ [0, *T*] *for all x* ∈ *L*2(Ω) *and continuous in x* ∈ *L*2(Ω) *for all t* ∈ [0, *T*]. *There exist two bounded measurable functions ki* : [0, *T*] → *R and two positive constants ci such that*

$$||h\_i(t, \mathbf{x})||\_2 \le |k\_i(t)| + c\_i ||\mathbf{x}(t)||\_{2\prime} \quad i = 1, 2. \tag{7}$$

**Assumption 3.** *<sup>M</sup>* <sup>=</sup> max{sup*t*∈[0,*T*] <sup>|</sup>*m*1(*t*)|, sup*t*∈[0,*T*] <sup>|</sup>*m*2(*t*)|}, *<sup>b</sup>* <sup>=</sup> max{*b*1, *<sup>b</sup>*2}.

**Assumption 4.** *<sup>K</sup>* <sup>=</sup> max{sup*t*∈[0,*T*] <sup>|</sup>*k*1(*t*)|, sup*t*∈[0,*T*] <sup>|</sup>*k*2(*t*)|}, *<sup>c</sup>* <sup>=</sup> max{*c*1, *<sup>c</sup>*2}.

**Assumption 5.** (*b* + *c*)*T* < 1.

Now, integrating the two random and stochastic differential Equations (1) and (2) (see [1,2,9–14]) and using the nonlocal conditions (3) and (4) the following Lemma can be proven.

**Lemma 1.** *The integral representations of the solutions of the nonlocal problems (1) and (2) with conditions (3) and (1) and (2) with conditions (4) are given by*

$$\mathbf{x}(t) \quad = \quad \mathbf{x}\_0 - \int\_0^\tau h\_1(s, \mathbf{y}(s)) d\mathcal{W}(s) + \int\_0^t f\_1(s, \mathbf{y}(\phi\_1(s))) ds,\tag{8}$$

$$y(t) = -y\_0 - \int\_0^{\eta} h\_2(s, \mathbf{x}(s)) ds + \int\_0^t f\_2(s, \mathbf{x}(\phi\_2(s))) dW(s). \tag{9}$$

*and*

$$\mathbf{x}(t) = \mathbf{x}\_0 - \int\_0^\tau h\_1(s, \mathbf{x}(s)) d\mathcal{W}(s) + \int\_0^t f\_1(s, y(\phi\_1(s))) ds,\tag{10}$$

$$y(t) = -y\_0 - \int\_0^{\eta} h\_2(s, y(s)) ds + \int\_0^t f\_2(s, \mathfrak{x}(\phi\_2(s))) dW(s). \tag{11}$$

*respectively.*
