On a Neutral Itô and Arbitrary (Fractional) Orders Stochastic Differential Equation with Nonlocal Condition

Abstract

In this paper, we are concerned with the combinations of the stochastic Itô-differential and the arbitrary (fractional) orders derivatives in a neutral differential equation with a stochastic, nonlinear, nonlocal integral condition. The existence of solutions will be proved. The sufficient conditions for the uniqueness of the solution will be given. The continuous dependence of the unique solution will be studied.

1. Introduction

The existence and uniqueness of solutions to stochastic differential equations driven by the Winner Processes have been studied by many authors (see [1,2,3,4,5] ).

In addition, the stochastic differential equations with nonlocal conditions and of fractional orders have been studied by some authors (see, for example, [6,7,8] and references therein).

The results are important since they cover nonlocal generalizations of differential SDEs, and more applications are arising in fields such as heat conduction, electromagnetic theory and dynamical systems and in materials with memory (see, e.g., [9,10,11,12,13,14,15,16,17]), optimal fractional problems and numerical models (see, e.g., [18,19,20,21]).

P. Balasubramaniam et al. [22] obtained sufficient conditions for the existence of a mild solution of the considered system by using analytic resolvent operators, the uniform continuity of the resolvent and Schauder fixed point theorem.

Here we study the existence of solutions of an It$\widehat{o}$ and arbitrary (fractional) orders stochastic nonlinear differential equation with nonlocal integral conditions containing the involved Caputo fractional order derivative. We combined two different senses of derivatives and stated the conditions for the existence of at least one solution.

Let $\left(W\right(t\left)\right),t\ge 0$ be a standard Brownian motion on a complete probability space $(\Omega ,ϝ,\mu ),$ where $\Omega$ is a sample space, $ϝ$ is a $\sigma -$algebra and $\mu$ is a probability measure.

Let $x(t;\omega )=x\left(t\right),t\in [0,T],\omega \in \Omega$ be a second order stochastic process, i.e., $E\left(x^2\left(t\right)\right)<+\infty ,t\in [0,T].$

Let $C=C([0,T],L_2(\Omega ))$ be the space of all second order mean square (m.s) continuous stochastic processes on $[0,T]$. The norm of $x\in C([0,T],L_2(\Omega ))$ is given by

$${\parallel x\parallel }_C=\underset{t\in [0,T]}{sup}{\parallel x\left(t\right)\parallel }_2,{\parallel x\left(t\right)\parallel }_2=\sqrt{E\left(x^2\left(t\right)\right)}.$$

Let $\alpha ,\beta \in (0,1]$ be such that $\alpha \ge \beta$ and $I=[0,T].$ In this paper we study the existence of solutions $x\in C(I,L_2(\Omega ))$ of the It$\widehat{o}—$arbitrary (fractional) orders stochastic nonlinear differential equation

$$d(\frac{dx\left(t\right)}{dt}-g(t,x\left(\varphi \left(t\right)\right))=f(t,D^{\alpha }x\left(t\right)))dW\left(t\right),t\in I$$

(1)

subject to

$$\begin{array}{ccc}\hfill \frac{dx}{dt}{\mid }_{t=0}& =& g(0,x(\varphi \left(0\right)\left)\right),\hfill \\ \hfill x\left(0\right)& +& {\int }_0^{\tau }h(s,D^{\beta }x\left(s\right))dW\left(s\right)=x_0\hfill \end{array}$$

(2)

where $x_0$ is a second-order random variable.

The existence of solutions $x\in C(I,L_2(\Omega ))$ of the problem (1)–(2) is proved. The sufficient conditions for the uniqueness of the solution will be given. The continuous dependence of the solution on the random variable $x_0$ and the random function h will be studied.

The definitions of arbitrary (fractional) integral and derivatives have been studied by [23].

Definition 1.

Let $x\in C(I,L_2(\Omega ))$ and $\alpha ,\beta \in (0,1].$ The stochastic fractional order integral $I^{\beta }x\left(t\right)$ is defined by

$$I^{\beta }x\left(t\right)={\int }_0^t\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}x\left(s\right)ds,$$

and the stochastic fractional order derivative is defined by

$$D^{\alpha }x\left(t\right)=I^{1-\alpha }\frac{dx}{dt}.$$

For the properties to stochastic fractional calculus, see [23].

Consider the following assumptions

(B1)

$\varphi :[0,T]\to [0,T]$ is a continuous function such that $\varphi \left(t\right)\le t.$

(B2)

$f:I\times L_2(\Omega )\to L_2(\Omega )$ is measurable in $t\in I$ for all $x\in L_2(\Omega )$ and continuous in $x\in L_2(\Omega )$ for all $t\in I$ and there exists a bounded measurable function $k:I\to R$ and a positive constant $b$ such that

$${\parallel f(t,x)\parallel }_2\le \left|k\left(t\right)\right|+{b\parallel x\left(t\right)\parallel }_2.$$

(3)

(B3)

$h:I\times L_2(\Omega )\to L_2(\Omega )$ is measurable in $t\in I$ for all $x\in L_2(\Omega )$ and continuous in $x\in L_2(\Omega )$ for all $t\in I$ and there exists a bounded measurable function $m:I\to R$ and a positive constant $c$ such that

$${\parallel h(t,x)\parallel }_2\le \left|m\left(t\right)\right|+{c\parallel x\left(t\right)\parallel }_2.$$

(4)

(B4)

$g:I\times L_2(\Omega )\to L_2(\Omega )$ is measurable in $t\in I$ for all $x\in L_2(\Omega )$ and continuous in $x\in L_2(\Omega )$ for all $t\in I$ and there exists a bounded measurable function $l:I\to R$ and a positive constant $q$ such that

$${\parallel g(t,x)\parallel }_2\le \left|l\left(t\right)\right|+{q\parallel x\left(t\right)\parallel }_2.$$

(5)

(B5)

$K={sup}_{t\in I}\left|k\left(t\right)\right|,M={sup}_{t\in I}\left|m\left(t\right)\right|,L={sup}_{t\in I}\left|l\left(t\right)\right|.$

(B6)

$$\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}\le 1.$$

2. Integral Representations of the Solution

Integrating Equation (1) we obtain ([24]),

$$\frac{dx\left(t\right)}{dt}=g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^{t}f(t,D^{\alpha }x\left(t\right)))dW\left(t\right),$$

then operating by $I^{1-\alpha }$, we obtain

$$D^{\alpha }x\left(t\right)=I^{1-\alpha }\frac{dx}{dt}=I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+I^{1-\alpha }{\int }_0^{t}f(s,D^{\alpha }x\left(s\right))dW\left(s\right).$$

Let

$$y\left(t\right)=D^{\alpha }x\left(t\right),$$

(6)

then we deduce that

$$\begin{array}{ccc}\hfill y\left(t\right)& =& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+I^{1-\alpha }{\int }_0^{t}f(s,y\left(s\right))dW\left(s\right)\hfill \\ & =& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{-\alpha }}{\Gamma (1-\alpha )}{\int }_0^{s}f(\theta ,y\left(\theta \right))dW\left(\theta \right)ds\hfill \\ & =& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{-\alpha }}{\Gamma (1-\alpha )}{\int }_0^s(f(\theta ,y\left(\theta \right)).\frac{dW}{d\theta })d\theta .ds\hfill \\ & =& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y\left(s\right))dW\left(s\right).\hfill \end{array}$$

(7)

Let $\alpha \ge \beta ,$ then $D^{\beta }x\left(t\right)=I^{\alpha -\beta }y\left(t\right).$ Now, integrating (6) we can obtain

$$x\left(t\right)=x_0-{\int }_0^{\tau }h(s,I^{\alpha -\beta }y\left(s\right))dW\left(s\right)+I^{\alpha }y\left(t\right).$$

(8)

Then the following lemma is proved.

Lemma 1.

The integral representations corresponding to the solution of the nonlocal problem (1)–(2) is given by

$$\begin{array}{ccc}\hfill x\left(t\right)& =& x_0-{\int }_0^{\tau }h(s,I^{\alpha -\beta }y\left(s\right))dW\left(s\right)+I^{\alpha }y\left(t\right),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill y\left(t\right)& =& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y\left(s\right))dW\left(s\right).\hfill \end{array}$$

(10)

3. Existence Theorem

Let $\Lambda =C(I,L_2(\Omega ))\times C(I,L_2(\Omega ))$ be the class of all ordered pairs $(x,y),x,y\in C$ with the norm

$${\parallel (x,y)\parallel }_{\Lambda }={\parallel x\parallel }_C+{\parallel y\parallel }_C=\underset{t\in I}{sup}{\parallel x\left(t\right)\parallel }_2+\underset{t\in I}{sup}{\parallel y\left(t\right)\parallel }_2.$$

(11)

Define the mapping $F(x,y)=(F_{1}y,F_2(x,y))$ where $F_{1}y,F_2(x,y)$ are given by the following stochastic integral equations

$$\begin{array}{ccc}\hfill F_{1}y\left(t\right)& =& x_0-{\int }_0^{\tau }h(s,I^{\alpha -\beta }y\left(s\right))dW\left(s\right)+I^{\alpha }y\left(t\right),\hfill \\ \hfill F_2(x,y)\left(t\right)& =& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y\left(s\right))dW\left(s\right).\hfill \end{array}$$

Consider the set Q such that

$$Q_{=}\{x,y\in C(I,L_2(\Omega )),(x,y)\in {\Lambda :\parallel (x,y)\parallel }_{\Lambda }={\{\parallel x\left(t\right)\parallel }_2+{\parallel y\left(t\right)\parallel }_2\}\le r\}.$$

Now, we have the following existence theorem

Theorem 1.

Let the assumptions (B1)–(B6) be satisfied, then there exists at least one solution $(x,y)\in \Lambda$ of the problem (9) and (10).

Proof. 

Let $(x,y)\in Q,$ then we have

$$\begin{array}{ccc}\hfill \parallel F_1{y\left(t\right)\parallel }_2& \le & \parallel x_0{\parallel }_2+\parallel {\int }_0^{\eta }h(s,I^{\alpha -\beta }y\left(s\right)){dW\left(s\right)\parallel }_2+{\parallel I^{\alpha }y\left(t\right)\parallel }_2\hfill \\ & \le & \parallel x_0{\parallel }_2+M\sqrt{T}+[\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel y\parallel }_C\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel F_2{(x,y)\left(t\right)\parallel }_2& \le & \parallel I^{1-\alpha }{g(t,x\left(\varphi \left(t\right)\right)))\parallel }_2+{\parallel {\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y\left(s\right))dW\left(s\right)\parallel }_2\hfill \\ & \le & \frac{T^{1-\alpha }}{\Gamma (2-\alpha )}{[L+q\parallel x\parallel }_C]\hfill \\ & +& \sqrt{{\int }_0^t{\parallel \frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y\left(s\right))\parallel }_2^{2}ds}\hfill \\ & \le & \frac{L{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{K{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }{\parallel x\parallel }_C}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }{\parallel y\parallel }_C}{\Gamma (2-\alpha )}.\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill {\parallel F(x,y)\parallel }_{\Lambda }& =& \parallel (F_{1}y,F_2(x,y)){\parallel }_{\Lambda }\hfill \\ & =& \parallel F_1{y\parallel }_C+{\parallel F_{2}x\parallel }_C\hfill \\ & \le & \parallel x_0{\parallel }_2+M\sqrt{T}+\frac{L{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{K{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}\hfill \\ & +& \frac{q{T}^{1-\alpha }{\parallel x\parallel }_C}{\Gamma (2-\alpha )}+[\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel y\parallel }_C\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \parallel x_0{\parallel }_2+M\sqrt{T}+\frac{L{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{K{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}\hfill \\ & +& [\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{[\parallel x\parallel }_C+{\parallel y\parallel }_C]\hfill \\ & \le & \parallel x_0{\parallel }_2+M\sqrt{T}+\frac{L{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{K{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}\hfill \\ & +& [\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]r=r\hfill \end{array}$$

where

$$r=\frac{\parallel x_0{\parallel }_2+M\sqrt{T}+\frac{L{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{K{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}}{1-[\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]},$$

then the class $\left\{F\right(x,y\left)\right\}$ is uniformly bounded and $F(x,y):Q\to Q.$

Let $(x,y)\in Q,t_1,t_2\in [0,T],t_1<t_2$ such that $|t_2-t_1|<\delta ,$ then

$$\begin{array}{ccc}\hfill F_{1}y\left(t_2\right)-F_{1}y\left(t_1\right)& =& {\int }_0^{t_2}\frac{{(t_2-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}y\left(s\right)ds-{\int }_0^{t_1}\frac{{(t_1-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}y\left(s\right)ds\hfill \\ & =& \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{t_1}[{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}y\left(s\right)ds+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{t_1}^{t_2}{(t_2-s)}^{\alpha -1}y\left(s\right)ds\hfill \end{array}$$

this implies that

$$\begin{array}{ccc}\hfill \parallel F_{1}y\left(t_2\right)-F_{1}y\left(t_1\right){\parallel }_2& \le & \frac{r}{\Gamma \left(\alpha \right)}[{\int }_0^{t_2}\frac{{(t_2-s)}^{1-\alpha }-{(t_1-s)}^{1-\alpha }}{{(t_2-s)}^{1-\alpha }{(t_1-s)}^{1-\alpha }}ds\hfill \\ & +& {\int }_{t_1}^{t_2}{(t_2-s)}^{\alpha -1}ds].\hfill \end{array}$$

(12)

In a similar way,

$$\begin{array}{ccc}\hfill \parallel F_2(x,y)\left(t_2\right)-F_2(x,y)\left(t_1\right){\parallel }_2& \le & \frac{L+qr}{\Gamma (1-\alpha )}[{\int }_0^{t_1}\frac{{(t_2-s)}^{\alpha }-{(t_1-s)}^{\alpha }}{{(t_2-s)}^{\alpha }{(t_1-s)}^{\alpha }}ds+{\int }_{t_1}^{t_2}{(t_2-s)}^{-\alpha }ds\}]\hfill \\ & +& \frac{K+br}{\Gamma (2-\alpha )}[\sqrt{{\int }_0^{t_1}{|{(t_2-s)}^{1-\alpha }-{(t_1-s)}^{1-\alpha }|}^{2}ds}\hfill \\ & +& \sqrt{{\int }_{t_1}^{t_2}{(t_2-s)}^{2-2\alpha }ds}].\hfill \end{array}$$

(13)

However,

$$\begin{array}{ccc}\hfill F(x\left(t_2\right),y\left(t_2\right))-F(x\left(t_1\right),y\left(t_1\right))& =& (F_{1}y\left(t_2\right),F_2(x,y)\left(t_2\right))-(F_{1}y\left(t_1\right),F_2(x,y)\left(t_1\right))\hfill \\ & =& \left(\right(F_{1}y\left(t_2\right)-F_{1}y\left(t_1\right),(F_2(x,y)\left(t_2\right)-F_2(x,y)\left(t_1\right)),\hfill \end{array}$$

then from (12) and (13), we deduce the equicontinuity of the class $\left\{F\right(x,y\left)\right(t\left)\right\}$ on $Q$.

Consider $(x_n,y_n)\in Q$ such that ${L.i.m}_{n\to \infty }(x_n,y_n)=(x,y)w.p.1$ where $L.i.m$ denotes the limit in the mean square sense of the continuous second-order process ([23,25,26]) then by the Arzela–Ascoli Theorem [25], the closure of $FQ$ is a compact subset of $\Lambda .$

Now applying stochastic Lebesgue dominated convergence Theorem [25], we can obtain

$$\begin{array}{ccc}\hfill {L.i.m}_{n\to \infty }F(x_n,y_n)& =& ({L.i.m}_{n\to \infty }{F}_1{y}_n,{L.i.m}_{n\to \infty }{F}_2(x_n,y_n))\hfill \\ & =& ({L.i.m}_{n\to \infty }\{x_0-{\int }_0^{\tau }{h}_1(s,I^{\alpha -\beta }{y}_n\left(s\right))dW\left(s\right)+I^{\alpha }{y}_n\left(t\right)\},\hfill \\ & & {L.i.m}_{n\to \infty }\{I^{1-\alpha }g(t,x_n\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{-\alpha }}{\Gamma (2-\alpha )}f(s,y_n\left(s\right))dW\left(s\right)\})\hfill \\ & =& (F_{1}y,F_{2}x)=F(x,y).\hfill \end{array}$$

This proves that the operator $F:Q\to Q$ is continuous, and then applying the Schauder Fixed Point Theorem [25], there exists at least one solution $(x,y)\in \Lambda$ of the problem (9) and (10) $x,y\in C(I,L_2(\Omega )).$ □

4. Uniqueness of the Solution

Consider the assumptions

(B2${}^{*}$)

$f:I\times L_2(\Omega )\to L_2(\Omega )$ is measurable in $t\in I$ for all $x\in L_2(\Omega )$ and satisfies the Lipschitz condition

$${\parallel f(t,u\left(t\right))-f(t,v\left(t\right))\parallel }_2\le b{\parallel u\left(t\right)-v\left(t\right)\parallel }_2.$$

(14)

(B3${}^{*}$)

$h:I\times L_2(\Omega )\to L_2(\Omega )$ is measurable in $t\in I$ for all $x\in L_2(\Omega )$ and satisfies the Lipschitz condition

$${\parallel h(t,u\left(t\right))-h(t,v\left(t\right))\parallel }_2\le c{\parallel u\left(t\right)-v\left(t\right)\parallel }_2.$$

(15)

(B4${}^{*}$)

$g:I\times L_2(\Omega )\to L_2(\Omega )$ is measurable in $t\in I$ for all $x\in L_2(\Omega )$ and satisfies the Lipschitz condition

$${\parallel g(t,u\left(t\right))-g(t,v\left(t\right))\parallel }_2\le q{\parallel u\left(t\right)-v\left(t\right)\parallel }_2.$$

(16)

Theorem 2.

Let the assumptions (B2${}^{*}$)–(B4${}^{*}$) and (B5) and (B6) be satisfied, and then the solution of problem (9) and (10) is unique.

Proof. 

Let $(x_1,y_1)$ and $(x_2,y_2)$ be two solutions of the problem (9) and (10) on the form

$$\begin{array}{ccc}\hfill (x\left(t\right),y\left(t\right))=(x_0& -& {\int }_0^{\tau }h(s,I^{\alpha -\beta }y\left(s\right))dW\left(s\right)+I^{\alpha }y\left(t\right),\hfill \\ \hfill y_0& -& I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y\left(s\right))dW\left(s\right)),\hfill \end{array}$$

then we can obtain

$$\begin{array}{ccc}\hfill \parallel x_1\left(t\right)-x_2{\left(t\right)\parallel }_2& \le & \parallel {\int }_0^{\tau }[h(s,I^{\alpha -\beta }{y}_2\left(s\right))-h(s,I^{\alpha -\beta }{y}_1\left(s\right))]{dW\left(s\right)\parallel }_2+{\parallel I^{\alpha }{y}_1\left(t\right)-I^{\alpha }{y}_2\left(t\right)\parallel }_2\hfill \\ & \le & [\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}\parallel y_1-y_2{\parallel }_C.\hfill \end{array}$$

Similarly, we can obtain

$$\begin{array}{ccc}\hfill \parallel y_1\left(t\right)-y_2{\left(t\right)\parallel }_2& \le & \parallel I^{1-\alpha }g(t,x_1\left(\varphi \left(t\right)\right))-I^{1-\alpha }g(t,x_2\left(\varphi \left(t\right)\right)){\parallel }_2\hfill \\ & +& \parallel {\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}[f(s,y_1\left(s\right))-f(s,y_2\left(s\right))]dW\left(s\right){)\parallel }_2\hfill \\ & \le & \frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}\parallel x_1-x_2{\parallel }_C+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}{\parallel y_1-y_2\parallel }_C.\hfill \end{array}$$

However,

$$\begin{array}{ccc}\hfill \parallel (x_1,y_1)-(x_2,y_2){\parallel }_{\Lambda }& =& \parallel (x_1-x_2){\parallel }_C+{\parallel (y_1-y_2)\parallel }_C\hfill \\ & \le & \frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}\parallel x_1-x_2{\parallel }_C+[\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel y_1-y_2\parallel }_C\hfill \\ & \le & [\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}][\parallel x_1-x_2{\parallel }_C+\parallel y_1-y_2{\parallel }_C]\hfill \\ & \le & [\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel (x_1,y_1)-(x_2,y_2)\parallel }_{\Lambda }.\hfill \end{array}$$

This implies that

$$(1-[\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]){\parallel (x_1,y_1)-(x_2,y_2)\parallel }_{\Lambda }\le 0.$$

Then

$$\parallel (x_1,y_1)-(x_2,y_2){\parallel }_{\Lambda }=0$$

and $(x_1,y_1)=(x_2,y_2)$ which proves that the solution of the problem (9) and (10) is unique. □

5. Continuous Dependence

We shall present the main definitions related to the concepts of continuous dependence of the solution [27].

Definition 2.

The solution $(x,y)\in \Lambda$ of the nonlocal problem (9) and (10) is continuously dependent (on the random initial value $x_0$) if for all $ϵ>0,$ there exists ${\delta }_1>0$ such that $\Vert x_0-x_0^{*}{\Vert }_2\le {\delta }_1$ implies that $\Vert x-x^{*}{\Vert }_C\le ϵ.$

For this, we have the following theorem.

Theorem 3.

The unique solution of the system (9) and (10) depends continuously on the random data $x_0.$

Proof. 

Let $(x^{*},y^{*})$ be the solution of the system

$$\begin{array}{ccc}\hfill x^{*}\left(t\right)& =& x_0^{*}-{\int }_0^{\tau }h(s,I^{\alpha -\beta }{y}^{*}\left(s\right))dW\left(s\right)+I^{\alpha }{y}^{*}\left(t\right)\hfill \\ \hfill y^{*}\left(t\right)& =& I^{1-\alpha }g(t,x^{*}\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y^{*}\left(s\right))dW\left(s\right),\hfill \end{array}$$

such that $\parallel x_0-x_0^{*}{\parallel }_2<{\delta }_1$. Then we have

$$\begin{array}{ccc}\hfill \parallel x\left(t\right)-x^{*}{\left(t\right)\parallel }_2& \le & \parallel x_0-x_0^{*}{\parallel }_2+\parallel {\int }_0^{\tau }[h(s,I^{\alpha -\beta }y\left(s\right))-h(s,I^{\alpha -\beta }{y}^{*}\left(s\right))]{dW\left(s\right)\parallel }_2+{\parallel I^{\alpha }y\left(t\right)-I^{\alpha }{y}^{*}\left(t\right)\parallel }_2\hfill \\ & \le & {\delta }_1+[\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel y_1-y_2\parallel }_C\hfill \end{array}$$

and

$$\parallel y-y^{*}{\parallel }_2\le \frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}\parallel x-x^{*}{\parallel }_C+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}{\parallel y-y^{*}\parallel }_C$$

and we can obtain that

$$\begin{array}{ccc}\hfill \parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }& =& \parallel x-x^{*}{\parallel }_C+{\parallel y-y^{*}\parallel }_C\hfill \\ & \le & {\delta }_1+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}{\parallel x-x^{*}\parallel }_C\hfill \\ & +& [\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}\parallel y-y^{*}{\parallel }_C\hfill \\ & \le & {\delta }_1+[\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel (x,y)-(x^{*},y^{*})\parallel }_{\Lambda }\hfill \end{array}$$

which gives our result

$$\parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }\le \frac{{\delta }_1}{(1-[\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}])}=ϵ.$$

□

Definition 3.

The solution $(x,y)\in \Lambda$ of the nonlocal problem (9) and (10) is continuously dependent (on the random function $h$) if for all $ϵ>0,$ there exists ${\delta }_2>0$ such that $\parallel h^{*}{(t,u\left(t\right))-h(t,u\left(t\right))\parallel }_2\le {\delta }_2$ implies that $\Vert x-x^{*}{\Vert }_C\le ϵ.$

Theorem 4.

The unique solution of the system (9) and (10) depends continuously on the random functions h.

Proof. 

Let $(x^{*},y^{*})$ be the solution of the system of stochastic integral Equations (9) and (10) such that

$$\begin{array}{ccc}\hfill x^{*}\left(t\right)& =& x_0-{\int }_0^{\tau }{h}^{*}(s,I^{\alpha -\beta }{y}^{*}\left(s\right))dW\left(s\right)+I^{\alpha }{y}^{*}\left(t\right)\hfill \\ \hfill y^{*}\left(t\right)& =& I^{1-\alpha }g(t,x^{*}\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,x^{*}\left(s\right))dW\left(s\right).\hfill \end{array}$$

Let $\parallel h^{*}{(t,u\left(t\right))-h(t,u\left(t\right))\parallel }_2\le {\delta }_2$ then

$$\begin{array}{ccc}\hfill \parallel x\left(t\right)-x^{*}{\left(t\right)\parallel }_2& \le & \parallel {\int }_0^{\tau }[h^{*}(s,I^{\alpha -\beta }{y}^{*}\left(s\right))-h(s,I^{\alpha -\beta }y\left(s\right))]{dW\left(s\right)\parallel }_2+{\parallel I^{\alpha }y\left(t\right)-I^{\alpha }{y}^{*}\left(t\right)\parallel }_2\hfill \\ & \le & {\delta }_2\sqrt{T}+[\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel y-y^{*}\parallel }_C.\hfill \end{array}$$

Similarly, we can obtain

$$\parallel y\left(t\right)-y^{*}{\left(t\right)\parallel }_2\le \frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}\parallel x-x^{*}{\parallel }_C+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}{\parallel y-y^{*}\parallel }_C$$

and

$$\begin{array}{ccc}\hfill \parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }& =& \parallel x-x^{*}{\parallel }_C+{\parallel y-y^{*}\parallel }_C\hfill \\ & \le & {\delta }_2\sqrt{T}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]\parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }.\hfill \end{array}$$

This implies that

$$\parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }\le \frac{{\delta }_2\sqrt{T}}{1-[\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]}=ϵ$$

which completes the proof. □

Theorem 5.

The unique solution of the system of stochastic integral Equations (9) and (10) depends continuously on the random functions g.

Proof. 

Let $(x^{*},y^{*})$ be the solutions of the system (9) and (10) such that

$$\begin{array}{ccc}\hfill x^{*}\left(t\right)& =& x_0-{\int }_0^{\tau }h(s,I^{\alpha -\beta }{y}^{*}\left(s\right))dW\left(s\right)+I^{\alpha }{y}^{*}\left(t\right)\hfill \\ \hfill y^{*}\left(t\right)& =& I^{1-\alpha }{g}^{*}(t,x^{*}\left(\varphi \left(t\right)\right))+{\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}f(s,y^{*}\left(s\right))dW\left(s\right).\hfill \end{array}$$

Let $\parallel g^{*}{(t,u\left(t\right))-g(t,u\left(t\right))\parallel }_2\le {\delta }_3$ then

$$\begin{array}{ccc}\hfill \parallel x\left(t\right)-x^{*}{\left(t\right)\parallel }_2& \le & \parallel {\int }_0^{\tau }[h(s,I^{\alpha -\beta }{y}^{*}\left(s\right))-h(s,I^{\alpha -\beta }y\left(s\right))]{dW\left(s\right)\parallel }_2+{\parallel I^{\alpha }y\left(t\right)-I^{\alpha }{y}^{*}\left(t\right)\parallel }_2\hfill \\ & \le & \sqrt{{\int }_0^{\tau }[c\parallel I^{\alpha -\beta }{y}^{*}-I^{\alpha -\beta }y{{\parallel }_2]}^{2}ds}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}{\parallel y-y^{*}\parallel }_C\hfill \\ & \le & [\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}]{\parallel y-y^{*}\parallel }_C\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel y\left(t\right)-y^{*}{\left(t\right)\parallel }_2& \le & \parallel I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))-I^{1-\alpha }{g}^{*}(t,x^{*}\left(\varphi \left(t\right)\right)){\parallel }_2\hfill \\ & +& \parallel {\int }_0^t\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}\left[f\right(s,y\left(s\right)-f(s,y^{*}\left(s\right)]{dW\left(s\right)\parallel }_2\hfill \\ & \le & \parallel I^{1-\alpha }g(t,x\left(\varphi \left(t\right)\right))-I^{1-\alpha }g(t,x^{*}\left(\varphi \left(t\right)\right)){\parallel }_2\hfill \\ & +& \parallel I^{1-\alpha }g(t,x^{*}\left(\varphi \left(t\right)\right))-I^{1-\alpha }{g}^{*}(t,x^{*}\left(\varphi \left(t\right)\right)){\parallel }_2\hfill \\ & +& \sqrt{{\int }_0^t[\parallel f(s,y\left(s\right))-f(s,y^{*}\left(s\right)){{\parallel }_2\frac{{(t-s)}^{1-\alpha }}{\Gamma (2-\alpha )}]}^{2}ds}\hfill \\ & \le & \frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}\parallel x-x^{*}{\parallel }_C+\frac{{\delta }_3{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}{\parallel y-y^{*}\parallel }_C.\hfill \end{array}$$

Now

$$\begin{array}{ccc}\hfill \parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }& =& \parallel x-x^{*}{\parallel }_C+{\parallel y-y^{*}\parallel }_C\hfill \\ & \le & \frac{{\delta }_3{T}^{1-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}{\parallel x-x^{*}\parallel }_C\hfill \\ & +& [\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}]{\parallel y-y^{*}\parallel }_C\hfill \\ & \le & \frac{{\delta }_3{T}^{1-\alpha }}{\Gamma (2-\alpha )}\hfill \\ & +& [\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}]{\parallel (x,y)-(x^{*},y^{*})\parallel }_{\Lambda }.\hfill \end{array}$$

This implies that

$$\parallel (x,y)-(x^{*},y^{*}){\parallel }_{\Lambda }\le \frac{\frac{T^{1-\alpha }}{\Gamma (2-\alpha )}{\delta }_3}{1-[\frac{c{T}^{\alpha -\beta +\frac{1}{2}}}{\Gamma (1+\alpha -\beta )}+\frac{T^{\alpha }}{\Gamma (1+\alpha )}+\frac{b{T}^{\frac{3}{2}-\alpha }}{\Gamma (2-\alpha )}+\frac{q{T}^{1-\alpha }}{\Gamma (2-\alpha )}]}=ϵ$$

which completes the proof. □

6. An Example

Consider the stochastic differential equation

$$d(\frac{dx\left(t\right)}{dt}-\frac{x\left(t\right)sint}{30(1+\parallel x\left(t\right){\parallel }_2)})=\frac{a\left(t\right)+D^{\frac{3}{4}}x\left(t\right)}{9(1+\parallel x\left(t\right){\parallel }_2)}dW\left(t\right),t\in (0,\frac{1}{2}]$$

(17)

subject to

$$\begin{array}{ccc}\hfill \frac{dx}{dt}{\mid }_{t=0}& =& 0,\hfill \\ \hfill x_0& =& {\int }_0^{\tau }\frac{e^{-s}{D}^{\frac{1}{2}}x\left(s\right)}{36+s^2}dW\left(s\right)\hfill \end{array}$$

(18)

where

$$\begin{array}{ccc}\hfill {\parallel f(t,x\left(t\right))\parallel }_2& \le & \frac{1}{120}\left[\right|a\left(t\right)|+{\parallel x\left(t\right)\parallel }_2{],\parallel g(t,x\left(t\right))\parallel }_2\le \frac{1}{30}{\parallel x\left(t\right)\parallel }_2,\hfill \\ \hfill \parallel h(t,x(t)\parallel & \le & \frac{{\parallel x\parallel }_2}{36}\hfill \end{array}$$

Easily, the problem (17) with nonlocal integral conditions (18) satisfies all the assumptions (B1)–(B6) of Theorem 1 with $b=\frac{1}{120},c=\frac{1}{36},q=\frac{1}{30}$ then there exists at least one solution to the problem (17) on $[0,\frac{1}{2}]$.

7. Conclusions

Here, we have combined the two senses of derivatives, the stochastic It$\widehat{o}$-differential and the arbitrary (fractional) orders derivative, for the second-order stochastic process. The existence of solutions has been proved. The sufficient conditions for the uniqueness of the solution have been given. The continuous dependence of the unique solution has been studied.