Approximation of Solutions of a Stochastic Differential Equation

Abstract

The existence, uniqueness, and convergence of approximate solutions of a stochastic differential equation with deviated argument is studied using analytic semigroup theory and fixed point method. Then we consider Faedo-Galerkin approximation of solution and prove some convergence results. We also study an example to illustrate our result.

1 Introduction

Fractional differential equations appear abundantly in the theory of fractals, visco-elasticity, seismology, polymers, etc. Stochastic evolution equations are natural generalizations of ordinary differential equations incorporating the random noise which causes fluctuations in deterministic models. For details refer [1]. In certain real-world problems, delay depends not only on the time but also on the unknown quantity as we can see in [2]. Das et al. [3, 4] can be referred for related work with deviated argument. Bahuguna et. al. [5] discussed the Faedo-Galerkin approximation of solution. So far the Faedo-Galerkin approximation of solution stochastic fractional differential equation with deviated argument is neglected in the literature. In an attempt to fill this gap we study the following stochastic fractional differential equation with deviated argument in a separable Hilbert space (H, (., .)).

$$\begin{aligned} ^cD^{\beta }_tu(t)+Au(t)= & {} f(u(t),u(h(u(t))))\frac{dw(t)}{dt},~~t\in [0,T]\nonumber \\ u(0)= & {} u_0\in H \end{aligned}$$

(1)

where $0<\beta <1$ and $0<T<\infty .$ $^cD^{\beta }_t$ denotes the Caputo fractional derivative of order $\beta$ and $A:D(A)\subset X\rightarrow H$ is a linear operator. A and the functions f, h are defined in the hypotheses $(H1)-(H3)$ of Sect. 2.

2 Preliminaries

Here we deal with two separable Hilbert spaces H and K.

• (H1) A is a closed, densely defined, self-adjoint operator with pure point spectrum $0\le \lambda _0\le \lambda _1\le \cdots \le \lambda _m\le \cdots$ with $\lambda _m\rightarrow \infty$ and $m\rightarrow \infty$ and corresponding complete orthonormal system of eigenfunctions ${\phi _j}$ such that

  $$A\phi _j=\lambda _j\phi _j ~ and ~<\phi _i,\phi _j>=\delta _{i,j} $$

• (H2) The function $f:[O,T]\times H_{\alpha }\times H_{\alpha -1}\rightarrow L(K,H)$ is continuous and $\exists$ constant $L_f$ such that

  $$\Vert f(u,u_1)-f(v,v_1)\Vert ^2_{Q}\le L_f[+\Vert u-v\Vert _{\alpha }+\Vert u_1-v_1\Vert _{\alpha -1}]$$

• (H3) The map $h:H_{\alpha }\times \mathcal {R}_+\rightarrow \mathcal {R}_{+}$ satisfies ${\Vert }h(u,)-h(v,)\Vert \le L_h(\Vert u-v{\Vert }_{\alpha } )$

If (H1) is satisfied then $-A$ is the infinitesimal generator of an analytic semigroup $\{e^{-tA}:t\ge 0\}$ in H. We also note that $\exists$ constant C such that $\Vert S(t)\Vert \le {Ce}^{\omega t}$ and constants $C_i$ ’s such that $\Vert \frac{d^i}{{dt}^i}S(t) \Vert \le C_{i}, \, t > 0,\, i=1,2.$ Also $\Vert A S(t)\Vert \le Ct^{-1}$ and $\Vert A^{\alpha }S(t)\Vert \le C_{\alpha }t^{-\alpha }$.

We define the space $H_{\alpha }$ as $D(A^{\alpha })$ endowed with the norm ${\Vert }.{\Vert }_{\alpha }.$ Let $({\varOmega },\mathfrak {F},P)$ be a complete probability space endowed with complete family of right continuous increasing sub $\sigma$—algebras $\{\mathfrak {F}_t,t\in J\}$ such that $\mathfrak {F}_t\subset \mathfrak {F}.$ A H—valued random variable is a $\mathcal {F}$—measurable process. We also assume that W is a Wiener process on K with covariance operator Q. Suppose Q is symmetric, positive, linear and bounded operator with $TrQ < \infty .$ Let $K_0=Q^{\frac{1}{2}}(K).$ The space $L_2^0=L_2(K_0,H_{\alpha })$ is a separable Hilbert space with norm $\Vert \psi \Vert _{L_2^0}=\Vert \psi Q^{\frac{1}{2}}\Vert _{L_2(K,H_{\alpha })}.$ Let $L_2({\varOmega },\mathfrak {F},P;H_{\alpha })\equiv L_2({\varOmega };H_{\alpha })$ be the Banach space of all strongly measurable, square integrable, $H_{\alpha }$—valued random variables equipped with the norm $\Vert u(.)\Vert ^2_{L_2}=E\Vert u(.;w)\Vert ^2_{H_{\alpha }}.$  $C^{\alpha }_T$ denotes the Banach space of all continuous maps from $J=(0,T]$ into $L_2({\varOmega };H_{\alpha })$ which satisfy ${sup}_{t\in J}E\Vert u(t)\Vert _{C^{\alpha }}^2<\infty .$ $L^0_2({\varOmega }, H_{\alpha })=\{f\in L_2 ({\varOmega }, H_{\alpha } ):f ~ is ~ \mathcal {F}_0-measurable\}$ denotes an important subspace. For $0\le \alpha < 1$ define

$$\begin{aligned} C^{\alpha -1}_T=\{u\in C^{\alpha }_T:\Vert u(t)-u(s)\Vert _{\alpha -1}\le L|t-s|,\forall t,s\in [0,T]\}. \end{aligned}$$

Now let us define mild solution of (1):

Definition 1

The mild solution of (1) is a continuous $\mathfrak {F}_t$ adapted stochastic process $u\in C^{\alpha }_T\cap C^{\alpha -1}_T$ which satisfies the following:

• 1. $u(t)\in H_{\alpha }$ has $C\grave{a}dl\grave{a}g$ paths on $t\in [0,T].$

• 2. $\forall t\in [0,T],~u(t)$ is the solution of the integral equation

$$\begin{aligned} u(t)=T_{\beta }(t)u_0\,+\,\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f(u(s),u(h(u(s),s)))dw(s),~t\in [0,T] \end{aligned}$$

(2)

where $S_{\beta }(t)=\int ^{\infty }_0\zeta _{\beta }(\theta )S(t^{\beta }\theta )d\theta ;$ and $T_{\beta }(t)=q\int ^{\infty }_0\theta \zeta _{\beta }(\theta )S(t^{\beta }\theta )d\theta ;$ $\zeta _{\beta }$ is a probability density function defined on $(0,\infty ),$ i.e. $\zeta _{\beta }(\theta )\ge 0,$ $\theta \in (0,\infty )$ and $\int ^{\infty }_0\zeta _{\beta }(\theta )d\theta =1.$ Also $\Vert T_{\beta }(t)u\Vert \le C\Vert u\Vert ,~ \Vert S_{\beta }(t)u\Vert \le \frac{\beta C}{{\varGamma }(1+\beta )}\Vert u\Vert ,~ \Vert A^{\alpha }S_{\beta }(t)u\Vert \le \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}t^{-\alpha \beta }\Vert u\Vert .$

Lemma 1

Let $f:J\times {\varOmega }\times {\varOmega }\rightarrow L_2^0$ be a strongly measurable mapping with $\int ^T_0 E\Vert f(t)\Vert ^p_{L_2^0}dt<\infty .$ Then

$$E\Vert \int ^t_0f(s)dw(s)\Vert ^p\le l_s\int ^t_0E\Vert f(s)\Vert ^p_{L^0_2}ds$$

$\forall t\in [0,T]$ and $p\ge 2$ where $l_s$ is a constant containing p and T.

$l_s$ is incorporated into the constants in the following sections.

3 Existence and Uniqueness of Approximate Solutions

In this section we consider a sequence of approximate integrals and establish the existence and uniqueness of solution for each of the approximate integral equations. For $0\le \alpha <1$ and $u\in C^{\alpha }_{T_0},$ the hypotheses $(H2)-(H3),$ imply that f(u(s), u(h(u(s), s))) is continuous on $[0,T_0].$ Therefore, $\exists$ a positive constant

$$N=2L_f[T_0^{\theta _1}+2R(1+LL_h)+LL_hT^{\theta _2}_0]+2N_0, ~~~N_0=E\Vert f(u_0,u_0)\Vert ^2$$

such that $\Vert f(s,u(s),u(h(u(s),s)))\Vert \le N,~~t\in [0,T].$ Choose $T_0,$ $0<T_0\le T$

such that

$$\left( \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}\right) ^2N\frac{T_0^{\beta (1-\alpha )-1}}{\beta (1-\alpha )-1}\le \frac{R}{4} ,$$

$$\begin{aligned} D\,=\,\left( \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}\right) ^22L_f\frac{T_0^{\beta (1-\alpha )-1}}{2\beta (1-\alpha )-1}\le 1\end{aligned}$$

(3)

Let

$$B_R=\{u\in \mathcal {C}^{\alpha }_{T_0}\cap \mathcal {C}^{\alpha -1}_{T_0}:u(0)=u_0,~~~\Vert u-u_0\Vert _{T_0,\alpha }\le R\}$$

It is easy to see that $B_R$ is a closed and bounded subset of $\mathcal {C}^{\alpha -1}_{T_0}$ and complete. Let us define the operator $\mathcal {F}_n:B_R:\rightarrow B_R$ by

$$\begin{aligned} (\mathcal {F}_nu)(t)=T_{\beta }(t)u_0\,+\,\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u(s),u(h(u(s),s)))dw(s).\quad \end{aligned}$$

(4)

Theorem 1

If the hypotheses $(H1),(H2)\ \text {and}\ (H3)$ are satisfied and $u_0\in L^0_2({\varOmega },X_{\alpha }),$ $0\le \alpha <1,$ then $\exists$ a unique $u_n\in B_R$ such that $\mathcal {F}_nu_n=u_n,$ $\forall$ $n=0,1,2,\cdots ,$ i.e., $u_n$ satisfies the approximate integral equation

$$\begin{aligned} u_n(t)=T_{\beta }(t)u_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(s,u_n(s),&u_n(h(u_n(s),s)))dw(s),\nonumber \\ {}&t\in [0,T]\end{aligned}$$

(5)

Proof

Step1 :  We need to show that $\mathcal {F}_nu\in \mathcal {C}^{\alpha -1}_{T_0},$ $\forall u\in \mathcal {C}^{\alpha -1}_{T_0}.$ It is easy to check that $\mathcal {F}_n:\mathcal {C}^{\alpha }_T\rightarrow \mathcal {C}^{\alpha }_T.$ If $u\in \mathcal {C}^{\alpha -1}_{T_0},$ $0<t_1<t_2<T_0$ and $0\le \alpha <1$ then

$$\begin{aligned}&E\Vert \mathcal {F}_nu(t_2)-\mathcal {F}_nu(t_1)\Vert ^2_{\alpha -1}\nonumber \\&\le 3E\Vert [T_{\beta }(t_2)-T_{\beta }(t_1)]u_0\Vert ^2_{\alpha -1}\nonumber \\&\quad +\,3E\Vert \int ^{t_2}_{t_1}(t_2-s)^{\beta -1}A^{\alpha -1}S_{\beta }(t_2-s)f_n(u(s),u(h(u(s)))dw(s)\Vert ^2_Q\nonumber \\&\quad +\,3E\Vert \int ^{t_1}_0A[(t_2-s)^{\beta -1}S_{\beta }(t_2-s)-(t_1-s)^{\beta -1}S_{\beta }(t_1-s)]\nonumber \\&\quad A^{\alpha -2}\times f_n(u(s),u(h(u(s))))dw(s)\Vert _Q\nonumber \\&\le 3E\Vert [T_{\beta }(t_2)-T_{\beta }(t_1)]u_0\Vert ^2_{\alpha -1}+3\frac{\beta ^2 C^2_{\alpha }{\varGamma }^2(2-\alpha )}{{\varGamma }^2(1+\beta (1-\alpha ))}\displaystyle \int ^{t_2}_{t_1}\Vert (t_2-s)^{2\beta (1-\alpha )-2}\Vert \nonumber \\&\quad \times \,\Vert A^{-1}\Vert ^2 E\Vert f_n(u(s),u(h(u(s),)))\Vert ^2ds\nonumber \\&\quad +\,3\int ^{t_1}_0\Vert A[(t_2-s)^{\beta -1}S_{\beta }(t_2-s)-(t_1-s)^{\beta -1}S_{\beta }(t_1-s)]\nonumber \\&\quad \times \,\Vert A^{\alpha -2}\Vert ^2 E\Vert f_n(u(s),u(h(u(s))))\Vert ^2ds\end{aligned}$$

(6)

$\forall u\in H,$ we can write

$$[S(t_2^{\beta }\theta )-S(t_1^{\beta }\theta )]u=\int ^{t_2}_{t_1}\frac{d}{dt}S(t^{\beta }\theta )udt=\int ^{t_2}_{t_1}\theta \beta t^{\beta -1}AS(t^{\beta }\theta )dt.$$

The first term of (6) can be estimated as follows:

$$\begin{aligned} \Vert [T_{\beta }(t_2)-T_{\beta }(t_1)]u_0\Vert ^2_{\alpha -1}&\le \left( \int ^{\infty }_0\zeta _{\beta }(\theta )\Vert S(t^{\beta }_2\theta )-S(t^{\beta }_1\theta )\Vert \Vert A^{\alpha -1}u_0\Vert d\theta \right) ^2\nonumber \\&\le \left( \int ^{\infty }_0\zeta _{\beta }(\theta )[\int ^{t_2}_{t_1}\Vert \frac{d}{dt}S(t^{\beta }\theta )\Vert dt]\Vert u_0\Vert _{\alpha }d\theta \right) ^2\nonumber \\&\le C^2_1\Vert u_0\Vert ^2_{\alpha -1}(t_2-t_1)^2\end{aligned}$$

(7)

For the second term of (6) we get the following estimate

$$\begin{aligned} \int ^{t_2}_{t_1}&(t_2-s)^{2\beta (1-\alpha )-2}E\Vert f_n(u(s),u(h(u(s))))\Vert ^2ds\nonumber \\ {}&\le \frac{N(t_2-t_1)^{2\beta (1-\alpha )-1}}{2\beta (1-\alpha )-1}\end{aligned}$$

(8)

For the third term we will use the following estimate

$$\begin{aligned} \int ^{t_1}_0&\Vert A[(t_2-s)^{\beta -1}S_{\beta }(t_2-s)-(t_1-s)^{\beta -1}S_{\beta }(t_1-s)]\Vert ^2\nonumber \\&\quad \times \Vert A^{\alpha -2}\Vert ^2E\Vert f_n(u(s),u(h(u(s))))\Vert ^2ds\nonumber \\&\le \int ^{t_1}_0\left( \int ^{\infty }_0\zeta _{\beta }(\theta )\Vert [\frac{d}{dt}S((t-s)^{\beta }\theta )|_{t=t_2}-\frac{d}{dt}S((t-s)^{\beta }\theta )|_{t=t_1}] \Vert d\theta \right) ^2\nonumber \\&\quad \times E\Vert f(u(s),u(h(u(s))))\Vert ^2ds\nonumber \\&\le \int ^{t_1}_0 \left( \int ^{\infty }_0\zeta _{\beta }(\theta )[\int ^{t_2}_{t_1}\Vert A^{\alpha -2}\frac{d^2}{dt^2}S((t-s)^{\beta }\theta )\Vert dt]d\theta \right) ^2Nds\nonumber \\&\le C_2^2\Vert A^{\alpha -2}\Vert ^2(t_2-t_1)^2NT_0 \end{aligned}$$

(9)

Hence from inequalities (7)–(9) we see that the map $\mathcal {F}_n:\mathcal {C}^{\alpha -1}_{T_0}\rightarrow \mathcal {C}^{\alpha -1}_{T_0}$ is well-defined. Now we prove that $\mathcal {F}_n:B_R\rightarrow B_R.$ So for $t\in [0,T_0]$ and $u\in B_R.$

$$\begin{aligned} E&\Vert (\mathcal {F}_nu)(t)-u_0\Vert ^2_{\alpha }\\ {}&\le 2E\Vert (T_{\beta }(t)-I)u_0\Vert ^2_{\alpha }\\ {}&\quad +\,2E\Vert \int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f(u(s),u(h(u(s))))dw(s)\Vert ^2_Q\\&\le 2E\Vert (T_{\beta }(t)-I)u_0\Vert ^2_{\alpha }+2\left( \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}\right) ^2\int ^{t}_{0}\Vert (t_2-s)^{2\beta (1-\alpha )-2}\Vert ^2\nonumber \\ {}&\quad \times \,E\Vert f_n(u(s),u(h(u(s))))\Vert ^2ds\\&\le \frac{R}{2}+2\left( \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}\right) ^2N\frac{T_0^{\beta (1-\alpha )-1}}{\beta (1-\alpha )-1}\le \frac{R}{2}+\frac{R}{2}=R \end{aligned}$$

Now we show that $\mathcal {F}_n$ is a contraction map by using (3) in last but one inequality. $\forall u,v\in B_R$

$$\begin{aligned}E\Vert (\mathcal {F}_nu)(t)&-(\mathcal {F}_nv)(t)\Vert ^2_{\alpha } =E\Vert \int ^t_0(t-s)^{\beta -1}A^{\alpha }S_{\beta }(t-s)\\ {}&\quad \times [f(u(s),u(h(u(s)))) -f(s,v(s),v(h(v(s),s)))dw(s)]\Vert ^2_Q\\&\le \left( \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}\right) ^2\int ^{t}_0(t_2-s)^{2\beta (1-\alpha )-2}\\ {}&\quad \times \,E\Vert f(u(s),u(h(u(s))))-f(v(s),v(h(v(s))))\Vert ^2ds\\&\le \left( \frac{\beta C_{\alpha }{\varGamma }(2-\alpha )}{{\varGamma }(1+\beta (1-\alpha ))}\right) ^22L_f(1+2LLh)\Vert u-v\Vert ^2_{\alpha }\frac{T^2\beta (1-\alpha )-1}{2\beta (1-\alpha )-1}\\ {}&\le \Vert u-v\Vert ^2_{\alpha }. \end{aligned}$$

This implies that $\exists$ a unique fixed point $u_n$ of $\mathcal {F}_n.$ Thus there a unique mild approximate solution of (1)

Lemma 2

Let $(H1)-(H3)$ hold. If $u_0\in L^0_2({\varOmega },D(A^{\alpha })),$ $\forall 0<\alpha <\eta <1,$ then $u_n(t)\in D(A^{\gamma })$ for all $t\in [0,T_0]$ with $0<\gamma <\eta <1.$ Also if $u_0\in D(A),$ then $u_n(t)\in D(A^{\gamma })$ $\forall t\in [0,T_0],$ where $0<\gamma <\eta <1.$

Proof

By Theorem (1) we get the existence of a unique $u_n\in B_R,$ satisfying (5). Theorem 2.6.13 of [6] implies for $t>0,~0\le \gamma <1,$ $S(t):H\rightarrow D(A^{\gamma })$ and for $0\le \gamma <\eta <1,$ $D(A^{\eta })\subset D(A^{\gamma }).$ It is easy to see that Holder continuity of $u_n$ can be proved using the similar arguments from (6) to (9). Also from Theorem 1.2.4 in [6], we have $S(t)u\in D(A)$ if $u\in D(A).$ The result follows from these facts and that $D(A)\subset D(A^{\gamma })$ for $0\le \gamma <1.$

Lemma 3

Let $(H1)-(H3)$ hold and $u_0\in L^0_2({\varOmega },X_{\alpha })$. Then for any $t_0\in (0,T_0]$ $\exists$ a constant $U_{t_0},$ independent of n such that $E\Vert u_n(t)\Vert ^2_{\gamma }\le U_{t_0}~~\forall t\in [t_0,T_0],~~n=1,2,\cdots .$ Also if $u_0\in L_2^0({\varOmega },D(A))$ then $\exists$ constant $U_0$ independent of n such that $E\Vert u_n(t)\Vert ^2_{\gamma }\le U_{0}~~\forall t\in [t_0,T_0],~~n=1,2,\cdots ,~~~\forall ~0<\gamma \le 1.$

Proof

Let $u_0\in L^0_2({\varOmega },H_{\alpha }).$ Applying $A^{\gamma }$ on both sides of (4)

$$\begin{aligned}&E\Vert u_n(t)\Vert ^2_{\gamma }\\ {}&\le 2E\Vert T_{\beta }(t)u_0\Vert _{\gamma }^2+2\Vert \int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u(s),u(h(u(s))))dw(s)\Vert ^2_Q\\&\le 2C_{\gamma }^2t_0^{-2\gamma \beta }\Vert u_0\Vert ^2+\left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{N(T_0)^{2\beta (1-\gamma )-1}}{2\beta (1-\gamma )-1}=U_{t_0}. \end{aligned}$$

Also if $u_0\in L_2^0({\varOmega },D(A)),$ then we have that $u_0\in L^0_2({\varOmega },D(A^{\gamma }))$ for $0\le \gamma <1.$ Hence,

$$\begin{aligned}&E\Vert u_n(t)\Vert ^2_{\gamma }\\ {}&\le 2E\Vert T_{\beta }(t)u_0\Vert _{\gamma }^2+2\Vert \int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u(s),u(h(u(s))))dw(s)\Vert ^2_Q\\&\le 2C^2\Vert u_0\Vert ^2+\left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{N(T_0)^{2\beta (1-\gamma )-1}}{2\beta (1-\gamma )-1}=U_0. \end{aligned}$$

Hence proved.

4 Convergence of Solutions

In this section the convergence of the solution $u_n\in H_{\alpha }$ of the approximate integral equation (5) to a unique solution u of (2), is discussed.

Theorem 2

Let the hypotheses $(H1)-(H3)$ hold and if $u_0\in L_2^0({\varOmega },H_{\alpha })$ then $\forall t_0\in (0,T],$

$$\displaystyle \lim _{m\rightarrow \infty }\displaystyle \sup _{\{n\ge M,t_0\le t\le T_0\}}\Vert u_n(t)-u_m(t)\Vert _{\alpha }=0.$$

Proof

Let $0<\alpha <\gamma <\eta .$ For $t_0\in (0,T_0]$

$$\begin{aligned} E&\Vert f_n(u_n(t),u_n(h(u_n(t))))-f_m(t,u_m(t),u_m(h(u_m(t))))\Vert ^2\nonumber \\&\le 2E\Vert f_n(u_n(t),u_n(h(u_n(t))))-f_n(t,u_m(t),u_m(h(u_m(t))))\Vert ^2\nonumber \\&\le 2E\Vert f_n(u_m(t),u_m(h(u_m(t))))-f_m(t,u_m(t),u_m(h(u_m(t))))\Vert ^2\nonumber \\&\le 2(2L_f(1+2LL_h)[E\Vert u_n-u_m\Vert ^2_{\alpha }+E\Vert (P^n-P^m)u_m(t)\Vert ^2_{\alpha }]) \end{aligned}$$

(10)

Now,

$$E\Vert (P^n-P^m)u_m(t)\Vert ^2\le E\Vert A^{\alpha -\gamma }(P^n-P^m)A^{\gamma }u_m(t)\Vert ^2\le \frac{1}{\lambda _m^{2(\gamma -\alpha )}}E\Vert A^{\gamma }u_m(t)\Vert ^2$$

Then we have

$$\begin{aligned}E&\Vert f_n(t,u_n(t),u_n(h(u_n(t))))-f_m(t,u_m(t),u_m(h(u_m(t))))\Vert ^2\\&\le 2\left( 2L_f(1+2LL_h)\left[ E\Vert u_n-u_m\Vert ^2_{\alpha }+\frac{1}{\lambda _m^{2(\gamma -\alpha )}}E\Vert A^{\gamma }u_m(t)\Vert ^2\right] \right) \end{aligned}$$

For $0<t'_0<t_0$

$$\begin{aligned} E\Vert u_n(t)&-u_m(t)\Vert ^2_{\alpha }\le 2\left( \int ^{t'_0}_0+\int ^t_{t'_0}\right) \Vert (t-s)^{\beta -1}A^{\alpha }S_{\beta }(t-s)\Vert ^2\nonumber \\ {}&\times \, E\Vert f_n(u_n(t),u_n(h(u_n(t))))-f_m(u_m(t),u_m(h(u_m(t))))\Vert ^2ds \end{aligned}$$

(11)

The estimate of first integral of the above inequality is

$$\begin{aligned} E\Vert u_n(t)&-u_m(t)\Vert ^2_{\alpha }\nonumber \\ {}&\le \int ^{t'_0}_0\Vert (t-s)^{\beta -1}A^{\alpha }S_{\beta }(t-s)\Vert ^2\nonumber \\ {}&\quad \times \, E\Vert f_n(u_n(t),u_n(h(u_n(t))))-f_m(u_m(t),u_m(h(u_m(t))))\Vert ^2ds\nonumber \\&\le \left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{2N(t_0-\delta _1t_0')^{2\beta (1-\gamma )-2}}{2\beta (1-\gamma )-1}t'_0,~~0<\delta <1 \end{aligned}$$

(12)

The estimate of second integral is

$$\begin{aligned} E\Vert u_n(t)&-u_m(t)\Vert ^2_{\alpha }\le \int _{t'_0}^t\Vert (t-s)^{\beta -1}A^{\alpha }S_{\beta }(t-s)\Vert ^2\nonumber \\ {}&\quad \times \, E\Vert f_n(u_n(t),u_n(h(u_n(t))))-f_m(u_m(t),u_m(h(u_m(t))))\Vert ^2ds\nonumber \\&\le \left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\int ^t_{t'_0}(t-s)^{2\beta (\alpha -1)-2}\nonumber \\&\quad \times \,4L_f(1+2LL_h)\left[ E\Vert u_n-u_m\Vert ^2_{\alpha }+\frac{E\Vert A^{\gamma }u_m(s)\Vert ^2}{\lambda ^2(\gamma -\alpha )}\right] ds\nonumber \\&\le 4L_f(1+2LL_h)\left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2[\int ^t_{t'_0}(t-s)^{2\beta (\alpha -1)-2}\nonumber \\&\quad \times \, E\Vert u_n-u_m\Vert ^2_{\alpha }ds+\frac{U_{t_0}}{\lambda _m^{2(\gamma -\alpha )}} \frac{T_0^{2\beta (1-\alpha )-1}}{2\beta (1-\alpha )-1}] \end{aligned}$$

(13)

Substituting inequalities (12), (13) into (11) we get

$$\begin{aligned}E\Vert u_n(t)&-u_m(t)\Vert ^2_{\alpha }\\ {}&\le \left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{4N(t_0-\delta _1t_0')^{2\beta (1-\gamma )-2}}{2\beta (1-\gamma )-1}t'_0\\&\quad +\,8L_f(1+2LL_h)\left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2[\int ^t_{t'_0}(t-s)^{2\beta (\alpha -1)-2}\\&\times \,E\Vert u_n-u_m\Vert ^2_{\alpha }ds+\frac{U_{t_0}}{\lambda _m^{2(\gamma -\alpha )}} \frac{T_0^{2\beta (1-\alpha )-1}}{2\beta (1-\alpha )-1}] \end{aligned}$$

By using Gronwall’s inequality, $\exists$ a constant D such that

$$\begin{aligned}E\Vert u_n(t)&-u_m(t)\Vert ^2_{\alpha }\le \left[ \left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{4N(t_0-\delta _1t_0')^{2\beta (1-\gamma )-2}}{2\beta (1-\gamma )-1}t'_0\right. \\&\left. +\,8L_f(1+2LL_h)\left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{U_{t_0}}{\lambda _m^{2(\gamma -\alpha )}} \frac{T_0^{2\beta (1-\alpha )-1}}{2\beta (1-\alpha )-1}\right] \times D \end{aligned}$$

Let $m\rightarrow \infty .$ Taking supremum over $[t_0,T_0]$ we get the following inequality:

$$\begin{aligned}E\Vert u_n(t)&-u_m(t)\Vert ^2_{\alpha }\le \left[ \left( \frac{\beta C_{\gamma }{\varGamma }(2-\gamma )}{{\varGamma }(1+\beta (1-\gamma ))}\right) ^2\frac{4N(t_0-\delta _1t_0')^{2\beta (1-\gamma )-2}}{2\beta (1-\gamma )-1}t'_0 \right] \times D \end{aligned}$$

Since $t_0'$ is arbitrary, the right-hand side can be made infinitesimally small by choosing $t'_0$ sufficiently small. Thus the lemma is proved.

Corollary 1

If $u_0\in D(A),$ then $\displaystyle \lim _{m\rightarrow \infty }\displaystyle \sup _{\{n\ge m,~0\le t\le T_0\}}E\Vert u_n(t)-u_m(t)\Vert ^2_{\alpha }=0$

Proof

By using Lemmas (2) and (3) we can take $t_0=0$ in the proof of Theorem (2) and hence the corollary follows.

Theorem 3

Let us assume that $(H1)-(H3)$ are satisfied and suppose $u_0\in L_2^0({\varOmega },X_{\alpha }).$ Then for $t\in [0,T_0],$ $\exists$ a unique function $u_n\in B_R$ where

$u_n(t)=T_{\beta }u_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u_n(s),u_n(h_n(u_n(s))))dw(s),$

and $u(t)\in B_R,$ where

$u(t)=T_{\beta }u_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f(u(s),u(h(u(s))))dw(s), t\in [0,T_0],$ such that $u_n\rightarrow u$ as $n\rightarrow \infty$ in $B_R$ and u satisfies (2) on $[0,T_0].$

Proof

By using the above Corollary, Theorems 1 and 2 it is to see that $\exists ~u(t)\in B_R$ such that

$\lim _{n\rightarrow \infty }E\Vert u_n(t)-u(t)\Vert ^2_{\alpha }=0$ on $[0,T_0].$ Now

$$\begin{aligned} E\Vert u_n(t)&-T_{\beta }u_0+\int ^t_{t_0}(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u_n(s),u_n(h_n(u_n(s))))dw(s)\Vert ^2\nonumber \\&\le E\Vert \int ^{t_0}_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u_n(s),u_n(h_n(u_n(s))))dw(s)\Vert ^2\nonumber \\&\le \left( \frac{\beta C}{{\varGamma }(1+\beta )}\right) ^2N\frac{T_0^{2\beta -2}}{2\beta -2}t_0\end{aligned}$$

(14)

Let $n\rightarrow \infty$ then

$E\Vert u_n(t)-T_{\beta }u_0+\int ^t_{t_0}(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u_n(s),u_n(h_n(u_n(s))))dw(s)\Vert ^2$

$\le \left( \frac{\beta C}{{\varGamma }(1+\beta )}\right) ^2N\frac{T_0^{2\beta -2}}{2\beta -2}t_0$ and since $t_0$ is arbitrary we conclude u(t) satisfies (2). Uniqueness follows easily from Theorems 1, 2 and Gronwall’s inequality.

4.1 Faedo-Galerkin Approximations

We know from the previous sections that for any $0\le T_0\le T$, we have a unique $u\in C_{T_0}^{\alpha }$ satisfying the integral equation

$u(t)=T_{\beta }u_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f(u(s),u(h(u(s))))dw(s),$ $t\in [0,T_0]$ Also, $\exists$ a unique solution $u_n\in C_{T_0}^{\alpha }$ of the approximate integral equation

$u_n(t)=T_{\beta }u_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f_n(u_n(s),u_n(h(u_n(s))))dw(s),$ $t\in [0,T_0].$

Faedo-Galerkin approximation ${\bar{u}}_n=P^n{u_n}$ is given by

$P^nu_n(t)={\bar{ u}}_n(t)=T_{\beta }(t)P^nu_0$

$~~~~~~~~~~~~~~~~~~+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)P^nf(u_n(s),u_n(h(u_n(s))))dw(s),t\in [0,T_0].$ If the solution u(t) to (2) exists on $[0,T_0]$ then it has the representation

$u(t)=\displaystyle \sum _{i=0}^{\infty }{\alpha }_{i}(t)\phi _i,$ where ${\alpha }_{i}(t)=(u(t),\phi _i)$ for $i=0,1,2,3,\cdots$ and

${\bar{u}}_n(t)=\displaystyle \sum _{i=0}^{n}{\alpha }_{i}^{n}(t)\phi _i,$ where ${\alpha }_{i}^{n}(t)=({{\bar{u}_n}}(t),\phi _i)$ for $i=0,1,2,3,\cdots$.

As a consequence of Theorems 1 and 2, we have the following result.

Theorem 4

Let us assume that $(H1)-(H3)$ are satisfied and suppose $u_0\in L_2^0({\varOmega },X_{\alpha }).$ Then for $t\in [0,T_0],$ $\exists$ a unique function $u_n\in B_R$ where

$u_n(t)=T_{\beta }P^nu_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)P^nf_n(u_n(s),u_n(h(u_n(s))))dw(s),$

and $u(t)\in B_R,$ where

$u(t)=T_{\beta }u_0+\int ^t_0(t-s)^{\beta -1}S_{\beta }(t-s)f(u(s),u(h(u(s))))dw(s), t\in [0,T_0],$ such that $u_n\rightarrow u$ as $n\rightarrow \infty$ in $B_R$ and u satisfies (2) on $[0,T_0].$

Now the convergence of $\alpha _i^n(t)\rightarrow \alpha _i(t)$ is shown. It is easily seen that

$A^{\alpha }\left[ u(t)-{\bar{u}}_n(t)\right] =A^{\alpha }\big [\displaystyle \sum _{i=0}^n\{{\alpha }_i(t)-{\alpha }_i^n(t)\}\phi _i\big ]+ A^{\alpha }\displaystyle \sum _{i=n+1}^{\infty }\alpha _i(t)\phi _i$

$~~~~~~~~~~~~~~~=\displaystyle \sum _{i=0}^{n}\lambda _i^{\alpha }\{{\alpha }_i(t)-{\alpha }_i^n(t)\}\phi _i+\displaystyle \sum ^{\infty }_{i=n+1}\lambda _i^{\alpha }\alpha _i(t)\phi _i.$ Thus we have

$E\Vert A^{\alpha }[u(t)-{\bar{u}}_n(t)\Vert ^2\ge \sum _{i=0}^{n}{\lambda }_i^{2\alpha }E|{\alpha }_i(t)-{\alpha }_i^n(t)|^2.$

Theorem 5

Let us assume $(H1)-(H3)$ hold.

   (i) If $u_0\in L^0_2({\varOmega },X_{\alpha })$ then $\displaystyle \lim _{n\rightarrow \infty }\displaystyle \sup _{t\in [t_0,T_0]}\left[ \displaystyle \sum _{i=0}^n\lambda _i(t)^{2\alpha }E\Vert \alpha _i(t)-\alpha _i^n(t)\Vert ^2\right] =0$

(ii) If $u_0\in L^0_2({\varOmega },D(A))$ then $\displaystyle \lim _{n\rightarrow \infty }\displaystyle \sup _{t\in [0,T_0]}\left[ \displaystyle \sum _{i=0}^n\lambda _i(t)^{2\alpha }E\Vert \alpha _i(t)-\alpha _i^n(t)\Vert ^2\right] =0$

Theorem 5 follows from the facts mentioned above the theorem.

Corollary 2

Let us assume $(H1)-(H3)$ hold.

(i) If $u_0\in L^0_2({\varOmega },X_{\alpha })$ then $\displaystyle \lim _{n\rightarrow \infty }\displaystyle \sup _{t\in [t_0,T_0],n\ge m}E\Vert A^{\alpha }[{\bar{u}}_n(t)-{\bar{u}}_m(t)]\Vert ^2=0$

(ii) If $u_0\in L^0_2({\varOmega },D(A))$ then $\displaystyle \lim _{n\rightarrow \infty }\displaystyle \sup _{t\in [0,T_0],n\ge m}E\Vert A^{\alpha }[{\bar{u}}_n(t)-{\bar{u}}_m(t)]\Vert ^2=0$

Proof

$$\begin{aligned}E\Vert A^{\alpha }[{\bar{u}}_n(t)-{\bar{u}}_m(t)]\Vert ^2&=E\Vert P^nu_n(t)-P^mu_m(t)\Vert ^2_{\alpha }\\&\le 2E\Vert P^n[u_n(t)-u_m(t)]\Vert ^2_{\alpha }+2E\Vert (P^n-P^m)y_m(t)\Vert ^2_{\alpha }\\&\le 2E\Vert [u_n(t)-u_m(t)]\Vert ^2_{\alpha }+2\frac{1}{\lambda _m^{\gamma -\alpha }}E\Vert A^{\gamma }u_m(t)\Vert ^2\end{aligned}$$

Then the result (i) follows from Theorem 2 and result (ii) follows from Corollary 1.

5 Example

Suppose for $t\ge 0,~x\in (0,1),0<\beta \le 1$

$$\begin{aligned} ^cD^{\beta }v_t(t,x)&=v_{xx}(t,x)+F(v(t,x),v(h(t,v(x))))\frac{dw(t)}{dt},\nonumber \\ v(t,x)&=v_0,~ t=0,~ x\in (0,1)~~~~and ~~~~v(t,0)=v(t,1)=0,~t\ge 0 \end{aligned}$$

(15)

Let F be an appropriate Holder continuous function satisfying (H2) in

$L_2^0(K,(0,1)).$ w is a standard $L_2(0,1)$ valued Weiner process. Let us define $A=-\frac{d^2}{dx^2},$ $f:=F,~v(x)=u(t)$ and let $D(A)=H^1_0(0,1)\cap H^2(0,1),$ $D(A^{1/2})=H_0^1(0,1).$ Then (15) can be reformulated into (1). Now from Theorems (1), (2) we can similarly prove the existence, uniqueness, and approximation of the mild solution of (15).