Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t),\,\,0<t<1\) with boundary conditions \(u(0)=0,\,\,u'(0)=u'(1)=0,\) where \(\lambda>0\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.
Fractional derivatives and fractional order differential equations have received an increasingly high interest and attention due to its physical and modeling applications in science, engineering and mathematics[1,2,3,4]. On the other hand, researches have shown that two-point boundary value problems have found their applications in theoretical physics, applied mathematics, optimization and control theory and engineering [5]. The use of second-order Boundary Value Problem (BVP) arises in several areas of engineering and applied sciences such as celestial mechanics, circuit theory, astrophysics, chemical kinetics, and biology [6].
They are particularly encountered in the study of following natural phenomena: in the study of surface-tension-induced flows of a liquid metal or semiconductor [7,8], used in modeling biological materials (elastic and hyperelastic materials) [9]. In [10], authors studied the existence and uniqueness of a nontrivial solution of a two-point boundary value problems. Stochastic nonlinear fractional order differential equation and its associated BVPs, on the other hand, will undoubtedly give more realistic models for the above natural occurrences.
Motivated by the above applications and the results of the papers [2,3,4], we study the white noise pertubation of a nonlinear BVP and consider the following BVP for a stochastic nonlinear fractional differential equation
Definition 1. We say that \(\{u(t)\}_{0\leq t\leq 1}\) is a mild solution to Equation (1) if \(a. s\), the following is satisfied \begin{eqnarray*} u(t)=\lambda\int_0^1 G(t,s)\sqrt{I^\beta[\sigma^2(s,u(s))]}\dot{w}(s)ds=\lambda\int_0^1 G(t,s)\sqrt{I^\beta[\sigma^2(s,u(s))]}dw(s), \end{eqnarray*} where \(G(t,s)\) is as defined in (3).
If \(\{u(t)\}_{0< t< 1}\) satisfies the additional condition \(\sup_{0\leq t\leq 1}{\mathbb E}|u(t)|^2< \infty,\) then we say that \(\{u(t)\}_{0\leq t\leq 1}\) is a random field solution to Equation (1).
Definition 2. The R-L fractional integral of order \(\beta>0\) of a given continuous function \(f:(0,\infty)\rightarrow\mathbb{R}\) is defined by \[I^\beta f(t)=\frac{1}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}f(s)ds,\] provided the right integral exists. Denote \(I^0 f(t)=f(t).\)
Definition 3. The R-L fractional derivative of order \(\alpha>0\) of a given continuous function \(f:(0,\infty)\rightarrow\mathbb{R}\) is defined by \[D^\alpha f(t)=\frac{1}{\Gamma(n-\alpha)}\bigg(\frac{d}{dt}\bigg)^n\int_0^t\frac{f(s)}{(t-s)^{\alpha-n+1}}ds,\] where \(n=[\alpha]+1\), provided the right integral converges.
Many authors studied different boundary value problems of nonlinear fractional order differential equations, see [1,11,12,13,14] and their references. Xu in [3] considered the following boundary value problem of a nonlinear fractional differential equation:
Lemma 1 ([3]). Given \(h\in C[0,1]\) and \(3< \alpha\leq 4\), the unique solution of \begin{equation*} \begin{cases} D^\alpha u(t)=h(t),\,\,0< t< 1\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{cases}\end{equation*} is \(u(t)=\displaystyle\int_0^1 G(t,s)h(s)ds\) where
Lemma 2 ([4]). The Green function \(G(t,s)\) in (2) has the following properties
Lemma 3([3]). The Green function \(G(t,s)\) defined by (2) satisfies the following conditions:
Lemma 4([2]). Suppose that \(\rho\in L^1[0,1]\). Then \(u(t)=\int_0^1 G(t,s)\rho(s)ds\) for \(t\in [0,1]\) is the unique solution of the following equation in \(C^1[0,1]\): \begin{equation*} \begin{cases} D^\alpha u(t)+\rho(t)=0,&\alpha\in(2,3)\\ u(0)=0,\,\, u'(0)=u'(1)=0,& \end{cases} \end{equation*} where
Lemma 5([2]). The function \(G(t,s)\) given in (3) has the following properties:
Remark 1.
Lemma 6([10]). Let \(y(t)\in X\), then the Boundary Value Problem \begin{eqnarray*} \left \{ \begin{array}{lll} u”(t)-y(t)=0,& 0< t< 1\\ u(0)=u(1)=0,& \end{array}\right. \end{eqnarray*} has a unique solution \(\displaystyle\int_0^1G(t,s)y(s)ds,\) where
Remark 2. The Green function has the following bound: \(\displaystyle\sup_{0\leq t,s\leq 1}G(t,s)\leq\frac{1}{2}.\)
Condition 1. Let there exist a nonnegative function \(p\in L^2[0,1],\) such that \(|\sigma(t,x)-\sigma(t,y)|\leq p(t)|x-y|,\,\,\forall\,t\in [0,1],\,x,y\in\mathbb{R},\) with \(\sigma(t,0)=0\) for convenience, and there exists \(t_0\in [0,1]\) such that \(p(t_0)\neq 0.\)
In particular, for \(p(t)=Lip_{\sigma}\) and for all \(t\in[0,1]\), we have the following:
Condition 2. There exist a finite positive constant \(Lip_{\sigma},\) such that for all \(x,\,y\in\mathbb{R}\), we have \( |\sigma(t,x)-\sigma(t,y)|\leq Lip_{\sigma}|x-y|, \) with \(\sigma(t,0)=0\) for convenience.
Now, define \(L^2({\mathbb P})\) norm of the solution \(u\) by \[\|u\|_2:=\bigg\{\sup_{0\leq t\leq 1}{\mathbb E}|u(t)|^2\bigg\}^{1/2}.\]
Theorem 1. Suppose \(\lambda0\). For a positive constant \(Lip_{\sigma}\) together with Condition 2, there exists a solution \(u\) for Equation (1) that is unique up to modification.
To proof the Theorem 2, let \(u(t)=\mathcal{A} u(t)\), where the operator \(\mathcal{A}\) is given by
\[\mathcal{A} u(t)=\lambda\int_0^1 G(t,s)\sqrt{I^\beta[\sigma^2(s,u(s))]}dw(s),\] and we will use the fixed point of \(\mathcal{A}\). The proof follows using the Lemma(s) below:Lemma 7. Given a random solution \(u\) such that \(\|u\|_2< \infty\) and Condition 2 holds. Then \[\|\mathcal{A} u\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u\|_2^2,\] where \(c_{\alpha,\beta}=\frac{1}{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}.\)
Proof. Take second moment of both sides and use Itó isometry together with Lemma 5(2) to obtain \begin{eqnarray*} \mathbb{E}|\mathcal{A} u(t)|^2&=& \lambda^2\int_0^1G^2(t,s){\mathbb E}|\sqrt{I^\beta[\sigma^2(s,u(s))]}|^2ds\\ &\leq&\lambda^2\int_0^1G^2(t,s)\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}{\mathbb E}|\sigma^2(r,u(r))|dr\bigg]ds\\ &\leq&\lambda^2Lip_{\sigma}^2\int_0^1G^2(t,s)\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}{\mathbb E}|u(r)|^2dr\bigg]ds\\ &\leq&\lambda^2Lip_{\sigma}^2\sup_{0\leq s\leq 1}G^2(t,s)\|u\|_2^2\int_0^1\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}dr\bigg]ds\\ &\leq&\lambda^2Lip_{\sigma}^2\bigg[\sup_{0\leq s\leq 1}G(t,s)\bigg]^2\|u\|_2^2\int_0^1\bigg[\frac{1}{\Gamma(\beta)}\int_0^s(s-r)^{\beta-1}dr\bigg]ds\\ &\leq&\frac{\lambda^2Lip_{\sigma}^2}{\Gamma^2(\alpha)\Gamma(\beta)}\|u\|_2^2\int_0^1\bigg[\int_0^s(s-r)^{\beta-1}dr\bigg]ds=\frac{\lambda^2Lip_{\sigma}^2}{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}\|u\|_2^2. \end{eqnarray*} Now, by taking supremum over \(t\in [0,1]\), the result follows.
Remark 3. The operator \(\mathcal{A}\) is a contraction for \(\frac{\lambda^2Lip_{\sigma}^2}{\beta(\beta+1)\Gamma(\beta)\Gamma^2(\alpha)}< 1\).
Lemma 8. Suppose \(u\) and \(v\) are random solutions such that \(\| u\|_{2}+\|v\|_{2}< \infty\) and Condition 2 holds. Then \[\|\mathcal{A} u-\mathcal{A} v\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u-v\|_2^2.\]
Proof. The proof follows similarly to the proof of Lemma 7.
Now, we present the proof of Theorem 1.
Proof of Theorem 1. From Lemma 7, we have \[\|u\|^2_2=\|\mathcal{A} u\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u\|_2^2,\] which follows that \[\|u\|^2_2\bigg[1-c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\bigg]\leq 0,\] and this shows that \(\|u\|_2< \infty\) whenever \(c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2< 1\) or \( \lambda< \frac{1}{\sqrt{c_{\alpha,\beta}}Lip_{\sigma}}\).
Similarly from Lemma 8, we have
\[\|u-v\|^2_2=\|\mathcal{A} u-\mathcal{A} v\|^2_2\leq c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\|u-v\|_2^2,\] and therefore \[\|u-v\|^2_2\bigg[1-c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2\bigg]\leq 0.\] This implies that \(\|u-v\|_2< 0\) if and only if \(c_{\alpha,\beta}\lambda^2Lip_{\sigma}^2< 1\) and thus \(\|u-v\|_2=0\Rightarrow u=v\). This shows the existence and uniqueness result by Banach's contraction principle.Definition 4. We say that \(\{u(t)\}_{0\leq t\leq 1}\) is a mild solution of Equation (6) if \(a. s\), \begin{equation*} u(t)=\lambda\int_0^1 G(t,s) \sigma(s,u(s))\dot{w}(s)ds=\lambda\int_0^1 G(t,s) \sigma(s,u(s))dw(s), \end{equation*} is satisfied, where \(G(t,s)\) is as given in Equation (5).
If \(\{u(t)\}_{0< t< 1}\) satisfies the additional condition \(\displaystyle\sup_{0\leq t\leq 1}{\mathbb E}|u(t)|^2< \infty,\) then we say that \(\{u(t)\}_{0\leq t\leq 1}\) is a random field solution to Equation (6).
Theorem 2. Suppose Condition 1 holds and there exists a positive constant \(\lambda^*\) such that for any \(0< \lambda\leq\lambda^*\), then Equation (6) has a unique solution.
To proof the Theorem 2, we define the operator
\[\mathcal{B} u(x,t)=\lambda\int_0^1 G(t,s) \sigma(s,u(s))dw(s),\] and use the fixed point of the operator \(\mathcal{B}\). The proof follows using the Lemma(s) below:Lemma 9. Given a random solution \(u\) such that \(\|u\|_2< \infty\) and Condition 1 holds. Then there exists a positive constant \(\lambda^*\) such that for \(0< \lambda\leq\lambda^*\), \[\|\mathcal{B} u\|_2\leq \frac{1}{2}\|u\|_2.\]
Proof. By Itó Isometry, we obtain \begin{eqnarray*} {\mathbb E}|\mathcal{B} u(t)|^2&\leq&\lambda^2\int_0^1 G^2(t,s) {\mathbb E}|\sigma(s,u(s))|^2ds\\ &\leq&\lambda^2\sup_{0\leq t,s\leq 1}G^2(t,s)\int_0^1 p^2(s) {\mathbb E}|u(s)|^2ds\\ &\leq&\lambda^2\bigg[\sup_{0\leq t,s\leq 1}G(t,s)\bigg]^2\int_0^1 p^2(s){\mathbb E}|u(s)|^2ds\\ &\leq&\frac{\lambda^2}{4}\|u\|_2^2\int_0^1 p^2(s) ds. \end{eqnarray*} Taking suprimum of both sides over \(t\in[0,1]\) and letting \(\lambda^*:=\displaystyle\bigg(\int_0^1 p^2(s) ds\bigg)^{-2}\) such that \(0< \lambda\leq\lambda^*\), we have \(\|\mathcal{B} u\|^2_2\leq \frac{1}{4}\|u\|_2^2.\)
Following similar steps of Lemma 9, we obtain the following result.
Lemma 10. Suppose \(u\) and \(v\) are random solutions such that \(\| u\|_{2}+\|v\|_{2}< \infty\) and Condition 1 holds. Then there exists a positive constant \(\lambda^*\) such that for \(0< \lambda\leq\lambda^*\), \[\|\mathcal{B} u-\mathcal{B} v\|_2\leq \frac{1}{2}\|u-v\|_2.\]
Proof of Theorem 2. Following Lemma 9 and Lemma 12, it is clear that \(\mathcal{B}\) is a contraction. Thus by Banach fixed point theorem, the existence of a unique solution for Equation (6) follows: Let \(u=\mathcal{B}\) and \[\|u\|^2_2=\|\mathcal{B} u\|^2_2\leq \frac{1}{4}\|u\|_2^2\Rightarrow \|u\|^2_2\left[1-\frac{1}{4}\right]\leq 0\Rightarrow \|u\|_2=0\Rightarrow u=0.\] Assume a nontrivial solution \(u\) of Equation (6) and we show that it is unique. Suppose for contradiction that there exists another solution \(v\) of Equation (6) such that \[\|u-v\|^2_2=\|\mathcal{B} u-\mathcal{B} v\|^2_2\leq \frac{1}{4}\|u-v\|_2^2,\] then \(\|u-v\|^2_2[1-\frac{1}{4}]\leq 0\), which follows that \(\|u-v\|=0\). Thus \(u=v\), a unique solution.
Next, we seek to establish the existence and uniqueness of solution for Equation (6) using Condition 2 as follows:
Theorem 3. Suppose \(\lambda< \frac{2}{Lip_{\sigma}}\), for positive constant \(Lip_{\sigma}\) together with Condition 2. Then there exists solution \(u\) that is unique up to modification.
Lemma 11. Given a random solution \(u\) such that \(\|u\|_2< \infty\) and Condition 2 holds. Then \(\|\mathcal{B} u\|_2\leq\frac{\lambda Lip_{\sigma}}{2}\|u\|_2.\)
Lemma 12. Suppose \(u\) and \(v\) are random solutions such that \(\| u\|_{2}+\|v\|_{2}< \infty\) and Condition 2 holds. Then \(\|\mathcal{B} u-\mathcal{B} v\|_2\leq\frac{\lambda Lip_{\sigma}}{2}\|u-v\|_2.\)
Proof of Theorem 3. By fixed point theorem we have \(u(t)=\mathcal{A} u(t)\) and \( \|u\|^2_{2}=\|\mathcal{B} u\|^2_{2}\leq \frac{\lambda^2Lip_{\sigma}^2}{4}\|u\|^2_{2}, \) which follows that \(\|u\|^2_{2}\left[1-\frac{\lambda^2Lip_{\sigma}^2}{4}\right]\leq 0\Rightarrow \|u\|_{2}< \infty \Leftrightarrow \lambda< \frac{2}{Lip_{\sigma}}.\)
Similarly, \(\|u-v\|^2_{2}=\|\mathcal{B} u-\mathcal{B} v\|^2_{2}\leq \frac{\lambda^2Lip_{\sigma}^2}{4}\|u-v\|^2_{2},\) thus \(\|u-v\|^2_{2}\left[1-\frac{\lambda^2Lip_{\sigma}^2}{4}\right]\leq 0\) and therefore \(\|u-v\|_{2}< 0\) if and only if \(\lambda< \frac{2}{Lip_{\sigma}}.\) Hence, the existence and uniqueness result follows by Banach's contraction principle.
