\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 235, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/235\hfil Random differential equations] {Solutions to fourth-order random differential equations with periodic boundary conditions} \author[X. Han, X. Ma, G. Dai \hfil EJDE-2012/235\hfilneg] {Xiaoling Han, Xuan Ma, Guowei Dai} \address{Xiaoling Han\newline Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China} \email{hanxiaoling@nwnu.edu.cn} \address{Xuan Ma \newline Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China} \email{sfsolo@163.com} \address{Guowei Dai \newline Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China} \email{daiguowei@nwnu.edu.cn} \thanks{Submitted October 17, 2012. Published December 21, 2012.} \thanks{Supported by grants 11101335 and 11261052 from the National Natural Science Foundation \hfill\break\indent of China} \subjclass[2000]{47H40, 47N20, 60H25} \keywords{Random differential equation; periodic boundary conditions; \hfill\break\indent random solution; extremal solutions} \begin{abstract} Existence of solutions and of extremal random solutions are proved for periodic boundary-value problems of fourth-order ordinary random differential equations. Our investigation is done in the space of continuous real-valued functions defined on closed and bounded intervals. Also we study the applications of the random version of a nonlinear alternative of Leray-Schauder type and an algebraic random fixed point theorem by Dhage. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Let $\mathbb{R}$ denote the real line and let $J =[0, 1]$, a closed and bounded interval in $\mathbb{R}$. Let $C^1(J, \mathbb{R})$ denote the class of real-valued functions defined and continuously on $J$. Given a measurable space $(\Omega, \mathcal{A})$ and a measurable function $x: \Omega \to AC^{3}(J, \mathbb{R})$, we consider a fourth-order periodic boundary-value problem of ordinary random differential equations (for short PBVP) $$\begin{gathered} x^{(4)}(t,\omega)=f(t,x(t,\omega),x''(t,\omega),\omega) \quad \text{a.e. } t\in J\\ x^{(i)}(0,\omega)=x^{(i)}(1,\omega),\quad i=0,1,2,3 \end{gathered} \label{e1}$$ for all $\omega\in \Omega$, where $f:J\times\mathbb{R}\times\mathbb{R}\times\Omega\to\mathbb{R}$. By a \emph{random solution} of equation \eqref{e1} we mean a measurable function $x:\Omega\to AC^{(3)}(J,\mathbb{R})$ that satisfies the equation \eqref{e1}, where $AC^{(3)}(J,\mathbb{R})$ is the space of real-valued functions whose 3rd derivative exists and is absolutely continuously differentiable on $J$. When the random parameter $\omega$ is absent, the random \eqref{e1} reduce to the fourth-order ordinary differential equations, $$\begin{gathered} x^{(4)}(t)=f(t,x(t),x''(t)) \quad\text{a.e. } t\in J\\ x^{(i)}(0)=x^{(i)}(1), \quad i = 0, 1, 2, 3 \end{gathered}\label{e2}$$ where, $f:J\times\mathbb{R}\to\mathbb{R}$. Equation \eqref{e2} has been studied by many authors for different aspects of solutions. See for example \cite{j1,j2,l3,l4,m1}. Only a few authors have studied the random periodic boundary-value problem, see \cite{d4,c1,z1}, Dhage \cite{d4} studied the periodic boundary-value problems for the random differential equation \begin{gather*} - x''(t,\omega)=f(t,x(t,\omega),\omega) \quad\text{a.e. } t \in J,\\ x(0,\omega)=x(2\pi,\omega), \quad x'(0,\omega)=x'(2\pi,\omega)\,. \end{gather*} In this article, we study the existence of solutions and the existence of extremal solutions for the fourth-order random equation \eqref{e1}, under suitable conditions. Our work relays on the random versions of fixed point theorems based on the theorems in \cite{d1,d2}. \section{Existence result} Let $E$ denote a Banach space with the norm $\|\cdot\|$ and let $Q:E\to E$. We further assume that the Banach space $E$ is separable; i.e., $E$ has a countable dense subset and let $\beta_{E}$ be the $\sigma$-algebra of Borel subsets of $E$. We say a mapping $x:\Omega\to E$ is measurable if for any $B\in\beta_{E}$, $$x^{-1}(B)=\{\omega\in\Omega: x(\omega)\in B\}\in\mathcal{A}.$$ To define integrals of sample paths of random process, it is necessary to define a map is jointly measurable, a mapping $x:\Omega\times E\to E$ is called $jointly measurable$, if for any $B\in\beta_{E}$, one has $$x^{-1}(B)=\{(\omega,x)\in\Omega\times E: x(\omega,x)\in B\} \in\mathcal{A}\times\beta_{E},$$ where $\mathcal{A}\times\beta_{E}$ is the direct product of the $\sigma$-algebras $\mathcal{A}$ and $\beta_{E}$ those defined in $\Omega$ and $E$ respectively. Let $Q:\Omega\times E\to E$ be a mapping. Then $Q$ is called a random operator if $Q(\omega,x)$ is measurable in $\omega$ for all $x\in E$ and it is expressed as $Q(\omega)x=Q(\omega,x)$. A random operator $Q(\omega)$ on $E$ is called continuous (resp. compact, totally bounded and completely continuous) if $Q(\omega,x)$ is continuous (resp. compact, totally bounded and completely continuous) in $x$ for all $\omega\in\Omega$. We could get more details of completely continuous random operators on Banach spaces and their properties in Itoh \cite{i1}. In this article, we use the following lemma in proving the main result of this paper, that lemma is an immediate corollary to the results in \cite{d1,d2}. \begin{lemma}[\cite{d4}] \label{lem2.1} Let $\mathcal{B}_{R}(0)$ and $\bar{\mathcal{B}}_{R}(0)$ be the open and closed balls centered at origin of radius $R$ in the separable Banach space $E$ and let $Q:\Omega\times\bar{\mathcal{B}}_{R}(0)\to E$ be a compact and continuous random operator. Further suppose that there does not exists an $u\in E$ with $\| u\|=R$ such that $Q(\omega)u=\alpha u$ for all $\alpha\in \Omega$, where $\alpha>1$. Then the random equation $Q(\omega)x=x$ has a random solution; i.e., there is a measurable function $\xi:\Omega\to \bar{\mathcal{B}}_{R}(0)$ such that $Q(\omega)\xi(\omega)=\xi(\omega)$ for all $\omega\in\Omega$. \end{lemma} \begin{lemma}[\cite{d4}] \label{lem2.2} Let $Q:\Omega\times E\to E$ be a mapping such that $Q(\cdot,x)$ is measurable for all $x\in E$ and $Q(\omega,\cdot)$ is continuous for all $\omega\in \Omega$. Then the map $(\omega,x)\to Q(\omega,x)$ is jointly measurable. \end{lemma} We need the following definitions in the sequel. \begin{definition} \label{def2.1}\rm A function $f:J\times\mathbb{R}\times\mathbb{R}\times\Omega\to\mathbb{R}$ is called random Carath\'eodory if \begin{itemize} \item the map $(t,\omega)\to f(t,x,y,\omega)$ is jointly measurable for all $(x,y)\in\mathbb{R}^{2}$, and \item the map $(x,y)\to f(t,x,y,\omega)$ is continuous for almost all $t\in J$ and $\omega\in \Omega$. \end{itemize} \end{definition} \begin{definition} \label{def2.2}\rm A function $f:J\times\mathbb{R}\times\mathbb{R}\times\Omega\to \mathbb{R}$ is called random $L^1$-Carath\'eodory if \begin{itemize} \item %[(3)] for each real number $r>0$ there is a measurable and bounded function $q_{r}:\Omega\to L^1(J,\mathbb{R})$ such that \begin{gather*} | f(t,x,y,\omega)| \leq q_{r}(t,\omega) \quad\text{a.e. } t\in J \end{gather*} whenever $| x|,| y| \leq r$, and for all $\omega\in\Omega$. \end{itemize} \end{definition} Now we seek the random solutions of \eqref{e1} in the Banach space $C(J,\mathbb{R})$ of continuous real-valued functions defined on $J$. We equip this space with the supremum norm $$\| x\|=\sup_{t\in J}| x(t)|.$$ It is know that the Banach space $C(J,\mathbb{R})$ is separable. We use $L^1(J,\mathbb{R})$ denote the space of Lebesque measurable real-valued functions defined on $J$, and the usual norm in $L^1(J,\mathbb{R})$ defined by $$\| x\|_{L^1}=\int_{0}^1| x(t)| dt.$$ For a given real number $M\in(0,4\pi^{4}), h\in C(J,\mathbb{R})$, consider the linear PBVP $$\begin{gathered} x^{(4)}(t)+Mx(t)=h(t) \quad t\in J\\ x^{(i)}(0)=x^{(i)}(1), \quad i = 0, 1, 2, 3. \end{gathered} \label{e3}$$ By the theorem of \cite{l2}, the unique solution of problem $$\begin{gathered} x^{(4)}(t)+Mx(t)=0 \quad t\in J\\ x^{(i)}(0)=x^{(i)}(1), \quad i = 0, 1, 2\\ x^{(3)}(0)-x^{(3)}(1)=1 \end{gathered}\label{e4}$$ has a unique solution $r(t)\in C^{4}(J,\mathbb{R})$ satisfying $r(t)>0$. Then the unique solution of \eqref{e3} is $$x(t)=\int^1_{0}G(t,s)h(s)ds, \label{e5}$$ where G(t,s)= \begin{cases} r(t-s), & 0\leq s\leq t\leq1; \\ r(1+t-s), & 0\leq t0$such that $$R>r_{M}\| \gamma(\omega)\|_{L^1}\psi(R)\label{e7}$$ for all$t\in J$and$\omega\in \Omega$, where$r_{M}=\max_{t\in[0,1]}r(t), r(t)$is in the Green's function \eqref{e6}. Then \eqref{e1} has a random solution defined on$J$. \end{theorem} \begin{proof} Set$E=C(J,\mathbb{R})$and define a mapping$Q:\Omega\times E\to E$by $$Q(\omega)x(t)=\int_{0}^1G(t,s)(f(s,x(s,\omega), x''(s,\omega),\omega)+Mx(s,\omega))ds \label{e8}$$ for all$t\in J$,$\omega\in\Omega$. Then the solutions of \eqref{e1} are fixed points of operator$Q$. Define a closed ball$\bar{\mathcal{B}}_{R}(0)$in$E$centered at origin 0 of radius$R$, where the real number$R$satisfies the inequality \eqref{e7}. We show that$Q$satisfies all the conditions of Lemma2.1 on$\bar{\mathcal{B}}_{R}(0)$. First we show that$Q$is a random operator in$\bar{\mathcal{B}}_{R}(0)$, since$f(t,x,x'',\omega)$is random Carath\'{e}odory and$x(t,\omega)$is measurable, the map$\omega\to f(t,x,x'',\omega)+Mx$is measurable. Similarly, the production$G(t,s)[f(s,x(s,\omega),x''(s,\omega),\omega)+Mx(s,\omega)]$of a continuous and measurable function is again measurable. Further, the integral is a limit of a finite sum of measurable functions, therefore, the map $$\omega\mapsto\int_{0}^1G(t,s)(f(s,x(s,\omega),x''(s,\omega),\omega) +Mx(s,\omega))ds=Q(\omega)x(t)$$ is measurable. As a result,$Q$is a random operator on$\Omega\times\bar{\mathcal{B}}_{R}(0)$into$E$. Next we show that the random operator$Q(\omega)$is continuous on$\bar{\mathcal{B}}_{R}(0)$. Let${x_{n}}$be a sequence of points in$\bar{\mathcal{B}}_{R}(0)$converging to the point$x$in$\bar{\mathcal{B}}_{R}(0). Then it is sufficiente to prove that $$\lim_{n\to\infty}Q(\omega)x_{n}(t)=Q(\omega)x(t) \quad\text{for all } t\in J, \omega\in\Omega.$$ By the dominated convergence theorem, we obtain \begin{align*} \lim_{n\to\infty}Q(\omega)x_{n}(t) &= \lim_{n\to\infty}\int_{0}^1G(t,s)(f(s,x_{n}(s,\omega),x_{n}''(s,\omega),\omega) +Mx_{n}(s,\omega))ds\\ &= \int_{0}^1G(t,s)\lim_{n\to\infty}[f(s,x_{n}(x,\omega),x''_{n}(s,\omega),\omega) +Mx_{n}(s,\omega)]ds\\ &= \int_{0}^1G(t,s)[f(s,x(s,\omega),x''(s,\omega),\omega)+Mx_{n}(s,\omega)]ds\\ &= Q(\omega)x(t) \end{align*} for allt\in J, \omega\in\Omega$. This shows that$Q(\omega)$is a continuous random operator on$\bar{\mathcal{B}}_{r}(0)$. Now we show that$Q(\omega)$is compact random operator on$\bar{\mathcal{B}}_{R}(0)$. To finish it, we should prove that$Q(\omega)(\bar{\mathcal{B}}_{r}(0))$is uniformly bounded and equi-continuous set in$E$for each$\omega\in \Omega$. Since the map$\omega\to \gamma(t,\omega)$is bounded and$L^{2}(J,\mathbb{R})\subset L^1(J,\mathbb{R})$, by (H$_2$), there is a constant$c$such that$\| \gamma(\omega)\|_{L^1}\leq c$for all$\omega\in\Omega$. Let$\omega\in \Omega$be fixed, then for any$x:\Omega\to\bar{\mathcal{B}}_{R}(0), one has \begin{align*} | Q(\omega)x(t)| &\leq \int_{0}^1G(t,s)| (f(s,x(s,\omega),x''(s,\omega),\omega)+Mx(s,\omega))| ds\\ &\leq \int_{0}^1G(t,s) \gamma(s,\omega)\psi(| x(s,\omega)|)ds\\ &\leq r_{M}c\psi(R) = K \end{align*} for allt\in J$and each$\omega \in \Omega$. This shows that$Q(\omega)(\bar{\mathcal{B}}_{R}(0))$is a uniformly bounded subset of$E$for each$\omega\in \Omega$. Next we show$Q(\omega)(\bar{\mathcal{B}}_{R}(0))$is an equi-continuous set in$E$. For any$x\in\bar{\mathcal{B}}_{R}(0)$,$t_1,t_2\in J, we have \begin{align*} | Q(\omega)x(t_1)-Q(\omega)x(t_2)| &\leq \int_{0}^1| (G(t_1,s)-G(t_2,s)) | \gamma(s,\omega)\psi(| x(s,\omega)|)ds\\ &\leq \int_{0}^1| (G(t_1,s)-G(t_2,s))| \gamma(s,\omega) \psi(R)ds, \end{align*} by h\"{o}lder inequality, \begin{align*} &| Q(\omega)x(t_1)-Q(\omega)x(t_2)| \\ &\leq \Big(\int_{0}^1| G(t_1,s)-G(t_2,s)|^{2}ds\Big)^{1/2} \Big(\int_{0}^1| \gamma(s,\omega)|^{2}ds\Big)^{1/2}\psi(R). \end{align*} Hence for allt_1,t_2\in J$, $$| Q(w)x(t_1)-Q(\omega)x(t_2)|\to 0 \quad\text{as } t_1\to t_2$$ uniformly for all$x\in\bar{\mathcal{B}}_{R}(0)$. Therefore,$Q(\omega)\bar{\mathcal{B}}_{R}(0)$is an equi-continuous set in$E$, then we know it is compact by Arzel\'{a}-Ascoli theorem for each$\omega\in \Omega$. Consequently,$Q(\omega)$is a completely continuous random operator on$\bar{\mathcal{B}}_{R}(0)$. Finally, we suppose there exists such an element$u$in$E$with$\| u\|=R$satisfying$Q(\omega)u(t)=\alpha u(t,\omega)$for some$\omega\in \Omega$, where$\alpha>1$. Now for this$\omega\in\Omega, we have \begin{align*} | u(t,\omega)| &\leq \frac{1}{\alpha}| Q(\omega)u(t)|\\ &\leq \int_{0}^1G(t,s)| f(s,u(s,\omega),u''(s,\omega),\omega)+Mu(s,\omega)| ds\\ &\leq r_{M}\int_{0}^1 \gamma(s,\omega)\psi(| u(s,\omega)|)ds\\ &\leq r_{M}\| \gamma(\omega)\|_{L^1}\psi(\| u(\omega)\|) \quad \text{for all }t\in J. \end{align*} Taking supremum overt$in the above inequality yields $$R = \| u(\omega)\| \leq r_{M}\| \gamma(\omega)\|_{L^1}\psi(R)$$ for some$\omega\in \Omega$. This contradicts to condition \eqref{e7}. Thus, all the conditions of Lemma2.1 are satisfied. Hence the random equation $$Q(\omega)x(t)=x(t,\omega)$$ has a random solution in$\bar{\mathcal{B}}_{R}(0)$; i.e., there is a measurable function$\xi:\Omega\to\bar{\mathcal{B}}_{R}(0)$such that$Q(\omega)\xi(t)=\xi(t,\omega)$for all$t\in J, \omega\in\Omega$. As a result, the random \eqref{e1} has a random solution defined on$J$. This completes the proof. \end{proof} \section{Extremal random solutions} It is sometimes desirable to know the realistic behavior of random solutions of a given dynamical system. Therefore, we prove the existence of extremal positive random solution of \eqref{e1} defined on$\Omega\times J$. We introduce an order relation$\leq$in$C(J,\mathbb{R})$with the help of a cone$K$defined by $$K=\{x\in C(J,\mathbb{R}): x(t)\geq 0 \text{ on } J\}.$$ Let$x,y\in X$, then$x\leq y$if and only if$y-x\in K$. Thus, we have $$x\leq y\; \Leftrightarrow\; x(t)\leq y(t)\text{ for all } t\in J.$$ It is known that the cone$K$is normal in$C(J,\mathbb{R})$. For any function$a,b:\Omega\to C(J,\mathbb{R})$we define a random interval$[a,b]$in$C(J,\mathbb{R})$by $[a,b] = \{x\in C(J,\mathbb{R}): a(\omega)\leq x\leq b(\omega)\, \forall \omega\in\Omega\} = \cap_{\omega\in\Omega}[a(\omega),b(\omega)].$ \begin{definition} \label{def3.1}\rm An operator$Q:\Omega\times E\to E$is called nondecreasing if$Q(\omega)x\leq Q(\omega)y$for all$\omega\in\Omega$, and for all$x,y\in E$for which$x\leq y$. \end{definition} We use the following random fixed point theorem of Dhage in what follows. \begin{lemma}[{Dhage \cite{d1}}] \label{lem3.1} Let$(\Omega,\mathcal{A})$be a measurable space and let$[a,b]$be a random order interval in the separable Banach space$E$. Let$Q:\Omega\times[a,b]\to[a,b]$be a completely continuous and nondecreasing random operator. Then$Q$has a minimal fixed point$x_{*}$and a maximal random fixed point$y^{*}$in$[a,b]$. Moreover, the sequences$\{Q(\omega)x_{n}\}$with$x_{0}=a$and$\{Q(\omega)y_{n}\}$with$y_{0}=b$converge to$x_{*}$and$y^{*}$respectively. \end{lemma} We need the following definitions in the sequel. \begin{definition} \label{def3.2}\rm A measurable function$\alpha:\Omega\to C(J,\mathbb{R})$is called a lower random solution of \eqref{e1} if \begin{gather*} \alpha ^{(4)}(t,\omega)\leq f(t,\alpha(t,\omega),\alpha(t,\omega),\omega) \quad\text{a.e. } t \in J.\\ \alpha^{(i)}(0,\omega)= \alpha^{(i)}(1,\omega), \quad i=0,1,2.\\ \alpha^{(3)}(0,\omega)\leq \alpha^{(3)}(1,\omega) \end{gather*} for all$\omega\in\Omega$. Similarly, a measurable function$\beta:\Omega\to C(J,\mathbb{R})$is called an upper random solution of \eqref{e1} if \begin{gather*} \beta^{(4)}(t,\omega)\geq f(t,\alpha(t,\omega),\alpha(t,\omega),\omega) \quad\text{a.e. } t \in J.\\ \beta^{(i)}(0,\omega)= \beta^{(i)}(1,\omega), \quad i=0,1,2.\\ \beta^{(3)}(0,\omega)\geq \beta^{(3)}(1,\omega) \end{gather*} for all$t\in J$and$ \omega\in\Omega$. \end{definition} \begin{definition} \label{def3.3}\rm A random solution$\theta$of \eqref{e1} is called maximal if for all random solutions of \eqref{e1}, one has$x(t,\omega)\leq \theta(t,\omega)$for all$t\in J$and$\omega\in\Omega$. A minimal random solution of \eqref{e1} on$J$is defined similarly, \end{definition} We consider the following set of assumptions: \begin{itemize} \item[(H3)] Problem \eqref{e1} has a lower random solution$\alpha$and upper random solution$\beta$with$\alpha\leq\beta$on$J$. \item[(H4)] For any$u_2,u_1\in[\alpha,\beta]$and$u_2>u_1$$$f(t,u_2,v,\omega)-f(t,u_1,v,\omega)\geq -M(u_1-u_2)$$ for a.e.$t\in[0,1]$and$\omega\in\Omega$. \item[(H5)] The function$q:J\times\Omega\to\mathbb{R}_{+}$defined by $$q(t,\omega)=| f(t,\alpha(t,\omega),\alpha''(t,\omega),\omega) +M\alpha(t,\omega)|+| f(t,\beta(t,\omega),\beta''(t,\omega),\omega) +M\beta(t,\omega)|$$ is Lebesgue integrable in$t$for all$\omega\in\Omega$. \end{itemize} Hypotheses (H3) holds, in particular, when there exist measurable functions$u,v:\Omega\to C(J,\mathbb{R})$such that for each$\omega\in\Omega$, $$u(t,\omega)\leq f(t,x,y,\omega)+Mx\leq v(t,\omega)$$ for all$t\in J$and$x\in \mathbb{R}$. In this case, the lower and upper random solutions of \eqref{e1} are given by $$\alpha(t,\omega)=\int_{0}^1G(t,s)u(s,\omega)ds$$ and $$\beta(t,\omega)=\int_{0}^1G(t,s)v(s,\omega)ds$$ respectively. The details about the lower and upper random solutions for different types of random differential equations could be found in \cite{l1}. Hypotheses (H4) is natural and used in several research papers. Finally, if$f$is$L^1$-Carath\'{e}odory on$\mathbb{R}\times\Omega$, then (H5) remains valid. \begin{theorem} \label{thm3.1} Assume that {\rm (H), (H3)--(H5)} hold, then \eqref{e1} has a minimal random solution$x_{*}(\omega)$and a maximal random solution$y^{*}(\omega)$defined on$J$. Moreover, $x_{*}(t,\omega)=\lim_{n\to\infty}x_{n}(t,\omega) ,\quad y^{*}(t,\omega)=\lim_{n\to\infty}y_{n}(t,\omega)$ for all$t\in J$and$\omega\in\Omega$, where the random sequences$\{x_{n}(\omega)\}$and$\{y_{n}(\omega)\}$are given by $x_{n+1}(t,\omega) = \int_{0}^1G(t,s)(f(s,x_{n}(s,\omega),x_{n}''(s,\omega) ,\omega)+Mx_{n}(s,\omega))ds$ for$n\geq 0$with$x_{0}=\alpha$, and $y_{n+1}(t,\omega) = \int_{0}^1G(t,s)(f(s,y_{n}(s,\omega),y_{n}''(s,\omega), \omega)+My_{n}(s,\omega))ds$ for$n\geq 0$with$y_{0}=\beta$, for all$t\in J$and$\omega\in\Omega$. \end{theorem} \begin{proof} We Set$E=C(J,\mathbb{R})$and define an operator$Q:\Omega\times[\alpha,\beta]\to E$by \eqref{e8}. We show that$Q$satisfies all the conditions of Lemma3.1 on$[\alpha,\beta]$. It can be shown as in the proof of Theorem 2.1 that$Q$is a random operator on$\Omega\times[\alpha,\beta]$. We show that it is nondecreasing random operator on$[\alpha,\beta]$. Let$x,y:\Omega\to[\alpha,\beta]$be arbitrary such that$x\leq y$on$\Omega. Then \begin{align*} &Q(\omega)y(t)-Q(\omega)x(t)\\ &=\int_{0}^1 G(t,s)\Big[\big(f(s,y(s,\omega),y''(s,\omega),\omega) -f(s,x(s,\omega),x''(s,\omega),\omega)\big)\\ & \quad +M(y(s,\omega)-x(s,\omega))\Big]ds\\ &\geq \int_{0}^1G(t,s)[(-M(y(s,\omega)-x(s,\omega)) +M(y(s,\omega)-x(s,\omega)]ds=0 \end{align*} for allt\in J$and$\omega\in\Omega$. As a result,$Q(\omega)x\leq Q(\omega)y$for all$\omega\in\Omega$and that$Q$is nondecreasing random operator on$[\alpha,\beta]. Now, by (H4), \begin{align*} \alpha(t,\omega) &\leq Q(\omega)\alpha(t)\\ &= \int_{0}^1G(t,s)[f(\alpha(s,\alpha'(s,\omega), \alpha''(s,\omega),\omega),\omega)+M\alpha(s,\omega)]ds\\ &\leq \int_{0}^1G(t,s)f(s,x'(s,\omega),x''(s,\omega),\omega)+Mx(s,\omega)ds\\ &= Q(\omega)x(t)\\ &\leq Q(\omega)\beta(t)\\ &= \int_{0}^1G(t,s)[f(\beta(s,\beta'(s,\omega),\beta''(s,\omega),\omega),\omega) +M\beta(s,\omega)]ds\\ &\leq \beta(t,\omega) \end{align*} for allt\in J$and$\omega\in\Omega$. As a result$Q$defines a random operator$Q:\Omega\times[\alpha,\beta]\to[\alpha,\beta]$. Then, since (H5) holds, we replace$\gamma(t,\omega)$and$\psi(r)$with$\gamma(t,\omega)=q(t,\omega)$for all$(t,\omega)\in J\times\Omega$and$\psi(R)=1$for all real number$R\geq 0$. Now it can be show as in the proof of Theorem 2.1 that the random operator$Q(\omega)$satisfies all the conditions of Lemma 3.1 and so the random operator equation$Q(\omega)x=x(\omega)$has a least and a greatest random solution in$[\alpha,\beta]$. Consequently, \eqref{e1} has a minimal and a maximal random solution defined on$J\$. The proof is complete. \end{proof} \begin{thebibliography}{99} \bibitem{c1} Y. Chang, Z. Zhao, Gaston M. N'Guerekata; \emph{Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space}, Adv. Difference Equ. 2011 (2011), 9-12. \bibitem{d1} B. C. Dhage; \emph{Some algebraic and topological random fixed point theorem with applications to nonlinear random intefralequations}, Tamkang J. Math. 35 (2004), 321-345. \bibitem{d2} B. C. Dhage; \emph{A random version of a Schaefer type fixed point theorem with applications to functional random integral equations}, Tamkang J. Math. 35 (2004), 197-205. \bibitem{d3} B. C. Dhage; \emph{Monotone iterative technique for Carath\'{e}odory theorem of nonlinear functional random integral equations}, Tamkang J. Math. 35 (2002), 341-351. \bibitem{d4} B. C. Dhage, S. V. Badgire, S. K. Ntouyas; \emph{Periodic boundary value problems of second order random differential equations}, Electron. J. Qaul. Theory Diff. Equ., 21 (2009), 1-14. \bibitem{i1} S. Itoh; \emph{Random fixed point theorems with applications to random differential equations in Banach spaces}, J. Math. Anal. Appl. 67 (1979), 261-273. \bibitem{j1} D. Jiang, W. Gao, A. Wan; \emph{A monotone method for constructing extremal solutions to fourth order periodic boundary value problems}. Appl. Math. Comput. 132 (2002), 411-421. \bibitem{j2} D.Jiang, Z. Liu, L. Zhang; \emph{Optimal existence theory for single and multiple positive solutions to fourth order periodic boundary value problems}. Nonlinear Anal. 7 (2006), 841-852. \bibitem{l1} G. S. Ladde, V. Lakshmikantham; \emph{Random Differential Inequalities}, Academic Press, New York, 1980. \bibitem{l2} Y. Li; \emph{Positive solution of higher order periodic boundary value problems}, Comput. Math. Appl. 48 (2004), 153-161. \bibitem{l3} Y. Li; \emph{Positive solutions of fourth order periodic boundary value problems}, Nonlinear. Anal. 54 (2003), 1069-1078. \bibitem{l4} R. Liu; \emph{The existence and multiplicity of positive solutions to fourth-order periodic boundary value problems}, Journal of Lanzhou University. 41 (1) (2005), 112-114. \bibitem{m1} R. Ma, C. Gao, Y. Chang; \emph{Existence of solutions of a discrete fourth-order boundary value problem}, Discrete dynamics in nature and society. 2010 (2010), 1-19. \bibitem{z1} Z. Zhao, Y. Chang, W. Li; \emph{Asymptotically almost periodic, almost periodic and pseudo-almost periodic mild solutions for neutral differential equations}, Nonlinear Anal. Real World Appl. 11 (2010), 3037-3044. \end{thebibliography} \end{document}