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On some common random fixed point theorems with PPF dependence in Banach spaces. (English) Zbl 1290.47055
Given two Banach spaces $E_1$ and $E_2$ and a measurable space $(\Omega, {\cal A})$, by a random operator it is meant a mapping $Q: \Omega\times E_1\to E_2$ such that $\omega\mapsto Q(\omega, x)$ is measurable for all $x\in E_1$. Let $a< b$ and $[a,b]$ be a bounded closed interval in $\Bbb R$. Given a Banach space $E$, the space of all $E$-valued continuous functions on $[a,b]$ is denoted by $E_0$ which is in turn a Banach space with the norm $\|x\|_{E_0}= \sup_{t\in [a,b]} \|x(t)\|_E$. A PPF random operator is, in fact, a random operator $Q:\Omega\times E_0\to E$ and a measurable function $\xi^*:\Omega\to E_0$ is a PPF random fixed point for $Q$ if $Q(\omega, \xi^*(\omega))= \xi^*(c, \omega)$ for some $c\in [a,b]$. The aim of the paper is to prove some PPF random fixed point theorems for a pair of continuous random operators in Banach spaces satisfying a more general contraction condition.
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34K10Boundary value problems for functional-differential equations
47H40Random operators (nonlinear)