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On some common random fixed point theorems with PPF dependence in Banach spaces. (English) Zbl 06174791
Given two Banach spaces ${E}_{1}$ and ${E}_{2}$ and a measurable space $\left({\Omega },𝒜\right)$, by a random operator it is meant a mapping $Q:{\Omega }×{E}_{1}\to {E}_{2}$ such that $\omega ↦Q\left(\omega ,x\right)$ is measurable for all $x\in {E}_{1}$. Let $a and $\left[a,b\right]$ be a bounded closed interval in $ℝ$. Given a Banach space $E$, the space of all $E$-valued continuous functions on $\left[a,b\right]$ is denoted by ${E}_{0}$ which is in turn a Banach space with the norm ${\parallel x\parallel }_{{E}_{0}}={sup}_{t\in \left[a,b\right]}{\parallel x\left(t\right)\parallel }_{E}$. A PPF random operator is, in fact, a random operator $Q:{\Omega }×{E}_{0}\to E$ and a measurable function ${\xi }^{*}:{\Omega }\to {E}_{0}$ is a PPF random fixed point for $Q$ if $Q\left(\omega ,{\xi }^{*}\left(\omega \right)\right)={\xi }^{*}\left(c,\omega \right)$ for some $c\in \left[a,b\right]$. The aim of the paper is to prove some PPF random fixed point theorems for a pair of continuous random operators in Banach spaces saitsfying a more general contraction condition.
##### MSC:
 34K10 Boundary value problems for functional-differential equations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H40 Random operators (nonlinear)
##### Keywords:
separable Banach space; random fixed point; PPF dependence