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Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x) is the value...

Let $(\Omega ,\Sigma )$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p>1$. We prove random fixed point theorems for a class of mappings $T\Omega \times C\to C$ satisfying: for each $x,y\in C$, $\omega \in \Omega$ and integer $n\ge 1$, $$\parallel T^n(\omega ,x)-T^n(\omega ,y)\parallel \le a\left(\omega \right)·\parallel x-y\parallel +b\left(\omega \right)\{\parallel x-T^n(\omega ,x)\parallel +\parallel y-T^n(\omega ,y)\parallel \}+c\left(\omega \right)\{\parallel x-T^n(\omega ,y)\parallel +\parallel y-T^n(\omega ,x)\parallel \},$$ where $a,b,c\Omega \to [0,\infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,·)$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p}$...