Random fixed point theorems for a certain class of mappings in banach spaces

Abstract

Let (Ω, Σ) 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: Ω × C → C satisfying: for each x, y ∈ C, ω ∈ Ω and integer n ≥ 1, $$\left\| {T^\user1{n} (\omega ,\user1{x}) - T^\user1{n} (\omega ,\user1{x})} \right\|$$ $$ \geqslant \user1{a}(\omega ) \cdot \left\| {\user1{x} - \user1{y}} \right\| + \user1{b}(\omega )\left\{ {\left\| {\user1{x} - T^\user1{n} (\omega ,\user1{x})} \right\| + \left\| {\user1{y} - T^\user1{n} (\omega ,\user1{y})} \right\|} \right\}$$ $$ + \user1{c}(\omega )\left\{ {\left\| {\user1{x} - T^\user1{n} (\omega ,\user1{y})} \right\| + \left\| {\user1{y} - T^\user1{n} (\omega ,\user1{x})} \right\|} \right\},$$ where a, b, c: Ω → [0, ∞) are functions satisfying certain conditions and T n(ω, x) is the value at x of the n-th iterate of the mapping T(ω, ·). 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 for 1 < p < ∞ and k ≥ 0. As a consequence of our main result, we also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41].