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

Let (, ) 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: × C C satisfying: for each x, y C, and integer n 1, where a, b, c: [0, ) are functions satisfying certain conditions and Tⁿ(, x) is the value at x of the n-th iterate of the mapping T(, ·). 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].