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 T ⁿ(, 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].