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,
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].
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Jung, J.S., Cho, Y.J., Kang, S.M. et al. Random fixed point theorems for a certain class of mappings in banach spaces. Czechoslovak Mathematical Journal 50, 379–396 (2000). https://doi.org/10.1023/A:1022483205068
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DOI: https://doi.org/10.1023/A:1022483205068