| Abstrakt: |
Abstract: 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 $n \ge1$, \aligned\Vert T^n&(\omega, x) - T^n(\omega, y) \Vert \le&a(\omega)\cdot\Vert x - y \Vert+ b(\omega)\{ \Vert x - T^n(\omega,x) \Vert+ \Vert y - T^n(\omega,y) \Vert\} &+ c(\omega)\{ \Vert x - T^n(\omega,y) \Vert+ \Vert y - T^n(\omega,x) \Vert\}, 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 $T(\omega,\cdot)$. 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 $1 < p < \infty$ and $k \ge0$. 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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