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Keywords:
$p$-uniformly convex Banach space; normal structure; asymptotic center; random fixed points; generalized random uniformly Lipschitzian mapping
Summary:
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 $T\: \Omega \times C \rightarrow C$ satisfying: for each $x, y \in C$, $\omega \in \Omega $ and integer $n \ge 1$, \[\Vert T^n(\omega , x) - T^n(\omega , y) \Vert \le a(\omega )\cdot \Vert x - y \Vert + b(\omega )\lbrace \Vert x - T^n(\omega ,x) \Vert + \Vert y - T^n(\omega ,y) \Vert \rbrace + c(\omega )\lbrace \Vert x - T^n(\omega ,y) \Vert + \Vert y - T^n(\omega ,x) \Vert \rbrace , \] where $a,b,c\: \Omega \rightarrow [0, \infty )$ 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 $k \ge 0$. 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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