Random fixed point theorems for multivalued nonlinear mapping (Q2755167)

Using the implicit iteration and selector theorems the authors prove some new random fixed point theorems for some classes of multivalued nonlinear random mappings. One of the presented results is the following. Let \(D\) be a nonempty bounded closed convex subset of a separable reflexive Banach space \(X\) and let \(T:\Omega\times D\times D\to\text{CB}(D)\) be a random non-expansive mapping, that is NEWLINE\[NEWLINEH(T(\omega,x,y),T(\omega,z,t))\leq\max\{\|x-y\|, \|y-t\|\}, \quad \forall x,y,z,t\in D,\;\omega\in\Omega,NEWLINE\]NEWLINE where \(\text{CB}(D)\) is a collection of nonempty closed bounded subsets in \(D\); \(H(A,B)\) is the Hausdorff metric. Moreover \(T(\Omega\times D\times D)\) is assumed to be compact and if an obtainable sequence \(\{x_{n}(\omega)\}\) of measurable mappings \(x_{n}:\Omega\to D\), \(x_{n}(\omega)\in T(\omega,x_{n}(\omega),x_{n-1}(\omega))\), \(n\geq 1\), has a convergent subsequence \(\{x_{n_{k}}(\omega)\}\), then \(\lim_{k}\|x_{n_{k}}(\omega)-x_{n_{k}-1}(\omega)\|=0\). Then the operator \(T\) has a random fixed point defined by the weak limit of a subsequence of the sequence of solutions to the equation \(x_{n}(\omega)\in T(\omega, x_{n}(\omega),x_{n-1}(\omega))\), \(n\geq 1\), with a given measurable mapping \(x_0:\Omega\to D\).