Random fixed points of \(K\)-set- and pseudo-contractive random maps
Let \(M\) be a measurable space, \(X\) a separable Banach space and \(C\) a weakly compact subset of \(X\). Assume that \(T: M\times C\to X\) is a continuous 1-set-contractive random operator. The problem of the existence of random fixed points of \(T\), i.e., the problem of the existence of solutions of the random equation \[ x- T(w,x)= 0,\quad w\in M,\quad x\in X, \] has been considered by many mathematicians. In the paper under review, the author discusses the stochastic solvability of the equation \[ f(w,x)- T(w,x)= 0,\quad w\in M,\quad x\in X, \] where \(f\), \(F\) are continuous random operators. Next, the main result is applied to establish some random fixed point theorems for various classes of \(k\)-set-contractive or so-called pseudo-contractive random operators. In the proofs, some results of Opial, Kuratowski, Ryll-Nardzewski, Petryshyn and Browder are used.
scientific article; zbMATH DE number 2102564