Periodic points of random multivalued operators (Q2866643)

By a (set-valued) random operator it is meant a (set-valued) mapping defined on the product space \(\Omega\times X\) (\(\Omega\) a measurable space and \(X\) a separable metric space) which is measurable with respect to the first variable for each fixed value of the second variable. The random operator \(T\) is said to be \(\varepsilon\)-contractive if \(x\neq y\) and \(d(x, y) <\varepsilon\), then \(d(T(.,x), T(., y)) <d(x,y)\) and \(\varepsilon\)-expansive if \(d(T(., x), T(., y))> d(x,y)\). The aim of the paper is to prove the existence of random periodic point for random \(\varepsilon\)-contractive operators on separable metric spaces. Then the authors apply the results to obtain random periodic points for random \(\varepsilon\)-expansive operators.