Random directional contractors and their applications (Q1091677)

In this paper the author introduces the concept of random directional contractors for set-valued random operators. This is a generalization of directional contractors introduced by \textit{M. Altman} [see Pac. J. Math. 62, 1-18 (1976; Zbl 0352.47027)] and random contractors studied by \textit{A. C. H. Lee} and \textit{W. J. Padgett} [see Nonlinear Anal., Theory, Methods Appl. 1, 175-185 (1977; Zbl 0347.60045)]. Using this concept and transfinite induction, he obtains several existence theorems for generalized equations of the following types: \(0\in P(\omega,x)\), u(\(\omega)\in P(\omega,x)\) and \(x\in P(\omega,x)\), where a set-valued random operator P and a random variable u are given. Some consequences for single-valued random operators are applied to nonlinear random integral and differential equations.