The Method of Random Contractors and Its Applications to Random Nonlinear Equations

Abstract

This chapter discusses the method of random contractors and its applications to random nonlinear equations. Random operator equations often arise in the formulation of mathematical models in the biological, engineering, and physical sciences, and such equations are frequently nonlinear by the nature of the problems. Therefore, the development of very general techniques for solving random nonlinear operator equations has been an active area of mathematical and probabilistic research over the past two decades. The chapter reviews the results on random contractors and their applications to highlight the existence, uniqueness, measurability, and approximation of solutions to random nonlinear operator equations. It presents basic definitions concerning random operators, random solutions to random operator equations, and some special spaces of functions that are useful in the study of random operator equations.