Research in probabilistic operator theory generally includes the solutions of random operator equations and random operator inclusion, random extension theorems, limit theorems, measure theoretic problems, spectral theory of random operators and semi groups of random operators and their properties. Various ideas associated with random fixed point theory are used to form a particularly elegant approach for the solution of nonlinear random systems. Now this theory has become full- fledged research area lying at the intersection of nonlinear analysis and probability theory. In this monograpgh those aspects of random solution of random operator equations and random operator inclusion, which fall within the scope of investigation of random fixed point are discussed.

Random Integral Equations

This monograph is devoted to the existence and stability (Ulam-Hyers-Rassias stability and asymptotic stability) of solutions for various classes of functional differential equations or inclusions involving the Hadamard or Hilfer fractional derivative. Some equations present delay which may be finite, infinite, or state-dependent. Others are subject to impulsive effect which may be fixed or non-instantaneous.Readers will find the book self-contained and unified in presentation. It provides the necessary background material required to go further into the subject and explores the rich research literature in detail. Each chapter concludes with a section devoted to notes and bibliographical remarks and all abstract results are illustrated by examples. The tools used include many classical and modern nonlinear analysis methods such as fixed-point theorems, as well as some notions of Ulam stability, attractivity and the measure of non-compactness as well as the measure of weak noncompactness. It is useful for researchers and graduate students for research, seminars, and advanced graduate courses, in pure and applied mathematics, physics, mechanics, engineering, biology, and all other applied sciences.