Fixed points and stability for a sum of two operators in locally convex spaces

George Lee Cain Jr. and Mohammed Zuhair Zaki Nashed

Vol. 39 (1971), No. 3, 581–592

DOI: 10.2140/pjm.1971.39.581

Abstract

Some fixed point theorems for a sum of two operators are proved, generalizing to locally convex spaces a fixed point theorem of M. A. Krasnoselskii, for a sum of a completely continuous and a contraction mapping, as well as some of its recent variants.

A notion of stability of solutions of nonlinear operator equations in linear topological spaces is formulated in terms of specific topologies on the set of nonlinear operators, and a theorem on the stability of fixed points of a sum of two operators is given. As a byproduct, sufficient conditions for a mapping to be open or to be onto are obtained.

Mathematical Subject Classification 2000

Primary: 47H10

Milestones

Received: 10 November 1970

Published: 1 December 1971