On a fixed point theorem of Krasnoselskii for locally convex spaces

Virinda Mohan Sehgal and S. P. Singh

Vol. 62 (1976), No. 2, 561–567

DOI: 10.2140/pjm.1976.62.561

Abstract

Let 𝒰 be a neighborhood basis of the origin consisting of absolutely convex open subsets of a separated locally convex topological vector space E and S a subset of E. Let a mapping f : S → E satisfy the condition: for each U ∈𝒰 and 𝜖 > 0, there exists a δ = δ(𝜖,U) > 0 such that if x,y ∈ S and x − y ∈ (𝜖 + δ)U, then f(x) − f(y) ∈ 𝜖U. In the present paper, sufficient conditions are given for the mapping f to have a fixed point in S. The result is extended to the sum of two mappings of Krasnoselskii type.

Mathematical Subject Classification 2000

Primary: 47H10

Milestones

Received: 21 May 1975

Published: 1 February 1976