Vol. 62, No. 2, 1976

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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

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
Received: 21 May 1975
Published: 1 February 1976
Virinda Mohan Sehgal
S. P. Singh