Some remarks on Krasnoselskii’s fixed point theorem. (English) Zbl 1072.47051
The authors give results dealing with the following theorem of Krasnoselskii. Theorem: Let \(C\) be a closed bounded convex nonempty subset of a Banach space \(X\). Suppose that \(Ax + By\) lies in \(C\) for all \(x,y\) in \(C\), where \(A\) is continuous and \(A(C)\) is contained in a compact set, \(B\) is a contraction mapping with constant \(k\), \(0 < k < 1\). Then there is an \(x\) in \(C\) with \(x = Ax + Bx\).
In the present paper, two other possibilities are discussed and interesting results with corollaries are proved. In the end, the problem of the existence of solutions for a system of equations is discussed by using Schaefer’s theorem.
In the present paper, two other possibilities are discussed and interesting results with corollaries are proved. In the end, the problem of the existence of solutions for a system of equations is discussed by using Schaefer’s theorem.
Reviewer: S. P. Singh (London, Ontario)