On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings

Abstract

The purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation T(u,C(u))=u using Kannan-type equicontractive mappings, T:A×C(A)¯→Y, where C is a compact mapping, A is a bounded, closed and convex subset of a Banach space Y. To achieve this objective, the authors have presented Sadovskii’s theorem, which utilizes the measure of noncompactness. The relevance of the obtained results has been illustrated through the consideration of various initial value problems.

Keywords

• Hausdorff measure of noncompactness
• compact mapping
• Kannan-type equicontraction
• initial value problem