Fixed point theorems for Kannan type mappings with applications to split feasibility and variational inequality problems

Abstract

The aim of this paper in to introduce a large class of mappings, called enriched Kannan mappings, that includes all Kannan mappings and some nonexpansive mappings. We study the set of fixed points and prove a convergence theorem for Kransnoselskij iteration used to approximate fixed points of enriched Kannan mappings in Banach spaces. We then extend further these mappings to the class of enriched Bianchini mappings. Examples to illustrate the effectiveness of our results are also given. As applications of our main fixed point theorems, we present two Kransnoselskij projection type algorithms for solving split feasibility problems and variational inequality problems in the class of enriched Kannan mappings and enriched Bianchini mappings, respectively.

On some enriched contractions in Banach spaces

“…Remark 3.8. One can perceive from the above remarks that Berinde and Pȃcurar's results regarding existence of fixed point of different types of enriched contractions [3][4][5] are particular cases of our results.…”

“…So it will be attractive to apply enrichment technique to the contractions by the above mentioned mathematicians. Berinde and Pȃcurar did this task for Kannan and Chatterjea mappings, see [3,4]. So it is open in the literature to apply enrichment technique to the contractions of Reich [12], Ćirić [8,9], Bianchini [6], Khan [11] etc.…”

Krasnoselskii iteration process for approximating fixed points of enriched generalized nonexpansive mappings in Banach spaces

“…And, they proved strong convergence theorem for the Krasnoselskii iteration used to approximate the fixed points of enriched Banach contractions. After, some authors [3][4][5] introduced the enriched Kannan mappings, enriched Chatterjea mappings and enriched nonexpansive mappings and they gave convergence results for Krasnoselskii iteration used to approximate fixed points of such mappings in Banach spaces. Definition 1 ([2]).…”

Fixed point theorems of enriched Ciric's type and enriched Hardy-Rogers contractions

“…In 2021, Berinde and Pacurar [6] introduced enriched Chatterjea contraction as a generalization of Chatterjea contraction. Many other contractions, such as enriched Kannan contraction [10] and enriched nonexpansive [5] have been established by various authors. The technique of enriching nonlinear mappings enables for the generalization of essential classes of nonlinear mappings in diverse spaces, including Hilbert spaces, Banach spaces and convex metric spaces.…”