Generalizations of Kannan and Reich Fixed Point Theorems, Using Sequentially Convergent Mappings and Subadditive Altering Distance Functions

Alireza Pourmoslemi,Shayesteh Rezaei,Mehdi Salimi,Tahereh Nazari

doi:10.3390/math8091432

Abstract

In this paper, first, using interpolative Kannan type contractions, a new fixed point theorem has been proved. Then, by applying sequentially convergent mappings and using subadditive altering distance functions, we generalize contractions in complete metric spaces. Finally, we investigate some fixed point theorems which are generalizations of Kannan and Reich fixed points.

Highlights

Introduction and PreliminariesFixed point theory is one of the most popular tools for solving optimization and approximation problems in nonlinear analysis
One such attempt is due to Reich, who proved that if θ is a self-map on a complete metric space (T, d) that satisfies d(θ (q), θ (s)) ≤ αd(q, θ (q)) + βd(s, θ (s)) + γd(q, s), (3)
In 2011, new generalizations of Kannan fixed point theorem on a complete metric space are investigated as follows: Theorem 1 ([8])

Summary

Introduction and Preliminaries

Fixed point theory is one of the most popular tools for solving optimization and approximation problems in nonlinear analysis. If the metric space (T , d) is complete the mapping satisfying (1) has a unique fixed point [1]. The significance of the Kannan fixed point theorem appeared in Subrahmanyam paper [4] He showed that a metric space is complete if and only if every Kannan type mapping has a fixed point. The Banach and Kannan fixed point theorems have been improved by various successful attempts One such attempt is due to Reich, who proved that if θ is a self-map on a complete metric space (T , d) that satisfies d(θ (q), θ (s)) ≤ αd(q, θ (q)) + βd(s, θ (s)) + γd(q, s),. In 2011, new generalizations of Kannan fixed point theorem on a complete metric space are investigated as follows: Theorem 1 ([8]). A mapping θ : T → T is an interpolative Kannan type contraction if there exists a constant 0 ≤ ζ < 1 and 0 < μ < 1 such that d(θ (q), θ (s)) ≤ ζ [d(q, θ (q))]μ .[d(s, θ (s))]1−μ , for all q, s ∈ T with q 6= θ (q)

Generalized Kannan and Reich Fixed Point Theorems

Conclusions