Set valued Reich type G -contractions in a complete metric space with graph
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Set valued Reich type G-contractions in a complete metric space with graph Pradip Debnath1
· Murchana Neog2 · Stojan Radenovi´c3
Received: 10 January 2019 / Accepted: 19 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, set valued Reich type G-contractions are introduced in the framework of a complete metric space equipped with graph. Existence and uniqueness of fixed points for these contractions are established. We provide the graph version of Reich’s theorems for set valued mappings. Further, Kannan’s theorem in this context has been stated as a consequence. Examples have been constructed to signify the validity of our results. Keywords Fixed point · Complete metric space · set valued mapping · Kannan type G-contraction · Reich type G-contraction Mathematics Subject Classification 47H10 · 54H25 · 54E50
1 Introduction with preliminaries A map satisfying the Banach Contraction Principle is necessarily continuous. In 1968, an interesting generalization of Banach’s Principle was put forward by Kannan [17] where the contractive self map did not have to be continuous to admit a fixed point. Another significant aspect of Kannan’s theorem was found by Subrahmanyam [28] that it gives characterization for the metric completeness. A necessary and sufficient condition for a metric space to be complete is that every Kannan type mapping on it has a fixed point. Due to such utilities
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Pradip Debnath [email protected] Murchana Neog [email protected] Stojan Radenovi´c [email protected]
1
Department of Applied Science and Humanities, Assam University, Silchar, Cachar, Assam 788011, India
2
Department of Mathematics, North Eastern Regional Institute of Science and Technology, Nirjuli, Arunachal Pradesh 791109, India
3
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
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P. Debnath et al.
Kannan’s theorem has been generalized for different purposes, e.g. see [6,7,18,20]. Recently, Gornicki [13] provided an elementary proof of original Kannan’s theorem. Below we list some important definitions and results that are necessary for our main results. Throughout N, R and R+ denote the set of natural numbers, set of real numbers and set of nonnegative real numbers respectively. In 1968, Kannan proved the following fixed point theorem. Theorem 1.1 [17] Consider the complete metric space (, δ). The mapping L : → is called a Kannan map if there exists some λ ∈ [0, 1) such that λ [δ(u, Lu) + δ(v, Lv)] 2 for all u, v ∈ . L has a unique fixed point. δ(Lu, Lv) ≤
Reich again generalized the Bananch’s Contraction Principle in the following manner and observed that Kannan’s theorem reduces to a particular case of it with suitable choice of the constants. Theorem 1.2 [25] Consider the complete metric space (, δ). Let L : → be a self-map satisfying the following property: δ(Lu, Lv) ≤ l · δ(u, Lu) + m · δ(v, Lv) + n · δ(u, v), for all u, v ∈ where l, m, n ∈ R+ satisfying l + m + n < 1. Then L has a unique fixed point.
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