Fixed points of Kannan contractive mappings in relational metric spaces
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ORIGINAL RESEARCH PAPER
Fixed points of Kannan contractive mappings in relational metric spaces Gopi Prasad1 Received: 4 June 2020 / Accepted: 14 September 2020 Ó Forum D’Analystes, Chennai 2020
Abstract In this paper, we prove an analogous version of fixed point theorem due to Kannan [Bull. Calcutta Math. Soc., 60, (1968), 71–76] via conditional contractive mappings and some other relational metrical notions. In the proof of present results, we utilize the notion of comparable mappings and some other well known classical fixed point theorems in the settings of relational and ordered metric spaces. We also, highlight the close connection of a-admissible mappings with the binary relation and partial order relation as well. Radically, these investigations open another new direction of metric fixed point theory for contractive type mappings. Moreover, non-trivial examples are given to demonstrate the importance and usefulness of such findings. Keywords Fixed point R-complete metric spaces Kannan contractive mapping
AMS Subject Classification: 47H10 54H25
1 Introduction After the famous Banach contraction principle [8] in 1922, the existence of fixed points for contractive mappings was presented by Kannan [15] in 1968. His fixed point results has laid down the foundation of modern fixed point theory for such types of mappings in the metric and Banach spaces. It is well known that every Banach contraction mapping is continuous however, Kannan [15] showed that a discontinuous function satisfying their contractive condition has unique fixed point and presented the following theorem : & Gopi Prasad [email protected] 1
Department of Mathematics, H.N.B. Garhwal University, Srinagar Garhwal, Uttarakhand, India
123
G. Prasad
Theorem 1.1 ([15]). Let (X, d) be a complete metric space and T be a self-mapping on X. If T is Kannan contraction i.e. there exist b 2 ½0; 12Þ such that dðTx; TyÞ b½dðx; TxÞ þ dðy; TyÞfor allx; y 2 X:
ð1:1Þ
Then T has a unique fixed point z 2 X and for each x 2 X the sequence of iterates fT n xg converges to z. It is important to note that the class of Kannan contractive mappings is independent to that of Banach contractions see, for instance ([18] and [27]). Thereafter many extensions and generalizations of the Kannan fixed point theorem were published and there exists an extensive literature on this theme but keeping in light of the necessity of the presentation of this article we merely refer to ([9, 10, 13, 18, 24, 25, 27, 34]). On the other hand recently, an analogue of the Banach contraction principle for monotone mappings equipped with partial order relation in metric spaces can be traced back to Turinici [30, 31] which was later generalized and extended by several mathematicians [1, 4, 11, 12, 19, 23, 26, 28, 32, 33]. The respective authors also presented several applications of these fixed point theorems to linear and nonlinear matrix equations and integro-differential equations. Meanwhile, one of the interesting metric fixed po
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