Fixed points of Suzuki contractive mappings in relational metric spaces
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Fixed points of Suzuki contractive mappings in relational metric spaces Gopi Prasad1
· Ramesh Chandra Dimri1 · Ayush Bartwal1
Received: 11 July 2019 / Accepted: 25 November 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, we prove an analogous version of fixed point theorem due to Suzuki (Proc Am Math Soc 136:1861–1869, 2008) using R-sequential limit property and some other relation theoretic metrical variants. The present results generalize well known recent results such as Paesano and Vetro (Topol Appl 159:911–920, 2012), Alam and Imdad (J Fixed Point Theory Appl 17(4):693–702, 2015) and besides many others. Radically, these investigations open another new direction of relational metric fixed point theory. We also present non-trivial example to show the validity and importance of such investigations. Keywords Binary relation · R-completeness · R-continuity. Mathematics Subject Classification 47H10 · 54H25
1 Introduction The classical Banach contraction principle [8] has numerous generalizations, extensions and applications. In 2008 Suzuki [28] introduced new generalization of the Banach contraction principle which characterizes the metric completeness of the underlying space. Thereafter several generalizations of the theorem due to Suzuki has been obtained by many researchers [9,14–16,27] in various metric spaces. In 2012 Paesano and Vetro [22] generalize the Suzukitype fixed point theorem in partial and partially ordered metric spaces. On the other hand, analogue of the Banach contraction principle for monotone mapping endowed with partial order relation in metric spaces can be traced back to Turinici [29, 32] which was later undertaken by several researchers [1–7,11–13,18,20–26,30,31]. They
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Gopi Prasad [email protected] Ramesh Chandra Dimri [email protected] Ayush Bartwal [email protected]
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Department of Mathematics, H.N.B. Garhwal University, Srinagar (Garhwal), Uttarakhand 246174, India
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G. Prasad et al.
also presented several applications to linear and nonlinear matrix equations and integrodifferential equations. Recently, Alam and Imdad [5] presented another new variant of Banach contraction principle on complete metric space endowed with a binary relation in which they used the relation theoretic analogues of certain involved metrical notions such as contraction, completeness, continuity etc. utilized by earlier authors, in fact under the universal relation such newly identified notions reduced to their corresponding usual notions. The main goal in this paper is to investigate Suzuki-type fixed point theorem in complete metric space using relation theoretic metrical notions. In this context, the contractive condition is relatively weaker than the usual contraction as it applies only on those elements which are related under the underlying binary relation rather than whole space. In this way these findings revel another direction of relational metric fixed-point theory.
2 Preliminaries Throughout this paper, R stands for a non-empty bin
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