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A Suzuki Type Unique Common Coupled Fixed Point Theorem in Metric Spaces

K.P.R.Rao1, K.R.K.Rao2, V.C.C.Raju3
  1. Professor, Dept. of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, A.P., India
  2. Assistant Professor, Dept. of Mathematics, VBIT, Ghatkesar-501 301, A.P., India
  3. Professor, Department of Mathematics, University of Botswana, Private Bag UB 00704, Gaborone, Botswana
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Abstract

In this article, we study a unique common coupled fixed point theorem of Suzuki type for Jungck type mappings in metric spaces. Our result generalize and modify several comparable results in the literature.

Keywords
Coupled fixed point, metric space, weakly compatible maps.

Mathematics Subject Classification
54H25, 47H10.

INTRODUCTION AND PRELIMINARIES
Banach contraction principle plays a very important role in nonlinear analysis and has many generalizations.Recently Suzuki [ 31 ] proved generalized versions of both Banach’s and Edelstein’s basic results and thus initiated a lot of work in this direction, for example refer [3,4,5,8,13,18-21,24-28,31,32] and the references in them .
In 2006, Bhaskar and Lakshmikantham [29]introduced the notion of a coupled fixed point in partially ordered metric spaces,also discussed some problems of the uniqueness of a coupled fixed point and applied their results to the problems of the existence and uniqueness of a solution for the periodic boundary value problems. Later several authors obtained coupled fixed point theorems in various spaces,for example refer [1,2,6,7,9-12,14-17,22,23,29,30,33-36] and the references in them.
The aim of this paper is to combine the ideas of coupled fixed points and Suzuki type fixed point theorems to obtain a unique common coupled fixed point theorem for Jungck type mappings in a metric space.
First we give the following theorem of Suzuki [31].
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