Communications in Nonlinear Analysis

Some remarks on tripled fixed point theorems for a sequence of mappings satisfying Geraghty contraction with applications

Document Type : Original Article

Authors

1 Department of Applied Sciences, NITTTR, Under Ministry of HRD, Govt. of India, Bhopal, (M.P.), India,462002.

2 Department of Mathematics, NRI Institute of Research & Technology, Bhopal M.P,India.

3 Zanjan Branch, Islamic Azad University, Zanjan, Iran

Abstract

The purpose of this paper is threefold. Firstly, we establish a tripled coincidence fixed point theorem for a sequence of mappings involving Geraghty contraction using compatibility and weakly reciprocally continuous
maps in the structure of partially ordered metric spaces. The technique used in A. Roldan et al. [9] and
in S. Radenovic [10] are not applicable to the presented theorems, we show that our results cannot be
obtained from the existing results in this eld of study and thus our results are completely new and give rise to
a new dimension. Secondly, the notable works due to V. Berinde [3], V. Lakshmikantam and L. Ciric [8] and
Babu and Subhashini [1] are generalized and extended. Finally, some sufficient conditions are given for the
uniqueness of a tripled common fixed point. Consequently, we point out some slip-ups in the main results
of R. Vats et al.[12] and present a furnished version of the same. Some illustrative examples to highlight
the realized improvements are also furnished. Moreover, existence and uniqueness for the solution of an
initial-boundary-value problem are discussed. On the other hand, as an application to establish existence and
uniqueness for the system of integral equations our results are utilized.

Keywords

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Communications in Nonlinear Analysis
Volume 3, Issue 2
Winter and Spring 2017
Pages 68-86