Research Article
Tripled Coincidence and Common Fixed Point Results for
Two Pairs of Hybrid Mappings
Marwan Amin Kutbi,
1
Jamshaid Ahmad,
2
Mujahid Abbas,
3
and Muhammad Arshad
4
1
Department of Mathematics, King Abdul Aziz University, Jeddah, Saudi Arabia
2
Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad 44000, Pakistan
3
Department of Mathematics & Applied Mathematics, University of Pretoria, Pretoria 002, South Africa
4
Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan
Correspondence should be addressed to Muhammad Arshad; marshad zia@yahoo.com
Received 26 August 2013; Accepted 24 December 2013; Published 30 January 2014
Academic Editor: Patricia J. Y. Wong
Copyright © 2014 Marwan Amin Kutbi et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Te tripled fxed point is a generalization of the well-known concept of “coupled fxed point.” In this paper, tripled coincidence
and common fxed point results for two hybrid pairs consisting of multivalued and single valued mappings on a metric space are
proved. We give examples to illustrate our results. In the process, several comparable coincidence and fxed point results in the
existing literature are improved, unifed, and generalized.
1. Introduction and Preliminaries
Te study of fxed points for multivalued contraction map-
pings using the Hausdorf metric was initiated by Nadler
Jr. [1]. Afer this, fxed point theory has been developed
further and applied to many disciplines to solve functional
equations. Banach contraction principle has been extended
in diferent directions. Some authors used generalized con-
tractions for multivalued mappings and hybrid pairs of single
and multi-valued mappings, while others used more general
spaces. Dhage [2, 3] established hybrid fxed point theorems
and obtained some applications of presented results. Gnana
Bhaskar and Lakshmikantham [4] introduced the notion of
a coupled fxed point and proved some coupled fxed point
results under certain contractive conditions in a complete
metric space endowed with a partial order. Tey applied their
results to study the existence and uniqueness of solution for
a periodic boundary value problem associated with a frst-
order ordinary diferential equation. Later, Lakshmikantham
and
´
Ciri´ c[5] established the existence of coupled coincidence
point results to generalize the results of Gnana Bhaskar and
Lakshmikantham [4]; Karapınar [6] generalized these results
on a complete cone metric space endowed with a partial
order. Recently, Berinde and Borcut [7, 8] introduced the
concept of a tripled fxed point for nonlinear contractive
mappings in partially ordered complete metric spaces and
obtained tripled coincidence and fxed point results for com-
muting maps. Hussain et al. [9, 10] obtained some coupled
and tripled coincidence results without compatibility. Ili´ c
et al. [11] obtained coupled coincidence and common fxed
point theorems for a hybrid pair of mappings. For other
related results in this direction, we refer to [12–16] and
references mentioned therein. Te purpose of this paper is to
obtain tripled coincidence and common fxed point results
for two hybrid pairs consisting of multivalued and single
valued mappings.
Let us recall some defnitions and well known results
needed in the sequel.
Let (,) be a metric space. For ∈ and ⊆,
we denote (,) = inf {(,) : ∈ }. Te set of all
nonempty bounded and closed subsets of is denoted by
(). Let be the Hausdorf metric induced by the metric
on ; that is,
(,)= max {sup
∈
(,), sup
∈
(,)}, (1)
for every ,∈().
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 803729, 11 pages
http://dx.doi.org/10.1155/2014/803729