Research Article
Existence of Strong Coupled Fixed Points for Cyclic Coupled
Ciric-Type Mappings
Xavier Udo-utun
Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria
Correspondence should be addressed to Xavier Udo-utun; xvior@yahoo.com
Received 28 July 2014; Accepted 16 October 2014; Published 23 November 2014
Academic Editor: Gabriel Nagy
Copyright © 2014 Xavier Udo-utun. 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.
In this short communication the concept of cyclic coupled Kannan-type contractions is generalized using a certain class of Ciric-
type mappings.
1. Introduction and Preliminaries
Te Banach contraction condition in a metric space (,)
given by (,) ≤ (,), 0 ≤ < 1, has so
many signifcant generalizations which include the class of
generalized contractions defned by Ciric [1] as follows. A
mapping :→ is called a generalized contraction if
and only if there exist nonnegative numbers . , , and such
that
(,)≤(,)+ (,)+ (,)
+[(,)+(,)];
sup {+++2}<1.
(1)
is called contractive if
(,)<(,). (2)
It is worth mentioning that the contractive condition (2)
restricts applications only to the class of continuous operators
while the contractive conditions (1) accommodate discontin-
uous operators as well. Te search for contractive conditions
that do not require continuity of operators culminated in 1969
with the appearance of the Kannan [2] contractive condition
below:
(, ) ≤ [ (,)+ (,)], 0≤<
1
2
. (3)
Te Chatterjea [3] contractive condition which followed is
independent of both the contractive condition (2) and the
Kannan condition (3) which in turn is independent of (2).
Consequently, unlike condition (2) the Kannan condition
(3) does not generalize the well-known Banach condition
above. In a frst attempt, the three contractive conditions were
combined by Zamfrescu [4] in one theorem to generalize
and extend the Banach fxed point theorem. Following Zam-
frescu Ciric unifed contractive conditions mentioned above
by introducing the larger and unifying class of operators
called quasi-contractions. is called a quasi-contraction
(
´
Ciri´ c[5]) if there exists ∈ (0, 1) such that
(, ) ≤ max [(,), (,), (,),
(,),(,)].
(4)
´
Ciri´ c[5] observed that the class of quasi-contractions
contains the class of generalized contractions as a proper
subclass. Rhoades [6] noted that the Zamfrescu result is
generalized by the Ciric contractive condition (4).
Tere have been numerous generalizations and exten-
sions of the Banach fxed point theorem in literature and they
are, basically, modifcations of those mentioned above. Very
recently Choudhury and Maity [7] introduced the concept
of cyclic coupled Kannan-type contractions and established
a strong cyclic coupled fxed point result below. We recall the
following defnition. Let and be two nonempty subsets
of a given set . A mapping :×→, such that
(, ) ∈ if ∈ and ∈ and (, ) ∈ if ∈ and
∈, is called a cyclic mapping with respect to and .
Hindawi Publishing Corporation
Journal of Operators
Volume 2014, Article ID 381685, 4 pages
http://dx.doi.org/10.1155/2014/381685