Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation
Math Sci
Coincidence and common fixed point results for b-quasi contractive mappings on metric spaces endowed with binary relation
M. I. Ayari 0 1 2 3
M. Berzig 0 1 2 3
I. Ke´dim 0 1 2 3
0 Mathematics Subject Classification 47H10
1 Carthage University, Faculte ́ des Sciences de Bizerte , 7021 Bizerte, Zarzouna , Tunisia
2 Tunis University, E ́ cole Nationale Supe ́rieure d'Inge ́nieurs de Tunis , 5 Avenue Taha Hussein, BP 56, Bab Manara, 1008 Tunis , Tunisia
3 Carthage University, Institu National Des Sciences Applique ́e et de Technologie de Tunis, Centre Urbain Nord , BP 676-1080 Tunis , Tunisia
Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations.
Fixed point theory; Ordered metric spaces; Transitive binary relation
-
54H25
Introduction
The Banach contraction principle [
1
] may be considered as
one of most powerful tools for establishing existence and
uniqueness of solutions for various non linear problems.
&
dðTx; TyÞ
q maxfdðx; yÞ; dðx; TxÞ; dðy; TyÞ; dðx; TyÞ; dðy; TxÞg;
for all x; y 2 X. It is worthwhile noting that the
quasi-contractive condition subsumes the original contraction and
may also be applied to non continuous mappings. Later,
Samet and Turinici [
3
], established an important fixed point
result where the contractive condition is required only on
binary related elements of X. Several known results from
literature can be derived from their main theorem, for
instance we cite those of Kannan [
4
], Chatterjea [
5
], Hardy
and Rogers [
6
], Ran and Reurings [
7
], Nieto and Lo´ pez [
8
],
C´ iric´ [
9
] and Kirk et al. [
10
]. More recently, there have been
some attempts to generalize the fixed point results for
metric spaces endowed with binary relation. For more
details, we refer the reader to [
11–19
].
In this paper, some coincidence point results in complete
metric spaces endowed with transitive binary relations are
first highlighted. Subsequently, the existence and
uniqueness of common fixed point theorem for two mappings are
established under some suitable conditions. Moreover,
some coincidence and common fixed point theorems
involving amorphous binary relation are proven. Finally, as
an application, existence study of solutions to some integral
ð1Þ
equation is presented. The paper is divided into five
sections. Section introduces the notation used herein, presents
a number of definitions and recalls some useful results.
Coincidence and common fixed point theorems are stated
in Sect. 1, while their proofs are the subject of Sect. 2.
Several consequences are subsequently derived in Sect. 3.
Finally, the existence of solutions for some class of
Urysohn integral equation is shown in Sect. 4.
Preliminaries
Let us introduce some definitions and recall some basic
preliminary results which will be needed in the following
sections. Throughout this paper, we denote by N the set of
0 . Let (X, d) be a
all positive integer and N0 ¼ N [ f g
metric space, R be a binary relation on X and g; T : X ! X
be two mappings. Denote by
Cðg; TÞ :¼ fx 2 X : gx ¼ Tx g:
the set of coincidence points of g and T. Let x0 2 X, and
suppose that T ðXÞ gðXÞ. Hence, we can choose x1 2 X
satisfying gx1 ¼ Tx0. Again from T ðXÞ gðXÞ; we can
choose x2 2 X satisfying gx2 ¼ Tx1: Continuing this
process, we construct a sequence fxng satisfying
gxnþ1 ¼ Txn for all n 2 N0:
ð2Þ
Denote by Oðx0Þ the set of all sequences fxng satisfying
(2). Define also the g-orbit set, at s ¼ fxkgk2N0 2 Oðx0Þ,
from p to q with q [ p [ 0, by Op;qðg; sÞ ¼ fgxp; gxpþ1;
. . .; gxqg. Similarly, define the infinite g-orbit set at s from
p by Op;1ðg; sÞ ¼ fgxp; gxpþ1; gxpþ2; . . .g.
Definition 1.1 We say that (X, d) is g-orbitally complete
if and only if for all x0 2 X and s 2 Oðx0Þ: every Cauchy
sequence in O0;1ðg; sÞ converges in X.
Definition 1.2 ([
11
]) A subset D of X is called
R-g-directed if for every x; y 2 D, there exists z 2 X such that
gxRgz and gyRgz.
Definition 1.3 ([
3
]) We say that ðX; d; RÞ is regular if for a
sequence fxng in X, if we have xnRxnþ1 for all n 2 N0 and
limn!1 dðxn; xÞ ¼ 0 for some x 2 X; then there exists a
subsequence fxnðkÞg of fxng such that xnðkÞRx for all k 2 N0.
Definition 1.4 ([
11
]) Let X be a non empty set, g; T :
X ! X be two mappings and R be a binary relation. We
say that T is g-comparative if
x; y 2 X;
gxRgy ¼) TxRTy:
Definition 1.5 Let b 2 ð0; þ1Þ and u : ½0; þ1Þ !
½0; þ1Þ be a function satisfying the properties:
(P1)
u is nondecreasing;
(P2)
(P3)
(P4)
limn!1 unbðtÞ ¼ 0 for all t [ 0, where unb denote
the n-th iterate of ub and ubðtÞ ¼ uðb tÞ;
there exists s 2 ð0; þ1Þ such that Pn1¼1 unbðsÞ
\1;
ðid ubÞ ubðtÞ ub ðid ubÞðtÞforallt 0;
where id : ½0; þ1Þ ! ½0; þ1Þ is the identity
func (...truncated)