Coupled fixed point theorems for

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Open Math. 2015; 13: 877–888
Open Mathematics
Open Access
Research Article
Jianhua Chen and Xianjiu Huang*
Coupled fixed point theorems for
.˛; '/g -contractive type mappings
in partially ordered G -metric spaces
DOI 10.1515/math-2015-0082
Received May 12, 2015; accepted September 28, 2015.
Abstract: In this paper, we introduce a new concept of .˛; '/g -contractive type mappings and establish coupled
coincidence and coupled common fixed point theorems for such mappings in partially ordered G-metric spaces. The
results on fixed point theorems are generalizations of some existing results. We also give some examples to illustrate
the usability of the obtained results.
Keywords: Partially ordered set, Coincidence point, Coupled fixed point, .˛; '/g -contractive type mappings,
G-metric space
MSC: 47H10, 54H25
1 Introduction and preliminaries
Fixed point theory is one of the most powerful and fruitful tools in nonlinear analysis, differential equation, and
economic theory and has been studied in many various metric spaces. Especially, in 2006, Mustafa and Sins [1]
introduced a generalized metric spaces which are called G-metric space. Following Mustafa and Sins’ work, many
authors developed and introduced various fixed point theorems in G-metric spaces (see [17–21]). Recently, some
authors have been interested in partially ordered G-metric spaces and proved some fixed theorems. Simultaneously,
fixed point theory has developed rapidly in partially ordered metric spaces [15, 16]. Fixed point theorems have also
been considered in partially ordered probabilistic metric spaces [7], in partially ordered cone metric spaces [12, 13],
and in partially ordered G-metric spaces [2–6, 8–11, 27, 28]. In particular, in [3], Bhaskar and Lakshmikantham
introduced notions of a mixed monotone mapping and a coupled fixed point, proved some coupled fixed point
theorems for mixed monotone mappings, and discussed the existence and uniqueness of solutions for periodic
boundary value problems. Afterwards, some coupled fixed point and coupled coincidence point results and their
applications have been established.
Some authors studied fixed point theorems for ˛- -contractive type mappings in various spaces. For example,
in [24], Samet et al. introduced ˛- -contractive type mappings and proved some fixed point theorem for such
mappings in metric spaces. In [25], Murseleen et al. introduced ˛- -contractive type mappings and proved some
fixed point theorem for such mappings in partially metric spaces. Recently, in [26], Ali et al. has introduced
˛- -contractive type mappings and proved some fixed point theorem for such mappings in uniform spaces.
Throughout this paper, let N denote the set of nonnegative integers, and RC be the set of positive real numbers.
Before giving our main results, we recall some basic concepts and results in G-metric spaces.
Jianhua Chen: Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, P. R. China and School of Mathematics
and Statistics, Central South University, Changsha, 410083, Hunan, P. R. China, E-mail: [email protected]
*Corresponding Author: Xianjiu Huang: Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, P. R. China,
E-mail: [email protected]
© 2015 Chen and Huang, published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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Definition 1.1 ([1]). Let X be a non-empty set, G W X X X ! RC be a function satisfying the following
properties:
(G1) G.x; y; z/ D 0 if x D y D z.
(G2) 0 < G.x; x; y/ for all x; y 2 X with x ¤ y.
(G3) G.x; x; y/ G.x; y; z/ for all x; y; z 2 X with y ¤ z.
(G4) G.x; y; z/ D G.x; z; y/ D G.y; z; x/ D : : :(symmetry in all three variables).
(G5) G.x; y; z/ G.x; a; a/ C G.a; y; z/ for all x; y; z; a 2 X (rectangle inequality).
Then the function G is called a generalized metric and the pair .X; G/ is called a G-metric space.
Definition 1.2 ([1]). Let .X; G/ be a G-metric space and let fxn g be a sequence of points of X. A point x 2 X
is said to be the limit of the sequence fxn g if lim G.xn ; xn ; xm / D 0, and one says the sequence fxn g is
n;m!1
G-convergent to x.
Thus, if xn ! x in G-metric space .X; G/ then, for any > 0, there exists a positive integer N such that
G.x; xn ; xm / < for all n; m > N .
On one hand, in [1], the authors have shown that the G-metric induces a Hausdorff topology, and the convergence
described in the above definition is relative to this topology. The topology being Hausdorff, a sequence can converge
at most to a point. On the other hand, the authors achieve some conclusions. In case of being tedious, we will not list
those conclusions which were obtained by [1].
Next, we begin with some definitions and conclusions which will be needed in the sequel.
Definition 1.3 ([3]). Let .X; / be a partially ordered set and let F W X X ! X . The mapping F is said to have
the mixed monotone property if F .x; y/ is monotone non-decreasing in x and is monotone non-increasing in y; that
is, for any x; y 2 X ,
x1 ; x2 2 X; x1 x2 ) F .x1 ; y/ F .x2 ; y/
and
y1 ; y2 2 X; y1 y2 ) F .x; y1 / F .x; y2 /:
Definition 1.4 ([3]). An element .x; y/ 2 X X is called a coupled fixed point of the mapping F W X X ! X if
x D F .x; y/ and y D F .y; x/:
Definition 1.5 ([8]). Let .X; / be a partially ordered set and F W X X ! X and g W X ! X be two mappings.
We say that F has the mixed-g-monotone property if F .x; y/ is g-monotone nondecreasing in x and it is g-monotone
nonincreasing in y, that is, for any x; y 2 X , we have:
x1 ; x2 2 X;
gx1 gx2 ) F .x1 ; y/ F .x2 ; y/
y1 ; y2 2 X;
gy1 gy2 ) F .x; y1 / F .x; y2 /:
and, respectively,
Definition 1.6 ([8]). An element .x; y/ 2 X X is called a coupled coincidence point of the mappings F W X X !
X and g W X ! X if
gx D F .x; y/ and gy D F .y; x/:
Definition 1.7 ([8]). We say that the mappings F W X X ! X and g W X ! X are commutative if
g.F .x; y// D F .gx; gy/ f or al l x; y 2 X:
In [8], Lakshmikantham and Ćirić considered the following class of functions. We denote by ˆ the set of functions
' W Œ0; C1/ ! Œ0; C1/ satisfying
(a) '
1
f0g D f0g.
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(b) '.t/ < t for all t > 0.
(c) lim '.r/ < t for all t > 0.
r!t C
Hence, it concluded that lim ' n .t / D 0.
n!1
Choudhury et al. [5] proved the following theorems.
Theorem 1.8. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is a
complete G-metric space. Let F W X X ! X be continuous having the mixed monotone property on X . Assume
that there exists k 2 Œ0; 1/ such that
k
.G.x; u; w/ C G.y; v; z//
(1)
2
for all x; y; z; u; v; w 2 X with w u x and y v z where either u ¤ w or v ¤ z. If there exist x0 ; y0 2 X
such that x0 F .x0 ; y0 / and y0 F .y0 ; x0 /, then F and g have a coupled coincidence point, that is, there exists
.x; y/ 2 X X such that x D F .x; y/ and y D F .y; x/.
G.F .x; y/; F .u; v/; F .w; z// Aydi et al. [15] proved the following theorems.
Theorem 1.9. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is
a complete G-metric space. Let F W X X ! X and g W X ! X be such that F is continuous and has the
mixed-g-monotone property. Assume there is a function ' 2 ˆ such that
G.gx; gu; gw/ C G.gy; gv; gz/
(2)
G.F .x; y/; F .u; v/; F .w; z// '
2
for all x; y; z; u; v; w 2 X with gw gu gx and gy gv gz. Suppose also that F .X X / g.X / and
g is continuous and commutes with F . If there exist x0 ; y0 2 X such that gx0 F .x0 ; y0 / and gy0 F .y0 ; x0 /,
then F and g have a coupled coincidence point, that is, there exists .x; y/ 2 X X such that gx D F .x; y/ and
gy D F .y; x/.
The purpose of this paper is to introduce a new concept of .˛; '/g -contractive type mappings and establish some
fixed theorems for such mappings in partially ordered G-metric spaces. Our results extend and generalize some
results obtained by [5] and [15]. Some examples are presented to support our main results.
2 Main results
In this section, we give coupled coincidence and coupled common fixed theorems for .˛; '/g -type contractive
mappings in partially ordered G-metric spaces. Our results extend some existing results in [5, 15]. Now, we give
the following definitions.
Definition 2.1. Let .X; G; / be a partially ordered G-metric space and F W X X ! X and g W X ! X be two
mappings. Then a map F is said to be .˛; '/g -contractive if there exist two functions ˛ W X 2 X 2 X 2 ! Œ0; C1/
and ' 2 ˆ such that
G.gx; gu; gz/ C G.gy; gv; gw/
(3)
˛..gx; gy/; .gu; gv/; .gz; gw//G.F .x; y/; F .u; v/; F .z; w// '
2
for all x; y; u; v; z; w 2 X with gx gu gz and gy gv gw.
Remark 2.2.
(1) If F
W X X
! X satisfies (1), then F is a .˛; '/g -contractive mapping, where
˛..gx; gy/; .gu; gv/; gz; gw// D 1 for all x; y; u; v; z; w 2 X , g D IX and '.t / D k t for all t 0
and some k 2 Œ0; 1/.
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(2) If ' satisfies (2), then F is a .˛; '/g -contractive mapping, where ˛..gx; gy/; .gu; gv/; .gz;
gw// D 1 for all x; y; u; v; z; w 2 X .
Definition 2.3. Let F W X X ! X , g W X ! X and ˛ W X 2 X 2 X 2 ! Œ0; C1/ be three mappings. Then F
is said to be g-˛-admissible if
˛..gx; gy/; .gu; gv/; .gz; gw// 1 ) ˛..F .x; y/; F .y; x//; .F .u; v/; F .v; u//; .F .z; w/; F .w; z/// 1;
for all x; y; u; v; z; w 2 X .
Before presenting our main fixed point results for .˛; '/g -contractive type mappings, we show some examples to
illustrate the potential applicability of Definition 2.3.
Example 2.4. Let X D Œ0; C1/. Define F W X X ! X , g W X ! X and ˛ W X 2 X 2 X 2 ! Œ0; C1/ by
gx D 12 x, F .x; y/ D x for all x; y 2 X and
(
2 if gx gy;
˛..gx; gy/; .gu; gv/; .gz; gw// D
0 others:
Then F is g-˛-admissible.
Example 2.5. Let X D Œ0; C1/. Define F W X X ! X , g W X ! X and ˛ W X 2 X 2 X 2 ! Œ0; C1/ by
gx D 12 x, F .x; y/ D x 2 for all x; y 2 X and
( x y
e 2 if gx gy;
˛..gx; gy/; .gu; gv/; .gz; gw// D
0
others:
Then F is g-˛-admissible.
Example 2.6. Let X D Œ0; C1/. Define F W X X ! X , g W X ! X and ˛ W X 2 X 2 X 2 ! Œ0; C1/ by
gx D e x , F .x; y/ D x 2 C y 2 for all x; y 2 X and
˛..gx; gy/; .gu; gv/; .gz; gw// D e x
Then F is g-˛-admissible.
Next, we prove our main results under .˛; '/g -contractive conditions.
Theorem 2.7. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is a
complete G-metric space. Let F W X X ! X and g W X ! X be such that F has the mixed-g-monotone property.
Assume there is a function ' 2 ˆ and ˛ W X 2 X 2 X 2 ! Œ0; C1/ for all x; y; z; u; v; w 2 X , the following
hold:
G.gx; gu; gz/ C G.gy; gv; gw/
(4)
˛..gx; gy/; .gu; gv/; .gz; gw//G.F .x; y/; F .u; v/; F .z; w// '
2
for all gw gu gx and gy gv gz. Suppose also that F .X X / g.X /, g is continuous and commutes
with F and
(i) F is g-˛-admissible,
(ii) there exist x0 ; y0 2 X such that
˛..gx0 ; gy0 /; .F .x0 ; y0 /; F .y0 ; x0 //; .F .x0 ; y0 /; F .y0 ; x0 /// 1
(5)
˛..gy0 ; gx0 /; .F .y0 ; x0 /; F .x0 ; y0 //; .F .y0 ; x0 /; F .x0 ; y0 /// 1;
(6)
and
(iii) F is continuous.
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If there exist x0 ; y0 2 X such that gx0 F .x0 ; y0 / and gy0 F .y0 ; x0 /, then F and g have a coupled
coincidence point, that is, there exists .x; y/ 2 X X such that gx D F .x; y/ and gy D F .y; x/.
Proof. Let x0 ; y0 2 X be such that gx0 F .x0 ; y0 / and gy0 F .y0 ; x0 /. Since F .X X / g.X /, we can
choose x1 ; y1 2 X such that gx1 D F .x0 ; y0 / and gy1 D F .y0 ; x0 /. Again since F .X X / g.X /, we can
choose x2 ; y2 2 X such that gx2 D F .x1 ; y1 / and gy2 D F .y1 ; x1 /. Since F has the mixed g-monotone property,
we have gx0 gx1 gx2 and gy2 gy1 gy0 . Continuing this process, we can construct two sequences fxn g
and fyn g in X such that
gxn D F .xn 1 ; yn 1 / gxnC1 D F .xn ; yn /
and
gynC1 D F .yn ; xn / gyn D F .yn
1 ; xn 1 /:
If for some n, we have .gxnC1 ; gynC1 / D .gxn ; gyn /, then F .xn ; yn / D gxn and F .yn ; xn / D gyn , that is, F
and g have a coincidence point. So from now on, we assume .gxnC1 ; gynC1 / ¤ .gxn ; gyn / for all n 2 N, that is,
we assume that either gxnC1 D F .xn ; yn / ¤ gxn or gynC1 D F .yn ; xn / ¤ gyn . Since F is g-˛ admissible, we
have
˛..gx0 ; gy0 /; .gx1 ; gy1 /; .gx1 ; gy1 // D ˛..gx0 ; gy0 /; .F .x0 ; y0 /; F .y0 ; x0 //; .F .x0 ; y0 /; F .y0 ; x0 /// 1
) ˛..F .x0 ; y0 /; F .y0 ; x0 //; .F .x1 ; y1 /; F .y1 ; x1 //; .F .x1 ; y1 /; F .y1 ; x1 ///
D ˛..gx1 ; gy1 /; .gx2 ; gy2 /; .gx2 ; gy2 // 1:
Thus, by mathematical induction, we have
˛..gxn ; gyn /; .gxnC1 ; gynC1 /; .gxnC1 ; gynC1 // 1
(7)
˛..gyn ; gxn /; .gynC1 ; gxnC1 /; .gynC1 ; gxnC1 // 1
(8)
and similarly,
for all n 2 N. Using (4) and (7), we obtain
G.gxn ; gxnC1 ; gxnC1 / D G.F .xn 1 ; yn 1 /; F .xn ; yn /; F .xn ; yn //
˛..gx
.xn 1 ; yn
n 1 ; gyn 1 /; .gxn ; gyn /; .gxn ; gyn //G.F G.gxn 1 ; gxn ; gxn / C G.gyn 1 ; gyn ; gyn /
:
'
2
1 /; F .xn ; yn /; F .xn ; yn //
(9)
Similarly, we have
G.gyn ; gynC1 ; gynC1 / D G.F .yn 1 ; xn 1 /; F .yn ; xn /; F .yn ; xn //
˛..gy
n 1 ; gxn 1 /; .gyn ; gxn /; .gyn ; gxn //; G.F.xn
G.gyn 1 ; gyn ; gyn / C G.gxn 1 ; gxn ; gxn /
'
:
2
1 ; yn 1 /; F .yn ; xn /; F .yn ; xn //
(10)
Adding (9) and (10), we get
G.gyn ; gynC1 ; gynC1 / C G.gxn ; gxnC1 ; gxnC1 /
G.gyn
'
2
1 ; gyn ; gyn /
C G.gxn
2
1 ; gxn ; gxn /
:
Repeating the above process, since ' is nondecreasing, we get
G.gxn ; gxnC1 ; gxnC1 / C G.gyn ; gynC1 ; gynC1 /
n G.gx0 ; gx1 ; gx1 / C G.gy0 ; gy1 ; gy1 /
'
2
2
for all n 2 N. Hence, for any > 0 there exists n./ 2 N such that
X
G.gx0 ; gx1 ; gx1 / C G.gy0 ; gy1 ; gy1 /
'n
< :
2
2
nn./
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Let n; m 2 N be such that m > n > n./. Then by using the triangle inequality, we have
m
X
G.gxn ; gxm ; gxm / C G.gyn ; gym ; gym /
G.gxk ; gxkC1 ; gxkC1 / C G.gyk ; gykC1 ; gykC1 /
2
2
kDn
m
1
X
G.gx0 ; gx1 ; gx1 / C G.gy0 ; gy1 ; gy1 /
'k
2
kDn
X
G.gx
;
gx
;
gx
0
1
1 / C G.gy0 ; gy1 ; gy1 /
'n
2
nn./
< :
2
This implies that G.gxn ; gxm ; gxm / C G.gyn ; gym ; gym / < 2 . Since
G.gxn ; gxm ; gxm / G.gxn ; gxm ; gxm / C G.gyn ; gym ; gym / <
2
and
;
2
and hencefgxn g and fgyn g are G-Cauchy sequences in the G-metric space .X; G/. Now, since .X; G/ is
G-complete, there are x; y 2 X such that fgxn g and fgyn g are respectively G-convergent to x and y, that is,
from the definition of limit, we have
G.gyn ; gym ; gym / G.gxn ; gxm ; gxm / C G.gyn ; gym ; gym / <
lim
n!C1
F .xn ; yn / D
lim
n!C1
g.xn / D x;
lim
n!C1
F .yn ; xn / D
lim
n!C1
g.yn / D y:
(11)
Since g is continuous and commutes with F , hence we have
gx D lim gF .xn ; yn / D lim F .gxn ; gyn / D F .x; y/;
(12)
gy D lim gF .yn ; xn / D lim F .gyn ; gxn / D F .y; x/:
(13)
n!1
n!1
and
n!1
n!1
The proof is completed.
Remark 2.8.
(1) In Theorem 2.7, taking ˛..gx; gy/; .gu; gv/; gz; gw// D 1, we can get Theorem 1.9 (Theorem 3.1 of [15]).
(2) If we take '.t / D k t and ˛..gx; gy/; .gu; gv/; gz; gw// D 1 for all k 2 Œ0; 1/ in Theorem 2.7, we can get the
following corollary.
Corollary 2.9. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is a
complete G-metric space. Let F W X X ! X and g W X ! X be such that F has the mixed-g-monotone property.
Assume there is k 2 Œ0; 1/ such that
G.F .x; y/; F .u; v/; F .w; z// k
G.gx; gu; gw/ C G.gy; gv; gz/
2
for all x; y; z; u; v; w 2 X with gw gu gx and gy gv gz. Suppose that F .X X / g.X / and g is
continuous and commutes with F and F is continuous. If there exist x0 ; y0 2 X such that gx0 F .x0 ; y0 / and
gy0 F .y0 ; x0 /, then F and g have a coupled coincidence point, that is, there exists .x; y/ 2 X X such that
gx D F .x; y/ and gy D F .y; x/.
Remark 2.10.
(1) In Corollary 2.9, taking g D IX , we can get Theorem 1.8 (Theorem 3.1 of [5]).
(2) If taking ˛..gx; gy/; .gu; gv/; gz; gw// D k1 for all k 2 .0; 1 in Theorem 2.7, we can get the following
corollary.
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Corollary 2.11. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is a
complete G-metric space. Let F W X X ! X and g W X ! X be such that F has the mixed-g-monotone property.
Assume there exists k 2 .0; 1 and ' 2 ˆ such that
G.gx; gu; gw/ C G.gy; gv; gz/
G.F .x; y/; F .u; v/; F .w; z// k'
2
for all x; y; z; u; v; w 2 X with gw gu gx and gy gv gz. Suppose that F .X X / g.X / and g is
continuous and commutes with F and F is continuous. If there exist x0 ; y0 2 X such that gx0 F .x0 ; y0 / and
gy0 F .y0 ; x0 /, then F and g have a coupled coincidence point, that is, there exists .x; y/ 2 X X such that
gx D F .x; y/ and gy D F .y; x/.
Let g D IX in Corollary 2.11, we can get the following corollary.
Corollary 2.12. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is
a complete G-metric space. Let F W X X ! X and g be continuous. Assume there exists k 2 .0; 1 such that
G.x; u; w/ C G.y; v; z/
G.F .x; y/; F .u; v/; F .w; z// k'
2
for all x; y; z; u; v; w 2 X with w u x and y v z. If there exist x0 ; y0 2 X such that x0 F .x0 ; y0 / and
y0 F .y0 ; x0 /, then F has a coupled fixed point, that is, there exists .x; y/ 2 X X such that x D F .x; y/ and
y D F .y; x/.
Let  denote all continuous functions
W Œ0; 1/ ! Œ0; 1/ satisfying lim
definition of , we can get the following corollary.
t !r
.t / > 0 for each r > 0. Using the
Corollary 2.13. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is a
complete G-metric space. Let F W X X ! X and g W X ! X be such that F has the mixed-g-monotone property.
Assume there exists 2  such that
G.F .x; y/; F .u; v/; F .w; z// G.gx; gu; gw/ C G.gy; gv; gz/
2
.
G.gx; gu; gw/ C G.gy; gv; gz/
/
2
for all x; y; z; u; v; w 2 X with gw gu gx and gy gv gz. Suppose that F .X X / g.X / and g is
continuous and commutes with F and F is continuous. If there exist x0 ; y0 2 X such that gx0 F .x0 ; y0 / and
gy0 F .y0 ; x0 /, then F and g have a coupled coincidence point, that is, there exists .x; y/ 2 X X such that
gx D F .x; y/ and gy D F .y; x/.
Proof. Let '.t/ D t
proof is completed.
.t /. Obviously, ' 2 ˆ. Hence, Corollary 2.13 satisfies all conditions of Theorem 2.7. The
In the next theorem, we use another condition instead of the continuity hypothesis of F so that we can illustrate that
the introduction of the function ˛ is useful.
Theorem 2.14. Let .X; / be a partially ordered set and suppose there is a G-metric G on X such that .X; G/ is a
complete G-metric space. Let F W X X ! X and g W X ! X be such that F has the mixed-g-monotone property.
Assume there is a function ' 2 ˆ and ˛ W X 2 X 2 X 2 ! Œ0; C1/ for all x; y; z; u; v; w 2 X, the following
hold:
G.gx; gu; gz/ C G.gy; gv; gw/
˛..gx; gy/; .gu; gv/; .gz; gw//G.F .x; y/; F .u; v/; F .z; w// '
2
for all gw gu gx and gy gv gz. Suppose also that F .X X / g.X /, g is continuous and
(i) conditions (i) and (ii) of Theorem 2.7 hold,
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(ii) if fgxn g and fgyn g are sequences in X such that
˛..gxn ; gyn /; .gxnC1 ; gynC1 /; .gxnC1 ; gynC1 // 1
and
˛..gyn ; gxn /; .gynC1 ; gxnC1 /; .gynC1 ; gxnC1 // 1;
for all n and lim gxn D x 2 X and lim gyn D y 2 X , then
n!1
n!1
˛..g.gxn /; g.gyn //; .gx; gy/; .gx; gy// 1
and
˛..g.gyn /; g.gxn //; .gy; gx/; .gy; gx// 1:
If there exist x0 ; y0 2 X such that gx0 F .x0 ; y0 / and gy0 F .y0 ; x0 /, then F and g have a coupled
coincidence point, that is, there exists .x; y/ 2 X X such that gx D F .x; y/ and gy D F .y; x/.
Proof. Proceeding along the same lines as in the proof of Theorem 2.7, we know that fgxn g and fgyn g are Cauchy
sequences in the complete G-metric space .X; G/. Then there exist x; y 2 X such that lim gxn D x and
n!1
lim gyn D y. On the other hand, from (7) and hypothesis (ii), we obtain
n!1
˛..g.gxn /; g.gyn //; .gx; gy/; .gx; gy// 1
(14)
˛..g.gyn /; g.gxn //; .gy; gx/; .gy; gx// 1
(15)
and similarly,
for all n 2 N. Using the triangle inequality, (14) and the property of '.t / < t for all t > 0, we get
G.gx; F .x; y/; F .x; y// G.gx; g.gxnC1 /; g.gxnC1 // C G.g.gxnC1 /; F .x; y/; F .x; y//
G.gx; g.gxnC1 /; g.gxnC1 // C G.F .gxn ; gyn /; F .x; y/; F .x; y//
G.gx; g.gxnC1 /; g.gxnC1 //
C˛..g.gxn /; g.gyn //; .gx; gy/; .gx;
gy//G.F .gxn ; gyn /; F .x; y/; F .x; y// G.g.gxn /; gx; gx/ C G.g.gyn /; gy; gy/
G.gx; g.gxnC1 /; g.gxnC1 // C '
2
G.g.gxn /; gx; gx/ C G.g.gyn /; gy; gy/
< G.gx; g.gxnC1 /; g.gxnC1 // C
2
Similarly, using (15), we obtain
G.gy; F .y; x/; F .y; x// G.gy; g.gynC1 /; g.gynC1 // C G.g.gynC1 /; F .y; x/; F .y; x//
G.gy; g.gynC1 /; g.gynC1 // C G.F .gyn ; gxn /; F .y; x/; F .y; x//
G.gx; g.gynC1 /; g.gynC1 //
C˛..g.gyn /; g.gxn //; .gy; gx/; .gy;
gx//G.F .gyn ; gxn /; F .y; x/; F .y; x// G.g.gyn /; gy; gy/ C G.g.gxn /; gx; gx/
G.gy; g.gynC1 /; g.gynC1 // C '
2
G.g.gyn /; gy; gy/ C G.g.gxn /; gx; gx/
< G.gy; g.gynC1 /; g.gynC1 // C
:
2
Taking the limit as n ! 1 in the above two inequalities, we get
G.gx; F .x; y/; F .x; y// D 0 and G.gy; F .y; x/; F .y; x// D 0:
Hence, F .x; y/ D gx and F .y; x/ D gy. Thus, F and g have a coupled coincidence point. The proof is completed.
Remark 2.15. In Corollaries 2.9–2.13, if we omit the continuity of F , then Corollaries 2.9–2.13 will not hold, since
by the proof of Theorem 2.7, we must use the continuity of F to get limit. But the function F in Theorem 2.14 is
not continuous because the function ˛ meets the conditions .i i / in Theorem 2.14 instead of the continuity of F .
Therefore, Theorem 2.14 is essentially different from Theorem 2.7. They are two parallel conclusions.
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Coupled fixed point theorems for .˛; '/g -contractive type mappings
Now, we shall prove the uniqueness of the coupled fixed point. Note that, if .X; / is a partially ordered set, then we
endow the product X X with the following partial order relation:
.x; y/; .u; v/ 2 X X; .x; y/ .u; v/ , x u; y v:
Theorem 2.16. In addition to the hypotheses of Theorem 2.7, suppose that for all .x; y/; .x ; y / 2 X X , there
exists .u; v/ 2 X X such that
˛..gx; gy/; .gu; gv/; .gu; gv// 1 and ˛..gx ; gy /; .gu; gv/; .gu; gv// 1
and also assume that .F .u; v/; F .v; u// is comparable with .F .x; y/F .y; x// and .F .x ; y /; F .y ; x //. Then
F and g have a unique coupled common fixed point, that is, there exists a unique .x; y/ 2 X X such that
x D gx D F .x; y/ and y D gy D F .y; x/:
(16)
Proof. From Theorem 2.7, the set of coupled coincidences is non-empty. We shall show that if .x; y/ and .x ; y /
are coupled coincidence points, that is, if g.x/ D F .x; y/, g.y/ D F .y; x/, g.x / D F .x ; y / and g.y / D
F .y ; x /, then gx D gx and gy D gy . By assumption, there exists .u; v/ 2 X X such that .F .u; v/; F .v; u//
is comparable with .F .x; y/; F .y; x// and .F .x ; y /; F .y ; x //. Without restriction to the generality, we can
assume that
.F .x; y/; F .y; x// .F .u; v/; F .v; u//
and
.F .x ; y /; F .y ; x // .F .u ; v /; F .v ; u //:
Put u0 D u, v0 D v, and choose u1 ; v1 2 X such that gu1 D F .u0 ; v0 /, gv1 D F .v0 ; u0 /. Then, similarly as in
the proof of Theorem 2.7, we can inductively define sequences .gun / and .gvn / in X by gunC1 D F .un ; vn / and
gvnC1 D F .vn ; un /. Further, let x0 D x; y0 D y; x0 D x ; y0 D y . and, in the same way, define the sequences
.gxn /, .gyn /, .gxn / and .gyn /. Since
.F .x; y/; F .y; x// D .gx1 ; gy1 / D .gx; gy/ .F .u; v/; F .v; u// D .gu1 ; gv1 /;
then, gx gu1 and gv1 gy. Since for all .x; y/; .x ; y / 2 X X , there exists .u; v/ 2 X X such that
˛..gx; gy/; .gu; gv/; .gu; gv// 1 and ˛..gx ; gy /; .gu; gv/; .gu; gv// 1:
(17)
Since F is g-˛-admissible, so from (17), we have
˛..gx; gy/; .gu; gv/; .gu; gv// 1 ) ˛..F .x; y/; F .y; x//; .F .u; v/; F .v; u//; .F .u; v/; F .u; v/// 1:
Since u D u0 and v D v0 , we get
˛..gx; gy/; .gu; gv/; .gu; gv// 1
) ˛..F .x; y/; F .y; x//; .F .u0 ; v0 /; F .v0 ; u0 //; .F .u0 ; v0 //; F .v0 ; u0 /// 1:
Thus,
˛..gx; gy/; .gu; gv/; .gu; gv// 1 ) ˛..gx; gy/; .gu1 ; gv1 /; .gu1 ; gv1 // 1:
Therefore, by mathematical induction, we obtain
˛..gx; gy/; .gun ; gvn /; .gun ; gvn // 1
(18)
for all n 2 N and similarly, ˛..gy; gx/; .gvn ; gun /; .gvn ; gun // 1. From (17) and (18), we get
G.gx; gunC1 ; gunC1 / D G.F .x; y/; F .un ; vn /; F .un ; vn //
˛..gx;
gy/; .gun ; gvn /; .gun ; gvn //G.F
.x; y/; F .un ; vn /; F .un ; vn //
G.gx; gun ; gun / C G.gy; gvn ; gvn /
'
:
2
(19)
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J. Chen, X. Huang
Similarly, we have
G.gy; gvnC1 ; gvnC1 / D G.F .y; x/; F .vn ; un /; F .vn ; un //
˛..gy;
gx/; .gvn ; gun /; .gvn ; gun //G.F .y; x/; F .vn ; un /; F .vn ; un //
G.gy; gvn ; gvn / C G.gx; gun ; gun /
:
'
2
(20)
Adding (19) and (20), we get
G.gx; gunC1 ; gunC1 / C G.gy; gvnC1 ; gvnC1 /
G.gx; gun ; gun / C G.gy; gvn ; gvn /
'
:
2
2
Thus,
G.gx; gu1 ; gu1 / C G.gy; gv1 ; gv1 /
G.gx; gunC1 ; gunC1 / C G.gy; gvnC1 ; gvnC1 /
'n
2
2
(21)
for each n 1. Letting n ! 1 in (21) and using the properties of ', we get
lim G.gx; gunC1 ; gunC1 / C G.gy; gvnC1 ; gvnC1 / D 0:
n!1
This implies
lim G.gx; gunC1 ; gunC1 / D 0 and
n!1
lim G.gy; gvnC1 ; gvnC1 / D 0:
(22)
lim G.gy ; gvnC1 ; gvnC1 / D 0:
(23)
n!1
Similarly, one can show that
lim G.gx ; gunC1 ; gunC1 / D 0 and
n!1
n!1
Therefore, from (22), (23) and the uniqueness of the limit, we can get
gx D gx and gy D gy :
(24)
Since gx D F .x; y/ and gy D F .y; x/, by commuting F with g, we have
g.gx/ D g.F .x; y// D F .gx; gy/ and g.g.y// D g.F .y; x// D F .gy; gx/:
(25)
Put gx D z and gy D w, then by (25), we get
gz D F .z; w/ and gw D F .w; z/:
(26)
Thus, .z; w/ is a coincidence point. Then by (24) with x D z and y D w, we have gx D gz and gy D gw, that
is,
gz D gx D z and gy D gw D w:
(27)
From (26) and (27), we get z D gz D F .z; w/ and w D gw D F .w; z/. Then, .z; w/ is a coupled fixed point
of F and g. To prove the uniqueness, assume that .p; q/ is another coupled fixed point. Then by (27), we have
p D gp D gz D z and q D gq D gw D w. The proof is completed.
In what follows, firstly, we give a linear example (i.e. trivial example) of F and g to illustrate the usefulness of
Theorem 2.7.
Example 2.17. Let X D Œ0; 1 and .X; / be a partially ordered set with the natural ordering of real numbers. Let
G.x; y; z/ D jx yj C jy zj C jz xj for all x; y; z 2 X. Then .X; G/ is a complete G-metric space. Define
a mapping F W X X ! X by F .x; y/ D xCy
for all x; y 2 X and g W X ! X by g.x/ D x for all x 2 X .
4
Consider a mapping ˛ W X 2 X 2 X 2 ! Œ0; C1/
(
1 if gx gy; gu gv; gz gw
˛..gx; gy/; .gu; gv/; .gz; gw// D
0 others
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Coupled fixed point theorems for .˛; '/g -contractive type mappings
887
for all x; y 2 X . Obviously, F is commuting with g. Then, we get
xCy uCv zCw
G..F .x; y//; F .u; v/; F .z; w// D G.
;
;
/
4
4
4
1
D j.x C y/ .u C v/j C j.u C v/ .z C w/j C j.z C w/ .x C y/j
4
1
jx uj C ju zj C jz xj C jy vj C jv wj C jw yj
4
and
G.gx; gu; gz/ C G.gy; gv; gw//
G.x; u; z/ C G.y; v; w//
D
2
2
jx uj C ju zj C jz xj C jy vj C jv wj C jw yj/
:
D
2
Therefore, (3) holds for '.t / D 12 t for all t > 0. and also the hypothesis of Theorem 2.7 is fulfilled. Then there exists
a coincide coupled point of F and g. In this case, (0,0) is a coincide coupled point of F and g.
Secondly, we give a nonlinear example (i.e. nontrivial example) of F and g to illustrate the usefulness of
Theorem 2.7.
Example 2.18. Let X D Œ0; 1 and .X; / be a partially ordered set with the natural ordering of real numbers. Let
G.x; y; z/ D jx yj C jy zj C jz xj for all x; y; z 2 X . Then .X; G/ is a complete G-metric space. Define a
2
mapping F W X X ! X by F .x; y/ D .xCy/
for all x; y 2 X and g W X ! X by g.x/ D x for all x 2 X .
24
Consider a mapping ˛ W X 2 X 2 X 2 ! Œ0; C1/
(
e x y if 0 x y 23
˛..gx; gy/; .gu; gv/; .gz; gw// D
0
others
for all x; y 2 X . Obviously, F is commuting with g. Then, we get
2
2
2
; .uCv/
; .zCw/
/
G..F .x; y//; F .u; v/; F .z; w// D G. .xCy/
6
6
24
1
2
D 24 j.x C y/
.u C v/2 j C j.u C v/2 .z C w/2 j C j.z C w/2 .x C y/2 j
1
24
j.x C y C u C v/.x C y .u C v//j
Cj.u C v C z C w/.u C v .z C w//j C j.z C w C x C y/.z C w .x C y//j
61 jx uj C ju zj C jz xj C jy vj C jv wj C jw yj :
and
G.gx; gu; gz/ C G.gy; gv; gw//
G.x; u; z/ C G.y; v; w//
D
2
2
jx uj C ju zj C jz xj C jy vj C jv wj C jw yj/
D
:
2
2
Therefore, (3) holds for '.t / D e33 t for all t > 0. and also the hypothesis of Theorem 2.7 is fulfilled. Then there
exists a coincide coupled point of F and g. In this case, .0; 0/ is a coincide coupled point of F and g.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final
manuscript.
Acknowledgement: The authors thank the editor and the referees for their valuable comments and suggestions.
This research has been supported by the National Natural Science Foundation of China (11461043, 11361042 and
11326099) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201003
and 20142BAB201005) and the Science and Technology Project of Educational Commission of Jiangxi Province,
China (GJJ11346).
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