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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Common coupled fixed point theorems in cone metric spaces
for w-compatible mappings
M. Abbas a, M. Ali Khan a, S. Radenovic´ b,*
a
b
Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia
a r t i c l e
i n f o
Keywords:
Coupled common fixed point
Coupled coincidence point
Coupled point of coincidence
Cone metric space
Normal and non-normal cone
a b s t r a c t
In this paper we introduce the concept of a w-compatible mappings to obtain coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in cone
metric space with a cone having non-empty interior. Coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend and unify several
well known comparable results in the literature. Results are supported by three examples.
Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries
Cone metric spaces were introduced by Huang and Zhang in [11], where they investigated the convergence in cone metric
spaces, introduced the notion of their completeness, and proved some fixed point theorems for contractive mappings on
these spaces. Recently, in [1–19] some common fixed point theorems have been proved for maps on cone metric spaces.
However, in [1,2,11–13], the authors usually obtain their results for normal cones. For more results in cone metric space
we, refer to ([4–7,18,20] and references mentioned therein). In this paper we do not impose the normality condition for
the cones. The only assumption is that the interior of the cone P is non-empty, so we use neither continuity of the vector
metric d, nor Sandwich Theorem.
Following definitions and results will be needed in the sequel.
Definition 1.1 [10]. Let E be a real Banach space. A subset P of E is called a cone if and only if:
(a) P is closed, non-empty and P – {h};
(b) a, b 2 R, a, b P 0, x, y 2 P imply that ax + by 2 P;
(c) P \ (P) = {h}.
Given a cone define a partial ordering with respect to P by x y if and only if y x 2 P. We shall write x y for
y x 2 IntP, where IntP stands for interior of P. Also we will use x y to indicate that x y and x – y.
There exist two kinds of cones (see [10]): normal with normal constant k P 1 and non-normal, that is which is not normal
cone. The cone P in normed space E is called normal whenever there is a number k > 0 such that for all x, y 2 E, h x y implies kxk 6 kkyk. The least positive number satisfying this norm inequality is called the normal constant of P ([10]). For details see also [21].
* Corresponding author.
E-mail addresses: [email protected] (M. Abbas), [email protected] (M. Ali Khan), [email protected], [email protected] (S. Radenovic´).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2010.05.042
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Definition 1.2 [11]. Let X be a non-empty set. Suppose that the mapping d: X X ? E satisfies:
(d1) h d(x, y) for all x, y 2 X and d(x, y) = h if and only if x = y;
(d2) d(x, y) = d(y, x) for all x, y 2 X;
(d3) d(x, y) d(x, z) + d(z, y) for all x, y, z 2 X.
Then d is called a cone metric on X and (X, d) is called a cone metric space. The concept of a cone metric space is more
general than that of a metric space.
Definition 1.3 [11]. Let (X, d) be a cone metric space, {xn} a sequence in X and x 2 X. For every c 2 E with h c, we say that
{xn} is
(c1) a Cauchy sequence if there is some k 2 N such that, for all n, m P k, d(xn, xm) c;
(c2) a convergent sequence if there is some k 2 N such that, for all n P k, d(xn, x) c. Then x is called limit of the sequence
{xn}.
Note that every convergent sequence in a cone metric space X is a Cauchy sequence. A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Let (X, d) be a cone metric space. Then the following properties are often used (particularly when dealing with cone metric
spaces in which the cone need not be normal):
(p1)
(p2)
(p3)
(p4)
(p5)
If
If
If
If
If
E is a real Banach space with a cone P and if a ha where a 2 P and h 2 [0, 1), then a = h.
h u c for each h c, then u = h.
a b + c for each h c, then a b.
u v and v w, then u w.
c 2 intP,0 an and an ? h, then there exists an k such that for all n > k we have an c.
For details about these properties see: [14–17,19].
It follows from (p5) that the sequence xn converges to x 2 X if d(xn, x) ? h as n ? 1 and xn is a Cauchy sequence if
d(xn, xm) ? h as n, m ? 1. In the case when the cone is not necessarily normal, we have only one half of the statements of
Lemma 1 and 4 from [11]. Also, in this case, the fact that d(xn, yn) ? d(x, y) if xn ? x and yn ? y is not applicable.
2. Main results
Bhashkar and Lakshmikantham in [8] introduced the concept of coupled fixed point of a mapping F: X X ? X and investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by
investigating the existence and uniqueness of solution for a periodic boundary value problem. Sabetghadam et al. in [22]
introduced this concept in cone metric spaces. They investigated some fixed point theorems in cone metric spaces which
in turn extends and modify Theorem 2.1 of [8]. Recently, Lakshmikantham and C´iric´ [9] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces.
Definition 2.1 [22]. An element (x, y) 2 X X is called a coupled fixed point of mapping F: X X ? X if x = F(x, y) and
y = F(y, x).
Inspired with Definition 2.1. Following is the concept of a coupled fixed point of a mapping F: X X ? X.
Definition 2.2. An element (x, y) 2 X X is called
(g1) a coupled coincidence point of mappings F: X X ? X and g: X ? X if g(x) = F(x, y) and g(y) = F(y, x), and (gx, gy) is
called coupled point of coincidence;
(g2) a common coupled fixed point of mappings F: X X ? X and g: X ? X if x = g(x) = F(x, y) and y = g(y) = F(y, x).
Note that if g is the identity mapping, then Definition 2.2 reduces to Definition 2.1. We introduce the following definition.
Definition 2.3. The mappings F: X X ? X and g: X ? X are called w-compatible if g(F(x, y)) = F(gx, gy) whenever g(x) = F(x, y)
and g(y) = F(y, x).
Now we prove our main result.
Theorem 2.4. Let (X, d) be a cone metric space with a cone P having non-empty interior, F: X X ? X and g: X ? X be mappings
satisfying
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dðFðx; yÞ; Fðu; v ÞÞ a1 dðgx; guÞ þ a2 dðFðx; yÞ; gxÞ þ a3 dðgy; g v Þ
þa4 dðFðu; v Þ; guÞ þ a5 dðFðx; yÞ; guÞ þ a6 dðFðu; v Þ; gxÞ;
for all x, y, u, v 2 X, where ai, i = 1, 2, . . . , 6 are nonnegative real numbers such that
plete subset of X, then F and g have a coupled coincidence point in X.
ð2:1Þ
P6
i¼1 ai
< 1. If F(X X) # g(X) and g(X) is com-
Proof. Let x0, y0 be any two arbitrary points in X. Set g(x1) = F(x0, y0) and g(y1) = F(y0, x0), this can be done because
F(X X) # g(X). Continuing this process we obtain two sequences {xn} and {yn} in X such that g(xn+1) = F(xn, yn) and
g(yn+1) = F(yn, xn). From (2.1), we have
dðgxn ; gxnþ1 Þ ¼ dðFðxn1 ; yn1 Þ; Fðxn ; yn ÞÞ
a1 dðgxn1 ; gxn Þ þ a2 dðFðxn1 ; yn1 Þ; gxn1 Þ þ a3 dðgyn1 ; gyn Þ þ a4 dðFðxn ; yn Þ; gxn Þ þ a5 dðFðxn1 ; yn1 Þ; gxn Þ
þ a6 dðFðxn ; yn Þ; gxn1 Þ
¼ a1 dðgxn1 ; gxn Þ þ a2 dðgxn ; gxn1 Þ þ a3 dðgyn1 ; gyn Þ þ a4 dðgxnþ1 ; gxn Þ þ a5 dðgxn ; gxn Þ þ a6 dðgxnþ1 ; gxn1 Þ
a1 dðgxn1 ; gxn Þ þ a2 dðgxn ; gxn1 Þ þ a3 dðgyn1 ; gyn Þ þ a4 dðgxnþ1 ; gxn Þ þ a6 dðgxnþ1 ; gxn Þ þ a6 dðgxn ; gxn1 Þ
¼ ða1 þ a2 þ a6 Þdðgxn1 ; gxn Þ þ a3 dðgyn1 ; gyn Þ þ ða4 þ a6 Þdðgxn ; gxnþ1 Þ;
from which it follows
ð1 a4 a6 Þdðgxn ; gxnþ1 Þ ða1 þ a2 þ a6 Þdðgxn1 ; gxn Þ þ a3 dðgyn1 ; gyn Þ:
ð2:2Þ
Similarly,
ð1 a4 a6 Þdðgyn ; gynþ1 Þ ða1 þ a2 þ a6 Þdðgyn1 ; gyn Þ þ a3 dðgxn1 ; gxn Þ:
ð2:3Þ
Because of the symmetry in (2.1),
dðgxnþ1 ; gxn Þ ¼ dðFðxn ; yn Þ; Fðxn1 ; yn1 ÞÞ
a1 dðgxn ; gxn1 Þ þ a2 dðFðxn ; yn Þ; gxn Þ þ a3 dðgyn ; gyn1 Þ þ a4 dðFðxn1 ; yn1 Þ; gxn1 Þ þ a5 dðFðxn ; yn Þ; gxn1 Þ
þ a6 dðFðxn1 ; yn1 Þ; gxn Þ
¼ a1 dðgxn ; gxn1 Þ þ a2 dðgxnþ1 ; gxn Þ þ a3 dðgyn ; gyn1 Þ þ a4 dðgxn ; gxn1 Þ þ a5 dðgxnþ1 ; gxn1 Þ þ a6 dðgxn ; gxn Þ
a1 dðgxn ; gxn1 Þ þ a2 dðgxnþ1 ; gxn Þ þ a3 dðgyn ; gyn1 Þ þ a4 dðgxn ; gxn1 Þ þ a5 dðgxnþ1 ; gxn Þ þ a5 dðgxn ; gxn1 Þ;
that is,
ð1 a2 a5 Þdðgxnþ1 ; gxn Þ ða1 þ a4 þ a5 Þdðgxn1 ; gxn Þ þ a3 dðgyn ; gyn1 Þ:
ð2:4Þ
Similarly,
ð1 a2 a5 Þdðgynþ1 ; gyn Þ ða1 þ a4 þ a5 Þdðgyn1 ; gyn Þ þ a3 dðgxn ; gxn1 Þ:
ð2:5Þ
Let dn = d(gxn, gxn+1) + d(gyn, gyn+1). Now, from (2.2) and (2.3) respectively (2.4) and (2.5) we obtain:
ð1 a4 a6 Þdn ða1 þ a2 þ a3 þ a6 Þdn1 ;
ð1 a2 a5 Þdn ða1 þ a3 þ a4 þ a5 Þdn1 :
ð2:6Þ
ð2:7Þ
Finally, from (2.6) and (2.7) we have
ð2 a2 a4 a5 a6 Þdn ð2a1 þ 2a3 þ a2 þ a4 þ a5 þ a6 Þdn1 ;
that is,
dn gdn1 ;
g¼
2a1 þ 2a3 þ a2 þ a4 þ a5 þ a6
< 1:
2 a2 a4 a5 a6
ð2:8Þ
Consequently, we have
h dn gdn1 gn d0 :
ð2:9Þ
If d0 = 0 then (x0, y0) is a coupled coincidence point of F and g. So let h d0. If m > n, we have
dðgxm ; gxn Þ dðgxm ; gxm1 Þ þ dðgxm1 ; gxm2 Þ þ þ dðgxnþ1 ; gxn Þ and
dðgym ; gyn Þ dðgym ; gym1 Þ þ dðgym1 ; gym2 Þ þ þ dðgynþ1 ; gyn Þ:
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Therefore,
dðgxm ; gxn Þ þ dðgym ; gyn Þ dm1 þ dm2 þ þ dn ðgm1 þ gm2 þ þ gn Þd0 From (p5) it follows that for h c and large n:
gn
1g
gn
1g
d0 ! h;
as n ! 1:
d0 c; thus, according to (p4), d(gxn, gxm) + d(gyn, gym) c. Hence, by
Definition 1.3. (c1) {d(gxn, gxm) + d(gyn, gym)} is a Cauchy sequence. Since, d(gxn, gxm) d(gxn, gxm) + d(gyn, gym) and d(gyn,
gym) d(gxn, gxm) + d(gyn, gym) then again by (p4), {gxn} and {gyn} are Cauchy sequences in g(X), so there exists x and y in X
such that gxn ? gx and gyn ? gy. Now, we prove that F(x, y) = gx and F(y, x) = gy. For that we have
dðFðx; yÞ; gxÞ dðFðx; yÞ; gxnþ1 Þ þ dðgxnþ1 ; gxÞ ¼ dðFðx; yÞ; Fðxn ; yn ÞÞ þ dðgxnþ1 ; gxÞ
a1 dðgx; gxn Þ þ a2 dðFðx; yÞ; gxÞ þ a3 dðgy; gyn Þ þ a4 dðFðxn ; yn Þ; gxn Þ þ a5 dðFðx; yÞ; gxn Þ þ a6 dðFðxn ; yn Þ; gxÞ
þ dðgxnþ1 ; gxÞ
¼ a1 dðgx; gxn Þ þ a2 dðFðx; yÞ; gxÞ þ a3 dðgy; gyn Þ þ a4 dðgxnþ1 ; gxn Þ þ a5 dðFðx; yÞ; gxn Þ þ a6 dðgxnþ1 ; gxÞ
þ dðgxnþ1 ; gxÞ
a1 dðgx; gxn Þ þ a2 dðFðx; yÞ; gxÞ þ a3 dðgy; gyn Þ þ a4 dðgxnþ1 ; gxÞ þ a4 dðgx; gxn Þ þ a5 dðFðx; yÞ; gxÞ
þ a5 dðgx; gxn Þ þ a6 dðgxnþ1 ; gxÞ þ dðgxnþ1 ; gxÞ;
which further implies that
dðFðx; yÞ; gxÞ a1 þ a4 þ a5
1 þ a4 þ a6
a3
dðgxn ; gxÞ þ
dðgxnþ1 ; gxÞ þ
dðgyn ; gyÞ:
1 a2 a5
1 a2 a5
1 a2 a5
ð2:10Þ
ð1a2 a5 Þc
ð1a2 a5 Þc
Since gxn ? gx and gyn ? gy then for hc there exists N 2 N such that dðgxn ; gxÞ 3ða
; dðgxnþ1 ; gxÞ 3ð1þa
and
1 þa4 þa5 Þ
4 þa6 Þ
2 a5 Þc
dðgyn ; gyÞ ð1a3a
, for all n P N. Thus,
3
dðFðx; yÞ; gxÞ c c c
þ þ ¼ c:
3 3 3
ð2:11Þ
Now, according to (p2) it follows that d(F(x, y), gx) = h, and hence F(x,y) = gx. Similarly, F(y, x) = gy. Hence (x, y) is coupled coincidence point of the mappings F and g. h
Corollary 2.5. Let (X, d) be cone metric space, F: X X ? X and g: X ? X be mappings satisfying
dðFðx; yÞ; Fðu; v ÞÞ a½dðgx; guÞ þ dðFðx; yÞ; gxÞ þ b½dðgy; g v Þ þ dðFðu; v Þ; guÞ þ c½dðFðx; yÞ; guÞ þ dðFðu; v Þ; gxÞ
for all x, y, u, v 2 X, where a, b, and c are nonnegative real numbers such that a þ b þ c < 12. If F(X X) # g(X) and g(X) is complete
subset of X, then F and g have a coupled coincidence point in X.
Corollary 2.6 (Theorem 2.2 of [22]). Let (X, d) be a complete cone metric space. Suppose F: X X ? X satisfies the following contractive condition for all x, y, u, v 2 X:
dðFðx; yÞ; Fðu; v ÞÞ kdðx; uÞ þ ldðy; v Þ;
where k, l are nonnegative constants with k + l < 1. Then F has a unique coupled fixed point.
Corollary 2.7 (Theorem 2.5 of [22]). Let (X, d) be a complete cone metric space. Suppose F: X X ? X satisfies the following contractive condition for all x, y, u, v 2 X:
dðFðx; yÞ; Fðu; v ÞÞ kdðFðx; yÞ; xÞ þ ldðFðu; v Þ; uÞ;
where k, l are nonnegative constants with k + l < 1. Then F has a unique coupled fixed point.
Corollary 2.8 (Theorem 2.6 of [22]). Let (X, d) be a complete cone metric space. Suppose F: X X ? X satisfies the following contractive condition for all x, y, u, v 2 X:
dðFðx; yÞ; Fðu; v ÞÞ kdðFðx; yÞ; uÞ þ ldðFðu; v Þ; xÞ;
where k, l are nonnegative constants with k + l < 1. Then F has a unique coupled fixed point.
We present now two examples showing that Theorem 2.4 is a proper extension of known results. In both examples, the
conditions of Theorem 2.4 are fulfilled. Note that in both examples, the main theorems from [22] cannot be applied. This
shows that Theorem 2.4 is more general, that is, the main results from [22] can be obtained as its special cases taking
g = iX, identity mapping of X, ai = k, aj = l for some i, j 2 {1, 2, 3, 4, 5, 6}.
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Example 2.9 (The case of non-normal cone). Let X = [0, 1), E ¼ C 1R ½0; 1 and P = {u 2 E:u(t) P 0, t 2 [0, 1]}. The mapping d:
X X ? E is defined in the following way: d(x, y) = jx yju, where u(t) = et. Clearly, (X, d) is a complete cone metric space.
Given the functions F: X X ? X and g: X ? X by
gðxÞ ¼
3x if x 2 ½0; 1;
2x if x 2 ð1; 1Þ;
and
(
Fðx; yÞ ¼
x
3
x
4
þ 3y if x 2 ½0; 1 and y 2 R;
þ 4y if x 2 ð1; 1Þ and y 2 R:
1
It is easy to verify that F and g satisfy all the conditions of Theorem 2.4, taking a1 ¼ a3 ¼ 25 ; a2 ¼ a4 ¼ 10
and a5 = a6 = 0. Moreover (0, 0) is common coupled coincidence point of F and g.
Example 2.10 (The case of normal cone). Let X = [0, 1). Assume E = R2 and P = {(x, y) 2 R2: x, y P 0}. Define d: X X ? E by
d(x, y) = (jx yj, jx yj). Clearly (X, d) is a complete cone metric space. Consider the mappings F: X X ? X and g: X ? X
defined by
gðxÞ ¼ 5x;
and
Fðx; yÞ ¼ x þ
j sin yj
:
3
7
1
It is easy to verify that F and g satisfy all the conditions of Theorem 2.4, taking a1 ¼ 15 ; a3 ¼ 10
; a5 ¼ a6 ¼ 25
and a2 = a4 = 0.
Moreover (0, 0) is common coupled coincidence point of F and g.
Theorem 2.11. Let F: X X ? X and g: X ? X be two mappings which satisfy all the conditions of Theorem 2.1. If F and g are wcompatible, then F and g have unique common coupled fixed point. Moreover, common fixed point of F and g is of the form (u, u) for
some u 2 X.
Proof. First we claim that coupled point of coincidence is unique. Suppose that (x, y), (x*, y*) 2 X X with g(x) = F(x, y),
g(y) = F(y, x) and g(x*) = F(x*, y*),g(y*) = F(y*, x*). Using (2.1), we get
dðgx; gx Þ ¼ dðFðx; yÞ; Fðx ; y ÞÞ
a1 dðgx; gx Þ þ a2 dðFðx; yÞ; gxÞ þ a3 dðgy; gy Þ þ a4 dðFðx ; y Þ; gx Þ þ a5 dðFðx; yÞ; gx Þ þ a6 dðFðx ; y Þ; gxÞ
¼ ða1 þ a5 þ a6 Þdðgx; gx Þ þ a3 dðgy; gy Þ;
and
dðgx; gx Þ ða1 þ a5 þ a6 Þdðgx; gx Þ þ a3 dðgy; gy Þ:
ð2:12Þ
Similarly
dðgy; gy Þ ða1 þ a5 þ a6 Þdðgy; gy Þ þ a3 dðgx; gx Þ:
ð2:13Þ
Thus
dðgx; gx Þ þ dðgy; gy Þ ða1 þ a3 þ a5 þ a6 Þðdðgx; gx Þ þ dðgy; gy ÞÞ:
Since a1 + a3 + a5 + a6 < 1, therefore by (p1), we have d(gx, gx*) + d(gy, gy*) = 0, which implies that gx = gx* and gy = gy*. Similarly we prove that gx = gy* and gy = gx*. Thus gx = gy. Therefore (gx, gx) is unique coupled point of coincidence of F and g.
Now, let g(x) = u. Then we have u = g(x) = F(x, x). By w- compatibility of F and g, we have
gðuÞ ¼ gðgðxÞÞ ¼ gðFðx; xÞÞ ¼ Fðgx; gxÞ ¼ Fðu; uÞ:
ð2:14Þ
Then (gu, gu) is coupled point of coincidence of F and g. Consequently gu = gx. Therefore u = gu = F(u, u). Hence (u, u) is unique
common coupled fixed point of F and g. h
Example 2.12. Let X = {(x, 0): x 2 [0, 1]} [ {(0, x): x 2 [0, 1]}. Assume E = R2 and P = {(x, y) 2 R2: x, y P 0}. Define d: X X ? E by
d(x, y) = (jx1 y1j, jx2 y2j), where x = (x1, x2) and y = (y1, y2). Clearly, (X, d) is a complete cone metric space. Consider mappings F: X X ? X and g: X ? X, given by
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gðxÞ ¼
ð0; tÞ if x ¼ ðt; 0Þ; t 2 ½0; 1;
ðt; 0Þ if x ¼ ð0; tÞ; t 2 ½0; 1;
and
Fððx1 ; x2 Þ; ðy1 ; y2 ÞÞ ¼
x x 1
2
;
:
8 8
Note that F and g satisfy all the conditions of Theorem 2.11 if we take ai ¼ 18 ; i ¼ 1; 6. Moreover (0, 0) is the unique common
coupled fixed point of F and g.
Theorem 2.13. Let (X, d) be cone metric space with a cone P having non-empty interior, F: X X ? X and g: X ? X be w-compatible mappings such that
dðFðx; yÞ; Fðu; v ÞÞ kU þ mV
ð2:15Þ
for all x, y, u, v 2 X, where,
U; V 2 Sx;y
u;v ¼ fdðgx; guÞ; dðgy; g v Þ; dðFðx; yÞ; gxÞ; dðFðx; yÞ; guÞ; dðFðu; v Þ; guÞg;
and k, m are nonnegative real numbers such that k + m < 1. If F(X X) # g(X) then F and g have unique common coupled fixed
point having the form (u, u) for some u 2 X.
Proof. Following similar arguments to those given in Theorem 2.4, we construct two sequences {xn} and {yn} in X such that
g(xn+1) = F(xn, yn) and g(yn+1) = F(yn, xn). Now, from (2.15), we have
dðgxn ; gxnþ1 Þ ¼ dðFðxn1 ; yn1 Þ; Fðxn ; yn ÞÞ kU þ mV;
ð2:16Þ
that is,
dðgyn ; gynþ1 Þ ¼ dðFðyn1 ; xn1 Þ; Fðyn ; xn ÞÞ kU þ mV;
where U; V 2
;yn1
Sxxn1
n ;yn
i.e., U; V 2
ð2:17Þ
;xn1
Syyn1
.
n ;xn
We have the following 12 cases:
(i):
(ii):
(iii):
(iv):
(v):
(vi):
(vii):
(viii):
(ix):
(x):
(xi):
(xii):
U = d(gxn1, gxn) and V = d(gxn1, gxn).
U = d(gxn1, gxn) and V = d(gyn1, gyn),
U = d(gxn1, gxn) and V = d(F((xn1, yn1), gxn1).
U = d(gxn1, gxn) and V = d(F((xn1, yn1), gxn).
U = d(gyn1, gyn) and V = d(gyn1, gyn).
U = d(gyn1, gyn) and V = d(F((xn1, yn1), gxn1).
U = d(gyn1, gyn) and V = d(F((xn1, yn1), gxn).
U = d(F(xn1, yn1), gxn1) and V = d(F((xn1, yn1), gxn1).
U = d(F(xn1, yn1), gxn1) and V = d(F((xn1, yn1), gxn).
U = d(F((xn1, yn1), gxn1) and V = d(F((xn, yn), gxn).
U = d(F((xn1, yn1), gxn) and V = d(F((xn1, yn1), gxn).
U = d(F((xn1, yn1), gxn) and V = d(F((xn, yn), gxn).
In the cases (i), (ii), (iii), (v), (vi), (viii) and (xi) according to (2.16) and (2.17) we obtain that
dðgxn ; gxnþ1 Þ þ dðgyn ; gynþ1 Þ ðm þ kÞðdðgxn1 ; gxn Þ þ dðgyn1 ; gyn ÞÞ:
Similarly, from (2.16) and (2.17) in the cases (iv), (vii), (ix), (x) and (xii) we again obtain that
dðgxn ; gxnþ1 Þ þ dðgyn ; gynþ1 Þ kðdðgxn1 ; gxn Þ þ dðgyn1 ; gyn ÞÞ;
k
where k 2 k; 1m
;m .
Note that for U 2 {d(gx, gu), d(gy, gv), d(F(x, y), gx)}, the case of taking V =d(F(u, v), gu) is superfluous because it is the same
as we take V = d(F(x, y), gx) in condition (2.15) which are the cases that have already been discussed. Thus we conclude that
dðgxn ; gxnþ1 Þ þ dðgyn ; gynþ1 Þ b½dðgxn1 ; gxn Þ þ dðgyn1 ; gyn Þ
for some b < 1 and for all n P 1. Following similar arguments to those given in Theorem 2.1, (x, y) is the common coupled
coincidence point of F and g, where gxn ? gx and gyn ? gy. Now we will prove that coupled point of coincidence is unique.
For that take (x, y), (x*, y*) 2 X X such that g(x) = F(x, y), g(y) = F(y, x) and g(x*) = F(x*, y*), g(y*) = F(y*, x*). Again using (2.15),
we get
dðgx; gx Þ ¼ dðFðx; yÞ; Fðx ; y ÞÞ kU þ mV;
ð2:18Þ
Please cite this article in press as: M. Abbas et al., Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. Comput. (2010), doi:10.1016/j.amc.2010.05.042
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7
that is,
dðgy; gy Þ ¼ dðFðy; xÞ; Fðy ; x ÞÞ kU þ mV;
Sx;y
x ;y
ð2:19Þ
Sy;x
y ;x .
where U; V 2
i.e., U; V 2
Again we have the following 12 cases:
(i):
(ii):
(iii):
(iv):
(v):
(vi):
(vii):
(viii):
(ix):
(x):
(xi):
(xii):
U = d(gx, gx*) and V = d(gx, gx*).
U = d(gx, gx*) and V = d(gy, gy*).
U = d(gx, gx*) and V = d(F(x, y), gx).
U = d(gx, gx*) and V = d(F(x, y), gx*).
U = d(gy, gy*) and V = d(gy, gy*).
U = d(gy, gy*) and V = d(F(x, y), gx).
U = d(gy, gy*) and V = d(F(x, y), gx*).
U = d(F(x, y), gx) and V = d(F(x, y), gx).
U = d(F(x, y), gx) and V = d(F(x, y), gx*).
U = d(F(x, y), gx) and V = d(F(x*, y*), gx*).
U = d(F(x, y), gx*) and V = d(F(x, y), gx*).
U = d(F(x, y), gx*) and V = d(F(x*, y*), gx*).
In the cases (i), (ii), (i), (v), (vii) and (xi) according to (2.18) and (2.19) we obtain that
dðgx; gx Þ þ dðgy; gy Þ ðk þ mÞðdðgx; gx Þ þ dðgy; gy ÞÞ:
Similarly, from (2.18) and (2.19) in the cases (iii), (viii), (ix), (x) and (xii) we again obtain that
dðgx; gx Þ þ dðgy; gy Þ kðdðgx; gx Þ þ dðgy; gy ÞÞ;
where k 2 {0,k,m}.
According to (p1) from all the above cases, we have d(gx, gx*) + d(gy, gy*) = 0, i.e., gx = gx* and gy = gy*. That is, (gx, gx) is the
unique common coupled point of coincidence. Since F and g are w-compatible maps, so we have,
gðgðxÞÞ ¼ Fðgx; gxÞ:
ð2:20Þ
Let u = g(x). By (2.20), we have, g(u) = F(u, u). Therefore (gu, gu) is a coupled point of coincidence of F and g. Consequently,
u = g(u) = F(u, u). Hence (u, u) is unique common coupled fixed point of F and g. h
Acknowledgement
S. Radenovic´ is thankful to the Ministry of Science and Environmental Protection of Serbia.
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