Coupled coincidence and common fixed point theorems for hybrid pair of mappings

RESEARCH Open Access

Coupled coincidence and common fixed pointtheorems for hybrid pair of mappingsMujahid Abbas1*, Ljubomir Ćirić2, Bosko Damjanović3 and Muhammad Ali Khan1

* Correspondence: [email protected] of Mathematics,Lahore University of ManagementSciences, Lahore - 54792, PakistanFull list of author information isavailable at the end of the article

Abstract

Bhaskar and Lakshimkantham proved the existence of coupled fixed point for asingle valued mapping under weak contractive conditions and as an application theyproved the existence of a unique solution of a boundary value problem associatedwith a first order ordinary differential equation. Recently, Lakshmikantham and Ćirićobtained a coupled coincidence and coupled common fixed point of two singlevalued maps. In this article, we extend these concepts to multi-valued mappings andobtain coupled coincidence points and common coupled fixed point theoremsinvolving hybrid pair of single valued and multi-valued maps satisfying generalizedcontractive conditions in the frame work of a complete metric space. Two examplesare presented to support our results.2000 Mathematics Subject Classification: 47H10; 47H04; 47H07.

Keywords: coupled common fixed point, coupled coincidence point, coupled pointof coincidence, w-compatible mappings, F-weakly commuting mappings

1 Introduction and preliminariesLet (X, d) be a metric space. For x Î X and A ⊆ X, we denote d(x, A) = inf{d(x, A): y ÎA}. The class of all nonempty bounded and closed subsets of X is denoted by CB(X).

Let H be the Hausdorff metric induced by the metric d on X, that is,

H(A, B) = max{supx∈A

d(x, B), supy∈B

d(y, A)},

for every A, B Î CB(X).

Lemma 1 [1]Let A, B Î CB(X), and a >1. Then, for every a Î A, there exists b Î B

such that d(a, b) ≤ aH(A, B).Lemma 2 [2]Let A, B Î CB(X), then for any a Î A, d(a, B) ≤ H(A, B).

Definition 3 Let X be a nonempty set, F : X × X ® 2X (collection of all nonempty subsets

of X) and g : X ® X. An element (x, y) Î X ×X is called (i) coupled fixed point of F if × Î F

(x, y) and y Î F(y, x) (ii) coupled coincidence point of a hybrid pair {F, g} if g(x) Î F(x, y)

and g(y) Î F(y, x) (iii) coupled common fixed point of a hybrid pair {F, g} if × = g(x) Î F(x,

y) and y = g(y) Î F(y, x).

We denote the set of coupled coincidence point of mappings F and g by C(F, g).

Note that if (x, y) Î C(F, g), then (y, x) is also in C(F, g).

Abbas et al. Fixed Point Theory and Applications 2012, 2012:4http://www.fixedpointtheoryandapplications.com/content/2012/1/4

© 2012 Abbas et al.; licensee Springer. This is an open access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

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Definition 4 Let F : X × X ® 2X be a multi-valued mapping and g be a self map on

X. The hybrid pair {F, g} is called w- compatible if g(F(x, y)) ⊆ F(gx, gy) whenever (x, y)

Î C(F, g).

Definition 5 Let F : X × X ® 2X be a multi-valued mapping and g be a self-mapping

on X. The mapping g is called F- weakly commuting at some point (x, y) Î X × X if g2

(x) Î F(gx, gy) and g2(y) Î F(gy, gx).

Bhaskar and Lakshmikantham [3] introduced the concept of coupled fixed point of a

mapping F from X ×X to X and established some coupled fixed point theorems in par-

tially ordered sets. As an application, they studied the existence and uniqueness of

solution for a periodic boundary value problem associated with a first order ordinary

differential equation. Ćirić et al. [4] proved coupled common fixed point theorems for

mappings satisfying nonlinear contractive conditions in partially ordered complete

metric spaces and generalized the results given in [3]. Sabetghadam et al. [5] employed

these concepts to obtain coupled fixed point in the frame work of cone metric spaces.

Lakshmikantham and Ćirić [4] introduced the concepts of coupled coincidence and

coupled common fixed point for mappings satisfying nonlinear contractive conditions

in partially ordered complete metric spaces. The study of fixed points for multi-valued

contractions mappings using the Hausdorff metric was initiated by Nadler [1] and

Markin [6]. Later, an interesting and rich fixed point theory for such maps was devel-

oped which has found applications in control theory, convex optimization, differential

inclusion and economics (see [7] and references therein). Klim and Wardowski [8] also

obtained existence of fixed point for set-valued contractions in complete metric spaces.

Dhage [9,10] established hybrid fixed point theorems and gave some applications (see

also [11]). Hong in his recent study [12] proved hybrid fixed point theorems involving

multi-valued operators which satisfy weakly generalized contractive conditions in

ordered complete metric spaces. The study of coincidence point and common fixed

points of hybrid pair of mappings in Banach spaces and metric spaces is interesting

and well developed. For applications of hybrid fixed point theory we refer to [13-16].

For a survey of fixed point theory and coincidences of multimaps, their applications

and related results, we refer to [16-22].

The aim of this article is to obtain coupled coincidence point and common fixed

point theorems for a pair of multi-valued and single valued mappings which satisfy

generalized contractive condition in complete metric spaces. It is to be noted that to

find coupled coincidence points, we do not employ the condition of continuity of any

mapping involved therein. Our results unify, extend, and generalize various known

comparable results in the literature.

2 Main resultsIn the following theorem we obtain coupled coincidence and common fixed point for

hybrid pair of mappings satisfying a generalized contractive condition.

Theorem 6 Let (X, d) be a metric space, F : X × X ® CB(X) and g : X ® X be map-

pings satisfying

H(F(x, y), F(u, v)) ≤ a1d(gx, gu) + a2d(F(x, y), gx) + a3d(gy, gv)

a4d(F(u, v), gu) + a5d(F(x, y), gu) + a6d(F(u, v), gx)(1)


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for all x, y, u, v Î X, where ai = ai (x, y, u, v), i = 1, 2, ..., 6, are nonnegative real

numbers such that

a1 + a2 + a3 + a4 + a5 + a6 ≤ h < 1, (2)

where h is a fixed number. If F(X ×X) ⊆ g(X) and g(X) is complete subset of X, then F

and g have coupled coincidence point. Moreover F and g have coupled common fixed

point if one of the following conditions holds.

(a) F and g are w- compatible, limn→∞ gnx = u and lim

n→∞ gny = v for some (x, y) Î C(F,

g), u, v Î X and g is continuous at u and v.

(b) g is F- weakly commuting for some (x, y) Î C(g, F), g2x = gx and g2y = gy.

(c) g is continuous at x, y for some (x, y) Î C(g, F) and for some u, v Î X,

limn→∞ gnv = yand lim

n→∞ gnv = y.

(d) g(C(g, F)) is singleton subset of C(g, F).

Proof. Let x0, y0 Î X be arbitrary. Then F (x0, y0) and F (y0, x0) are well defined.

Choose gx1 Î F (x0, y0) and gy1 Î F (y0, x0). This can be done because F (X × X) ⊆ g

(X). If a1 = a2 = a3 = a4 = a5 = a6 = 0, then

d(gx1, F(x1, y1)) ≤ H(F(x0, y0), F(x1, y1)) = 0.

Hence d(gx1, F (x1, y1)) = 0. Since F (x1, y1) is closed, gx1 Î F (x1, y1). Similarly gy1 ÎF (y1, x1). Thus (x1, y1) is a coupled coincidence point of {F, g} and so we finish the

proof. Now assume that ai >0, for some i = 1, ..., 6. Then h >0 and so there exist z1 ÎF (x1, y1) and z2 Î F (y1, x1) such that

d(gx1, z1) ≤ H(F(x0, y0), F(x1, y1)) +h2

,

d(gy1, z2) ≤ H(F(y0, x0), F(y1, x1)) +h2

.

Since F(X × X) ⊆ g(X), there exist x2 and y2 in X such that z1 = gx2 and z2 = gy2.

Thus

d(gx1, gx2) ≤ H(F(x0, y0), F(x1, y1)) +h2

,

d(gy1, gy2) ≤ H(F(y0, x0), F(y1, x1)) +h2

.

Continuing this process, one obtains two sequences {xn} and {yn} in X such that

gxn+1 ∈ F(xn, yn) and gyn+1 ∈ F(yn, xn),

d(gxn, gxn+1) ≤ H(F(xn−1, yn−1), F(xn, yn)) +hn

2,

d(gyn, gyn+1) ≤ H(F(yn−1, xn−1), F(yn, xn)) +hn

2.


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From (1), we have

d(gxn, gxn+1)

≤ H(F(xn−1, yn−1), F(xn, yn)) +hn

2≤ a1d(gxn−1, gxn) + a2d(F(xn−1, yn−1), gxn−1) + a3d(gyn−1, gyn) + a4d(F(xn, yn), gxn)

+a5d(F(xn−1, yn−1), gxn) + a6d(F(xn, yn), gxn−1) +hn

2≤ a1d(gxn−1, gxn) + a2d(gxn, gxn−1) + a3d(gyn−1, gyn) + a4d(gxn+1, gxn)

+a6d(gxn+1, gxn−1) +hn

2≤ a1d(gxn−1, gxn) + a2d(gxn, gxn−1) + a3d(gyn−1, gyn) + a4d(gxn+1, gxn)

+a6d(gxn+1, gxn) + a6d(gxn, gxn−1) +hn

2

= (a1 + a2 + a6)d(gxn−1, gxn) + a3d(gyn−1, gyn) + (a4 + a6)d(gxn, gxn+1) +hn

2,

and it follows that

(1 − a4 − a6) d(gxn, gxn+1) ≤ (a1 + a2 + a6) d(gxn−1, gxn) + a3d(gyn−1, gyn) +hn

2. (3)

Similarly it can be shown that,

(1 − a4 − a6) d(gyn, gyn+1) ≤ (a1 + a2 + a6) d(gyn−1, gyn) + a3d(gxn−1, gxn) +hn

2. (4)

Again,

d(gxn+1, gxn)

= H(F(xn, yn), F(xn−1, yn−1)) +hn

2≤ a1d(gxn, gxn−1) + a2d(F(xn, yn), gxn) + a3d(gyn, gyn−1) + a4d(F(xn−1, yn−1), gxn−1)

+a5d(F(xn, yn), gxn−1) + a6d(F(xn−1, yn−1), gxn) +hn

2≤ a1d(gxn, gxn−1) + a2d(gxn+1, gxn) + a3d(gyn, gyn−1) + a4d(gxn, gxn−1)

+a5d(gxn+1, gxn−1) +hn

2≤ a1d(gxn, gxn−1) + a2d(gxn+1, gxn) + a3d(gyn, gyn−1) + a4d(gxn, gxn−1)

+a5d(gxn+1, gxn) + a5d(gxn, gxn−1) +hn

2.

Hence,

(1 − a2 − a5) d(gxn+1, gxn) ≤ (a1 + a4 + a5) d(gxn−1, gxn) + a3d(gyn, gyn−1) +hn

2(5)

and

(1 − a2 − a5) d(gyn+1, gyn) ≤ (a1 + a4 + a5) d(gyn−1, gyn) + a3d(gxn, gxn−1) +hn

2. (6)

Let

δn = d(gxn, gxn+1) + d(gyn, gyn+1).


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Now, from (3) and (4), and respectively (5) and (6), we obtain:

(1 − a4 − a6) δn ≤ (a1 + a2 + a3 + a6) δn−1 +hn

2, (7)

(1 − a2 − a5) δn ≤ (a1 + a3 + a4 + a5) δn−1 +hn

2. (8)

Adding (7) and (8) we get

(2 − a2 − a4 − a5 − a6) δn ≤ (2a1 + a2 + 2a3 + a4 + a5 + a6) δn−1 + hn. (9)

Since by (2), a1 + a2 + a3 + a4 + a5 + a6 ≤ h <1, so we have

2a1 + a2 + 2a3 + a4 + a5 + a6 = 2(a1 + a2 + a3 + a4 + a5 + a6) − a2 − a4 − a5 − a6

≤ 2h − (a2 + a4 + a5 + a6)

≤ 2h − h(a2 + a4 + a5 + a6)

= h (2 − a2 − a4 − a5 − a6) .

Thus from (9) we get

(2 − a2 − a4 − a5 − a6) δn ≤ h (2 − a2 − a4 − a5 − a6) δn−1 + hn.

Hence, as 1/(2 - a2 - a4 - a5 - a6) <1,

δn ≤ hδn−1 + hn.

Thus we have

δn ≤ h(hδn−2 + hn−1) + hn = h2δn−2 + 2hn.

Continuing this process we obtain

δn ≤ hnδ0 + nhn. (10)

By the triangle inequality and (10), for m, n Î N with m > n, we have

d(gxn, gxm+n) + d(gyn, gym+n)

≤ d(gxn, gxn+1) + d(gxn+1, gxn+2) + · · · + d(gxn+m−1, gxm+n)

+d(gyn, gyn+1) + d(gyn+1, gyn+2) . . . + d(gyn+m−1, gym+n)

≤ (hnδ0 + nhn) + (hn+1δ0 + (n + 1)hn+1) + · · · + (hn+m−1δ0 + (n + m − 1)hn+m−1)

+(hnδ0 + nhn) + (hn+1δ0 + (n + 1)hn+1) + · · · + (hn+m−1δ0 + (n + m − 1)hn+m−1).

Thus

d(gxn, gxm+n) + d(gyn, gym+n) ≤n+m−1∑

i=nδ0hi +

n+m−1∑i=n

ihi.

Since h <1, we conclude that {gxn} and {gyn} are Cauchy sequences in g(X). Since g(X)

is complete, there exist x, y Î X such that gxn ® gx and gyn ® gy. Then from (1), we

obtain


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d(F(x, y), gx)

≤ d(F(x, y), gxn+1) + d(gxn+1, gx)

≤ H(F(x, y), F(xn, yn)) + d(gxn+1, gx)

≤ a1d(gx, gxn) + a2d(F(x, y), gx) + a3d(gy, gyn) + a4d(F(xn, yn), gxn)

+a5d(F(x, y), gxn) + a6d(F(xn, yn), gx) + d(gxn+1, gx)

≤ a1d(gx, gxn) + a2d(F(x, y), gx) + a3d(gy, gyn) + a4d(gxn+1, gxn)

+a5d(F(x, y), gxn) + a6d(gxn+1, gx) + d(gxn+1, gx).

On taking limit as n ® ∞, we have

d(F(x, y), gx) ≤ (a2 + a5)d(F(x, y), gx),

which implies that d(F(x, y), gx) = 0 and hence F(x, y) = gx. Similarly, F(y, x) = gy.

Hence (x, y) is coupled coincidence point of the mappings F and g. Suppose now that

(a) holds. Then for some (x, y) Î C(F, g), limn→∞ gnx = u and lim

n→∞ gny = v, where u, v Î

X. Since g is continuous at u and v, so we have that u and v are fixed points of g. As F

and g are w- compatible, gnx Î C(F, g) for all n ≥ 1 and gnx Î F(gn-1x, gn-1y).

Using (1), we obtain,

d(gu, F(u, v)) ≤ d(gu, gnx) + d(gnx, F(u, v))

≤ d(gu, gnx) + H(F(gn−1x, gn−1y), F(u, v))

≤ d(gu, gnx) + a1d(gnx, gu) + a2d(F(gn−1x, gn−1y), gnx) + a3d(gny, gv) + a4d(F(u, v), gu)

+ a5d(gnx, gu) + a6d(F(u, v), gnx).

On taking limit as n ® ∞, we have

d(gu, F(u, v)) ≤ (a4 + a6)d(gu, F(u, v)),

which implies d(gu, F(u, v)) = 0 and hence gu Î F(u, v). Similarly, gv Î F(v, u). Con-

sequently u = gu ÎF(u, v) and v = gv Î F(v, u). Hence (u, v) is a coupled fixed point of

F and g. Suppose now that (b) holds.

If for some (x, y) Î C(F, g), g is F- commuting, g2x = gx and g2y = gy, then gx = g2x

Î F(gx, gy) and gy = g2y Î F(gy, gx). Hence (gx, gy) is a coupled fixed point of F and g.

Suppose now that (c) holds and assume that for some (x, y) Î C(g, F) and for some u,

v Î X, limn→∞ gnu = x and lim

n→∞ gnv = y. By the continuity of g at x and y, we get x = gx Î

F(x, y) and y = gy Î F(y, x). Hence (x, y) is coupled fixed point of F and g. Finally, sup-

pose that (d) holds. Let g(C(F, g)) = {(x, x). Then {x} = {gx} = F(x, x). Hence (x, x) is

coupled fixed point of F and g. ■Now we present following example to support our Theorem 8.

Example 9. Let X = [1, 5] and F : X × X ® CB(X), g : X ® X be defined as follows:

F(x, y) = [2, 3] for all x, y ∈ X,

g(x) = 5 − 35

x, for all x ∈ X.

Then H(F(x, y), F(u, v)) = 0 for all x, y, u, v Î X. Therefore, F and g satisfy (1) for

any ai Î [0, 1), i = 1, 2, ..., 6. Also (4, 5) Î X × X is a coupled coincidence point of

hybrid pair {F, g}. Note that F and g do not satisfy anyone of the conditions from (a)-

(d) of Theorem 8 and do not have a coupled common fixed point.

If in Theorem 8 g = I (I = the identity mapping), then we have the following result.


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Corollary 10. Let (X, d) be a complete metric space, F : X × X ® CB(X) be a map-

ping satisfying

H(F(x, y), F(u, v)) ≤ a1d(x, u) + a2d(F(x, y), x) + a3d(y, v)

+a4d(F(u, v), u) + a5d(F(x, y), u) + a6d(F(u, v), x)

for all x, y, u, v Î X, where ai = ai(x, y, u, v), i = 1, 2, ..., 6, satisfy (2). Then F has a

coupled fixed point.

Corollary 11. Let (X, d) be a metric space, F : X × X ® CB(X) and g : X ® X be

mappings satisfying

H((F(x, y), F(u, v)) ≤ k2

[d(gx, gu) + d(gy, gv)] (11)

for all x, y, u, v Î X, where k Î [0, 1). 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. Moreover, F and g have a

coupled common fixed point if anyone of the conditions (a)-(d) of Theorem 8 holds.

Example 12. Let X = [0, 1], F : X × X ® CB(X) and g : X ® X be given as

F(x, y) = [0,sin x + sin y

8] for all x, y ∈ X,

and

g(x) =x

2for all x ∈ X.

Case (i) If sin x + sin y = sin u + sin v, then

H(F(x, y), F(u, v)) = 0.

Case (ii) If sin x + sin y ≠ sin u + sin v, then

H(F(x, y), F(u, v)) =18

∣∣(sin x + sin y) − (sin u + sin v)

∣∣≤ 1

8

(|sin x − sin u| +∣∣sin y − sin v

∣∣)

≤ 18

(|x − u| +∣∣y − v

∣∣)

≤ 316

(|x − u| +∣∣y − v

∣∣)

=38

(∣∣∣ x

2− u

2

∣∣∣ +∣∣∣ y

2− v

2

∣∣∣)

=38

(d(gx, gu), d(gy, gv)) =34

2[d(gx, gu) + d(gy, gv)].

Therefore F and g satisfy all the conditions of Corollary 11 with k = 34. Moreover, (0,

0) is a coupled common fixed point of F and g.

Corollary 13. Let (X, d) be a complete metric space, F : X × X ® CB(X) be a map-

ping satisfying

H((F(x, y), F(u, v)) ≤ k2

[d(x, u) + d(y, v)]

for all x, y, u, v Î X, where k Î [0, 1), then F has a coupled fixed point in X.


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Theorem 14. Let (X, d) be a metric space. Suppose that the mappings F : X × X ®CB(X) and g : X ® X satisfy

H(F(x, y), F(u, v)) ≤ h max{d(gx, gu), d(gy, gv), d(F(x, y), gx),

d(F(x, y), gu) + d(F(u, v), gx)2

, d(F(u, v), gu)} (12)

for all x, y, u, v Î X, where h Î [0, 1). If F(X × X) ⊆ g(X) and g(X) is a complete sub-

set of X, then F and g have a coupled coincidence point in X. Moreover, F and g have a

coupled common fixed point if one of the conditions (a)-(d) of Theorem 8 holds.

Proof. Let x0 and y0 be two arbitrary points in X. Choose gx1 Î F (x0, y0) and gy1 ÎF (y0, x0). This can be done because F(X × X) ⊆ g(X). If h = 0, then

d(gx1, F(x1, y1)) ≤ H(F(x0, y0), F(x1, y1)) = 0

gives that d(gx1, F (x1, y1)) = 0, and gx1 Î F (x1, y1). Similarly gy1 Î F (y1, x1). Hence

(x1, y1) is a coupled coincidence point of {F, g}. Now assume that h >0. Set k =1√h.

Then k >1 and so there exists z1 Î F(x1, y1) and z2 Î F(y1, x1) such that gx2 Î F (x1,

y1), gy2 Î F (y1, x1) and such that

d(gx1, z1) ≤ kH(F(x0, y0), F(x1, y1)),

d(gy1, z2) ≤ kH(F(y0, x0), F(y1, x1)).

Since F(X × X) ⊆ g(X), there exist x2 and y2 in X such that z1 = gx2 and z2 = gy2.

Also,

d(gx1, gx2) ≤ kH(F(x0, y0), F(x1, y1)),

d(gy1, gy2) ≤ kH(F(y0, x0), F(y1, x1)).

Continuing this process, one obtains two sequences {xn} and {yn} in X such that gxn+1Î F (xn, yn), gyn+1 Î F (yn, xn) and

d(gxn, gxn+1) ≤ kH(F(xn−1, yn−1), F(xn, yn)),

d(gyn, gyn+1) ≤ kH(F(yn−1, xn−1), F(yn, xn)).

For each n, using (12), we have

d(gxn, gxn+1)

≤ kH(F(xn−1, yn−1), F(xn, yn))

≤√

h max{d(gxn−1, gxn), d(gyn−1, gyn), d(F(xn−1, yn−1), gxn−1),

d(F(xn−1, yn−1), gxn) + d(F(xn, yn), gxn−1)2

, d(F(xn, yn), gxn)}

≤√

h max{

d(gxn−1, gxn), d(gyn−1, gyn),d(gxn+1, gxn−1)

2, d(gxn+1, gxn)

}

≤√

h max{

d(gxn−1, gxn), d(gyn−1, gyn),d(gxn+1, gxn) + d(gxn, gxn−1)

2, d(gxn+1, gxn)

}

=√

h max{d(gxn−1, gxn), d(gyn−1, gyn), d(gxn+1, gxn)

}.

Hence, if we suppose that d(gxn, gxn+1) ≤ √hd(gxn, gxn+1), then d(gxn, gxn+1) = 0.

Therefore,

d(gxn, gxn+1)√

h max{d(gxn−1, gxn), d(gyn−1, gyn)}. (13)


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Similarly,

d(gyn, gyn+1) ≤ √h max{d(gyn−1, gyn), d(gxn−1, gxn)}. (14)

Using (13) and (14), we obtain

d(gxn, gxn+1) ≤ (√

h)nδ

and

d(gyn, gyn+1) ≤ (√

h)nδ,

where δ = max{d(gx0, gx1),d(gy0, gy1)}. Thus for m, n Î N with m > n,

d(gxn, gxm+n) ≤ d(gxn, gxn+1) + d(gxn+1, gxn+2) + · · · + d(gxn+m−1, gxm+n)

+ (√

h)nδ + (√

h)n+1δ + · · · + (√

h)n+m−1δ.

Therefore

d(gxn, gxm+n) ≤n+m−1∑

i=n

(√

h)iδ.

Hence we conclude that {gxn} is a Cauchy sequence in g(X). Similarly we obtain that

{gyn} is a Cauchy sequence in g(X). Since g(X) is complete, so there exists x, y Î X

such that gxn ® gx and gyn ® gy.

Thus from (12),

d(F(x, y), gxn+1) ≤ H(F(x, y), F(xn, yn))

≤ h max{d(gx, gxn

), d

(gy, gyn

), d

(F(x, y), gx

),

d(F(x, y), gxn) + d(F(xn, yn), gx)2

, d(F(xn, yn), gxn)}

≤ h max{d(gx, gxn), d(gy, gyn), d(F(x, y), gx),

d(F(x, y), gxn) + d(gxn+1, gx)2

, d(gxn+1, gxn)}

On taking limit as n ® ∞, we obtain

d(F(x, y), gx) ≤ hd(F(x, y), gx),

which implies d(F(x, y), gx) = 0. As F(x, y) is closed, so gx Î F(x, y). Similarly, gy Î F

(y, x). Therefore (x, y) is a coupled coincidence point of F and g.

(a) Suppose that limn→∞ gx = u and lim

n→∞ gy = v, for some (x, y) Î C(F, g); u, v Î X.

Since g is continuous at u and v, so u and v are fixed points of g. Also since F and

g are w- compatible, gn(x) Î C(F, g) for all n ≥ 1 and gn(x) Î F(gn-1(x), gn-1(y)).

Using (12), we obtain


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d(gu, F(u, v)) ≤ d(gu, gn(x)) + d(gn(x), F(u, v))

≤ d(gu, gn(x)) + H(F(gn−1x, gn−1y), F(u, v))

≤ d(gu, gn(x)) + h max{d(g(gn−1x), gu), d(g(gn−1y), gv),

d(F(gn−1x, gn−1y), g(gn−1x)),

d(F(gn−1x, gn−1y), gu) + d(F(u, v), g(gn−1x)2

, d(F(u, v), gu)}

≤ d(gu, gn(x)) + h max{d(gnx, gu) + d(gny, gv),

d(gnx, gu) + d(F(u, v), gnx)2

, d(F(u, v), gu)}

.

Hence, taking limit as n ® ∞, we get

d(gu, F(u, v)) ≤ hd(gu, F(u, v)).

Hence d(gu, F(u, v)) = 0 and therefore gu Î F(u, v). Similarly, gv Î F(v, u). Conse-

quently u = gu Î F(u, v) and v = gv Î F(v, u). Hence (u, v) is a coupled fixed point

of F and g.

If the pair {F, g} satisfies condition (b)-(d) of Theorem 8, then result follow using

arguments similar to those given in the proof of Theorem 8. ■

Author details1Department of Mathematics, Lahore University of Management Sciences, Lahore - 54792, Pakistan 2Faculty ofMechanical Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia 3Department of Mathematics, Faculty ofAgriculture, Nemanjina 6, 11 000 Belgrade, Serbia

Authors’ contributionsAll authors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 3 October 2011 Accepted: 9 January 2012 Published: 9 January 2012

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doi:10.1186/1687-1812-2012-4Cite this article as: Abbas et al.: Coupled coincidence and common fixed point theorems for hybrid pair ofmappings. Fixed Point Theory and Applications 2012 2012:4.

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Coupled coincidence and common fixed point theorems for hybrid pair of mappings

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