JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 134712 10.1155/2013/134712 134712 Research Article A Unique Coupled Common Fixed Point Theorem for Symmetric (φ,ψ)-Contractive Mappings in Ordered G-Metric Spaces with Applications Jain Manish 1 Taş Kenan 2 Ferreira Antonio J. M. 1 Department of Mathematics Ahir College Rewari 123401 India 2 Department of Mathematics and Computer Science Cankaya University Ankara Turkey cankaya.edu.tr 2013 8 12 2013 2013 21 07 2013 19 10 2013 2013 Copyright © 2013 Manish Jain and Kenan Taş. This 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.

We establish the existence and uniqueness of coupled common fixed point for symmetric (φ,ψ)-contractive mappings in the framework of ordered G-metric spaces. Present work extends, generalize, and enrich the recent results of Choudhury and Maity (2011), Nashine (2012), and Mohiuddine and Alotaibi (2012), thereby, weakening the involved contractive conditions. Our theoretical results are accompanied by suitable examples and an application to integral equations.

1. Introduction and Preliminaries

The structure of G-metric spaces introduced by Mustafa and Sims  is a generalization of metric spaces. The theory of fixed points in this generalized structure was initiated by Mustafa et al. , in which Banach contraction principle was established in G-metric spaces. After that different authors proved several fixed point results in this space. References  are some examples of these works.

Definition 1 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let X be nonempty set, and let G:X×X×XR+ be a function satisfying the following properties:

G(x,y,z)=0 if x=y=z,

0<G(x,x,y) for all x,yX with xy,

G(x,x,y)G(x,y,z) for all x,y,zX with zy,

G(x,y,z)=G(x,z,y)=G(y,z,x)= (symmetry in all three variables),

G(x,y,z)G(x,a,a)+G(a,y,z) for all x,y,z,aX (rectangle inequality).

Then the function G is called a G-metric on X and the pair (X,G) is a called a G-metric space.

Definition 2 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let (X,G) be a G-metric space, and let {xn} be a sequence of points of X. A point xX is said to be the limit of the sequence {xn} if limn, mG(x,xn,xm)=0, and then we say that the sequence {xn} is G-convergent to x.

Thus, if xnx in G-metric space (X,G) then, for any ε>0, there exists a positive integer N such that G(x,xn,xm)<ε for all n,mN.

In , 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. This is a Hausdorff topology, so a sequence can converge at most to one point.

Definition 3 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let (X,G) be a G-metric space. A sequence {xn} is called G-Cauchy if, for every ε>0, there is a positive integer N such that G(xn,xm,xl)<ε for all n,m,lN; that is, G(xn,xm,xl)0 as n,m,l.

Lemma 4 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

If (X,G) is a G-metric space, then the following are equivalent:

{xn} is G-convergent to x,

G(xn,xn,x)0 as n,

G(xn,x,x)0 as n,

G(xm,xn,x)0 as m,n.

Lemma 5 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

If (X,G) is a G-metric space, then the following are equivalent:

the sequence {xn} is G-Cauchy;

for every ε>0, there exists a positive integer N such that G(xn,xm,xm)<ε for all n,m>N.

Lemma 6 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

If (X,G) is a G-metric space, then G(x,y,y)2G(y,x,x) for all x,yX.

Lemma 7.

If (X,G) is a G-metric space, then G(x,x,y)G(x,x,z)+G(z,z,y) for all x,y,zX.

Definition 8 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let (X,G), (X,G) be two G-metric spaces. Then a function f:XX is G-continuous at a point xX if and only if it is G-sequentially continuous at x; that is, whenever {xn} is G-convergent to x, {f(xn)} is G-convergent to f(x).

Lemma 9 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let (X,G) be a G-metric space; then the function G(x,y,z) is jointly continuous in all three of its variables.

Definition 10 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

A G-metric space (X,G) is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in (X,G) is convergent in X.

Definition 11 (see [<xref ref-type="bibr" rid="B23">10</xref>]).

Let (X,G) be a G-metric space. A mapping F:X×XX is said to be continuous if for any two G-convergent sequences {xn} and {yn} converging to x and y, respectively, {F(xn,yn)} is G-convergent to F(x,y).

Recently, fixed point theorems under different contractive conditions in metric spaces endowed with the partial ordering have been studied by various authors. Works noted in  are some examples in this direction. Bhaskar and Lakshmikantham  introduced the notion of coupled fixed points and proved some coupled fixed point theorems for a mapping having mixed monotone property. The work  was illustrated by proving the existence and uniqueness of the solution for a periodic boundary value problem.

Lakshmikantham and C´iric´  extended the notion of mixed monotone property by introducing the notion of mixed g-monotone property in partially ordered metric spaces.

Definition 12 (see [<xref ref-type="bibr" rid="B10">11</xref>]).

Let (X,) be a partially ordered set and F:X×XX. The mapping F is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and monotone nonincreasing in y; that is, for any x,yX, (1)x1,x2X,x1x2  implies  F(x1,y)F(x2,y),y1,y2X,y1y2  implies  F(x,y1)F(x,y2).

Definition 13 (see [<xref ref-type="bibr" rid="B11">12</xref>]).

Let (X,) be a partially ordered set F:X×XX, and g:XX. We say that the mapping F has the mixed g-monotone property if F(x,y) is monotone g-nondecreasing in its first argument and monotone g-nonincreasing in its second argument; that is, for any x,yX, (2)x1,x2X,gx1gx2  implies  F(x1,y)F(x2,y),y1,y2X,gy1gy2  implies  F(x,y1)F(x,y2).

Definition 14 (see [<xref ref-type="bibr" rid="B10">11</xref>]).

An element (x,y)X×X is called a coupled fixed point of the mapping F:X×XX if F(x,y)=x and F(y,x)=y.

Definition 15 (see [<xref ref-type="bibr" rid="B11">12</xref>]).

An element (x,y)X×X is called a coupled coincidence point of the mappings F:X×XX and g:XX if F(x,y)=gx and F(y,x)=gy.

Definition 16 (see [<xref ref-type="bibr" rid="B11">12</xref>]).

An element (x,y)X×X is called a coupled common fixed point of the mappings F:X×XX and g:XX if x=gx=F(x,y) and y=gy=F(y,x).

Definition 17 (see [<xref ref-type="bibr" rid="B11">12</xref>]).

The mappings F:X×XX and g:XX are called commutative if (3)gF(x,y)=F(gx,gy), for all x,yX.

Let (X,) be a partially ordered set, and let G be a G-metric on X such that (X,G) is a complete G-metric space.

Choudhury and Maity  established some coupled fixed point theorems for the mixed monotone mapping F:X×XX under a contractive condition of the form (4)G(F(x,y),F(u,v),F(w,z))k2[G(x,u,w)+G(y,v,z)], where k[0,1).

Various authors extended and generalized the results of Choudhury and Maity  under different contractive conditions in G-metric spaces. For more works, one can see . Nashine  generalized and extended the contractive condition (4) and thereby obtained the coupled coincidence points for a pair of commuting mappings under the following contraction: (5)G(F(x,y),F(u,v),F(w,z))k[G(gx,gu,gw)+G(gy,gv,gz)], where k[0,1/2).

Mohiuddine and Alotaibi  further generalized the contraction (4) by considering the following more general contractive condition: (6)φ(G(F(x,y),F(u,v),F(w,z)))  12φ(G(x,u,w)+G(y,v,z))-ψ(G(x,u,w)+G(y,v,z)2), where φ,ψ:[0,)[0,) are functions satisfying some appropriate conditions mentioned in .

Interestingly, for φ(t)=t, ψ(t)=((1-k)/2)t, with 0k<1, condition (6) reduces to (4).

On the other hand, Karapinar et al.  improved various results present in the literature of coupled fixed point theory of G-metric spaces by considering a generalized ϕ-contraction. Assigning the value kt to ϕ(t) with k[0,1) for t>0, Karapinar et. al.  in an alternative way generalized the contraction (4) for a pair of commutative mappings as follows: (7)G(F(x,y),F(u,v),F(w,z))+G(F(x,y),F(u,v),F(w,z))k[G(gx,gu,gw)+G(gy,gv,gz)], where k[0,1).

Present work extend and generalize several results present in the literature of fixed point theory of G-metric spaces. Our result directly derive a result of Karapinar et. al. . We give suitable examples to show how our results generalize and enrich the well-known results of Choudhury et al. , Nashine , and Mohiuddine and Alotaibi  by significantly weakening the involved contractive conditions. The effectiveness of the present work is shown by suitable examples and an application to the integral equations.

2. Main Results

Before proving our results, we need the following.

Denote by Φ the class of all functions φ:[0,)[0,) with the following properties:

φ is continuous and nondecreasing;

φ(t)<t for all t>0;

φ(t+s)φ(t)+φ(s) for all t,s[0,).

We note that (φi) and (φii) imply φ(t)=0 if and only if t=0.

Denote by Ψ the class of all functions ψ:[0,)[0,) with the following properties:

limtrψ(t)>0 for all r>0;

limt0+ψ(t)=0.

Some examples of φ(t) are kt (where k>0), and t/(t+1), t/(t+2) and examples of ψ(t) are kt (where k>0), ln(2t+1)/2.

Let (X,G) be a G-metric space, and let F:X×XX, g:XX be two mappings. We say that F and g are symmetric (φ,ψ)-contractive mappings on X if there exist φΦ and ψΨ such that (8)φ(+  G(F(y,x),F(v,u),F(z,w)))2-1(G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w)))2-1)φ(G(gx,gu,gw)+G(gy,gv,gz)2)-ψ(G(gx,gu,gw)+G(gy,gv,gz)2) for all x,y,u,v,w,zX.

Our first result is the following.

Theorem 18.

Let (X,) be a partially ordered set, and suppose that there exists a G-metric G on X such that (X,G) is a complete G-metric space. Let F and g be symmetric (φ,ψ)-contractive mappings on X with gxgugw and gygvgz (or gxgugw and gygvgz) such that F has the mixed g-monotone property. Assume that F(X×X)g(X); both the mappings F and g commutes and are continuous.

If there exist two elements x0,y0X with gx0F(x0,y0) and gy0F(y0,x0) (or gx0F(x0,y0) and gy0F(y0,x0)), then there exist x,yX such that gx=F(x,y) and gy=F(y,x); that is, F and g have a coupled coincidence point in X.

Proof.

Without loss of generality, assume that there exist x0,y0X such that gx0F(x0,y0), gy0F(y0,x0). Since F(X×X)g(X), we can choose x1,y1X such that gx1=F(x0,y0),gy1=F(y0,x0). Again we can choose x2,y2X such that gx2=F(x1,y1), gy2=F(y1,x1).

Continuing this process, we can construct sequences {gxn} and {gyn} in X such that (9)gxn+1=F(xn,yn),gyn+1=F(yn,xn),for  all  n0. We will prove, for all n0, that (10)gxngxn+1,gyngyn+1. Since gx0F(x0,y0), gy0F(y0,x0), and gx1=F(x0,y0), gy1=F(y0,x0), we have gx0gx1,  gy0gy1; that is, (10) holds for n=0.

Suppose that (10) holds for some n>0; that is, gxngxn+1, gyngyn+1. As F has the mixed g-monotone property, using (9), we have (11)gxn+1=F(xn,yn)F(xn+1,yn)F(xn+1,yn+1)=gxn+2,gyn+1=F(yn,xn)F(yn+1,xn)F(yn+1,xn+1)=gyn+2. Then by mathematical induction, it follows that (10) holds for all n0.

If for some n0, we have (gxn+1,gyn+1)=(gxn,gyn), and then F(xn,yn)=gxn and F(yn,xn)=  gyn; that is, F and g have a coupled coincidence point. So now onwards, we suppose that (gxn+1,gyn+1)(gxn,gyn) for all n0; that is, we suppose that either gxn+1=F(xn,yn)gxn or gyn+1=F(yn,xn)gyn.

Using (8)–(10), we have (12)φ(G(gxn+1gxn+1,gxn)+G(gyn+1,gyn+1gyn)2)=φ((G(F(xn,yn),F(xn,yn),F(xn-1,yn-1))+G(F(yn,xn),F(yn,xn),F(yn-1,xn-1)))2-1)φ(G(gxn,gxn,gxn-1)+G(gyn,gyn,gyn-1)2)-ψ(G(gxn,gxn,gxn-1)+G(gyn,gyn,gyn-1)2). Since ψ is nonnegative, using (12), we get (13)φ(G(gxn+1,gxn+1,gxn)+G(gyn+1,gyn+1,gyn)2)φ(G(gxn,gxn,gxn-1)+G(gyn,gyn,gyn-1)2). By monotonicity of φ, we get (14)G(gxn+1,gxn+1,gxn)+G(gyn+1,gyn+1,gyn)2G(gxn,gxn,gxn-1)+G(gyn,gyn,gyn-1)2. Let δn=(G(gxn+1,gxn+1,gxn)+G(gyn+1,gyn+1,gyn))/2; then {δn} is a monotone decreasing sequence. Therefore, there exists some δ0 such that (15)Limnδn=limn[G(gxn+1,gxn+1,gxn)+G(gyn+1,gyn+1,gyn)2]=δ. We claim that δ=0.

On the contrary, suppose that δ>0.

Taking limit as n on both sides of (12) and using the properties of φ and ψ, we have (16)φ(δ)=limnφ(δn)limn[φ(δn-1)-ψ(δn-1)]=φ(δ)-limδn-1δψ(δn-1)<φ(δ),a  contradiction. Thus, δ=0; that is, (17)limnδn=limn[G(gxn+1,gxn+1,gxn)+G(gyn+1,gyn+1,gyn)2]=0. Next, we shall show that {gxn} and {gyn} are G-Cauchy sequences.

If possible, suppose that at least one of {gxn} and {gyn} is not a G-Cauchy sequence. Then there exists an ε>0 for which we can find subsequences {gxn(k)}, {gxm(k)} of {gxn} and {gyn(k)}, {gym(k)} of {gyn} with n(k)>m(k)k such that (18)rk=G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k))2ε. Further, corresponding to m(k), we can choose n(k) in such a way that it is the smallest integer with n(k)>m(k)k and satisfies (18). Then, (19)G(gxn(k)-1,gxn(k)-1,gxm(k))+G(gyn(k)-1,gyn(k)-1,gym(k))2<ε. Using (18), (19) and Lemma 7, we get (20)εrk=G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k))2{G(gxn(k),gxn(k)gxn(k)-1)+G(gxn(k)-1,gxn(k)-1,gxm(k))+G(gyn(k),gyn(k),gyn(k)-1)+G(gyn(k)-1,gyn(k)-1,gym(k))}2-1<(G(gxn(k),gxn(k),gxn(k)-1)+G(gyn(k),gyn(k),gyn(k)-1))2-1+ε. Letting k and using (17), we have (21)limkrk=limk[+G(gyn(k),gyn(k),gym(k)))(2)-1(G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k)))2-1]=ε. Using Lemma 6 and Lemma 7, we get (22)G(gxn(k),gxn(k),gxm(k))G(gxn(k),gxn(k),gxn(k)+1)+G(gxn(k)+1,gxn(k)+1,gxm(k))2G(gxn(k)+1,gxn(k)+1,gxn(k))+G(gxn(k)+1,gxn(k)+1,gxm(k)+1)+G(gxm(k)+1,gxm(k)+1,gxm(k)). Similarly, we can obtain (23)G(gyn(k),gyn(k),gym(k))2G(gyn(k)+1,gyn(k)+1,gyn(k))+G(gyn(k)+1,gyn(k)+1,gym(k)+1)+G(gym(k)+1,gym(k)+1,gym(k)). Now, for all k0, using (22)-(23) in (18), we get (24)rk=G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k))22δn(k)+δm(k)+(G(gxn(k)+1,gxn(k)+1,gxm(k)+1)+G(gyn(k)+1,gyn(k)+1,gym(k)+1))2-1. By monotonicity of φ and property (φiii), we have (25)φ(rk)  φ(2-12δn(k)+δm(k)+(G(gxn(k)+1,gxn(k)+1,gxm(k)+1)+G(gyn(k)+1,gyn(k)+1,gym(k)+1))2-1)2φ(δn(k))+φ(δm(k))+φ×(2-1(G(gxn(k)+1,gxn(k)+1,gxm(k)+1)+G(gyn(k)+1,gyn(k)+1,gym(k)+1))2-1). Also, since n(k)>m(k), gxn(k)gxm(k) and gyn(k)gym(k), using (8) and (9), we have (26)φ(2-1(G(gxn(k)+1,gxn(k)+1,gxm(k)+1)+G(gyn(k)+1,gyn(k)+1,gym(k)+1))2-1)=φ(2-1(G(F(xn(k),yn(k)),F(xn(k),yn(k)),F(xm(k),ym(k)))+G(F(yn(k),xn(k)),F(yn(k),xn(k)),F(ym(k),xm(k))))2-1)φ(2-1(G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k)))2-1)-ψ(2-1(G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k)))2-1)=φ(rk)-ψ(rk). Combining (25) and (26), we obtain that (27)φ(rk)2φ(δn(k))+φ(δm(k))+φ(rk)-ψ(rk). On letting k, using (17), (21) and continuity of φ, we get (28)φ(ε)2φ(0)+φ(0)+φ(ε)-limkψ(rk)=2φ(0)+φ(0)+φ(ε)-limrkεψ(rk)<φ(ε),a  contradiction. Therefore both {gxn} and {gyn} are G-Cauchy sequences in X. Now, since the G-metric space (X,G) is G-complete, there exist x,y in X such that the sequences {gxn} and {gyn} are, respectively, G-convergent to x and y, and then by Lemma 4, we have (29)limnG(gxn,gxn,x)=limnG(gxn,x,x)=0,limnG(gyn,gyn,y)=limnG(gyn,y,y)=0. Using the G-continuity of g, and Definition 8, we get (30)limnG(ggxn,ggxn,gx)=limnG(ggxn,gx,gx)=0,(31)limnG(ggyn,ggyn,gy)=limnG(ggyn,gy,gy)=0. Since gxn+1=F(xn,yn) and gyn+1=F(yn,xn), hence using commutativity of F and g we obtain (32)ggxn+1=gF(xn,yn)=F(gxn,gyn),(33)ggyn+1=gF(yn,xn)=F(gyn,gxn). Since the mapping F is G-continuous and the sequences {gxn} and {gyn} are, respectively, G-convergent to x and y, hence using Definition 11, the sequence {F(gxn,gyn)} is G-convergent to F(x,y). By uniqueness of limit and using (30), and (32) we get F(x,y)=gx. Similarly, we can show that F(y,x)=gy. Hence, (x,y)X×X is a coupled coincidence point of F and g.

In the next theorem, we omit the continuity hypotheses of the mapping F along with the commutativity of mappings F and g. We need the following definition.

Definition 19.

Let (X,) be a partially ordered set, and suppose that there exists a G-metric G on X. We say that (X,G,) is regular if the following conditions hold:

if a nondecreasing sequence {xn}X such that xnx, then xnx for all n,

if a nonincreasing sequence {yn}X such that yny, then yyn for all n.

Theorem 20.

Let (X,) be a partially ordered set, and suppose that there exists a G-metric G on X. Let F and g be symmetric (φ,ψ)-contractive mappings on X with gxgugw and gygvgz (or gxgugw and gygvgz) such that F has the mixed g-monotone property. Assume that (X,G,) is regular. Suppose that (g(X),G) is G-complete and F(X×X)g(X). Suppose that there exist x0,y0X with gx0F(x0,y0) and gy0F(y0,x0)  (or  gx0F(x0,y0) and gy0F(y0,x0)); then F and g have a coupled coincidence point in X; that is, there exist x,yX such that gx=F(x,y) and gy=F(y,x).

Proof.

Proceeding exactly as in Theorem 18, we have that {gxn} and {gyn} are G-Cauchy sequences in the complete G-metric space (g(X),G). Then there exist x,yX such that gxngx and gyngy; that is, (34)LimnG(gxn,gx,gx)=limnG(gxn,gxn,gx)=0,limnG(gyn,gy,gy)=limnG(gyn,gyn,gy)=0. Since {gxn} is nondecreasing and {gyn} is nonincreasing, using the regularity of (X,G,), we have gxngx and gygyn for all n0. Using (8), we get (35)φ(2-1(G(F(x,y),gxn+1,gxn+1)+G(F(y,x),gyn+1,gyn+1))2-1)=φ(2-1(G(F(x,y),F(xn,yn),F(xn,yn))+G(F(y,x),F(yn,xn),F(yn,xn)))2-1)φ(G(gx,gxn,gxn)+G(gy,gyn,gyn)2)-ψ(G(gx,gxn,gxn)+G(gy,gyn,gyn)2). On letting n and using (34) and the properties of φ and ψ, we obtain that (36)φ(limn((G(F(x,y),gxn+1,gxn+1)+G(F(y,x),gyn+1,gyn+1))2-1)limn((G(F(x,y),gxn+1,gxn+1))φ(limn(G(gx,gxn,gxn)+G(gy,gyn,gyn)2))-limnψ(G(gx,gxn,gxn)+G(gy,gyn,gyn)2)=0, which yields (37)limn(2-1(G(F(x,y),gxn+1,gxn+1)+G(F(y,x),gyn+1,gyn+1))2-1)=0. Hence we can obtain that (38)limnG(F(x,y),gxn+1,gxn+1)=0=limnG(F(y,x),gyn+1,gyn+1). On the other hand, by condition (G5), we have (39)G(F(x,y),gx,gx)+G(F(y,x),gy,gy){G(F(x,y),gxn+1,gxn+1)+G(gxn+1,gx,gx)}+{G(F(y,x),gyn+1,gyn+1)+G(gyn+1,gy,gy)}. Letting n in (39) and using (34)–(38), we have (40)G(F(x,y),gx,gx)+G(F(y,x),gy,gy)=0. Thus F(x,y)=gx and gy=F(y,x). Therefore, we proved that (x,y) is a coupled coincidence point of F and g.

Next we give an example in support of Theorem 18 that shows that Theorem 18 is more general than Theorem 3.1 in , since the contractive condition (8) is more general than (5).

Example 21.

Let X=. Then (X,) is a partially ordered set with the natural ordering of real numbers. Let G:X×X×XR+ be defined by (41)G(x,y,z)=|x-y|+|y-z|+|z-x|hhhhhhhhhhhhhhhhhhhfor  x,y,zX. Then (X,G) is a complete G-metric space.

Define F:X×XX by F(x,y)=(x-2y)/8, (x,y)X×X and g:XX by g(x)=x/2, xX.

Clearly F(X×X)g(X), F and g are continuous, F has the mixed g-monotone property, and the pair (F,g) is commutative and satisfies condition (8) but does not satisfy the condition (5). Assume, to the contrary, that there exists some k[0,1/2) such that (5) holds. Then, we must have (42)(|x-2y8-u-2v8|+|u-2v8-w-2z8|+|w-2z8-x-2y8|)k[(|x2-u2|+|u2-w2|+|w2-x2|)+(|y2-v2|+|v2-z2|+|z2-y2|)]=k2[+(|y-v|+|v-z|+|z-y|)](|x-u|+|u-w|+|w-x|)+(|y-v|+|v-z|+|z-y|)] for all xuw and yvz. Taking x=u=w, v<z in the last inequality and setting ρ=|v-z|/2+|y-v|/2+|z-y|/2, we obtain (43)ρ2kρ,ρ>0, which implies that 1/2k, a contradiction since k[0,1/2). Hence F and g do not satisfy (5).

Indeed, for xuw and yvz, we have (44)|x-2y8-u-2v8|18|x-u|+14|y-v|,|u-2v8-w-2z8|18|u-w|+14|v-z|,|w-2z8-x-2y8|18|w-x|+14|z-y|,|y-2x8-v-2u8|18|y-v|+14|x-u|,|v-2u8-z-2w8|18|v-z|+14|u-w|,|z-2w8-y-2x8|18|z-y|+14|w-x|. By summing up the above six inequalities, we get exactly (8) with φ(t)=(1/3)t, ψ(t)=(1/12)t. Also, x0=-1, y0=1 are the two points in X such that gx0F(x0,y0) and gy0F(y0,x0). Now F, g, φ, and ψ satisfy all the conditions of Theorem 18; by Theorem 18, we obtain that F and g have a coupled coincidence point (0,0), but Theorem 3.1 in  cannot be applied to F and g in this example.

Now, putting g=IX (the identity map of X) in the previous results, we obtain the following.

Corollary 22.

Let (X,) be a partially ordered set, and let G be a G-metric on X. Let F:X×XX be a mapping satisfying (8) (with g=IX); that is, (45)φ(2-1(G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w)))2-1)φ(G(x,u,w)+G(y,v,z)2)-ψ(G(x,u,w)+G(y,v,z)2) for all x,y,u,v,w,zX with xuw and yvz (or xuw and yvz). Assume that (X,G) is complete and F has the mixed monotone property. Also suppose that either

F is continuous, or

(X,,G) is regular.

If there exist two elements x0,y0X with x0F(x0,y0) and y0F(y0,x0)  (or x0F(x0,y0) and y0F(y0,x0)), then there exist x,yX such that x=F(x,y) and y=F(y,x); that is, F has a coupled fixed point in X.

The following example shows that the contractive condition (45) is more general than the contractive conditions (4) and (6).

Example 23.

Let X=. Then (X,) is a partially ordered set with the natural ordering of real numbers. Let G:X×X×XR+ be defined by (46)G(x,y,z)=|x-y|+|y-z|+|z-x|hhhhhhhhhhhhhhhhhhhfor  x,y,zX. Then (X,G) is a complete G-metric space.

Define F:X×XX by F(x,y)=(x-5y)/10, (x,y)X×X.

Then F is continuous and satisfies the mixed monotone property. We note that F satisfies condition (45) but does not satisfy the conditions (4) and (6). Indeed, assume that there exists k[0,1), such that (4) holds. Then, we must have (47)|x-5y10-u-5v10|+|u-5v10-w-5z10|+|w-5z10-x-5y10|k2{+(|y-v|+|v-z|+|z-y|)}(|x-u|+|u-w|+|w-x|)+(|y-v|+|v-z|+|z-y|)}hhhhhhhhhhhxuw,yvz, by which, for x=u=w, we get (48)|y-v|+|v-z|+|z-y|k(|y-v|+|v-z|+|z-y|), which for v<z implies 1k, a contradiction, since k[0,1). Hence F does not satisfy (4).

Next, we prove that (6) is not satisfied, either. Assume, to the contrary, that there exist functions φ and ψ satisfying appropriate conditions as in  such that (6) holds. This means that (49)φ(|x-5y10-u-5v10|+|u-5v10-w-5z10|+|w-5z10-x-5y10|)12φ(+|y-v|+|v-z|+|z-y|)|x-u|+|u-w|+|w-x|+|y-v|+|v-z|+|z-y|)-ψ(2-1(|x-u|+|u-w|+|w-x|+|y-v|+|v-z|+|z-y|)2-1), for all xuw and yvz. Taking x=u=w, v<z in the previous inequality and setting α=(|y-v|+|v-z|+|z-y|)/2, we obtain (50)φ(α)12φ(2α)-ψ(α),α>0. Since φ satisfy the subadditive property, we have (1/2)φ(2α)φ(α), and therefore, we deduce that, for all α>0, ψ(α)0; that is, ψ(α)=0, which contradicts the definition of ψ.

This shows that F does not satisfy (6).

Finally, we prove that (45) holds. Indeed, for xuw and yvz, we have (51)|x-5y10-u-5v10|110|x-u|+12|y-v|,|u-5v10-w-5z10|110|u-w|+12|v-z|,|w-5z10-x-5y10|110|w-x|+12|z-y|,|y-5x10-v-5u10|110|y-v|+12|x-u|,|v-5u10-z-5w10|110|v-z|+12|u-w|,|z-5w10-y-5x10|110|z-y|+12|w-x|. By summing up the above six inequalities, we get exactly (45) with φ(t)=(1/2)t, ψ(t)=(1/5)t. Also, x0=-1, y0=1 are the two points in X such that x0F(x0,y0) and y0F(y0,x0).

By Corollary 22, we obtain that F has a coupled fixed point (0,0), but Theorems 3.1 and 3.2 in  and Theorem 3.1 in  cannot be applied to F in this example.

Corollary 24.

Let (X,) be a partially ordered set, and let G be a G-metric on X. Let F:X×XX be a mapping having mixed monotone property on X. Suppose there exists ψΨ such that (52)[G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w))][G(x,u,w)+G(y,vz)]-2ψ(G(x,u,w)+G(y,v,z)2) for all x,y,u,v,w,zX with xuw and yvz (or xuw and yvz). Assume that (X,G) is complete, and also suppose that either

F is continuous, or

(X,,G) is regular.

If there exist two elements x0,y0X with x0F(x0,y0) and y0F(y0,x0) (or x0F(x0,y0) and y0F(y0,x0)), then there exist x,yX such that x=F(x,y) and y=F(y,x); that is, F has a coupled fixed point in X.

Proof.

Note that if ψΨ, then for all r>0, rψΨ. Now divide (52) by 4 and take φ(t)=(1/2)t, t[0,); then condition (52) reduces to (8) with ψ1=(1/2)ψ and g(x)=x; and hence by Theorem 18 and Theorem 20, we obtain Corollary 24.

The following result provides us the recent result of Karapinar et. al. [32, Corollary 2.5].

Theorem 25 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Let (X,) be a partially ordered set, and suppose that there exists a G-metric G on X such that (X,G) is a complete G-metric space. Let F:X×XX and g:XX be two mappings such that F has the mixed g-monotone property on X and F(X×X)g(X). Suppose that there exists a real number k[0,1) such that (53)G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w))k[G(gx,gu,gw)+G(gy,gv,gz)] for all x,y,u,v,w,zX with gxgugw and gygvgz (or gxgugw and gygvgz). Also suppose that either

F and g are continuous, (X,G) is complete, and g commutes with F, or

(g(X),G) is complete and (X,,G) is regular.

If there exist two elements x0,y0X with gx0F(x0,y0) and gy0F(y0,x0) (or gx0F(x0,y0) and gy0F(y0,x0)), then there exist x,yX such that gx=F(x,y) and gy=F(y,x).

Proof.

Define φ,ψ:[0,)[0,), φ(t)=t/2 and ψ(t)=(1-k)(t/2), 0k<1. Then (53) holds. Hence the result follows from Theorem 18 or Theorem 20.

Remark 26.

The choice of functions F and g in Example 21 shows that Theorem 25 is more general than Theorem 3.1 in , since the contractive condition (53) is more general than (5). Indeed, the contractive condition (5) does not hold for the choice of functions F and g, but (53) holds exactly for k=3/4 with x0=-1 and y0=1 and yields (0,0) as the coupled coincidence point of F and g.

Now, putting g=IX (the identity map of X) in Theorem 25, we obtain the following.

Corollary 27.

Let (X,) be a partially ordered set, and let G be a G-metric on X such that (X,G) is a complete G-metric space. Let F:X×XX be a mapping having mixed monotone property on X. Suppose there exists a real number k[0,1) such that (54)G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w))k[G(x,u,w)+G(y,v,z)] for all x,y,u,v,w,zX with xuw and yvz (or xuw and yvz). Suppose that either

F is continuous, or

(X,,G) is regular.

If there exist two elements x0,y0X with x0F(x0,y0) and y0F(y0,x0) (or x0F(x0,y0) and y0F(y0,x0)), then there exist x,yX such that x=F(x,y) and y=F(y,x); that is, F has a coupled fixed point in X.

Remark 28.

The choice of function F in Example 23 shows that Corollary 27 is more general than Theorem 3.1 and Theorem 3.2 in , since the contractive condition (54) is more general than (4). Indeed the contractive condition (4) does not hold for the choice of function F, but (54) holds exactly for k=3/5 with x0=-1, y0=1.

Next we prove the existence and uniqueness of the coupled common fixed point for our main result.

Theorem 29.

In addition to the hypotheses of Theorem 18, suppose that for every (x,y),(x*,y*)X×X, there exists a (u,v)X×X such that (F(u,v),F(v,u)) is comparable to (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)X×X such that x=g(x)=F(x,y) and y=g(y)=F(y,x).

Proof.

From Theorem 18, the set of coupled coincidences is nonempty. In order to prove the theorem, we shall first show that if (x,y) and (x*,y*) are coupled coincidence points, that is, if g(x)=F(x,y), g(y)=F(y,x) and g(x*)=F(x*,y*), g(y*)=F(y*,x*), then (55)g(x)=g(x*),g(y)=g(y*). By assumption, there is (u,v)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*)). Put u0=u, v0=v and choose u1,v1X, so that gu1=F(u0,v0), gv1=F(v0,u0).

Then, as in the proof of Theorem (8), we can inductively define the sequences {gun} and {gvn} such that gun+1=F(un,vn) and gvn+1=F(vn,un).

Further, set x0=x, y0=y, x0*=x*, and y0*=y*, and on the same way define the sequences {gxn}, {gyn} and {gxn*}, {gyn*}. Then, it is easy to show that (56)gxn+1=F(xn,yn),gyn+1=F(yn,xn),gxn+1*=F(xn*,yn*),gyn+1*=F(yn*,xn*),hhhhhhhhhhhhhhhhhhhhhhhfor  all  n0. Since (F(u,v),F(v,u))=(gu1,gv1) and (F(x,y),F(y,x))=(gx1,gy1)=(gx,gy) are comparable, then gu1gx and gv1gy. It is easy to show that (gun,gvn) and (gx,gy) are comparable; that is, gungx and gvngy for all n1. Thus from (8), we have (57)φ(G(gun+1,gun+1,gx)+G(gvn+1,gvn+1,gy)2)=φ((G(F(un,vn),F(un,vn),F(x,y))+G(F(vn,un),F(vn,un),F(y,x)))2-1)φ(G(gun,gun,gx)+G(gvn,gvn,gy)2)-ψ(G(gun,gun,gx)+G(gvn,gvn,gy)2). Since ψ is nonnegative, we have (58)φ(G(gun+1,gun+1,gx)+G(gvn+1,gvn+1,gy)2)φ(G(gun,gun,gx)+G(gvn,gvn,gy)2). By monotonicity of φ, we have (59)G(gun+1,gun+1,gx)+G(gvn+1,gvn+1,gy)2G(gun,gun,gx)+G(gvn,gvn,gy)2. Thus, the sequence {γn  } defined by γn  =(G(gun,gun,gx)+G(gvn,gvn,gy))/2 is monotonically decreasing, so there exists some γ0 such that limnγn=γ.

We shall show that γ=0. Suppose, to the contrary, that γ>0. Then taking limit as n in (57) and using the continuity of φ, we have (60)φ(γ)φ(γ)-limγnγψ(γn)<φ(γ),a  contradiction. Thus, γ=0; that is, limnγn=0.

Hence, it follows that gungx, gvngy.

Similarly, we can show that gungx*, gvngy*.

By uniqueness of limit, it follows that gx=gx* and gy=gy*. Thus, we proved (55).

Since gx=F(x,y), gy=F(y,x), and the pair (F,g) is commuting, it follows that (61)ggx=gF(x,y)=F(gx,gy),ggy=gF(y,x)=F(gy,gx). Denote gx=z, gy=w. Then from (61), we have (62)gz=F(z,w),gw=F(w,z). Thus (z,w) is a coupled coincidence point.

Then from (55) with x*=z and y*=w, it follows that gz=gx and gw=gy; that is, (63)gz=z,gw=w. Combining (62) and (63), we obtain (64)z=gz=F(z,w),w=gw=F(w,z). Therefore, (z,w) is the coupled common fixed point of F and g.

To prove the uniqueness, assume that (p,q) is another coupled common fixed point. Then by (55), we have p=gp=gz=z and q=gq=gw=w.

Theorem 30.

In addition to the hypotheses of Theorem 20, suppose that, for every (x,y),(x*,y*)X×X, there exists a (u,v)X×X such that (F(u,v),F(v,u)) is comparable to (F(x,y),F(y,x)) and (F(x*,y*),F(y*,x*)). If F and g commute, then F and g have a unique coupled common fixed point; that is, there exists a unique (x,y)X×X such that x=g(x)=F(x,y) and y=g(y)=F(y,x).

Proof.

Proceeding exactly as in Theorem 29 result follows immediately.

3. Applications to Integral Equations

Motivated by the works of Aydi et al.  and Luong and Thuan , in this section, we study the existence of solutions to nonlinear integral equations using some of our main results.

Consider the integral equations in the following system: (65)x(t)=p(t)+0TS(t,s)[f(s,x(s))+k(s,y(s))]ds,y(t)=p(t)+0TS(t,s)[f(s,y(s))+k(s,x(s))]ds.

Let Θ denote the class of functions θ:[0,)[0,) which satisfies the following conditions:

θ is increasing;

there exists ψΨ such that θ(r)=(r/2)-ψ(r/2) for all r[0,).

For example, θ1(x)=αx (where 0α1/2), θ2(x)=x2/2(x+1) are some members of Θ.

We shall analyze the system (65) under the following assumptions:

f,k:[0,T]× are continuous;

p:[0,T] is continuous;

S:[0,T]×[0,) is continuous;

there exist λ>0 and θΘ such that for all x,y, yx, (66)0f(s,y)-f(s,x)λθ(y-x),0k(s,x)-k(s,y)λθ(y-x);

we suppose that (67)3λsupt[0,T]0TS(t,s)ds12.

there exist continuous functions α,β:[0,T] such that (68)α(t)p(t)+0TS(t,s)(f(s,α(s))+k(s,β(s)))ds,β(t)p(t)+0TS(t,s)(f(s,β(s))+k(s,α(s)))ds.

Consider the space X=C([0,T],) of continuous functions defined on [0,T] endowed with the (G-complete) G-metric given by (69)G(u,v,w)=supt[0,T]|u(t)-v(t)|+supt[0,T]|v(t)-w(t)|+supt[0,T]|w(t)-u(t)|u,v,wX. Endow X with the partial order given by x,yX, xyx(t)y(t) for all t[0,T]. Also, we may adjust as in  to prove that (X,G,) is regular.

Theorem 31.

Under assumptions (i)–(vi), the system (65) has a solution in X2=(C([0,T],))2.

Proof.

Consider the operator F:X×XX defined by (70)F(x,y)(t)=p(t)+0TS(t,s)[f(s,x(s))+k(s,y(s))]dsfor  t[0,T],x,yX. First, we shall prove that F has the mixed monotone property.

In fact, for x1x2 and t[0,T], we have (71)F(x2,y)(t)-F(x1,y)(t)=0TS(t,s)[f(s,x2(s))-f(s,x1(s))]ds. Taking into account that x1(t)x2(t) for all t[0,T], so by (iv), f(s,x2(s))f(s,x1(s)). Then F(x2,y)(t)F(x1,y)(t) for all t[0,T]; that is, (72)F(x1,y)F(x2,y). Similarly, for y1y2 and t[0,T], we have (73)F(x,y1)(t)-F(x,y2)(t)=0TS(t,s)[k(s,y1(s))-k(s,y2(s))]ds. Having y1(t)y2(t), so by (iv), k(s,y1(s))k(s,y2(s)). Then F(x,y1)(t)F(x,y2)(t) for all t[0,T]; that is, (74)F(x,y1)F(x,y2). Therefore, F has the mixed monotone property.

First we estimate the quantity G(F(x,y),F(u,v),F(w,z)) for all x,y,u,v,w,zX with xuw and yvz. Since F has the mixed monotone property, we have (75)F(w,z)F(u,v)F(x,y). We obtain (76)G(F(x,y),F(u,v),F(w,z))=supt[0,T]|F(x,y)(t)-F(u,v)(t)|+supt[0,T]|F(u,v)(t)-F(w,z)(t)|+supt[0,T]|F(w,z)(t)-F(x,y)(t)|=supt[0,T](F(x,y)(t)-F(u,v)(t))+supt[0,T](F(u,v)(t)-F(w,z)(t))+supt[0,T](F(x,y)(t)-F(w,z)(t)). Also for all t[0,T], from (iv), we have (77)F(x,y)-F(u,v)=0TS(t,s)[f(s,x(s))-f(s,u(s))]ds+0TS(t,s)[k(s,y(s))-k(s,v(s))]dsλ0TS(t,s)[θ(x(s)-u(s))+θ(v(s)-y(s))]ds. Since the function θ is increasing xuw, and yvz, we have (78)θ(x(s)-u(s))θ(suptI|x(t)-u(t)|),θ(v(s)-y(s))θ(suptI|v(t)-y(t)|). Hence by (77), we obtain (79)|F(x,y)-F(u,v)|λ0TS(t,s)[θ(suptI|x(t)-u(t)|)+θ(suptI|v(t)-y(t)|)]ds, as all the quantities on the right hand side of (77) are nonnegative, so (79) is justified.

Similarly, we can obtain that (80)|F(x,y)-F(w,z)|λ0TS(t,s)[θ(suptI|x(t)-w(t)|)+θ(suptI|z(t)-y(t)|)]ds,(81)|F(u,v)-F(w,z)|λ0TS(t,s)[θ(suptI|u(t)-w(t)|)+θ(suptI|z(t)-v(t)|)]ds. Summing (79), (80), and (81) and then taking the supremum with respect to t, we get (82)G(F(x,y),F(u,v),F(w,z))λsupt[0,T]0TS(t,s)ds·[θ(suptI|x(t)-u(t)|)+θ(suptI|x(t)-w(t)|)+θ(suptI|u(t)-w(t)|)]+λsupt[0,T]0TS(t,s)ds·[θ(suptI|v(t)-y(t)|)+θ(suptI|z(t)-y(t)|)+θ(suptI|z(t)-v(t)|)]. Further, since θ is increasing, so that we have (83)θ(suptI|x(t)-u(t)|)θ(G(x,u,w)),θ(suptI|x(t)-w(t)|)θ(G(x,u,w)),θ(suptI|u(t)-w(t)|)θ(G(x,u,w)). Similarly, we have (84)θ(suptI|v(t)-y(t)|)θ(G(y,v,z)),θ(suptI|z(t)-y(t)|)θ(G(y,v,z)),θ(suptI|z(t)-v(t)|)θ(G(y,v,z)). Then by (82), we have (85)G(F(x,y),F(u,v),F(w,z))λsupt[0,T]0TS(t,s)ds  ·3θ(G(x,u,w))+λsupt[0,T]×0TS(t,s)ds·3θ(G(y,v,z))=3λsupt[0,T]×0TS(t,s)ds·(θ(G(x,u,w))+θ(G(y,v,z))). Similarly, we can obtain that (86)G(F(y,x),F(v,u),F(z,w))3λsupt[0,T]×0TS(t,s)ds·(θ(G(x,u,w))+θ(G(y,v,z))). Summing (85) and (86), dividing by 2, and using (v), we get (87)(G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w)))2-13λsupt[0,T]×0TS(t,s)ds·(θ(G(x,u,w))+θ(G(y,v,z))).(θ(G(x,u,w))+θ(G(y,v,z)))2. Since θ is increasing, we have (88)θ(G(x,u,w))θ(G(x,u,w)+G(y,v,z)),θ(G(y,v,z))θ(G(x,u,w)+G(y,v,z)) and so (89)(θ(G(x,u,w))+θ(G(y,v,z)))2θ(G(x,u,w)+G(y,v,z))=G(x,u,w)+G(y,v,z)2-ψ(G(x,u,w)+G(y,v,z)2) by definition of θ. Thus, using (87) and (89), we finally get (90)(G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w)))2-1G(x,u,w)+G(y,v,z)2-ψ(G(x,u,w)+G(y,v,z)2), or (91)G(F(x,y),F(u,v),F(w,z))+G(F(y,x),F(v,u),F(z,w))G(x,u,w)+G(y,v,z)-2ψ(G(x,u,w)+G(y,v,z)2), which is just the contractive condition (52) in Corollary 24.

Let α, β be the functions appearing in assumption (vi); we get (92)αF(α,β),βF(β,α). Applying Corollary 24, we deduce the existence of x,yX such that (93)x=F(x,y),y=F(y,x); that is, (x,y) is a solution of the system (65).

4. Conclusion

In the setup of ordered G-metric spaces, we established some coupled coincidence and common coupled fixed point theorems for the mixed g-monotone mappings satisfying symmetric (φ,ψ)-contractive conditions. We accompanied our theoretical results by an applied example and an application to integral equations. Contractive conditions presented in this paper extend, complement, and unify the contractions in [10, 31, 33, 34] as well as several other contractions as in relevant items from the reference section of this paper and in the literature in general.

Mustafa Z. Sims B. A new approach to generalized metric spaces Journal of Nonlinear and Convex Analysis 2006 7 2 289 297 MR2254125 Mustafa Z. Obiedat H. Awawdeh F. Some fixed point theorem for mapping on complete G-metric spaces Fixed Point Theory and Applications 2008 2008 2-s2.0-53849097017 10.1155/2008/189870 189870 Mustafa Z. Shatanawi W. Bataineh M. Fixed point theorems on uncomplete G-metric spaces Journal of Mathematics and Statistics 2008 4 4 196 201 Mustafa Z. Shatanawi W. Bataineh M. Existence of fixed point results in G-metric spaces International Journal of Mathematics and Mathematical Sciences 2009 2009 10 2-s2.0-69249215420 10.1155/2009/283028 283028 Mustafa Z. Sims B. Fixed point theorems for contractive mappings in complete G-Metric spaces Fixed Point Theory and Applications 2009 2009 2-s2.0-68349122777 10.1155/2009/917175 917175 Abbas M. Rhoades B. E. Common fixed point results for noncommuting mappings without continuity in generalized metric spaces Applied Mathematics and Computation 2009 215 1 262 269 2-s2.0-67949104949 10.1016/j.amc.2009.04.085 Saadati R. Vaezpour S. M. Vetro P. Rhoades B. E. Fixed point theorems in generalized partially ordered G-metric spaces Mathematical and Computer Modelling 2010 52 5-6 797 801 2-s2.0-77954177283 10.1016/j.mcm.2010.05.009 Chugh R. Kadian T. Rani A. Rhoades B. E. Property P in G-metric spaces Fixed Point Theory and Applications 2010 2010 2-s2.0-77956761968 10.1155/2010/401684 401684 Abbas M. Nazir T. Radenović S. Some periodic point results in generalized metric spaces Applied Mathematics and Computation 2010 217 8 4094 4099 2-s2.0-78650024191 10.1016/j.amc.2010.10.026 Choudhury B. S. Maity P. Coupled fixed point results in generalized metric spaces Mathematical and Computer Modelling 2011 54 1-2 73 79 2-s2.0-79955482531 10.1016/j.mcm.2011.01.036 Bhaskar T. G. Lakshmikantham V. Fixed point theorems in partially ordered metric spaces and applications Nonlinear Analysis: Theory, Methods and Applications 2006 65 7 1379 1393 2-s2.0-33745215115 10.1016/j.na.2005.10.017 Lakshmikantham V. Ćirić L. Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces Nonlinear Analysis: Theory, Methods and Applications 2009 70 12 4341 4349 2-s2.0-63749106584 10.1016/j.na.2008.09.020 Choudhury B. S. Kundu A. A coupled coincidence point result in partially ordered metric spaces for compatible mappings Nonlinear Analysis: Theory, Methods and Applications 2010 73 8 2524 2531 2-s2.0-77955467920 10.1016/j.na.2010.06.025 Harjani J. Lpez B. Sadarangani K. Fixed point theorems for mixed monotone operators and applications to integral equations Nonlinear Analysis: Theory, Methods and Applications 2011 74 5 1749 1760 2-s2.0-78651351359 10.1016/j.na.2010.10.047 Luong N. V. Thuan N. X. Coupled fixed points in partially ordered metric spaces and application Nonlinear Analysis: Theory, Methods and Applications 2011 74 3 983 992 2-s2.0-78149282220 10.1016/j.na.2010.09.055 Samet B. Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces Nonlinear Analysis: Theory, Methods and Applications 2010 72 12 4508 4517 2-s2.0-77950460808 10.1016/j.na.2010.02.026 Ran A. C. M. Reurings M. C. B. A fixed point theorem in partially ordered sets and some applications to matrix equations Proceedings of the American Mathematical Society 2004 132 5 1435 1443 2-s2.0-2142762916 10.1090/S0002-9939-03-07220-4 Nieto J. J. Rodríguez-López R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations Order 2005 22 3 223 239 2-s2.0-33644688928 10.1007/s11083-005-9018-5 Nieto J. J. Rodríguez-López R. Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations Acta Mathematica Sinica, English Series 2007 23 12 2205 2212 2-s2.0-36448985642 10.1007/s10114-005-0769-0 Ćirić L. Cakić N. Rajović M. Ume J. S. Monotone generalized nonlinear contractions in partially ordered metric spaces Fixed Point Theory and Applications 2008 2008 11 2-s2.0-63849179332 10.1155/2009/131294 131294 Harjani J. Sadarangani K. Fixed point theorems for weakly contractive mappings in partially ordered sets Nonlinear Analysis: Theory, Methods and Applications 2009 71 7-8 3403 3410 2-s2.0-67349191471 10.1016/j.na.2009.01.240 Harjani J. Sadarangani K. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations Nonlinear Analysis: Theory, Methods and Applications 2010 72 3-4 1188 1197 2-s2.0-71549146236 10.1016/j.na.2009.08.003 Karapinar E. Couple fixed point theorems for nonlinear contractions in cone metric spaces Computers and Mathematics with Applications 2010 59 12 3656 3668 2-s2.0-77953231875 10.1016/j.camwa.2010.03.062 Aydi H. Karapınar E. Shatanawi W. Tripled coincidence point results for generalized contractions in ordered generalized metric spaces Fixed Point Theory and Applications 2012 2012, article 101 10.1186/1687-1812-2012-101 MR2946508 Jain M. Tas K. Kumar S. Gupta N. Coupled common fixed points involving a (ϕ,ψ)—contractive condition for mixed g-monotone operators in partially ordered metric spaces Journal of Inequalities and Applications 2012 2012, article 285 Abbas M. Khan A. R. Nazir T. Coupled common fixed point results in two generalized metric spaces Applied Mathematics and Computation 2011 217 13 6328 6336 2-s2.0-79952003710 10.1016/j.amc.2011.01.006 Aydi H. Damjanović Bosko B. Samet B. Shatanawi W. Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces Mathematical and Computer Modelling 2011 54 9-10 2443 2450 2-s2.0-80051596394 10.1016/j.mcm.2011.05.059 Shatanawi W. Abbas M. Samet B. Common coupled fixed points for mapping satisfying (ψ,ϕ)-weakly contractive condition in generalized metric spaces submitted for publication. In press Luong N. V. Thuan N. X. Coupled fixed point theorems in partially ordered G-metric spaces Mathematical and Computer Modelling 2012 55 3-4 1601 1609 2-s2.0-84855199648 10.1016/j.mcm.2011.10.058 Aydi H. Postolache M. Shatanawi W. Coupled fixed point results for (ψ, φ)-weakly contractive mappings in ordered G-metric spaces Computers and Mathematics with Applications 2012 63 1 298 309 2-s2.0-83655163959 10.1016/j.camwa.2011.11.022 Nashine H. K. Coupled common fixed point results in ordered G-metric spaces Journal of Nonlinear Science and Its Applications 2012 5 1 1 13 MR2909233 Karapinar E. Kaymakcalan B. Tas K. On coupled fixed point theorems on partially ordered G-metric spaces Journal of Inequalities and Applications 2012 2012, article 200 Mohiuddine S. A. Alotaibi A. On coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces Abstract and Applied Analysis 2012 2012 15 897198 MR2999919 10.1155/2012/897198 Ding H. S. Karapinar E. A note on some coupled fixed-point theorems on G-metric spaces Journal of Inequalities and Applications 2012 2012, article 170 Luong N. V. Thuan N. X. Coupled fixed points in partially ordered metric spaces and application Nonlinear Analysis: Theory, Methods and Applications 2011 74 3 983 992 2-s2.0-78149282220 10.1016/j.na.2010.09.055