# A Unique Coupled Common Fixed Point Theorem for Symmetric(φ,ψ)-Contractive Mappings in Ordered G-Metric Spaces with Applications

Authors: Manish Jain; Kenan Taş
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction and Preliminaries

The structure ofG-metric spaces introduced by Mustafa and Sims [1] is a generalization of metric spaces. The theory of fixed points in this generalized structure was initiated by Mustafa et al. [2], in which Banach contraction principle was established in G-metric spaces. After that different authors proved several fixed point results in this space. References [3–9] are some examples of these works.Definition 1 (see [1]).
LetX be nonempty set, and let G:X×X×X→R+ be a function satisfying the following properties:(G1)
G(x,y,z)=0 if x=y=z,(G2)
0<G(x,x,y) for all x,y∈X with x≠y,(G3)
G(x,x,y)≤G(x,y,z) for all x,y,z∈X with z≠y,(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯ (symmetry in all three variables),(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z) for all x,y,z,a∈X (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 [1]).
Let(X,G) be a G-metric space, and let {xn} be a sequence of points of X. A point x∈X is said to be the limit of the sequence {xn} if limn,m→∞G(x,xn,xm)=0, and then we say that the sequence {xn} is G-convergent to x.
Thus, ifxn→x 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,m≥N.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. This is a Hausdorff topology, so a sequence can converge at most to one point.Definition 3 (see [1]).
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,l≥N; that is, G(xn,xm,xl)→0 as n,m,l→∞.Lemma 4 (see [1]).
If(X,G) is a G-metric space, then the following are equivalent:(1)
{xn} is G-convergent to x,(2)
G(xn,xn,x)→0 as n→∞,(3)
G(xn,x,x)→0 as n→∞,(4)
G(xm,xn,x)→0 as m,n→∞.Lemma 5 (see [1]).
If(X,G) is a G-metric space, then the following are equivalent:(1)
the sequence{xn} is G-Cauchy;(2)
for everyε>0, there exists a positive integer N such that G(xn,xm,xm)<ε for all n,m>N.Lemma 6 (see [1]).
If(X,G) is a G-metric space, then G(x,y,y)≤2G(y,x,x) for all x,y∈X.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,z∈X.Definition 8 (see [1]).
Let(X,G), (X′,G′) be two G-metric spaces. Then a function f:X→X′ is G-continuous at a point x∈X 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 [1]).
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 [1]).
AG-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 [10]).
Let(X,G) be a G-metric space. A mapping F:X×X→X 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 [11–25] are some examples in this direction. Bhaskar and Lakshmikantham [11] introduced the notion of coupled fixed points and proved some coupled fixed point theorems for a mapping having mixed monotone property. The work [11] was illustrated by proving the existence and uniqueness of the solution for a periodic boundary value problem.Lakshmikantham andC´iric´ [12] extended the notion of mixed monotone property by introducing the notion of mixed g-monotone property in partially ordered metric spaces.Definition 12 (see [11]).
Let(X,≤) be a partially ordered set and F:X×X→X. 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,y∈X,
(1)x1,x2∈X,x1≤x2impliesF(x1,y)≤F(x2,y),y1,y2∈X,y1≤y2impliesF(x,y1)≥F(x,y2).Definition 13 (see [12]).
Let(X,≤) be a partially ordered set F:X×X→X, and g:X→X. 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,y∈X,
(2)x1,x2∈X,gx1≤gx2impliesF(x1,y)≤F(x2,y),y1,y2∈X,gy1≤gy2impliesF(x,y1)≥F(x,y2).Definition 14 (see [11]).
An element(x,y)∈X×X is called a coupled fixed point of the mapping F:X×X→X if F(x,y)=x and F(y,x)=y.Definition 15 (see [12]).
An element(x,y)∈X×X is called a coupled coincidence point of the mappings F:X×X→X and g:X→X if F(x,y)=gx and F(y,x)=gy.Definition 16 (see [12]).
An element(x,y)∈X×X is called a coupled common fixed point of the mappings F:X×X→X and g:X→X if x=gx=F(x,y) and y=gy=F(y,x).Definition 17 (see [12]).
The mappingsF:X×X→X and g:X→X are called commutative if
(3)gF(x,y)=F(gx,gy),
for all x,y∈X.
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 [10] established some coupled fixed point theorems for the mixed monotone mapping F:X×X→X 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 [10] under different contractive conditions in G-metric spaces. For more works, one can see [26–34]. Nashine [31] 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 [33] 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 [33].Interestingly, forφ(t)=t, ψ(t)=((1-k)/2)t, with 0≤k<1, condition (6) reduces to (4).On the other hand, Karapinar et al. [32] 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. [32] 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 ofG-metric spaces. Our result directly derive a result of Karapinar et. al. [32]. We give suitable examples to show how our results generalize and enrich the well-known results of Choudhury et al. [10], Nashine [31], and Mohiuddine and Alotaibi [33] 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 byΦ the class of all functions φ:[0,∞)→[0,∞) with the following properties:(φi)
φ is continuous and nondecreasing;(φii)
φ(t)<t for all t>0;(φiii)
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 byΨ the class of all functions ψ:[0,∞)→[0,∞) with the following properties:(ψi)
limt→rψ(t)>0 for all r>0;(ψii)
limt→0+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×X→X, g:X→X be two mappings. We say that F and g are symmetric (φ,ψ)-contractive mappings on X if there exist φ∈Φ and ψ∈Ψ 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,z∈X.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 gx≥gu≥gw and gy≤gv≤gz (or gx≤gu≤gw and gy≥gv≥gz) 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 elementsx0,y0∈X with gx0≤F(x0,y0) and gy0F(y0,x0) (or gx0≥F(x0,y0) and gy0≤F(y0,x0)), then there exist x,y∈X 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 existx0,y0∈X such that gx0≤F(x0,y0), gy0≥F(y0,x0). Since F(X×X)⊆g(X), we can choose x1,y1∈X such that gx1=F(x0,y0),gy1=F(y0,x0). Again we can choose x2,y2∈X 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),foralln≥0.
We will prove, for all n≥0, that
(10)gxn≤gxn+1,gyn≥gyn+1.
Since gx0≤F(x0,y0), gy0≥F(y0,x0), and gx1=F(x0,y0), gy1=F(y0,x0), we have gx0≤gx1,gy0≥gy1; that is, (10) holds for n=0.
Suppose that (10) holds for some n>0; that is, gxn≤gxn+1, gyn≥gyn+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 n≥0.
If for somen≥0, 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 n≥0; 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)2≤G(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 δ≥0 such that
(15)Limn→∞δn=limn→∞[G(gxn+1,gxn+1,gxn)+G(gyn+1,gyn+1,gyn)2]=δ.
We claim that δ=0.
On the contrary, suppose thatδ>0.
Taking limit asn→∞ on both sides of (12) and using the properties of φ and ψ, we have
(16)φ(δ)=limn→∞φ(δn)≤limn→∞[φ(δn-1)-ψ(δn-1)]=φ(δ)-limδn-1→δψ(δn-1)<φ(δ),acontradiction.
Thus, δ=0; that is,
(17)limn→∞δn=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))2≥ε.
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)ε≤rk=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)limk→∞rk=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 k≥0, using (22)-(23) in (18), we get
(24)rk=G(gxn(k),gxn(k),gxm(k))+G(gyn(k),gyn(k),gym(k))2≤2δn(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)-limk→∞ψ(rk)=2φ(0)+φ(0)+φ(ε)-limrk→εψ(rk)<φ(ε),acontradiction.
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)limn→∞G(gxn,gxn,x)=limn→∞G(gxn,x,x)=0,limn→∞G(gyn,gyn,y)=limn→∞G(gyn,y,y)=0.
Using the G-continuity of g, and Definition 8, we get
(30)limn→∞G(ggxn,ggxn,gx)=limn→∞G(ggxn,gx,gx)=0,(31)limn→∞G(ggyn,ggyn,gy)=limn→∞G(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 mappingF 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:(i)
if a nondecreasing sequence{xn}⊆X such that xn→x, then xn≤x for all n,(ii)
if a nonincreasing sequence{yn}⊆X such that yn→y, then y≤yn 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 gx≥gu≥gw and gy≤gv≤gz (or gx≤gu≤gw and gy≥gv≥gz) 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,y0∈X with gx0≤F(x0,y0) and gy0≥F(y0,x0)  (orgx0≥F(x0,y0) and gy0≤F(y0,x0)); then F and g have a coupled coincidence point in X; that is, there exist x,y∈X such that gx=F(x,y) and gy=F(y,x).Proof.
Proceeding exactly as in Theorem18, we have that {gxn} and {gyn} are G-Cauchy sequences in the complete G-metric space (g(X),G). Then there exist x,y∈X such that gxn→gx and gyn→gy; that is,
(34)Limn→∞G(gxn,gx,gx)=limn→∞G(gxn,gxn,gx)=0,limn→∞G(gyn,gy,gy)=limn→∞G(gyn,gyn,gy)=0.
Since {gxn} is nondecreasing and {gyn} is nonincreasing, using the regularity of (X,G,≤), we have gxn≤gx and gy≤gyn for all n≥0. 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))-limn→∞ψ(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)limn→∞G(F(x,y),gxn+1,gxn+1)=0=limn→∞G(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 Theorem18 that shows that Theorem 18 is more general than Theorem 3.1 in [31], since the contractive condition (8) is more general than (5).Example 21.
LetX=ℝ. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Let G:X×X×X→R+ be defined by
(41)G(x,y,z)=|x-y|+|y-z|+|z-x|hhhhhhhhhhhhhhhhhhhforx,y,z∈X.
Then (X,G) is a complete G-metric space.
DefineF:X×X→X by F(x,y)=(x-2y)/8, (x,y)∈X×X and g:X→X by g(x)=x/2, x∈X.
ClearlyF(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 x≥u≥w and y≤v≤z. 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)ρ2≤kρ,ρ>0,
which implies that 1/2≤k, a contradiction since k∈[0,1/2). Hence F and g do not satisfy (5).
Indeed, forx≥u≥w and y≤v≤z, 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 gx0≤F(x0,y0) and gy0≥F(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 [31] cannot be applied to F and g in this example.
Now, puttingg=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×X→X 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,z∈X with x≥u≥w and y≤v≤z (or x≤u≤w and y≥v≥z). Assume that (X,G) is complete and F has the mixed monotone property. Also suppose that either(a)
F is continuous, or(b)
(X,≤,G) is regular.If there exist two elements x0,y0∈X with x0≤F(x0,y0) and y0≥F(y0,x0)  (or x0≥F(x0,y0) and y0≤F(y0,x0)), then there exist x,y∈X 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.
LetX=ℝ. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Let G:X×X×X→R+ be defined by
(46)G(x,y,z)=|x-y|+|y-z|+|z-x|hhhhhhhhhhhhhhhhhhhforx,y,z∈X.
Then (X,G) is a complete G-metric space.
DefineF:X×X→X by F(x,y)=(x-5y)/10, (x,y)∈X×X.
ThenF 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|)}hhhhhhhhhhhx≥u≥w,y≤v≤z,
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 1≤k, 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 [33] 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 x≥u≥w and y≤v≤z. 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 thatF does not satisfy (6).
Finally, we prove that (45) holds. Indeed, for x≥u≥w and y≤v≤z, 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 x0≤F(x0,y0) and y0≥F(y0,x0).
By Corollary22, we obtain that F has a coupled fixed point (0,0), but Theorems 3.1 and 3.2 in [10] and Theorem 3.1 in [33] 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×X→X be a mapping having mixed monotone property on X. Suppose there exists ψ∈Ψ 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,z∈X with x≥u≥w and y≤v≤z (or x≤u≤w and y≥v≥z). Assume that (X,G) is complete, and also suppose that either(i)
F is continuous, or(ii)
(X,≤,G) is regular.If there exist two elements x0,y0∈X with x0≤F(x0,y0) and y0≥F(y0,x0) (or x0≥F(x0,y0) and y0≤F(y0,x0)), then there exist x,y∈X such that x=F(x,y) and y=F(y,x); that is, F has a coupled fixed point in X.Proof.
Note that ifψ∈Ψ, then for all r>0, rψ∈Ψ. 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 [32]).
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×X→X and g:X→X 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,z∈X with gx≥gu≥gw and gy≤gv≤gz (or gx≤gu≤gw and gy≥gv≥gz). Also suppose that either(i)
F and g are continuous, (X,G) is complete, and g commutes with F, or(ii)
(g(X),G) is complete and (X,≤,G) is regular.If there exist two elements x0,y0∈X with gx0≤F(x0,y0) and gy0≥F(y0,x0) (or gx0≥F(x0,y0) and gy0≤F(y0,x0)), then there exist x,y∈X such that gx=F(x,y) and gy=F(y,x).Proof.
Defineφ,ψ:[0,∞)→[0,∞), φ(t)=t/2 and ψ(t)=(1-k)(t/2), 0≤k<1. Then (53) holds. Hence the result follows from Theorem 18 or Theorem 20.Remark 26.
The choice of functionsF and g in Example 21 shows that Theorem 25 is more general than Theorem 3.1 in [31], 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, puttingg=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×X→X 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,z∈X with x≥u≥w and y≤v≤z (or x≤u≤w and y≥v≥z). Suppose that either(i)
F is continuous, or(ii)
(X,≤,G) is regular.If there exist two elements x0,y0∈X with x0≤F(x0,y0) and y0≥F(y0,x0) (or x0≥F(x0,y0) and y0≤F(y0,x0)), then there exist x,y∈X 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 functionF in Example 23 shows that Corollary 27 is more general than Theorem 3.1 and Theorem 3.2 in [10], 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 Theorem18, 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 Theorem18, 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,v1∈X, 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, setx0=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*),hhhhhhhhhhhhhhhhhhhhhhhforalln≥0.
Since (F(u,v),F(v,u))=(gu1,gv1) and (F(x,y),F(y,x))=(gx1,gy1)=(gx,gy) are comparable, then gu1≥gx and gv1≤gy. It is easy to show that (gun,gvn) and (gx,gy) are comparable; that is, gun≥gx and gvn≤gy for all n≥1. 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)2≤G(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 γ≥0 such that limn→∞γn=γ.
We shall show thatγ=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)-limγn→γψ(γn)<φ(γ),acontradiction.
Thus, γ=0; that is, limn→∞γn=0.
Hence, it follows thatgun→gx, gvn→gy.
Similarly, we can show thatgun→gx*, gvn→gy*.
By uniqueness of limit, it follows thatgx=gx* and gy=gy*. Thus, we proved (55).
Sincegx=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 Theorem20, 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 Theorem29 result follows immediately.


## 3. Applications to Integral Equations

Motivated by the works of Aydi et al. [24] and Luong and Thuan [35], 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.LetΘ denote the class of functions θ:[0,∞)→[0,∞) which satisfies the following conditions:(i)
θ is increasing;(ii)
there existsψ∈Ψ such that θ(r)=(r/2)-ψ(r/2) for all r∈[0,∞).For example, θ1(x)=αx (where 0≤α≤1/2), θ2(x)=x2/2(x+1) are some members of Θ.We shall analyze the system (65) under the following assumptions:(i)
f,k:[0,T]×ℝ→ℝ are continuous;(ii)
p:[0,T]→ℝ is continuous;(iii)
S:[0,T]×ℝ→[0,∞) is continuous;(iv)
there existλ>0 and θ∈Θ such that for all x,y∈ℝ, y≥x,
(66)0≤f(s,y)-f(s,x)≤λθ(y-x),0≤k(s,x)-k(s,y)≤λθ(y-x);(v)
we suppose that(67)3λsupt∈[0,T]∫0TS(t,s)ds≤12.(vi)
there exist continuous functionsα,β:[0,T]→ℝ 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],ℝ) 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,w∈X.
Endow X with the partial order ≤ given by x,y∈X, x≤y⟺x(t)≤y(t) for all t∈[0,T]. Also, we may adjust as in [18] to prove that (X,G,≤) is regular.Theorem 31.
Under assumptions (i)–(vi), the system (65) has a solution in X2=(C([0,T],ℝ))2.Proof.
Consider the operatorF:X×X→X defined by
(70)F(x,y)(t)=p(t)+∫0TS(t,s)[f(s,x(s))+k(s,y(s))]dsfort∈[0,T],x,y∈X.
First, we shall prove that F has the mixed monotone property.
In fact, forx1≤x2 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 y1≤y2 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 quantityG(F(x,y),F(u,v),F(w,z)) for all x,y,u,v,w,z∈X with x≥u≥w and y≤v≤z. 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))]ds≤λ∫0TS(t,s)[θ(x(s)-u(s))+θ(v(s)-y(s))]ds.
Since the function θ is increasing x≥u≥w, and y≤v≤z, we have
(78)θ(x(s)-u(s))≤θ(supt∈I|x(t)-u(t)|),θ(v(s)-y(s))≤θ(supt∈I|v(t)-y(t)|).
Hence by (77), we obtain
(79)|F(x,y)-F(u,v)|≤λ∫0TS(t,s)[θ(supt∈I|x(t)-u(t)|)+θ(supt∈I|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)|≤λ∫0TS(t,s)[θ(supt∈I|x(t)-w(t)|)+θ(supt∈I|z(t)-y(t)|)]ds,(81)|F(u,v)-F(w,z)|≤λ∫0TS(t,s)[θ(supt∈I|u(t)-w(t)|)+θ(supt∈I|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(supt∈I|x(t)-u(t)|)+θ(supt∈I|x(t)-w(t)|)+θ(supt∈I|u(t)-w(t)|)]+λsupt∈[0,T]∫0TS(t,s)ds·[θ(supt∈I|v(t)-y(t)|)+θ(supt∈I|z(t)-y(t)|)+θ(supt∈I|z(t)-v(t)|)].
Further, since θ is increasing, so that we have
(83)θ(supt∈I|x(t)-u(t)|)≤θ(G(x,u,w)),θ(supt∈I|x(t)-w(t)|)≤θ(G(x,u,w)),θ(supt∈I|u(t)-w(t)|)≤θ(G(x,u,w)).
Similarly, we have
(84)θ(supt∈I|v(t)-y(t)|)≤θ(G(y,v,z)),θ(supt∈I|z(t)-y(t)|)≤θ(G(y,v,z)),θ(supt∈I|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)))2-1≤3λsupt∈[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)))2≤θ(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)))2-1≤G(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.
Letα, β be the functions appearing in assumption (vi); we get
(92)α≤F(α,β),β≥F(β,α).
Applying Corollary 24, we deduce the existence of x,y∈X 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 orderedG-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.



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*Source: 134712-2013-12-08.xml*