# Some Fixed Point Theorems in OrderedG-Metric Spaces and Applications

Authors: W. Shatanawi
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study a number of fixed point results for the two weakly increasing mappingsf and g with respect to partial ordering relation ⪯ in generalized metric spaces. An application to integral equation is given.


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## Body

## 1. Introduction

The existence of fixed points in partially ordered sets has been at the center of active research. In fact, the existence of fixed point in partially ordered sets has been investigated in [1]. Moreover, Ran and Reurings [1] applied their results to matrix equations. Some generalizations of the results of [1] are given in [2–6]. In [6], O’Regan and Petruşel gave some existence results for the Fredholm and Volterra type.The notion ofG-metric space was introduced by Mustafa and Sims [7] as a generalization of the notion of metric spaces. Mustafa et al. studied many fixed point results in G-metric space [8–10] (also see [11–15]). In fact the study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity. The following definition is introduced by Mustafa and Sims [7].Definition 1.1 (see [7]).
LetX be a 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.Then the function G is called a generalized metric, or, more specifically, a G-metric on X, and the pair (X,G) is called a G-metric space.Definition 1.2 (see [7]).
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 we say that the sequence {xn} is G-convergent to x or {xn}G-converges to x.Thus,xn→x in a G-metric space (X,G) if for any ɛ>0, there exists k∈N such that G(x,xn,xm)<ɛ for all m,n≥k.Proposition 1.3 (see [7]).
Let(X,G) be 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(xn,xm,x)→0 as n,m→+∞.Definition 1.4 (see [7]).
Let(X,G) be a G-metric space, a sequence {xn} is called G-Cauchy if for every ɛ>0, there is k∈N such that G(xn,xm,xl)<ɛ, for all n,m,l≥k; that is G(xn,xm,xl)→0 as n,m,l→+∞.Proposition 1.5 (see [7]).
Let(X,G) be a G-metric space. Then the following are equivalent: (1)
the sequence{xn} is G-Cauchy;(2)
for everyϵ>0, there is k∈N such that G(xn,xm,xm)<ϵ, for all n,m≥k.Definition 1.6 (see [7]).
Let(X,G) and (X′,G′) be G-metric spaces, and let f:(X,G)→(X′,G′) be a function. Then f is said to be G-continuous at a point a∈X if and only if for every ɛ>0, there is δ>0 such that x,y∈X and G(a,x,y)<δ implies G′(f(a),f(x),f(y))<ɛ. A function f is G-continuous at X if and only if it is G-continuous at all a∈X.Proposition 1.7 (see [7]).
Let(X,G) be a G-metric space. Then the function G(x,y,z) is jointly continuous in all three of its variables.EveryG-metric on X will define a metric dG on X by(1.1)dG(x,y)=G(x,y,y)+G(y,x,x),∀x,y∈X.
For a symmetric G-metric space,(1.2)dG(x,y)=2G(x,y,y),∀x,y∈X.
However, if G is not symmetric, then the following inequality holds:(1.3)32G(x,y,y)≤dG(x,y)≤3G(x,y,y),∀x,y∈X.The following are examples ofG-metric spaces.Example 1.8 (see [7]).
Let(R,d) be the usual metric space. Define Gs by
(1.4)Gs(x,y,z)=d(x,y)+d(y,z)+d(x,z),
for all x,y,z∈R. Then it is clear that (R,Gs) is a G-metric space.Example 1.9 (see [7]).
LetX={a,b}. Define G on X×X×X by
(1.5)G(a,a,a)=G(b,b,b)=0,G(a,a,b)=1,G(a,b,b)=2
and extend G to X×X×X by using the symmetry in the variables. Then it is clear that (X,G) is a G-metric space.Definition 1.10 (see [7]).
AG-metric space (X,G) is called G-complete if every G-Cauchy sequence in (X,G) is G-convergent in (X,G).The notion of weakly increasing mappings was introduced in by Altun and Simsek [16].Definition 1.11 (see [16]).
Let(X,⪯) be a partially ordered set. Two mappings F,G:X→X are said to be weakly increasing if Fx⪯GFx and Gx⪯FGx, for all x∈X.Two weakly increasing mappings need not be nondecreasing.Example 1.12 (see [16]).
LetX=R, endowed with the usual ordering. Let F,G:X→X defined by
(1.6)Fx={x,0≤x≤1,0,1<x<+∞,gx={x,0≤x≤1,0,1<x<+∞.
Then F and G are weakly increasing mappings. Note that F and G are not nondecreasing.The aim of this paper is to study a number of fixed point results for two weakly increasing mappingsf and g with respect to partial ordering relation (⪯) in a generalized metric space.


## 2. Main Results

Theorem 2.1.
Let(X,⪯) be a partially ordered set and suppose that there exists G-metric in X such that (X,G) is G-complete. Let f,g:X→X be two weakly increasing mappings with respect to ⪯. Suppose there exist nonnegative real numbers a, b, and c with a+2b+2c<1 such that
(2.1)G(fx,gy,gy)≤aG(x,y,y)+b[G(x,fx,fx)+G(y,gy,gy)]+c[G(x,gy,gy)+G(y,fx,fx)],(2.2)G(gx,fy,fy)≤aG(x,y,y)+b[G(x,gx,gx)+G(y,fy,fy)]+c[G(x,fy,fy)+G(y,gx,gx)],
for all comparative x,y∈X. If f or g is continuous, then f and g have a common fixed point u∈X.Proof.
By inequality (2.2), we have
(2.3)G(gy,fx,fx)≤aG(y,x,x)+b[G(y,gy,gy)+G(x,fx,fx)]+c[G(y,fx,fx)+G(x,gy,gy)].
If X is a symmetric G-metric space, then by adding inequalities (2.1) and (2.3), we obtain
(2.4)G(fx,gy,gy)+G(gy,fx,fx)≤a[G(x,y,y)+G(y,x,x)]+2b[G(x,fx,fx)+G(y,gy,gy)]+2c[G(x,gy,gy)+G(y,fx,fx)],
which further implies that
(2.5)dG(fx,fy)≤adG(x,y)+b[dG(x,fx)+dG(y,gy)]+c[dG(x,gy)+dG(y,fx)],
for all x,y∈X with 0≤a+2b+2c<1 and the fixed point of f and g follows from [2].
Now ifX is not a symmetric G-metric space. Then by the definition of metric (X,dG) and inequalities (2.1) and (2.3), we obtain(2.6)dG(fx,gy)=G(fx,gy,gy)+G(gy,fx,fx)≤a[G(x,y,y)+G(x,x,y)]+2b[G(x,fx,fx)+G(y,gy,gy)]+2c[G(x,gy,gy)+G(y,fx,fx)]≤adG(x,y)+2b[23dG(x,fx)+23dG(y,gy)]+2c[23dG(x,gy)+23dG(y,fx)]=adG(x,y)+43b[dG(x,fx)+dG(y,gy)]+43c[dG(x,gy)+dG(y,fx)],
for all x∈X. Here, the contractivity factor a+(8/3)b+(8/3)c may not be less than 1.
Therefore metric gives no information. In this case, for givenx0∈X, choose x1∈X such that x1=fx0. Again choose x2∈X such that gx1=x2. Also, we choose x3∈X such that x3=fx2. Continuing as above process, we can construct a sequence {xn} in X such that x2n+1=fx2n, n∈N∪{0} and x2n+2=gx2n+1, n∈N∪{0}. Since f and g are weakly increasing with respect to ⪯, we have (2.7)x1=fx0⪯g(fx0)=gx1=x2⪯f(gx1)=fx2=x3⪯g(fx2)=gx3=x4⪯⋯.
Thus from (2.1), we have
(2.8)G(x2n+1,x2n+2,x2n+2)=G(fx2n,gx2n+1,gx2n+1)≤aG(x2n,x2n+1,x2n+1)+b[G(x2n,fx2n,fx2n)+G(x2n+1,gx2n+1,gx2n+1)]+c[G(x2n,gx2n+1,gx2n+1)+G(x2n+1,fx2n,fx2n)]=aG(x2n,x2n+1,x2n+1)+b[G(x2n,x2n+1,x2n+1)+G(x2n+1,x2n+2,x2n+2)]+c[G(x2n,x2n+2,gx2n+2)+G(x2n+1,x2n+1,x2n+1)]=(a+b)G(x2n,x2n+1,x2n+1)+bG(x2n+1,x2n+2,x2n+2)+cG(x2n,x2n+2,x2n+2).
By (G5), we have
(2.9)G(x2n+1,x2n+2,x2n+2)≤a+b+c1-b-cG(x2n,x2n+1,x2n+1).
Also, we have
(2.10)G(x2n,x2n+1,x2n+1)=G(gx2n-1,fx2n,fx2n)≤aG(x2n-1,x2n,x2n)+b[G(x2n-1,gx2n-1,gx2n-1)+G(x2n,fx2n,fx2n)]+c[G(x2n-1,fx2n,fx2n)+G(x2n,gx2n-1,gx2n-1)]=aG(x2n-1,x2n,x2n)+b[G(x2n-1,x2n,x2n)+G(x2n,x2n+1,x2n+1)]+c[G(x2n-1,x2n+1,gx2n+1)+G(x2n,x2n,x2n)]=(a+b)G(x2n-1,x2n,x2n)+bG(x2n,x2n+1,x2n+1)+cG(x2n-1,x2n+1,x2n+1).
By (G5), we get
(2.11)G(x2n,x2n+1,x2n+1)≤a+b+c1-b-cG(x2n-1,x2n,x2n).
Let
(2.12)k=a+b+c1-b-c.
Then by (2.9) and (2.11), we have
(2.13)G(xn,xn+1,xn+1)≤kG(xn-1,xn,xn),∀n∈N.
Thus, if x0=x1, we get G(xn,xn+1,xn+1)=0 for each n∈N. Hence xn=x0 for each n∈N. Therefore {xn} is G-Cauchy. So we may assume that x0≠x1. Let n,m∈N with m>n. By axiom (G5) of the definition of G-metric space, we have
(2.14)G(xn,xm,xm)≤G(xn,xn+1,xn+1)+G(xn+1,xn+2,xn+2)+⋯+G(xm-1,xm,xm).
By (2.13), we get
(2.15)G(xn,xm,xm)≤knG(x0,x1,x1)+kn+1G(x0,x1,x1)+⋯+km-1G(x0,x1,x1)≤kn1-kG(x0,x1,x1).
On taking limit m,n→∞, we have
(2.16)lim⁡m,n→∞G(xn,xm,xm)=0.
So we conclude that (xn) is a Cauchy sequence in X. Since X is G-complete, then it yields that (xn) and hence any subsequence of (xn) converges to some u∈X. So that, the subsequences (x2n+1)=(fx2n) and (x2n+2)=(gx2n+1) converge to u. First suppose that f is G-continuous. Since (x2n) converges to u, we get (fx2n) converges fu. By the uniqueness of limit we get fu=u. Claim: gu=u.
Sinceu⪯u, by inequality (2.1), we have (2.17)G(u,gu,gu)=G(fu,gu,gu)≤aG(u,u,u)+b[G(u,fu,fu)+G(u,gu,gu)]+c[G(u,gu,gu)+G(u,fu,fu)]≤(b+c)G(u,gu,gu).
Since b+c<1, we get G(u,gu,gu)=0. Hence gu=u. If g is G-continuous, by similar argument as above we show that g and f have a common fixed point.Theorem 2.2.
Let(X,⪯) be a partially ordered set and suppose that there exists G-metric in X such that (X,G) is G-complete. Let f,g:X→X be two weakly increasing mappings with respect to ⪯. Suppose there exist nonnegative real numbers a, b, and c with a+2b+2c<1 such that
(2.18)G(fx,gy,gy)≤aG(x,y,y)+b[G(x,fx,fx)+G(y,gy,gy)]+c[G(x,gy,gy)+G(y,fx,fx)],G(gx,fy,fy)≤aG(x,y,y)+b[G(x,gx,gx)+G(y,fy,fy)]+c[G(x,fy,fy)+G(y,gx,gx)],
for all comparative x,y∈X. Assume that X has the following property:(P)
If(xn) is an increasing sequence converges to x in X, then xn⪯x for all n∈N.Then f and g have a common fixed point u∈X.Proof.
As in the proof of Theorem3.1, we construct an increasing sequence (xn) in X such that x2n+1=fx2n and x2n+2=gx2n+1. Also, we can show (xn) is G-Cauchy. Since X is G-complete, there is u∈X such that (xn) is converges to u∈X. Thus (x2n),(x2n+1),(fx2n) and (gx2n+1) converge to u. Since X satisfies property (P), we get that xn⪯u, for all n∈N. Thus x2n and u are comparative. Hence by inequality (2.1), we have
(2.19)G(fx2n,gu,gu)≤aG(x2n,u,u)+b[G(x2n,fx2n,fx2n)+G(u,gu,gu)]+c[G(x2n,gu,gu)+G(u,fx2n,fx2n)].
On letting n→+∞, we get
(2.20)G(u,gu,gu)≤(b+c)G(u,gu,gu).
Since b+c<1, we get G(u,gu,gu)=0. Hence gu=u. By similar argument, we may show that u=fu.Corollary 2.3.
Let(X,⪯) be a partially ordered set, and suppose that (X,G) is a G-complete metric space. Let f:X→X be a continuous mapping such that fx⪯f(fx), for all x∈X. Suppose there exist nonnegative real numbers a, b and c with a+2b+2c<1 such that
(2.21)G(fx,fy,fy)≤aG(x,y,y)+b[G(x,fx,fx)+G(y,fy,fy)]+c[G(x,fy,fy)+G(y,fx,fx)],
for all comparative x,y∈X. Then f has a fixed point u∈X.Proof.
It follows from Theorem2.1 by taking g=f.Corollary 2.4.
Let(X,⪯) be a partially ordered set and suppose that there exists G-metric in X such that (X,G) is G-complete. Let f:X→X be a mapping such that fx⪯f(fx) for all x∈X. Suppose there exist nonnegative real numbers a, b and c with a+2b+2c<1 such that
(2.22)G(fx,fy,fy)≤aG(x,y,y)+b[G(x,fx,fx)+G(y,fy,fy)]+c[G(x,fy,fy)+G(y,fx,fx)]
for all comparative x,y∈X. Assume that X has the following property:(P)
If(xn) is an increasing sequence converges to x in X, then xn⪯x for all n∈N.Then f has fixed point u∈X.Proof.
It follows from Theorem2.2 by taking g=f.


## 3. Application

Consider the integral equation:(3.1)u(t)=∫0TK(t,s,u(s))ds+g(t),t∈[0,T],
where T>0. The aim of this section is to give an existence theorem for a solution of the above integral equation using Corollary 2.4 This section is related to those [16–19].LetX=C([0,T]) be the set of all continuous functions defined on [0,T]. Define (3.2)G:X×X×X→R+
by (3.3)G(x,y,z)=sup⁡t∈[0,T]|x(t)-y(t)|+sup⁡t∈[0,T]|x(t)-z(t)|+sup⁡t∈[0,T]|y(t)-z(t)|.
Then (X,G) is a G-complete metric space. Define an ordered relation ⪯ on X by (3.4)x⪯yiffx(t)≤y(t),∀t∈[0,T].
Then (X,⪯) is a partially ordered set. The purpose of this section is to give an existence theorem for solution of integral equation on (3.1). This section is inspired in [17–19].Theorem 3.1.
Suppose the following hypotheses hold:(1)
K:[0,T]×[0,T]×R→R and g:R→R are continuous,(2)
for eacht,s∈[0,T], one has
(3.5)K(t,s,u(t))≤K(t,s,∫0TK(s,τ,u(τ))dτ+g(s)),(3)
there exists a continuous functionG:[0,T]×[0,T]→[0,+∞] such that
(3.6)|K(t,s,u)-K(t,s,v)|≤G(t,s)|u-v|,
for each comparable u,v∈R and each t,s∈[0,T],(4)
supt∈[0,T]∫0TG(t,s)ds≤r for some r<1.Then the integral equation (3.1) has a solution u∈C([0,T]).Proof.
DefineS:C([0,T])→C([0,T]) by
(3.7)Sx(t)=∫0TK(t,s,x(s))ds+g(t),t∈[0,T].
Now, we have
(3.8)Sx(t)=∫0TK(t,s,x(s))ds+g(t)≤∫0TK(t,s,∫0TK(s,τ,x(τ))dτ+g(s))ds+g(t)=∫0TK(t,s,Sx(s))ds+g(t)=S(Sx(t)).
Thus, we have Sx⪯S(Sx), for all x∈C([0,T]).
Forx,y∈C([0,T]) with x⪯y, we have (3.9)G(Sx,Sy,Sy)=2sup⁡t∈[0,T]|Sx(t)-Sy(t)|=2sup⁡t∈[0,T]|∫0TK(t,s,x(s))-K(t,s,y(s))ds|≤2sup⁡t∈[0,T]∫0T|K(t,s,x(s))-K(t,s,y(s))|ds≤2sup⁡t∈[0,T]∫0TG(t,s)|x(s)-y(s)|ds≤2sup⁡t∈[0,T]|x(t)-y(t)|sup⁡t∈[0,T]∫0TG(t,s)ds=G(x,y,y)sup⁡t∈[0,T]∫0TG(t,s)ds.
By using hypotheses (4), there is r∈[0,1) such that
(3.10)sup⁡t∈[0,T]∫0TG(t,s)ds<r.
Thus, we have G(Sx,Sy,Sy)≤rG(x,y,y). Thus all the required hypotheses of Corollary 2.4 are satisfied. Thus there exist a solution u∈C([0,T]) of the integral equation (3.1).



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