# PropertyP and Some Fixed Point Results on (ψ,ϕ)-Weakly Contractive G-Metric Spaces

Authors: Mona Khandaqji; Sharifa Al-Sharif; Mohammad Al-Khaleel
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We prove some fixed point results for(ψ,ϕ)-weakly contractive maps in G-metric spaces, we show that these maps satisfy property P. The results presented in this paper generalize several well-known comparable results in the literature.


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## Body

## 1. Introduction

Metric fixed point theory is an important mathematical discipline because of its applications in areas such as variational and linear inequalities, optimization, and approximation theory. Generalizations of metric spaces were proposed by Gähler [1, 2] (called 2 metricspaces). In 2005, Mustafa and Sims [3] introduced a new structure of generalized metric spaces, which are called G-metric spaces as a generalization of metric space (X,d), to develop and introduce a new fixed point theory for various mappings in this new structure. Many papers dealing with fixed point theorems for mappings satisfying different contractive conditions on G-metric spaces can be found in [4–16]. Let T be a self-map of a complete metric space (X,d) with a nonempty fixed point set F(T). Then T is said to satisfy property P if F(T)=F(Tn) for each n∈N. However, the converse is false. For example, consider X=[0,1] and T defined by Tx=1-x. Then T has a unique fixed point at x=1/2, but every even iterate of T is the identity map, which has every point of [0,1] as a fixed point. On the other hand, if X=[0,π], Tx=cosx, then every iterate of T has the same fixed point as T (see [17, 18]). Jeong and Rhoades [17] showed that maps satisfying many contractive conditions have property P. An interesting fact about maps satisfying property P is that they have no nontrivial periodic points. Papers dealing with property P are those in [17–19]. In this paper, we will prove some general fixed point theorems for (ψ,ϕ)-weakly contractive maps in G-metric spaces, and then we show that these maps satisfy property P.Now we give first in what follows preliminaries and basic definitions which will be used throughout the paper.


## 2. Preliminaries

Consistent with Mustafa and Sims [3], the following definitions and results will be needed in the sequel.Definition 2.1.
LetX be a nonempty set and G:X×X×X→R+ satisfy the following properties:(G1)
G(x,y,z)=0 if x=y=z=0 (coincidence),(G2)
0<G(x,x,y), for all x,y∈X, where 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(p{x,y,z}), where p is a permutation of x,y,z (symmetry),(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z), for all x,y,z,a∈X (rectangle inequality).Then, the functionG is called the G-metric on X, and the pair (X,G) is called the G-metric space.Definition 2.2.
AG-metric is said to be symmetric if G(x,y,y)=G(y,x,x) for all x,y∈X.Proposition 2.3.
Every G -metric space(X,G) will define a metric space (X,dG) by dG(x,y)=G(x,y,y)+G(y,x,x), for all x,y∈X.Definition 2.4.
Let(X,G) be a G-metric space, and (xn) be a sequence of points in X. Then,(i)
a pointx∈X is said to be the limit of the sequence (xn) if
(2.1)G(xn,xm,x)⟶0,(asn,m⟶∞),
and we say that the sequence (xn) is G convergent to x (we say xn→(G)x),(ii)
A sequence(xn) is said to be G-Cauchy if
(2.2)G(xn,xm,xl)⟶0,(asn,m,l⟶∞),(iii)
(X,G) is called a complete G-metric space if every G-Cauchy sequence in X is G converge in X.Proposition 2.5.
Let(X,G) be a G-metric space, then the following are equivalent:(1)
xn→(G)x,(2)
G(xn,xn,x)→0(asn→∞),(3)
G(xn,x,x)→0(asn→∞),(4)
G(xn,xm,x)→0(asn→∞).Proposition 2.6.
Let(X,G) be a G-metric space, then the following are equivalent:(1)
(xn) is be G-Cauchy in X,(2)
G(xn,xm,xm)→0(asn,m→∞).Proposition 2.7.
Let(X,G) be a G-metric space. Then, for any x,y,z,a∈X, it follows that:(i)
IfG(x,y,z)=0, then x=y=z=0,(ii)
G(x,y,z)≤G(x,x,y)+G(x,x,z),(iii)
G(x,x,y)≤2G(y,x,x),(iv)
G(x,y,z)≤G(x,a,z)+G(a,y,z),(v)
G(x,y,z)≤G(x,a,a)+G(y,a,a)+G(z,a,a).


## 3. Main Results

Throughout the paper,N denotes the set of all natural numbers.Definition 3.1 (see [20]).
A functionψ:[0,∞)→[0,∞) is called altering distance if the following properties are satisfied:(1)
ψ is continuous and increasing,(2)
ψ(t)=0 if and only if t=0.The altering distance functions alter the metric distance between two points and enable us to deal with relatively new classes of fixed points problems.Theorem 3.2.
Let(X,G) be a complete G-metric space. Let f be a self-map on X satisfying the following:
(3.1)ψ(G(fx,fy,fz))≤ψ(max{G(x,y,z),G(x,fx,fx),G(y,fy,fy),G(z,fz,fz),αG(fx,fx,y)+(1-α)(G(fy,fy,z)),βG(x,fx,fx)+(1-β)(G(y,fy,fy)),})-ϕ(max{G(x,y,z),G(x,fx,fx),G(y,fy,fy),G(z,fz,fz),αG(fx,fx,y)+(1-α)(G(fy,fy,z)),βG(x,fx,fx)+(1-β)(G(y,fy,fy)),}),
for all x,y,z∈X, where 0<α,β<1, ψ is an altering distance function, and ϕ:[0,∞)→[0,∞) is a continuous function with ϕ(t)=0 if and only if t=0. Then, f has a unique fixed point (say u), where f is G continuous at u.Proof.
Fixx0∈X. Then construct a sequence {xn} by xn+1=fxn=fnx0. We may assume that xn≠xn+1 for each n∈N∪{0}. Since, if there exist n∈N such that xn=xn+1, then xn is a fixed point of f.
From (3.1), substituting x≡xn-1, y=z≡xn then, for all n∈N,
(3.2)ψ(G(xn,xn+1,xn+1))≤ψ(max{G(xn-1,xn,xn),G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn+1,xn+1),αG(xn,xn,xn)+(1-α)(G(xn+1,xn+1,xn)),βG(xn-1,xn,xn)+(1-β)(G(xn+1,xn+1,xn))})-ϕ(max{G(xn-1,xn,xn),G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn+1,xn+1),αG(xn,xn,xn)+(1-α)(G(xn+1,xn+1,xn)),βG(xn-1,xn,xn)+(1-β)(G(xn+1,xn+1,xn))})≤ψ(max{G(xn-1,xn,xn),G(xn,xn+1,xn+1),βG(xn-1,xn,xn)+(1-β)(G(xn+1,xn+1,xn))})-ϕ(max{G(xn-1,xn,xn),G(xn,xn+1,xn+1),βG(xn-1,xn,xn)+(1-β)(G(xn+1,xn+1,xn))}).
Let Mn=max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}. Then, (3.2) gives
(3.3)ψ(G(xn,xn+1,xn+1))≤ψ(Mn)-ϕ(Mn).
We have two cases, either Mn=G(xn,xn+1,xn+1) or Mn=G(xn-1,xn,xn). Suppose that, for some n∈N0,Mn=G(xn,xn+1,xn+1). Then, we have
(3.4)ψ(G(xn,xn+1,xn+1))≤ψ(G(xn,xn+1,xn+1))-ϕ(G(xn,xn+1,xn+1)).
Therefore, ψ(G(xn,xn+1,xn+1))=0. Hence xn=xn+1. This is a contradiction since the xn's are distinct.
Thus,Mn=G(xn,xn+1,xn+1), and (3.2) becomes
(3.5)ψ(G(xn,xn+1,xn+1))≤ψ(G(xn-1,xn,xn))-ϕ(G(xn-1,xn,xn))≤ψ(G(xn-1,xn,xn)).
But ψ is an increasing function. Thus, from (3.5), we get
(3.6)G(xn,xn+1,xn+1)≤G(xn-1,xn,xn),∀n∈N.
Therefore, {G(xn,xn+1,xn+1),n∈N∪{0}} is a positive nonincreasing sequence. Hence there exists r≥0 such that
(3.7)lim⁡n→∞G(xn,xn+1,xn+1)=r.
Letting n→∞, and using (3.5) and the continuity of ψ and ϕ, we get
(3.8)ψ(r)≤ψ(r)-ϕ(r).
Hence, ϕ(r)=0, therefore r=0, which implies that
(3.9)lim⁡n→∞G(xn,xn+1,xn+1)=0.
Consequently, for a given ε>0, there is an integer n0 such that
(3.10)G(xn,xn+1,xn+1)<ε2,∀n>n0.
For m,n∈N with m>n, we claim that
(3.11)G(xn,xm,xm)<ε2,∀m>n>n0.
To show (3.11), we use induction on m. Inequality (3.11) holds for m=n+1 from (3.10). Assume (3.11) holds for m=k, that is,
(3.12)G(xn,xk,xk)<ε2,∀n>n0.
For all n>n0, take m=k+1. Using (G5) in Definition 2.1 and inequalities (3.10), (3.12), we get
(3.13)G(xn,xk+1,xk+1)≤G(xn,xn+1,xn+1)+G(xn+1,xk+1,xk+1)≤G(xn,xn+1,xn+1)+G(xn,xk,xk)<ε.
By induction on m, we conclude that
(3.14)G(xn,xm,xm)<ε2,∀m>n>n0.
We conclude from Proposition 2.6 that {xn} is a G-Cauchy sequence in X. From the completeness of X, there exists u in X such that xn→(G)u. For n∈N, we have
(3.15)ψ(G(fu,fu,xn))=ψ(G(fu,fu,fxn-1))≤ψ(max{G(u,u,xn-1),G(u,fu,fu),G(u,fu,fu),G(xn-1,xn,xn),αG(fu,fu,u)+(1-α)(G(fu,fu,xn-1)),βG(u,fu,fu)+(1-β)(G(u,fu,fu))})-ϕ(max{G(u,u,xn-1),G(u,fu,fu),G(u,fu,fu),G(xn-1,xn,xn),αG(fu,fu,u)+(1-α)(G(fu,fu,xn-1)),βG(u,fu,fu)+(1-β)(G(u,fu,fu))})≤ψ(max{G(u,u,xn-1),G(u,fu,fu),G(xn-1,xn,xn),αG(fu,fu,u)+(1-α)(G(fu,fu,xn-1))})-ϕ(max{G(u,u,xn-1),G(u,fu,fu),G(xn-1,xn,xn),αG(fu,fu,u)+(1-α)(G(fu,fu,xn-1))}).
Letting n→∞, and using the fact that ψ is continuous and G is continuous on its variables, we get that G(u,fu,fu)=0. Hence fu=u. So u is a fixed point of f. Now, to show uniqueness, let v be another fixed point of f with v≠u. Therefore,
(3.16)ψ(G(u,u,v))=ψ(G(fu,fu,fv))≤ψ(max{G(u,u,v),G(u,fu,fu),G(u,fu,fu),G(v,fv,fv),αG(fu,fu,u)+(1-α)(G(fu,fu,v)),βG(u,fu,fu)+(1-β)(G(v,fv,fv))})-ϕ(max{G(u,u,v),G(u,fu,fu),G(u,fu,fu),G(v,fv,fv),αG(fu,fu,u)+(1-α)(G(fu,fu,v)),βG(u,fu,fu)+(1-β)(G(v,fv,fv))})=ψ(max{G(u,u,v),(1-α)G(u,u,v)})-ϕ(max{G(u,u,v),(1-α)G(u,u,v)})=ψ(G(u,u,v))-ϕ(G(u,u,v)).
Hence,
(3.17)ψ(G(u,u,v))≤ψ(G(u,u,v))-ϕ(G(u,u,v)).
This implies that ϕ(G(u,u,v))=0, then G(u,u,v)=0 and u=v.
Now to show thatf is G continuous at u, let {xn} be a sequence in X with limit u(i.e.,xn→(G)u). Using (3.1), we have
(3.18)ψ(G(fxn,u,u))=ψ(G(fxn,fu,fu))≤ψ(max{G(xn,u,u),G(xn,fxn,fxn),G(u,fu,fu),G(u,fu,fu),αG(fxn,fxn,u)+(1-α)(G(fu,fu,u)),βG(xn,fxn,fxn)+(1-β)(G(u,fu,fu))})-ϕ(max{G(xn,u,u),G(xn,fxn,fxn),G(u,fu,fu),G(u,fu,fu),αG(fxn,fxn,u)+(1-α)(G(fu,fu,u)),βG(xn,fxn,fxn)+(1-β)(G(u,fu,fu))})=ψ(max{G(xn,u,u),αG(fxn,fxn,u),βG(xn,fxn,fxn)})-ϕ(max{G(xn,u,u),αG(fxn,fxn,u),βG(xn,fxn,fxn)})≤ψ(max{G(xn,u,u),αG(xn+1,xn+1,u),βG(xn,xn+1,xn+1)})≤ψ(max{G(xn,u,u),αG(xn+1,xn+1,u),βG(xn,u,u)+βG(u,xn+1,xn+1)})≤ψ(max{G(xn,u,u),G(xn+1,xn+1,u),βG(xn,u,u)+βG(u,xn+1,xn+1)}).
But ψ is an increasing function, thus from (3.18), we get
(3.19)G(fxn,u,u)≤max⁡{G(xn,u,u),G(xn+1,xn+1,u),βG(xn+1,u,u)+βG(u,xn+1,xn+1)}.
Therefore, limn→∞G(fxn,u,u)=0.Corollary 3.3.
LetT be a self-map on a complete G-metric space X satisfying the following for all x,y,z∈X:(3.20)G(fx,fy,fz)≤λmax⁡{G(x,y,z),G(x,fx,fx),G(y,fy,fy),G(z,fz,fz),αG(fx,fx,y)+(1-α)(G(fy,fy,z)),βG(x,fx,fx)+(1-β)(G(y,fy,fy))},
where 0<α,β,λ<1,ψ is an altering distance function, and ϕ:[0,∞)→[0,∞) is a continuous function with ϕ(t)=0 if and only if t=0. Then f has a unique fixed point (say u), and f is G continuous at u.Proof.
We get the result by takingψ(t)=t and ϕ(t)=t-λt, then apply Theorem 3.2.Corollary 3.4.
Let(X,G) be a complete G-metric space. Let f be a self-map on X satisfying the following:
(3.21)G(fx,fy,fz)≤λmax⁡{G(x,y,z),G(x,fx,fx),G(y,fy,fy),G(z,fz,fz),12(G(fx,fx,y)+(G(fy,fy,z))),12(G(x,fx,fx)+G(y,fy,fy))},
for all x,y,z∈X where 0<λ<1,ψ is an altering distance function and, ϕ:[0,∞)→[0,∞) is a continuous function with ϕ(t)=0 if and only if t=0. Then f has a unique fixed point (say u), and f is G continuous at u.Proof.
We get the result by takingψ(t)=t and ϕ(t)=t-λt, α=β=1/2 in Theorem 3.2.Corollary 3.5.
Let(X,G) be a complete G-metric space. Let f be a self-map on X satisfying the following:
(3.22)G(fx,fy,fz)≤λmax⁡{G(x,y,z),G(x,fx,fx),G(y,fy,fy),G(z,fz,fz),13G(fx,fx,y)+23G(fy,fy,z),13G(fx,fx,x)+23G(fy,fy,y)},
for all x,y,z∈X, where 0<λ<1,ψ is an altering distance function, and ϕ:[0,∞)→[0,∞) is a continuous function with ϕ(t)=0 if and only if t=0. Then f has a unique fixed point (say u) and f is G continuous at u.Proof.
We get the result by takingψ(t)=t and ϕ(t)=t-λt, α=β=1/3 in Theorem 3.2.Theorem 3.6.
Under the condition of Theorem3.2, f has property P.Proof.
From Theorem3.2, f has a fixed point. Therefore F(fn)≠φ for each n∈N. Fix n>1, and assume that u∈F(fn). We claim that u∈F(f). To prove the claim, suppose that u≠fu. Using (3.1), we have
(3.23)ψ(G(u,fu,fu))=ψ(G(fnu,fn+1u,fn+1u))=ψ(G(ffn-1u,ffnu,ffnu))≤ψ(max{G(fn-1u,u,u),G(u,fu,fu),αG(u,u,u)+(1-α)(G(fu,fu,u))βG(fn-1u,u,u)+(1-β)(G(u,fu,fu))})-ϕ(max{G(fn-1u,u,u),G(u,fu,fu),αG(u,u,u)+(1-α)(G(fu,fu,u))βG(fn-1u,u,u)+(1-β)(G(u,fu,fu))})=ψ(max{G(fn-1u,u,u),G(u,fu,fu)})-ϕ(max{G(fn-1u,u,u),G(u,fu,fu)}).
Letting M=max{G(fn-1u,u,u),G(u,fu,fu)}, we deduce form (3.23),
(3.24)ψ(G(u,fu,fu))≤ψ(M)-ϕ(M).
If M=G(u,fu,fu), then
(3.25)ψ(G(u,fu,fu))≤ψ(G(u,fu,fu))-ϕ(G(u,fu,fu)),
hence, ϕ(G(u,fu,fu))=0. By a property of ϕ, we deduce that G(u,fu,fu)=0, therefore, u=fu. This is a contradiction. On the other hand, if M=G(fn-1u,u,u), then (3.1) gives that
(3.26)ψ(G(fnu,fn+1u,fn+1u))=ψ(G(u,fu,fu))≤ψ(G(fn-1u,u,u))-ϕ(G(fn-1u,u,u))=ψ(G(fn-1u,fnu,fnu))-ϕ(G(fn-1u,fnu,fnu))≤ψ(G(fn-2u,fn-1u,fn-1u))-ϕ(G(fn-2u,fn-1u,fn-1u))-ϕ(G(fn-1u,fnu,fnu))≤⋯≤ψ(G(u,fu,fu))-∑k=0n-1‍ϕ(G(fn-k-1u,fn-ku,fn-ku)).
Therefore,
(3.27)∑k=0n-1‍ϕ(G(fn-k-1u,fn-ku,fn-ku))=0,
which implies that ϕ(G(fn-k-1u,fn-ku,fn-ku))=0, for all (0≤k≤n-1). Thus, ϕ(G(u,fu,fu))=0, and by a property of ϕ, we have u=fu. This is a contradiction.
Therefore,u∈F(f), and f has property P. 							(3.28)LetMα,β(x,y,z)=max⁡{G(x,y,z),G(x,fx,fx),G(y,fy,fy),G(z,fz,fz),αG(fx,fx,y)+(1-α)(G(fy,fy,z)),βG(x,fx,fx)+(1-β)(G(y,fy,fy)),},
where α,β∈(0,1].Example 3.7.
LetX=[0,1] and G(x,y,z)=max{|x-y|,|y-z|,|z-x|} be a G-metric on X. Define f:X→X by f(x)=x/8. We take ψ(t)=t and ϕ(t)=7/8t, for t∈[0,∞) and α,β∈(0,1]. So that
(3.29)ψ(Mα,β(x,y,z))-ϕ(Mα,β(x,y,z))=18Mα,β(x,y,z).
We have
(3.30)G(fx,fy,fz)=max{|x8-y8|,|y8-z8|,|z8-x8|}=18max{|x-y|,|y-z|,|z-x|}=18G(x,y,z)≤18Mα,β(x,y,z)=ψ(Mα,β(x,y,z))-ϕ(Mα,β(x,y,z)).


## 4. Applications

Denote byΛ the set of functions λ:[0,∞)→[0,∞) satisfying the following hypotheses.(1)
λ is a Lebesgue integral mapping on each compact of [0,∞).(2)
For everyε>0, we have ∫0t‍λ(s)ds>0.It is an easy matter to see that the mapping ψ:[0,∞)→[0,∞), defined by ψ(t)=∫0t‍λ(s)ds, is an altering distance function. Now, we have the following result.Theorem 4.1.
Let(X,G) be a complete G-metric space. Let f be a self-map on X satisfying the following:
(4.1)∫0G(fx,fy,fz)‍λ(s)ds≤∫0Mα,β(x,y,z)‍λ(s)ds-∫0Mα,β(x,y,z)‍μ(s)ds,
where λ, μ∈Λ and α,β∈(0,1]. Then f has a unique fixed point.Proof.
It follows from Theorem by takingψ(t)=∫0t‍λ(s)ds and ϕ(t)=∫0t‍μ(s)ds.



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