# Periodic Points and Fixed Points for the Weaker(ϕ,φ)-Contractive Mappings in Complete Generalized Metric Spaces

Authors: Chi-Ming Chen; W. Y. Sun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce the notion of weaker(ϕ,φ)-contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.


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## Body

## 1. Introduction and Preliminaries

Let(X,d) be a metric space, D a subset of X and f:D→X a map. We say f is contractive if there exists α∈[0,1) such that, for all x,y∈D,
(1.1)d(fx,fy)≤α⋅d(x,y).
The well-known Banach’s fixed point theorem asserts that if D=X, f is contractive and (X,d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] introduced the notion of Φ-contraction. A mapping f:X→X on a metric space is called Φ-contraction if there exists an upper semicontinuous function Φ:[0,∞)→[0,∞) such that(1.2)d(fx,fy)≤Φ(d(x,y))∀x,y∈X.In 2000, Branciari [3] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involving three terms instead of two. Later, many authors worked on this interesting space (e.g., [4–9]).Let(X,d) be a generalized metric space. For γ>0 and x∈X, we define
(1.3)Bγ(x)∶={y∈X∣d(x,y)<γ}.
Branciari [3] also claimed that {Bγ(x):γ>0,x∈X} is a basis for a topology on X, d is continuous in each of the coordinates and a generalized metric space is a Hausdorff space. We recall some definitions of a generalized metric space, as follows.Definition 1.1 (see [3]).
LetX be a nonempty set and d:X×X→[0,∞) a mapping such that for all x,y∈X and for all distinct point u,v∈X each of them different from x and y, one has(i)
d(x,y)=0 if and only if x=y;(ii)
d(x,y)=d(y,x);(iii)
d(x,y)≤d(x,u)+d(u,v)+d(v,y) (rectangular inequality). Then (X,d) is called a generalized metric space (or shortly g.m.s).Definition 1.2 (see [3]).
Let(X,d) be a g.m.s, {xn} a sequence in X, and x∈X. We say that {xn} is g.m.s convergent to x if and only if d(xn,x)→0 as n→∞. We denote by xn→x as n→∞.Definition 1.3 (see [3]).
Let(X,d) be ag.m.s, {xn} a sequence in X, and x∈X.We say that {xn} is g.m.sCauchy sequence if and only if for each ε>0,there exists n0∈ℕ such thatd(xm,xn)<ε for all n>m>n0.Definition 1.4 (see [3]).
Let(X,d) be ag.m.s. Then X is called completeg.m.sif everyg.m.sCauchy sequence isg.m.sconvergent in X.In this paper, we also recall the notion of Meir-Keeler function (see [10]). A function ϕ:[0,∞)→[0,∞) is said to be a Meir-Keeler function if for each η>0, there exists δ>0 such that for t∈[0,∞) with η≤t<η+δ, we have ϕ(t)<η. Generalization of the above function has been a heavily investigated branch research. Particularly, in [11, 12], the authors proved the existence and uniqueness of fixed points for various Meir-Keeler-type contractive functions. We now introduce the notion of weaker Meir-Keeler function ϕ:[0,∞)→[0,∞), as follows.Definition 1.5.
We callϕ:[0,∞)→[0,∞) a weaker Meir-Keeler function if for each η>0, there exists δ>0 such that for t∈[0,∞) with η≤t<η+δ, there exists n0∈ℕ such that ϕn0(t)<η.


## 2. Main Results

In the paper, we denote byΦ the class of functions ϕ:[0,∞)→[0,∞) satisfying the following conditions:(ϕ1) ϕ:[0,∞)→[0,∞) is a weaker Meir-Keeler function;(ϕ2) ϕ(t)>0 for t>0, ϕ(0)=0;(ϕ3) for all t∈(0,∞), {ϕn(t)}n∈ℕ is decreasing;(ϕ4) if limn→∞tn=γ, then limn→∞ϕ(tn)≤γ.And we denote byΘ the class of functions φ:[0,∞)→[0,∞) satisfying the following conditions:(φ1) φ is continuous;(φ2) φ(t)>0 for t>0 and φ(0)=0.Our main result is the following.Theorem 2.1.
Let(X,d) be a Hausdorff and complete g.m.s, and let  f:X→X be a function satisfying
(2.1)d(fx,fy)≤ϕ(d(x,y))-φ(d(x,y))
for all x,y∈X and ϕ∈Φ, φ∈Θ. Then f has a periodic point μ in X, that is, there exists μ∈X such that μ=fpμ for some p∈ℕ.Proof.
Givenx0, define a sequence {xn} in X by
(2.2)xn+1=fxnforn∈N∪{0}.Step 1.
We will prove that(2.3)lim⁡n→∞d(xn,xn+1)=0,lim⁡n→∞d(xn,xn+2)=0.
Using inequality (2.1), we have that for each n∈ℕ∪{0}(2.4)d(xn,xn+1)=d(fxn-1,fxn)≤ϕ(d(xn-1,xn))-φ(d(xn-1,xn))≤ϕ(d(xn-1,xn)),
and so
(2.5)d(xn,xn+1)≤ϕ(d(xn-1,xn))≤ϕ(ϕ(d(xn-2,xn-1)))=ϕ2(d(xn-2,xn-1))≤⋯≤ϕn(d(x0,x1)).
Since {ϕn(d(x0,x1))}n∈ℕ is decreasing, it must converge to some η≥0. We claim that η=0. On the contrary, assume that η>0. Then by the definition of weaker Meir-Keeler function ϕ, there exists δ>0 such that for x0,x1∈X with η≤d(x0,x1)<δ+η, there exists n0∈ℕ such that ϕn0(d(x0,x1))<η. Since limn→∞ϕn(d(x0,x1))=η, there exists p0∈ℕ such that η≤ϕp(d(x0,x1))<δ+η, for all p≥p0. Thus, we conclude that ϕp0+n0(d(x0,x1))<η. So we get a contradiction. Therefore, limn→∞ϕn(d(x0,x1))=0, that is,
(2.6)lim⁡n→∞d(xn,xn+1)=0.
Using inequality (2.1), we also have that for each n∈ℕ(2.7)d(xn,xn+2)=d(fxn-1,fxn+1)≤ϕ(d(xn-1,xn+1))-φ(d(xn-1,xn+1))≤ϕ(d(xn-1,xn+1)),
and so
(2.8)d(xn,xn+2)≤ϕ(d(xn-1,xn+1))≤ϕ(ϕ(d(xn-2,xn)))=ϕ2(d(xn-2,xn))≤⋯≤ϕn(d(x0,x2)).
Since {ϕn(d(x0,x2))}n∈ℕ is decreasing, by the same proof process, we also conclude
(2.9)lim⁡n→∞d(xn,xn+2)=0.
Next, we claim that{xn} is g.m.s Cauchy. We claim that the following result holds.Step 2.
Claim that for everyε>0, there exists n∈ℕ such that if p,q≥n then d(xp,xq)<ε.
Suppose the above statement is false. Then there existsϵ>0 such that for any n∈ℕ, there are pn,qn∈ℕ with pn>qn≥n satisfying
(2.10)d(xqn,xpn)≥ϵ.
Further, corresponding to qn≥n, we can choose pn in such a way that it the smallest integer with pn>qn≥n and d(xqn,xpn)≥ϵ. Therefore, d(xqn,xpn-1)<ϵ. By the rectangular inequality and (2.3), we have
(2.11)ϵ≤d(xpn,xqn)≤d(xpn,xpn-2)+d(xpn-2,xpn-1)+d(xpn-1,xqn)<d(xpn,xpn-2)+d(xpn-2,xpn-1)+ϵ.
Let n→∞. Then we get
(2.12)lim⁡n→∞d(xpn,xqn)=ϵ.
On the other hand, we have
(2.13)d(xpn,xqn)≤d(xpn,xpn-1)+d(xpn-1,xqn-1)+d(xqn-1,xqn),d(xpn-1,xqn-1)≤d(xpn-1,xpn)+d(xpn,xqn)+d(xqn,xqn-1).
Let n→∞. Then we get
(2.14)lim⁡n→∞d(xpn-1,xqn-1)=ϵ.
Using inequality (2.1), we have
(2.15)d(xpn,xqn)=d(fxpn-1,fxqn-1)≤ϕ(d(xpn-1,xqn-1))-φ(d(xpn-1,xqn-1)).
Letting n→∞, using the definitions of the functions ϕ and φ, we have
(2.16)ϵ≤ϵ-φ(ϵ),
which implies that φ(ϵ)=0. By the definition of the function φ, we have ϵ=0. So we get a contradiction. Therefore {xn} is g.m.s Cauchy.Step 3.
We claim thatf has a periodic point in X.
Suppose, on contrary,f has no periodic point. Then {xn} is a sequence of distinct points, that is, xp≠xq for all p,q∈ℕ with p≠q. By Step 2, since X is complete g.m.s, there exists ν∈X such that xn→ν. Using inequality (2.1), we have
(2.17)d(fxn,fν)≤ϕ(d(xn,ν))-φ(d(xn,ν)).
Letting n→∞, we have
(2.18)d(fxn,fν)⟶0,asn⟶∞,
that is,
(2.19)xn+1=fxn⟶fν,asn⟶∞.
As (X,d) is Hausdorff, we have ν=fν, a contradiction with our assumption that f has no periodic point. Therefore, there exists ν∈X such that ν=fp(ν) for some p∈ℕ. So f has a periodic point in X.Following Theorem2.1, it is easy to get the below fixed point result.Theorem 2.2.
Let(X,d) be a Hausdorff and complete g.m.s, and let  f:X→X be a function satisfying
(2.20)d(fx,fy)≤ϕ(d(x,y))-φ(d(x,y))
for all x,y∈X, where ϕ∈Φ with 0<ϕ(t)<t for all t>0, and φ∈Θ. Then f has a unique fixed point in X.Proof.
From Theorem2.1, we conclude that f has a periodic point ν∈X, that is, there exists ν∈X such that ν=fp(ν) for some p∈ℕ. If p=1, then we complete the proof, that is, ν is a fixed point of f. If p>1, then we will show that μ=fp-1ν is a fixed point of f. Suppose that it is not the case, that is, fp-1ν≠fpν. Then Using inequality (2.1), we have
(2.21)d(ν,fν)=d(fpν,fp+1ν)≤ϕ(d(fp-1ν,fpν))-φ(d(fp-1ν,fpν))<ϕ(d(fp-1ν,fpν))≤d(fp-1ν,fpν).
Using inequality (2.1), we also have
(2.22)d(fp-1ν,fpν)≤ϕ(d(fp-2ν,fp-1ν))-φ(d(fp-2ν,fp-1ν))≤ϕ(d(fp-2ν,fp-1ν))≤d(fp-2ν,fp-1ν).
Continuing this process, we conclude that
(2.23)d(ν,fν)<d(fp-1ν,fpν)≤d(fp-2ν,fp-1ν)≤⋯≤d(ν,fν),
which implies a contradiction. Thus, μ=fp-1ν is a fixed point of f.
Finally, to prove the uniqueness of the fixed point, supposeμ,ν are fixed points of f. Then,
(2.24)d(μ,ν)=d(fμ,fν)≤ϕ(d(μ,ν))-φ(d(μ,ν)),
which implies that d(μ,ν)=0, that is, μ=ν. So we complete the proof.



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*Source: 856974-2012-02-20.xml*