# Best Periodic Proximity Points for Cyclic Weaker Meir-Keeler Contractions

Authors: Chiming Chen; Chingju Lin
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is to present the existence of the best period proximity point for cyclic weaker Meir-Keeler contractions and asymptotic cyclic weaker Meir-Keeler contractions in metric spaces.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, byℝ+ we denote the set of all nonnegative numbers, while ℕ is the set of all natural numbers. Let A and B be nonempty subsets of a metric space (X,d). Consider a mapping f:A∪B→A∪B, f is called a cyclic map if f(A)⊆B and f(B)⊆A. A point x in A is called a best proximity point of f in A if d(x,fx)=d(A,B) is satisfied, where d(A,B)=inf{d(x,y):x∈A,y∈B}, and x∈A is called a best periodic proximity point of f in A if d(x,f2κ+1x)=d(A,B) is satisfied, for some κ∈ℕ∪{0}. In 2005, Eldred et al. [1] proved the existence of a best proximity point for relatively nonexpansive mappings using the notion of proximal normal structure. In 2006, Eldred and Veeramani [2] proved the following existence theorem.Theorem 1.1 (see Theorem  3.10 in [2]).
LetA and B be nonempty closed convex subsets of a uniformly convex Banach space. Suppose f:A∪B→A∪B is a cyclic contraction, that is, f(A)⊆B and f(B)⊆A, and there exists k∈(0,1) such that
(1.1)d(fx,fy)≤kd(x,y)+(1-k)d(A,B)foreveryx∈A,y∈B.
Then there exists a unique best proximity point in A. Further, for each x∈A, {f2nx} converges to the best proximity point.In this paper, we also recall the notion of Meir-Keeler type mapping. A mappingψ:ℝ+→ℝ+ is said to be a Meir-Keeler-type mapping (see [3]) if for each η>0, there exists δ>0 such that for t∈ℝ+ with η≤t<η+δ, we have ψ(t)<η.In the recent, Eldred et al. [1] introduced the below notion of cyclic Meir-Keeler contraction.Definition 1.2 (see [1]).
Let(X,d) be a metric space, and let A and B be nonempty subsets of X. Then f:A∪B→A∪B is called a cyclic Meir-Keeler contraction if the following are satisfied: (i)
f(A)⊂B and f(B)⊂A;(ii)
for everyɛ>0, there exists δ>0 such that(1.2)d(x,y)<d(A,B)+ɛ+δimpliesd(fx,fy)<d(A,B)+ɛ
for all x∈A and y∈B.
In the recent, Di Bari et al. [4] proved the following best proximity point theorem.Theorem 1.3 (see [4]).
LetX be a uniformly convex Banach space, and let A and B be nonempty subsets of X. Suppose A is closed and convex and f:A∪B→A∪B is a cyclic Meir-Keeler contraction. Then there exists a unique best proximity point in A. Further, for each x∈A, {f2nx} converges to best proximity point.Later, many authors studied this subject, and many results on best proximity points are proved. (see, e.g., [5–10]). In this study, we will introduce the new concepts of cyclic weaker Meir-Keeler contractions and asymptotic cyclic weaker Meir-Keeler contractions in metric spaces, and the purpose of this paper is to present the existence of the best period proximity point for these contractions.


## 2. The Best Periodic Proximity Points for Cyclic Weaker Meir-Keeler Contractions

In this section, we first introduce the below notions of the weaker Meir-Keeler-type mapping,φ-mapping, and cyclic weaker Meir-Keeler contraction in metric spaces.Definition 2.1.
Let(X,d) be a metric space, and φ:ℝ+→ℝ+. Then φ is called a weaker Meir-Keeler-type mapping in X if for each η>0, there exists δ>0 such that for x,y∈X with η≤d(x,y)<δ+η, there exists n0∈ℕ such that φn0(d(x,y))<η.The following provides an example of a weaker Meir-Keeler-type mapping that is not a Meir-Keeler-type mapping in a metric space(X,d).Example 2.2.
LetX=ℝ2, and we define d:X×X→ℝ+ by
(2.1)d(x,y)=|x1-y1|+|x2-y2|∀x=(x1,x2),y=(y1,y2)∈X.
If φ:ℝ+→ℝ+,
(2.2)φ(t)={0ift≤1,2tif1<t<2,1ift≥2,
wheret=d(x,y), x,y∈X, then φ is a weaker Meir-Keeler-type mapping that is not a Meir-Keeler-type mapping in X.Definition 2.3.
Let(X,d) be a metric space. A mapping φ:ℝ+→ℝ+ is called a φ-mapping in X if the mapping φ:ℝ+→ℝ+ satisfies the following conditions:(φ1)
φ is a weaker Meir-Keeler-type mapping in X;(φ2)
for allt>0, {φn(t)}n∈ℕ is nonincreasing;(φ3)
for allt>0, φ(t)>0 and φ(0)=0.
The following provides two examples of aφ-mapping.Example 2.4.
LetX=ℝ2,and we define d:X×X→ℝ+ by
(2.3)d(x,y)=|x1-y1|+|x2-y2|∀x=(x1,x2),y=(y1,y2)∈X.
Let φ:ℝ+→ℝ+ be
(2.4)φ(t)=12t∀t∈R+.
Then φ:ℝ+→ℝ+ is a φ-mapping in X.Example 2.5.
LetX=[0,4], and we define d:X×X→ℝ+ by
(2.5)d(x,y)=|x-y|∀x,y∈X.
If φ:ℝ+→ℝ+,
(2.6)φ(t)={34t2t1if0≤t≤1,if1<t<2,if2≤t≤4,
where t=d(x,y), x,y∈X, then φ is a φ-mapping in X.Definition 2.6.
Let(X,d) be a metric space, and let A and B be nonempty subsets of X. Then f:A∪B→A∪B is called a cyclic weaker Meir-Keeler contraction if the following conditions hold:(1)
f(A)⊂B and f(B)⊂A;(2)
there is aφ-mapping φ:ℝ+→ℝ+ in X such that for all n∈ℕ and x∈A, y∈B with d(x,y)-d(A,B)>0,(2.7)d(fnx,fny)-d(A,B)<φn(d(x,y)-d(A,B)),d(x,y)-d(A,B)=0impliesd(fnx,fny)-d(A,B)=0.
The following provides an example of a cyclic weaker Meir-Keeler contraction.Example 2.7.
LetA=[-2,0] and B=[0,2] in the metric space (ℝ,d), where d(x,y)=|x-y|. Define
(2.8)f(x)=-x4∀x∈A∪B.
Let φ:ℝ+→ℝ+ be defined by
(2.9)φ(t)={34tif0≤t≤1,2tif1<t<2,1if2≤t≤4,
where t=d(x,y), x∈A, y∈B. Then all conditions (1) and (2) of Definition 2.6 and therefore f are a cyclic weaker Meir-Keeler contraction. Notice that d(A,B)=0.
Now, we are in this position to state the following results.Lemma 2.8.
Let(X,d) be a metric space, and let A, B be nonempty subsets of X. Suppose f:A∪B→A∪B is a cyclic weaker Meir-Keeler contraction. Then limn→∞d(fnx,fn+1x)=d(A,B) holds.
Proof.
Sincef:A∪B→A∪B is a cyclic weaker Meir-Keeler contraction, there is a φ-mapping φ:ℝ+→ℝ+ in X such that(2.10)d(fnx,fny)-d(A,B)<φn(d(x,y)-d(A,B)),
for alln∈ℕandx∈A,y∈B.
Since{φn(d(x,y))}n∈ℕ is nonincreasing, hence we also conclude {φn(d(x,y)-d(A,B))}n∈ℕ is nonincreasing, and it must converge to some η≥0. We claim that η=0. On the contrary, assume that η>0. By the definition of the weaker Meir-Keeler-type mapping φ, corresponding to η use, there exists δ>0 such that for x,y∈X with η≤d(x,y)-d(A,B)<δ+η, there exists n0∈ℕ such that φn0(d(x,y)-d(A,B))<η. Since limn→∞φn(d(x,y)-d(A,B))=η, there exists m0∈ℕ such that η≤φm(d(x,y)-d(A,B))<δ+η, forall m≥m0. Thus, we conclude that φm0+n0(d(x,y)-d(A,B))<η. So we get a contradiction. So limn→∞φn(d(x,y)-d(A,B))=0, and so limn→∞d(fnx,fny)-d(A,B)=0, that is, limn→∞d(fnx,fny)=d(A,B). Thus, we also conclude that limn→∞d(fnx,fn+1x)=d(A,B).Applying above Lemma2.8, it is easy to conclude the following theorem.Theorem 2.9.
Let(X,d) be a metric space, and let A,B be nonempty subsets of X. Suppose f:A∪B→A∪B is a cyclic weaker Meir-Keeler contraction and if for some x∈A, the sequence {f2n+1x} converges to x¯∈A, then x¯ is a best periodic proximity point of f in A.
Proof.
By the definition of the weaker Meir-Keeler-type mappingφ:ℝ+→ℝ+ in X, there exists n0∈ℕ such that φn0(η)≤η for each η>0. Since {f2n+1x} converges tox¯∈A, corresponding to above n0 use, we have
(2.11)d(A,B)≤d(x¯,f2n0+1x¯)≤d(x¯,f2n+1x)+d(f2n+1x,f2n0+1x¯)-d(A,B)+d(A,B)≤d(x¯,f2n+1x)+φ2n0+1(d(f2(n-n0)x,x¯)-d(A,B))+d(A,B)≤d(x¯,f2n+1x)+φ2n0(d(f2(n-n0)x,x¯)-d(A,B))+d(A,B)≤d(x¯,f2n+1x)+d(f2(n-n0)x,x¯)-d(A,B)+d(A,B)≤d(x¯,f2n+1x)+d(f2(n-n0)x,f2(n-n0)+1x)+d(f2(n-n0)+1x,x¯),
Letting n→∞. Then d(A,B)=d(x¯,f2n0+1x¯). Thus x¯ is a best period proximity point off in A.


## 3. The Best Periodic Proximity Points for Asymptotic Cyclic Weaker Meir-Keeler Contractions

In this section, we introduce the below notions of the asymptotic cyclic weaker Meir-Keeler-type sequence and asymptotic cyclic weaker Meir-Keeler contraction in a metric space(X,d).Definition 3.1.
Let(X,d) be a metric space. A sequence {φn|φn:ℝ+→ℝ+}n∈ℕ in X is called an asymptotic weaker Meir-Keeler-type sequence if {φn|φn:ℝ+→ℝ+}n∈ℕ satisfies the following conditions:(C1)
for eachη>0, there exists δ>0 such that for x,y∈X with η≤d(x,y)<δ+η, there exists 2n0∈ℕ such that φ2n0(d(x,y))<η;(C2)
for alln∈ℕ and t>0, {φn(t)}n∈ℕ is nonincreasing;(C3)
for alln∈ℕ, φn(0)=0 and φn(t)>0, t>0.Example 3.2.
LetX=ℝ2 and we define d:X×X→ℝ+ by
(3.1)d(x,y)=|x1-y1|+|x2-y2|∀x=(x1,x2),y=(y1,y2)∈X.
Let φn:ℝ+→ℝ+ be
(3.2)φn(t)=12nt∀t∈R+,n∈N,
where t=d(x,y), x,y∈X. Then {φn∣φn:ℝ+→ℝ+}n∈ℕ is an asymptotic weaker Meir-Keeler-type sequence in a metric space (X,d).Definition 3.3.
Let(X,d) be a metric space, and let A and B be nonempty subsets of X. Then f:A∪B→A∪B is an asymptotic cyclic weaker Meir-Keeler contraction if the following conditions hold:
(1)
f(A)⊂B and f(B)⊂A;(2)
there is an asymptotic weaker Meir-Keeler-type sequence{φn|φn:ℝ+→ℝ+}n∈ℕ such that for all n∈ℕ and x∈A, y∈B with d(x,y)-d(A,B)>0,(3.3)d(fnx,fny)-d(A,B)<φn(d(x,y)-d(A,B)),d(x,y)-d(A,B)=0impliesd(fnx,fny)-d(A,B)=0.
Now, we are in this position to state the following results.Lemma 3.4.
Let(X,d) be a metric space and A,B nonempty subsets of X. Suppose f:A∪B→A∪B is an asymptotic cyclic weaker Meir-Keeler contraction. Then limn→∞d(fnx,fn+1x)=d(A,B) holds.
Proof.
Sincef:A∪B→A∪B is an asymptotic cyclic weaker Meir-Keeler contraction, there is an asymptotic weaker Meir-Keeler-type sequence {φn|φn:ℝ+→ℝ+}n∈ℕ such that(3.4)d(fnx,fny)-d(A,B)<φn(d(x,y)-d(A,B)),
for all n∈ℕ and x∈A, y∈B.
Since{φn(d(x,y))}n∈ℕ is nonincreasing, hence we also conclude {φn(d(x,y)-d(A,B))}n∈ℕ is nonincreasing, and it must converge to some η≥0. We claim that η=0. On the contrary, assume that η>0. By the definition of asymptotic weaker Meir-Keeler-type sequence, corresponding to η use, there exists δ>0 such that for x,y∈X with η≤d(x,y)-d(A,B)<δ+η, there exists 2n0∈ℕ such that φ2n0(d(x,y)-d(A,B))<η. Since limn→∞φn(d(x,y)-d(A,B))=η, there exists m0∈ℕ such that η≤φm(d(x,y)-d(A,B))<δ+η, for all m≥m0. Thus, we conclude that ψm0+2n0(d(x,y)-d(A,B))<η. So we get a contradiction. Therefore, limn→∞φn(d(x,y)-d(A,B))=0, and so limn→∞d(fnx,fny)-d(A,B)=0, that is, limn→∞d(fnx,fny)=d(A,B). Thus, we also conclude that limn→∞d(fnx,fn+1x)=d(A,B).Applying above Lemma3.4, we are easy to conclude the following theorem.Theorem 3.5.
Let(X,d) be a metric space and A, B nonempty subsets of X. Suppose f:A∪B→A∪B is an asymptotic cyclic weaker Meir-Keeler contraction, and if for some x∈A, the sequence {f2n+1x} converges to x¯∈A, then x¯ is a best periodic proximity point of f in A.
Proof.
By the definition of the asymptotic weaker Meir-Keeler-type sequence{φn|φn:ℝ+→ℝ+}n∈ℕ, thus there exists 2n0∈ℕ such that φ2n0(η)≤η for each η>0. Since {f2n+1x} converges to x¯∈A, corresponding to above 2n0 use, we have
(3.5)d(A,B)≤d(x¯,f2n0+1x¯)≤d(x¯,f2n+1x)+d(f2n+1x,f2n0+1x¯)-d(A,B)+d(A,B)≤d(x¯,f2n+1x)+φ2n0+1(d(f2(n-n0)x,x¯)-d(A,B))+d(A,B)≤d(x¯,f2n+1x)+φ2n0(d(f2(n-n0)x,x¯)-d(A,B))+d(A,B)≤d(x¯,f2n+1x)+d(f2(n-n0)x,x¯)-d(A,B)+d(A,B)≤d(x¯,f2n+1x)+d(f2(n-n0)x,f2(n-n0)+1x)+d(f2(n-n0)+1x,x¯).
Letting n→∞. Then d(A,B)=d(x¯,f2n0+1x¯). Thus x¯ is a best period proximity point of f in A.



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*Source: 782389-2012-05-31.xml*