# On Best Proximity Point Results for Some Type of Mappings

Authors: Vahid Parvaneh; Mohammad Reza Haddadi; Hassen Aydi
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghty- and Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction principle (BCP) in metric spaces has been generalized and extended in various ways. As a generalization of the BCP, Geraghty [1] proved the following.Theorem 1 [1].
LetX,d be a complete metric space and let T:X⟶X be an operator. Suppose that there exists β:0,+∞⟶0,1 satisfying the condition
(1)βtn⟶1impliestn⟶0,asn⟶+∞.
IfT satisfies the following inequality
(2)dTx,Ty≤βdx,ydx,y,forallx,y∈X,then T has a unique fixed point.The following are two examples of Geraghty functionsβ.
(3)βt=e−tet,ift∈0,∞,0,ift=0,aβt=11+t2,ift∈0,∞,0,ift=0.One of the important extensions of the BCP was given by Caristi [2].Theorem 2 [2].
LetΓ be a self-mapping on a complete metric space X,d. Assume that there is a bounded below and lower semicontinuous function ψ:X⟶ℝ so that
(4)dx,Γx≤ψx−ψΓxfor all x∈X. Then, Γ possesses a fixed point.On the other hand, Kirk et al. [3] in 2003 introduced the notion of a cyclic representation.Definition 3 [3].
LetA and B be nonempty subsets of a metric space X,d and T:A∪B⟶A∪B. Then, Tis called a cyclic map if TA⊆B and TB⊆A.The following interesting theorem for a cyclic map was given in [3].Theorem 4 [3].
LetA and B be nonempty closed subsets of a complete metric space X,d. Suppose that T:A∪B⟶A∪B is a cyclic map such that
(5)dTx,Ty≤kdx,yfor all x∈A and y∈B, where k∈0,1 is a constant. Then, T has a unique fixed point u and u∈A∩B.LetM and N be nonempty sets of a metric space X,d. Given a map, Γ:M⟶N. An element x∈M is called a best proximity point of Γ if
(6)dx,Γx=dM,N≕infdm,n,m,∈M,n∈N.The set of best proximity points ofΓ is denoted by PΓM,N. The research of best proximity points is meaningful in optimization. The problem of existence of best proximity points in uniformly convex Banach spaces and in metric spaces as well as the convergence of sequences to such points has been focused on and successfully solved in some classic pioneering works (see [4]).Definition 5 [5].
LetX,d be a metric space and A and B be subsets of X. A map T:A∪B⟶A∪B is said to be a cyclic contractive map if it satisfies
(7)TA⊂B,TB⊂A;(i)
dTx,Ty≤kdx,y+1−kdistA,B, for all x∈A and y∈B.Eldred and Veeramani [5] extended Theorem 4 to include the case A∩B=∅, by the following existence result of a best proximity point.Theorem 6 [5].
LetA and B be nonempty closed subsets of a metric space X and let T:A∪B⟶A∪B be a cyclic contraction map. If either A or B is boundedly compact, then there exists x∈A∪B such that dx,Tx=distA,B.A convenience attention has been recently devoted to the research on existence and uniqueness of best proximity points of self-mappings, as well as, to the investigation of associated relevant properties, for instance, stability of the iterations. The various related performed researches include the cases of cyclicϕ-contractions [6, 7], cyclic Meir-Keeler contractions [8], weak cyclic Bianchini contractions [9], weak cyclic Kannan contractions [10], p-cyclic summing iterated contractions [11], and MF-cyclic contractions with Property UC [12]. Some contractive conditions and related properties under general contractive conditions including some proximal rational types have been also investigated [13]. In this paper, we ensure the existence of best proximity points for Geraghty and Caristi type contraction mappings.


## 2. A Best Proximity Point Result for Geraghty-Type Contractions

In this section, we introduce cyclic Geraghty contraction maps and give new conditions for existence and uniqueness of a best proximity point.Definition 7.
LetX be a complete metric space and M and N be subsets of X. A map Γ:M∪N⟶M∪N is a cyclic Geraghty contraction map if there exists β∈F such that
(8)ΓM⊂N,ΓN⊂M;(i)
dΓx,Γy≤βdx,ydx,y for all x,y∈M∪Nwhere F is the set of functions β:0,∞⟶0,1 so thatif tn∈0,dM,N and βΓn⟶1, then tn⟶0;if tn∈dM,N,∞ and βΓn⟶1, then tn⟶dM,N.We give the following theorem (comparable to Theorem 3.1 of [1]).Theorem 8.
LetM and N be closed subsets of a complete metric space X such that diamM,diamN<dM,N. Suppose Γ:M∪N⟶M∪N is a cyclic Geraghty contraction map. Then, PΓM,N≠0. Further, if x0∈M and xn+1=Γxn, then x2n converges to a best proximity point.Proof.
Fixx∈M∪N and define a sequence xn in M∪N by xn=Γnx, n∈N0. First, we show that limn→∞dxn,xn+1=dM,N. We have
(9)dxn,xn+1=dΓxn−1,Γxn≤βdxn−1,xndxn−1,xn.
Hence,dxn,xn+1 is monotonic decreasing and bounded below. So,
(10)limn→∞dxn,xn+1exists. Let limn→∞dxn,xn+1=δ. It is clear that dM,N≤δ. Assume that δ>dM,N. We have
(11)δ=limn→∞dxn,xn+1≤limn→∞βdxn−1,xndxn−1,xn<limn→∞dxn−1,xn=δ,so δ=dM,N.
Hence,limn→∞dxn,xn+1=dM,N. We shall show that dx2n,x2n+2⟶0 and dx2n−1,x2n+1⟶0. We have
(12)dx2n,x2n+2=dΓx2n−1,Γx2n+1≤βdx2n−1,Γx2n+1dx2n−1,Γx2n+1.
That is,dx2n,x2n+2 is monotonic decreasing and bounded below. Hence, limn→∞dx2n,x2n+2 exists.
Letlimn→∞dx2n,x2n+2=δ. Assume that δ>0. One writes
(13)δ=limn→∞dx2n,x2n+2≤limn→∞βdx2n−1,x2n+1dx2n−1,x2n+1<limn→∞dx2n−2,x2n=δ,so δ=0. Hence, dx2n,x2n+2⟶0. Similarly, we have dx2n−1,x2n+1⟶0. Also, x2n is a Cauchy sequence. Assume that x2n is not Cauchy. Then,
(14)limsupn,m→∞dx2n,x2m>0.
By using the triangular inequality,(15)dx2n,x2m≤dx2n,x2n+2+dx2n+2,x2m+2+dx2m,x2m+2.
Hence, we have(16)dx2n,x2m≤dx2n,x2n+2+βdx2n+1,x2m+1dx2n+1,x2m+1+dx2m,x2m+2≤dx2n,x2n+2+βdx2n+1,x2m+1βdx2n,x2mdx2n,x2m+dx2m,x2m+2,which gives us
(17)dx2n,x2m≤1−βdx2n+1,x2m+1βdx2n,x2m−1dx2n,x2n+2+dx2m,x2m+2.
Sincelimsupn,m→∞dx2n,x2m>0 and limsupn,m→∞dx2n,x2n+2=0, we get
(18)limsupn,m→∞1−βdx2n+1,x2m+1βdx2n,x2m−1=∞.
Observe thatlimsupn,m→∞βdx2n,x2m=1. Taking into account that β∈F, we get limsupn,m→∞dx2n,x2m=0 and this contradicts our assumption. Hence, x2n is a Cauchy sequence in M. Because x2n is Cauchy, X is complete, and M is closed; limn→∞x2n=x∈M. Now,
(19)dM,N≤dx,x2n−1≤dx,x2n+dx2n,x2n−1.
Thus, we havedx,x2n−1 converges to dM,N. Since
(20)dM,N≤dx2n,Γx≤βdx2n−1,xdx2n−1,x,therefore,
(21)dM,N≤limn→∞dx2n,Γx≤limn→∞βdx2n−1,xdx2n−1,x=dM,N.
Thus,dx,Γx=dM,N.Theorem 9.
LetM and N be two nonempty closed and convex subsets of a uniformly convex Banach space X such that diamM<dM,N. Suppose Γ:M∪N⟶M∪N is a cyclic Geraghty contraction map. Then, there exists a unique x∈M such that x−Γx=dM,N. Further, if x0∈M and xn+1=Γxn, then x2n converges to the best proximity point.Proof.
By Theorem8, PΓM,N≠0. Suppose x,y∈PΓM,N such that x≠y. Since x−Γx=dM,N and y−Γy=dM,N were necessarily uniformly convexity of X, Γ2x=x and Γ2y=y. Since x≠y, we have dM,N<Γx−y and so βΓx−yΓx−y<Γx−y. Therefore, x−Γy=Γ2x−Γy≤βΓx−yΓx−y<Γx−y. Similarly, Γx−y<x−Γy; that is, it is a contradiction. Therefore, x=y.Example 1.
LetM and N be subsets of ℝ2 defined by
(22)M=x,0:x≥1,N=0,y:y≥1.Suppose(23)Γx,y=ln1+y,ln1+x(24)βΓ=1+ln1+t,0≤t<dM,N,tdM,N,t>dM,N.Here,dM,N=2. For x,0,y,0∈M, we have
(25)Γx,0−Γy,0=0,ln1+x−ln1+y=ln1+x1+y≤1+ln1+x−y=βx,0−y,0x,0−y,0.Forx,0∈M and 0,y∈N, we have
(26)Γx,0−Γ0,y=ln1+x,ln1+y=ln1+x2+ln1+y2≤x2+y2≤x,0−0,ydM,Nx,0−0,y=βx,0−0,yx,0−0,y.Then,Γ is a cyclic Geraghty contraction on M∪N. Also, 0,1−Γ1,0=2=dM,N.


## 3. A Best Proximity Point Result for Caristi-Type Mappings

Recently, Du [14] established a direct proof of Caristi’s fixed point theorem without using Zorn’s lemma. In this section, we introduce a generalization of Caristi’s fixed point theorem and provide a proof without using Zorn’s lemma. We start with the following definition.Definition 10.
LetM and N be nonempty subsets of a metric space X,d. A map Γ:M⟶N is called a semicontraction if for all u∈Mandξ∈N, we have dΓu,ξ≤du,ξ.Our related result is as follows.Theorem 11.
LetX,d be a complete metric space and M,N be a pair of nonempty closed subsets of X such that M is boundedly compact. Also, let Γ:M⟶Na semicontraction map and is a bounded below and lower semicontinuous function χ:M⟶−∞,∞. Assume that for all u∈M with infx∈Mχx<χu, there is v∈M so that
(27)dΓu,v>dM,N,dΓu,v≤χu−χv.
Then, there isu0∈M so that χu0=infx∈Mχx.Proof.
Assume thatinfx∈Mχx<χy for every y∈M. Given μ0∈M. Then, infx∈Mχx<χμ0. We have Γμ0∈N. Therefore, there is μ1∈M such that
(28)dμ1,Γμ0≤χμ1−χμ0.
Let us define inductively a sequenceμn⊆Sn, where
(29)Sn≔w∈M∣dΓμn−1,w≤χμn−1−χw,so that
(30)χμn<infw∈Snχτ+1n.
Therefore,(31)dΓμn−1,μn≤χμn−1−χμn.
Sinceμn⊆M and M is boundedly compact, μn has a convergent subsequence to x∈M. Suppose μnk⟶x, dΓμnk,x≤χμnk−χx. Since Γx∈N, there is z∈M so that dΓx,z>dM,N and dΓx,z≤χx−χz. Therefore,
(32)χz≤χx−dΓx,z≤χx−dΓx,z+χμnk−χx−dΓμnk,x=χμnk−dΓx,z+dΓμnk,x≤χμnk−dΓx,z+dΓμnk,Γx≤χμnk−dΓμnk,z
We find thatz∈Snk. By (30), we get
(33)χμnk−1nk<infw∈Snkχτ≤χz.
Thus,(34)χz<χx≤limk→∞χμnk≤χz.
It is a contradiction, so there isu0∈M such that χu0=infx∈Mχx.Definition 12.
LetM and N be nonempty subsets of a metric space X. The mapping Γ:M∪N⟶M∪N is a cyclic semicontraction if
(i)
ΓM⊂N and ΓN⊂M(ii)
For allμ∈M and ξ∈N, we have dΓμ,ξ≤dμ,ξ
For example, letM and N be subsets of ℝ2 defined by
(35)M=x,0:x≥1,N=0,y:y≥1.
SupposeΓx,y=y,x, then Γ is cyclic semicontraction on M∪N and 0,1−Γ1,0=dM,N.Theorem 13.
LetM,N be a pair of nonempty closed subsets of X such that M is boundedly compact. Suppose that Γ:M∪N⟶M∪N is a cyclic semicontraction map and there exists a bounded below and lower semicontinuous function χ:M⟶−∞,∞ so that
(36)dμ,Γμ≤χμ−χΓμfor all μ∈M∪N. Then, Γ admits a best proximity point.Proof.
The functionχ is proper, so there is ξ∈M with χξ<∞. Put
(37)H=μ∈M∣dΓξ,μ≤χξ−χμ.
Letμ∈H. Since dμ,Γμ≤χμ−χΓμ, we have χΓμ≤χμ. Thus,
(38)χΓ2μ≤χΓμ−dΓμ,Γ2μ≤χΓμ−dΓμ,Γ2μ+χξ−χμ−dΓξ,μ=χξ−dΓμ,Γ2μ+dΓξ,μ=χξ−dΓμ,Γ2μ+dΓξ,Γμ,bydefinitionofacyclicsemicontraction≤χξ−dΓ2μ,Γξ.
Hence,(39)dΓ2μ,Γξ≤χξ−χΓ2μ,and so Γ2μ∈C. Assume that dμ,Γμ>dM,N for all μ∈M. Then, for each μ∈H, there is τ∈H so that
(40)μ≠τ,dΓμ,τ≤χμ−χτ.
By Theorem11, there is μ0∈C so that χμ0=infμ∈Hχμ. Thus, for above μ0∈H, we obtain
(41)0≤dM,N<dΓμ,Γ2μ≤χμ−χΓ2μ≤χΓ2μ−χΓ2μ=0,a contradiction and so Γ has a best proximity point.From now on,M and N are nonempty subsets of a Banach space X. For x0,y0∈M×N, the inward sets of x0,y0 relative to M×N are as follows:
(42)INx0=ty+1−tx0,y∈N,t≥0,IMy0=Γx+1−ty0,x∈M,t≥0.For example, letM and N be subsets of R defined by
(43)M=1,2,N=−2,−1.Then, forx,y∈M×N, we have
(44)INx=−∞,x,IMy=y,+∞.We now define cyclic weakly inward mappings.Definition 14.
GivenΓ:M∪N⟶X . Such Γ is said to be
(i)
Cyclic inward ifΓx,Γy∈INx×IMx for all x,y∈M×N(ii)
Cyclic weakly inward ifΓx,Γy∈INx¯×IMx¯ for all x,y∈M×N(iii)
Cyclic weakly inward contraction if it is cyclic weakly inward anddΓx,Γy≤kdx,y for all x,y∈M×N, where k∈0,1.Theorem 15.
LetM and N be closed and convex. Then Γ:M∪N⟶X is cyclic weakly inward if
(45)limh→0+d1−hϑ+hΓϑ,Mh=dM,N,ϑ∈N,(46)limh→0+d1−hϑ+hΓϑ,Nh=dM,N,ϑ∈M.Proof.
Suppose that ((45), (46)) hold. Fix ϑ∈M, given ε>0. Choose λ∈0,1 and y∈N so that
(47)1−λx+λΓϑ−y≤dM,N+λε.
Thus,(48)Γϑ−1−λ−1ϑ+λ−1y≤λ−1dM,N+ε.
Consequently,Γϑ∈INϑ¯. Similarly, for y∈N, Γy∈IMy¯.
Conversely, assume thatΓ is cyclic weakly inward; that is, Γϑ,Γy∈INϑ¯×IMϑ¯ for all ϑ,y∈M×N. Given ε>0. There exists b∈INϑ so that
(49)b−Γϑ≤ε.
SinceM is convex, there is λ0>0 so that
(50)1−λϑ+λb∈M,0<λ≤λ0.
Therefore,(51)d1−λx+λΓϑ,Mλ≤1−λϑ+λΓϑ−1−λϑ+λbλ≤ε.
The condition (45) holds. Similarly, the condition (46) is satisfied.Theorem 16.
LetM and N be closed and convex. Let Γ:M∪N⟶X be a cyclic weakly inward contraction mapping. If Mis boundedly compact, then Γ has a unique best proximity point in M.Proof.
Letk0<k<1 denote a Lipschitz constant of Γ. Choose ε>0 so that k<1−ε/1+ε. By Theorem 15, (45) is verified. Let ϑ∈M with −Γϑ>dM,N, then there is λ∈0,1 so that
(52)d1−λϑ+λΓϑ,N<λεϑ−Γϑ.
By the definition of a dance, there isb∈N so that
(53)1−λϑ+λΓϑ−b<λεϑ−Γϑ.
Hence,(54)λεϑ−Γϑ>ϑ−b−λϑ−Γϑ≥ϑ−b−λϑ−Γϑ,and so
(55)b−Γb≤b−1−λϑ+λΓϑ+1−λϑ+λΓϑ−ΓϑΓϑ−Γb≤λεϑ−Γϑ+1−λϑ−Γϑ+kϑ−b=ϑ−Γϑ+ε−1λϑ−Γϑ+1−ε1+εϑ−b−1−ε1+ε−kϑ−b<ϑ−Γϑ−1−ε1+ε−kϑ−b.
If for allϑ∈M, ϑ−Γϑ>dM,N, then denote b∈N as above by fϑ, where f:M⟶N is a mapping. Put
(56)χϑ=1−ε1+ε−k−1ϑ−Γϑ.
Note thatχ:M∪N⟶R+ is continuous and the following
(57)ϑ−fϑ<χϑ−χfϑholds. Due to Theorem 13, f admits a best proximity point, which contradicts (57).



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