# The Existence Theorems of an Optimal Approximate Solution for Generalized Proximal Contraction Mappings

Authors: Wutiphol Sintunavarat; Poom Kumam
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Recently, Basha (2011) established the best proximity point theorems for
proximal contractions of the first and second kinds which are extension of Banach's contraction
principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the
notions of proximal contractions of the first and second kinds which are more general than the
notion of self-contractions, establish the existence of an optimal approximate solution theorems
for these non-self-mappings, and also give examples to validate our main results.


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## Body

## 1. Introduction

Since Banach’s contraction principle [1] first appeared, several authors have generalized this principle in different directions. However, they have shown the existence of a fixed point for self-mappings. One of the most interesting results on Banach’s contraction principle is the case of non-self-mappings. In fact, for any nonempty closed subsets A and B of a complete metric space (X,d), a contractive non-self-mapping T:A→B does not necessarily have a fixed point Tx=x. In this case, a best proximity point, that is, a point x∈A for which d(x,Tx)=d(A,B):=inf{d(x,y):x∈Ay∈B} represents an optimal approximate solution to the equation Tx=x. It is well known that a best proximity point reduces to a fixed point if the underlying mapping is assumed to be a self-mapping. Consequently, best proximity point theorems are improvement of Banach’s contraction principle in case of non-self-mappings.A classical best approximation theorem was introduced by Fan [2]. Afterward, several authors including Prolla [3], Reich [4], and Sehgal and Singh [5, 6] have derived extensions of Fan’s Theorem in many directions. Other works of the existence of a best proximity point for contractive mappings can be found in [7–13]. On the other hand, many best proximity point theorems for set-valued mappings have been established in [14–19]. In particular, Eldred et al. [20] have obtained best proximity point theorems for relatively nonexpansive mappings.Recently, Basha [21] gave necessary and sufficient conditions to claim the existence of best proximity point for proximal contraction of first and second kinds which are non-self-mapping analogues of contraction self-mappings, and they also established some best proximity theorems. Afterward, several mathematicians extended and improved these results in many ways (see in [22–25]).The purpose of this paper is to extend and generalize the class of proximal contraction of first and second kinds which are different from another type in the literature. For such mappings, we seek the necessary condition for these classes to have best proximity points and also give some examples to illustrate our main results. The results of this paper are generalizations of results of Basha in [21] and some results of the fundamental metrical fixed point and best proximity point theorems in the literature.


## 2. Preliminaries

Throughout this paper, suppose thatA and B are nonempty subsets of a metric space (X,d). We use the following notations:
(1)d(A,B):=inf{d(x,y):x∈Aandy∈B},A0:={x∈A:d(x,y)=d(A,B)forsomey∈B},B0:={y∈B:d(x,y)=d(A,B)forsomex∈A}.Remark 1.
It is easy to see thatA0 and B0 are nonempty whenever A∩B≠∅. Further, if A and B are closed subsets of a normed linear space such that d(A,B)>0, then A0⊆Bdr(A) and B0⊆Bdr(B), where Bdr(A) is a boundary of A.Definition 2 (see [21]).
A mappingT:A→B is called a proximal contraction of the first kind if there exists α∈[0,1) such that, for all a,b,x,y∈A,
(2)d(a,Tx)=d(A,B),d(b,Ty)=d(A,B)⟹d(a,b)≤αd(x,y).Remark 3.
IfT is self-mapping, then T is a proximal contraction of the first kind deduced to T which is a contraction mapping. But a non-self-proximal contraction is not necessarily a contraction.Definition 4 (see [21]).
A mappingT:A→B is said to be a proximal contraction of the second kind if there exists α∈[0,1) such that, for all a,b,x,y∈A,
(3)d(a,Tx)=d(A,B),d(b,Ty)=d(A,B)⟹d(Ta,Tb)≤αd(Tx,Ty).
The necessary condition for a self-mappingT to be a proximal contraction of the second kind is that
(4)d(TTx,TTy)≤αd(Tx,Ty)
for all x, y in the domain of T. Therefore, every contraction self-mapping is a proximal contraction of the second kind, but the converse is not true (see Example 5).Example 5.
Considerℝ endowed with the Euclidean metric. Let the self-mapping T:[0,1]→[0,1] be defined as follows:
(5)T(x)={0ifxisrational,1otherwise.
It is easy to prove that T is a proximal contraction of the second kind. However, T is not a contraction mapping.The above example also exhibits that a self-mapping, that is, a proximal contraction of the second kind, is not necessarily continuous.Definition 6.
LetS:A→B and T:B→A be mappings. The pair (S,T) is said to be(1)
a cyclic contractive pair ifd(A,B)<d(x,y)⇒d(Sx,Ty)<d(x,y) for all x∈A and y∈B;(2)
a cyclic expansive pair ifd(A,B)<d(x,y)⇒d(Sx,Ty)>d(x,y) for all x∈A and y∈B;(3)
a cyclic inequality pair ifd(A,B)<d(x,y)⇒d(Sx,Ty)≠d(x,y) for all x∈A and y∈B.Definition 7.
LetS:A→B and T:B→A be mappings. The pair (S,T) is said to satisfy min-max condition if, for all x∈A and y∈B,
(6)d(A,B)<d(x,y)⟹min(Sx,Ty)≠max(Sx,Ty),
where min(Sx,Ty) and max(Sx,Ty) are defined by
(7)min(Sx,Ty)=min{d(x,y),d(x,Sx),d(y,Ty),d(Sx,Ty),d(x,STy),d(y,TSx),d(Sx,TSx),d(Ty,STy),d(TSx,STy)},max(Sx,Ty)=max{d(x,y),d(x,Sx),d(y,Ty),d(x,Ty),d(y,Sx),d(Sx,Ty),d(x,TSx),d(y,STy),d(x,STy),d(y,TSx),d(Sx,TSx),d(Ty,STy),d(TSx,STy)}.
We observe that the cyclic contractive pairs, cyclic expansive pairs, and cyclic inequality pairs satisfy the min-max condition.Definition 8.
LetT:A→B a mapping and g:A→A be an isometry. The mapping T is said to preserve isometric distance with respect to g if
(8)d(Tgx,Tgy)=d(Tx,Ty)
for all x,y∈A.Definition 9.
A pointx∈A is said to be a best proximity point of a mapping T:A→B if it satisfies the condition that
(9)d(x,Tx)=d(A,B).
Observe that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.Definition 10.
A is said to be approximatively compact with respect to B if every sequence {xn} in A satisfies the condition that d(y,xn)→d(y,A) for some y∈B has a convergent subsequence.Remark 11.
Any nonempty subset of metric space(X,d) is approximatively compact with respect to itself.


## 3. Main Results

In this section, we introduce the notions of generalized proximal contraction mappings of the first and second kinds which are different from another type in the literature. We also give the existence theorems of an optimal approximate solution for these mappings.Definition 12.
LetA, B be nonempty subset of metric space (X,d),  T:A→B and 𝒦:A→[0,1). A mapping T is said to be a generalized proximal contraction of the first kind with respect to 𝒦 if
(10)d(a,Tx)=d(A,B),d(b,Ty)=d(A,B)⟹d(a,b)≤𝒦(x)d(x,y)
for all a,b,x,y∈A.Remark 13.
If we take𝒦(x)=α for all x∈A, where α∈[0,1), then a generalized proximal contraction of the first kind with respect to 𝒦 reduces to a proximal contraction of the first kind (Definition 2). In case of a self-mapping, it is apparent that the class of contraction mapping is contained in the class of generalized proximal contraction of the first kind with respect to 𝒦 mapping.Now, we give an example to claim that the class of proximal contraction mapping of the first kind is a proper subclass of the class of generalized proximal contractions of the first kind with respect to𝒦 mapping.Example 14.
Consider the metric spaceℝ2 with Euclidean metric. Let A={(0,y):-1<y<1} and B={(1,y):-1<y<1}. Define a mapping T:A→B as follows:
(11)T((0,y))=(1,y22)
for all (0,y)∈A.
It is easy to check that there is noα∈[0,1) satisfing
(12)d(a,Tx)=d(b,Ty)=d(A,B)⟹d(a,b)≤αd(x,y)
for all a,b,x,y∈A. Therefore, T is not a proximal contraction of the first kind.
Consider a function𝒦:A→[0,1) defined by
(13)𝒦((0,y))=|y|+12.
Next, we claim that T is a generalized proximal contraction of the first kind with respect to 𝒦.
If(0,y1),(0,y2)∈A such that
(14)d(a,T((0,y1)))=d(A,B)=1,d(b,T((0,y2)))=d(A,B)=1,
for all a,b∈A, then we have
(15)a=(0,y122),b=(0,y222).
Therefore, it follows that
(16)d(a,b)=d((0,y122),(0,y222))=|y122-y222|=(|y1+y2|2)|y1-y2|≤(|y1|+|y2|2)|y1-y2|≤(|y1|+12)|y1-y2|=𝒦((0,y1))d((0,y1),(0,y2)).
This implies that T is a generalized proximal contraction of the first kind with respect to 𝒦.Definition 15.
LetA, B be nonempty subset of metric space (X,d),  T:A→B and 𝒦:A→[0,1). A mapping T is said to be a generalized proximal contraction of the second kind with respect to 𝒦 if
(17)d(a,Tx)=d(A,B),d(b,Ty)=d(A,B)⟹d(a,b)≤𝒦(x)d(Tx,Ty)
for all a,b,x,y∈A.Clearly, a proximal contraction of the second kind (Definition4) is a generalized proximal contraction of the second kind.Next, we extend the results of Basha [21] and many results in the literature.Theorem 16.
Let(X,d) a complete metric space and A, B be nonempty closed subsets of X such that A0 and B0 are nonempty. Suppose that T:A→B, g:A→A, and 𝒦:A→[0,1) are mappings satisfying the following conditions:(a)
T is a continuous generalized proximal contraction of first kind with respect to 𝒦;(b)
T(A0)⊆B0 and A0⊆g(A0);(c)
g is an isometry;(d)
𝒦(x)≤𝒦(y), whenever d(gx,Ty)=d(A,B).
Then there exists a unique pointx∈A such that d(gx,Tx)=d(A,B).Proof.
Letx0 be a fixed element in A0. From T(A0)⊆B0 and A0⊆g(A0), it follows that there exists a point x1∈A0 such that
(18)d(gx1,Tx0)=d(A,B).
Again, since Tx1∈T(A0)⊆B0 and A0⊆g(A0), there exists a point x2∈A0 such that
(19)d(gx2,Tx1)=d(A,B).
Continuing this process, we can construct the sequence {xn} in A0 such that
(20)d(gxn,Txn-1)=d(A,B)
for all n∈ℕ. Since T is a generalized proximal contraction of the first kind with respect to 𝒦, it follows that
(21)d(gxn+1,gxn)≤𝒦(xn)d(xn,xn-1)
for all n∈ℕ. Also, since g is an isometry, we have
(22)d(xn+1,xn)≤𝒦(xn)d(xn,xn-1)
for all n∈ℕ. By using (20) and (d), we have
(23)d(xn+1,xn)≤𝒦(xn)d(xn,xn-1)≤𝒦(xn-1)d(xn,xn-1)≤𝒦(xn-2)d(xn,xn-1)⋮≤𝒦(x0)d(xn,xn-1)
for all n∈ℕ. By repeating (23), we get
(24)d(xn+1,xn)≤(𝒦(x0))nd(x1,x0)
for all n∈ℕ. Now, we let k:=𝒦(x0)∈[0,1). For positive integers m and n with n>m, it follows from (24) that
(25)d(xn,xm)≤d(xn,xn-1)+d(xn-1,xn-2)+⋯+d(xm+1,xm)≤kn-1d(x1,x0)+kn-2d(x1,x0)+⋯+kmd(x1,x0)≤(km1-k)d(x1,x0).
Since k∈[0,1), we have (km/(1-k))d(x1,x0)→0 as m→∞, which implies that {xn} is a Cauchy sequence in X. Since X is complete, it follows that the sequence {xn} converges to point x∈X. Since T and g are continuous, we get
(26)d(gx,Tx)=limn→∞d(gxn+1,Txn)=d(A,B).
Next, we suppose that x* is another point in X such that
(27)d(gx*,Tx*)=d(A,B).
Since T is a generalized proximal contraction of the first kind with respect to 𝒦, by using (26) and (27), we get
(28)d(gx,gx*)≤𝒦(x)d(x,x*).
Since g is an isometry, it follows that
(29)d(x,x*)≤𝒦(x)d(x,x*),
which implies that x=x*. This completes the proof.Now, we give an example to illustrate Theorem16.Example 17.
Consider the complete metric spaceℝ2 with Euclidean metric. Let A={(0,y):-1≤y≤1} and B={(1,y):-1≤y≤1}. Define two mappings T:A→B and g:A→A as follows:
(30)T((0,y))=(1,y24),g((0,y))=(0,-y)
for all (0,y)∈A. Then it is easy to see that d(A,B)=1, A0=A, B0=B, and the mapping g is an isometry.
Consider a function𝒦:A→[0,1) defined by
(31)𝒦((0,y))=|y|+14.
Next, we claim thatT is a generalized proximal contraction of the first kind with respect to 𝒦. If (0,y1),(0,y2)∈A such that
(32)d(a,T((0,y1)))=d(A,B)=1,d(b,T((0,y2)))=d(A,B)=1,
for all a,b∈A, then we have
(33)a=(0,y124),b=(0,y224).
Therefore, it follows that
(34)d(a,b)=d((0,y124),(0,y224))=|y124-y224|=(|y1+y2|4)|y1-y2|≤(|y1|+|y2|4)|y1-y2|≤(|y1|+14)|y1-y2|=𝒦((0,y1))d((0,y1),(0,y2)).
This implies that the non-self-mapping T is a generalized proximal contraction of the first kind with respect to 𝒦. It is easy to see that 𝒦(x)≤𝒦(y) whenever d(gx,Ty)=d(A,B). Moreover, since T is continuous and g is an isometry, all the conditions of Theorem 16 are satisfied, and so T has a unique element (0,0)∈A such that
(35)d(g((0,0)),T((0,0)))=d(A,B).Corollary 18 (see [21, Theorem 3.3]).
Let(X,d) be a complete metric space and A, B nonempty closed subsets of X such that A0 and B0 are nonempty. Suppose that T:A→B and g:A→A are mappings satisfying the following conditions:(a)
T is a continuous proximal contraction of the first kind;(b)
T(A0)⊆B0 and A0⊆g(A0);(c)
g is an isometry.
Then there exists a unique elementx∈A such that d(gx,Tx)=d(A,B).Proof.
Since a proximal contraction of the first kind is a special case of a generalized proximal contraction of the first kind, we can prove this result by applying Theorem16.In Theorem16, if g is the identity mapping, then it yields the following best proximity point theorem.Corollary 19.
Let(X,d) a complete metric space and A, B be nonempty closed subsets of X such that A0 and B0 are nonempty. Suppose that T:A→B and 𝒦:A→[0,1) are mappings satisfying the following conditions:(a)
T is a continuous generalized proximal contraction of first kind with respect to 𝒦;(b)
T(A0)⊆B0;(c)
𝒦(x)≤𝒦(y), whenever d(x,Ty)=d(A,B).
ThenT has a unique best proximity point in A.Corollary 20 (see [21, Corollary 3.4]).
Let(X,d) be a complete metric space and A, B nonempty closed subsets of X such that A0 and B0 are nonempty. Let T:A→B be a mapping satisfying the following conditions:(a)
T is a continuous proximal contraction of the first kind;(b)
T(A0)⊆B0.
ThenT has a unique best proximity point in A.Proof.
Since a proximal contraction of the first kind is a special case of a generalized proximal contraction of the first kind with respect to𝒦, we can prove this result by applying Corollary 19.Next, we prove the second main result for generalized proximal contraction of the second kind with respect to𝒦 mapping.Theorem 21.
Let(X,d) a complete metric space and A, B be nonempty closed subsets of X such that A is approximatively compact with respect to B. Suppose that A0 and B0 are nonempty and T:A→B, g:A→A, and 𝒦:A→[0,1) are mappings satisfying the following conditions:(a)
T is a continuous generalized proximal contraction of the second kind with respect to 𝒦;(b)
T(A0)⊆B0 and A0⊆g(A0);(c)
g is an isometry;(d)
T preserves isometric distance with respect to g;(e)
𝒦(x)≤𝒦(y), whenever d(gx,Ty)=d(A,B).
Then there exists a pointx∈A such that d(gx,Tx)=d(A,B). Moreover, if x* is another point in A for which d(gx*,Tx*)=d(A,B), then Tx=Tx*.Proof.
As in the proof of Theorem16, for fixed x0∈A0, we can define a sequence {xn} in A0 such that
(36)d(gxn,Txn-1)=d(A,B)
for all n∈ℕ. Since T is a generalized proximal contraction of the second kind with respect to 𝒦, it follows that
(37)d(Tgxn+1,Tgxn)≤𝒦(xn)d(Txn,Txn-1).
Since T preserves isometric distance with respect to g, we have
(38)d(Txn+1,Txn)≤𝒦(xn)d(Txn,Txn-1)
for all n∈ℕ. By using (36) and (e), we have
(39)d(Txn+1,Txn)≤𝒦(xn)d(Txn,Txn-1)≤𝒦(xn-1)d(Txn,Txn-1)≤𝒦(xn-2)d(Txn,Txn-1)⋮≤𝒦(x0)d(Txn,Txn-1)
for all n∈ℕ. By repeating (39), we get
(40)d(Txn+1,Txn)≤(𝒦(x0))nd(Tx1,Tx0)
for all n∈ℕ. Now, we let k:=𝒦(x0)∈[0,1). For positive integers m and n with n>m, it follows from (40) that
(41)d(Txn,Txm)≤d(Txn,Txn-1)+d(Txn-1,Txn-2)+⋯+d(Txm+1,Txm)≤kn-1d(Tx1,Tx0)+kn-2d(Tx1,Tx0)+⋯+kmd(Tx1,Tx0)≤(km1-k)d(Tx1,Tx0).
Since k∈[0,1), we have (km/(1-k))d(Tx1,Tx0)→0 as m→∞, which implies that {Txn} is a Cauchy sequence in B. By completeness of B⊆X, there exists a point y∈B such that Txn→y as n→∞. By (36) and the triangle inequality, we have
(42)d(y,A)≤d(y,gxn)≤d(y,Txn-1)+d(Txn-1,gxn)=d(y,Txn-1)+d(A,B)≤d(y,Txn-1)+d(y,A).
Letting n→∞ in (42), we get d(y,gxn)→d(y,A). Since A is approximatively compact with respect to B, it follows that {gxn} has a convergence subsequence {gxnk}; say gxnk→z∈A as k→∞. Thus we have
(43)d(z,y)=limk→∞d(gxnk,Txnk-1)=d(A,B),
which implies that z∈A0. Since A0⊆g(A0), we have z=gx for some x∈A0. Therefore, gxnk→gx as k→∞. Since g is an isometry, we get xnk→x as k→∞. By the continuity of T, we have Txnk→Tx as k→∞ and then y=Tx. From (43), we can conclude that
(44)d(gx,Tx)=d(A,B).
Next, we suppose thatx* is another point in X such that
(45)d(gx*,Tx*)=d(A,B).
Since T is a generalized proximal contraction of the second kind with respect to 𝒦, by the virtue of (44) and (45), we get
(46)d(Tgx,Tgx*)≤𝒦(x)d(Tx,Tx*).
Since T preserves isometric distance with respect to g, it follows that
(47)d(Tx,Tx*)≤𝒦(x)d(Tx,Tx*),
which implies that Tx=Tx*. This completes the proof.Corollary 22 (see [21, Theorem 3.1]).
Let(X,d) be a complete metric space and A, B nonempty closed subsets of X such that A is approximatively compact with respect to B. Suppose that A0 and B0 are nonempty and T:A→B and g:A→A are mappings satisfying the following conditions:(a)
T is a continuous proximal contraction of the second kind;(b)
T(A0)⊆B0 and A0⊆g(A0);(c)
g is an isometry;(d)
T preserves isometric distance with respect to g.
Then there exists a pointx∈A such that d(gx,Tx)=d(A,B). Moreover, if x* is another point in A for which d(gx*,Tx*)=d(A,B), then Tx=Tx*.Proof.
Since a proximal contraction of the second kind is a special case of a generalized proximal contraction of the second kind with respect to𝒦, we can prove this result by applying Theorem 21.Corollary 23.
Let(X,d) be a complete metric space and A, B nonempty closed subsets of X such that A is approximatively compact with respect to B. Suppose that A0 and B0 are nonempty and T:A→B and 𝒦:A→[0,1) are mappings satisfying the following conditions:(a)
T is a continuous generalized proximal contraction of the second kind with respect to 𝒦;(b)
T(A0)⊆B0;(c)
𝒦(x)≤𝒦(y), whenever d(x,Ty)=d(A,B).
ThenT has a best proximity point. Moreover, if x* is another best proximity point of T, then Tx=Tx*.Proof.
We can prove this result by applying Theorem21 with g=IA, where IA is an identity mapping on A.Corollary 24 (see [21, Corollary 3.2]).
Let(X,d) be a complete metric space and A, B nonempty closed subsets of X such that A is approximatively compact with respect to B. Suppose that A0 and B0 are nonempty and T:A→B is mapping satisfying the following conditions:(a)
T is a continuous generalized proximal contraction of the second kind;(b)
T(A0)⊆B0.
ThenT has a best proximity point. Moreover, if x* is another best proximity point of T, then Tx=Tx*.Proof.
Since a proximal contraction of the second kind is a special case of a generalized proximal contraction of the second kind, we can prove this result by applying Corollary23.Here, we give the last result in this work.Theorem 25.
Let(X,d) be a complete metric space, A and B nonempty closed subsets of X, and 𝒦:A∪B→[0,1). Suppose that S:A→B is a mapping satisfying
(48)d(Sx,Sy)≤𝒦(x)d(x,y)
for all x,y∈A. Then the following holds.(A)
There exists a nonexpansive mappingT:B→A such that (S,T) satisfies the min-max condition whenever S has a best proximity point.(B)
If there exists a nonexpansive mappingT:B→A such that (S,T) satisfies the min-max condition and 𝒦(Sx)≤𝒦(x) and 𝒦(Tx)≤𝒦(x) for all x∈A, then S has a best proximity point.(C)
For two any best proximity pointsz and z* of   S, we have
(49)d(z,z*)≤(21-𝒦(z))d(A,B).Proof.
(A) Let S has a best proximity point a∈A. We define a mapping T:B→A by Ty=a for all y∈B. Clearly, T is a nonexpansive mapping. It follows from the definition of T that
(50)d(Ty,STy)=d(a,Sa)=d(A,B)
for all y∈B. Thus we can conclude that min(Sx,Ty)=d(A,B) for all x∈A and y∈B.
Next, we show that(S,T) satisfies the min-max condition. Suppose that x∈A and y∈B such that d(A,B)<d(x,y). Then we have
(51)min(Sx,Ty)=d(A,B)<d(x,y)≤max(Sx,Ty),
which implies that the pair (S,T) satisfies the min-max condition. Therefore, we can find a nonexpansive mapping T:B→A such that (S,T) satisfies the min-max condition.
(B) Fix x0∈A and define a sequence {xn} in A∪B by
(52)x2n-1=Sx2n-2,x2n=Tx2n-1
for all n∈ℕ. Since T is nonexpansive, it follows from (48) that
(53)d(x2n-2,x2n)=d(Tx2n-3,Tx2n-1)≤d(x2n-3,x2n-1)=d(Sx2n-4,Sx2n-2)≤𝒦(x2n-4)d(x2n-4,x2n-2)=𝒦(Tx2n-5)d(x2n-4,x2n-2)≤𝒦(x2n-5)d(x2n-4,x2n-2)=𝒦(Sx2n-6)d(x2n-4,x2n-2)≤𝒦(x2n-6)d(x2n-4,x2n-2)⋮≤𝒦(x0)d(x2n-4,x2n-2)
for all n∈ℕ. By repeating the above argument, we have
(54)d(x2n-2,x2n)≤(𝒦(x0))n-1d(x0,x2)
for all n∈ℕ, which implies that the sequence {x2n} is a Cauchy sequence in X. A similar argument asserts that the sequence {x2n-1} is a Cauchy sequence in X. By the completeness of X, we conclude that {x2n} converges to a point a∈A and {x2n-1} converges to a point b∈B. Since S is continuous, {Sx2n} converges to Sa, which implies that {x2n-1} converges to Sa. Thus Sa=b.
Similarly, it is easy to check thatTb=a. Therefore, we have
(55)TSa=Tb=a,STb=Sa=b.
Now, we can conclude that
(56)min(Sa,Tb)=d(a,b)=max(Sa,Tb).
By the virtue of the min-max condition of (S,T), we get d(a,b)≤d(A,B). Since d(A,B)≤d(a,b), we have d(a,b)=d(A,B). Therefore, we have
(57)d(a,Sa)=d(a,b)=d(A,B),
which implies that S has a best proximity point in A.
(C) Let z and z* be best proximity points of S. Then d(z,Sz)=d(A,B) and d(z*,Sz*)=d(A,B). Using the triangle inequality and (48), we have
(58)d(z,z*)≤d(z,Sz)+d(Sz,Sz*)+d(Sz*,z*)≤𝒦(z)d(z,z*)+2d(A,B).
This implies that d(z,z*)≤(2/(1-𝒦(z)))d(A,B). This completes the proof.Corollary 26 (see [21, Theorem 3.6]).
Let(X,d) be a complete metric space and A and B nonempty closed subsets of X. Suppose that S:A→B is a contraction mapping. Then S has a best proximity point if and only if there exists a nonexpansive mapping T:B→A such that (S,T) satisfies the min-max condition.
Moreover,d(z,z*)≤(2/(1-α))d(A,B) for some α∈[0,1) and any two best proximity points z and z* of S.Proof.
SinceS is a contraction mapping, we have d(Sx,Sy)≤αd(x,y) for some α∈[0,1) and all x,y∈A. Now, we can prove this result by applying Theorem 25 with a function 𝒦:A∪B→[0,1) defined by 𝒦(x)=α for all x∈A∪B.



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*Source: 375604-2013-09-01.xml*