# Best Proximity Points for Some Classes of Proximal Contractions

Authors: Maryam A. Alghamdi; Naseer Shahzad; Francesca Vetro
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Given a self-mappingg:A→A and a non-self-mapping T:A→B, the
aim of this work is to provide sufficient conditions for the existence of a
unique point x∈A, called g-best proximity point, which satisfies dgx,Tx=dA,B. In so doing, we provide a useful answer for the
resolution of the nonlinear programming problem of globally minimizing the
real valued function x→dgx,Tx, thereby getting an optimal
approximate solution to the equation Tx=gx. An iterative algorithm is
also presented to compute a solution of such problems. Our results
generalize a result due to Rhoades (2001) and hence such
results provide an extension of Banach's contraction principle to the case
of non-self-mappings.


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## Body

## 1. Introduction

A fundamental result in the fixed point theory is the Banach contraction principle, which has various nontrivial implications in many branches of pure and applied sciences.LetA and B be nonempty subsets of a metric space (X,d). We say that a non-self-mapping T:A→B is a contraction if there exists k∈[0,1) such that, for all x,y∈X,
(1)d(Tx,Ty)≤kd(x,y).The Banach contraction principle asserts that if a self-mappingT:X→X is a contraction and (X,d) is complete, then T has a unique fixed point x∈X. This result was extended to other important classes of mappings and has numerous applications. For some important and interesting generalizations of Banach contraction principle, one can refer to [1, 2]. The following notion of weakly contractive self-mapping was introduced by Alber and Guerre-Delabriere in [3].Definition 1 (see [3]).
Let(X,d) be a metric space and let A be a nonempty subset of X. A self-mapping T:A→A is said to be weakly contractive if
(2)d(Tx,Ty)≤d(x,y)-ψ(d(x,y)),
for all x,y∈A, where ψ:[0,+∞)→[0,+∞) is a continuous and nondecreasing function such that ψ is positive on (0,+∞), ψ(0)=0 and limt→+∞ψ(t)=+∞. If A is bounded, then the infinity condition can be omitted.Since all contractions are weakly contractive with the functionψ(t)=(1-k)t, the above theorem extends Banach contraction principle. In fact, the class of weakly contractive mappings lies between the classes of mappings called contraction ones and contractive ones (d(Tx,Ty)<d(x,y), for all x,y∈X with x≠y).Generally, the solution of the equationTx=x, where T:A→X is a non-self-mapping, is called a fixed point of T. Hence, the condition T(A)∩A≠∅ is necessary for the existence of a fixed point of T. Clearly, when T(A)∩A=∅, we have d(x,Tx)>0, for all x∈A. In such a situation it is natural to search for a point x∈A such that x the is closest to Tx in some sense. The following well-known best approximation theorem, due to Fan [4], explores the existence of an approximate solution to the equation Tx=x.Theorem 2 (see [4]).
LetA be a nonempty compact convex subset of a normed linear space X and let T:A→X be a continuous mapping. Then there exists x∈A such that ∥x-Tx∥=d(Tx,A).The pointx∈A in Theorem 2 is called a best approximant of T in A. Again, let A,B be nonempty subsets of a metric space (X,d) and let T:A→B be a non-self-mapping. A point x0∈A is called a best proximity point of T if d(x0,Tx0)=d(A,B). Some interesting results in approximation theory can be found in [4–23].The aim of this paper is to prove some best proximity point theorems for proximal contractions which are extensions of Banach contraction principle to the case of non-self-mappings. Precisely, given a self-mappingg:A→A and a non-self-mapping T:A→B, this work focuses on g-best proximity point theorems for some classes of proximal contractions and a new family of mappings known as g-weak contractions. In fact, we provide sufficient conditions for the existence of a unique point x∈A, called g-best proximity point, which satisfies the condition d(gx,Tx)=d(A,B). Further, an iterative algorithm is furnished to determine an optimal approximate solution in the guise of a g-best proximity point. As a consequence, one can compute an optimal approximate solution to some coincidence point equations.


## 2. Preliminaries

Letℝ+ denote the set of all positive real numbers and ℕ denote the set of all positive integers. Let A,B be two nonempty subsets of a metric space (X,d). Let us fix the following notation which will be needed throughout this paper:
(3)A0={x∈A:d(x,y)=d(A,B)forsomey∈B},B0={y∈B:d(x,y)=d(A,B)forsomex∈A},
where d(A,B)=inf{d(x,y):x∈Aandy∈B}. In [11], the authors discussed sufficient conditions which guarantee the nonemptiness of A0 and B0. Also, in [20], the authors proved that A0 is contained in the boundary of A.We denote byΨ the set of nondecreasing functions ψ:[0,+∞)→[0,+∞) satisfying the following condition:(ψ1)
limn→+∞ψn(t)=0, for all t>0, where ψn is the nth iterate of ψ.Note that ifψ∈Ψ, then the following conditions hold:(ψ2)
ψ(t)<t, for all t>0; ψ(0)=0;  ψ is continuous at t=0.We denote byΦ the set of nondecreasing functions ψ:[0,+∞)→[0,+∞) such that ψ(t)=0 if and only if t=0 and with Φc={ψ∈Φ:ψiscontinuousatt=0}.Definition 3 (see [21]).
LetA and B be two nonempty subsets of a metric space (X,d). A non-self-mapping T:A→B is said to be a proximal ψ-contraction of the first kind if
(4)d(u,Tx)=d(A,B)=d(v,Ty)⟹d(u,v)≤ψ(d(x,y)),
for all u,v,x,y∈A, where ψ∈Ψ. If ψ(t)=αt for some α∈[0,1), then T is said to be a proximal contraction of the first kind.Definition 4 (see [21]).
LetA and B be two nonempty subsets of a metric space (X,d). A non-self-mapping T:A→B is said to be a proximal ψ-contraction of the second kind if
(5)d(u,Tx)=d(A,B)=d(v,Ty)⟹d(Tu,Tv)≤ψ(d(Tx,Ty)),
for all u,v,x,y∈A, where ψ∈Ψ. If ψ(t)=αt for some α∈[0,1), then T is said to be a proximal contraction of the second kind.Definition 5 (see [14]).
LetA and B be two nonempty subsets of a metric space (X,d). A non-self-mapping T:A→B is said to be a weak proximal ψ-contraction of the first kind if
(6)d(u,Tx)=d(A,B)=d(v,Ty)⟹d(u,v)≤d(x,y)-ψ(d(x,y)),
for all u,v,x,y∈A, where ψ∈Φ.Definition 6 (see [14]).
LetA and B be two nonempty subsets of a metric space (X,d). A non-self-mapping T:A→B is said to be a weak proximal ψ-contraction of the second kind if
(7)d(u,Tx)=d(A,B)=d(v,Ty)⟹d(Tu,Tv)≤d(Tx,Ty)-ψ(d(Tx,Ty)),
for all u,v,x,y∈A, where ψ∈Φ.An example of a non-self-mappingT that is weak proximal ψ-contraction of the first and second kinds can be found in [14].The following result is a best proximity point theorem for weak proximalψ-contraction of the first and second kinds.Theorem 7 (see [14, Theorem 3.1]).
LetA and B be closed subsets of a complete metric space (X,d) such that A0 and B0 are nonvoid. Suppose that the mappings g:A→A and T:A→B satisfy the following conditions:(a)
T is a weak proximal ψ-contraction of the first and second kinds;(b)
g is an isometry;(c)
T(A0)⊆B0;(d)
A0⊆g(A0);(e)
T preserves the isometric distance with respect to g.
Then, there exists a unique elementx* in A such that d(gx*,Tx*)=d(A,B). Further, for any fixed element x0 in A0, the iterative sequence {xn}, defined by d(gxn+1,Txn)=d(A,B) for every n∈ℕ∪{0}, converges to the element x*.Note that in Theorem7, Sadiq Basha assumes that the function ψ∈Φ is continuous such that limt→+∞ψ(t)=+∞.Let us define the notion of non-self-g-weakly contractive mappings as follows.Definition 8.
Let(X,d) be a metric space, let A,B be two nonempty subsets of X, and let g:A→A. A non-self-mapping T:A→B is said to be a g-weakly contractive mapping if there exists ψ∈Φc such that
(8)d(Tx,Ty)≤d(gx,gy)-ψ(d(gx,gy)),
for all x,y∈A.Note that(9)d(Tx,Ty)≤d(gx,gy)-ψ(d(gx,gy))<d(gx,gy)
if x,y∈A with gx≠gy; that is, T is a g-contractive mapping.Sankar Raj, in [22], introduced the notion called P-property, which was used to prove an extended version of Banach contraction principle.Definition 9.
Let(A,B) be a pair of nonempty subsets of a metric space (X,d) with A0≠∅.  (i)
The pair(A,B) is said to have the P-property if and only if d(x1,y1)=d(A,B)=d(x2,y2) implies d(x1,x2)=d(y1,y2), where x1,x2∈A0 and y1,y2∈B0 (see [22]).(ii)
The pair(A,B) is said to have the weak P-property if and only if d(x1,y1)=d(A,B)=d(x2,y2) implies d(x1,x2)≤d(y1,y2), where x1,x2∈A0 and y1,y2∈B0 (see [24]).
It is easy to see that, for any nonempty subsetA of X, the pair (A,A) has the P-property.Definition 10.
LetA and B be two nonempty subsets of a metric space (X,d). Let g:A→A be a self-mapping and T:A→B a non-self-mapping. Then(i)
g∈𝒢A if g is continuous and d(x,y)≤d(gx,gy), for all x,y∈A;(ii)
T∈𝒯g if d(Tx,Ty)≤d(Tgx,Tgy) for all x,y∈A;(iii)
T is said to preserve (isometric) distance with respect to g if d(Tgx,Tgy)=d(Tx,Ty), for every x,y∈A (see [9]).


## 3. Best Proximity Point Theorems for Proximal Contractions

In this section, we establish some results of best proximity point for proximalψ-contractions and weak proximal ψ-contractions.Theorem 11.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that A0 is nonempty and closed. Assume also that the mappings T:A→B and g:A→A  satisfy the following conditions:(a)
T is a proximal ψ-contraction of the first kind;(b)
g∈𝒢A0;(c)
T(A0)⊆B0;(d)
A0⊆g(A0).
Then there exists a unique pointx∈A0 such that d(gx,Tx)=d(A,B). Moreover, for every x0∈A0 there exists a sequence {xn}⊆A such that d(gxn+1,Txn)=d(A,B) for every n∈ℕ∪{0} and xn→x.Proof.
Letx0∈A0. Since T(A0)⊆B0 and A0⊆g(A0), there exists x1∈A0 such that
(10)d(gx1,Tx0)=d(A,B).
Again, forx1∈A0, there exists x2∈A0 such that
(11)d(gx2,Tx1)=d(A,B).
By repeating this process, forxn∈A0, we can find xn+1∈A0 such that
(12)d(gxn+1,Txn)=d(A,B),∀n∈ℕ.
SinceT is a proximal ψ-contraction of the first kind and g∈𝒢A0, we have
(13)d(xn+1,xn)≤d(gxn+1,gxn)≤ψ(d(xn,xn-1))
for every n∈ℕ∪{0}. Since ψ is nondecreasing, we get by induction that
(14)d(xn+1,xn)≤ψn(d(x1,x0)).
By the definition ofψ, letting n→+∞, we obtain that
(15)limn→+∞d(xn+1,xn)=0.
We now prove that{xn} is a Cauchy sequence. Given that ε>0 there exists n(ε)∈ℕ such that
(16)d(xn,xn+1)<ε-ψ(ε),∀n≥n(ε).
Now, fixm≥n(ε) and we prove that
(17)d(xm,xn+1)<ε,∀n≥m.
Note that (17) holds if n=m, by (16). Assume that (17) holds for some n≥m. Since T is a proximal ψ-contraction of the first kind,
(18)d(xm,xn+2)≤d(xm,xm+1)+d(xm+1,xn+2)≤d(xm,xm+1)+d(gxm+1,gxn+2)≤d(xm,xm+1)+ψ(d(xm,xn+1))<ε-ψ(ε)+ψ(ε)=ε.
This implies that (17) holds, for all n≥m, and hence
(19)limm→+∞d(xm,xn+1)=0.
That is, {xn} is a Cauchy sequence. By the completeness of X and since A0 is closed, we have xn→x∈A0. Moreover, by the continuity of g, we have gxn→gx and thus gx∈A0, since gxn∈A0, for all n∈ℕ. On the other hand, since x∈A0 and T(A0)⊆B0, there exists z∈A such that
(20)d(z,Tx)=d(A,B).
Clearly z∈A0. Again, since T is a proximal ψ-contraction of the first kind, we get
(21)d(z,gxn+1)≤ψ(d(x,xn))≤d(x,xn),
for all n∈ℕ. Letting n→+∞, we obtain that d(z,gxn+1)→0 and then z=gx. This implies that
(22)d(gx,Tx)=d(A,B).
To prove the uniqueness, letx* be another point in A0 such that
(23)d(gx*,Tx*)=d(A,B).
Ifx≠x*, since g∈𝒢A0 and T is a proximal ψ-contraction of the first kind, we get
(24)d(x,x*)≤d(gx,gx*)≤ψ(d(x,x*))<d(x,x*),
which is a contradiction; thus we have x=x*.Remark 12.
If in Theorem11 we assume g∈𝒢A, then we get that there exists a unique x∈A such that d(gx,Tx)=d(A,B).From Theorem11 and the above remark, we obtain the following corollary.Corollary 13 (see [9, Theorem 3.1]).
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that A0 is nonempty and closed. Assume also that the mappings T:A→B and g:A→A  satisfy the following conditions:(a)
T is a proximal contraction of the first kind;(b)
g is an isometry;(c)
T(A0)⊆B0;(d)
A0⊆g(A0).
Then there exists a unique pointx∈A such that d(gx,Tx)=d(A,B). Moreover, for every x0∈A0 there exists a sequence {xn}⊆A such that d(gxn+1,Txn)=d(A,B) for every n∈ℕ∪{0} and xn→x.If in Theorem11 the mapping g is the identity on A, then we get the following corollary.Corollary 14.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that A0 is nonempty and closed. Let T:A→B satisfy the following conditions:(a)
T is a proximal ψ-contraction of the first kind;(b)
T(A0)⊆B0.
Then there exists a unique pointx∈A such that d(x,Tx)=d(A,B). Moreover, for every x0∈A0 there exists a sequence {xn}⊆A such that d(xn+1,Txn)=d(A,B) for every n∈ℕ∪{0} and xn→x.The following theorem is our main result for proximalψ-contractions of the second kind.Theorem 15.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that T(A0) is nonempty and closed. Assume also that the mappings T:A→B and g:A→A  satisfy the following conditions:(a)
T is a proximal ψ-contraction of the second kind;(b)
T∈𝒯g;(c)
T(A0)⊆B0;(d)
A0⊆g(A0).
Then there exists a pointx∈A such that d(gx,Tx)=d(A,B). Moreover, if T is injective, then the point x such that d(gx,Tx)=d(A,B) is unique.Proof.
Similar to the proof of Theorem11, we can find a sequence {xn}⊆A0 such that
(25)d(gxn+1,Txn)=d(A,B),∀n∈ℕ∪{0}.
SinceT is a proximal ψ-contraction of the second kind, we have
(26)d(Tgxn+1,Tgxn)≤ψ(d(Txn,Txn-1))
for every n∈ℕ. Since T∈𝒯g, we get
(27)d(Txn+1,Txn)≤ψ(d(Txn,Txn-1))
for every n∈ℕ. Since ψ is nondecreasing, we get by induction that
(28)d(Txn+1,Txn)≤ψn(d(Tx1,Tx0)).
By definition ofψ, letting n→+∞, we obtain that
(29)limn→+∞d(Txn+1,Txn)=0.
Similar to the proof of Theorem11, we prove that {Txn} is a Cauchy sequence. By the completeness of X and since T(A0) is closed, we have Txn→Tu∈B0. Moreover, there exists z∈A0 such that
(30)d(z,Tu)=d(A,B).
SinceA0⊆g(A0), we obtain that z=gx for some x∈A0, and then
(31)d(gx,Tu)=d(A,B).
Again, sinceT is a proximal ψ-contraction of the second kind, we get
(32)d(Tx,Txn+1)≤d(Tgx,Tgxn+1)≤ψ(d(Tu,Txn))≤d(Tu,Txn).
Lettingn→+∞, we obtain that d(Tx,Txn+1)→0 and hence Tx=Tu. This implies that
(33)d(gx,Tx)=d(A,B).
To prove the uniqueness, letx* be another point in A such that
(34)d(gx*,Tx*)=d(A,B).
Ifx≠x*, since T∈𝒯g is injective, we deduce
(35)d(Tx,Tx*)≤d(Tgx,Tgx*)≤ψ(d(Tx,Tx*))<d(Tx,Tx*),
which is a contradiction; thus we have Tx=Tx* and hence x=x*.From Theorem15, we deduce the following corollary.Corollary 16 (see [15, Theorem 3.2]).
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that T(A0) is nonempty and closed. Assume also that the mappings T:A→B and g:A→A  satisfy the following conditions:(a)
T is a proximal contraction of the second kind;(b)
g is an isometry;(c)
T preserves isometric distance with respect to g;(d)
T(A0)⊆B0;(e)
A0⊆g(A0).
Then there exists a pointx∈A such that d(gx,Tx)=d(A,B). Moreover, if z∈A is another point for which d(gz,Tz)=d(A,B), then Tx=Tz.If in Theorem15 the mapping g is the identity on A, then we get the following corollary.Corollary 17.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that T(A0) is nonempty and closed. Let T:A→B satisfy the following conditions:(a)
T is a proximal ψ-contraction of the second kind;(b)
T(A0)⊆B0.
Then there exists a pointx∈A such that d(x,Tx)=d(A,B). Moreover, if T is injective on A, then the point x such that d(x,Tx)=d(A,B) is unique.The following is a theorem for weak proximalψ-contractions of the first kind.Theorem 18.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that A0 is nonempty and closed. Assume also that the mappings T:A→B and g:A→A  satisfy the following conditions:(a)
T is a weak proximal ψ-contraction of the first kind;(b)
g∈𝒢A0;(c)
T(A0)⊆B0;(d)
A0⊆g(A0).
Then there exists a unique pointx∈A0 such that d(gx,Tx)=d(A,B). Moreover, for every x0∈A0 there exists a sequence {xn}⊆A such that d(gxn+1,Txn)=d(A,B) for every n∈ℕ∪{0} and xn→x.Proof.
Letx0∈A0. Since T(A0)⊆B0 and A0⊆g(A0), there exists x1∈A0 such that
(36)d(gx1,Tx0)=d(A,B).
Again, forx1∈A0, there exists x2∈A0 such that
(37)d(gx2,Tx1)=d(A,B).
By repeating this process, forxn∈A0, we can find xn+1∈A0 such that
(38)d(gxn+1,Txn)=d(A,B),∀n∈ℕ∪{0}.
SinceT is a weak proximal ψ-contraction of the first kind and g∈𝒢A0, we have
(39)d(xn+1,xn)≤d(gxn+1,gxn)≤d(xn,xn-1)-ψ(d(xn,xn-1))≤d(xn,xn-1),
for every n∈ℕ. Let tn=d(xn,xn+1); then {tn} is a bounded nonincreasing sequence of nonnegative real numbers. Therefore, {tn} converges to t, where t≥0. Now let us claim that t=0. Suppose that t>0. Since ψ∈Φ, we get 0<ψ(t)≤ψ(tn), for all n∈ℕ. Then, we have
(40)tn=d(xn,xn+1)≤d(gxn,gxn+1)≤d(xn-1,xn)-ψ(d(xn-1,xn))=tn-1-ψ(tn-1)≤tn-1-ψ(t).
Inductively we obtaintn+p≤tn-pψ(t), which is a contradiction for p large enough. Therefore t=0 and hence d(xn,xn+1)→0.
Now let us claim that{xn} is a Cauchy sequence. Suppose it is not. Then there exist ε>0 and subsequences {xmk},{xnk} of {xn} such that
(41)rk=d(xmk,xnk)≥ε,d(xmk,xnk-1)<ε,
and nk>mk≥k, for all k∈ℕ. Therefore,
(42)ε≤rk≤d(xmk,xnk-1)+d(xnk-1,xnk)<ε+tnk-1.
By lettingk→+∞, we have
(43)limk→+∞rk=ε.
Since(44)d(gxmk+1,Txmk)=d(A,B),d(gxnk+1,Txnk)=d(A,B),
and T is a weak proximal ψ-contraction of the first kind, we obtain that
(45)d(xmk+1,xnk+1)≤d(gxmk+1,gxnk+1)≤d(xmk,xnk)-ψ(d(xmk,xnk)).
Thus,(46)ε≤rk≤d(xmk,xmk+1)+d(xmk+1,xnk+1)+d(xnk+1,xnk)=tmk+tnk+d(xmk+1,xnk+1)≤tmk+tnk+d(xmk,xnk)-ψ(d(xmk,xnk))≤tmk+tnk+d(xmk,xnk)-ψ(ε).
Lettingk→+∞, we have ε≤ε-ψ(ε), which is a contradiction. Therefore, {xn} is a Cauchy sequence. By the completeness of X and since A0 is closed, we have xn→x∈A0. Moreover, by the continuity of g, we have gxn→gx and thus gx∈A0, since gxn∈A0, for all n∈ℕ.
On the other hand, sincex∈A0 and T(A0)⊆B0, there exists z∈A0 such that
(47)d(z,Tx)=d(A,B).
Again, sinceT is a weak proximal ψ-contraction of the first kind, we get
(48)d(z,gxn+1)≤d(x,xn)-ψ(d(x,xn))≤d(x,xn).
Lettingn→+∞, we obtain that d(z,gxn+1)→0 and then z=gx. This implies that
(49)d(gx,Tx)=d(A,B).
To prove the uniqueness, letx* be another point in A0 such that
(50)d(gx*,Tx*)=d(A,B).
Ifx≠x*, since g∈𝒢A0 and T is a weak proximal ψ-contraction of the first kind, we get
(51)d(x,x*)≤d(gx,gx*)≤d(x,x*)-ψ(d(x,x*))<d(x,x*),
which is a contradiction; thus we have x=x*.Remark 19.
If in Theorem18 we assume g∈𝒢A, then we get that there exists a unique x∈A such that d(gx,Tx)=d(A,B).If we takeg as the identity mapping on A in Theorem 18, then we get the following corollary, which extends a result of Rhoades [25] to non-self-mappings.Corollary 20.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that A0 is nonempty and closed. Let T:A→B satisfy the following conditions:(a)
T is a weak proximal ψ-contraction of the first kind;(b)
T(A0)⊆B0.
Then there exists a unique pointx∈A0 such that d(x,Tx)=d(A,B). Moreover, for every x0∈A0 there exists a sequence {xn}⊆A such that d(xn+1,Txn)=d(A,B) for every n∈ℕ∪{0} and xn→x.The following theorem is our main result for weak proximalψ-contractions of the second kind.Theorem 21.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that T(A0) is nonempty and closed. Assume also that the mappings T:A→B and g:A→A  satisfy the following conditions:(a)
T is a weak proximal ψ-contraction of the second kind;(b)
T∈𝒯g;(c)
T(A0)⊆B0;(d)
A0⊆g(A0).
Then there exists a pointx∈A such that d(gx,Tx)=d(A,B). Moreover, if T is injective on A, then the point x such that d(gx,Tx)=d(A,B) is unique.Proof.
Similar to the proof of Theorem18, we can find a sequence {xn}⊆A0 such that
(52)d(gxn+1,Txn)=d(A,B),∀n∈ℕ∪{0}.
SinceT is a weak proximal ψ-contraction of the second kind, we have
(53)d(Tgxn+1,Tgxn)≤d(Txn,Txn-1)-ψ(d(Txn,Txn-1))≤d(Txn,Txn-1)
for every n∈ℕ. Since T∈𝒯g, we get
(54)d(Txn+1,Txn)≤d(Tgxn+1,Tgxn)≤d(Txn,Txn-1)
for every n∈ℕ. Let tn=d(Txn,Txn+1); then {tn} is a bounded nonincreasing sequence of nonnegative real numbers. Therefore, {tn} converges to t, where t≥0. Now let us claim that t=0. Suppose that t>0. Since ψ∈Φ, we get 0<ψ(t)≤ψ(tn), for all n∈ℕ. Then, we have
(55)tn=d(Txn,Txn+1)≤d(Tgxn,Tgxn+1)≤d(Txn-1,Txn)-ψ(d(Txn-1,Txn))=tn-1-ψ(tn-1)≤tn-1-ψ(t).
Inductively we obtaintn+p≤tn-pψ(t), which is a contradiction for p large enough. Therefore t=0 and hence d(Txn,Txn+1)→0.
Now let us claim that{Txn} is a Cauchy sequence. Suppose it is not. Then there exist ε>0 and subsequences {Txmk},{Txnk} of {Txn} such that
(56)rk=d(Txmk,Txnk)≥ε,d(Txmk,Txnk-1)<ε,
and nk>mk≥k, for all k∈ℕ. Therefore, we get
(57)ε≤rk≤d(Txmk,Txnk-1)+d(Txnk-1,Txnk)<ε+tnk-1.
By lettingk→+∞, we have
(58)limk→+∞rk=ε.
Since(59)d(gxmk+1,Txmk)=d(A,B),d(gxnk+1,Txnk)=d(A,B),
and T is a weak proximal ψ-contraction of the second kind, we obtain that
(60)d(Txmk+1,Txnk+1)≤d(Tgxmk+1,Tgxnk+1)≤d(Txmk,Txnk)-ψ(d(Txmk,Txnk)).
Thus,(61)ε≤rk≤d(Txmk,Txmk+1)+d(Txmk+1,Txnk+1)+d(Txnk+1,Txnk)=tmk+tnk+d(Txmk+1,Txnk+1)≤tmk+tnk+d(Txmk,Txnk)-ψ(d(Txmk,Txnk))≤tmk+tnk+d(Txmk,Txnk)-ψ(ε).
Lettingk→+∞, we have ε≤ε-ψ(ε), which is a contradiction. Therefore, {Txn} is a Cauchy sequence. By the completeness of X and since T(A0) is closed, we have Txn→Tu∈B0. Moreover, there exists z∈A0 such that
(62)d(z,Tu)=d(A,B).
SinceA0⊆g(A0), we obtain that z=gx for some x∈A0, and then
(63)d(gx,Tu)=d(A,B).
Again, sinceT is a weak proximal ψ-contraction of the second kind, we get
(64)d(Tx,Txn+1)≤d(Tgx,Tgxn+1)≤d(Tu,Txn)-ψ(d(Tu,Txn))≤d(Tu,Txn).
Lettingn→+∞, we obtain that d(Tx,Txn+1)→0 and hence Tx=Tu. This implies that
(65)d(gx,Tx)=d(A,B).
To prove the uniqueness, letx* be another point in A such that
(66)d(gx*,Tx*)=d(A,B).
Ifx≠x*, since T∈𝒯g is injective on A, we have
(67)d(Tx,Tx*)≤d(Tgx,Tgx*)≤d(Tx,Tx*)-ψ(d(Tx,Tx*))<d(Tx,Tx*)
which is a contradiction; thus we have Tx=Tx* and hence x=x*.If in Theorem21 the mapping g is the identity on A, we get the following corollary.Corollary 22.
LetA and B be two nonempty subsets of a complete metric space (X,d). Suppose that T(A0) is nonempty and closed. Let T:A→B satisfy the following conditions:(a)
T is a weak proximal ψ-contraction of the second kind;(b)
T(A0)⊆B0.
Then there exists a pointx∈A such that d(x,Tx)=d(A,B). Moreover, if T is injective on A, then the point x such that d(x,Tx)=d(A,B) is unique.


## 4. Best Proximity Point Theorem forg-Weak Contractions

The following result is a best proximity point theorem forg-weak contractions. Recall that a non-self-mapping T:A→B is g-weakly contractive if there exists ψ∈Φc such that d(Tx,Ty)≤d(gx,gy)-ψ(d(gx,gy)), for all x,y∈A, where g:A→A.Theorem 23.
LetA and B be closed subsets of a complete metric space (X,d) such that A0,B0≠∅ and the pair (A,B) has the weak P-property. Suppose that the mappings g:A→A and T:A→B satisfy the following conditions:(a)
T is a g-weak contraction;(b)
T(A0)⊂B0;(c)
A0⊂g(A0).
Then, there exists an elementx*∈A0 such that d(gx*,Tx*)=d(A,B). Further, if g is one to one then we have a unique element x*∈A such that d(gx*,Tx*)=d(A,B).Proof.
Letx0 be an element of A0. In light of the fact that T(A0)⊂B0 and A0⊂g(A0), it is ensured that there exists an element x1∈A0 such that
(68)d(gx1,Tx0)=d(A,B).
Again, in view of the fact thatT(A0)⊂B0 and A0⊂g(A0), it is guaranteed that there exists an element x2∈A0 such that
(69)d(gx2,Tx1)=d(A,B).
Continuing this process, we can find a sequence{xn} in A0 such that
(70)d(gxn,Txn-1)=d(A,B),∀n∈ℕ.
Since(A,B) has the weak P-property, we conclude that
(71)d(gxn,gxn+1)≤d(Txn-1,Txn),∀n∈ℕ.
Now, asT is a g-weak contraction, we get
(72)d(gxn,gxn+1)≤d(Txn-1,Txn)≤d(gxn-1,gxn)-ψ(d(gxn-1,gxn)),
where ψ∈Φc (see Definition 8). If we set tn=d(gxn,gxn+1), then {tn} is a nonincreasing sequence of nonnegative real numbers and hence converges. Let t≥0 be the limit of the sequence {tn}. Now let us claim that t=0. Suppose that t>0. Since ψ is a nondecreasing function, we deduce that ψ(tn)≥ψ(t)>0, for all n∈ℕ. Then for any positive integer n, by (72), we get that
(73)tn+1≤tn-ψ(t).
Now, for alln>t1/ψ(t), by (73), we obtain that
(74)tn+1≤t1-nψ(t)<0,
a contradiction. Therefore t=0 and hence the sequence {d(gxn,gxn+1)} converges to 0. As
(75)d(gxn,gxn+1)≤d(Txn-1,Txn)≤d(gxn-1,gxn),
we deduce that the sequence {d(Txn-1,Txn)} converges to 0. Now, let us prove that {Txn} is a Cauchy sequence. Let ε>0 be given and we choose a positive integer n(ε) such that
(76)d(Txn,Txn+1)≤min{ε2,ψ(ε2)},
for all n≥n(ε). Fix n≥n(ε) and let
(77)A(n,ε):={x∈A:d(Txn,Tx)≤ε}.
Now, it is asserted that ifx∈A(n,ε) and u∈A is such that d(gu,Tx)=d(A,B), then u∈A(n,ε). First, we note that as d(gxn+1,Txn)=d(A,B), then by the weak P-property d(gxn+1,gu)≤d(Txn,Tx). Two cases will be considered to establish this fact. Precisely, if d(gxn+1,gu)≤ε/2, then it follows that
(78)d(Txn,Tu)≤d(Txn,Txn+1)+d(Txn+1,Tu)≤ε2+d(gxn+1,gu)-ψ(d(gxn+1,gu))≤ε2+d(gxn+1,gu)≤ε.
On the other hand ifε/2<d(gxn+1,gu)≤ε, then it follows that
(79)d(Txn,Tu)≤d(Txn,Txn+1)+d(Txn+1,Tu)≤ψ(ε2)+d(gxn+1,gu)-ψ(d(gxn+1,gu))≤ψ(ε2)+d(gxn+1,gu)-ψ(ε2)=d(gxn+1,gu)≤ε.
So,u∈A(n,ε). Now, we prove that
(80)xn+m∈A(n,ε),
for all m≥1. From xn∈A(n,ε) and d(gxn+1,Txn)=d(A,B), we deduce that xn+1∈A(n,ε); that is (80) holds for m=1. Now, we assume that (80) holds for some m≥1. From, xn+m∈A(n,ε) and d(gxn+m+1,Txn+m)=d(A,B), we deduce that xn+m+1∈A(n,ε); that is (80) holds for m+1 and hence for all m≥1. Thus, it follows that {Txn} is a Cauchy sequence. From the completeness of the space X, the sequence {Txn} converges to some element y*∈B. From d(gxn+1,gxm+1)≤d(Txn,Txm), we deduce that {gxn} is also a Cauchy sequence. As A is a complete subspace of X, then there exists z∈A such that gxn→z. Therefore, we have
(81)d(z,y*)=limn→+∞d(gxn+1,Txn)=d(A,B),
and so z∈A0. In light of the fact that A0 is contained in g(A0), there is x*∈A0 such that z=gx*. Since T(A0)⊂B0, there exists an element x¯∈A0 such that
(82)d(gx¯,Tx*)=d(A,B).
In view of the fact thatT is a g-weak contraction and (A,B) has the weak P-property and the continuity of ψ at t=0, we get
(83)d(gxn+1,gx¯)≤d(Txn,Tx*)≤d(gxn+1,gx*)-ψ(d(gxn+1,gx*)).
Lettingn→+∞, it follows that gx¯=gx*. Thus, we conclude that d(gx*,Tx*)=d(A,B).
To assert the uniqueness, let us assume thatz*∈A is another element such that d(gz*,Tz*)=d(A,B). Then
(84)d(gx*,gz*)≤d(Tx*,Tz*)≤d(gx*,gz*)-ψ(d(gx*,gz*)),
from which it follows that gx*=gz* and hence z*∈g-1gx*. If g is one to one then we deduce the uniqueness.Remark 24.
From the proof of Theorem23, we obtain that the method for getting the sequence {gxn}, that is the relation d(gxn,gxn+1)=d(Txn-1,Txn), also gives an iterative algorithm for computing solutions of coincidence equations.If in Theorem23 the mapping g is the identity on A, then yields the following result which is a generalization of a result due to Rhoades [25] to non-self-mappings.Corollary 25.
LetA and B be closed subsets of a complete metric space (X,d) such that A0,B0≠∅ and the pair (A,B) has the weak P-property. Suppose that the mapping T:A→B satisfies the following conditions:(i)
T is a g-weak contraction;(ii)
T(A0)⊂B0.
Then, there exists a unique elementx*∈A such that d(x*,Tx*)=d(A,B). Further, for any fixed element x0∈A0, the iterative sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to the element x*.Example 26.
ConsiderX=ℝ2 with the usual metric. Let us define
(85)A:={(x,y)∈ℝ2:x=0,y≥0},B:={(x,y)∈ℝ2:x=1,y≥0}.
ThenA and B are nonempty closed subsets of X and A0=A and B0=B. Note that d(A,B)=1. Let g:A→A and T:A→B be defined as g(0,x)=(0,2x) and T(0,x)=(1,x/(1+x)). Define ψ:[0,+∞)→[0,+∞) by ψ(t)=t2/(1+t), for all t≥0. Then, T is a g-weak contraction. As (A,B) has the weak P-property and g is one to one, we obtain that (0,0)∈A is the unique g-best proximity point of T; that is, d(g(0,0),T(0,0))=d(A,B).The following example shows that the weakP-property in Theorem 23 cannot be relaxed; that is, a g-weakly contractive mapping T:A→B may not have a g-best proximity point in A if the pair (A,B) does not have the weak P-property, where A and B are nonempty closed subsets of a complete metric space X.Example 27.
ConsiderX=ℝ with the usual metric, A={-10,10} and B={-2,2}. Then A and B are nonempty closed subsets of X with A0=A and B0=B. Note that d(A,B)=8. Let T:A→B be a mapping given by T(-10)=2 and T(10)=-2. It is easy to see that T:A→B is a contraction mapping with T(A0)⊂B0 and hence it is g-weakly contractive, where g is the identity mapping. Since d(x,Tx)=12>8=d(A,B), for all x∈A, then T has no g-best proximity points. It is worth noting that the pair (A,B) does not have the weak P-property.



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*Source: 713252-2013-09-29.xml*