# Best Proximity Point Results for Modifiedα-ψ-Proximal Rational Contractions

Authors: N. Hussain; M. A. Kutbi; P. Salimi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We first introduce certain new concepts ofα-η-proximal admissible and α-η-ψ-rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to
illustrate the usability of the obtained results.


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## Body

## 1. Introduction and Preliminaries

Let(X,d) be a metric space and T be a self-mapping defined on a subset of X. Fixed point theory is an important tool for solving equations of the kind Tx=x, whose solutions are the fixed points of the mapping T. Many problems arising in different areas of mathematics, such as optimization, variational analysis, and differential equations, can be modeled as fixed point equations of the form Tx=x. On the other hand, if T is not a self-mapping, the equation Tx=x could have no solutions and, in this case, it is of a certain interest to determine an element x that is in some sense closest to Tx. One of the most interesting results in this direction is due to Fan [1] and can be stated as follows.Theorem F.
LetK be a nonempty compact convex subset of a normed space E and T:K→E be a continuous nonself-mapping. Then there exists an x such that ∥x-Tx∥=d(K,Tx)=inf{∥Tx-u∥:u∈K}.Many generalizations and extensions of this result appeared in the literature (see [2–6] and, references therein).LetA and B be two nonempty subsets of a metric space (X,d). A best proximity point of a nonself-mapping T:A→B is a point x∈A satisfying the equality d(x,Tx)=d(A,B), where d(A,B)=inf{d(a,b):a∈A,b∈B}. Though best approximation theorems ensure the existence of approximation solutions, such results need not yield optimal solutions. But best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. For more details on this approach, we refer the reader to [5, 7–32].The aim of this paper is to introduce certain new concepts ofα-η-proximal admissible and α-η-ψ-rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to illustrate the usability of the obtained results.Now we give some basic notations and definitions that will be used in the sequel.LetA and B be two nonempty subsets of a metric space (X,d). We denote by A0 and B0 the following sets:
(1)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∈A,y∈B}. For the map T:A→B, we define the set of all best proximity points of T by
(2)Best(T)={x∈A:d(x,Tx)=d(A,B)}.Definition 1.
Let(X,d) be a metric space and let A and B be two nonempty subsets of X. Then B is said to be approximatively compact with respect to A if every sequence {yn} in B, satisfying the condition d(x,yn)→d(x,B) for some x in A, has a convergent subsequence.Obviously, every set is approximatively compact with respect to itself.Very recently, Nashine et al. [22] introduced rational proximal contraction of the first and second kinds as follows.Definition 2.
LetA and B be nonempty subsets of a metric space (X,d). Then T:A→B is said to be a rational proximal contraction of the first kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d<1, such that, for all x1,x2,u1,u2∈A,
(3)d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(u1,u2)kkkkkkkkk≤ad(x1,x2)kkkkkkkkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkkkkkkkk+c[d(x1,u1)+d(x2,u2)]kkkkkkkkk+d[d(x1,u2)+d(x2,u1)].Definition 3.
LetA and B be nonempty closed subsets of a metric space (X,d). Then T:A→B is said to be a rational proximal contraction of the second kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d<1, such that, for all x1,x2,u1,u2∈A,
(4)d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(Tu1,Tu2)kkkkkkkkk≤ad(Tx1,Tx2)kkkkkkkkk+b[1+d(Tx1,Tu1)]d(Tx2,Tu2)1+d(Tx1,Tx2)kkkkkkkkk+c[d(Tx1,Tu1)+d(Tx2,Tu2)]kkkkkkkkk+d[d(Tx1,Tu2)+d(Tx2,Tu1)].
Note that a rational proximal contraction of the second kind is not necessarily a rational proximal contraction of the first kind; for examples, see [22].Definition 4 (see [28]).
LetT be a self-mapping on a metric space (X,d) and α:X×X→[0,+∞) be a function. We say that T is α-admissible mapping if
(5)x,y∈X,α(x,y)≥1⟹α(Tx,Ty)≥1.Recently, Jleli and Samet [15] introduced new concepts of α-proximal admissible and α-ψ-proximal contractive type mappings as follows.Definition 5 (see [15]).
LetT:A→B, α:A×A→[0,∞). We say that T is α-proximal admissible if
(6)α(x1,x2)≥1,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B),kkkkkkkkk⟹α(u1,u2)≥1
for all x1,x2,u1,u2∈A.Clearly, ifA=B, then α-proximal admissible map T  reduces to α-admissible map.Definition 6 (see [15]).
A nonself-mappingT:A→B is said to be α-ψ-proximal contraction if
(7)α(x,y)d(Tx,Ty)≤ψ(d(x,y))
for all x,y∈A, α:A×A→[0,∞), ψ∈Ψ.Salimi et al. [27] modified the concept of α-admissible mappings as follows.Definition 7.
LetT be a self-mapping on a metric space (X,d) and α,η:X×X→[0,+∞) be two functions. We say that T is α-admissible mapping with respect to η if
(8)x,y∈X,α(x,y)≥η(x,y)⟹α(Tx,Ty)≥η(Tx,Ty).
Note that if we take η(x,y)=1, then this definition reduces to Definition 4. Also, if we take α(x,y)=1, then we say that T is η-subadmissible mapping.For the examples ofα-admissible mappings with respect to η, we refer to [27] and the examples in the next section.


## 2. Best Proximity and Fixed Point Results in Metric Spaces

First we modify the notion ofα-proximal admissible mapping as follows.Definition 8.
LetT:A→B and α,η:A×A→[0,∞) be functions. We say that T is α-proximal admissible with respect to η if, for all x1,x2,u1,u2∈A,
(9)α(x1,x2)≥η(x1,x2),d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B),kkkkkkkkk⟹α(u1,u2)≥η(u1,u2).
Note that if we take η(x,y)=1 for all x,y∈A, then this definition reduces to Definition 5. In case α(x,y)=1 for all x,y∈A, then we shall say that T is η-proximal subadmissible mapping.Clearly, ifA=B, then the previous definition reduces to Definition 7.Definition 9.
LetA and B be nonempty subsets of a metric space (X,d). Then T:A→B is said to be α-η-rational proximal contraction of the first kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d<1, such that, for all x1,x2,u1,u2∈A,
(10)α(x1,x2)≥η(x1,x2)d(u1,Tx1)=d(A,B)d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(u1,u2)kkkkkkkkk≤ad(x1,x2)kkkkkkkkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkkkkkkkk+c[d(x1,u1)+d(x2,u2)]kkkkkkkkk+d[d(x1,u2)+d(x2,u1)].
In case η(x,y)=1 for all x,y∈A, then T:A→B is said to be α-rational proximal contraction of the first kind.Definition 10.
LetA and B be nonempty closed subsets of a metric space (X,d). Then T:A→B is said to be a α-η-rational proximal contraction of the second kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d<1, such that, for all x1,x2,u1,u2∈A,
(11)α(x1,x2)≥η(x1,x2),d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(Tu1,Tu2)kkkkkkkkk≤ad(Tx1,Tx2)kkkkkkkkk+b[1+d(Tx1,Tu1)]d(Tx2,Tu2)1+d(Tx1,Tx2)kkkkkkkkk+c[d(Tx1,Tu1)+d(Tx2,Tu2)]kkkkkkkkk+d[d(Tx1,Tu2)+d(Tx2,Tu1)].
In case η(x,y)=1 for all x,y∈A, then T:A→B is said to be α-rational proximal contraction of the second kind.We are ready to prove the following best proximity point result forα-η-rational proximal contraction of the first kind.Theorem 11.
LetA and B be nonempty closed subsets of a complete metric space (X,d) and let B be approximatively compact with respect to A. Assume that α,η:A×A→[0,∞) are functions, A0 and B0 are nonempty, and T:A→B is an α-η-rational proximal contraction of the first kind which satisfies the following assertions:(i)
T(A0)⊆B0,(ii)
T is α-proximal admissible with respect to η,(iii)
there exist elementsx0 and x1 in A0 such that
(12)d(x1,Tx0)=d(A,B),α(x0,x1)≥η(x0,x1),(iv)
if{xn} is a sequence in A such that α(xn,xn+1)≥η(xn,xn+1) and xn→x∈A as n→∞, then α(xn,x)≥η(xn,x) for all n∈ℕ.Then there exists z∈A0, such that
(13)d(z,Tz)=d(A,B).
Moreover, if α(x,y)≥η(x,y) for all x,y∈Best(T), then z is unique best proximity point of T.Proof.
By (iii) there exist elementsx0 and x1 in A0, such that
(14)d(x1,Tx0)=d(A,B),α(x0,x1)≥η(x0,x1).
On the other hand, T(A0)⊆B0, then and there exists x2∈A0, such that
(15)d(x2,Tx1)=d(A,B).
Now, since, T is α-η-proximal admissible, then we have α(x1,x2)≥η(x1,x2). That is,
(16)d(x2,Tx1)=d(A,B),α(x1,x2)≥η(x1,x2).
Again, since T(A0)⊆B0, there exists x3∈A0, such that
(17)d(x3,Tx2)=d(A,B).Thus,(18)d(x2,Tx1)=d(A,B),d(x3,Tx2)=d(A,B),α(x1,x2)≥η(x1,x2)
together with T is α-η-proximal admissible imply that α(x2,x3)≥η(x2,x3). Hence,
(19)d(x3,Tx2)=d(A,B),α(x2,x3)≥η(x2,x3).
Continuing this process, we get
(20)d(xn+1,Txn)=d(A,B),α(xn,xn+1)≥η(xn,xn+1),∀n∈ℕ∪{0}.
Since T is α-η-rational proximal contraction of the first kind, then we have
(21)d(xn,xn+1)≤ad(xn-1,xn)+b[1+d(xn-1,xn)]d(xn,xn+1)1+d(xn-1,xn)+c[d(xn-1,xn)+d(xn,xn+1)]+d[d(xn-1,xn+1)+d(xn,xn)]≤ad(xn-1,xn)+bd(xn,xn+1)+c[d(xn-1,xn)+d(xn,xn+1)]+d[d(xn-1,xn)+d(xn,xn+1)],
which implies
(22)d(xn,xn+1)≤hd(xn-1,xn),
where h=(a+c+d)/(1-b-c-d)<1. That is, {xn} is a Cauchy sequence in A and since (X,d) is a complete metric space and A is closed, so there exists an element z∈A such that xn→z as n→∞. Also, we have
(23)d(z,B)≤d(z,Txn)≤d(z,xn+1)+d(xn+1,Txn)=d(z,xn+1)+d(A,B)≤d(z,xn+1)+d(z,B).
Taking limit as n→∞ in the previous inequality, we have
(24)limn→∞d(z,Txn)=d(z,B).
As B is approximatively compact with respect to A, so the sequence {Txn} has a subsequence {Txnk} that converges to some y∈B. Hence,
(25)d(z,y)=limn→∞d(xnk+1,Txnk)=d(A,B)
and so z∈A0. Now, since T(A0)⊆B0, then, d(w,Tz)=d(A,B) for some w∈A. From (iv) and (66), we have α(xn,z)≥η(xn,z) for all n∈ℕ. Therefore, we proved that
(26)α(xn,z)≥η(xn,z),d(w,Tz)=d(A,B),d(xn+1,Txn)=d(A,B)
for all n∈ℕ. Since T is a α-η-rational proximal contraction of the first kind, so we have
(27)d(w,xn+1)≤ad(z,xn)+b[1+d(z,w)]d(xn,xn+1)1+d(z,xn)+c[d(z,w)+d(xn,xn+1)]+d[d(z,xn+1)+d(xn,w)].
Taking limit as n→∞ in the previous inequality, we get
(28)d(w,z)≤(c+d)d(w,z).
As c+d<1, so w=z. This implies that
(29)d(z,Tz)=d(w,Tz)=d(A,B).
Assume thaty* is another best proximity point of T such that α(z,y*)≥η(z,y*). That is,
(30)α(z,y*)≥η(z,y*),d(z,Tx*)=d(A,B),d(y*,Ty*)=d(A,B).
Now, since T is α-η-rational proximal contraction of the first kind, so we have
(31)d(z,y*)≤ad(z,y*)+b[1+d(z,z)]d(y*,y*)1+d(z,y*)+c[d(z,z)+d(y*,y*)]+d[d(z,y*)+d(y*,z)],
which implies that d(z,y*)≤(a+2d)d(z,y*). As a+2d<1, so z=y*. That is, z is a unique best proximity point of T.By takingη(x,y)=1 in Theorem 11, we deduce the following corollary.Corollary 12.
LetA and B be nonempty closed subsets of a complete metric space (X,d) such that B is approximatively compact with respect to A. Assume that α:A×A→[0,∞), A0 and B0 are nonempty, and T:A→B is an α-rational proximal contraction of the first kind satisfying the following assertions:  (i)
T(A0)⊆B0,(ii)
T is α-proximal admissible,(iii)
there exist elementsx0 and x1 in A0, such that
(32)d(x1,Tx0)=d(A,B),α(x0,x1)≥1,(iv)
if{xn} is a sequence in A such that α(xn,xn+1)≥1 and xn→x∈A as n→∞, then α(xn,x)≥1 for all n∈ℕ. Then there exists z∈A0, such that
(33)d(z,Tz)=d(A,B).
Moreover, if α(x,y)≥1 for all x,y∈Best(T), then z is unique best proximity point of T.If in the previous corollary we takeα(x,y)=1, then we obtain the following result.Corollary 13 (see [22, Theorem 3.1]).
LetA and B be nonempty closed subsets of a complete metric space (X,d) and let B be approximatively compact with respect to A. Assume that A0 and B0 are nonempty and T:A→B is a rational proximal contraction of the first kind with T(A0)⊆B0. Then there exists a unique z∈A0, such that
(34)d(z,Tz)=d(A,B).
Further, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to z.Example 14.
LetX=ℝ and d(x,y)=|x-y| be metric on X. Suppose A=(-∞,-1] and B=[5/4,+∞). Define T:A→B by
(35)Tx={-x3+54,ifx∈(-∞,-2)∖{-3},94,ifx=-3,-14x+1,ifx∈[-2,-1].
Also, define α:X2→[0,∞) by
(36)α(x,y)={3,ifx,y∈[-2,-1],0,otherwise.
Clearly, d(A,B)=9/4. Now, we have
(37)A0={94x∈A:d(x,y)=d(A,B)ggg=94forsomey∈B}={-1},B0={94y∈B:d(x,y)=d(A,B)hhh=94forsomex∈A}={54}.
Also, T(A0)⊆B0. Let
(38)α(x1,x2)≥1,d(u1,Tx1)=d(A,B)=94,d(u2,Tx2)=d(A,B)=94.
Then,
(39)x1,x2∈[-2,-1],d(u1,Tx1)=94,d(u2,Tx2)=94.
Note that Tw∈[5/4,3/2] for all w∈[-2,-1]. Hence, u1=u2=-1. That is, α(u1,u2)≥1. That is, T is a α-proximal admissible mapping. Also, assume that α(xn,xn+1)≥1 for all n∈ℕ∪{0} and xn→x as n→∞. Therefore, {xn}⊆[-2,-1] and then x∈[-2,-1]. That is, α(xn,x)≥1 for all n∈ℕ∪{0}. Further, d(-1,T(-1))=d(A,B)=9/4 and α(-1,-1)≥1.
Again, assume that(40)α(x1,x2)≥1,d(u1,Tx1)=d(A,B)=94,d(u2,Tx2)=d(A,B)=94.
Then x1,x2∈[-2,-1] and u1=u2=-1. Hence,
(41)d(u1,u2)=0≤ad(x1,x2)kkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkk+c[d(x1,u1)+d(x2,u2)]kkk+d[d(x1,u2)+d(x2,u1)].
Thus, all of the conditions of Corollary 12 (Theorem 11) hold and there exists a unique z=-1∈A0, such that
(42)d(-1,T(-1))=d(A,B)=94.Example 15.
LetX=ℝ and d(x,y)=|x-y| be metric on X. Suppose A=[0,1]∪{2,20} and B=[0,1/4]∪{2,20}. Define T:A→B by
(43)Tx={20,ifx=2,2,ifx=20,14x,ifx∈[0,1].
Also, define α:X2→[0,∞) by
(44)α(x,y)={1,ifx,y∈[0,1],0,otherwise.
Clearly, d(A,B)=0. Now, we have
(45)A0={x∈A:d(x,y)=d(A,B)=0gggforsome∈B(x,y)}=B,B0={y∈B:d(x,y)=d(A,B)=0gggforsomex∈A(x,y)}=B.
Also, T(A0)⊆B0. Let
(46)α(x1,x2)≥1,d(u1,Tx1)=d(A,B)=0,d(u2,Tx2)=d(A,B)=0.
Then, x1,x2∈[0,1] and u1=Tx1 and u2=Tx2. Note that Tw∈[0,1] for all w∈[0,1]; Hence, u1,u2∈[0,1] that is, α(u1,u2)≥1. That is, T is a α-proximal admissible mapping. Also, assume that α(xn,xn+1)≥1 for all n∈ℕ∪{0} and xn→x as n→∞. Therefore, {xn}⊆[0,1] and then x∈[0,1]. That is, α(xn,x)≥1 for all n∈ℕ∪{0}. Further, d(0,T(0))=d(A,B)=0 and α(0,0)≥1
Leta=1/2, b=1/6, and c=d=1/24.
Again, assume that(47)α(x1,x2)≥1,d(u1,Tx1)=d(A,B)=0,d(u2,Tx2)=d(A,B)=0.
Thenx1,x2∈[0,1] and u1=Tx1=(1/4)x1 and u2=Tx2=(1/4)x2. Hence,
(48)d(u1,u2)=14|x-y|≤12|x-y|≤12d(x1,x2)d(x1,x2)+16[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)+124[d(x1,u1)+d(x2,u2)]+124[d(x1,u2)+d(x2,u1)].
All of the conditions of Corollary 12 (Theorem 11) hold and there exists a unique z=0∈A0, such that
(49)d(0,T(0))=d(A,B)=0.
Let(50)d(0,T0)=d(A,B)=0,d(2,T20)=d(A,B)=0,
imply
(51)d(0,20)≤12d(0,2)+16[1+d(0,0)]d(2,20)1+d(0,2)+124[d(0,0)+d(2,20)]+124[d(0,20)+d(2,0)].
Then,
(52)20≤1+1+1824+2224,
which is a contradiction. Therefore, Corollary 32 [22, Theorem 3.1] cannot be applied here.If in Theorem11 we take α(x,y)=1, then we obtain the following result.Corollary 16.
LetA and B be nonempty closed subsets of a complete metric space (X,d) and let B be approximatively compact with respect to A. Assume that η:A×A→[0,∞), A0 and B0 are nonempty, and T:A→B is a nonself-mapping satisfying the following assertions:  (i)
T(A0)⊆B0,(ii)
T is η-proximal subadmissible,(iii)
there exist elementsx0 and x1 in A0, such that
(53)d(x1,Tx0)=d(A,B),η(x0,x1)≤1,(iv)
(54)η(x1,x2)≤1,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(u1,u2)kkkkkkkkk≤ad(x1,x2)kkkkkkkkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkkkkkkkk+c[d(x1,u1)+d(x2,u2)]kkkkkkkkk+d[d(x1,u2)+d(x2,u1)],(v)
if{xn} is a sequence in A such that η(xn,xn+1)≤1 and xn→x∈A as n→∞, then η(xn,x)≤1 for all n∈ℕ, where a+b+2c+2d<1. Then there exists z∈A0, such that,
(55)d(z,Tz)=d(A,B).
Moreover, if η(x,y)≤1 for all x,y∈Best(T), then z is unique.The following are immediate consequences of Theorem11.Theorem 17.
LetX be a complete metric space and let T:X→X be a mapping satisfying the following assertions:  (i)
T is α-admissible with respect to η,(ii)
there exists elementx0 in X, such that
(56)α(x0,Tx0)≥η(x0,Tx0),(iii)
if{xn} is a sequence in X such that α(xn,xn+1)≥η(xn,xn+1) and xn→x∈X as n→∞, then α(xn,x)≥η(xn,x) for all n∈ℕ,(iv)
(57)α(x1,x2)≥η(x1,x2)⟹d(Tx1,Tx2)≤ad(x1,x2)kk+b[1+d(x1,Tx1)]d(x2,Tx2)1+d(x1,x2)kk+c[d(x1,Tx1)+d(x2,Tx2)]kk+d[d(x1,Tx2)+d(x2,Tx1)],where a+b+2c+2d<1. Then T has a unique fixed point in X.If in Theorem17 we take η(x,y)=1, then we obtain the following result.Theorem 18.
LetX be a complete metric space and let T:X→X be a mapping satisfying the following assertions:  (i)
T is α-admissible,(ii)
there exists elementx0 in X, such that,
(58)α(x0,Tx0)≥1(iii)
if{xn} is a sequence in X such that α(xn,xn+1)≥1 and xn→x∈X as n→∞, then α(xn,x)≥1 for all n∈ℕ,(iv)
(59)α(x1,x2)≥1⟹d(Tx1,Tx2)≤ad(x1,x2)kk+b[1+d(x1,Tx1)]d(x2,Tx2)1+d(x1,x2)kk+c[d(x1,Tx1)+d(x2,Tx2)]kk+d[d(x1,Tx2)+d(x2,Tx1)], where a+b+2c+2d<1. Then T has a unique fixed point in X.If in Theorem17 we take α(x,y)=1, then we obtain the following result.Theorem 19.
LetX be a complete metric space and T:X→X be a mapping satisfying the following assertions:(i)
T is η-subadmissible,(ii)
there exists elementx0 in X, such that
(60)η(x0,Tx0)≤1,(iii)
if{xn} is a sequence in X such that η(xn,xn+1)≤1 and xn→x∈X as n→∞, then η(xn,x)≤1 for all n∈ℕ,(iv)
(61)η(x1,x2)≤1⟹d(Tx1,Tx2)≤ad(x1,x2)kk+b[1+d(x1,Tx1)]d(x2,Tx2)1+d(x1,x2)kk+c[d(x1,Tx1)+d(x2,Tx2)]kk+d[d(x1,Tx2)+d(x2,Tx1)],where a+b+2c+2d<1. Then T has a unique fixed point in X.If in Theorem18 we take α(x,y)=1, then we obtain the following fixed point result for rational contraction of first kind.Theorem 20.
LetX be a complete metric space and let T:X→X be a mapping satisfying the following rational inequality:
(62)d(Tx1,Tx2)≤ad(x1,x2)+b[1+d(x1,Tx1)]d(x2,Tx2)1+d(x1,x2)+c[d(x1,Tx1)+d(x2,Tx2)]+d[d(x1,Tx2)+d(x2,Tx1)],
where a+b+2c+2d<1. Then T has a unique fixed point in X.We now establish best proximity point result forα-η-rational proximal contraction of the second kind.Theorem 21.
LetA and B be nonempty closed subsets of a complete metric space (X,d) such that A is approximatively compact with respect to B. Assume that α,η:A×A→[0,∞), A0 and B0 are nonempty, and T:A→B is a continuous α-η-rational proximal contraction of the second kind, such that(i)
T(A0)⊆B0,(ii)
T is α-proximal admissible with respect to η,(iii)
There exist elementsx0 and x1 in A0, such that
(63)d(x1,Tx0)=d(A,B),α(x0,x1)≥η(x0,x1). Then there exists x∈Best(T) and, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, and Tx=Tz for all x,z∈Best(T) when α(x,y)≥η(x,y) for all x,y∈Best(T).Proof.
Following the same lines of the proof of Theorem11, there exists a sequence {xn}∈A0, such that
(64)d(xn+1,Txn)=d(A,B),α(xn,xn+1)≥η(xn,xn+1),∀n∈ℕ∪{0}.
Since T is a α-η-rational proximal contraction of the second kind, we get
(65)d(Txn,Txn+1)≤ad(Txn-1,Txn)+b[1+d(Txn-1,Txn)]d(Txn,Txn+1)1+d(Txn-1,Txn)+c[d(Txn-1,Txn)+d(Txn,Txn+1)]+d[d(Txn-1,Txn+1)+d(Txn,Txn)]≤ad(Txn-1,Txn)+bd(Txn,Txn+1)+c[d(Txn-1,Txn)+d(Txn,Txn+1)]+d[d(Txn-1,Txn)+d(Txn,Txn+1)],
which implies
(66)d(Txn,Txn+1)≤hd(Txn-1,Txn),
where h=(a+c+d)/(1-b-c-d)<1. That is, {Txn} is a Cauchy sequence and since (X,d) is a complete metric space and B is closed, so there exists an element y*∈B such that Txn→y* as n→∞. Also, we have
(67)d(y*,A)≤d(y*,xn+1)≤d(y*,Txn)+d(Txn,xn+1)=d(y*,Txn)+d(A,B)≤d(y*,Txn)+d(y*,A).
Taking limit as n→∞ in the previous inequality, we have
(68)limn→∞d(y*,xn)=d(y*,A).
Since A is approximatively compact with respect to B, so the sequence, {xn} has a subsequence {xnk} that converges to some x*∈A. Now, by applying continuity of T, we get
(69)d(x*,Tx*)=limk→∞d(xnk+1,Txnk)=d(A,B).
That is, x*∈Best(T). Now, assume that z* is a another best proximity point of T. That is, d(z*,Tz*)=d(A,B). Now, since, T is a α-η-rational proximal contraction of the second kind and α(x,y)≥η(x,y) for all x,y∈Best(T), then
(70)d(Tx*,Tz*)≤ad(Tx*,Tz*)+b[1+d(Tx*,Tx*)]d(Tz*,Tz*)1+d(Tx*,Tz*)+c[d(Tx*,Tx*)+d(Tz*,Tz*)]+d[d(Tx*,Tz*)+d(Tz*,Tx*)].
This implies that
(71)d(Tx*,Tz*)≤(c+d)d(Tx*,Tz*).
And, hence, d(Tx*,Tz*)=0 gives us Tx*=Tz*.Corollary 22.
LetA and B be nonempty closed subsets of a complete metric space (X,d) such that A is approximatively compact with respect to B. Assume that α:A×A→[0,∞), A0 and B0 are nonempty, and T:A→B is a continuous mapping, such that (i)
T(A0)⊆B0,(ii)
T is α-proximal admissible,(iii)
there exist elementsx0 and x1 in A0 such that
(72)d(x1,Tx0)=d(A,B),α(x0,x1)≥1,(iv)
(73)α(x1,x2)≥1,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(Tu1,Tu2)kkkkkkkkk≤ad(Tx1,Tx2)kkkkkkkkk+b[1+d(Tx1,Tu1)]d(Tx2,Tu2)1+d(Tx1,Tx2)kkkkkkkkk+c[d(Tx1,Tu1)+d(Tx2,Tu2)]kkkkkkkkk+d[d(Tx1,Tu2)+d(Tx2,Tu1)],where a+b+2c+2d<1. Then there exists x∈Best(T) and, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, and Tx=Tz for all x,z∈Best(T) when α(x,y)≥1 for all x,y∈Best(T).If in the previous corollary we takeα(x,y)=1, then we have the following result.Corollary 23 (see [22, Theorem 3.2]).
LetA and B be nonempty closed subsets of a complete metric space (X,d) such that A is approximatively compact with respect to B. Assume that A0 and B0 are nonempty and T:A→B is a continuous rational proximal contraction of the second kind, such that T(A0)⊆B0. Then there exists x∈Best(T) and, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, and Tx=Tz for all x,z∈Best(T).Corollary 24.
LetA and B be nonempty closed subsets of a complete metric space (X,d) such that A is approximatively compact with respect to B. Assume that η:A×A→[0,∞), A0 and B0 are nonempty and T:A→B is a continuous mapping, such that T(A0)⊆B0 and T is η-proximal subadmissible, such that
(74)η(x1,x2)≤1d(u1,Tx1)=d(A,B)d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(Tu1,Tu2)kkkkkkkkk≤ad(Tx1,Tx2)kkkkkkkkk+b[1+d(Tx1,Tu1)]d(Tx2,Tu2)1+d(Tx1,Tx2)kkkkkkkkk+c[d(Tx1,Tu1)+d(Tx2,Tu2)]kkkkkkkkk+d[d(Tx1,Tu2)+d(Tx2,Tu1)]
for all x1,x2,u1,u2∈A, where a+b+2c+2d<1. Then there exists x∈Best(T) and, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, and Tx=Tz for all x,z∈Best(T) when η(x,y)≤1 for all x,y∈Best(T).The following are immediate consequences of Theorem21.Theorem 25.
LetX be a complete metric space and let T:X→X be a continuous mapping satisfying the following assertions:  (i)
T is α-admissible with respect to η,(ii)
there exists elementx0 in X, such that
(75)α(x0,Tx0)≥η(x0,Tx0),(iii)
(76)α(x1,x2)≥η(x1,x2)⟹d(T2x1,T2x2)≤ad(Tx1,Tx2)kk+b[1+d(Tx1,T2x1)]d(Tx2,T2x2)1+d(Tx1,Tx2)kk+c[d(Tx1,T2x1)+d(Tx2,T2x2)]kk+d[d(Tx1,T2x2)+d(Tx2,T2x1)],where a+b+2c+2d<1. Then T has a unique fixed point in X.If in Theorem25 we take η(x,y)=1, then we obtain the following result.Theorem 26.
LetX be a complete metric space and let T:X→X be a continuous mapping satisfying the following assertions:  (i)
T is α-admissible,(ii)
there exists elementx0 in X, such that
(77)α(x0,Tx0)≥1(iii)
(78)α(x1,x2)≥1⟹d(T2x1,T2x2)≤ad(Tx1,Tx2)kk+b[1+d(Tx1,T2x1)]d(Tx2,T2x2)1+d(Tx1,Tx2)kk+c[d(Tx1,T2x1)+d(Tx2,T2x2)]kk+d[d(Tx1,T2x2)+d(Tx2,T2x1)], where a+b+2c+2d<1. Then T has a unique fixed point in X.If in Theorem25 we take α(x,y)=1, then we obtain the following result.Theorem 27.
LetX be a complete metric space and let T:X→X be a continuous mapping satisfying the following assertions:(i)
T is η-subadmissible(ii)
there exists elementx0 in X, such that
(79)η(x0,Tx0)≤1,(iii)
(80)η(x1,x2)≤1⟹d(T2x1,T2x2)≤ad(Tx1,Tx2)kkkk+b[1+d(Tx1,T2x1)]d(Tx2,T2x2)1+d(Tx1,Tx2)kk+c[d(Tx1,T2x1)+d(Tx2,T2x2)]kk+d[d(Tx1,T2x2)+d(Tx2,T2x1)], where a+b+2c+2d<1. Then T has a unique fixed point in X.If in Theorem26 we take α(x,y)=1, then we obtain the following result.Theorem 28.
LetX be a complete metric space and let T:X→X be a continuous mapping satisfying the following rational inequality:
(81)d(T2x1,T2x2)≤ad(Tx1,Tx2)+b[1+d(Tx1,T2x1)]d(Tx2,T2x2)1+d(Tx1,Tx2)+c[d(Tx1,T2x1)+d(Tx2,T2x2)]+d[d(Tx1,T2x2)+d(Tx2,T2x1)],
where a+b+2c+2d<1. Then T has a unique fixed point in X.Our next best proximity point result is aboutα-η-rational proximal contraction of the first and second kinds where we consider only completeness of (X,d) without assuming continuity of the mapping T and approximative compactness of A and B.Theorem 29.
LetA and B be nonempty closed subsets of a complete metric space (X,d). Assume that α,η:A×A→[0,∞), A0 and B0 are nonempty, and T:A→B is α-η-rational proximal contraction of the first and second kinds, such that  (i)
T(A0)⊆B0,(ii)
T is α-proximal admissible with respect to η,(iii)
there exists elementsx0 and x1 in A0 such that
(82)d(x1,Tx0)=d(A,B),α(x0,x1)≥η(x0,x1),(iv)
if{xn} is a sequence in A such that α(xn,xn+1)≥η(xn,xn+1) and xn→x∈A as n→∞, then α(xn,x)≥η(xn,x) for all n∈ℕ. Then there exists unique x∈Best(T). Also, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, whenever α(x,y)≥η(x,y) for all x,y∈Best(T).Proof.
As in proof of Theorem11, there exists a sequence {xn}∈A0, such that
(83)d(xn+1,Txn)=d(A,B),α(xn,xn+1)≥η(xn,xn+1),∀n∈ℕ∪{0},
and the sequence {xn} is a Cauchy sequence and so converges to some x*∈A. Also, by proof of Theorem 21, we obtain that the sequence {Txn} is a Cauchy sequence and converges to some y*∈B. Hence, we have
(84)d(x*,y*)=limn→∞d(xn,Txn)=d(A,B).
That is, x*∈A0. Since TA0⊆B0, so d(u,Tx*)=d(A,B) for some u∈A. Thus we have d(xn+1,Txn)=d(A,B) and d(u,Tx*)=d(A,B) and so by (iv) this implies that α(xn,x*)≥η(xn,x*) for all n≥0. Now, since T is a α-η-rational proximal contraction of the first kind, we get
(85)d(xn+1,u)≤ad(xn,x*)+b[1+d(xn,xn+1)]d(x*,u)1+d(xn,x*)+c[d(xn,xn+1)+d(x*,u)]+d[d(xn,u)+d(x*,xn+1)].
Taking limit asn→∞ in the previous inequality, we get
(86)d(x*,u)≤(b+c+d)d(x*,u),
which implies that d(x*,u)=0. That is, x*=u. Hence, d(x*,Tx*)=d(A,B). Further, following similar proof of Theorem 11 we can deduce the uniqueness of best proximity point of T.Ifη(x,y)=1 for all x,y∈A in the previous theorem, we obtain the following result.Corollary 30.
LetA and B be nonempty closed subsets of a complete metric space (X,d). Assume that α:A×A→[0,∞), A0 and B0 are nonempty, and T:A→B is α-rational proximal contraction of the first and second kinds, such that (i)
T(A0)⊆B0,(ii)
T is α-proximal admissible,(iii)
there exists elementsx0 and x1 in A0, such that
(87)d(x1,Tx0)=d(A,B),α(x0,x1)≥1,(iv)
if{xn} is a sequence in A such that α(xn,xn+1)≥η(xn,xn+1) and xn→x∈A as n→∞, then α(xn,x)≥1 for all n∈ℕ. Then there exists unique x∈Best(T). Also, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, whenever α(x,y)≥1 for all x,y∈Best(T).Corollary 31.
LetA and B be nonempty closed subsets of a complete metric space (X,d) such that B is approximatively compact with respect to A. Assume that η:A×A→[0,∞) and a+b+2c+2d<1. Let A0 and B0 be nonempty and let T:A→B be a nonself-mapping such that T(A0)⊆B0 and T is η-proximal subadmissible mapping such that
(88)η(x1,x2)≤1,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(u1,u2)kkkkkkkkk≤ad(x1,x2)kkkkkkkkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkkkkkkkk+c[d(x1,u1)+d(x2,u2)]kkkkkkkkk+d[d(x1,u2)+d(x2,u1)],η(x1,x2)≤1,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(Tu1,Tu2)kkkkkkkkk≤ad(Tx1,Tx2)kkkkkkkkk+b[1+d(Tx1,Tu1)]d(Tx2,Tu2)1+d(Tx1,Tx2)kkkkkkkkk+c[d(Tx1,Tu1)+d(Tx2,Tu2)]kkkkkkkkk+d[d(Tx1,Tu2)+d(Tx2,Tu1)]
for all x1, x2, u1, u2. Then there exists unique x∈Best(T). Also, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, where η(x,y)≤1 for all x,y∈Best(T).Corollary 32 (see [22, Theorem 3.3]).
LetA and B be nonempty closed subsets of a complete metric space (X,d). Assume that A0 and B0 are nonempty and T:A→B is rational proximal contraction of the first and second kinds, such that T(A0)⊆B0. Then there exists unique x∈Best(T). Also, for any fixed x0∈A0, the sequence {xn} defined by d(xn+1,Txn)=d(A,B), converges to x.Remark 33.
(1) Similarly we may obtain many results as an immediate consequence of Theorem29.
(2) Ifb=0 in our results (Theorem 11–Corollary 32), we get the modified and improved versions of recent results in [26].


## 3. Best Proximity and Fixed Point Results in Partially Ordered Metric Spaces

The aim of this section is to deduce main results (Theorems 3.1–3.3 [22]) in the context of partially ordered metric spaces. Moreover, we obtain certain recent fixed point results as corollaries in partially ordered metric spaces. Existence of best proximity and fixed points in partially ordered metric spaces has been considered recently by many authors (see, [7, 8, 11, 25, 33]).Definition 34 (see [25]).
A mappingT:A→B is said to be proximally order preserving if and only if it satisfies the condition that
(89)kkkkkkx1⪯x2,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkl⟹u1⪯u2
for all x1,x2,u1,u2∈A.Clearly, ifA=B, then proximally order-preserving map T reduces to nondecreasing map.Definition 35.
LetA and B be nonempty closed subsets of a complete partially ordered metric space (X,d,⪯). Then T:A→B is said to be an ordered rational proximal contraction of the first kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d<1, such that
(90)x1⪯x2,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(u1,u2)kkkkkkkkk≤ad(x1,x2)kkkkkkkkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkkkkkkkk+c[d(x1,u1)+d(x2,u2)]kkkkkkkkk+d[d(x1,u2)+d(x2,u1)].Definition 36.
LetA and B be nonempty closed subsets of a complete partially ordered metric space (X,d,⪯). Then T:A→B is said to be an ordered rational proximal contraction of the second kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d<1, such that
(91)x1⪯x2,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(Tu1,Tu2)kkkkkkkkk≤ad(Tx1,Tx2)kkkkkkkkk+b[1+d(Tx1,Tu1)]d(Tx2,Tu2)1+d(Tx1,Tx2)kkkkkkkkk+c[d(Tx1,Tu1)+d(Tx2,Tu2)]kkkkkkkkk+d[d(Tx1,Tu2)+d(Tx2,Tu1)].Theorem 37.
LetA and B be nonempty closed subsets of a complete partially ordered metric space (X,d,⪯) and let B be approximatively compact with respect to A. Assume that A0 and B0 are nonempty and T:A→B is an ordered rational proximal contraction of the first kind which satisfies the following assertions:  (i)
T(A0)⊆B0,(ii)
T is proximally order-preserving,(iii)
there exist elementsx0 and x1 in A0, such that
(92)d(x1,Tx0)=d(A,B),x0⪯x1,(iv)
if{xn} is a nondecreasing sequence in A such that xn→x∈A as n→∞, then xn⪯x for all n∈ℕ. Then there exists z∈A0, such that,
(93)d(z,Tz)=d(A,B).Proof.
Defineα:A×A→[0,+∞) by
(94)α(x,y)={1,iffx⪯y0,otherwise.
At first, we prove that T is α-proximal admissible mapping. For this, assume that
(95)α(x,y)≥1,d(u,Tx)=d(A,B),d(v,Ty)=d(A,B).
So,
(96)x⪯y,d(u,Tx)=d(A,B),d(v,Ty)=d(A,B).
Now since, T is a proximally order preserving, so, u⪯v. That is, α(u,v)≥1 which implies that T is α-proximal admissible.
By (iii), we have(97)d(x1,Tx0)=d(A,B),α(x0,x1)≥1.
Further as T is an ordered rational proximal contraction, we have
(98)α(x1,x2)≥1,d(u1,Tx1)=d(A,B),d(u2,Tx2)=d(A,B)kkkkkkkkk⟹d(u1,u2)kkkkkkkkk≤ad(x1,x2)kkkkkkkkk+b[1+d(x1,u1)]d(x2,u2)1+d(x1,x2)kkkkkkkkk+c[d(x1,u1)+d(x2,u2)]kkkkkkkkk+d[d(x1,u2)+d(x2,u1)],
which implies that T:A→B is α-rational proximal contraction of the first kind. Assume that α(xn,xn+1)≥1 for all n∈ℕ such that xn→x as n→∞. Then xn⪯xn+1 for all n∈ℕ. Hence, by (iv) we get xn⪯x for all n∈ℕ and so α(xn,x)≥1 for all n∈ℕ. That is, all conditions of Corollary 12 hold and consequently there exists z∈A0, such that
(99)d(z,Tz)=d(A,B).Similarly, we can prove following best proximity point result in partially ordered metric space.Theorem 38.
LetA and B be nonempty closed subsets of a complete partially ordered metric space (X,d,⪯) and B be approximatively compact with respect to A. Assume that A0 and let B0 are nonempty and T:A→B is a continuous ordered rational proximal contraction of the second kind, such that(i)
T(A0)⊆B0,(ii)
T is proximally order-preserving,(iii)
there exist elementsx0 and x1 in A0 such that
(100)d(x1,Tx0)=d(A,B),x0⪯x1,where a+b+2c+2d<1. Then there exists x∈Best(T) and, for any fixed x0∈A0, the sequence {xn}, defined by d(xn+1,Txn)=d(A,B), converges to x, and Tx=Tz for all x,z∈Best(T) when x⪯y for all x,y∈Best(T).Theorem 39.
LetA and B be nonempty closed subsets of a complete partially ordered metric space (X,d,⪯). Assume that A0 and B0 are nonempty and T:A→B is a continuous ordered rational proximal contraction of the first and second kind, such that(i)
T(A0)⊆B0,(ii)
T is proximally order-preserving,(iii)
there exist elementsx0 and x1 in A0, such that
(101)d(x1,Tx0)=d(A,B),x0⪯x1,(iv)
if{xn} is a nondecreasing sequence in A such that xn→x∈A as n→∞, then xn⪯x for all n∈ℕ. Then there exists z∈A0, such that
(102)d(z,Tz)=d(A,B).Further, we can easily deduce the following recent fixed point results from Theorem37.Theorem 40.
Let(X,d,⪯) be a complete partially ordered metric space. Assume that T:X→X is self-mapping on X which satisfies the following assertions:  (i)
T is nondecreasing mapping,(ii)
there existsx0∈X such that, x0⪯Tx0,(iii)
if{xn} is nondecreasing sequence in X such that xn→x∈X as n→∞, then xn⪯x for all n∈ℕ,(iv)
(103)d(Tx1,Tx2)≤ad(x1,x2)+b[1+d(x1,Tx1)]d(x2,Tx2)1+d(x1,x2)+c[d(x1,Tx1)+d(x2,Tx2)]+d[d(x1,Tx2)+d(x2,Tx1)]for all x1⪯x2∈X. Then T has a fixed point.If we putc=d=0 in the previous theorem, we obtain the following recent results.Corollary 41 (see [11, Theorems 2 and 3]).
Let(X,d,⪯) be a complete partially ordered metric space. Assume that T:X→X is self-mapping on X which satisfies the following assertions:  (i)
T is nondecreasing mapping,(ii)
there existsx0∈X such that x0⪯Tx0,(iii)
ifT is continuous or {xn} is a nondecreasing sequence in X such that xn→x∈X as n→∞, then xn⪯x for all n∈ℕ,(iv)
(104)d(Tx1,Tx2)≤ad(x1,x2)+b[1+d(x1,Tx1)]d(x2,Tx2)1+d(x1,x2)  for all x1⪯x2 in X, where a+b<1. Then T has a fixed point.Remark 42.
(1) Similarly we may obtain many results as an immediate consequence of Theorems38 and 39.
(2) If in Corollary41 we put b=0, then Theorems 2.1 and 2.2 of [23] are obtained. If a=0 in Corollary 41, we get Theorem 5 in [11].



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