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Best proximity point theorems for generalized contractions in partially ordered metric spaces
Fixed Point Theory and Applications volumeÂ 2013, ArticleÂ number:Â 83 (2013)
Abstract
The purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, our Poperator technique, which changes a nonself mapping to a selfmapping, plays an important role. Some recent results in this area have been improved.
MSC:47H05, 47H09, 47H10.
1 Introduction and preliminaries
Let A and B be nonempty subsets of a metric space (X,d). An operator T:A\xe2\u2020\u2019B is said to be contractive if there exists k\xe2\u02c6\u02c6[0,1) such that d(Tx,Ty)\xe2\u2030\xa4kd(x,y) for any x,y\xe2\u02c6\u02c6A. The wellknown Banach contraction principle is as follows: Let (X,d) be a complete metric space, and let T:X\xe2\u2020\u2019X be a contraction of X into itself. Then T has a unique fixed point in X.
In the sequel, we denote by Î“ the functions \mathrm{\xce\xb2}:[0,\mathrm{\xe2\u02c6\u017e})\xe2\u2020\u2019[0,1) satisfying the following condition:
In 1973, Geraghty introduced the Geraghtycontraction and obtained Theorem 1.2 as follows.
Definition 1.1 [1]
Let (X,d) be a metric space. A mapping T:X\xe2\u2020\u2019X is said to be a Geraghtycontraction if there exists \mathrm{\xce\xb2}\xe2\u02c6\u02c6\mathrm{\xce\u201c} such that for any x,y\xe2\u02c6\u02c6X,
Theorem 1.2 [1]
Let (X,d) be a complete metric space, and let T:X\xe2\u2020\u2019X be an operator. Suppose that there exists \mathrm{\xce\xb2}\xe2\u02c6\u02c6\mathrm{\xce\u201c} such that for any x,y\xe2\u02c6\u02c6X,
Then T has a unique fixed point.
Obviously, Theorem 1.2 is an extensive version of the Banach contraction principle. Recently, the generalized contraction principle has been studied by many authors in metric spaces or more generalized metric spaces. Some results have been got in partially ordered metric spaces as follows.
Theorem 1.3 [2]
Let (X,\xe2\u2030\xa4) be a partially ordered set, and suppose that there exists a metric d such that (X,d) is a complete metric space. Let f:X\xe2\u2020\u2019X be an increasing mapping such that there exists an element {x}_{0}\xe2\u02c6\u02c6X with {x}_{0}\xe2\u2030\xa4f({x}_{0}). Suppose that there exists \mathrm{\xce\xb2}\xe2\u02c6\u02c6\mathrm{\xce\u201c} such that
Assume that either f is continuous or X is such that if an increasing sequence {x}_{n}\xe2\u2020\u2019x\xe2\u02c6\u02c6X, then {x}_{n}\xe2\u2030\xa4x, âˆ€n. Besides, if for each x,y\xe2\u02c6\u02c6X, there exists z\xe2\u02c6\u02c6X which is comparable to x and y, then f has a unique fixed point.
Theorem 1.4 [3]
Let (X,\xe2\u2030\xa4) be a partially ordered set, and suppose that there exists a metric d\xe2\u02c6\u02c6X such that (X,d) is a complete metric space. Let f:X\xe2\u2020\u2019X be a continuous and nondecreasing mapping such that
where \mathrm{\xcf\u02c6}:[0,\mathrm{\xe2\u02c6\u017e})\xe2\u2020\u2019[0,\mathrm{\xe2\u02c6\u017e}) is a continuous and nondecreasing function such that Ïˆ is positive in (0,\mathrm{\xe2\u02c6\u017e}), \mathrm{\xcf\u02c6}(0)=0 and {lim}_{t\xe2\u2020\u2019\mathrm{\xe2\u02c6\u017e}}\mathrm{\xcf\u02c6}(t)=\mathrm{\xe2\u02c6\u017e}. If there exists {x}_{0}\xe2\u02c6\u02c6X with {x}_{0}\xe2\u2030\xa4f({x}_{0}), then f has a fixed point.
Definition 1.5 [4]
An altering distance function is a function \mathrm{\xcf\u02c6}:[0,\mathrm{\xe2\u02c6\u017e})\xe2\u2020\u2019[0,\mathrm{\xe2\u02c6\u017e}) which satisfies

(i)
Ïˆ is continuous and nondecreasing.

(ii)
\mathrm{\xcf\u02c6}(t)=0 if and only if t=0.
Theorem 1.6 [4]
Let X be a partially ordered set, and suppose that there exists a metric d in X such that (X,d) is a complete metric space. Let T:X\xe2\u2020\u2019X be a continuous and nondecreasing mapping such that
where Ïˆ is an altering distance function and \mathrm{\xcf\u2022}:[0,\mathrm{\xe2\u02c6\u017e})\xe2\u2020\u2019[0,\mathrm{\xe2\u02c6\u017e}) is a continuous function with the condition \mathrm{\xcf\u02c6}(t)>\mathrm{\xcf\u2022}(t) for all t>0. If there exists {x}_{0}\xe2\u02c6\u02c6X such that {x}_{0}\xe2\u2030\xa4T{x}_{0}, then T has a fixed point.
Theorem 1.7 [5]
Let (X,\xe2\u2030\xa4) be a partially ordered set, and suppose that there exists a metric d\xe2\u02c6\u02c6X such that (X,d) is a complete metric space. Let f:X\xe2\u2020\u2019X be a continuous and nondecreasing mapping such that
where Ïˆ and Ï• are altering distance functions. If there exists {x}_{0}\xe2\u02c6\u02c6X with {x}_{0}\xe2\u2030\xa4f({x}_{0}), then f has a fixed point.
In 2012, Caballero et al. introduced a generalized Geraghtycontraction as follows.
Definition 1.8 [6]
Let A, B be two nonempty subsets of a metric space (X,d). A mapping T:A\xe2\u2020\u2019B is said to be a Geraghtycontraction if there exists \mathrm{\xce\xb2}\xe2\u02c6\u02c6\mathrm{\xce\u201c} such that for any x,y\xe2\u02c6\u02c6A,
Now we need the following notations and basic facts. Let A and B be two nonempty subsets of a metric space (X,d). We denote by {A}_{0} and {B}_{0} the following sets:
where d(A,B)=inf\{d(x,y):x\xe2\u02c6\u02c6A\text{and}y\xe2\u02c6\u02c6B\}.
In [7], the authors give sufficient conditions for when the sets {A}_{0} and {B}_{0} are nonempty. In [8], the authors prove that any pair (A,B) of nonempty, closed convex subsets of a uniformly convex Banach space satisfies the Pproperty.
Definition 1.9 [9]
Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with {A}_{0}\xe2\u2030\mathrm{\xe2\u02c6\dots}. Then the pair (A,B) is said to have the Pproperty if and only if for any {x}_{1},{x}_{2}\xe2\u02c6\u02c6{A}_{0} and {y}_{1},{y}_{2}\xe2\u02c6\u02c6{B}_{0},
Let A, B be two nonempty subsets of a complete metric space, and consider a mapping T:A\xe2\u2020\u2019B. The best proximity point problem is whether we can find an element {x}_{0}\xe2\u02c6\u02c6A such that d({x}_{0},T{x}_{0})=min\{d(x,Tx):x\xe2\u02c6\u02c6A\}. Since d(x,Tx)\xe2\u2030\yen d(A,B) for any x\xe2\u02c6\u02c6A, in fact, the optimal solution to this problem is the one for which the value d(A,B) is attained.
In [6], the authors give a generalization of Theorem 1.2 by considering a nonself mapping, and they get the following theorem.
Theorem 1.10 [6]
Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that {A}_{0} is nonempty. Let T:A\xe2\u2020\u2019B be a Geraghtycontraction satisfying T({A}_{0})\xe2\u0160\u2020{B}_{0}. Suppose that the pair (A,B) has the Pproperty. Then there exists a unique {x}^{\xe2\u02c6\u2014} in A such that d({x}^{\xe2\u02c6\u2014},T{x}^{\xe2\u02c6\u2014})=d(A,B).
Inspired by [6], the purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, a series of best proximity point problems can be solved by our Poperator technique, which changes a nonself mapping to a selfmapping. Some recent results in this area have been improved.
2 Main results
Before giving our main results, we first introduce the weak Pmonotone property.
Weak Pmonotone property Let (A,B) be a pair of nonempty subsets of a partially ordered metric space (X,d) with {A}_{0}\xe2\u2030\mathrm{\xe2\u02c6\dots}. Then the pair (A,B) is said to have the weak Pmonotone property if for any {x}_{1},{x}_{2}\xe2\u02c6\u02c6{A}_{0} and {y}_{1},{y}_{2}\xe2\u02c6\u02c6{B}_{0},
furthermore, {y}_{1}\xe2\u2030\yen {y}_{2} implies {x}_{1}\xe2\u2030\yen {x}_{2}.
Now we are in a position to give our main results.
Theorem 2.1 Let (X,\xe2\u2030\xa4) be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\xe2\u2030\mathrm{\xe2\u02c6\dots}. Let f:A\xe2\u2020\u2019B be an increasing mapping with f({A}_{0})\xe2\u0160\u2020{B}_{0}, and let there exist \mathrm{\xce\xb2}\xe2\u02c6\u02c6\mathrm{\xce\u201c} such that
Assume that either f is continuous or that {\stackrel{\xc2\xaf}{A}}_{0} is such that if an increasing sequence {x}_{n}\xe2\u2020\u2019x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}, then {x}_{n}\xe2\u2030\xa4x, âˆ€n. Suppose that the pair (A,B) has the weak Pmonotone property. And for some {x}_{0}\xe2\u02c6\u02c6{A}_{0}, there exists {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{\stackrel{\xcb\u2020}{x}}_{0})=d(A,B) and {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u2030\xa4f({x}_{0}). Besides, if for each x,y\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}, there exists z\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0} which is comparable to x and y, then there exists an {x}^{\xe2\u02c6\u2014} in A such that d({x}^{\xe2\u02c6\u2014},f{x}^{\xe2\u02c6\u2014})=d(A,B).
Proof We first prove that {B}_{0} is closed. Let \{{y}_{n}\}\xe2\u0160\u2020{B}_{0} be a sequence such that {y}_{n}\xe2\u2020\u2019q\xe2\u02c6\u02c6B. It follows from the weak Pmonotone property that
as n,m\xe2\u2020\u2019\mathrm{\xe2\u02c6\u017e}, where {x}_{n},{x}_{m}\xe2\u02c6\u02c6{A}_{0} and d({x}_{n},{y}_{n})=d(A,B), d({x}_{m},{y}_{m})=d(A,B). Then \{{x}_{n}\} is a Cauchy sequence so that \{{x}_{n}\} converges strongly to a point p\xe2\u02c6\u02c6A. By the continuity of a metric d, we have d(p,q)=d(A,B), that is, q\xe2\u02c6\u02c6{B}_{0}, and hence {B}_{0} is closed.
Let {\stackrel{\xc2\xaf}{A}}_{0} be the closure of {A}_{0}, we claim that f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u0160\u2020{B}_{0}. In fact, if x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}\xe2\u02c6\u2013{A}_{0}, then there exists a sequence \{{x}_{n}\}\xe2\u0160\u2020{A}_{0} such that {x}_{n}\xe2\u2020\u2019x. By the continuity of f and the closeness of {B}_{0}, we have fx={lim}_{n\xe2\u2020\u2019\mathrm{\xe2\u02c6\u017e}}f{x}_{n}\xe2\u02c6\u02c6{B}_{0}. That is, f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u0160\u2020{B}_{0}.
Define an operator {P}_{{A}_{0}}:f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u2020\u2019{A}_{0}, by {P}_{{A}_{0}}y=\{x\xe2\u02c6\u02c6{A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property and f is increasing, we have
for any y\xe2\u2030\yen x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}. Obviously, {P}_{{A}_{0}}f is increasing. Let {x}_{n}, x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}, {x}_{n}\xe2\u2020\u2019x, when f is continuous, then we have
Then {P}_{{A}_{0}}f is continuous. When {\stackrel{\xc2\xaf}{A}}_{0} is such that if an increasing sequence {x}_{n}\xe2\u2020\u2019x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}, then {x}_{n}\xe2\u2030\xa4x (âˆ€n), we need not prove the continuity of {P}_{{A}_{0}}f. For some {x}_{0}\xe2\u02c6\u02c6{A}_{0}, there exists {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{\stackrel{\xcb\u2020}{x}}_{0})=d(A,B) and {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u2030\xa4f({x}_{0}). That is,
By the weak Pmonotone property, we have {P}_{{A}_{0}}f{x}_{0}\xe2\u2030\yen {x}_{0}.
This shows that {P}_{{A}_{0}}f:{\stackrel{\xc2\xaf}{A}}_{0}\xe2\u2020\u2019{\stackrel{\xc2\xaf}{A}}_{0} is a contraction satisfying all the conditions in Theorem 1.3. Using Theorem 1.3, we can get {P}_{{A}_{0}}f has a unique fixed point {x}^{\xe2\u02c6\u2014}. That is, {P}_{{A}_{0}}f{x}^{\xe2\u02c6\u2014}={x}^{\xe2\u02c6\u2014}\xe2\u02c6\u02c6{A}_{0}, which implies that
Therefore, {x}^{\xe2\u02c6\u2014} is the unique one in {A}_{0} such that d({x}^{\xe2\u02c6\u2014},f{x}^{\xe2\u02c6\u2014})=d(A,B).â€ƒâ–¡
Theorem 2.2 Let (X,\xe2\u2030\xa4) be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\xe2\u2030\mathrm{\xe2\u02c6\dots}. Let f:A\xe2\u2020\u2019B be a continuous and nondecreasing mapping with f({A}_{0})\xe2\u0160\u2020{B}_{0}, and let f satisfy
where \mathrm{\xcf\u02c6}:[0,\mathrm{\xe2\u02c6\u017e})\xe2\u2020\u2019[0,\mathrm{\xe2\u02c6\u017e}) is a continuous and nondecreasing function such that Ïˆ is positive in (0,\mathrm{\xe2\u02c6\u017e}), \mathrm{\xcf\u02c6}(0)=0 and {lim}_{t\xe2\u2020\u2019\mathrm{\xe2\u02c6\u017e}}\mathrm{\xcf\u02c6}(t)=\mathrm{\xe2\u02c6\u017e}. Suppose that the pair (A,B) has the weak Pmonotone property. If for some {x}_{0}\xe2\u02c6\u02c6{A}_{0}, there exists {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{\stackrel{\xcb\u2020}{x}}_{0})=d(A,B) and {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u2030\xa4f({x}_{0}), then there exists an {x}^{\xe2\u02c6\u2014} in A such that d({x}^{\xe2\u02c6\u2014},f{x}^{\xe2\u02c6\u2014})=d(A,B).
Proof In Theorem 2.1, we have proved that {B}_{0} is closed and f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u0160\u2020{B}_{0}. Now we define an operator {P}_{{A}_{0}}:f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u2020\u2019{A}_{0} by {P}_{{A}_{0}}y=\{x\xe2\u02c6\u02c6{A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property, by the definition of f, we have
for any y\xe2\u2030\yen x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}. Obviously, {P}_{{A}_{0}}f is continuous and nondecreasing. For some {x}_{0}\xe2\u02c6\u02c6{A}_{0}, there exists {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{\stackrel{\xcb\u2020}{x}}_{0})=d(A,B) and {\stackrel{\xcb\u2020}{x}}_{0}\xe2\u2030\xa4f({x}_{0}). That is,
By the weak Pmonotone property, we have {P}_{{A}_{0}}f{x}_{0}\xe2\u2030\yen {x}_{0}.
This shows that {P}_{{A}_{0}}f:{\stackrel{\xc2\xaf}{A}}_{0}\xe2\u2020\u2019{\stackrel{\xc2\xaf}{A}}_{0} is a contraction satisfying all the conditions in Theorem 1.4. Using Theorem 1.4, we can get {P}_{{A}_{0}}f has a fixed point {x}^{\xe2\u02c6\u2014}. That is, {P}_{{A}_{0}}f{x}^{\xe2\u02c6\u2014}={x}^{\xe2\u02c6\u2014}\xe2\u02c6\u02c6{A}_{0}, which implies that
Therefore, {x}^{\xe2\u02c6\u2014} is the one in {A}_{0} such that d({x}^{\xe2\u02c6\u2014},f{x}^{\xe2\u02c6\u2014})=d(A,B).â€ƒâ–¡
Theorem 2.3 Let X be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\xe2\u2030\mathrm{\xe2\u02c6\dots}. Let T:A\xe2\u2020\u2019B be a continuous and nondecreasing mapping with T({A}_{0})\xe2\u0160\u2020{B}_{0}, and let T satisfy
where Ïˆ is an altering distance function and \mathrm{\xcf\u2022}:[0,\mathrm{\xe2\u02c6\u017e})\xe2\u2020\u2019[0,\mathrm{\xe2\u02c6\u017e}) is a continuous function with the condition \mathrm{\xcf\u02c6}(t)>\mathrm{\xcf\u2022}(t) for all t>0. Suppose that the pair (A,B) has the weak Pmonotone property. If for some {x}_{0}\xe2\u02c6\u02c6{A}_{0}, there exists {y}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\xe2\u2030\xa4T{x}_{0}, then there exists an {x}^{\xe2\u02c6\u2014} in A such that d({x}^{\xe2\u02c6\u2014},T{x}^{\xe2\u02c6\u2014})=d(A,B).
Proof In Theorem 2.1, we have proved that {B}_{0} is closed and f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u0160\u2020{B}_{0}. Define an operator {P}_{{A}_{0}}:T({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u2020\u2019{A}_{0} by {P}_{{A}_{0}}y=\{x\xe2\u02c6\u02c6{A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property and Ïˆ is nondecreasing, by the definition of T, we have
for any y\xe2\u2030\yen x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}. Since
this shows that {P}_{{A}_{0}}T is continuous and nondecreasing. Because there exist {x}_{0}\xe2\u02c6\u02c6{A}_{0} and {y}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\xe2\u2030\xa4T{x}_{0}, by the weak Pmonotone property, we have {x}_{0}\xe2\u2030\xa4{P}_{{A}_{0}}T{x}_{0}.
This shows that {P}_{{A}_{0}}T:{\stackrel{\xc2\xaf}{A}}_{0}\xe2\u2020\u2019{\stackrel{\xc2\xaf}{A}}_{0} is a contraction from a complete metric subspace {\stackrel{\xc2\xaf}{A}}_{0} into itself and satisfies all the conditions in Theorem 1.6. Using Theorem 1.6, we can get {P}_{{A}_{0}}T has a fixed point {x}^{\xe2\u02c6\u2014}. That is, {P}_{{A}_{0}}T{x}^{\xe2\u02c6\u2014}={x}^{\xe2\u02c6\u2014}\xe2\u02c6\u02c6{A}_{0}, which implies that
Therefore, {x}^{\xe2\u02c6\u2014} is the one in {A}_{0} such that d({x}^{\xe2\u02c6\u2014},T{x}^{\xe2\u02c6\u2014})=d(A,B).â€ƒâ–¡
Theorem 2.4 Let (X,\xe2\u2030\xa4) be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\xe2\u2030\mathrm{\xe2\u02c6\dots}. Let T:A\xe2\u2020\u2019B be a continuous and nondecreasing mapping with T({A}_{0})\xe2\u0160\u2020{B}_{0}, and let T satisfy
where Ïˆ and Ï• are altering distance functions. Suppose that the pair (A,B) has the weak Pmonotone property. If for some {x}_{0}\xe2\u02c6\u02c6{A}_{0}, there exists {y}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\xe2\u2030\xa4T{x}_{0}, then there exists an {x}^{\xe2\u02c6\u2014} in A such that d({x}^{\xe2\u02c6\u2014},T{x}^{\xe2\u02c6\u2014})=d(A,B).
Proof In Theorem 2.1, we have proved that {B}_{0} is closed and f({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u0160\u2020{B}_{0}. Define an operator {P}_{{A}_{0}}:T({\stackrel{\xc2\xaf}{A}}_{0})\xe2\u2020\u2019{A}_{0} by {P}_{{A}_{0}}y=\{x\xe2\u02c6\u02c6{A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property and Ïˆ is nondecreasing, by the definition of T, we have
for any y\xe2\u2030\yen x\xe2\u02c6\u02c6{\stackrel{\xc2\xaf}{A}}_{0}. Since
This shows that {P}_{{A}_{0}}T is continuous and nondecreasing. Because there exist {x}_{0}\xe2\u02c6\u02c6{A}_{0} and {y}_{0}\xe2\u02c6\u02c6{B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\xe2\u2030\xa4T{x}_{0}, by the weak Pmonotone property, we have {x}_{0}\xe2\u2030\xa4{P}_{{A}_{0}}T{x}_{0}.
This shows that {P}_{{A}_{0}}T:{\stackrel{\xc2\xaf}{A}}_{0}\xe2\u2020\u2019{\stackrel{\xc2\xaf}{A}}_{0} is a contraction from a complete metric subspace {\stackrel{\xc2\xaf}{A}}_{0} into itself and satisfies all the conditions in Theorem 1.7. Using Theorem 1.7, we can get {P}_{{A}_{0}}T has a fixed point {x}^{\xe2\u02c6\u2014}. That is, {P}_{{A}_{0}}T{x}^{\xe2\u02c6\u2014}={x}^{\xe2\u02c6\u2014}\xe2\u02c6\u02c6{A}_{0}, which implies that
Therefore, {x}^{\xe2\u02c6\u2014} is the one in {A}_{0} such that d({x}^{\xe2\u02c6\u2014},T{x}^{\xe2\u02c6\u2014})=d(A,B).â€ƒâ–¡
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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Zhang, J., Su, Y. & Cheng, Q. Best proximity point theorems for generalized contractions in partially ordered metric spaces. Fixed Point Theory Appl 2013, 83 (2013). https://doi.org/10.1186/16871812201383
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DOI: https://doi.org/10.1186/16871812201383