AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 189567 10.1155/2013/189567 189567 Research Article A Note on Some Best Proximity Point Theorems Proved under P-Property Abkar Ali 1 Gabeleh Moosa 2 Jleli Mohamed 1 Department of Mathematics Imam Khomeini International University Qazvin 34149 Iran ikiu.ac.ir 2 Department of Mathematics Ayatollah Boroujerdi University Borujerd Iran 2013 4 11 2013 2013 20 07 2013 29 09 2013 2013 Copyright © 2013 Ali Abkar and Moosa Gabeleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We show that some recent results concerning the existence of best proximity points can be obtained from the same results in fixed point theory.

1. Introduction

Let A and B be two nonempty subsets of a metric space (X,d). In this paper, we adopt the following notations and definitions: (1)D(x,B)=inf{d(x,y):yB},xX,A0={xA:d(x,y)=dist(A,B),  for  some  yB},B0={yB:d(x,y)=dist(A,B),  for  some  xA}.

The notion of best proximity point is defined as follows.

Definition 1.

Let A and B be nonempty subsets of a metric space (X,d) and T:AB a non-self-mapping. A point x*A is called a best proximity point of T if d(x*,Tx*)=dist(A,B), where (2)dist(A,B)=inf{d(x,y):(x,y)A×B}.

Similarly, for a multivalued non-self-mapping T:A2B, where (A,B) is a nonempty pair of subsets of a metric space (X,d), a point x*A is a best proximity point of T provided that D(x*,Tx*)=dist(A,B).

Recently, the notion of P-property was introduced in  as follows.

Definition 2 (see [<xref ref-type="bibr" rid="B11">1</xref>]).

Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with A0. The pair (A,B) is said to have P-property if and only if (3)d(x1,y1)=dist(A,B)d(x2,y2)=dist(A,B)d(x1,x2)=d(y1,y2), where x1,x2A0 and y1,y2B0.

By using this notion, some best proximity point results were proved for various classes of non-self-mappings. Here, we state some of them.

Theorem 3 (see [<xref ref-type="bibr" rid="B11">1</xref>]).

Let (A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:AB be a weakly contractive non-self-mapping; that is, (4)d(Tx,Ty)d(x,y)-ϕ(d(x,y))x,yA, where ϕ:[0,)[0,) is a continuous and nondecreasing function such that ϕ is positive on (0,), ϕ(0)=0, and limtϕ(t)=. Assume that the pair (A,B) has the P-property and T(A0)B0. Then, T has a unique best proximity point.

Theorem 4 (see [<xref ref-type="bibr" rid="B1">2</xref>]).

Let (A,B) be a pair of nonempty closed subsets of a Banach space X such that A is compact and A0 is nonempty. Let T:AB be a nonexpansive mapping; that is, (5)Tx-Tyx-y,  x,yA. Assume that the pair (A,B) has the P-property and T(A0)B0. Then, T has a best proximity point.

Theorem 5 (see [<xref ref-type="bibr" rid="B10">3</xref>]).

Let (A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:AB be a Meir-Keeler non-self-mapping; that is, for all x,yA and for any ε>0, there exists δ(ε)>0 such that (6)εd(x,y)<ε+δimplies  d(Tx,Ty)ε. Assume that the pair (A,B) has the P-property and T(A0)B0. Then, T has a unique best proximity point.

Theorem 6 (see [<xref ref-type="bibr" rid="B2">4</xref>]).

Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A0 and (A,B) satisfies the P-property. Let T:A2B be a multivalued contraction non-self-mapping; that is, (7)H(Tx,Ty)αd(x,y), for some α(0,1) and for all x,yA. If  Tx is bounded and closed in B for all xA and Tx0 is included in B0 for each x0A0, then T has a best proximity point in A.

Theorem 7 (see [<xref ref-type="bibr" rid="B4">5</xref>]).

Let (A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:AB be a Geraghty-contraction non-self-mapping; that is, (8)d(Tx,Ty)β(d(x,y)),d(x,y),  x,yA, where β:[0,)[0,1) is a function which satisfies the following condition: (9)β(tn)1tn0. Suppose that the pair (A,B) has the P-property and T(A0)B0. Then, T has a unique best proximity point.

2. Main Result

In this section, we show that the existence of a best proximity point in the main theorems of  can be obtained from the existence of the fixed point for a self-map. We begin our argument with the following lemmas.

Lemma 8 (see [<xref ref-type="bibr" rid="B5">6</xref>]).

Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A0 is nonempty and (A,B) has the P-property. Then, (A0,B0) is a closed pair of subsets of X.

Lemma 9.

Let (A,B) be a pair of nonempty closed subsets of a metric space (X,d) such that A0 is nonempty. Assume that the pair (A,B) has the P-property. Then there exists a bijective isometry g:A0B0 such that d(x,gx)=dist(A,B).

Proof.

Let xA0; then there exists an element yB0 such that (10)d(x,y)=dist(A,B). Assume that there exists another point y´B0 such that (11)d(x,y´)=dist(A,B). By the fact that (A,B) has the P-property, we conclude that y=y´. Consider the non-self-mapping g:A0B0 such that d(x,gx)=dist(A,B). Clearly, g is well defined. Moreover, g is an isometry. Indeed, if x1,x2A0, then (12)d(x1,gx1)=dist(A,B),d(x2,gx2)=dist(A,B). Again, since (A,B) has the P-property, (13)d(x1,x2)=d(gx1,gx2); that is, g is an isometry.

Here, we prove that the existence and uniqueness of the best proximity point in Theorem 3 are a sample result of the existence of fixed point for a weakly contractive self-mapping.

Theorem 10.

Let (A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:AB be a weakly contractive mapping. Assume that the pair (A,B) has the P-property and T(A0)B0. Then, T has a unique best proximity point.

Proof.

Consider the bijective isometry g:A0B0 as in Lemma 9. Since T(A0)B0, for the self-mapping g-1T:A0A0, we have (14)d(g-1(Tx),g-1(Ty))=d(Tx,Ty)φ(d(x,y)), for all x,yA0 which implies that the self-mapping g-1T is weakly contractive. Note that A0 is closed by Lemma 8. Thus, g-1T has a unique fixed point . Suppose that x*A0 is a unique fixed point of the self-mapping g-1T; that is, g-1T(x*)=x*. So, Tx*=gx*, and then (15)d(x*,Tx*)=d(x*,gx*)=dist(A,B), from which it follows that x*A0 is a unique best proximity point of the non-self weakly contractive mapping T.

Remark 11.

By a similar argument, using the fact that every nonexpansive self-mapping defined on a nonempty compact and convex subset of a Banach space has a fixed point, we conclude Theorem 4. Also, the existence and uniqueness of best proximity point for Meir-Keeler non-self-mapping T (Theorem 5) follow from the Meir-Keeler's fixed point theorem (). Moreover, in Theorem 6, Nadler's fixed point theorem () ensures the existence of a best proximity point for multivalued non-self mapping T. Finally, Theorem 7 due to Caballero et al., is obtained from Geraghty's fixed point theorem ().

Raj V. S. A best proximity point theorem for weakly contractive non-self-mappings Nonlinear Analysis. Theory, Methods & Applications 2011 74 14 4804 4808 10.1016/j.na.2011.04.052 MR2810719 ZBL1228.54046 Abkar A. Gabeleh M. Best proximity points of non-self mappings TOP 2013 21 2 287 295 2-s2.0-84858066181 10.1007/s11750-012-0255-7 Samet B. Some results on best proximity point theorem Journal of Optimization Theory and Applications 2013 159 1 281 291 10.1007/s10957-013-0269-9 MR3103299 Abkar A. Gabeleh M. The existence of best proximity points for multivalued non-self-mappings Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales A 2013 107 2 319 325 10.1007/s13398-012-0074-6 Caballero J. Harjani J. Sadarangani K. A best proximity point theorem for Geraghty-contractions Fixed Point Theory and Applications 2012 2012, article 231 10.1186/1687-1812-2012-231 Gabeleh M. Proximal weakly contractive and proximal nonexpansive non-self-mappings in metric and Banach spaces Journal of Optimization Theory and Applications 2013 158 2 615 625 10.1007/s10957-012-0246-8 MR3084393 ZBL06210557 Rhoades B. E. Some theorems on weakly contractive maps Nonlinear Analysis, Theory, Methods and Applications 2001 47 4 2683 2693 2-s2.0-0035420958 10.1016/S0362-546X(01)00388-1 Meir A. Keeler E. A theorem on contraction mappings Journal of Mathematical Analysis and Applications 1969 28 326 329 MR0250291 10.1016/0022-247X(69)90031-6 ZBL0194.44904 Nadler, S. B. Jr. Multi-valued contraction mappings Pacific Journal of Mathematics 1969 30 475 488 MR0254828 10.2140/pjm.1969.30.475 ZBL0187.45002 Geraghty M. A. On contractive mappings Proceedings of the American Mathematical Society 1973 40 604 608 MR0334176 10.1090/S0002-9939-1973-0334176-5 ZBL0245.54027