# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Best proximity point theorems generalizing the contraction principle. (English) Zbl 1238.54021

Let $A$ and $B$ be non-void subsets of a metric space $\left(X,d\right)$ and $d\left(A,B\right)=\text{inf}\left\{d\left(x,y\right):x\in A$ and $y\in B\right\}$. An element $x\in A$ is said to be a best proximity point of the mapping $S:A\to B$ if $d\left(x,Sx\right)=d\left(A,B\right)$.

Given non-void closed subsets $A$ and $B$ of a complete metric space, a contraction non-self-mapping $S:A\to B$ is improbable to have a fixed point. So, it is quite natural to seek an element $x$ such that $d\left(x,Sx\right)$ is minimal, which implies that $x$ and $Sx$ are in close proximity to each other. The fact that $d\left(x,Sx\right)$ is at least $d\left(A,B\right)$, best proximity point theorems guarantee the existence of an element $x$ such that $d\left(x,Sx\right)=d\left(A,B\right)$. The famous Banach contraction principle asserts that every contraction self-mapping on a complete metric space has a unique point. This article explores some interesting generalizations of the contraction principle to the case of non-self-mappings. The proposed extensions are presented as best proximity point theorems for non-self-proximal contractions.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E50 Complete metric spaces 41A65 Abstract approximation theory
##### References:
 [1] Eldred, A. A.; Veeramani, P.: Existence and convergence of best proximity points, J. math. Anal. appl. 323, 1001-1006 (2006) · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081 [2] Wlodarczyk, K.; Plebaniak, R.; Banach, A.: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear anal. 70, 3332-3341 (2009) · Zbl 1182.54024 · doi:10.1016/j.na.2008.11.020 [3] Wlodarczyk, K.; Plebaniak, R.; Banach, A.: Erratum to: ”best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces”, [Nonlinear anal. 70 (2009) 3332-3341, doi:10.1016/j.na.2008.04.037], Nonlinear anal. 71, 3583-3586 (2009) [4] Wlodarczyk, K.; Plebaniak, R.; Obczynski, C.: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear anal. 72, 794-805 (2010) · Zbl 1185.54020 · doi:10.1016/j.na.2009.07.024 [5] Fan, K.: Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112, 234-240 (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225 [6] Prolla, J. B.: Fixed point theorems for set valued mappings and existence of best approximations, Numer. funct. Anal. optim. 5, 449-455 (1982–1983) · Zbl 0513.41015 · doi:10.1080/01630568308816149 [7] Reich, S.: Approximate selections, best approximations, fixed points and invariant sets, J. math. Anal. appl. 62, 104-113 (1978) · Zbl 0375.47031 · doi:10.1016/0022-247X(78)90222-6 [8] Sehgal, V. M.; Singh, S. P.: A generalization to multifunctions of Fan’s best approximation theorem, Proc. amer. Math. soc. 102, 534-537 (1988) · Zbl 0672.47043 · doi:10.2307/2047217 [9] Sehgal, V. M.; Singh, S. P.: A theorem on best approximations, Numer. funct. Anal. optim. 10, 181-184 (1989) · Zbl 0635.41022 · doi:10.1080/01630568908816298 [10] Vetrivel, V.; Veeramani, P.; Bhattacharyya, P.: Some extensions of Fan’s best approximation theorem, Numer. funct. Anal. optim. 13, 397-402 (1992) · Zbl 0763.41026 · doi:10.1080/01630569208816486 [11] Basha, S. Sadiq; Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres, J. approx. Theory 103, 119-129 (2000) · Zbl 0965.41020 · doi:10.1006/jath.1999.3415 [12] Kirk, W. A.; Reich, S.; Veeramani, P.: Proximinal retracts and best proximity pair theorems, Numer. funct. Anal. optim. 24, 851-862 (2003) · Zbl 1054.47040 · doi:10.1081/NFA-120026380