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Best proximity point theorems generalizing the contraction principle. (English) Zbl 1238.54021

Let \(A\) and \(B\) be non-void subsets of a metric space \((X,d)\) and \(d(A,B)= \text{inf}\{d(x,y): x\in A\) and \(y\in B\}\). An element \(x\in A\) is said to be a best proximity point of the mapping \(S: A\to B\) if \(d(x,Sx)= d(A,B)\).
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 \(d(x,Sx)\) is minimal, which implies that \(x\) and \(Sx\) are in close proximity to each other. The fact that \(d(x,Sx)\) is at least \(d(A,B)\), best proximity point theorems guarantee the existence of an element \(x\) such that \(d(x,Sx)= d(A,B)\). 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 (topological aspects)
54E50 Complete metric spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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[1] Eldred, A. A.; Veeramani, P., Existence and Convergence of best proximity points, J. Math. Anal. Appl., 323, 1001-1006 (2006) · Zbl 1105.54021
[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
[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) · Zbl 1171.54311
[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
[5] Fan, K., Extensions of two fixed point theorems of F.E. Browder, Math. Z., 112, 234-240 (1969) · Zbl 0185.39503
[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
[7] Reich, S., Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl., 62, 104-113 (1978) · Zbl 0375.47031
[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
[9] Sehgal, V. M.; Singh, S. P., A theorem on best approximations, Numer. Funct. Anal. Optim., 10, 181-184 (1989) · Zbl 0635.41022
[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
[11] Sadiq Basha, S.; Veeramani, P., Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103, 119-129 (2000) · Zbl 0965.41020
[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
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