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A best proximity pair theorem

100%

Shahzad N.

Demonstratio Mathematica

2007

tom Vol. 40, nr 4

869-874

The aim of this notę is to obtain a best proximity pair theorem which contains a recent result of Kirk, Reich and Veeramani (Numer. Funct. Anal. Optim., 24 (2003), 851-862) as a special case.

Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings

72%

Gabeleh M. , Künzi H.-P. A.

Demonstratio Mathematica

2020

tom Vol. 53, nr 1

38--43

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

Coincidence points an best proximity pair results for R-subweakly commuting multimaps

72%

O'Regan D. , Shahzad N.

Demonstratio Mathematica

2006

tom Vol. 39, nr 4

845-854

Some coincidence point theorems for R-subweakly commuting mappings satisfying a general contractive condition are proved. As applications, some best proximity pair results are also obtained and several related results in the literature are extended to a new class of noncommuting mappings.