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

• Autorzy: Gabeleh Moosa , Künzi Hans-Peter A.

Identyfikatory

DOI

10.1515/dema-2020-0005

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

best proximity pair; best proximity point; uniformly convex Banach space; noncyclic contraction; cyclic contraction

PL

twierdzenie o bliskości par; przestrzeń Banacha; przestrzeń jednostajnie wypukła

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2020
• Tom: Vol. 53, nr 1
• Strony: 38--43
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Gabeleh Moosa

• Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran

autor

Künzi Hans-Peter A.

• Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa

Bibliografia

• [1] M. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), 3665–3671.
• [2] M. De la Sen, Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings, Fixed Point Theory Appl. 2010 (2010), DOI: 10.1155/2010/572057.
• [3] M. De la Sen and R. P. Agarwal, Some fixed point-type results for a class of extended cyclic self-mappings with a more general contractive condition, Fixed Point Theory Appl. 2011 (2011), DOI: 10.1186/1687-1812-2011-59.
• [4] S. Karpagam and S. Agrawal, Best proximity point theorems for p-cyclic Meir-Keeler contractions, Fixed Point TheoryAppl. 2009 (2009), DOI: 10.1155/2009/197308.
• [5] M. Petric and B. Zlatanov, Best proximity points for p-cyclic summing iterated contractions, Filomat 32 (2018), 3275–3287.
• [6] A. A. Eldred, W. A. Kirk, and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), 283–293.
• [7] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), 1001–1006.
• [8] M. Gabeleh, Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), 227–239.
• [9] M. Gabeleh, Common best proximity pairs in strictly convex Banach spaces, Georgian Math. J. 24 (2017), 363–372.
• [10] A. Fernández-León and M. Gabeleh, Best proximity pair theorems for noncyclic mappings in Banach and metric spaces, Fixed Point Theory 17 (2016), no. 1, 63–84.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).