Weak proximal normal structure and coincidence quasi-best proximity points





pointwise cyclic-noncyclic pairs, weak proximal normal structure, coincidence quasi-best proximity point


We introduce the notion of pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. We study the best proximity point problem for this class of mappings. We also study the same problem for the class of pointwise noncyclic-noncyclic relatively nonexpansive pairs involving orbits. Finally, under the assumption of weak proximal normal structure, we prove a coincidence quasi-best proximity point theorem for pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. Examples are provided to illustrate the observed results.


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Author Biographies

Farhad Fouladi, Imam Khomeini International University

Department of Pure Mathemathics,Faculty of Science

Ali Abkar, Imam Khomeini International University

Department of Pure Mathemathics, Faculty of Science

Erdal Karapinar, Thu Dau Mot University

ETSI Division of Applied Mathematics


A. Abkar and M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theorey. Appl. 150 (2011), 188-193. https://doi.org/10.1007/s10957-011-9810-x

A.Abkar and M. Norouzian, Coincidence quasi-best proximity points for quasi-cyclic-noncyclic mappings in convex metric spaces, Iranian Journal of Mathematical Sciences and Informatics, to appear.

M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), 3665-3671. https://doi.org/10.1016/j.na.2008.07.022

M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk USSR 59 (1948), 837-840 (in Russian).

M. De la Sen, Some results on fixed and best proximity points of multivalued cyclic self mappings with a partial order, Abst. Appl. Anal. 2013 (2013), Article ID 968492, 11 pages. https://doi.org/10.1155/2013/968492

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. 59 (2011), 14 pages. https://doi.org/10.1186/1687-1812-2011-59

C. Di Bari, T. Suzuki and C. Verto, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008), 3790-3794. https://doi.org/10.1016/j.na.2007.10.014

A. A. Eldred, W. A. Kirk and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), 283-293. https://doi.org/10.4064/sm171-3-5

R. Espinola, M. Gabeleh and P. Veeramani, On the structure of minimal sets of relatively nonexpansive mappings, Numer. Funct. Anal. Optim. 34 (2013), 845-860. https://doi.org/10.1080/01630563.2013.763824

A. F. Leon and M. Gabeleh, Best proximity pair theorems for noncyclic mappings in Banach and metric spaces, Fixed Point Theory 17 (2016), 63-84.

M. Gabeleh, A characterization of proximal normal structure via proximal diametral sequences, J. Fixed Point Theory Appl. 19 (2017), 2909-2925. https://doi.org/10.1007/s11784-017-0460-y

M. Gabeleh, O. Olela Otafudu and N. Shahzad, Coincidence best proximity points in convex metric spaces, Filomat 32 (2018), 1-12. https://doi.org/10.2298/FIL1801001D

M. Gabeleh, H. Lakzian and N.Shahzad, Best proximity points for asymptotic pointwise contractions, J. Nonlinear Convex Anal. 16 (2015), 83-93.

E. Karapinar, Best proximity points of Kannan type cyclic weak φ-contractions in ordered metric spaces, An. St. Univ. Ovidius Constanta. 20 (2012), 51-64. https://doi.org/10.2478/v10309-012-0055-y

H. Aydi, E. Karapinar, I. M. Erhan and P. Salimi, Best proximity points of generalized almost -ψ Geraghty contractive non-self mappings, Fixed Point Theory Appl. 2014:32 (2014). https://doi.org/10.1186/1687-1812-2014-32

N. Bilgili, E. Karapinar and K. Sadarangani, A generalization for the best proximity point of Geraghty-contractions, J. Ineqaul. Appl. 2013:286 (2013). https://doi.org/10.1186/1029-242X-2013-286

E. Karapinar and I. M. Erhan, Best proximity point on different type contractions, Appl. Math. Inf. Sci. 3, no. 3 (2011), 342-353.

E. Karapinar, Fixed point theory for cyclic weak $phi$-contraction, Appl. Math. Lett. 24, no. 6 (2011), 822-825. https://doi.org/10.1186/1687-1812-2011-69

E. Karapinar, G. Petrusel and K. Tas, Best proximity point theorems for KT-types cyclic orbital contraction mappings, Fixed Point Theory 13, no. 2 (2012), 537-546. https://doi.org/10.1186/1687-1812-2012-42

W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345

W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), 851-862. https://doi.org/10.1081/NFA-120026380

U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005), 89-128. https://doi.org/10.1090/S0002-9947-04-03515-9

V. Pragadeeswarar and M. Marudai, Best proximity points: approximation and optimization in partially ordered metric spaces, Optim. Lett. 7 (2013), 1883-1892. https://doi.org/10.1007/s11590-012-0529-x

T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topological Methods in Nonlin. Anal. 8 (1996), 197-203. https://doi.org/10.12775/TMNA.1996.028

T. Suzuki, M. Kikkawa and C. Vetro, The existence of best proximity points in metric spaces with to property UC, Nonlinear Anal. 71 (2009), 2918-2926. https://doi.org/10.1016/j.na.2009.01.173




How to Cite

F. Fouladi, A. Abkar, and E. Karapinar, “Weak proximal normal structure and coincidence quasi-best proximity points”, Appl. Gen. Topol., vol. 21, no. 2, pp. 331–347, Oct. 2020.