Best Proximity Pair Theorems for Noncyclic Mappings in Banach and Metric Spaces

Let A and B be two nonempty subsets of a metric space X. A mapping T : A[B ! A[B is said to be noncyclic if T(A) A and T(B) B. For such a mapping, a pair (x; y) 2 A B such that Tx = x, Ty = y and d(x; y) = dist(A;B) is called a best proximity pair. In this paper we give some best proximity pair results for noncyclic mappings under certain contractive conditions

Best proximity point for proximal Berinde nonexpansive mappings on starshaped sets

summary:In this paper, we introduce the new concept of proximal mapping, namely proximal weak contractions and proximal Berinde nonexpansive mappings. We prove the existence of best proximity points for proximal weak contractions in metric spaces, and for proximal Berinde nonexpansive mappings on starshape sets in Banach spaces. Examples supporting our main results are also given. Our main results extend and generalize some of well-known best proximity point theorems of proximal nonexpansive mappings in the literatures

Weak proximal normal structure and coincidence quasi-best proximity points

[EN] 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.Fouladi, F.; Abkar, A.; Karapinar, E. (2020). Weak proximal normal structure and coincidence quasi-best proximity points. Applied General Topology. 21(2):331-347. https://doi.org/10.4995/agt.2020.13926OJS331347212A. 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-xA.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.2008.07.022M. 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/10.1155/2013/968492M. 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/10.1186/1687-1812-2011-59C. Di Bari, T. Suzuki and C. 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Shahzad, Coincidence best proximity points in convex metric spaces, Filomat 32 (2018), 1-12. https://doi.org/10.2298/FIL1801001DM. 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-yH. 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/10.1186/1687-1812-2014-32N. 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-2013-286E. Karapinar and I. M. Erhan, Best proximity point on different type contractions, Appl. Math. Inf. 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On a new variant of F-contractive mappings with application to fractional differential equations

The present article intends to prove the existence of best proximity points (pairs) using the notion of measure of noncompactness. We introduce generalized classes of cyclic (noncyclic) F-contractive operators, and then derive best proximity point (pair) results in Banach (strictly convex Banach) spaces. This work includes some of the recent results as corollaries. We apply these conclusions to prove the existence of optimum solutions for a system of Hilfer fractionaldifferential equations

Best proximity point theorems for α-nonexpansive mappings in Banach spaces

In this paper, we discuss sufficient and necessary conditions for the existence of best proximity points for non-self-a-nonexpansive mappings in Banach spaces. We obtain convergence results under some assumptions, and we prove the existence of common best proximity points for a family of non-self-a-nonexpansive mappings