Some Results on Fixed and Best Proximity Points of Multivalued

This paper is devoted to investigate the fixed points and best proximity points of multivalued cyclic self-mappings on a set of subsets of complete metric spaces endowed with a partial order under a generalized contractive condition involving a Hausdorff distance. The existence and uniqueness of fixed points of both the cyclic self-mapping and its associate composite self-mappings on each of the subsets are investigated, if the subsets in the cyclic disposal are nonempty, bounded and of nonempty convex intersection. The obtained results are extended to the existence of unique best proximity points in uniformly convex Banach spaces.Spanish Government DPI2012-30651; Basque Government IT378-10 and SAIOTEK S-PE12UN015; UPV/EHU UFI 2011/0

Existence of fixed point and best point of proximity for multifunctional non self mappings in a partial metric space

In this paper we give some theorems of existence of best points of proximity for a multifunctional non-self-mapping in a partial metric space and some approximations on the sets of the best points of proximity. Other results are also given.The authors are partially supported by operator laboratory, Eloued, Algeria.Publisher's Versio

On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.The first author thanks the Spanish Ministry of Economy and Competitiveness for partial support of this work through Grant DPI2012-30651. He also thanks the Basque Government for its support through Grant IT378-10, and to the University of Basque Country by its support through Grant UFI 11/07

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. Verto, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008), 3790-3794. https://doi.org/10.1016/j.na.2007.10.014A. 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/sm171-3-5R. 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/10.1080/01630563.2013.763824A. 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-yM. 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/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. 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/10.1186/1687-1812-2011-69E. 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/10.1186/1687-1812-2012-42W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345W. 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-120026380U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005), 89-128. https://doi.org/10.1090/S0002-9947-04-03515-9V. 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-xT. 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.1996.028T. 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.2009.01.17

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

On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-called r-weaker Meir-Keeler or r,r0-stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed

On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.The first author thanks the Spanish Ministry of Economy and Competitiveness for partial support of this work through Grant DPI2012-30651. He also thanks the Basque Government for its support through Grant IT378-10, and to the University of Basque Country by its support through Grant UFI 11/07

Generalization of best proximity points theorem for non-self proximal contractions of first kind

The primary objective of this paper is the study of the generalization of some results given by Basha (Numer. Funct. Anal. Optim. 31:569-576, 2010). We present a new theorem on the existence and uniqueness of best proximity points for proximal ?-quasi-contractive mappings for non-self-mappings S: M? N and T: N? M. Furthermore, as a consequence, we give a new result on the existence and uniqueness of a common fixed point of two self mappings. - 2019, The Author(s).Scopu

On Proximal contractions via implicit relations and Best proximity points

In this paper, we employ two types of implicit relations to define some new kind of proximal contractions and study about their best proximity points. More precisely, we use two class of functions $\mathcal{A}$ and $\mathcal{A}'$ to explore proximal $\mathcal{A}$, $\mathcal{A}'$-contractions of first and second type and strong proximal $\mathcal{A}$, $\mathcal{A}'$-contractions. We investigate the existence of best proximity point results of the same. It is worth mentioning that the well-known results of Sadiq Basha [J. Approx. Theory, $2011$] on proximal contractions are the special cases of our obtained results. We authenticate our results by suitable examples.Comment: 16 page