Best proximity points of contractive mappings on a metric space with a graph and applications

[EN] We establish an existence and uniqueness theorem on best proximity point for contractive mappings on a metric space endowed with a graph. As an application of this theorem, we obtain a result on the existence of unique best proximity point for uniformly locally contractive mappings. Moreover, our theorem subsumes and generalizes many recent fixed point and best proximity point results.The first author is thankful to University Grants Commission F.2 − 12/2002(SA − I), New Delhi, India for the financial support.Sultana, A.; Vetrivel, V. (2017). Best proximity points of contractive mappings on a metric space with a graph and applications. Applied General Topology. 18(1):13-21. https://doi.org/10.4995/agt.2017.3424SWORD1321181Dinevari, T., & Frigon, M. (2013). Fixed point results for multivalued contractions on a metric space with a graph. Journal of Mathematical Analysis and Applications, 405(2), 507-517. doi:10.1016/j.jmaa.2013.04.014Fan, K. (1969). Extensions of two fixed point theorems of F. E. Browder. Mathematische Zeitschrift, 112(3), 234-240. doi:10.1007/bf01110225Jachymski, J. (2007). The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 136(04), 1359-1373. doi:10.1090/s0002-9939-07-09110-1Kim, W. K., & Lee, K. H. (2006). Existence of best proximity pairs and equilibrium pairs. Journal of Mathematical Analysis and Applications, 316(2), 433-446. doi:10.1016/j.jmaa.2005.04.053Kim, W. K., Kum, S., & Lee, K. H. (2008). On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Analysis: Theory, Methods & Applications, 68(8), 2216-2227. doi:10.1016/j.na.2007.01.057Kirk, W. A., Reich, S., & Veeramani, P. (2003). Proximinal Retracts and Best Proximity Pair Theorems. Numerical Functional Analysis and Optimization, 24(7-8), 851-862. doi:10.1081/nfa-120026380Máté, L. (1993). The Hutchinson-Barnsley theory for certain non-contraction mappings. Periodica Mathematica Hungarica, 27(1), 21-33. doi:10.1007/bf01877158Nieto, J. J., & Rodríguez-López, R. (2005). Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22(3), 223-239. doi:10.1007/s11083-005-9018-5Pragadeeswarar, V., & Marudai, M. (2012). Best proximity points: approximation and optimization in partially ordered metric spaces. Optimization Letters, 7(8), 1883-1892. doi:10.1007/s11590-012-0529-xRan, A. C. M., & Reurings, M. C. B. (2004). Proceedings of the American Mathematical Society, 132(05), 1435-1444. doi:10.1090/s0002-9939-03-07220-4Sultana, A., & Vetrivel, V. (2014). Fixed points of Mizoguchi–Takahashi contraction on a metric space with a graph and applications. Journal of Mathematical Analysis and Applications, 417(1), 336-344. doi:10.1016/j.jmaa.2014.03.015Vetrivel, V., & Sultana, A. (2014). On the existence of best proximity points for generalized contractions. Applied General Topology, 15(1), 55. doi:10.4995/agt.2014.222

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

Fixed point results for generalized cyclic contraction mappings in partial metric spaces

Rus (Approx. Convexity 3:171–178, 2005) introduced the concept of cyclic contraction mapping. P˘acurar and Rus (Nonlinear Anal. 72:1181–1187, 2010) proved some fixed point results for cyclic φ-contraction mappings on a metric space. Karapinar (Appl. Math. Lett. 24:822–825, 2011) obtained a unique fixed point of cyclic weak φ- contraction mappings and studied well-posedness problem for such mappings. On the other hand, Matthews (Ann. New York Acad. Sci. 728:183–197, 1994) introduced the concept of a partial metric as a part of the study of denotational semantics of dataflow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In this paper, we initiate the study of fixed points of generalized cyclic contraction in the framework of partial metric spaces. We also present some examples to validate our results.S. Romaguera acknowledges the support of the Ministry of Science and Innovation of Spain, grant MTM2009-12872-C02-01.Abbas, M.; Nazir, T.; Romaguera Bonilla, S. (2012). Fixed point results for generalized cyclic contraction mappings in partial metric spaces. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 106(2):287-297. https://doi.org/10.1007/s13398-011-0051-5S2872971062Abdeljawad T., Karapinar E., Tas K.: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 24(11), 1894–1899 (2011). doi: 10.1016/j.aml.2011.5.014Altun, I., Erduran A.: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. article ID 508730 (2011). doi: 10.1155/2011/508730Altun I., Sadarangani K.: Corrigendum to “Generalized contractions on partial metric spaces” [Topology Appl. 157 (2010), 2778–2785]. Topol. Appl. 158, 1738–1740 (2011)Altun I., Simsek H.: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 1, 1–8 (2008)Altun I., Sola F., Simsek H.: Generalized contractions on partial metric spaces. Topol. Appl. 157, 2778–2785 (2010)Aydi, H.: Some fixed point results in ordered partial metric spaces. arxiv:1103.3680v1 [math.GN](2011)Boyd D.W., Wong J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)Bukatin M., Kopperman R., Matthews S., Pajoohesh H.: Partial metric spaces. Am. Math. Monthly 116, 708–718 (2009)Bukatin M.A., Shorina S.Yu. et al.: Partial metrics and co-continuous valuations. In: Nivat, M. (eds) Foundations of software science and computation structure Lecture notes in computer science vol 1378., pp. 125–139. Springer, Berlin (1998)Derafshpour M., Rezapour S., Shahzad N.: On the existence of best proximity points of cyclic contractions. Adv. Dyn. Syst. Appl. 6, 33–40 (2011)Heckmann R.: Approximation of metric spaces by partial metric spaces. Appl. Cat. Struct. 7, 71–83 (1999)Karapinar E.: Fixed point theory for cyclic weak $${\phi}$$ -contraction. App. Math. Lett. 24, 822–825 (2011)Karapinar, E.: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011,4 (2011). doi: 10.1186/1687-1812-2011-4Karapinar E.: Weak $${\varphi}$$ -contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math. Aeterna. 1(4), 237–244 (2011)Karapinar E., Erhan I.M.: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 24, 1894–1899 (2011)Karpagam S., Agrawal S.: Best proximity point theorems for cyclic orbital Meir–Keeler contraction maps. Nonlinear Anal. 74, 1040–1046 (2011)Kirk W.A., Srinavasan P.S., Veeramani P.: Fixed points for mapping satisfying cylical contractive conditions. Fixed Point Theory. 4, 79–89 (2003)Kosuru, G.S.R., Veeramani, P.: Cyclic contractions and best proximity pair theorems). arXiv:1012.1434v2 [math.FA] 29 May (2011)Matthews S.G.: Partial metric topology. in: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728, 183–197 (1994)Neammanee K., Kaewkhao A.: Fixed points and best proximity points for multi-valued mapping satisfying cyclical condition. Int. J. Math. Sci. Appl. 1, 9 (2011)Oltra S., Valero O.: Banach’s fixed theorem for partial metric spaces. Rend. Istit. Mat. Univ. Trieste. 36, 17–26 (2004)Păcurar M., Rus I.A.: Fixed point theory for cyclic $${\phi}$$ -contractions. Nonlinear Anal. 72, 1181–1187 (2010)Petric M.A.: Best proximity point theorems for weak cyclic Kannan contractions. Filomat. 25, 145–154 (2011)Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. (2010, article ID 493298, 6 pages).Romaguera, S.: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl. (2011). doi: 10.1016/j.topol.2011.08.026Romaguera S., Valero O.: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 19, 541–563 (2009)Rus, I.A.: Cyclic representations and fixed points. Annals of the Tiberiu Popoviciu Seminar of Functional equations. Approx. Convexity 3, 171–178 (2005), ISSN 1584-4536Schellekens M.P.: The correspondence between partial metrics and semivaluations. Theoret. Comput. Sci. 315, 135–149 (2004)Valero O.: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Top. 6, 229–240 (2005)Waszkiewicz P.: Quantitative continuous domains. Appl. Cat. Struct. 11, 41–67 (2003

Best proximity point results for modified α-proximal C-contraction mappings

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