Common fixed points for set-valued contraction on a metric space with graph

In this article, we derive a common fixed point result for a pair of single valued and set-valued mappings on a metric space having graphical structure. In this case, the set-valued map is assumed to be closed valued instead of closed and bounded valued. Several results regarding common fixed points and fixed points follow from the main theorem of this article. By applying our theorem, we deduce the convergence of the iterates for a nonlinear $q$-analogue Bernstein operator. Furthermore, we establish sufficient criteria for the occurrence of a solution to a fractional differential equation.Comment: Keywords: Common fixed points; Coincidence points; Graph; Fractional differential equation; $q$-analogue Bernstein operato

Mizoguchi-Takahashi local contractions to Feng-Liu contractions

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.Comment: Keywords: Fixed points; Set-valued map; metrically convex metric space; uniformly local contraction

On the existence of best proximity points for generalized contractions

[EN] In this article we establish the existence of a unique best proximity point for some generalized non self contractions on a metric space in a simpler way using a geometric result. Our results generalize some recent best proximity point theorems and several fixed point theorems proved by various authors.The authors are grateful to the referees for their valuable comments and suggestions to improve this manuscript. The first author is thankful to University Grants Commission F.2 − 12/2002(SA − I), New Delhi, India for the financial supportSultana, A.; Vetrivel, V. (2014). On the existence of best proximity points for generalized contractions. Applied General Topology. 15(1):55-63. https://doi.org/10.4995/agt.2014.2221SWORD5563151Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Sankar Raj, V., & Veeramani, P. (2009). Best proximity pair theorems for relatively nonexpansive mappings. Applied General Topology, 10(1), 21-28. doi:10.4995/agt.2009.1784Anuradha, J., & Veeramani, P. (2009). Proximal pointwise contraction. Topology and its Applications, 156(18), 2942-2948. doi:10.1016/j.topol.2009.01.017Abkar, A., & Gabeleh, M. (2011). Global Optimal Solutions of Noncyclic Mappings in Metric Spaces. Journal of Optimization Theory and Applications, 153(2), 298-305. doi:10.1007/s10957-011-9966-4Eldred, A. A., & Veeramani, P. (2006). Existence and convergence of best proximity points. Journal of Mathematical Analysis and Applications, 323(2), 1001-1006. doi:10.1016/j.jmaa.2005.10.081Amini-Harandi, A. (2012). Best proximity points for proximal generalized contractions in metric spaces. Optimization Letters, 7(5), 913-921. doi:10.1007/s11590-012-0470-zKirk, 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-120026380Granas, A., & Dugundji, J. (2003). Fixed Point Theory. Springer Monographs in Mathematics. doi:10.1007/978-0-387-21593-8Rhoades, B. E. (1977). A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society, 226, 257-257. doi:10.1090/s0002-9947-1977-0433430-4Fan, K. (1969). Extensions of two fixed point theorems of F. E. Browder. Mathematische Zeitschrift, 112(3), 234-240. doi:10.1007/bf01110225Kim, 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.2007.01.05

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