Fixed points and coupled fixed points in partially ordered ν-generalized metric spaces

[EN] New fixed point and coupled fixed point theorems in partially ordered ν-generalized metric spaces are presented. Since the product of two ν-generalized metric spaces is not in general a ν-generalized metric space, a different approach is needed than in the case of standard metric spaces.Abtahi, M.; Kadelburg, Z.; Radenovic, S. (2018). Fixed points and coupled fixed points in partially ordered ν-generalized metric spaces. Applied General Topology. 19(2):189-201. doi:10.4995/agt.2018.7409SWORD189201192M. Abtahi, Fixed point theorems for Meir-Keeler type contractions in metric spaces, Fixed Point Theory 17, no. 2 (2016), 225-236.M. Abtahi, Z. Kadelburg and S. Radenovic, Fixed points of Ciric-Matkowski-type contractions in $nu$-generalized metric spaces, Rev. Real Acad. Cienc. Exac. Fis. Nat. Ser. A, Mat. 111, no. 1 (2017), 57-64.B. Alamri, T. Suzuki and L. A. Khan, Caristi's fixed point theorem and Subrahmanyam's fixed point theorem in $nu$-generalized metric spaces, J. Function Spaces, 2015, Art. ID 709391, 6 pp.V. Berinde and M. Pacurar, Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory Appl. (2012) 2012:115. https://doi.org/10.1186/1687-1812-2012-115T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65, no. 7 (2006), 1379-1393. https://doi.org/10.1016/j.na.2005.10.017A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31-37.Lj. B. Ciric, A new fixed-point theorem for contractive mappings, Publ. Inst. Math. (N.S) 30 (44) (1981), 25-27.Z. Kadelburg and S. Radenovic, On generalized metric spaces: A survey, TWMS J. Pure Appl. Math. 5 (2014), 3-13.Z. Kadelburg and S. Radenovic, Fixed point results in generalized metric spaces without Hausdorff property, Math. Sciences 8:125 (2014). https://doi.org/10.1007/s40096-014-0125-6R. Kannan, Some results on fixed points-II, Amer. Math. Monthly 76 (1969), 405-408.W. A. Kirk and N. Shahzad, Generalized metrics and Caristi's theorem, Fixed Point Theory Appl. 2013:129 (2013). https://doi.org/10.1186/1687-1812-2013-129V. Lakshmikantham and Lj. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009), 4341-4349. https://doi.org/10.1016/j.na.2008.09.020N. V. Luong and N. X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011), 983-992. https://doi.org/10.1016/j.na.2010.09.055J. J. Nieto and R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, Engl. Ser. 23, no. 12 (2007), 2205-2212. https://doi.org/10.1007/s10114-005-0769-0P. D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 64 (2006), 546-557. https://doi.org/10.1016/j.na.2005.04.044B. Samet, Discussion on 'A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces' by A. Branciari, Publ. Math. Debrecen 76 (2010), 493-494.B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517. https://doi.org/10.1016/j.na.2010.02.026I. R. Sarma, J. M. Rao and S. S. Rao, Contractions over generalized metric spaces, J. Nonlinear Sci. Appl. 2 (2009), 180-182. https://doi.org/10.22436/jnsa.002.03.06T. Suzuki, Generalized metric spaces do not have the compatible topology, Abstr. Appl. Anal., 2014, Art. ID 458098, 5 pp.T. Suzuki, B. Alamri and L. A. Khan, Some notes on fixed point theorems in v-generalized metric spaces, Bull. Kyushu Inst. Tech. Pure Appl. Math. 62 (2015), 15-23.M. Turinici, Functional contractions in local Branciari metric spaces, Romai J. 8 (2012),189-199

Unique Solution of a Coupled Fractional Differential System Involving Integral Boundary Conditions from Economic Model

We study the existence and uniqueness of the positive solution for the fractional differential system involving the Riemann-Stieltjes integral boundary conditions , , , , , and , where , , and and are the standard Riemann-Liouville derivatives, and are functions of bounded variation, and and denote the Riemann-Stieltjes integral. Our results are based on a generalized fixed point theorem for weakly contractive mappings in partially ordered sets

Effective Choice and Boundedness Principles in Computable Analysis

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles on closed sets which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. Well-known omniscience principles from constructive mathematics such as $LPO$ and $LLPO$ can naturally be considered as Weihrauch degrees and they play an important role in our classification. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example

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