Abstract metric spaces and Hardy-Rogers-type theorems

The purpose of the present paper is to establish coincidence point theorem for two mappings and fixed point theorem for one mapping in abstract metric space which satisfy contractive conditions of Hardy-Rogers type. Our results generalize fixed point theorems of Nemytzki [V.V. Nemytzki, Fixed point method in analysis, Uspekhi Mat. Nauk 1 (1936) 141-174], Edelstein I M. Edelstein, On fixed and periodic point under contractive mappings, J. Lond. Math. Soc. 37 (1962) 74-79] and Huang, Zhang [LG. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2)(2007) 1468-1476] from abstract metric spaces to symmetric spaces (Theorem 2.1) and to metric spaces (Theorem 2.4, Corollaries 2.6-2.8). Two examples are given to illustrate the usability of our results

Generalized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalized F-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.The authors are very grateful to the referees for their valuable comments and suggestions, and, in particular, to one of them for calling our attention on the crucial fact stated in the first part of Remark 5 and for the elegant reformulation of Theorem 13 stated in Remark 14. Salvador Romaguera acknowledges the support of the Universitat Politecnica de Valencia, Grant PAID-06-12-SP20120471.Abbas, M.; Ali, B.; Romaguera Bonilla, S. (2014). Generalized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces. Abstract and Applied Analysis. 2014:1-5. https://doi.org/10.1155/2014/391952S152014Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3, 133-181. doi:10.4064/fm-3-1-133-181Arandjelović, I., Kadelburg, Z., & Radenović, S. (2011). Boyd–Wong-type common fixed point results in cone metric spaces. Applied Mathematics and Computation, 217(17), 7167-7171. doi:10.1016/j.amc.2011.01.113Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Huang, L.-G., & Zhang, X. (2007). Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332(2), 1468-1476. doi:10.1016/j.jmaa.2005.03.087Rakotch, E. (1962). A note on contractive mappings. Proceedings of the American Mathematical Society, 13(3), 459-459. doi:10.1090/s0002-9939-1962-0148046-1Tarafdar, E. (1974). An approach to fixed-point theorems on uniform spaces. Transactions of the American Mathematical Society, 191, 209-209. doi:10.1090/s0002-9947-1974-0362283-5Dix, J. G., & Karakostas, G. L. (2009). A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems. Nonlinear Analysis: Theory, Methods & Applications, 71(9), 3872-3880. doi:10.1016/j.na.2009.02.057Latrach, K., Aziz Taoudi, M., & Zeghal, A. (2006). Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations. Journal of Differential Equations, 221(1), 256-271. doi:10.1016/j.jde.2005.04.010Meinardus, G. (1963). Invarianz bei linearen Approximationen. Archive for Rational Mechanics and Analysis, 14(1), 301-303. doi:10.1007/bf00250708Habiniak, L. (1989). Fixed point theorems and invariant approximations. Journal of Approximation Theory, 56(3), 241-244. doi:10.1016/0021-9045(89)90113-5Hicks, T. ., & Humphries, M. . (1982). A note on fixed-point theorems. Journal of Approximation Theory, 34(3), 221-225. doi:10.1016/0021-9045(82)90012-0Singh, S. . (1979). An application of a fixed-point theorem to approximation theory. Journal of Approximation Theory, 25(1), 89-90. doi:10.1016/0021-9045(79)90036-4Subrahmanyam, P. . (1977). An application of a fixed point theorem to best approximation. Journal of Approximation Theory, 20(2), 165-172. doi:10.1016/0021-9045(77)90070-3Wardowski, D. (2012). Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-94Abbas, M., Ali, B., & Romaguera, S. (2013). Fixed and periodic points of generalized contractions in metric spaces. Fixed Point Theory and Applications, 2013(1), 243. doi:10.1186/1687-1812-2013-24