## Best proximity points of generalized semicyclic impulsive self-mappings: applications to impulsive differential and difference equations.(English)Zbl 1357.54032

Summary: This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54E40 Special maps on metric spaces
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### References:

 [1] Saewan, S.; Kanjanasamranwong, P.; Kuman, P.; Cho, Y. J., The modified Mann type iterative algorithm for a countable family of totally quasi-phi-asymptotically nonexpansive mappings by the hybrid generalized f-projection method, Fixed Point Theory and Applications, 2013, article 63, (2013) · Zbl 1423.47049 [2] Yao, Y.; Noor, M. A.; Liou, Y.-C.; Kang, S. M., Iterative algorithms for general multivalued variational inequalities, Abstract and Applied Analysis, 2012, (2012) · Zbl 1232.49012 [3] De la Sen, M., Stable iteration procedures in metric spaces which generalize a Picard-type iteration, Fixed Point Theory and Applications, 2010, (2010) · Zbl 1203.54039 [4] Inchan, I., Viscosity iteration method for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Applied Mathematics and Computation, 219, 6, 2949-2959, (2012) · Zbl 1309.47076 [5] Sahu, D. R.; Kang, S. M.; Sagar, V., Approximation of common fixed points of a sequence of nearly nonexpansive mappings and solutions of variational inequality problems, Journal of Applied Mathematics, 2012, (2012) · Zbl 1251.65084 [6] Sahu, D. R.; Liu, Z.; Kang, S. M., Existence and approximation of fixed points of nonlinear mappings in spaces with weak uniform normal structure, Computers & Mathematics with Applications, 64, 4, 672-685, (2012) · Zbl 1252.47059 [7] Ratchagit, M.; Ratchagit, K., Asymptotic stability and stabilization of fixed points for iterative sequence, International Journal of Research and Reviews in Computer Science, 2, 4, 987-989, (2011) [8] Cho, Y. J.; Kim, J. K.; Kang, S. M., Fixed Point Theory and Applications, 3, (2002), Nova Publishers [9] Rus, I. A., Cyclic representations and fixed points, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, 3, 171-178, (2005) · Zbl 1463.54126 [10] Cirić, L. B., Generalized contractions and fixed-point theorems, Publications de l Institut Mathématique, 12, 26, 19-26, (1971) · Zbl 0234.54029 [11] Eldred, A. A.; Veeramani, P., Existence and convergence of best proximity points, Journal of Mathematical Analysis and Applications, 323, 2, 1001-1006, (2006) · Zbl 1105.54021 [12] Karpagam, S.; Agrawal, S., Best proximity point theorems for $$p$$-cyclic Meir-Keeler contractions, Fixed Point Theory and Applications, 2009, (2009) · Zbl 1172.54028 [13] Kirk, W. A.; Srinivasan, P. S.; Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 1, 79-89, (2003) · Zbl 1052.54032 [14] Karapınar, E.; Erhan, I. N., Cyclic contractions and fixed point theory, Filomat, 26, 4, 777-782, (2012) · Zbl 1289.47106 [15] Karapınar, E., Best proximity points of cyclic mappings, Applied Mathematics Letters, 25, 11, 1761-1766, (2012) · Zbl 1269.47040 [16] Karapınar, E.; Nashine, H. K., Fixed point theorem for cyclic Chatterjea type contractions, Journal of Applied Mathematics, 2012, (2012) · Zbl 1251.54049 [17] Aydi, H.; Karapınar, E., A fixed point result for Boyd-Wong cyclic contractions in partial metric spaces, International Journal of Mathematics and Mathematical Sciences, 2012, (2012) · Zbl 1250.54042 [18] Sanhan, W.; Mongkolkeha, C.; Kumam, P., Generalized proximal $$\psi$$-contraction mappings and best proximity points, Abstract and Applied Analysis, 2012, (2012) · Zbl 1387.54033 [19] De la Sen, M., Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings, Fixed Point Theory and Applications, 2010, (2010) · Zbl 1194.54059 [20] De la Sen, M., On a general contractive condition for cyclic self-mappings, Journal of Applied Mathematics, 2011, (2011) · Zbl 1235.54031 [21] Du, W.-S., New cone fixed point theorems for nonlinear multivalued maps with their applications, Applied Mathematics Letters, 24, 2, 172-178, (2011) · Zbl 1218.54037 [22] Singh, S. L.; Mishra, S. N.; Jain, S., Round-off stability for multi-valued maps, Fixed Point Theory and Applications, 2012, article 12, (2012) · Zbl 1273.65067 [23] Singh, S. L.; Mishra, S. N.; Chugh, R.; Kamal, R., General common fixed point theorems and applications, Journal of Applied Mathematics, 2012, (2012) · Zbl 1276.54040 [24] Laowang, W.; Panyanak, B., Common fixed points for some generalized multivalued nonexpansive mappings in uniformly convex metric spaces, Fixed Point Theory and Applications, 2011, article 20, (2011) · Zbl 1395.54050 [25] Khandani, H.; Vaezpour, S. M.; Sims, B., Common fixed points of generalized multivalued contraction on complete metric spaces, Journal of Computational Analysis and Applications, 13, 6, 1025-1038, (2011) · Zbl 1223.54059 [26] Nashine, H. K.; Shatanawi, W., Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces, Computers & Mathematics with Applications, 62, 4, 1984-1993, (2011) · Zbl 1231.65100 [27] Shatanawi, W.; Postolache, M., Common fixed point results of mappings for nonlinear contraction of cyclic form in ordered metric spaces, Fixed Point Theory and Applications, 2013, article 60, (2013) · Zbl 1286.54053 [28] Hussain, N.; Pathak, H. K., Common fixed point and approximation results for $$H$$-operator pair with applications, Applied Mathematics and Computation, 218, 22, 11217-11225, (2012) · Zbl 1298.54028 [29] Nashine, H. K.; Khan, M. S., An application of fixed point theorem to best approximation in locally convex space, Applied Mathematics Letters, 23, 2, 121-127, (2010) · Zbl 1200.47076 [30] Latif, A.; Kutbi, M. A., Fixed points for $$w$$-contractive multimaps, International Journal of Mathematics and Mathematical Sciences, 2009, (2009) · Zbl 1167.54314 [31] Husain, T.; Latif, A., Fixed points of multivalued nonexpansive maps, International Journal of Mathematics and Mathematical Sciences, 14, 3, 421-430, (1991) · Zbl 0736.54030 [32] Khan, M. S., Common fixed point theorems for multivalued mappings, Pacific Journal of Mathematics, 95, 2, 337-347, (1981) · Zbl 0419.54030 [33] Reich, S., Some remarks concerning contraction mappings, Canadian Mathematical Bulletin, 14, 121-124, (1971) · Zbl 0211.26002 [34] Pyatyshev, I. A., An example of boundedly approximatively compact set which is not locally compact, Russian Mathematical Surveys, 62, 5, 1007, (2007) · Zbl 1147.46019 [35] Vasilév, A. I., The bounded compactness of sets in linear metric spaces, Mathematical Notes of the Academy of Sciences of the USSR, 11, 6, 396-401, (1972) · Zbl 0245.46005 [36] Karapınar, E., On best proximity points of psi-Geraghty contractions, Fixed Point Theory and Applications, 2013, article 200, (2013) · Zbl 1295.41037 [37] Gabeleh, M.; Shahzad, N., Existence and convergence theorems of best proximity points, Journal of Applied Mathematics, 2013, (2013) · Zbl 1266.47076 [38] Lin, L.-J.; Du, W.-S., Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces, Journal of Mathematical Analysis and Applications, 323, 1, 360-370, (2006) · Zbl 1101.49022 [39] Du, W.-S., The existence of cone critical point and common fixed point with applications, Journal of Applied Mathematics, 2011, (2011) · Zbl 1304.54077 [40] De la Sen, M., Total stability properties based on fixed point theory for a class of hybrid dynamic systems, Fixed Point Theory and Applications, 2009, (2009) · Zbl 1189.34023 [41] Shen, T.; Yuan, Z., Stability criterion for a class of fixed-point digital filters using two’s complement arithmetic, Applied Mathematics and Computation, 219, 9, 4880-4883, (2013) · Zbl 1417.93242 [42] Nashine, H. K.; Pathak, R.; Somvanshi, P. S.; Pantelic, S.; Kumam, P., Solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic $$(\phi, \psi, \theta)$$-contraction, Advances in Difference Equations, 2013, article 106, (2013) · Zbl 1381.45023 [43] Solmaz, S.; Shorten, R.; Wulff, K.; Ó Cairbre, F., A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control, Automatica, 44, 9, 2358-2363, (2008) · Zbl 1153.93496 [44] De La Sen, M.; Luo, N., A note on the stability of linear time-delay systems with impulsive inputs, IEEE Transactions on Circuits and Systems, 50, 1, 149-152, (2003) · Zbl 1368.93272 [45] De la Sen, M.; Luo, N., On the uniform exponential stability of a wide class of linear time-delay systems, Journal of Mathematical Analysis and Applications, 289, 2, 456-476, (2004) · Zbl 1046.34086 [46] de la Sen, M.; Ibeas, A., On the global asymptotic stability of switched linear time-varying systems with constant point delays, Discrete Dynamics in Nature and Society, 2008, (2008) · Zbl 1166.34040
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