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New existence results for fixed point problem and minimization problem in compact metric spaces. (English) Zbl 1474.54163

Summary: We first present some new existence theorems for fixed point problem and minimization problem in compact metric spaces without assuming that mappings possess convexity property. Some applications of our results to new fixed point theorems for nonself mappings in the setting of strictly convex normed linear spaces and usual metric spaces are also given.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
49J27 Existence theories for problems in abstract spaces
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[1] Banach, S., Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fundamenta Mathematicae, 3, 133-181 (1922)
[2] Edelstein, M., An extension of Banach’s contraction principle, Proceedings of the American Mathematical Society, 12, 7-10 (1961) · Zbl 0096.17101
[3] Hyers, D. H.; Isac, G.; Rassias, T. M., Topics in Nonlinear Analysis and Applications (1997), Singapore: World Scientific, Singapore · Zbl 0878.47040 · doi:10.1142/9789812830432
[4] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama, Japan: Yokohama Publishers, Yokohama, Japan · Zbl 0997.47002
[5] Petruşel, A.; Sîntămărian, A., Single-valued and multi-valued Caristi type operators, Publicationes Mathematicae Debrecen, 60, 1-2, 167-177 (2002) · Zbl 1003.47041
[6] Kada, O.; Suzuki, T.; Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Mathematica Japonica, 44, 2, 381-391 (1996) · Zbl 0897.54029
[7] Suzuki, T., Generalized distance and existence theorems in complete metric spaces, Journal of Mathematical Analysis and Applications, 253, 2, 440-458 (2001) · Zbl 0983.54034 · doi:10.1006/jmaa.2000.7151
[8] Suzuki, T., Generalized Caristi’s fixed point theorems by Bae and others, Journal of Mathematical Analysis and Applications, 302, 2, 502-508 (2005) · Zbl 1059.54031 · doi:10.1016/j.jmaa.2004.08.019
[9] Feng, Y.; Liu, S., Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, Journal of Mathematical Analysis and Applications, 317, 1, 103-112 (2006) · Zbl 1094.47049 · doi:10.1016/j.jmaa.2005.12.004
[10] 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 · doi:10.1016/j.jmaa.2005.10.005
[11] Lin, L.-J.; Du, W.-S., Some equivalent formulations of the generalized Ekeland’s variational principle and their applications, Nonlinear Analysis: Theory, Methods & Applications, 67, 1, 187-199 (2007) · Zbl 1111.49013 · doi:10.1016/j.na.2006.05.006
[12] Lin, L.-J.; Du, W.-S., On maximal element theorems, variants of Ekeland’s variational principle and their applications, Nonlinear Analysis: Theory, Methods & Applications, 68, 5, 1246-1262 (2008) · Zbl 1133.58006 · doi:10.1016/j.na.2006.12.018
[13] Lin, L.-J.; Du, W.-S., Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems, Journal of Global Optimization, 40, 4, 663-677 (2008) · Zbl 1218.49014 · doi:10.1007/s10898-007-9146-0
[14] Du, W.-S., On some nonlinear problems induced by an abstract maximal element principle, Journal of Mathematical Analysis and Applications, 347, 2, 391-399 (2008) · Zbl 1148.49013 · doi:10.1016/j.jmaa.2008.06.020
[15] Du, W.-S., On Caristi type maps and generalized distances with applications, Abstract and Applied Analysis, 2013 (2013) · Zbl 1292.54022 · doi:10.1155/2013/407219
[16] Du, W.-S.; Karapinar, E., A note on Caristi-type cyclic maps: related results and applications, Fixed Point Theory and Applications, 2013, article 344 (2013) · Zbl 1334.54063 · doi:10.1186/1687-1812-2013-344
[17] Wodarczyk, K.; Plebaniak, R., Maximality principle and general results of ekeland and caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances, Fixed Point Theory and Applications, 2010 (2010) · Zbl 1201.54039 · doi:10.1155/2010/175453
[18] Karapinar, E., Generalizations of Caristi Kirk’s theorem on partial metric spaces, Fixed Point Theory and Applications, 2011, article 4 (2011) · Zbl 1281.54027
[19] Kirk, W. A.; Shahzad, N., Generalized metrics and Caristi’s theorem, Fixed Point Theory and Applications, 2013, article 129 (2013) · Zbl 1295.54060
[20] Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Academy of Sciences of the United States of America, 54, 1041-1044 (1965) · Zbl 0128.35801 · doi:10.1073/pnas.54.4.1041
[21] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[22] Kirk, W. A., Remarks on pseudo-contractive mappings, Proceedings of the American Mathematical Society, 25, 820-823 (1970) · Zbl 0203.14603 · doi:10.1090/S0002-9939-1970-0264481-X
[23] Kirk, W. A., Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings, Pacific Journal of Mathematics, 38, 89-94 (1971) · Zbl 0218.47025 · doi:10.2140/pjm.1971.38.89
[24] Assad, N. A.; Kirk, W. A., Fixed point theorems for set-valued mappings of contractive type, Pacific Journal of Mathematics, 43, 553-562 (1972) · Zbl 0239.54032 · doi:10.2140/pjm.1972.43.553
[25] Reich, S., Fixed points of condensing functions, Journal of Mathematical Analysis and Applications, 41, 460-467 (1973) · Zbl 0252.47062 · doi:10.1016/0022-247X(73)90220-5
[26] Ding, X. P.; He, Y. R., Fixed point theorems for metrically weakly inward set-valued mappings, Journal of Applied Analysis, 5, 2, 283-293 (1999) · Zbl 0949.47045 · doi:10.1515/JAA.1999.283
[27] Alghamdi, M. A.; Berinde, V.; Shahzad, N., Fixed points of multivalued nonself almost contractions, Journal of Applied Mathematics, 2013 (2013) · Zbl 1271.54071 · doi:10.1155/2013/621614
[28] Du, W.-S.; Karapınar, E.; Shahzad, N., The study of fixed point theory for various multivalued non-self-maps, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.54054 · doi:10.1155/2013/938724
[29] Du, W.-S., A note on approximate fixed point property and Du-Karapinar-Shahzad’s intersection theorems, Journal of Inequalities and Applications, 2013, article 506 (2013) · Zbl 1489.54109 · doi:10.1186/1029-242X-2013-506
[30] Caristi, J., Fixed point theorems for mappings satisfying inwardness conditions, Transactions of the American Mathematical Society, 215, 241-251 (1976) · Zbl 0305.47029 · doi:10.1090/S0002-9947-1976-0394329-4
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