×

zbMATH — the first resource for mathematics

Fixed point theory for multivalued \({\phi}\)-contractions. (English) Zbl 1315.47041
Summary: The purpose of this paper is to present a fixed point theory for multivalued \(\phi\)-contractions using the following concepts: fixed points, strict fixed points, periodic points, strict periodic points, multivalued Picard and weakly Picard operators; data dependence of the fixed point set, sequences of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, limit shadowing property of a multivalued operator, set-to-set operatorial equations and fractal operators. Our results generalize some recent theorems given in [A. Petruşel and I. A. Rus, in: Proceedings of the 9th international conference on fixed point theory and its applications (ICFPTA), Changhua, Taiwan, July 16–22, 2009. Yokohama: Yokohama Publishers. 161–175 (2010; Zbl 1225.54026)].

MSC:
47H04 Set-valued operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
47H14 Perturbations of nonlinear operators
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Rus IA: Generalized Contractions and Applications. Cluj University Press; 2001. · Zbl 0968.54029
[2] Nadler, SB, Periodic points of multi-valued \(ε\)-contractive maps, Topol Methods Nonlinear Anal, 22, 399-409, (2003) · Zbl 1042.54011
[3] Covitz, H; Nadler, SB, Multivalued contraction mappings in generalized metric spaces, Israel J Math, 8, 5-11, (1970) · Zbl 0192.59802
[4] Frigon, M, Fixed point and continuation results for contractions in metric and gauge spaces, Banach Center Publ, 77, 89-114, (2007) · Zbl 1122.47045
[5] Jachymski, J; Józwik, I; Jachymski, J (ed.); Reich, S (ed.), Nonlinear contractive conditions: a comparison and related problems, No. 77, 123-146, (2007) · Zbl 1149.47044
[6] Lazăr, T; O’Regan, D; Petruşel, A, Fixed points and homotopy results for ćirić-type multivalued operators on a set with two metrics, Bull Korean Math Soc, 45, 67-73, (2008) · Zbl 1153.47047
[7] Lazăr, TA; Petruşel, A; Shahzad, N, Fixed points for non-self operators and domain invariance theorems, Nonlinear Anal, 70, 117-125, (2009) · Zbl 1183.47052
[8] Meir, A; Keeler, E, A theorem on contraction mappings, J Math Anal Appl, 28, 326-329, (1969) · Zbl 0194.44904
[9] Mizoguchi, N; Takahashi, W, Fixed point theorems for multivalued mappings on complete metric spaces, J Math Anal Appl, 141, 177-188, (1989) · Zbl 0688.54028
[10] Petruşel, A, Generalized multivalued contractions, Nonlinear Anal, 47, 649-659, (2001) · Zbl 1042.47520
[11] Petruşel, A; Rus, IA, Fixed point theory for multivalued operators on a set with two metrics, Fixed Point Theory, 8, 97-104, (2007) · Zbl 1133.47036
[12] Rhoades, BE, Some theorems on weakly contractive maps, Nonlinear Anal, 47, 2683-2693, (2001) · Zbl 1042.47521
[13] Smithson, RE, Fixed points for contractive multifunctions, Proc Am Math Soc, 27, 192-194, (1971) · Zbl 0213.24501
[14] Tarafdar, E; Yuan, GXZ, Set-valued contraction mapping principle, Appl Math Letter, 8, 79-81, (1995) · Zbl 0837.54011
[15] Xu, HK, \(ε\)-chainability and fixed points of set-valued mappings in metric spaces, Math Japon, 39, 353-356, (1994) · Zbl 0813.54011
[16] Xu, HK, Metric fixed point theory for multivalued mappings, Diss Math, 389, 39, (2000) · Zbl 0972.47041
[17] Yuan GXZ: KKM Theory and Applications in Nonlinear Analysis. Marcel Dekker, New York; 1999. · Zbl 0936.47034
[18] Rus, IA; Petruşel, A; Sîntămărian, A, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal, 52, 1947-1959, (2003) · Zbl 1055.47047
[19] Petruşel, A; Rus, IA; Llorens Fuster, E (ed.); Garcia Falset, J (ed.); Sims, B (ed.), Multivalued Picard and weakly Picard operators, 207-226, (2004) · Zbl 1091.47047
[20] Lim, TC, On fixed point stability for set-valued contractive mappings with applications to generalized differential equations, J Math Anal Appl, 110, 436-441, (1985) · Zbl 0593.47056
[21] Markin, JT, Continuous dependence of fixed points sets, Proc Am Math Soc, 38, 545-547, (1973) · Zbl 0278.47036
[22] Saint-Raymond, J, Multivalued contractions, Set-Valued Anal, 2, 559-571, (1994) · Zbl 0820.47065
[23] Fraser, RB; Nadler, SB, Sequences of contractive maps and fixed points, Pac J Math, 31, 659-667, (1969) · Zbl 0186.56502
[24] Papageorgiou, NS, Convergence theorems for fixed points of multifunctions and solutions of differential inclusions in Banach spaces, Glas Mat Ser III, 23, 247-257, (1988) · Zbl 0673.34016
[25] Rus, IA, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10, 305-320, (2009) · Zbl 1204.47071
[26] Petruşel, A; Rus, IA; Cârjă, O (ed.); Vrabie, II (ed.), Well-posedness of the fixed point problem for multivalued operators, 295-306, (2007) · Zbl 1169.47037
[27] Petruşel, A; Rus, IA; Yao, JC, Well-posedness in the generalized sense of the fixed point problems, Taiwan J Math, 11, 903-914, (2007) · Zbl 1149.54022
[28] Glăvan, V; Guţu, V; Barbu, V (ed.); Lasiecka, I (ed.); Tiba, D (ed.); Varsan, C (ed.), On the dynamics of contracting relations, 179-188, (2003) · Zbl 1030.37011
[29] Nadler, SB, Multivalued contraction mappings, Pac J Math, 30, 475-488, (1969) · Zbl 0187.45002
[30] Andres, J, Some standard fixed-point theorems revisited, Atti Sem Mat Fis Univ Modena, 49, 455-471, (2001) · Zbl 1065.47052
[31] De Blasi, FS, Semifixed sets of maps in hyperspaces with applications to set differential equations, Set-Valued Anal, 14, 263-272, (2006) · Zbl 1110.47040
[32] Andres, J; Fišer, J, Metric and topological multivalued fractals, Internat J Bifur Chaos Appl Sci Engrg, 14, 1277-1289, (2004) · Zbl 1057.28003
[33] Barnsley MF: Fractals Everywhere. Academic Press, Boston; 1988. · Zbl 0691.58001
[34] Hutchinson, JE, Fractals and self-similarity, Indiana Univ Math J, 30, 713-747, (1981) · Zbl 0598.28011
[35] Jachymski, J, Continuous dependence of attractors of iterated function systems, J Math Anal Appl, 198, 221-226, (1996) · Zbl 0862.54033
[36] Lasota, A; Myjak, J, Attractors of multifunctions, Bull Polish Acad Sci Math, 48, 319-334, (2000) · Zbl 0962.28004
[37] Petruşel, A, Singlevalued and multivalued Meir-Keeler type operators, Revue D’Analse Num et de Th de l’Approx Tome, 30, 75-80, (2001) · Zbl 1074.47511
[38] Yamaguti, M; Hata, M; Kigani, J, Mathematics of fractals, No. 167, (1997), RI
[39] Andres, J; Górniewicz, L, On the Banach contraction principle for multivalued mappings, 1-23, (2001) · Zbl 1019.47042
[40] Chifu, C; Petruşel, A, Multivalued fractals and multivalued generalized contractions, Chaos Solit Fract, 36, 203-210, (2008) · Zbl 1131.28005
[41] Kirk WA, Sims B, (eds): Handbook of Metric Fixed Point Theory. Kluwer, Dordrecht; 2001. · Zbl 0970.54001
[42] Petruşel, A, Multivalued weakly Picard operators and applications, Sci Math Japon, 59, 169-202, (2004) · Zbl 1066.47058
[43] Rus IA, Petruşel A, Petruşel G: Fixed Point Theory. Cluj University Press, Cluj-Napoca; 2008. · Zbl 1171.54034
[44] Aubin JP, Frankowska H: Set-Valued Analysis. Birkhauser, Basel; 1990. · Zbl 0713.49021
[45] Beer G: Topologies on Closed and Closed Convex Sets. Kluwer, Dordrecht; 1993. · Zbl 0792.54008
[46] Ayerbe Toledano YM, Dominguez Benavides T, López Acedo L: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser Verlag, Basel; 1997. · Zbl 0885.47021
[47] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990. · Zbl 0708.47031
[48] Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings. Kluwer, Dordrecht; 1999. · Zbl 0937.55001
[49] Hu S, Papageorgiou NS: Handbook of Multivalued Analysis, Vol. I and II. Kluwer, Dordrecht; 1997.
[50] Rus IA, Petruşel A, Petruşel G: Fixed Point Theory 1950-2000: Romanian Contributions. House of the Book of Science, Cluj-Napoca; 2002. · Zbl 1005.54037
[51] Granas A, Dugundji J: Fixed Point Theory. Springer, Berlin; 2003. · Zbl 1025.47002
[52] Takahashi, W, Nonlinear functional analysis, (2000), Yokohama
[53] Rus, IA, Weakly Picard mappings, Comment Math University Carolinae, 34, 769-773, (1993) · Zbl 0787.54045
[54] Rus, IA, Picard operators and applications, Sci Math Japon, 58, 191-219, (2003) · Zbl 1031.47035
[55] Rus, IA; Şerban, MA; Breckner, W (ed.), Some generalizations of a Cauchy lemma and applications, 173-181, (2008) · Zbl 1224.54104
[56] Węgrzyk, R, Fixed point theorems for multifunctions and their applications to functional equations, Dissertationes Math (Rozprawy Mat.), 201, 28, (1982)
[57] Petruşel, A; Rus, IA; Lin, LJ (ed.); Petruşel, A (ed.); Xu, HK (ed.), The theory of a metric fixed point theorem for multivalued operators, 161-175, (2010) · Zbl 1225.54026
[58] Reich, S, Fixed point of contractive functions, Boll Un Mat Ital, 5, 26-42, (1972) · Zbl 0249.54026
[59] Reich, S, A fixed point theorem for locally contractive multivalued functions, Rev Roumaine Math Pures Appl, 17, 569-572, (1972) · Zbl 0239.54033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.