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The multicores in metric spaces and their application in fixed point theory. (English) Zbl 1253.55002
In the present paper the author defines notions of different types of so-called multicores and proves a result concerning the existence of fixed points for a certain class of multivalued maps (being admissible, compact and defined on a metric space). In fact it is shown that a multimap \(\varphi\) defined on a metric space \(X\) can be studied within the Lefschetz theory provided the closure of its image \(\varphi(X)\) is a multicore in an appropriate sense. There are no applications illustrating and justifying the presented approach for any concrete problem.
55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54C60 Set-valued maps in general topology
47H10 Fixed-point theorems
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[1] Agarwal, G. P., O’Regan, D.: A note on the Lefschetz fixed point theorem for admissible spaces. Bull. Korean Math. Soc. 42, 2 (2005), 307-313. · Zbl 1089.47040 · doi:10.4134/BKMS.2005.42.2.307
[2] Andres, J., Górniewicz, L.: Topological principles for boundary value problems. Kluwer, Dordrecht, 2003. · Zbl 1029.55002
[3] Bogatyi, S. A.: Approximative and fundamental retracts. Math. USSR Sb. 22 (1974), 91-103. · Zbl 0301.54042 · doi:10.1070/SM1974v022n01ABEH001687
[4] Borsuk, K.: Theory of Shape. Lect. Notes Ser. 28, Matematisk Institut, Aarhus Universitet, Aarhus, 1971. · Zbl 0232.55021
[5] Clapp, M. H.: On a generalization of absolute neighborhood retracts. Fund. Math. 70 (1971), 117-130. · Zbl 0231.54012 · eudml:214314
[6] Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. New Jersey Princeton University Press, Princeton, 1952. · Zbl 0047.41402
[7] Fournier, G., Granas, A.: The Lefschetz fixed point theorem for some classes of non-metrizable spaces. J. Math. Pures et Appl. 52 (1973), 271-284. · Zbl 0294.54034
[8] Górniewicz, L.: Topological methods in fixed point theory of multi-valued mappings. Springer, New York-Berlin-Heidelberg, 2006.
[9] Górniewicz, L., Rozpłoch-Nowakowska, D.: The Lefschetz fixed point theory for morphisms in topological vector spaces. Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center 20 (2002), 315-333. · Zbl 1031.55002
[10] Górniewicz, L., Ślosarski, M.: Once more on the Lefschetz fixed point theorem. Bull. Polish Acad. Sci. Math. 55 (2007), 161-170. · Zbl 1122.55002 · doi:10.4064/ba55-2-7
[11] Górniewicz, L., Ślosarski, M.: Fixed points of mappings in Klee admissible spaces. Control and Cybernetics 36, 3 (2007). · Zbl 1194.47062 · eudml:209520
[12] Granas, A.: Generalizing the Hopf-Lefschetz fixed point theorem for non-compact ANR’s. Symp. Inf. Dim. Topol., Baton-Rouge, 1967.
[13] Granas, A., Dugundji, J.: Fixed Point Theory. Springer, Berlin-Heidelberg-New York, 2003. · Zbl 1025.47002
[14] Jaworowski, J.: Continuous Homology Properties of Approximative Retracts. Bulletin de L’Academie Polonaise des Sciences 18, 7 (1970). · Zbl 0199.26004
[15] Krathausen, C., Müller, G., Reinermann, J., Schöneberg, R.: New Fixed Point Theorems for Compact and Nonexpansive Mappings and Applications to Hammerstein Equations. Sonderforschungsbereich 72, Approximation und Optimierung, Universität Bonn, preprint no. 92, 1976.
[16] Kryszewski, W.: The Lefschetz type theorem for a class of noncompact mappings. Suppl. Rend. Circ. Mat. Palermo 14, 2 (1987), 365-384. · Zbl 0641.55002 · eudml:221305
[17] Leray, J., Schauder, J.: Topologie et \(\acute{e}\)quations fonctionnelles. Ann. Sci. Ecole Norm. Sup. 51 (1934). · Zbl 0009.07301 · numdam:ASENS_1934_3_51__45_0 · eudml:81511
[18] Peitgen, H. O.: On the Lefschetz number for iterates of continuous mappings. Proc. AMS 54 (1976), 441-444. · Zbl 0316.55006 · doi:10.2307/2040837
[19] Skiba, R., Ślosarski, M.: On a generalization of absolute neighborhood retracts. Topology and its Applications 156 (2009), 697-709. · Zbl 1166.54008 · doi:10.1016/j.topol.2008.09.007
[20] Ślosarski, M.: On a generalization of approximative absolute neighborhood retracts. Fixed Point Theory 10, 2 (2009). · Zbl 1185.55002 · www.math.ubbcluj.ro
[21] Ślosarski, M.: Fixed points of multivalued mappings in Hausdorff topological spaces. Nonlinear Analysis Forum 13, 1 (2008), 39-48. · Zbl 1295.54091
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