The multicores in metric spaces and their application in fixed point theory

Ślosarski, Mirosław

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 1, pp. 75-94

Abstract (VO)

This paper discusses the notion, the properties and the application of multicores, i.e. some compact sets contained in metric spaces.

Détail

Classification : 47H10, 54C55, 54H25, 55M20
Keywords: Lefschetz number; fixed point; topological vector spaces; Klee admissible spaces; absolute neighborhood multi-retracts; approximative absolute neighborhood multi-retracts; multicore

Bibliographie

[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. | DOI | MR | Zbl

[2] Andres, J., Górniewicz, L.: Topological principles for boundary value problems. Kluwer, Dordrecht, 2003.

[3] Bogatyi, S. A.: Approximative and fundamental retracts. Math. USSR Sb. 22 (1974), 91–103. | DOI | MR

[4] Borsuk, K.: Theory of Shape. Lect. Notes Ser. 28, Matematisk Institut, Aarhus Universitet, Aarhus, 1971. | MR | Zbl

[5] Clapp, M. H.: On a generalization of absolute neighborhood retracts. Fund. Math. 70 (1971), 117–130. | MR | Zbl

[6] Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. New Jersey Princeton University Press, Princeton, 1952. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[10] Górniewicz, L., Ślosarski, M.: Once more on the Lefschetz fixed point theorem. Bull. Polish Acad. Sci. Math. 55 (2007), 161–170. | DOI | MR | Zbl

[11] Górniewicz, L., Ślosarski, M.: Fixed points of mappings in Klee admissible spaces. Control and Cybernetics 36, 3 (2007). | MR

[12] Granas, A.: Generalizing the Hopf–Lefschetz fixed point theorem for non-compact ANR’s. In: Symp. Inf. Dim. Topol., Baton-Rouge, 1967.

[13] Granas, A., Dugundji, J.: Fixed Point Theory. Springer, Berlin–Heidelberg–New York, 2003. | MR | Zbl

[14] Jaworowski, J.: Continuous Homology Properties of Approximative Retracts. Bulletin de L’Academie Polonaise des Sciences 18, 7 (1970). | MR | Zbl

[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. | MR | Zbl

[17] Leray, J., Schauder, J.: Topologie et $\acute{e}$quations fonctionnelles. Ann. Sci. Ecole Norm. Sup. 51 (1934).

[18] Peitgen, H. O.: On the Lefschetz number for iterates of continuous mappings. Proc. AMS 54 (1976), 441–444. | MR | Zbl

[19] Skiba, R., Ślosarski, M.: On a generalization of absolute neighborhood retracts. Topology and its Applications 156 (2009), 697–709. | MR | Zbl

[20] Ślosarski, M.: On a generalization of approximative absolute neighborhood retracts. Fixed Point Theory 10, 2 (2009). | MR | Zbl

[21] Ślosarski, M.: Fixed points of multivalued mappings in Hausdorff topological spaces. Nonlinear Analysis Forum 13, 1 (2008), 39–48. | MR