Abstract
Applications of topological characteristics of nonlinear (one-valued and multi-valued) maps are well-known efficient tools for the investigation of solvability for various problems of the theory of differential equations and of optimal control theory. In this paper, a construction of one such characteristic is proposed: this is the degree of condensing multi-valued perturbations of maps of class (S)+. Principal properties of the characteristic are studied. The considered characteristic is applied for the investigation of a class of controllable systems.
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References
E. S. Baranovskiĭ, “A relation theorem for the α-condition and the map properness,” Tr. Mat. Fak. Voronezh. Gos. Univ. (N.S.), No. 10, 15–17 (2006).
E. S. Baranovskiĭ, “On applications of topological degrees to the investigation of a structure of the solution set for a class of inclusions,” Vestnik Voronezh. Univ. Ser. Fiz. Mat., No. 1, 112–120 (2007).
E. S. Baranovskiĭ and V. G. Zvyagin, “The construction of the degree of a class of multivalued perturbations of the operators, satisfying the α-condition,” Nonlinear Bound. Value Probl., 16, 107–117 (2006).
Yu.G. Borisovich, B. D. Gel’man, A.D. Myshkis, and V. V. Obukhovskiĭ, Introduction to the Theory of Multi-valued Maps and Differential Inclusions [in Russian], URSS, Moscow (2005).
K. Borsuk, Theory of Retracts [Russian translation], Mir, Moscow (1971).
V. T. Dmitrienko and V.G. Zvyagin, “Homotopy classification of a class of continuous mappings,” Math. Notes, 31, 404–410 (1982).
P. Dzekka, V.G. Zvyagin, and V.V. Obukhovskiĭ, “On oriented coincidence index for nonlinear Fredholm inclusions,” Dokl. Math., 73, No. 1, 63-66 (2006).
B. D. Gel’man, “Topological properties of the set of fixed points of multivalued mappings,” Sb. Math., 188, No. 12, 1761–1782 (1997).
L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht–Boston–London (1999).
L. Gorniewicz, A. Granas, and W. Kryszewski, “On the homotopy method in the fixed point index theory for multi-mappings of compact ANR’s,” J. Math. Anal. Appl., 161, No. 2, 457–473 (1991).
M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin–New York (2001).
M. A. Krasnosel’skiĭ and P. P. Zabreiĭko, Geometric Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).
A. D. Myshkis, “Generalizations of the theorem on a stationary point of a dynamical system inside a closed trajectory,” Mat. Sb., 34 (76), 525–540 (1954).
V. Obukhovskii, P. Zecca, and V. Zvyagin, “An oriented index for nonlinear Fredholm inclusions with nonconvex-valued perturbations,” Abstr. Appl. Anal., 2006, Art. ID 51794, 1–21 (2006).
I. V. Skrypnik, Methods for Studying Nonlinear Elliptic Boundary Value Problems [in Russian], Nauka, Moscow (1990).
V. Zvyagin, “On the theory of generalized condensing perturbations of continuous mappings,” Springer Lecture Notes in Math., 1108, 173–193 (1984).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 35, Proceedings of the Fifth International Conference on Differential and Functional Differential Equations. Part 1, 2010.
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Zvyagin, V.G., Baranovskii, E.S. Topological degree of condensing multi-valued perturbations of the (S)+-class maps and its applications. J Math Sci 170, 405–422 (2010). https://doi.org/10.1007/s10958-010-0094-8
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DOI: https://doi.org/10.1007/s10958-010-0094-8