Geodesic
Topological characteristics of multi-valued maps and Lipschitzian functionals
Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 717-739.

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This paper deals with the operator inclusion $0\in F(x)+N_Q(x)$, where $F$ is a multi-valued map of monotonic type from a reflexive space $V$ to its conjugate $V^*$ and $N_Q$ is the cone normal to the closed set $Q$, which, generally speaking, is not convex. To estimate the number of solutions of this inclusion we introduce topological characteristics of multi-valued maps and Lipschitzian functionals that have the properties of additivity and homotopy invariance. We prove some infinite-dimensional versions of the Poincaré–Hopf theorem. 
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V. S. Klimov. Topological characteristics of multi-valued maps and Lipschitzian functionals. Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 717-739. http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a4/

[1] S. I. Pokhozhaev, “Solvability of nonlinear equations with odd operators”, Funct. Anal. Appl., 1:3 (1967), 227–233 | DOI | MR | Zbl

[2] F. E. Browder, “Pseudo-monotone operators and the direct method of the calculus of variations”, Arch. Ration. Mech. Anal., 38:4 (1970), 268–277 | DOI | MR | Zbl

[3] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Gauthier-Villars, Paris, 1969 | MR | MR | Zbl | Zbl

[4] I. V. Skrypnik, Nelineinye ellipticheskie uravneniya vysshego poryadka, Naukova dumka, Kiev, 1973 | MR | Zbl

[5] N. A. Bobylev, S. V. Emelyanov, S. G. Korovin, Geometricheskie metody v variatsionnykh zadachakh, Magistr, M., 1998

[6] A. D. Ioffe, V. M. Tikhomirov, Teoriya ekstremalnykh zadach, Nelineinyi analiz i ego prilozheniya, Nauka, M., 1974 | MR | Zbl

[7] Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, V. V. Obukhovskii, “Topological methods in the fixed-point theory of multi-valued maps”, Russian Math. Surveys, 35:1 (1980), 65–143 | DOI | MR | Zbl

[8] Yu. G. Borisovich, B. D. Gelman, A. D. Myshkis, V. V. Obukhovskii, Vvedenie v teoriyu mnogoznachnykh otobrazhenii, Izd-vo Voronezhskogo un-ta, Voronezh, 1986 | MR | Zbl

[9] F. H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1983 | MR | MR | Zbl | Zbl

[10] M. A. Krasnoselskii, P. P. Zabreiko, Geometricheskie metody nelineinogo analiza, Nelineinyi analiz i ego prilozheniya, Nauka, M., 1975 | MR

[11] J.-P. Aubin, I. Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics, Wiley, New York, 1984 | MR | MR | Zbl

[12] V. S. Klimov, N. V. Senchakova, “On the relative rotation of multivalued potential vector fields”, Math. USSR-Sb., 74:1 (1993), 131–144 | DOI | MR | Zbl | Zbl

[13] V. S. Klimov, “Nontrivial solutions of equations with potential operators”, Siberian Math. J., 21:6 (1981), 773–786 | DOI | MR | Zbl

[14] V. S. Klimov, “Topological characteristics of non-smooth functionals”, Izv. Math., 62:5 (1998), 969–984 | DOI | MR | Zbl

[15] V. S. Klimov, “Infinite-dimensional version of the Poincaré–Hopf theorem and homological characteristics of functionals”, Sb. Math., 192 (2001), 49–64 | DOI | MR | Zbl

[16] V. V. Fedorchuk, V. V. Filippov, Obschaya topologiya. Osnovnye konstruktsii, Uchebnoe posobie, Izd-vo MGU, M., 1988 | Zbl

[17] W. S. Massey, Homology and cohomology theory. An approach based on Alexander–Spanier cochains, Textbooks in Pure and Applied Mathematics, 46, Marcel Dekker, New York–Basel, 1978 | MR | MR | Zbl | Zbl

[18] S. Eilenberg, N. Steenrod, Foundations of algebraic topology, Princeton Mathematical Series, 15, Princeton University Press, Princeton, 1952 | MR | Zbl

[19] O. A. Ladyzhenskaya, “On the determination of minimal global attractors for the Navier–Stokes and other partial differential equations”, Russian Math. Surveys, 42:6 (1987), 27–73 | DOI | MR | Zbl

[20] V. V. Prasolov, Elementy kombinatornoi i differentsialnoi topologii, Izd-vo MTsNMO, M., 2004

[21] F. P. Vasilev, Metody resheniya ekstremalnykh zadach. Zadachi minimizatsii v funktsionalnykh prostranstvakh, regulyarizatsiya, approksimatsiya, Nauka, M., 1981 | MR | Zbl

[22] P. D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions, Birkhäuser, Boston, MA, 1985 | MR | MR | Zbl | Zbl

[23] V. S. Klimov, “Lower bounds for the number of solutions of some boundary-value problems”, Differ. Equ., 20:3 (1984), 373–379 | MR | Zbl

[24] V. S. Klimov, “The number of solutions to boundary value problems of the nonlinear theory of shells”, Soviet Phys. Dokl., 24:7 (1979), 580–582 | MR | Zbl

[25] V. S. Klimov, “Infinite-dimensional version of Morse theory for Lipschitz functionals”, Sb. Math., 193:6 (2002), 889–906 | DOI | MR | Zbl