On the structure of the solution set for differential inclusions in a Banach space

A. A. Tolstonogov

Sbornik. Mathematics, Tome 46 (1983) no. 1, pp. 1-15

Résumé

In this article a differential inclusion $\dot x\in\Gamma(t,x)$ is considered, where the mapping $\Gamma$ takes values in the family of all nonempty compact convex subsets of a Banach space, is upper semicontinuous with respect to $x$ for almost every $t$, and has a strongly measurable selection for every $x$. Under certain compactness conditions on $\Gamma$ proofs are given for a theorem on the existence of solutions, a theorem on the upper semicontinuous dependence of solutions on the initial conditions, and an analogue of the Kneser–Hukuhara theorem on connectedness of the solution set. Bibliography: 20 titles.

Bibliographie

[1] Sadovskii B. N., “Predelno kompaktnye i uplotnyayuschie operatory”, UMN, 27:1 (173) (1972), 81–146 | MR | Zbl

[2] Lakshmikantham V., Existence and comparison results for differential equations (Int. Conf. Diff. Equat., 1974), New York, 1975 | MR | Zbl

[3] Deimling K., Ordinary differential equations in Banach spaces, V. 596, Lecture Notes Math., Springer-Verlag, 1977 | MR | Zbl

[4] Szufla S., “Measure of non-compactness and ordinary differential equations in Banach spaces”, Bull. Acad. Polon. Sci., ser. math., 19:9 (1971), 831–835 | MR | Zbl

[5] Rzymowski W., Bull. Acad. Polon. Sci., ser. math., 19:4 (1971), On the existence of solutions of the equation $x= f(t,x)$ in Banach space | MR

[6] Goebel K., Rzymowski W., “An existence theorem for the equations $x=f(t,x)$ in Banach space”, Bull. Acad. Polon. Sci., ser. math., 18:7 (1970), 367–370 | MR | Zbl

[7] Kamenskii M. N., “K teoreme Peano v beskonechnomernykh prostranstvakh”, Matem. zametki, 11:5 (1972), 569–575 | MR

[8] Pianigiani G., “Existence of solutions for ordinary differential equations in Banach spaces”, Bull. Acad. Polon. Sci., ser. math., 23:8 (1975), 853–857 | MR | Zbl

[9] Filippov A. F., “Differentsialnye uravneniya s mnogoznachnoi razryvnoi pravoi chastyu”, DAN SSSR, 151:1 (1963), 65–68 | Zbl

[10] Davy J. L., “Properties of the solutions set of a generalized differential equation”, Bull. Austral. Math. Soc., 6:3 (1972), 379–398 | DOI | MR | Zbl

[11] De Blasi F. S., “Existence and stability of solutions for autonomous multivalued differential equations in Banach space”, Atti Accad. Naz. Lincei, Rend. CI. Sci. fis., math., nat., 60:6 (1976), 767–774 | MR | Zbl

[12] Szufla S., “Structure of solutions set of ordinary differential equations in Banach space”, Bull. Acad. Polon. Sci., ser. math., 21:1 (1973), 141–144 | MR | Zbl

[13] Szufla S., “Some properties of the solutions set of ordinary differential equations”, Bull. Acad. Polon. Sci., ser. math., 22:7 (1974), 675–678 | MR | Zbl

[14] Kuratovskii K., Topologiya, T. 1, Mir, M., 1966 | MR

[15] Aumann R. J., “Integrals of set-valued functions”, J. Math. Anal. and Appl., 12:1 (1965), 1–12 | DOI | MR | Zbl

[16] Kuratovskii K., Topologiya, T. 2, Mir, M., 1966 | MR

[17] Himmelberg C. J., “Measurable relations”, Fundam. Math., 87:1 (1975), 53–72 | MR | Zbl

[18] Diestel J., “Remarks on weak compactness in $L_1(\mu,X)$”, Glasgow Math., 18:1 (1977), 87–91 | DOI | MR | Zbl

[19] Eisenfeld J., Lakshmikantham V., Bernfeld S. R., “On the construction of a norm associated with the measure of noncompactness”, Nonlinear Analysis. Theory, Methods, Appl., 1:1 (1976), 49–54 | DOI | MR | Zbl

[20] Himmelberg C. Y., “Precompact contraction of metric uniformities and the continuity of $F(t,x)$”, Rend. Sem. Math. Univ. Padova, 50 (1973), 185–188 | MR