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Properties of the solution set of a generalized differential equation

Published online by Cambridge University Press:  17 April 2009

J.L. Davy
J.L. Davy
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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We prove that the solution set of a generalized differential equation is connected and that points on the boundary of the solution funnel are peripherally attainable. This is done without the additional assumption of continuity in the state variable required in previous results. The result on upper semicontinuity of the solution set with respect to initial conditions is extended to include variations of initial time.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 3 , June 1972 , pp. 379 - 398
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Aumann, Robert J., “Integrals of set-valued functions”, J. Math. Anal. Appl. 12 (1965), 1–12.CrossRefGoogle Scholar
[2]Berge, Claude, Espaces topologiques: fonctions multivoques (Dunod, Paris, 1959).Google Scholar
[3]Castaing, Charles, “Sur les équations différentielles multivoques”, C.R. Acad. Sci. Paris, Sér. A-B 263 (1966), A63–A66.Google Scholar
[4]Coppel, W.A., Stability and asymptotic behaviour of differential equations (D.C. Heath, Boston, 1965).Google Scholar
[5]Dunford, Nelson and Schwartz, Jacob T., Linear operators, Part I (Interscience [John Wiley & Sons], New York, London, 1958).Google Scholar
[6]Filippov, A.F., “Differential equations with many-valued discontinuous right-hand side”, Soviet Math. Dokl. 4 (1963), 941–945.Google Scholar
[7]Kikuchi, Norio, “Control problems of contingent equation”, Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A 3 (1967), 85–99.Google Scholar
[8]Kikuchi, Norio, “On some fundamental theorems of contingent equations in connection with the control problems”, Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A 3 (1967), 177–201.Google Scholar
[9]Kikuchi, Norio, “On contingent equations satisfying the Carathéodory type conditions”, Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A 3 (1968), 361–371.CrossRefGoogle Scholar
[10]Lasota, A., Opial, Z., “An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations”, Bull. Acad. Polon. Sci., Ser. Sci. math. astron. phys. 13 (1965), 781–786.Google Scholar
[11]Pliś, A., “Measurable orientor fields”, Bull. Acad. Polon. Sci., Ser. Sci. math. astron. phys. 13 (1965) 565–569.Google Scholar
[12]Wazewski, T., “On an optimal control problem”, Differential equations and their applications (Proc. Conf., Prague, 1962, 229–242). (Publ. House Czechoslovak Acad. Sci., Prague; Academic Press, New York, London, 1963.)Google Scholar