Himmelberg, C. J.; van Vleck, F. S. Existence of solutions for generalized differential equations with unbounded right-hand side. (English) Zbl 0582.34002 J. Differ. Equations 61, 295-320 (1986). This paper is concerned with the existence of solutions to the initial value problem for generalized differential equations (orientor fields) when the right-hand side, F(t,x), may be unbounded. Two global and one local existence theorems are established when F satisfies Carathéodory type conditions involving weak integral boundedness conditions. The multiple-valued function F(t,\(\cdot)\) is assumed to have closed graph and to be lower semicontinuous at each point x where F(t,x) is not convex. Cited in 10 Documents MSC: 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations Keywords:initial value problem; orientor fields; Carathéodory type conditions; weak integral boundedness conditions PDFBibTeX XMLCite \textit{C. J. Himmelberg} and \textit{F. S. van Vleck}, J. Differ. Equations 61, 295--320 (1986; Zbl 0582.34002) Full Text: DOI References: [1] Antosiewicz, H. A.; Cellina, A., Continuous selections and differential relations, J. Differential Equations, 19, 386-398 (1975) · Zbl 0279.54007 [2] Bressan, A., On differential relations with lower continuous right-hand side: An existence theorem, J. Differential Equations, 37, 89-97 (1980) · Zbl 0418.34017 [3] Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations (1965), Heath: Heath Boston · Zbl 0154.09301 [4] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions, (Lecture Notes in Mathematics, Vol. 580 (1977), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0346.46038 [5] Davy, J. L., Properties of the solution set of a generalized differential equation, Bull. Austral. Math. Soc., 6, 379-398 (1972) · Zbl 0239.49022 [6] Dauer, J. P.; Van Vleck, F. S., Measurable selectors of multifunctions and applications, Math. Systems Theory, 7, 367-376 (1974) · Zbl 0305.49011 [7] Filippov, A. F., Math. Notes, 608-611 (1971), English transl. · Zbl 0265.34074 [8] Himmelberg, C. J., Measurable relations, Fund. Math., 87, 53-72 (1975) · Zbl 0296.28003 [9] Himmelberg, C. J.; Van Vleck, F. S., An extension of Brunovsky’s Scorza Dragoni type theorem for unbounded set-valued functions, Mat. Casopis Sloven. Akad. Vied., 26, 47-52 (1976) · Zbl 0328.28004 [10] Jacobs, M. Q., On the approximation of integrals of multivalued functions, SIAM J. Control, 7, 158-177 (1969) · Zbl 0176.07302 [11] Kuratowski, K., Topology II (1968), Academic Press: Academic Press New York [12] Kaczynski, H.; Olech, C., Existence of solutions of orientor fields with non-convex right-hand side, Ann. Polon. Math., 19, 61-66 (1974) · Zbl 0285.34008 [13] Lasota, A.; Opial, A., An application of the Kakutani-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 13, 781-786 (1965) · Zbl 0151.10703 [14] Łojasiewicz, S., The existence of solutions for lower semicontinuous orientor fields, Bull. Acad. Polon. Sci. Sér. Sci. Math., 28, 483-487 (1980) · Zbl 0483.49028 [15] Łojasiewicz, S., Some theorems of Scorza-Dragoni type for multifunctions with applications to the problem of existence of solutions for differential multivalued equations, (preprint No. 255 (1982), Institute of Mathematics, Polish Academy of Sciences) · Zbl 0492.34012 [16] McShane, E. J., Integration (1957), Princeton University Press: Princeton University Press Princeton · Zbl 0323.60058 [17] Olech, C., Existence of solutions of non-convex orientor fields, Boll. Un. Mat. Ital., 11, 189-197 (1975) · Zbl 0322.34002 [19] Wagner, D., Survey of measurable selection theorems, SIAM J. Control Optim., 15, 859-903 (1977) · Zbl 0407.28006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.