×

Differential inclusions in Banach spaces. (English) Zbl 0609.34013

Let F be a multifunction from an open set \(\Omega\subset {\mathbb{R}}\times E\) (E a Banach space) to the space \({\mathcal K}_ c(E)\) of all nonempty bounded closed subsets of E, endowed with the Pompeiu-Hausdorff metric. In this paper, the authors establish the existence of solutions for the Cauchy problem \(\dot x\in F(t,x)\), \(x(t_ 0)=x_ 0\) under hypotheses on F that exclude compactness entirely. In fact, they suppose that for each (t,x)\(\in \Omega\), the closed convex hull of F(t,x) has a nonempty interior. If, in addition, the multifunction \(F: \Omega \to {\mathcal K}_ c(E)\) is continuous and the Banach space E is reflexive and separable, then they prove that the above Cauchy problem has solutions.
Reviewer: N.L.Maria

MSC:

34A60 Ordinary differential inclusions
34G10 Linear differential equations in abstract spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antosiewicz, A.; Cellina, A., Continuous selections and differential relations, J. Differential Equations, 19, 386-398 (1975) · Zbl 0279.54007
[2] Castaing, C.; Valadier, M., Équations differentielles multivoques dans les espaces vectoriels localement convexes, Rev. Française Informat. Recherche Opérationnelle, 3, 3-16 (1969) · Zbl 0186.21004
[3] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions, (Lecture Notes in Mathematics, Vol. 580 (1977), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0346.46038
[4] Cellina, A., On the differential inclusion \(ϵ\) [−1, 1], Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIII, 69, 1-6 (1980) · Zbl 0922.34009
[5] Daures, J. P., Contribution à l’étude des équations differentielles multivoques dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. A-B, 270, 269-272 (1970)
[6] De Blasi, F. S.; Pianigiani, G., A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces, Funkcial. Ekvac. (2), 25, 153-162 (1982) · Zbl 0535.34009
[7] De Blasi, F. S.; Pianigiani, G., Remarks on Hausdorff continuous multifunctions and selections, Comment. Mat. Univ. Carolin., 24, 533-561 (1983) · Zbl 0548.54011
[8] Filippov, A. F., The existence of solutions of generalized differential equations, Math. Notes, 10, 608-611 (1971) · Zbl 0265.34074
[9] Godunov, A. M., The Peano’s theorem in Banach spaces, Funkcional. Anal. i Priložen, 9, 59-60 (1974), [Russian]
[10] Hermes, H., The generalized differential equation \(ϵR(t, x)\), Adv. in Math., 4, 149-169 (1970) · Zbl 0191.38803
[11] Kaczyński, H.; Olech, C., Existence of solutions of orientor fields with non-convex right-hand side, Ann. Polon. Math., 29, 61-66 (1974) · Zbl 0285.34008
[12] Muhsinov, A. M., On differential inclusions in Banach spaces, Soviet Math. Dokl., 15, 1122-1125 (1974) · Zbl 0313.34069
[13] Lindenstrauss, J., On operators which attain their norm, Israel J. Math., 1, 139-148 (1963) · Zbl 0127.06704
[14] Tolstogonov, A. A., On differential inclusions in Banach spaces, Soviet. Math. Dokl., 20, 186-190 (1979) · Zbl 0439.34052
[15] De Blasi, F. S.; Pianigiani, G., The Baire category method in existence problems for a class of multivalued differential equations with nonconvex right hand side, Funkcial. Ekvac. (2), 28, 139-156 (1985) · Zbl 0584.34007
[16] Deimling, K., Ordinary Differential Equations in Banach Spaces, (Lecture Notes in Mathematics, Vol. 596 (1977), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0555.60036
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.