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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

34A60 Ordinary differential inclusions
34G10 Linear differential equations in abstract spaces
Full Text: DOI
[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, () · 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, () · Zbl 0555.60036
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