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Viability theorems for a class of differential-operator inclusions. (English) Zbl 0694.34011
Discussed in this paper is the following differential inclusion \[ x'(t)+Ax(t)\in F(x(t))\quad x(0)=x_ 0\in K,\quad x(t)\in K\quad \forall t\geq 0 \] where K is compact in a Banach space X, A is the infinitesimal generator of a compact differential semigroup of bounded linear operators, and \(F: K\to 2^ x\setminus \emptyset\) is upper semicontinuous with compact convex values. It is shown that a natural tangential condition is necessary and sufficient for the existence of a global solution to this problem.
Reviewer: S.Hu

MSC:
34A60 Ordinary differential inclusions
47E05 General theory of ordinary differential operators
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[1] Aubin, J.P, Domaines de viabilité des inclusions différentielles-opérationnelles, Rapport du centre de recherches mathématiques. univ. de montréal, CRM-1389, (1986)
[2] Aubin, J.P; Cellina, A, Differential inclusions, (1984), Springer-Verlag Berlin/Heidelberg/New York/Tokyo
[3] Aubin, J.P; Ekeland, I, Applied nonlinear analysis, (1984), Wiley New York
[4] Clarke, F.H, Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0727.90045
[5] Haddad, G, Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. math., 39, 83-100, (1981) · Zbl 0462.34048
[6] Martin, R.H, Differential equations on closed subsets of a Banach space, Trans. am. math. soc., 179, 399-414, (1973) · Zbl 0293.34092
[7] Martin, R.H, Nonlinear operators and differential equations in Banach spaces, (1976), Wiley New York
[8] Pavel, N.H, Invariant sets for a class of semilinear equations of evolution, Nonlinear anal., 1, 187-196, (1977) · Zbl 0344.45001
[9] Pavel, N.H, Semilinear equations with dissipative time-dependent domain perturbations, Israel J. math., 46, 103-122, (1983) · Zbl 0535.35035
[10] Pazy, A, Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York/Berlin/Heidelberg/Tokyo · Zbl 0516.47023
[11] Shuzhong, Shi, Théorèmes de viabilité pour LES inclusions aux dérivées partielles, C. R. acad. sci. Paris, Sér. I math., 303, 11-14, (1986) · Zbl 0591.47038
[12] Shuzhong, Shi, Nagumo type condition for partial differential inclusions, Nonlinear anal., 12, 951-967, (1988) · Zbl 0654.49016
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