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A survey of local existence theories for abstract nonlinear initial value problems. (English) Zbl 0678.34005
Nonlinear semigroups, partial differential equations and attractors, Proc. Symp., Washington/DC 1987, Lect. Notes Math. 1394, 136-184 (1989).
Summary: [For the entire collection see Zbl 0673.00012.]
This paper surveys the abstract theories concerning local-in-time existence of solutions to differential inclusions, \(u'(t)\in F(t,u(t))\), in a Banach space. Three main approaches assume generalized compactness, isotonicity in an ordered Banach space, or dissipativeness. We consider different notions of “solution”, and also the importance of assuming or not assuming that F(t,x) is continuous in x. Other topics include Carathéodory conditions, uniqueness, semigroups, semicontinuity, subtangential conditions, limit solutions, continuous dependence of u on F, and bijections between u and F.

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A60 Ordinary differential inclusions
34A34 Nonlinear ordinary differential equations and systems