Abstract
This paper surveys the abstract theories concerning local-in-time existence of solutions to differential inclusions, u′(t)∈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.
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Schechter, E. (1989). A survey of local existence theories for abstract nonlinear initial value problems. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1394. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086759
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