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A survey of local existence theories for abstract nonlinear initial value problems

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Nonlinear Semigroups, Partial Differential Equations and Attractors

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1394))

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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  1. Mathematics Department, Vanderbilt University, Station B, Box 21, 37235, Nashville, Tennessee, U.S.A.

    Eric Schechter

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  1. Eric Schechter
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Tepper L. Gill Woodford William Zachary

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© 1989 Springer-Verlag

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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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  • DOI: https://doi.org/10.1007/BFb0086759

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51594-4

  • Online ISBN: 978-3-540-46679-6

  • eBook Packages: Springer Book Archive

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