Differential equations on closed subsets of a Banach space

Lakshmikantham, V.; Mitchell, A. Richard; Mitchell, Roger W.

doi:10.1090/S0002-9947-1976-0402224-7

Differential equations on closed subsets of a Banach space
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by V. Lakshmikantham, A. Richard Mitchell and Roger W. Mitchell PDF

Trans. Amer. Math. Soc. 220 (1976), 103-113 Request permission

Abstract:

The problem of existence of solutions to the initial value problem $x’ = f(t,x),x({t_0}) = {x_0} \in F$, where $f \in C[[{t_0},{t_0} + a] \times F,E]$, F is a locally closed subset of a Banach space E is considered. Nonlinear comparison functions and dissipative type conditions in terms of Lyapunov-like functions are employed. A new comparison theorem is established which helps in surmounting the difficulties that arise in this general setup.

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