Abstract
Let X be a Banach space, D⊂X, f: [0,∞)xD→X continuous and ω-periodic. In this paper we consider various conditions on D and f sufficient for existence of an ω-periodic solution of the differential equation u′=f(t,u). In the main, we shall assume that D is closed bounded and convex and f satisfies a boundary condition at δD such that D is flow invariant for u′=f(t,u). The map f is assumed to be either compact or dissipative or a certain perturbation of such maps.
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Deimling, K. Periodic solutions of differential equations in Banach spaces. Manuscripta Math 24, 31–44 (1978). https://doi.org/10.1007/BF01168561
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DOI: https://doi.org/10.1007/BF01168561