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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BROWDER, F.E.: Periodic solutions of nonlinear equations in infinite dimensional spaces. Lect. Diff. Eqs. (ed. K. AZIZ) Vol. 1,71–96. Van Nostrand, New York 1969
CLARKSON, J.A.: Uniformly convex spaces. Trans. Amer. Math. Soc.40, 396–414 (1936)
DEIMLING, K.: Ordinary differential equations in Banach spaces. Lect. Notes596, Springer Verlag 1977
---: Open problems in the theory of differential equations in Banach spaces (submitted)
KENMOCHI, N.; TAKAHASHI, T.: On the global existence of solutions of differential equations on closed sub-sets of a Banach space. Proc. Jap. Acad.51, 520–525 (1975)
MASSERA, J.L.; SCHÄFFER, J.J.: Linear differential equations and function spaces. Acad. Press, New York 1966
SCHMITT, K.; VOLKMANN, P.: Boundary value problems for second order differential equations in convex sub-sets of a Banach space. Trans. Amer. Math. Soc.218, 397–405 (1976)
VIDOSSICH, G.: On the structure of periodic solutions of differential equations. J. Diff. Eqs.21, 263–278 (1976)
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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