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Periodic solutions of time-dependent, semilinear evolution equations of compact type

Published online by Cambridge University Press:  14 November 2011

Herbert C. Sager
Herbert C. Sager
Affiliation:
Department of Mathematics, University of the Witwatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa
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Synopsis

We establish the existence of solutions in a weak sense of

where t Є J = [0, T] and′ = d/dt. It is supposed that the unbounded, linear operators A(t) generate analytic and compact semigroups on a Hilbert space H and that B(t, x) are bounded linear operators. The function f(t, x) with values in H may have asymptotically sublinear growth.

We prove the existence of a periodic solution with the help of Schauder’s fixed point theorem.

Accordingly, we first verify that the corresponding linearized version of (0.1),

has a unique solution for each square integrable ψ(t), provided that the homogeneous problem has only the zero solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 95 , Issue 1-2 , 1983 , pp. 7 - 24
Copyright
Copyright © Royal Society of Edinburgh 1983

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