The search for almost periodic solutions of any dissipative equation of the form(1.1), in which p(t) is an almost periodic function, has come to be closely linked up with a number of standard « convergence » restrictions on f, g′, g″ and k (see, for example, and). The object of the present paper is to show that as far as the existence, alone, of an almost periodic solution of(1.1) is concerned these « convergence » restrictions on f, g′, g″ and k are quite unnecessary. The first result (Theorem 1) shows in fact that the conditions(1.2) alone are quite sufficient for the existence of an almost periodic solution of(1.1); and Theorem 2 extends this result (though under stronger conditions on f and g) to the case in which the forcing function depends on x and x as well.
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Partially supported by N.S.F. research project G-57 at The University of Michigan.
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Ezeilo, J.O.C. On the existence of almost periodic solutions of some dissipative second order differential equations. Annali di Matematica 65, 389–405 (1964). https://doi.org/10.1007/BF02418235
- Differential Equation
- Periodic Solution
- Periodic Function
- Strong Condition
- Order Differential Equation