On almost periodic solutions of logistic delay differential equations with almost periodic time dependence

Abstract

In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451-458].In this paper, we study the dynamics of logistic delay differential equations of the forṁwhere [·] denotes the greatest integer function, f (x) is continuously differentiable for x > 0, f (0) = 0, f (x) > 0 for x > 0, and a(t) and b(t) are positive almost periodic functions. For the special case of b ≡ 1, the existence, uniqueness, and asymptotic stability of an almost periodic solution of (1) have been shown by G. Seifert [14]. In the present paper, we will show that not only do Seifert's results hold for general b, but also the modules of almost periodic solutions can be characterized. We further solve an open problem of Seifert. In [14], Seifert showed

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“…The interested reader is referred to [16,21,22,23,31,35,37,39,40,41,50,51] etc. for some classical results and new trends on periodic solutions of delay differential equations, and to [19,30,38,46,55,56,57,59] and references therein for typical results on almost periodic solutions.…”

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