Qualitative analysis of a nonautonomous nonlinear delay differential equation

Abstract

This paper is devoted to the systematic study of some qualitative properties of solutions of a nonautonomous nonlinear delay equation, which can be utilized to model single population growths. Various results on the boundedness and oscillatory behavior of solutions are presented. A detailed analysis of the global existence of periodic solutions for the corresponding autonomous nonlinear delay equation is given. Moreover, sufficient conditions are obtained for the solutions to tend to the unique positive equilibrium.

Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology

“…In other words, there is an urgent need to investigate the global qualitative behavior of solutions of these equations. Several attempts along this line are documented in [5,9,16,22] for differential delay models in population dynamics. Related works for systems of nonlinear (autonomous and nonautonomous) differentia] delay equations can be found in [17 19, 21].…”

Solvability of a neutral differential equation with deviated argument

“…The result obtained here generalizes several ones obtained earlier by many authors (cf. [1,2,4,5,8,[11][12][13][14][15]18], for instance).…”

On Oscillation of a Generalized Logistic Equation with Several Delays

“…Then as in case m = 1 [16,17] the solution of (3), (26) is positive. A positive solution y of (3) is said to be oscillatory about K if there exists a sequence t n t n → ∞, such that y t n − K = 0 n = 1 2 ; y is said to be nonoscillatory about K if there exists T ≥ t 0 such that y t − K > 0 for t ≥ T .…”

“…Remark. If m = 1 and the parameters of (3) are continuous functions then the results of Theorems 11 and 12 were first obtained in [16] (see also [17]). …”

“…Publications [7][8][9][10][11][12][13][14][15][16][17][18] are devoted to various generalizations of logistic equation (1). For example, in [9,16,17] the authors considered an equationẏ t = r t y t 1 − y h t K 1 − y h t K α−1 (2) where α < 1 (sublinear case) or α > 1 (superlinear case).…”