Abstract
In this paper, a class of first-order neutral differential equations with time-varying delays and coefficients is considered. Some results on the existence of positive almost periodic solutions for the equations are obtained by using the contracting mapping principle and the differential inequality technique. In addition, an example is given to illustrate our results.
Highlights
IntroductionIn recent years, there has been increasing interest in the existence and stability of almost periodic type solutions for firstorder functional differential equations in population models [8,9,10,11,12]
In recent years, the following first-order neutral differential equations x(t) – P(t)x(t – r) = –Q(t)x(t) + f t, x(t – r) (1.1)and x(t) – cx t – τ (t) = –Q(t)x(t) + f t, x t – τ (t) (1.2)have been extensively used to describe the dynamic behaviors for the blood cell production models, population models, and control models
Motivated by the above discussions, in this paper we aim to establish some sufficient conditions on the existence of positive almost periodic solutions of the following first-order neutral differential equations with time-varying delays and coefficients: x(t) – P(t)x t – τ1(t) = –Q(t)x(t) + f t, x t – τ2(t), (1.4)
Summary
In recent years, there has been increasing interest in the existence and stability of almost periodic type solutions for firstorder functional differential equations in population models [8,9,10,11,12]. Motivated by the above discussions, in this paper we aim to establish some sufficient conditions on the existence of positive almost periodic solutions of the following first-order neutral differential equations with time-varying delays and coefficients: x(t) – P(t)x t – τ1(t) = –Q(t)x(t) + f t, x t – τ2(t) ,. The contributions of this paper can be summarized as follows: (1) In this manuscript, all delays and coefficients of (1.4) are time-varying, and (1.1) and (1.2) are special cases of (1.4); (2) The sufficient conditions for the existence of positive almost periodic solution are derived in terms of its coefficients without (1.3), which has not been investigated till now. We refer the reader to [13, 14]
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