Abstract
Using Krasnoselskii’s fixed point theorem and dichotomy theory, we prove the existence of periodic solutions for differential equations with multiple delays of the form $x'(t)+ cx'(t-\tau) = A(t)x(t) +f(t, x(t-\alpha_{1}(t)),\ldots, x(t-\alpha_{m}(t)))$ , where the parameter $c \ll1$ is a small perturbation for a delayed forced term. Moreover, we discuss the convergence of these solutions to a solution of the unperturbed problem as $c \rightarrow0$ .
Highlights
Delay differential equations are of interest in many areas of applications, such as population dynamics, drug administration, automatic control, laser optics, neural networks, economics and others
Using Krasnoselskii’s fixed point theorem and dichotomy theory, we prove the existence of periodic solutions for differential equations with multiple delays of the form x (t) + cx (t – τ ) = A(t)x(t) + f (t, x(t – α1(t)), . . . , x(t – αm(t))), where the parameter c 1 is a small perturbation for a delayed forced term
1 Introduction Delay differential equations are of interest in many areas of applications, such as population dynamics, drug administration, automatic control, laser optics, neural networks, economics and others
Summary
Delay differential equations are of interest in many areas of applications, such as population dynamics, drug administration, automatic control, laser optics, neural networks, economics and others (see for example [ – ]). Using Krasnoselskii’s fixed point theorem and dichotomy theory, we prove the existence of periodic solutions for differential equations with multiple delays of the form x (t) + cx (t – τ ) = A(t)x(t) + f (t, x(t – α1(t)), . Several results on the existence and uniqueness of bounded solutions, periodic solutions and almost periodic solutions of both linear and nonlinear differential equations are obtained under the assumption that the associated homogeneous linear equation
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