Abstract
Abstract In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.
Highlights
In the study of almost periodic di erential equations, many useful methods have been developed in the classical references such as Hale and Lunel [1], Fink [2], Yoshizawa [3], and Hino et al [4].Di erential equations with piecewise constant arguments are usually referred to as a hybrid system, and could model certain harmonic oscillators with almost periodic forcing
In this paper, we describe a method to solve the problem of nding periodic solutions for secondorder neutral delay-di erential equations with piecewise constant arguments of the form x (t) + px (t − ) = qx([t]) + f (t), where [·] denotes the greatest integer function, p and q are nonzero real or complex constants, and f (t) is complex valued periodic function
By applying the well-known properties of linear system in algebra, all existence conditions are described for n-periodic solutions that yields explicit formula for the solutions of (1)
Summary
In the study of almost periodic di erential equations, many useful methods have been developed in the classical references such as Hale and Lunel [1], Fink [2], Yoshizawa [3], and Hino et al [4]. Di erential equations with piecewise constant arguments are usually referred to as a hybrid system, and could model certain harmonic oscillators with almost periodic forcing. A recently published paper [9] has studied the di erential equation of the form x (t) + px (t − ) = qx( [ t + ]) + f (t),. In this paper we study certain functional di erential equation of neutral delay type with piecewise constant arguments of the form x (t) + px (t − ) = qx([t]) + f (t),
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