Almost periodic solutions of second-order neutral delay-differential equations with piecewise constant arguments

In this paper, we present an existence theorem of almost periodic solutions of second-order neutral delay-differential equations with piecewise constant arguments of the form (x(t)+x(t−1))″=qx([t])+f(t), where [·] denotes the greatest integer function, q is a nonzero constant, and f(t) is almost periodic.     

Bifurcation of special periodic solutions to delay-differential equations

Summary We consider a class of forced delay differential equations in which the delay is given by /2 and investigate the problem of finding its special periodic solutions. We first approximate these by a Rayleigh-Ritz-Galerkin sequence. Our second method introduces an averaged model thought to give a qualitative approximation to the solution behavior. The effects of cubic and quintic nonlinearities are compared.

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Zusammenfassung Wir untersuchen eine Klasse von forcierten Differentialgleichungen mit Zeitverzögerung /2 und suchen nach speziellen periodischen Lösungen. Zunächst approximieren wir Lösungen durch eine Folge von Rayleigh-Ritz-Galerkin-Näherungen. Als zweite Methode benutzen wir ein Mittelungsverfahren zur qualitativen Approximation des Lösungsverhaltens. Die Effekte von Nichtlinearitäten dritter und fünfter Ordnung werden verglichen.

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Supported in part by the DAAD/CONICYT Professor Exchange Program.

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Supported in part by CONICYT (Grant 89-576) and University of Concepción (Grant 20-12-17).

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Supported in part by DICYT at the University of Santiago (Grant 16-14).     

Stability of periodic solutions of state-dependent delay-differential equations

We consider a class of autonomous delay-differential equations

     

Almost periodic solutions for nonlinear duffing equations

The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain an intersting result on the existence of almost periodic solutions. This work is supported by NSF of China, No.19401013     

Almost periodic solutions for abstract impulsive differential equations

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.     

Almost periodic solutions of affine ito equations

We discuss the problem of the existence of almost periodic in distribution solutions of affine stochastic differential equations with almost periodic coefficients. We prove that if the linear part of the affine equation is exponentially stable in mean square then the unique continuous L² -bounded solution of the affine system has the onedimensional distributions almost periodic. An analogous result is shown for the asymptotic almost periodic case     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

Almost periodic solutions of neutral functional differential equations

We study a nonlinear neutral functional differential equation. Applying the properties of almost periodic function and exponential dichotomy of linear system as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of almost periodic solution of the equations.     

Almost periodic solutions of linear partial differential equations

Let L be an arbitrary linear partial differential operator and let f be an almost periodic function for t in R^m. In this paper we present sufficient conditions that a bounded solution u of Lu = f be almost periodic. Our work generalizes the theorem of Sibuya [5] for Poisson's equation and the theorems of Favard [3] and Bochner [1] for ordinary differential equations.     

Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups.     

Almost periodic solutions for some integrodifferential equations with infinite delay

By using a Liapunov functional, the conditions of existence and uniqueness of almost periodic solutions for some integrodifferential equations are obtained.