Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument

Certain almost periodic forced perturbed systems with piecewise argument are considered in this paper. By using the contraction mapping principle and some new analysis technique, some sufficient conditions are obtained for the existence and uniqueness of almost periodic solution of these systems. Furthermore, we study the harmonic and subharmonic solutions of these systems. The obtained results generalize the previous known results such as [A.M. Fink, Almost Periodic Differential Equation, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin, 1974; C.Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, 1992 (in Chinese); Z.S. Lin, The existence of almost periodic solution of linear system, Acta Math. Sinica 22 (5) (1979) 515-528 (in Chinese); C.Y. He, Existence of almost periodic solutions of perturbation systems, Ann. Differential Equations 9 (2) (1992) 173-181; Y.H. Xia, M. Lin, J. Cao, The existence of almost periodic solutions of certain perturbation system, J. Math. Anal. Appl. 310 (1) (2005) 81-96]. Finally, a tangible example and its numeric simulations show the feasibility of our results, the comparison between non-perturbed system and perturbed system, the relation between systems with and without piecewise argument.     

Pseudo almost periodic solutions of differential equations with piecewise constant arguments

In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems.     

Positive pseudo almost periodic solutions for a class of differential iterative equations with biological background

The article considers a class of differential iterative equations with biological background. We establish some sufficient conditions for the existence of positive pseudo almost periodic solutions by applying exponential dichotomy and contraction mapping principle. The obtained results extend some known ones.     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences.     

On a new almost periodic type solution of a class of singularly perturbed differential equations with piecewise constant argument

A new class of ergodic sequences, pseudo almost periodic sequence, is introduced, and the existence of pseudo almost periodic sequence to difference equation is studied. Based on these, we investigate the existence of pseudo almost periodic solutions for a nonautonomous, singularly perturbed differential equations with piecewise constant argument.     

Existence of almost periodic solutions of second order neutral delay differential equations with piecewise constant argument

By constructing almost periodic sequence solutions to difference equations, the existence of almost periodic solutions of neutral delay differential equations with piecewise constant argument

Pseudo almost periodic solutions for a class of first order differential iterative equations

This paper is concerned with a class of first order differential iterative equations. Under proper conditions, we employ a novel argument to establish a criterion on the existence of pseudo almost periodic solutions. The obtained result complements with some existing ones.     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [t]

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences     

A new almost periodic type of solutions of second order neutral delay differential equations with piecewise constant argument

A definition of pseudo almost periodic sequence is given and the existence of pseudo almost periodic sequence to difference equation is studied. Based on these, the existence of pseudo almost periodic solutions to neutral delay differential equations with piecewise constant argument is investigated     

On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument

We study the spectrum containment of almost periodic solution to neutral delay differential equations with piecewise constant argument (EPCA, for short). We find an important property, which is different from that given by Cartwright for ordinary differential equations (ODE). Some known (periodic solution) results would be expanded. As a corollary, it is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. This new phenomenon is due to the piecewise constant argument and illustrates a crucial difference between ODE and EPCA.     

Weighted pseudo almost periodic solutions of N-th order neutral differential equations with piecewise constant arguments

In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations.     

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

We consider the existence and stability of an almost periodic solution of the following hybrid system:

(1)

where if θ_i≤t<θ_i+1,i=…−2,−1,0,1,2,…, is an identification function, θ_i is a strictly ordered sequence of real numbers, unbounded on the left and on the right, p_j,j=1,2,…,m, are fixed integers, and the linear homogeneous system associated with (1) satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.     

Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant arguments

Existence criteria are proved for the periodic solutions of a first order nonlinear differential equation with piecewise constant arguments.     

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

In this paper, we present a theorem on the almost periodic solutions of second-order neutral equations with piecewise constant arguments of the form

(x(t)+px(t−1))^″=qx([t])+f(t),${(x\left(t\right)+px(t-1))}^{″}=qx\left(\left[t\right]\right)+f\left(t\right)\text{,}$

The almost periodic type difference equations

In this paper, we study the existence of almost periodic type solutions of difference equations by means of exponential dichotomy and trichotomy of linear difference equations.     

Periodic solutions of a neutral differential equation with piecewise constant arguments

Existence criteria are proved for the periodic solutions of a neutral differential equation with piecewise constant arguments.     

Pseudo S-asymptotically ω-periodic solution for a differential equation with piecewise constant argument in a Banach space

ABSTRACT

In this paper, we give sufficient conditions for the existence and uniqueness of Pseudo S-Asymptotically ω-periodic solutions for a differential equation with piecewise constant argument in a Banach space. This result is obtained using the Pseudo S-asymptotically ω-periodic sequences.     

一类时滞三阶微分方程概周期解的存在性

运用二分性及压缩映射原理,研究一类时滞三阶微分方程概周期解的存在性,得到此类微分方程的概周期解存在的充分性定理．