A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument

In this paper, the spectrum relation of almost periodic solution for the equation (x(t) + px(t − 1))″ = qx([t]) + f(t) is investigated. Although this has been discussed in an article, some counterexamples are constructed to show that some part of the spectrum inclusion in that article is not correct. The key point which causes such problem is found out. A new statement is formulated and proved.     

On the module containment of the almost periodic solution for a class of differential equations with piecewise constant delays

In the first part of this paper, we obtain a new property on the module containment for almost periodic functions. Based on it, we establish the module containment of an almost periodic solution for a class of differential equations with piecewise constant delays. In the second part, we investigate the existence, uniqueness and exponential stability of a positive almost periodic and quasi-periodic solution for a certain class of logistic differential equations with a piecewise constant delay. The module containment for the almost periodic solution is established.     

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

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

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 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.     

On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function.     

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.     

Pseudo almost periodic solutions for equation with piecewise constant argument

By using the roughness theory of exponential dichotomies and the contraction mapping, some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument

     

The existence of pseudo-almost periodic solutions of third-order neutral differential equations with piecewise constant argument

In this paper, we present some existence theorems for pseudo-almost periodic solutions of differential equations with piecewise constant argument by means of pseudo-almost periodic solutions of relevant difference equations.     

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.     

On almost periodic solutions of logistic delay differential equations with almost periodic time dependence

In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458].     

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 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.     

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 the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument

We study the spectrum containment of almost periodic solution to second order differential equations with piecewise constant argument. Some known (periodic, quasi-periodic) results would be expanded. As a corollary, it is shown that such equations with periodic perturbations possess a quasi-periodic solution and no periodic solution. This phenomena is due to the piecewise constant argument. The results are extended to nonlinear equations.     

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.     

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.     

Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments

In this paper we obtain the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument. We derive comparison results for the treated linear operator by studying the sign of the obtained Green's function.