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Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. (English) Zbl 1024.34068

The author gives conditions for a second-order scalar neutral functional-differential equation of the form

$\frac{{d}^{2}}{d{t}^{2}}\left(x\left(t\right)+px\left(t-1\right)\right)=qx\left(2\left[\frac{t+1}{2}\right]\right)+g\left(t,x\left(t\right),x\left(\left[t\right]\right)\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

to have a unique pseudo almost-periodic (PAP) solution; here, $p$ and $q$ are nonzero constants, $g:{ℝ}^{3}\to ℝ$ is PAP in $t$ uniformly on ${ℝ}^{2}$, and $\left[·\right]$ denotes the greatest integer function. By definition, a function $f:ℝ\to ℝ$ is PAP if $f={f}_{1}+{f}_{2}$, where $f$ is almost-periodic (in the sense of Bohr) and ${f}_{2}$ is continuous and bounded on $ℝ$ and ${\left(2T\right)}^{-1}{\int }_{-T}^{T}|{f}_{2}\left(t\right)|dt\to 0$ as $T\to \infty$. The method consists of first obtaining conditions under which the linear equation

$\left({d}^{2}/d{t}^{2}\right)\left(x\left(t\right)+px\left(t-1\right)\right)=qx\left(2\left[\left(t+1\right)/2\right]\right)+f\left(t\right),$

where $f$ is PAP, has a unique PAP solution, and then using this equation to define a map of the Banach space of PAP functions into itself, and showing that this map will under suitable Lipschitz conditions on $g$ be a contraction. Since these equations involve unknown functions with piecewise constant arguments, the solutions on $ℤ$, the set of all integers, can be determined in terms of difference equations, and the basic idea is then to show that such difference equations have PAP sequence solutions (on $ℤ$), and that these lead to PAP solutions on $ℝ$. The concept of pseudo-periodic function is also introduced and a result on the existence of such solutions is given.

The reviewer would have been interested in some remarks concerning any applications for which equation (1) is a model and also the significance of the concept of PAP functions in such applications.

##### MSC:
 34K14 Almost and pseudo-periodic solutions of functional differential equations 34K13 Periodic solutions of functional differential equations