*(English)*Zbl 1024.34068

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

to have a unique pseudo almost-periodic (PAP) solution; here, $p$ and $q$ are nonzero constants, $g:{\mathbb{R}}^{3}\to \mathbb{R}$ is PAP in $t$ uniformly on ${\mathbb{R}}^{2}$, and $[\xb7]$ denotes the greatest integer function. By definition, a function $f:\mathbb{R}\to \mathbb{R}$ 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 $\mathbb{R}$ and ${\left(2T\right)}^{-1}{\int}_{-T}^{T}\left|{f}_{2}\left(t\right)\right|dt\to 0$ as $T\to \infty $. The method consists of first obtaining conditions under which the linear equation

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 $\mathbb{Z}$, 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 $\mathbb{Z}$), and that these lead to PAP solutions on $\mathbb{R}$. 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 |