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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 \[ {d^2\over dt^2} (x(t)+ px(t- 1))= qx\Biggl(2\Biggl[{t+ 1\over 2}\Biggr]\Biggr)+ g(t, x(t), x([t]))\tag{1} \] 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 \([\cdot]\) 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 \((2T)^{-1}\int^T_{-T}|f_2(t)|dt\to 0\) as \(T\to\infty\). The method consists of first obtaining conditions under which the linear equation \[ (d^2/dt^2)(x(t)+ px(t- 1))= qx(2[(t+ 1)/2])+ f(t), \] 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-almost periodic solutions to functional-differential equations
34K13 Periodic solutions to functional-differential equations
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