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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