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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, $g: \bbfR^3\to \bbfR$ is PAP in $t$ uniformly on $\bbfR^2$, and $[\cdot]$ denotes the greatest integer function. By definition, a function $f: \bbfR\to\bbfR$ 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 $\bbfR$ 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 $\bbfZ$, 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 $\bbfZ$), and that these lead to PAP solutions on $\bbfR$. 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.

34K14Almost and pseudo-periodic solutions of functional differential equations
34K13Periodic solutions of functional differential equations
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