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, and are nonzero constants, is PAP in uniformly on , and denotes the greatest integer function. By definition, a function is PAP if , where is almost-periodic (in the sense of Bohr) and is continuous and bounded on and as . The method consists of first obtaining conditions under which the linear equation
where 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 be a contraction. Since these equations involve unknown functions with piecewise constant arguments, the solutions on , 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 ), and that these lead to PAP solutions on . 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.