Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments

Abstract In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.


Introduction
In the study of almost periodic di erential equations, many useful methods have been developed in the classical references such as Hale and Lunel [1], Fink [2], Yoshizawa [3], and Hino et al. [4].
Di erential equations with piecewise constant arguments are usually referred to as a hybrid system, and could model certain harmonic oscillators with almost periodic forcing. For some excellent works in this eld we refer the reader to [5,6] and references therein, and for a survey of work on di erential equations with piecewise constant arguments we refer the reader to [7,8].
A recently published paper [9] has studied the di erential equation of the form where [·] denotes the greatest integer function, p and q are nonzero constants, and f (t) is a n-periodic continuous function. The n-periodic solvable problem (1) is reduced to the study a system of n + linear equations. Furthermore, by applying the well-known properties of linear system in algebra, all existence conditions are described for n-periodic solutions that yields explicit formula for the solutions of (1). In this paper we study certain functional di erential equation of neutral delay type with piecewise constant arguments of the form where p and q are nonzero real or complex constants, f (t) is a complex valued n-periodic continuous function de ned on R.
The papers [5,6] have investigated the existence of almost periodic solutions of (2), when f is an almost periodic function, while Nth-order di erential equation of neutral delay type are studied in [10,11]. In these works some theorems for the existence and uniqueness of the almost periodic solutions have been obtained. However, there are some incorrectness of uniqueness results given in [5,[10][11][12].
In the present paper, by adapting the method in [9], we give all exact conditions for the uniqueness, in niteness and emptiness of n-periodic solutions of (2), in the case when f is n-periodic. We give explicit formula for the exact solutions of the equation. For some functions f , we also show that equation (2) may have in nitely many -periodic solutions.
Throughout this paper, we use the notations R for the set of reals and Z for the set of integers.

De nition of periodic solution
A solution of (2) is de ned in [13] as follows

De nition 2.1.
A function x is called a solution of (2) if the following conditions are satis ed: (i) x and x are continuous on R; (ii) the second-order derivative of x(t) exists everywhere, with possible exception at points t = n, n ∈ Z, where one-sided second-order derivatives of x(t) exist; (iii) x satis es (2) on each interval (n, n + ) with integer n ∈ Z.
Comments: The importance of the di erentiability condition imposed on x is given in De nition 13 of [13], since we deal with second-order di erential equations. This condition may admit uniqueness condition of periodic solution of (2), however it is a su cient condition for the uniqueness condition (see Theorem 3.2 (ii) of this paper). Equation (2) may have in nite number of periodic continuous functions, satisfying (2) on each interval (n, n + ) and may not have derivative at each point n, n ∈ Z (see Examples 1 and 2 below). Moreover, if we omit the continuity condition of x on R, then the uniqueness of periodic solution of (2) does not hold, and well-posedness of (2) is not true. In the de nition of solution, the continuity condition of x on R is omitted in many works (see, for example, [5], [11] and [12]) consequently, the uniqueness of pseudo periodic and hence almost periodic solution does not hold. We illustrate this in the following two examples.
Example 1. Let f (t) = − π cos πt, p = , q = . Then the function satis es (2) with the function Q de ned on [ , ] as where α is any number. One can easily check that x(t) is continuous on R. But x (t) is discontinuous at k ∈ Z when α ≠ − . If α = − , x(t) is the exact solution of (2) (see Figures 1 and 2).
Example 2. Let f (t) = π cos πt and p ≠ , − p + q ≠ . Then, by the de nition of solution given in the papers [5], [10] and [11], the 2-periodic functions are solutions of the equation (2), where α is any number and the function Q de ned on [ , ] as Note that x( ) = xα( ), which gives continuity of xα on R.
It has been claimed in the papers [5], [10] and [11] that if |λ i | ≠ , i = , , , then equation (2) has a unique solution, where λ i , i = , , , are the eigenvalues of the matrix For the case, when p = , q = , the matrix A has a form One can easily check that the eigenvalues λ i , i = , , are reals and |λ i | ≠ , i = , , . Moreover, the conditions of the Main results of these papers are satis ed. Example 2 shows incorrectness of the results Theorem 2.11 in [10], Theorem 3.3 in [11] and Theorem 1 in [5], which claim uniqueness of the almost periodic solutions of (2).

and -periodic solutions
In this section we give a method of nding periodic solutions of (2) and their existence conditions. Let f be n-periodic continuous function. We consider two cases n = and n = in this section before considering the general case n in Section 4.
The case n = . We seek a function x as a -periodic function that solves (2).
Substitute this into (2) and assuming p ≠ , we get Integrating (6) twice on [ , t], t < , we obtain where The function Q on [ , ) can be represented as and This shows that the right-hand side of (7) contains only unknown numbers x( ), x( ) and x ( ). Since x and x are continuous and periodic, they should satisfy x( ) = x( ), x ( ) = x ( ). To nd x( ), x( ) and x ( ) we apply (7) to get the system of equations Taking into account the last system of equations yields The determinant D(p, q) of this system is Since D(p, q) ≠ , we get Theorem 3.1. Let p − ≠ and f be -periodic continuous function. Then equation (2) has a unique -periodic solution having the form (7), where (x( ), x( ), x ( )) is the unique solution of (9).
Example 3. We consider equation (2) with p = , q = and -periodic function It can be shown from (8) that . The graph of 2-periodic solution (7) is shown in Figure 3.   (2) is The case n = . Let a function x be a -periodic function. From (2) we have In the last equation we have used the fact that x (t + ) = x (t − ). The system of equations (10) of x (t − ), x (t) and x (t + ) is solvable i From this system of equations, assuming p ≠ − , we get Integrating (11) two times on [ , t], t < , we have where The function Q on [ , ) can be presented by , Hence the right-hand side of (12) contains only unknown variables x( ), x( ), x( ) and x ( ). Since x and x are continuous and periodic, they should satisfy Using these equations and (12) we have Note that Therefore, we have Let D (p, q) be the determinant of the matrix It can be shown that Summarizing these results, we get where (x( ), x( ), x( ), x ( )) is an eigenvector of A corresponding to , α is any number.  (2) does not have any -periodic solution.

Remark 3.1. We emphasize again that in the de nition of solution, it is important to have the continuity condition of its derivative. If we omit this condition, we must remove the equality x ( ) = x ( ) in (13). Let p = .
Then the equation (13) is equivalent to × system where α is any number. Since the determinant D(q) of the system of equations (15) is has a unique solution (x( ), x( ), x( )) when D(q) ≠ . Therefore, for any continuous -periodic f and q with q( + q + q ) ≠ the continuous function satis es (2). In particular, the functions {xα} are well-de ned for q and -periodic functions f satisfying the conditions of the Theorem 2.1 in [12], which claim on uniqueness of almost periodic solutions. This example shows that the Theorem 2.1 in [12] is incorrect.
We denote by L the class of continuous -periodic functions f with F ( ) = F ( ) = F ( ) = F ( ) = , where

Remark 3.2. Theorem (3.2) (ii) shows that the di erentiability of a solution of (2) does not ensure uniqueness of the -periodic and hence almost periodic solutions of this equation. We can observe the existence of in nite
are -periodic solutions of (2), where α is any nonzero constant, ,

n-periodic solutions
We next solve equation (2), where f is periodic with positive integer period n ≥ . It is clear that to seek a function x as a periodic function, we assume that x(t) = x(t + n).
It follows from (2) and periodicity of x(t) that Assuming the right-hand sides of (18) are known, we consider this system of equations with respect to It is solvable if and only if ∆(p) ≠ , where ∆(p) = det P and P is the n × n matrix Observe that ∆(p) = det P = p n − (− ) n .
Assuming p n ≠ (− ) n , we can nd x (t) from (18), i.e., where ∆(p; t) = det Q and Q is the n × n matrix Taking into Q (t) = Q n+ (t), the equation (19) represents Integrating (20) we get where We set Then the function Qn on [ , n) can be represented by , , We remark that the functions Qn and Fn are twice di erentiable on (n − , n) and there exist one-sided second derivatives at t = n − and t = n.