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

Definition 2.1

A function x is called a solution of (2) if the following conditions are satisfied:

1. x and x′ are continuous on R;

2. 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;

3. x satisfies (2) on each interval (n, n + 1) with integer n ∈ Z.

Comments: The importance of the differentiability condition imposed on x is given in Definition 13 of [13], since we deal with second-order differential equations. This condition may admit uniqueness condition of periodic solution of (2), however it is a sufficient condition for the uniqueness condition (see Theorem 3.2 (ii) of this paper). Equation (2) may have infinite number of periodic continuous functions, satisfying (2) on each interval (n, n + 1) 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 definition 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) = −3π² cos π t, p = 2, q = 4. Then the function

satisfies (2) with the function Q defined on [0, 2] 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 ≠ −6. If α = −6, x(t) is the exact solution of (2) (see Figures 1 and 2).

Figure 1

The graph of x(t) when α = −10.

Figure 2

The graph of x(t) when α = −6.

Example 2

Let f(t) = 1π2 cos π t and p² ≠ 1, 8 − 8 p + q ≠ 0. Then, by the definition 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 defined on [0, 2] as

Note that x(0) = x_α(2), which gives continuity of x_α on R.

It has been claimed in the papers [5], [10] and [11] that if ∣λ_i∣ ≠ 1, i = 1, 2, 3, then equation (2) has a unique solution, where λ_i, i = 1, 2, 3, are the eigenvalues of the matrix

For the case, when p = 12, q = 3, the matrix A has a form

One can easily check that the eigenvalues λ_i, i = 1, 2, 3 are reals and ∣λ_i∣ ≠ 1, i = 1, 2, 3. Moreover, the conditions of the Main results of these papers are satisfied. 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).

Theorem 3.1

Let p² − 1 ≠ 0 and f be 2-periodic continuous function. Then equation (2) has a unique 2-periodic solution having the form (7), where (x(0), x(1), x′(0)) is the unique solution of (9).

Example 3

We consider equation (2) with p = 3, q = 1 and 2-periodic function f(t)=t,t∈[0,1),2−t,t∈[1,2]. It can be shown from (8) that

Solving (9), we obtained x(0)=−9748,x(1)=−9548,x′(0)=−1192. The graph of 2-periodic solution (7) is shown in Figure 3.

Figure 3

The graph of 2-periodic solution x(t) of equation (2) for Example 3.

Example 4

Let p = 2i, q = 1 and f(t) = e^iπt. For this case, it can be shown that the 2-periodic solution of equation (2) is

The case n = 3. Let a function x be a 3-periodic function. From (2) we have

In the last equation we have used the fact that x″(t + 2) = x″(t − 1). The system of equations (10) of x″(t − 1), x″(t) and x″(t + 1) is solvable iff

From this system of equations, assuming p³ ≠ −1, we get

Integrating (11) two times on [0, t], t < 3, we have

where

The function Q₃ on [0, 3) can be presented by

where Y(s) = p²x([s + 1]) − px([s + 2]) + x([s]),

Hence the right-hand side of (12) contains only unknown variables x(0), x(1), x(2) and x′(0). Since x and x′ are continuous and periodic, they should satisfy

Using these equations and (12) we have

Note that

Therefore, we have

Let D₃(p, q) be the determinant of the matrix

It can be shown that

Summarizing these results, we get

Theorem 3.2

Let p³ + 1 ≠ 0 and f be a 3-periodic continuous function.

1. If D₃(p, q) ≠ 0, then equation (2) has a unique 3-periodic solution having the form (12), where (x(0), x(1), x(2), x′(0)) is the unique solution of (14).

2. If D₃(p, q) = 0 and F₃(1) = F₃(2) = F₃(3) = F3′ (3) = 0, then equation (2) has infinite number of 3-periodic solutions having the form

   x(t)=α(x(0)+x′(0)t+q1+p3Q3(t))+F3(t),

   where (x(0), x(1), x(2), x′(0)) is an eigenvector of A corresponding to 0, α is any number.

3. If D₃(p, q) = 0 and the rank[A] < rank[A, F^T], F = (F₃(1), F₃(2), F₃(3), F3′ (3)), then equation (2) does not have any 3-periodic solution.

Remark 3.1

We emphasize again that in the definition of solution, it is important to have the continuity condition of its derivative. If we omit this condition, we must remove the equality x′(0) = x′(3) in (13). Let p = 1. Then the equation (13) is equivalent to 3 × 3 system

where α is any number. Since the determinant D(q) of the system of equations (15) is

(15) has a unique solution (x(0), x(1), x(2)) when D(q) ≠ 0. Therefore, for any continuous 3-periodic f and q with q(9+3q+q24)≠0 the continuous function

satisfies (2). In particular, the functions {x_α} are well-defined for q and 3- 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 𝓛₀ the class of continuous 3-periodic functions f with F₃(1) = F₃(2) = F₃(3) = F3′ (3) = 0, where

Remark 3.2

Theorem (3.2) (ii) shows that the differentiability of a solution of (2) does not ensure uniqueness of the 3-periodic and hence almost periodic solutions of this equation. We can observe the existence of infinite number of 3-periodic solutions for the case when f belongs to 𝓛₀. Let f(t) = f_k(t), f_k(t) = cos 2kπt, k = 1, 2, … and p = 1, q = −6. Then F3(t)=11+p21(2kπ)2(1−cos⁡2kπt) and hence f_k ∈ 𝓛₀. The matrix A is represented by

One can cheek that, det(A) = D₃(1, −6) = 0 and the number 0 is an eigenvalue of A with multiplicity 2. The corresponding eigenvectors are x1=(−12,12,0,1), x₂ = (−2, 1, 1, 0). By Theorem 3.2 the functions

are 3-periodic solutions of (2), where α is any nonzero constant,