Introduction

We studied the problem of constructing approximations to the 𝑇-periodic solution [1,2] 𝑧(𝑡, 𝜀) ∶ 𝑧(•, 𝜀) ∈ 𝐶¹[0, 𝑇], 𝑧(𝑡,•) ∈ 𝐶[0, 𝜀₀] of a system of differential equations with concentrated delay 𝑑𝑧(𝑡, 𝜀)/𝑑𝑡 = 𝐴(𝑡)𝑧(𝑡, 𝜀) + 𝐵(𝑡)𝑧(𝑡 − ∆, 𝜀) + 𝑓(𝑡) + 𝜀 𝑍(𝑧(𝑡, 𝜀), 𝑧(𝑡 − ∆, 𝜀), 𝑡, 𝜀).

(

)1

The solution of the periodic problem for equation (1) is sought in a small neighborhood of the Tperiodic solution Where 𝐴(𝑡), 𝐵(𝑡) are continuous 𝑇-periodic (n × n)-matrices, 𝑓(𝑡) is continuous 𝑇-periodic vectorfunction, 𝑍(𝑧(𝑡, 𝜀), 𝑧(𝑡 − ∆, 𝜀), 𝑡, 𝜀) is nonlinear vector function, which is analytic in a small neighborhood of the generating problem (2), continuous and 𝑇-periodic with the respect to the variable t, and also analytic with respect to the small parameter ε on the interval [0, 𝜀₀]. As is known, in the critical case [2], specifically, in the presence of 𝑇-periodic solutions of the homogeneous part 𝑑𝑧 0 /𝑑𝑡 = 𝐴(𝑡)𝑧 0 (𝑡) + 𝐵(𝑡)𝑧 0 (𝑡 − Δ)

𝑧₀(t) ∈ ℂ 1 [0, 𝑇]

of system (2), and in the case of constant matrices 𝐴(𝑡) ≡ 𝐴 and 𝐵(𝑡) ≡ 𝐵, with the presence of purely imaginary roots of the characteristic equation the generating periodic problem for equation ( 2) is solvable not for all vector functions 𝑓(𝑡). In the critical case, the adjoint system has a family of 𝑇-periodic solutions of the form Periodic problem for the equation ( 2) is solvable iff

Here 𝐻 𝑟 (𝑡) is (n × r)-matrix formed by r linearly independent 𝑇-periodic solutions of the adjoint system. Let us assume condition (4) is satisfied; in this case, the general solution of the generating 𝑇periodic problem for equation (2) has the form where 𝐺[𝑓(𝑠)](𝑡) is a particular solution of the generating 𝑇-periodic problem for equation (2), 𝑋 𝑟 (𝑡) is (n × r)-matrix formed by r linearly independent 𝑇-periodic solutions of the system (2). To construct a particular solution 𝐺[𝑓(𝑠)](𝑡) of the generating 𝑇-periodic problem for equation (2), provided its solvability, the method of least squares [6] is applicable.

The necessary and sufficient conditions for solvability

Similarly to [2], we obtain the necessary condition for the solvability of the 𝑇-periodic problem for equation (2).

Lemma. Let us assume that for the generating periodic problem for equation (2), a critical case occurs, and the solvability condition (4) is satisfied. In this case, the periodic problem for equation (2) has a family of T-periodic solutions in the form𝑧 0 (𝑡, 𝑐 𝑟 ) = 𝑋 𝑟 (𝑡)𝑐 𝑟 + 𝐺[𝑓(𝑠)](𝑡), 𝑐 𝑟 ∈ ℝ 𝑟 .

Let us also assume that the T-periodic problem for equation (1) has a T-periodic solution𝑧(𝑡, 𝜀) ∶ 𝑧(•, 𝜀) ∈ ℂ¹[0, 𝑇], 𝑧(𝑡,•) ∈ ℂ[0, 𝜀₀],

which, at 𝜀 = 0, transforms unto the generating solution 𝑧(𝑡, 0) = 𝑧 0 (𝑡, 𝑐 𝑟 * ). Under these conditions, the vector 𝑐 𝑟 * ∈ ℝ 𝑟 satisfies the equation for the generating amplitudes

𝐹(𝑐 𝑟 𝜆 𝑗 = ±𝑖𝑘 𝑗 𝑇, 𝑖 = −1, 𝑗 ∈ ℕ det 𝐴 + 𝐵𝑒 −𝜆Δ − 𝜆𝐼 𝑛 = 0, 𝑑𝑦 𝑡 /𝑑𝑡 = −𝐴 * 𝑡 𝑦 𝑡 − 𝐵 * 𝑡 𝑦(𝑡 + Δ) 𝑦 𝑡, 𝑐 𝑟 = 𝐻 𝑟 𝑡 𝑐 𝑟 , 𝑐 𝑟 ∈ ℝ 𝑟 . 𝐻 𝑟 * 𝑠 𝑓 𝑠 𝑑𝑠 = 0. 𝑇 0 (4) 𝑧 0 𝑡, 𝑐 𝑟 = 𝑋 𝑟 𝑡 𝑐 𝑟 + 𝐺 𝑓 𝑠 𝑡 , 𝑐 𝑟 ∈ ℝ 𝑟 ,

We will refer to equation ( 5) as the equation for the generating amplitudes of the nonlinear periodic boundary value problem for equation (1). The roots 𝑐 𝑟 * ∈ ℝ 𝑟 of equation ( 5) determine the generating solutions 𝑧 0 (𝑡, 𝑐 𝑟 * ), in a neighborhood of which the sought solutions to the original nonlinear Tperiodic boundary value problem for equation (1) may exist. However, if equation ( 5) has no real solutions for 𝑐 𝑟 * ∈ ℝ 𝑟 , then the original nonlinear 𝑇-periodic boundary value problem for equation (1) has no sought-after solutions.

Let us denote an (r × r)-matrix

B 0 ≔ ∫ H r * (s)[A 1,0 (s)X r (s) + A 0,1 (s)X r (s − Δ)]ds; T 0 here A 1,0 (t) = ∂Z(z(t, ε), z(t − Δ, ε), t, ε) ∂z(t, ε) | z(t, ε) = z 0 (t, c r * ) z(t − Δ, ε) = z 0 (t − Δ, c r * ) and A 0,1 (t) = ∂Z(z(t, ε), z(t − Δ, ε), t, ε) ∂z(t − Δ, ε) | z(t, ε) = z 0 (t, c r * ) z(t − Δ, ε) = z 0 (t − Δ, c r * )

Are (n × n)-matrices. The traditional solvability condition for the nonlinear periodic boundary value problem for equation (1) in a small neighborhood of the generating solution z 0 (t, c r * ) is the requirement for the simplicity of the roots [2,6] det B 0 ≠ 0 of equation ( 5) for the generating amplitudes. We will demonstrate that the requirement for the simplicity of the roots of equation ( 5) for the generating amplitudes is a sufficient condition for the solvability of the nonlinear periodic boundary value problem for equation (1) in a small neighborhood of the generating solution 𝑧 0 (𝑡, 𝑐 𝑟 * ) = 𝑋 𝑟 𝑐 𝑟 * + 𝐺[𝑓(𝑠)](𝑡), 𝑐 𝑟 * ∈ ℝ 𝑟 .

In the article [6], we found constructive necessary and sufficient conditions for solvability, along with a scheme for constructing solutions of the nonlinear 𝑇-periodic boundary value problem for equation (1). Based on the method of simple iterations, we developed a convergent iterative scheme to find approximations to the solutions of this problem. However, in the process of constructing solutions to the nonlinear 𝑇-periodic boundary value problem for equation (1) using the least squares method, the issue of impossibility of finding solutions in terms of elementary functions arises, which, in turn, leads to significant errors in solving nonlinear boundary value problems.

Furthermore, the construction of solutions for nonlinear boundary value problems using the method of simple iterations [2] and the least squares method is significantly complicated by the computation of derivatives of nonlinearities. Given this, simplifying the computation of nonlinear derivatives and the potential to find solutions for nonlinear boundary value problems, including periodic boundary value problems, in elementary functions can be achieved using the Adomian decomposition method [7,8].

Additionally, the use of the Adomian decomposition method significantly simplifies the proof of convergence of iterative schemes for constructing solutions to nonlinear boundary value problems. An example of such simplification will be provided below. Thus, the purpose of this article is to find constructive solvability conditions and a scheme for constructing solutions to the nonlinear 𝑇-periodic boundary value problem for equation (1) using the Adomian decomposition method. Fixing one of the solutions of equation ( 5), we approach the problem of finding analytical solutions for the nonlinear 𝑇-periodic boundary value problem for equation (1) in a small neighborhood of the generating solution 𝑧 0 (𝑡, 𝑐 𝑟 * ). We seek the solution of the nonlinear 𝑇-periodic boundary value problem for equation (1) in the critical case in the form 𝑧(𝑡, 𝜀) ∶= 𝑧 0 (𝑡, 𝑐 𝑟 ) + 𝑢 1 (𝑡, 𝜀) + … + 𝑢 𝑘 (𝑡, 𝜀) + … . 𝐵 0 = 𝐹 1 ′ (𝑐 1 (𝜀)), 𝛿 1 (𝑐 𝑟 * , 𝜀) = ∫ 𝐻 𝑟 * (𝑠) [𝐴 1,0 (𝑠)𝑢 1 (1) (𝑠, 𝜀) + 𝐴 0,1 (𝑠)𝑢 1 (1) (𝑠 − Δ, 𝜀)] 𝑑𝑠. 𝑇 0 Therefore, assuming the simplicity of the roots of the equation for the generating amplitudes (5), we obtain a unique solution to the boundary value problem in the first approximation 𝑢 1 (𝑡, 𝜀) = 𝑋 𝑟 (𝑡)𝑐 1 (𝜀) + 𝑢 1 (1) (𝑡, 𝜀), 𝑐 1 (𝜀) = −𝐵 0 −1 ∫ 𝐻 𝑟 * (𝑠) [𝐴 1,0 (𝑠)𝑢 1 (1) (𝑠, 𝜀) + 𝐴 0,1 (𝑠)𝑢 1 (1) Δ 0 (0, 1) ≈ 0, 0904 837, Δ 1 (0,1) ≈ 0, 0213 474, Δ 2 (0, 1) ≈ 0, 00 469 105, Δ 3 (0,1) ≈ 0, 00 112 528; Δ 0 (0, 01) ≈ 0, 0099 005, Δ 1 (0,01) ≈ 0, 000 222 616, Δ 2 (0, 01) ≈ 4, 97 520 × 10 −6 , Δ 3 (0,01) ≈ 1, 21 626 × 10 −7 .

Denote (n × n) -matrices A 2,0 (t, u 1 (𝑡, 𝜀)) ≔ = 𝜕 𝜕𝑧(𝑡, 𝜀) [ ∂Z(z(t, ε), z(t − Δ, ε), t, ε) ∂z(t, ε) 𝑢 1 (𝑡, 𝜀)]| z(t, ε) = z 0 (t, c r * ) z(t − Δ, ε) = z 0 (t − Δ, c r * ), A 1,1 (t, u 1 (𝑡 − Δ, 𝜀)) ≔ = 𝜕 𝜕𝑧(𝑡, 𝜀) [ ∂Z(z(t, ε), z(t − Δ, ε), t, ε) ∂z(t − Δ, ε) 𝑢 1 (𝑡 − Δ, 𝜀)]| z(t, ε) = z 0 (t, c r * ) z(t − Δ, ε) = z 0 (t − Δ, c r * ), and A 0,2 (t, u 1 (𝑡 − Δ, 𝜀)) ≔ = 𝜕 𝜕𝑧(𝑡 − Δ, 𝜀) [ ∂Z(z(t, ε), z(t − Δ, ε), t, ε) ∂z(t − Δ,

The research scheme proposed in the article for investigating solvability conditions and constructing approximations to the periodic solution of equation ( 1) can be transferred to matrix boundary value problems, including those with concentrated delay [13 -16].