Adomian Decomposition Method in The Theory of Nonlinear Periodic Boundary Value Problems with Delay Peter Benner 1, Sergey Chuiko ,1,2 and Viktor Chuiko 3 1 Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse, 1, Magdeburg, 39106, Germany; 2 Donbass State Pedagogical University, Donetsk region, Slavyansk, st. General Batyuk, 19, Slaviansk, 84116, Ukraine; 3 Taras Shevchenko National University of Kyiv, Volodymyrska St, 64/13, Kyiv, 01601, Ukraine. Abstract Among numerous studies of functional-differential equations, research on periodic boundary value problems for differential equations with concentrated delay holds a special place. This is primarily due to the wide application of periodic boundary value problems for differential equations with concentrated delay in physics, economics [3], biology [4], and mechanics [5]. By applying the Adomian decomposition method, we have derived the necessary and sufficient conditions for the existence of solutions to the weakly nonlinear periodic boundary value problem for a system of differential equations with concentrated delay in the critical case. Keywords 1 Functional-differential equations, differential equations with concentrated delay, periodic boundary value problems, weakly nonlinear boundary value problems, Adomian decomposition method. 1. Introduction We studied the problem of constructing approximations to the 𝑇-periodic solution [1, 2] 𝑧(𝑑, πœ€) ∢ 𝑧(βˆ™, πœ€) ∈ 𝐢¹[0, 𝑇], 𝑧(𝑑,βˆ™) ∈ 𝐢[0, πœ€β‚€] of a system of differential equations with concentrated delay 𝑑𝑧(𝑑, πœ€)/𝑑𝑑 = 𝐴(𝑑)𝑧(𝑑, πœ€) + 𝐡(𝑑)𝑧(𝑑 βˆ’ βˆ†, πœ€) + 𝑓(𝑑) + πœ€ 𝑍(𝑧(𝑑, πœ€), 𝑧(𝑑 βˆ’ βˆ†, πœ€), 𝑑, πœ€). (1) The solution of the periodic problem for equation (1) is sought in a small neighborhood of the T- periodic solution 𝑧₀(t) ∈ β„‚1 [0, 𝑇] of the generating system 𝑑𝑧₀/𝑑𝑑 = 𝐴(𝑑)𝑧₀(𝑑) + 𝐡(𝑑)𝑧₀(𝑑 βˆ’ βˆ†) + 𝑓(𝑑), βˆ† ∈ ℝ¹. (2) Where 𝐴(𝑑), 𝐡(𝑑) are continuous 𝑇-periodic (n Γ— n)-matrices, 𝑓(𝑑) is continuous 𝑇-periodic vector- function, 𝑍(𝑧(𝑑, πœ€), 𝑧(𝑑 βˆ’ βˆ†, πœ€), 𝑑, πœ€) 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 Ξ΅ on the interval [0, πœ€β‚€]. As is known, in the critical case [2], specifically, in the presence of 𝑇-periodic solutions Dynamical System Modeling and Stability Investigation (DSMSI-2023), December 19-21, 2023, Kyiv, Ukraine EMAIL: benner@mpi-magdeburg.mpg.de (A. 1); chujko-slav@ukr.net (A. 2); vitya.chuyko@gmail.com (A. 3) ORCID: 0000-0003-3362-4103 (A. 1); 0000-0001-7186-0129 (A. 2); 0009-0000-8450-9979 (A. 3) ©️ 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) CEUR Workshop ceur-ws.org ISSN 1613-0073 98 Proceedings 𝑧0 𝑑, π‘π‘Ÿ = π‘‹π‘Ÿ 𝑑 π‘π‘Ÿ , π‘π‘Ÿ ∈ β„π‘Ÿ of the homogeneous part 𝑑𝑧0 /𝑑𝑑 = 𝐴(𝑑)𝑧0 (𝑑) + 𝐡 (𝑑)𝑧0 (𝑑 βˆ’ Ξ”) (3) of system (2), and in the case of constant matrices 𝐴(𝑑) ≑ 𝐴 and 𝐡(𝑑) ≑ 𝐡, with the presence of purely imaginary roots πœ†π‘— = Β±π‘–π‘˜π‘— 𝑇, 𝑖 = βˆ’1, 𝑗 ∈ β„• of the characteristic equation det 𝐴 + 𝐡𝑒 βˆ’πœ†Ξ” βˆ’ πœ†πΌπ‘› = 0, the generating periodic problem for equation (2) is solvable not for all vector functions 𝑓(𝑑). In the critical case, the adjoint system 𝑑𝑦 𝑑 /𝑑𝑑 = βˆ’π΄βˆ— 𝑑 𝑦 𝑑 βˆ’ π΅βˆ— 𝑑 𝑦(𝑑 + Ξ”) has a family of 𝑇-periodic solutions of the form 𝑦 𝑑, π‘π‘Ÿ = π»π‘Ÿ 𝑑 π‘π‘Ÿ , π‘π‘Ÿ ∈ β„π‘Ÿ . Periodic problem for the equation (2) is solvable iff 𝑇 π»π‘Ÿβˆ— 𝑠 𝑓 𝑠 𝑑𝑠 = 0. (4) 0 Here π»π‘Ÿ (𝑑) is (n Γ— 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 𝑇- periodic problem for equation (2) has the form 𝑧0 𝑑, π‘π‘Ÿ = π‘‹π‘Ÿ 𝑑 π‘π‘Ÿ + 𝐺 𝑓 𝑠 𝑑 , π‘π‘Ÿ ∈ β„π‘Ÿ , where 𝐺[𝑓 (𝑠)](𝑑) is a particular solution of the generating 𝑇-periodic problem for equation (2), π‘‹π‘Ÿ (𝑑) is (n Γ— r)- matrix formed by r linearly independent 𝑇-periodic solutions of the system (2). To construct a particular solution 𝐺[𝑓(𝑠)](𝑑) of the generating 𝑇-periodic problem for equation (2), provided its solvability, the method of least squares [6] is applicable. 2. 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 form 𝑧0 (𝑑, π‘π‘Ÿ ) = π‘‹π‘Ÿ (𝑑)π‘π‘Ÿ + 𝐺[𝑓 (𝑠)](𝑑), π‘π‘Ÿ ∈ β„π‘Ÿ . Let us also assume that the T-periodic problem for equation (1) has a T-periodic solution 𝑧(𝑑, πœ€) ∢ 𝑧(βˆ™, πœ€) ∈ β„‚ΒΉ[0, 𝑇], 𝑧(𝑑,βˆ™) ∈ β„‚[0, πœ€β‚€], which, at πœ€ = 0, transforms unto the generating solution 𝑧(𝑑, 0) = 𝑧0 (𝑑, π‘π‘Ÿβˆ— ). Under these conditions, the vector π‘π‘Ÿβˆ— ∈ β„π‘Ÿ satisfies the equation for the generating amplitudes 𝑇 𝐹 (π‘π‘Ÿβˆ— ) ∢= ∫ π»π‘Ÿβˆ— (𝑠)𝑍(𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑠, πœ€)𝑑𝑠 = 0. (5) 0 99 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 π‘π‘Ÿβˆ— ∈ β„π‘Ÿ of equation (5) determine the generating solutions 𝑧0 (𝑑, π‘π‘Ÿβˆ— ), in a neighborhood of which the sought solutions to the original nonlinear T- periodic boundary value problem for equation (1) may exist. However, if equation (5) has no real solutions for π‘π‘Ÿβˆ— ∈ β„π‘Ÿ , then the original nonlinear 𝑇-periodic boundary value problem for equation (1) has no sought-after solutions. Let us denote an (r Γ— r)- matrix T B0 ≔ ∫ Hrβˆ— (s)[A1,0 (s)Xr (s) + A 0,1 (s)Xr (s βˆ’ Ξ”)]ds; 0 here βˆ‚Z(z(t, Ξ΅), z(t βˆ’ Ξ”, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβˆ— ) A1,0 (t) = | βˆ‚z(t, Ξ΅) z(t βˆ’ Ξ”, Ξ΅) = z0 (t βˆ’ Ξ”, crβˆ— ) and βˆ‚Z(z(t, Ξ΅), z(t βˆ’ Ξ”, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβˆ— ) A0,1 (t) = | βˆ‚z(t βˆ’ Ξ”, Ξ΅) z(t βˆ’ Ξ”, Ξ΅) = z0 (t βˆ’ Ξ”, crβˆ— ) Are (n Γ— n)- matrices. The traditional solvability condition for the nonlinear periodic boundary value problem for equation (1) in a small neighborhood of the generating solution z0 (t, crβˆ— ) is the requirement for the simplicity of the roots [2, 6] det B0 β‰  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 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) = π‘‹π‘Ÿ π‘π‘Ÿβˆ— + 𝐺[𝑓 (𝑠)](𝑑), π‘π‘Ÿβˆ— ∈ β„π‘Ÿ . 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 𝑧0 (𝑑, π‘π‘Ÿβˆ— ). We seek the solution of the nonlinear 𝑇-periodic boundary value problem for equation (1) in the critical case in the form 𝑧(𝑑, πœ€) ∢= 𝑧0 (𝑑, π‘π‘Ÿ ) + 𝑒1 (𝑑, πœ€) + … + π‘’π‘˜ (𝑑, πœ€) + … . 100 The nonlinear vector function 𝑍(𝑧(𝑑, πœ€), 𝑧(𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€) is analytic in a small neighborhood of the generating solution of problem (2); therefore, in the specified neighborhood, occurs an expansion in the form [7, p. 502] 𝑍(𝑧(𝑑, πœ€), 𝑧(𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€) = 𝑍(𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑑, 0) + 𝑍1 (𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑, πœ€), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€) + … + + π‘π‘˜ (𝑧0 (𝑑, π‘π‘Ÿ ), 𝑒1 (𝑑, πœ€), … , π‘’π‘˜ (𝑑, πœ€), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑 βˆ’ Ξ”, πœ€), … , π‘’π‘˜ (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€) + … . βˆ— First approximation to solution of nonlinear 𝑇- periodic boundary value problem for equation (1) in critical case 𝑧1 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€), 𝑒1 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐1 (πœ€) + (𝑑, πœ€), 𝑐1 (πœ€) ∈ β„π‘Ÿ determines the solution of the 𝑇- periodic boundary value problem for equation 𝑑𝑒1 (𝑑, πœ€) = 𝐴(𝑑)𝑒1 (𝑑, πœ€) + 𝐡(𝑑)𝑒1 (𝑑 βˆ’ Ξ”, πœ€) + πœ€π‘(𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑑, 0); 𝑑𝑑 Here (1) 𝑒1 (𝑑, πœ€) = πœ€πΊ[𝑍(𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑠, 0)](𝑑) is a particular solution of this problem. The solvability of the 𝑇-periodic boundary value problem in the first approximation is guaranteed by choosing the root π‘π‘Ÿβˆ— of equation (5) for the generating amplitudes of the nonlinear periodic boundary value problem for equation (1). The second approximation to the solution of the nonlinear 𝑇-periodic boundary value problem for equation (1) in the critical case (1) 𝑧2 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€) + 𝑒2 (𝑑, πœ€), 𝑒2 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐2 (πœ€) + 𝑒2 (𝑑, πœ€), 𝑐2 (πœ€) ∈ β„π‘Ÿ determines the solution of the 𝑇-periodic boundary value problem for the equation 𝑑𝑒2 (𝑑, πœ€) = 𝐴(𝑑)𝑒2 (𝑑, πœ€) + 𝐡(𝑑)𝑒2 (𝑑 βˆ’ Ξ”, πœ€) + 𝑑𝑑 +πœ€π‘1 (𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑, πœ€), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€); Here (1) 𝑒2 (𝑑, πœ€) = πœ€πΊ[𝑍1 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€)](𝑑) is a particular solution of this problem. The solvability of the 𝑇-periodic boundary value problem in the second approximation guarantees the solvability of the equation 𝑇 𝐹1 (𝑐1 (πœ€)) ≔ ∫0 π»π‘Ÿβˆ— (𝑠)𝑍1 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€)𝑑𝑠 = 0. Unlike the equation for the generating amplitudes, the last equation is linear: 𝐹1 (𝑐1 (πœ€)) = 𝐡0 𝑐1 (πœ€) + 𝛿1 (π‘π‘Ÿβˆ— , πœ€) = 0, and also solvable, provided the roots of the equation for the generating amplitudes are simple; here 𝐡0 = 𝐹1β€² (𝑐1 (πœ€)) ∈ β„π‘ŸΓ—π‘Ÿ , 𝛿1 (π‘π‘Ÿβˆ— ) ≔ 𝐹1 (𝑐1 (πœ€)) βˆ’ 𝐡0 𝑐1 (πœ€). In order to prove this, let us denote the vector function [8, 9] 𝑣(𝑑, πœ‡) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + πœ‡π‘’1 (𝑑, πœ€) + … + πœ‡π‘˜ π‘’π‘˜ (𝑑, πœ€) + … ; in this case 𝑇 𝐹1 (𝑐1 (πœ€)) ≔ ∫0 π»π‘Ÿβˆ— (𝑠)𝑍1 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑠, 0, 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), πœ€)𝑑𝑠 ≔ 𝑇 = ∫0 π»π‘Ÿβˆ— (𝑠)π‘πœ‡β€² (𝑣(𝑠, πœ‡), 𝑣 (𝑠 βˆ’ Ξ”, πœ‡), 𝑠, πœ€)𝑑𝑠| πœ‡=0= 𝑇 =∫0 π»π‘Ÿβˆ— (𝑠)[𝐴1,0 (𝑠)𝑒1 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒1 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠, and thus 101 𝑇 (1) (1) 𝐡0 = 𝐹1β€² (𝑐1 (πœ€)), 𝛿1 (π‘π‘Ÿβˆ— , πœ€) = ∫ π»π‘Ÿβˆ— (𝑠) [𝐴1,0 (𝑠)𝑒1 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒1 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠. 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 (1) 𝑒1 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐1 (πœ€) + 𝑒1 (𝑑, πœ€), 𝑇 (1) (1) 𝑐1 (πœ€) = βˆ’π΅0βˆ’1 ∫ π»π‘Ÿβˆ— (𝑠) [𝐴1,0 (𝑠)𝑒1 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒1 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠. 0 Third approximation to solution of nonlinear 𝑇- periodic boundary value problem for equation (1) in critical case 𝑧3 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€) + 𝑒2 (𝑑, πœ€) + 𝑒3 (𝑑, πœ€), (1) 𝑒3 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐3 (πœ€) + 𝑒3 (𝑑, πœ€), 𝑐3 (πœ€) ∈ β„π‘Ÿ determines the solution of the 𝑇-periodic boundary value problem for the equation 𝑑𝑒3 (𝑑, πœ€) = 𝐴(𝑑)𝑒3 (𝑑, πœ€) + 𝐡(𝑑)𝑒3 (𝑑 βˆ’ Ξ”, πœ€) + 𝑑𝑑 +πœ€π‘2 (𝑧0 (𝑑, π‘π‘Ÿ ), 𝑒1 (𝑑, πœ€), 𝑒2 (𝑑, πœ€), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑 βˆ’ Ξ”, πœ€), 𝑒2 (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€); βˆ— Here (1) 𝑒3 (𝑑, πœ€) = = πœ€πΊ[𝑍2 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑒2 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑒2 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€)](𝑑) is a particular solution of this problem. The solvability of the 𝑇-periodic boundary value problem in the second approximation guarantees the solvability of the equation 𝑇 𝐹2 (𝑐2 (πœ€), πœ€) ≔ ∫ π»π‘Ÿβˆ— (𝑠) Γ— 0 Γ— 𝑍2 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑒2 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑒2 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€)𝑑𝑠 = 0. Unlike the equation for the generating amplitudes, the last equation is linear: 𝐹2 (𝑐2 (πœ€)) = 𝐡0 𝑐2 (πœ€) + 𝛿2 (π‘π‘Ÿβˆ— , 𝑐1 (πœ€), πœ€) = 0, and also solvable, provided that the roots of the equation for the generating amplitudes are simple. Here 𝐡0 = 𝐹2β€² (𝑐2 (πœ€)) ∈ β„π‘ŸΓ—π‘Ÿ , 𝛿2 (π‘π‘Ÿβˆ— , 𝑐1 (πœ€), πœ€) ≔ 𝐹2 (𝑐2 (πœ€)) βˆ’ 𝐡0 𝑐2 (πœ€). Denote (n Γ— n) – matrices A2,0 (t, u1 (𝑑, πœ€)) ≔ πœ• βˆ‚Z(z(t, Ξ΅), z(t βˆ’ Ξ”, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβˆ— ) = [ 𝑒1 (𝑑, πœ€)]| πœ•π‘§(𝑑, πœ€) βˆ‚z(t, Ξ΅) z(t βˆ’ Ξ”, Ξ΅) = z0 (t βˆ’ Ξ”, crβˆ— ), A1,1 (t, u1 (𝑑 βˆ’ Ξ”, πœ€)) ≔ πœ• βˆ‚Z(z(t, Ξ΅), z(t βˆ’ Ξ”, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβˆ— ) = [ 𝑒1 (𝑑 βˆ’ Ξ”, πœ€)]| πœ•π‘§(𝑑, πœ€) βˆ‚z(t βˆ’ Ξ”, Ξ΅) z(t βˆ’ Ξ”, Ξ΅) = z0 (t βˆ’ Ξ”, crβˆ— ), and A0,2 (t, u1 (𝑑 βˆ’ Ξ”, πœ€)) ≔ πœ• βˆ‚Z z t, Ξ΅ , z t βˆ’ Ξ”, Ξ΅), t, Ξ΅) ( ( ) ( z(t, Ξ΅) = z0 (t, crβˆ— ) = [ 𝑒1 (𝑑 βˆ’ Ξ”, πœ€)]| πœ•π‘§(𝑑 βˆ’ Ξ”, πœ€) βˆ‚z(t βˆ’ Ξ”, Ξ΅) z(t βˆ’ Ξ”, Ξ΅) = z0 (t βˆ’ Ξ”, crβˆ— ), Indeed, 1 𝑇 βˆ— 𝐹2 (𝑐2 (πœ€), πœ€) = ∫ 𝐻 (𝑠)π‘πœ‡β€²β€²2 (𝑣(𝑠, πœ‡), 𝑣(𝑠 βˆ’ Ξ”, πœ‡), 𝑠, πœ€)𝑑𝑠| = 2! 0 π‘Ÿ πœ‡=0 𝑇 = ∫ π»π‘Ÿβˆ— (𝑠)[𝐴1,0 (𝑠)𝑒2 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒2 (𝑠 βˆ’ Ξ”, πœ€)]𝑑𝑠 + 0 102 1 𝑇 βˆ— + ∫ 𝐻 (𝑠)[𝐴2,0 (𝑠, 𝑒1 (𝑠, πœ€)) + 2𝐴1,1 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€) + 2! 0 π‘Ÿ +𝐴0,2 (𝑠, 𝑒1 (𝑠 βˆ’ Ξ”, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€)]𝑑𝑠, thus 𝐡0 = 𝐹2β€² (𝑐2 (πœ€), πœ€), furthermore 𝑇 (1) (1) 𝛿2 (π‘π‘Ÿβˆ— , 𝑐1 (πœ€), πœ€) = ∫ π»π‘Ÿβˆ— (𝑠) [𝐴1,0 (𝑠)𝑒2 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒2 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠 + 0 1 𝑇 + ∫ π»π‘Ÿβˆ— (𝑠)[𝐴2,0 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠, πœ€) + 2𝐴1,1 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€) + 2! 0 𝐴0,2 (𝑠, 𝑒1 (𝑠 βˆ’ Ξ”, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€)]𝑑𝑠. Thus, provided that the roots of the equation for the generating amplitudes (5) are simple, we obtain a unique solution to the boundary value problem in the second approximation (1) 𝑒2 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐2 (πœ€) + 𝑒2 (𝑑, πœ€), 𝑇 (1) (1) 𝑐2 (πœ€) = βˆ’π΅0βˆ’1 ∫ π»π‘Ÿβˆ— (𝑠) [𝐴1,0 (𝑠)𝑒2 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒2 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠 βˆ’ 0 𝑇 1 βˆ’ 𝐡0βˆ’1 ∫ π»π‘Ÿβˆ— (𝑠)[𝐴2,0 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠, πœ€) + 2𝐴1,1 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€) + 2! 0 +𝐴0,2 (𝑠, 𝑒1 (𝑠 βˆ’ Ξ”, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€)]𝑑𝑠. Solvability of the 𝑇-periodic boundary value problem in π‘˜ + 1 approximation guarantees the solvability of the equation 𝑇 πΉπ‘˜+1 (π‘π‘˜ (πœ€)) ≔ ∫ π»π‘Ÿβˆ— (𝑠)π‘π‘˜ (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), … , π‘’π‘˜+1 (𝑠, πœ€), 0 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), … , π‘’π‘˜+1 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€) 𝑑𝑠 = 0. The sequence of approximations to the solution of the nonlinear 𝑇-periodic boundary value problem for equation (1) in the critical case is determined by the iterative scheme (1) 𝑧1 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€), 𝑒1 (𝑑) = π‘‹π‘Ÿ (𝑑)𝑐1 (πœ€) + 𝑒1 (𝑑, πœ€), (1) u1 (𝑑, πœ€) = πœ€πΊ[𝑍(𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑠, 0)](𝑑), 𝑇 (1) (1) 𝑐1 (πœ€) = βˆ’π΅0βˆ’1 ∫ π»π‘Ÿβˆ— (𝑠) [𝐴1,0 (𝑠)𝑒1 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒1 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠, 0 (1) 𝑧2 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€) + 𝑒2 (𝑑, πœ€), 𝑒2 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐2 (πœ€) + 𝑒2 (𝑑, πœ€), (1) 𝑒2 (𝑑, πœ€) = πœ€πΊ[𝑍1 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€))](𝑑), 𝑇 (1) (1) 𝑐2 (πœ€) = βˆ’π΅0βˆ’1 ∫ π»π‘Ÿβˆ— (𝑠) [𝐴1,0 (𝑠)𝑒2 (𝑠, πœ€) + 𝐴0,1 (𝑠)𝑒2 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠 βˆ’ 0 𝑇 1 βˆ’ 𝐡0βˆ’1 ∫ π»π‘Ÿβˆ— (𝑠)[𝐴2,0 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠, πœ€) + 2𝐴1,1 (𝑠, 𝑒1 (𝑠, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€) + 2! 0 +𝐴0,2 (𝑠, 𝑒1 (𝑠 βˆ’ Ξ”, πœ€))𝑒1 (𝑠 βˆ’ Ξ”, πœ€)] 𝑑𝑠, …, π‘§π‘˜+1 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€) + … + π‘’π‘˜+1 (𝑑, πœ€), (6) 103 (1) π‘’π‘˜+1 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)π‘π‘˜+1 (πœ€) + π‘’π‘˜+1 (𝑑, πœ€), π›Ώπ‘˜+1 (π‘π‘Ÿβˆ— , 𝑐1 (πœ€), πœ€) = πΉπ‘˜+1 (𝑐2 (πœ€)) βˆ’ 𝐡0 π‘π‘˜+1 (πœ€), (1) π‘’π‘˜+1 (𝑑, πœ€) = πœ€πΊ[π‘π‘˜ (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), … , π‘’π‘˜ (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑠, πœ€, 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), … , π‘’π‘˜ (𝑠 βˆ’ Ξ”, πœ€))](𝑑), π‘π‘˜+1 (πœ€) = βˆ’π΅0βˆ’1 π›Ώπ‘˜+1 (π‘π‘Ÿβˆ— , π‘π‘˜+1 (πœ€), πœ€), π‘˜ = 1,2, … . Theorem. In the critical case, the periodic problem for equation (2) with concentrated delay, under condition (4), has an r-parametric family of solutions 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) = π‘‹π‘Ÿ (𝑑)π‘π‘Ÿ + 𝐺[𝑓(𝑠)](𝑑), π‘π‘Ÿ ∈ β„π‘Ÿ . Assuming det 𝐡0 β‰  0 the simplicity of the roots of the equation (5) for the generating amplitudes in a small neighborhood of the generating solution 𝑧0 (𝑑, π‘π‘Ÿβˆ— ), the nonlinear periodic boundary value problem for equation (1) has a unique solution 𝑧(𝑑, πœ€) ∢ 𝑧(βˆ™, πœ€) ∈ 𝐢¹[0, 𝑇], 𝑧(𝑑,βˆ™) ∈ 𝐢[0, πœ€β‚€]. The sequence of approximations to the solution of the nonlinear periodic boundary value problem for equation (1) is determined by the iterative scheme (6). If there exists a constant 0 < 𝛾 < 1, for which the inequalities ‖𝑒1 (𝑑, πœ€)β€– ≀ 𝛾‖𝑧0 (𝑑, π‘π‘Ÿβˆ— )β€–, β€–π‘’π‘˜+1 (𝑑, πœ€)β€– ≀ π›Ύβ€–π‘’π‘˜ (𝑑, πœ€)β€–, π‘˜ = 1,2, … (7) hold, then the iterative scheme (6) converges to the solution of the nonlinear periodic boundary value problem for equation (1) with concentrated delay. 3. Finding approximations to the periodic solution of the equation modeling a non-isothermal chemical reaction Let us apply the iterative scheme (6) in order to find approximations to the periodic solution of the equation with concentrated delay, which models a non-isothermal chemical reaction [10, 11]. Example. The conditions of the proven theorem hold in the case of a 2πœ‹-periodic boundary value problem with concentrated delay 𝑑𝑧(𝑑, πœ€)/𝑑𝑑 = 𝐴(𝑑)𝑧(𝑑, πœ€) + 𝐡(𝑑)𝑧(𝑑 βˆ’ Ξ”, πœ€) + 𝑓 (𝑑) + πœ€π‘(𝑧(𝑑, πœ€), 𝑧(𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€); (8) here 0 1 cos 𝑑 πœ‹ 𝐴(𝑑) ≔ ( ) , 𝐡(𝑑) ≔ 0, 𝑓 (𝑑) ≔ ( ), Ξ” ≔ , 0 0 0 2 and also πœ€ βˆ’ 1 π‘₯(𝑑, πœ€) 𝑍(𝑧(𝑑, πœ€), 𝑧(𝑑 βˆ’ Ξ”, πœ€), πœ€) ≔ (1 + π‘₯ (𝑑, πœ€))𝑒 1+𝑦(π‘‘βˆ’Ξ”,πœ€) ( ) , 𝑧(𝑑, πœ€) ≔ ( ). 1 𝑦(𝑑, πœ€) For the generating periodic problem for equation (8), there is a critical case [2, 12], and condition (4) is satisfied, therefore, it is solvable: 𝑧0 (𝑑, π‘π‘Ÿ ) = π‘‹π‘Ÿ (𝑑)π‘π‘Ÿ + 𝐺[𝑓 (𝑠)](𝑑), π‘π‘Ÿ ∈ ℝ1 ; here 1 sin 𝑑 π‘‹π‘Ÿ (𝑑) = ( ) , 𝐺[𝑓 (𝑠)](𝑑) = ( ). 0 0 104 The equation for the generating amplitudes (5) in the case of a problem of finding a periodic solution for equation (8) has a simple root π‘π‘Ÿβˆ— = 1, which determines the generating solution 1 + sin 𝑑 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) = βˆ’ ( ). 0 The matrix 𝐡0 = 2πœ‹ is non-singular, so according to the proven theorem, the 2πœ‹-periodic problem for equation (8) with concentrated delay is uniquely solvable. Thus, we obtain the first approximation (1) 𝑧1 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€), 𝑒1 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐1 (πœ€) + 𝑒1 (𝑑, πœ€), 𝑐1 (πœ€) = βˆ’πœ€ to the solution of the periodic problem for equation (8); here (1) 1 βˆ’ sin 𝑑 βˆ’ cos 𝑑 𝑒1 (𝑑, πœ€) = πœ€πΊ[𝑍(𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑠, 0)](𝑑) = πœ€ ( ), βˆ’ cos 𝑑 and also 1 𝑍(𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑑, 0) = (1 + π‘π‘Ÿβˆ— + sin 𝑑 ) ( ) . 1 Similarly, we obtain the second approximation to the solution of the nonlinear periodic boundary value problem for equation (8) in the critical case (1) 𝑧2 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€) + 𝑒2 (𝑑, πœ€), 𝑒2 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐2 (πœ€) + 𝑒2 (𝑑, πœ€); here (1) 𝑒2 (𝑑, πœ€) = πœ€πΊ[𝑍1 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— )), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€)](𝑑) = sin 𝑑 + 3 cos 𝑑 βˆ’ 3 7πœ€ 2 = πœ€2 ( ), 𝑐2 (πœ€) = . 2 cos 𝑑 βˆ’ sin 𝑑 2 and also 𝑍1 (𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑, πœ€), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€) = 1 = (1 + 𝑐1 (πœ€) βˆ’ 2 sin 𝑑 βˆ’ cos 𝑑 ) ( ). 1 In the same way, we obtain the third approximation to the solution of the nonlinear periodic boundary value problem for equation (8) in the critical case 65πœ€ 3 𝑧3 (𝑑, πœ€) ≔ 𝑧0 (𝑑, π‘π‘Ÿβˆ— ) + 𝑒1 (𝑑, πœ€) + 𝑒2 (𝑑, πœ€) + 𝑒3 (𝑑, πœ€), 𝑐3 (πœ€) = βˆ’ , 8 (1) (1) 𝑒3 (𝑑, πœ€) = π‘‹π‘Ÿ (𝑑)𝑐3 (πœ€) + 𝑒3 (𝑑, πœ€), 𝑒3 (𝑑, πœ€) = = πœ€πΊ[𝑍2 (𝑧0 (𝑠, π‘π‘Ÿβˆ— ), 𝑒1 (𝑠, πœ€), 𝑒2 (𝑠, πœ€), 𝑧0 (𝑠 βˆ’ Ξ”, π‘π‘Ÿβˆ— )), 𝑒1 (𝑠 βˆ’ Ξ”, πœ€), 𝑒2 (𝑠 βˆ’ Ξ”, πœ€), 𝑠, πœ€)](𝑑) = πœ€ 3 53 βˆ’ 52 cos 𝑑 βˆ’ cos 2𝑑 + 12 sin 𝑑 + 2 sin 2𝑑 = ( ); 8 2(16 sin 𝑑 + sin 2𝑑 βˆ’ 10 cos 𝑑) and also 𝑍2 (𝑧0 (𝑑, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑, πœ€), 𝑒2 (𝑑, πœ€), 𝑧0 (𝑑 βˆ’ Ξ”, π‘π‘Ÿβˆ— ), 𝑒1 (𝑑 βˆ’ Ξ”, πœ€), 𝑒2 (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€) = πœ€2 1 = (7 βˆ’ 2𝑐2 (πœ€) βˆ’ 8 cos 𝑑 βˆ’ cos 2𝑑 βˆ’ 5 sin 𝑑) ( ). 2 1 For the obtained approximations to the periodic solution of equation (8), the inequalities ‖𝑒1 (𝑑, πœ€)β€– ≀ 𝛾 ‖𝑧0 (𝑑, π‘π‘Ÿβˆ— )β€–, β€–π‘’π‘˜+1 (𝑑, πœ€)β€– ≀ 𝛾 β€–π‘’π‘˜ (𝑑, πœ€)β€–, 𝛾 β‰ˆ 0, 131 256, π‘˜ = 0,1,2, hold, indicating the practical convergence of the obtained approximations to the periodic solution of the equation (8) for πœ€ ∈ [0, πœ€0 ], πœ€0 β‰ˆ 0, 25. The accuracy of the obtained approximations to the periodic solution of equation (8) is determined by the residuals Ξ”π‘˜ (πœ€) ≔ ||π‘‘π‘§π‘˜ (𝑑, πœ€)/𝑑𝑑 βˆ’ 𝐴(𝑑)π‘§π‘˜ (𝑑, πœ€) βˆ’ 𝐡 (𝑑)π‘§π‘˜ (𝑑 βˆ’ Ξ”, πœ€) βˆ’ 𝑓 (𝑑) βˆ’ βˆ’πœ€π‘(π‘§π‘˜ (𝑑, πœ€), π‘§π‘˜ (𝑑 βˆ’ Ξ”, πœ€), 𝑑, πœ€)|| , π‘˜ = 0, 1, 2, 3. β„‚[0; 2πœ‹] In particular, 105 Ξ”0 (0, 1) β‰ˆ 0, 0904 837, Ξ”1 (0,1) β‰ˆ 0, 0213 474, Ξ”2 (0, 1) β‰ˆ 0, 00 469 105, Ξ”3 (0,1) β‰ˆ 0, 00 112 528; Ξ”0 (0, 01) β‰ˆ 0, 0099 005, Ξ”1 (0,01) β‰ˆ 0, 000 222 616, Ξ”2 (0, 01) β‰ˆ 4, 97 520 Γ— 10βˆ’6 , Ξ”3 (0,01) β‰ˆ 1, 21 626 Γ— 10βˆ’7 . 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]. 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