Method in The Theory of Nonlinear Periodic Boundary Value Problems with Delay

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. Functional-differential equations, differential equations with concentrated delay, periodic boundary value weakly nonlinear decomposition method.

Keywords

• problems

-

of the generating system

1. Introduction

We studied the problem of constructing approximations to the -periodic solution [1, 2] ( , ) ∶ (∙, ) ∈ ¹[0, ], ( ,∙) ∈ [0, ₀] of a system of differential equations with concentrated delay ( , )/ = ( ) ( , ) + ( ) ( − ∆, ) + ( ) + ( ( , ), ( − ∆, ), , ). ( 1 ) The solution of the periodic problem for equation ( 1 ) is sought in a small neighborhood of the T ₀(t) ∈ ℂ1[0, ] ₀/

= ( ) ₀( ) + ( ) ₀( − ∆) + ( ), ∆ ∈ ℝ¹. ( 2 ) Where ( ), ( ) are continuous -periodic (n × n)-matrices, ( ) is continuous -periodic vectorfunction, ( ( , ), ( − ∆, ), , ) is nonlinear vector function, which is analytic in a small neighborhood of the generating problem ( 2 ), continuous and -periodic with the respect to the variable t, and also analytic with respect to the small parameter ε on the interval [0, ₀]. As is known, in the critical case [2], specifically, in the presence of -periodic solutions

2023 Copyright for this paper by its authors. CEUR

ceur-ws.org of the homogeneous part purely imaginary roots of the characteristic equation of system ( 2 ), and in the case of constant matrices ( ) ≡ and ( ) ≡ , with the presence of the generating periodic problem for equation ( 2 ) is solvable not for all vector functions ( ). In the critical case, the adjoint system has a family of -periodic solutions of the form Periodic problem for the equation ( 2 ) is solvable iff 0 ,

= , ∈ ℝ 0/ = ( ) 0( ) + ( ) 0( − Δ) ( 3 ) = ± , =

−1, ∈ ℕ det + − Δ −

= 0, / = − ∗

− ∗ ( + Δ) , = ,

∈ ℝ . 0 ∗ = 0.

( 4 ) Here ( ) is (n × r)- matrix formed by r linearly independent -periodic solutions of the adjoint system. Let us assume condition ( 4 ) is satisfied; in this case, the general solution of the generating periodic problem for equation ( 2 ) has the form 0 , = + , ∈ ℝ , where [ ( )]( ) is a particular solution of the generating -periodic problem for equation ( 2 ), ( ) is (n × r)- matrix formed by r linearly independent -periodic solutions of the system ( 2 ). To construct a particular solution [ ( )]( ) of the generating -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 -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

0( , ) = ( ) + [ ( )]( ), ∈ ℝ .

Let us also assume that the T-periodic problem for equation ( 1 ) has a T-periodic solution which, at = 0, transforms unto the generating solution ( , ) ∶ (∙, ) ∈ ℂ¹[0, ], ( ,∙) ∈ ℂ[0, ₀],

( , 0) = 0( , ∗).

Under these conditions, the vector ∗ ∈ ℝ

satisfies the equation for the generating amplitudes ( ∗) ∶= ∫ ∗( ) ( 0( , ∗), 0( − Δ, ∗), , ) = 0.

( 5 ) 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 ∗ ∈ ℝ of equation ( 5 ) determine the generating solutions 0( , ∗), 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 ∗ ∈ ℝ , then the original nonlinear -periodic boundary value problem for equation ( 1 ) has no sought-after solutions. here and

Let us denote an (r × r)- matrix

T B0 ≔ ∫ Hr∗(s)[A1,0(s)Xr(s) + A0,1(s)Xr(s − Δ)]ds;

0 A1,0(t) = A0,1(t) = ∂Z(z(t, ε), z(t − Δ, ε), t, ε)

∂z(t, ε) ∂Z(z(t, ε), z(t − Δ, ε), t, ε) 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 z0(t, cr∗) is the requirement for the simplicity of the roots [2, 6]

det B0 ≠ 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

0( , ∗) = ∗ + [ ( )]( ), ∗ ∈ ℝ .

In the article [6], we found constructive necessary and sufficient conditions for solvability, along with a scheme for constructing solutions of the nonlinear -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 -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 -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 -periodic boundary value problem for equation ( 1 ) in a small neighborhood of the generating solution 0( , ∗). We seek the solution of the nonlinear -periodic boundary value problem for equation ( 1 ) in the critical case in the form ( , ) ∶= 0( , ) + 1( , ) + … + ( , ) + … . The nonlinear vector function ( ( , ), ( − Δ, ), , ) 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]

( ( , ), ( − Δ, ), , ) + ( 0( , ∗), 1( , ), … , ( , ), 0( − Δ, ∗), 1( − Δ, ), … , ( − Δ, ), , ) + … . = ( 0( , ∗), 0( − Δ, ∗), , 0) + 1( 0( , ∗), 1( , ), 0( − Δ, ∗), 1( − Δ, ), , ) + … + First approximation to solution of nonlinear - periodic boundary value problem for equation ( 1 ) in 1( , ) ≔ 0( , ∗)+ 1( , ), 1( , ) = ( ) 1( ) + ( , ), 1( ) ∈ ℝ determines the solution of the - periodic boundary value problem for equation

Here here in this case and thus

1( 1 )( , ) = [ ( 0( , ∗), 0( − Δ, ∗), , 0)]( ) is a particular solution of this problem. The solvability of the -periodic boundary value problem in the first approximation is guaranteed by choosing the root ∗ 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 -periodic boundary value problem for equation ( 1 ) in the critical case determines the solution of the -periodic boundary value problem for the equation 2( , ) ≔ 0( , ∗)+ 1( , ) + 2( , ), 2( , ) = ( ) 2( ) + 2( 1 )( , ), 2( ) ∈ ℝ 2( , )

= ( ) 2( , )+ ( ) 2( − Δ, )+ + 1( 0( , ∗), 1( , ), 0( − Δ, ∗), 1( − Δ, ), , ); 2( 1 )( , ) = [ 1( 0( , ∗), 1( , ), 0( − Δ, ∗), 1( − Δ, ), , )]( ) = ( ) 1( , )+ ( ) 1( − Δ, ) + ( 0( , ∗), 0( − Δ, ∗), , 0); is a particular solution of this problem. The solvability of the -periodic boundary value problem in the second approximation guarantees the solvability of the equation

1( 1( )) ≔ ∫0 ∗( ) 1( 0( , ∗), 1( , ), 0( − Δ, ∗), 1( − Δ, ), , ) = 0.

Unlike the equation for the generating amplitudes, the last equation is linear:

and also solvable, provided the roots of the equation for the generating amplitudes are simple; 1( 1( )) = 0 1( )+ 1( ∗, ) = 0, 0 = 1′( 1( )) ∈ ℝ × , 1( ∗) ≔ 1( 1( )) − 0 1( ).

In order to prove this, let us denote the vector function [8, 9]

( , ) ≔ 0( , ∗)+ 1( , )+ … + ( , ) + … ; 1( 1( )) ≔ ∫0 ∗( ) 1( 0( , ∗), 1( , ), 0( − Δ, ∗), , 0, 1( − Δ, ), ) ≔ = ∫0 ∗( ) ′ ( ( , ), ( − Δ, ), , ) | =0= =∫0 ∗( )[ 1,0( ) 1( , )+ 0,1( ) 1( − Δ, )] , 0 3( , ) 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( , ) = ( ) 1( ) + 1( 1 )( , ),

Third approximation to solution of nonlinear - periodic boundary value problem for equation ( 1 ) in critical case 3( , ) ≔ 0( , ∗)+ 1( , ) + 2( , )+ 3( , ),

3( , ) = ( ) 3( )+ 3( 1 )( , ), 3( ) ∈ ℝ determines the solution of the -periodic boundary value problem for the equation = ( ) 3( , )+ ( ) 3( − Δ, )+ + 2( 0( , ∗), 1( , ), 2( , ), 0( − Δ, ∗), 1( − Δ, ), 2( − Δ, ), , ); Here

3( 1 )( , ) = = [ 2( 0( , ∗), 1( , ), 2( , ), 0( − Δ, ∗), 1( − Δ, ), 2( − Δ, ), , )]( ) is a particular solution of this problem. The solvability of the -periodic boundary value problem in the second approximation guarantees the solvability of the equation 2( 2( ), ) ≔ ∫ ∗( )× 0 × 2( 0( , ∗), 1( , ), 2( , ), 0( − Δ, ∗), 1( − Δ, ), 2( − Δ, ), , ) = 0.

Unlike the equation for the generating amplitudes, the last equation is linear:

2( 2( )) = 0 2( ) + 2( ∗, 1( ), ) = 0, and also solvable, provided that the roots of the equation for the generating amplitudes are simple.

Here

0 = 2′( 2( )) ∈ ℝ × , 2( ∗, 1( ), ) ≔ 2( 2( )) − 0 2( ).

Denote (n × n) – matrices

A2,0(t,u1( , )) ≔ =

( , ) ∂Z(z(t,ε), z(t − Δ, ε), t, ε) ∂z(t,ε) 1( , )]|

z(t,ε) = z0(t, cr∗) z(t − Δ, ε) = z0(t − Δ, cr∗),

A1,1(t,u1( − Δ, )) ≔ = ( , ) [ ∂Z(z(t,ε), z(t − Δ, ε), t, ε) ∫ ∗( ) ′′2( ( , ), ( − Δ, ), , ) | = 0 = = ∫ ∗( )[ 1,0( ) 2( , ) + 0,1( ) 2( − Δ, )] + and Indeed, 2! 0 1 2! 0 0 0 0 0 2( ∗, 1( ), ) = ∫ ∗( )[ 1,0( ) 2

( 1 )( , )+ 0,1( ) 2( 1 )( − Δ, )] + + ∫ ∗( )[ 2,0( , 1( , )) 1( , ) + 2 1,1( , 1( , )) 1( − Δ, )+

0,2( , 1( − Δ, )) 1( − Δ, )] .

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

2( , ) = ( ) 2( ) + 2( 1 )( , ), 2( ) = − 0−1 ∫ ∗( )[ 1,0( ) 2

( 1 )( , ) + 0,1( ) 2( 1 )( − Δ, )] − − 2! 0−1 ∫ ∗( )[ 2,0( , 1( , )) 1( , ) + 2 1,1( , 1( , )) 1( − Δ, )+ 0

+ 0,2( , 1( − Δ, )) 1( − Δ, )] .

Solvability of the -periodic boundary value problem in + 1 approximation guarantees the solvability of the equation 0 +1( ( )) ≔ ∫ ∗( ) ( 0( , ∗), 1( , ), … , +1( , ),

0( − Δ, ∗), 1( − Δ, ), … , +1( − Δ, ), , ) = 0.

The sequence of approximations to the solution of the nonlinear -periodic boundary value problem for equation ( 1 ) in the critical case is determined by the iterative scheme 1( , ) ≔ 0( , ∗)+ 1( , ), 1( ) = ( ) 1( ) + 1( 1 )( , ),

u1( 1 )( , ) = [ ( 0( , ∗), 0( − Δ, ∗), , 0)]( ), thus furthermore + ∫ ∗( )[ 2,0( , 1( , )) + 2 1,1( , 1( , )) 1( − Δ, )+ + 0,2( , 1( − Δ, )) 1( − Δ, )] ,

0 = 2′( 2( ), ), 1( ) = − 0−1 ∫ ∗( )[ 1,0( ) 1 ( 1 )( − Δ, )] , 2( , ) ≔ 0( , ∗)+ 1( , )+ 2( , ), 2( , ) = ( ) 2( )+ 2( 1 )( , ), 2( 1 )( , ) = [ 1( 0( , ∗), 1( , ), 0( − Δ, ∗), 1( − Δ, ), , ))]( ), 2( ) = − 0−1 ∫ ∗( )[ 1,0( ) 2

( 1 )( , ) + 0,1( ) 2( 1 )( − Δ, )] − − 2! 0−1 ∫ ∗( )[ 2,0( , 1( , )) 1( , ) + 2 1,1( , 1( , )) 1( − Δ, )+ 0

+ 0,2( , 1( − Δ, )) 1( − Δ, )] , …, +1( , ) ≔ 0( , ∗)+ 1( , )+ … + +1( , ), ( 6 ) ( 1 ) ( , ) = [ ( 0( , ∗), 1( , ), … , ( , ),

+1 0( − Δ, ∗), , , 1( − Δ, ), … , ( − Δ, ))]( ), +1( ) = − 0−1 +1( ∗, +1( ), ), = 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

0( , ∗) = ( ) + [ ( )]( ), ∈ ℝ . problem for equation ( 1 ) has a unique solution Assuming det 0 ≠ 0 the simplicity of the roots of the equation ( 5 ) for the generating amplitudes in a small neighborhood of the generating solution 0( , ∗), the nonlinear periodic boundary value ( , ) ∶ (∙, ) ∈ ¹[0, ], ( ,∙) ∈ [0, ₀].

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 < < 1, for which the inequalities ‖ 1( , )‖ ≤ ‖ 0( , ∗)‖, ‖ +1( , )‖ ≤ ‖ ( , )‖, = 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 2 -periodic boundary value problem with concentrated delay For the generating periodic problem for equation ( 8 ), there is a critical case [2, 12], and condition ( 4 ) is satisfied, therefore, it is solvable: 0( , ) = ( ) + [ ( )]( ), ∈ ℝ1; ( ) = ( 10 ) , [ ( )]( ) = (si0n ). and also here and also and also 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 ∗ = 1, which determines the generating solution 0( , ∗) = − (1 + sin ).

0 The matrix 0 = 2 is non-singular, so according to the proven theorem, the 2 -periodic problem for equation ( 8 ) with concentrated delay is uniquely solvable. Thus, we obtain the first approximation 1( , ) ≔ 0( , ∗)+ 1( , ), 1( , ) = ( ) 1( ) + 1( 1 )( , ), 1( ) = − to the solution of the periodic problem for equation ( 8 ); here 1( 1 )( , ) = [ ( 0( , ∗), 0( − Δ, ∗), , 0)]( ) = (1 − sin − cos ), − cos 2( , ) ≔ 0( , ∗)+ 1( , ) + 2( , ), 2( , ) = ( ) 2( ) + 2( 1 )( , ); 2( 1 )( , ) = [ 1( 0( , ∗), 1( , ), 0( − Δ, ∗)), 1( − Δ, ), , )]( ) = 7 2 = 2 (sin + 3cos − 3), 2( ) = .

2cos − sin 2 1( 0( , ∗), 1( , ), 0( − Δ, ∗), 1( − Δ, ), , ) = = (1 + 1( ) − 2sin − cos )( 1 ).

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 3( , ) ≔ 0( , ∗)+ 1( , ) + 2( , )+ 3( , ), 3( ) = − = [ 2( 0( , ∗), 1( , ), 2( , ), 0( − Δ, ∗)), 1( − Δ, ), 2( − Δ, ), , )]( ) = 3 53 − 52cos − cos 2 + 12sin + 2sin 2 ) ; = 8 ( 2(16sin + sin2 − 10cos ) = 2 2( 0( , ∗), 1( , ), 2( , ), 0( − Δ, ∗), 1( − Δ, ), 2( − Δ, ), , ) = 2 (7 − 2 2( ) − 8 cos − cos 2 − 5 sin )( 1 ).

1 For the obtained approximations to the periodic solution of equation ( 8 ), the inequalities ‖ 1( , )‖ ≤ ‖ 0( , ∗)‖, ‖ +1( , )‖ ≤ ‖ ( , )‖, ≈ 0, 131 256, = 0,1,2, hold, indicating the practical convergence of the obtained approximations to the periodic solution of the equation ( 8 ) for

∈ [0, 0], 0 ≈ 0, 25.

The accuracy of the obtained approximations to the periodic solution of equation ( 8 ) is determined by the residuals Δ ( ) ≔ || ( , )/ − ( ) ( , ) − ( ) ( − Δ, ) − ( )−

− ( ( , ), ( − Δ, ), , )||ℂ[0;2 ] , = 0, 1, 2, 3.

In particular, 105

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].

The authors of the article express their sincere gratitude to the Managing Director of the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Professor Peter Benner, for his support and discussion of the obtained results.

4. References

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