Asymptotic solutions of singularly perturbed integro-di ﬀ erential systems with rapidly oscillating coe ﬃ cients in the case of a simple spectrum

: In this paper, we consider a system with rapidly oscillating coe ﬃ cients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop the algorithm of Lomov’s the regularization method for such systems and to identify the inﬂuence of the integral term on the asymptotics of the solution of the original problem. The case of identical resonance is considered, i.e. the case when an integer linear combination of the eigenvalues of a rapidly oscillating coe ﬃ cient coincides with the points of the spectrum of the limit operator is identical on the entire considered time interval. In addition, the case of coincidence of the eigenvalue of a rapidly oscillating coe ﬃ cient with the points of the spectrum of the limit operator is excluded. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require a more thorough analysis and are not considered in this paper.


Introduction
When studying various applied problems related to the properties of media with a periodic structure, it is necessary to study differential equations with rapidly changing coefficients. Equations of this type are often found, for example, in electrical systems under the influence of high frequency external forces. The presence of such forces creates serious problems for the numerical integration of the corresponding differential equations. Therefore, asymptotic methods are usually applied to such equations, the most famous of which are the Feshchenko -Shkil -Nikolenko splitting method [9][10][11][12]23] and the Lomov's regularization method [18,20,21]. The splitting method is especially effective when applied to homogeneous equations, and in the case of inhomogeneous differential equations, the Lomov regularization method turned out to be the most effective. However, both of these methods were developed mainly for singularly perturbed equations that do not contain an integral operator. The transition from differential equations to integro-differential equations requires a significant restructuring of the algorithm of the regularization method. The integral term generates new types of singularities in solutions that differ from the previously known ones, which complicates the development of the algorithm for the regularization method. The splitting method, as far as we know, has not been applied to integro-differential equations. In this article, the Lomov's regularization method [1-8, 13-17, 19, 24] is generalized to previously unexplored classes of integro-differential equations with rapidly oscillating coefficients and rapidly decreasing kernels of the form is the scalar function, A (t) and B (t) are (2 × 2)-matrices, moreover A (t) = 0 1 −ω 2 (t) 0 , ω (t) > 0, β (t) > 0 is the frequency of the rapidly oscillating cosine, z 0 = z 0 1 , z 0 2 , ε > 0 is a small parameter. It is precisely such a system in the case β (t) = 2γ (t) , B (t) = 0 0 1 0 and in the absence of an integral term was considered in [18,20,21].
To regularize the integral term, we introduce a class M ε , asymptotically invariant with respect to the operator Jz (see [18], p. 62]). We first consider the space of vector functions z (t, τ) , representable by sums where the asterisk * above the sum sign indicates that in it the summation for |m| ≥ 2 occurs only over nonresonant multi-indices m = (m 1 , ..., m 5 ) , i.e. over m 5 i=0 Γ i . Note that in (2.4) the degree of the polynomial with respect to exponentials e τ j depends on the element z. In addition, the elements of the space U depend on bounded in ε > 0 constants σ 1 = σ 1 (ε) and σ 2 = σ 2 (ε) , which do not affect the development of the algorithm described below, therefore, henceforth, in the notation of element (2.4) of this space U, we omit the dependence on σ = (σ 1 , σ 2 ) for brevity . We show that the class M ε = U| τ=ψ(t)/ε is asymptotically invariant with respect to the operator J.
The image of the operator J on the element (2.4) of the space U has the form: Here it is taken into account that (m − e 5 , λ (s)) 0, since by definition of the space U, multi-indices m Γ 5 . This means that the image of the operator J on the element (2.4) of the space U is represented as a series It is easy to show (see, for example, [22], pages 291-294) that this series converges asymptotically as ε → +0 (uniformly in t ∈ [t 0 , T ]). This means that the class M ε is asymptotically invariant (as ε → +0) with respect to the operator J.
We introduce the operators R ν : U → U, acting on each element z (t, τ) ∈ U of the form (2.4) according to the law: Let nowz (t, τ, ε) be an arbitrary continuous function converging as ε → +0 (uniformly in (t, τ) ∈ [t 0 , T ]× τ : Reτ j ≤ 0, j = 1, 5 ). Then the image Jz (t, τ, ε) of this function is decomposed into an asymptotic series This equality is the basis for introducing an extension of the operator J on series of the form (2.6): Although the operatorJ is formally defined, its usefulness is obvious, since in practice it is usual to construct the N-th approximation of the asymptotic solution of the problem (2.1), in which only N-th partial sums of the series (2.6) will take part, which have not formal, but true meaning. Now we can write down a problem that is completely regularized with respect to the original problem (2.1): were the operatorJ has the form (2.7).

Iterative problems and their solvability in the space U
Substituting the series (2.6) into (2.8) and equating the coefficients for the same powers of ε, we obtain the following iterative problems: τ) is a well-known vector-function of the space U, z * is a well-known constant vector of a complex space C 2 , and the operator R 0 has the form (see (2.5 In the future we need the λ j (t)-eigenvectors of the matrix A (t) : as well asλ j (t)-eigenvectors of the matrix A * (t) : These vectors form a biorthogonal system, i.e.
We introduce the scalar product (for each t ∈ [t 0 , T ]) in the space U : where we denote by ( * , * ) the ordinary scalar product in a complex space C 2 . We prove the following statement.
Theorem 1. Let conditions 1) and 2) are satisfied and the right-hand side 2) belongs to the space U. Then for the solvability of the system (3.2) in U it is necessary and sufficient that the identities are fulfilled. Proof. We will determine the solution to the system (3.2) in the form of an element (2.4) of the space U: Equating here separately the free terms and coefficients at the same exponents, we obtain the following systems of equations: Due to the invertibility of the matrix A (t), the system (3.5 0 ) has the solution −A −1 (t) H 0 (t). Since λ 5 (t) = µ (t) is a real function, and the eigenvalues of the matrix A (t) are purely imaginary, the matrix λ 5 (t) I − A (t) is invertible and therefore the system (3.5 5 ) can be written as since λ 3 (t) , λ 4 (t) do not belong to the spectrum of the matrix A (t) . Systems (3. +h 12 (t) ϕ 2 (t) e τ 1 + +h 21 (t) ϕ 1 (t) e τ 2 + z 5 (t) e τ 5 + where α k (t) ∈ C ∞ [t 0 , T ] , C 1 are arbitrary functions, k = 1, 2, z 0 (t) = −A −1 H 0 (t), z 5 (t) is the solution of the integral system (3.6) and the notations are introduced:

Unique solvability of the general iterative problem in the space U. The remainder term theorem
We proceed to the description of the conditions for the unique solvability of the system (3.2) in the space U. Along with the problem (3.2), we consider the system where z = z (t, τ) is the solution (3.9) of the system (3.2), Q (t, τ) ∈ U is the known function of the space U. The right-hand side of this system: may not belong to the space U, if z = z (t, τ) ∈ U. Since − ∂z ∂t , R 1 z, Q (t, τ) ∈ U, then this fact needs to be checked for the function therefore, the right-hand side G (t, τ) = Z (t, τ) − ∂z ∂t + R 1 z + Q (t, τ) of the system (19) also does not belong to the space U. Then, according to the well-known theory (see [18], p. 234), it is necessary to embed ∧ : G (t, τ) →Ĝ (t, τ) the right-hand side G (t, τ) of the system (4.1) in the space U. This operation is defined as follows.
We now turn to the proof of the following statement.
Taking into account that under conditions (4.2), scalar multiplication by vector functions χ k (t) e τ k , containing only exponentials e τ k , k = 1, 2, it is necessary to keep in the expression G (t, τ) only terms with exponents e τ 1 and e τ 2 . Then it follows from (**) that conditions (4.2) are written in the form where the functions w m j (α 1 (t) , α 2 (t) , t) , j = 1, 2, depend on α 1 (t) and α 2 (t) in a linear way. Performing scalar multiplication here, we obtain linear ordinary differential equations with respect to the functions α k (t) , k = 1, 2, involved in the solution (3.9) of the system (3.2). Attaching the initial conditions α k (t 0 ) = (z * , χ k (t 0 )) , k = 1, 2, calculated earlier to them, we find uniquely functions α k (t) , and, therefore, construct a solution (3.9) to the problem (3.2) in the space U in a unique way. The theorem 2 is proved. As mentioned above, the right-hand sides of iterative problems (3.1 k ) (if them solve sequentially) may not belong to the space U. Then, according to [18] (p. 234), the right-hand sides of these problems must be embedded into the U, according to the above rule. As a result, we obtain the following problems: Lz k (t, τ) = − ∂z k−1 ∂t + g(t) 2 (e τ 3 σ 1 + e τ 4 σ 2 ) B (t) z k−1 ∧ +R k z 0 + ... + R 1 z k−1 , z k (t 0 , 0) = 0, k ≥ 1, (3.1 k ) (images of linear operators ∂ ∂t and R ν do not need to be embedded in the space U, since these operators act from U to U). Such a replacement will not affect the construction of an asymptotic solution to the original problem (1.1) (or its equivalent problem (2.1)), so on the narrowing τ = ψ(t) ε the series of problems (3.1 k ) will coincide with the series of problems 3.1 k (see [18], pp. 234-235]. Applying Theorems 1 and 2 to iterative problems 3.1 k , we find their solutions uniquely in the space U and construct series (2.6). As in [18] (pp. 63-69), we prove the following statement.
was the case in ordinary integro-differential equations). The presence of a rapidly oscillating coefficient