Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump

is examined.The derivations prove that a unique pair (y (t, ?̃? (ε) , ε) , ?̃? (ε)) exists, in which componentsy(t, ?̃? (ε) , ε) and ?̃?(ε) satisfy the equation L ε y = h(t)λ and boundary value conditions L i y + σ i λ = a i , i = 1, n + 1. The issues of limit transfer of the perturbed problem solution to the unperturbed problem solution as a small parameter approaches zero and the existence of the initial jump phenomenon are studied. This research is conducted in two stages. In the first stage, the Cauchy function and boundary functions are introduced. Then, on the basis of the introduced Cauchy function and boundary functions, the solution of the restoration problem L ε y = h (t) λ, L i y+ σ i λ = a i , i = 1, n + 1 is obtained from the position of the singularly perturbed problem with the initial jump.Through this process, the formula of the initial jump and the asymptotic estimates of the solution of the considered boundary value problem are identified.


Introduction
One of the fundamental theorems of singular perturbations theories is Tikhonov's theorem [1,2] on the limit transition that establishes the limit equations, expressing the relations between the solutions of a degenerate problem and an original singularly perturbed initial problem, and this theorem allows us to obtain the leading member of asymptotics.
For a wide class of the singularly perturbed problems, effective asymptotic methods were developed, allowing for the development of uniform approximations with any level of precision.The methods of Višik and Lyusternik [3] and Vasilyeva [4] were the first methods to be developed, which are called methods of boundary functions.Butuzov [5] developed a method of angular boundary functions, which was a significant development for the boundary functions method.
Each of these methods has a certain area of applicability; that is, they are successful in solving some problems and are invalid when solving other problems.For example, there were some fundamental difficulties in the realization of the boundary functions method in problems with initial jumps.The beginning of the mathematical solution of the initial jump phenomenon was considered in the studies of Višik and Lyusternik [6] and Kasymov [7], in which the method of zone integration for nonlinear singularly perturbed initial tasks with unbounded initial data when a small parameter approaches zero.The research efforts of Višik, Lyusternik, and Kasymov were continued in [8,9] and other studies.
Simultaneously, there were problems in practice that extended beyond the scope of traditional research, in which ready-made asymptotic methods were inapplicable and required modification or generalization.For example, in the study of Neǐmark and Smirnova [10], a new rationale of the physical and mathematical nature of the Painleve paradox was introduced, which enriched the possible types of movements and dynamics of the system as a whole.At this point, we saw the races and contrasting structures.Butuzov and Vasiliyeva studied mathematical solutions of the question of contrast structures [11], and the phenomenon of the initial jump requires additional mathematical research.
Boundary value problems for ordinary differential equations containing parameters in the right-hand side and in the boundary conditions were examined [12,13].In these studies, a restoration problem of the right-hand side of the differential equations and boundary conditions is solved with the wellknown structure of the differential equation and additional information.
The following natural generalization in this direction is to study the solutions of singularly perturbed boundary value problems with an additional parameter having the initial jump phenomenon.Such a study has not yet been reported.This study considers such problems.It studies the problems of solution building, restoring the right-hand side of the equation and boundary conditions of the perturbed problem, the limit solution passing of the perturbed problem solution to the unperturbed problem solution.

Statement of the Problem
with nonseparated boundary conditions where > 0 is a small parameter,  is an unknown parameter,   , have a rank equal to , where  is the maximum number of linearly independent rows of matrix .To the certainty, let the first  rows of the matrix be linearly independent.Then, the linear forms   ,  = 1, , as the functions of  () (0, ),  = 0,   and  () (1, ),  = 0,   are linearly independent of each other.
The following study is conducted according to a specific rule.In the first stage, we build the recovery problem solutions (1) and (2) from the position of a singularly perturbed problem with the initial jump on the basis of initial and boundary functions.In the next stage, we study the asymptotic behavior of the boundary value problem solutions (1) and (2).

Fundamental Solution System
Along with (1), we consider the following corresponding homogeneous perturbed equation: We seek a fundamental solution system of (10) in the following form: where   () represents unknown functions to be found and () = − 1 ().
By substituting (11) into (10) and by matching the coefficients of like powers of  on both sides of the resulting relation, we obtain a sequence of equations for all terms in expansion (11).For our aim, however, it suffices to consider the zero approximation.So, for the zero approximation (for  0 (),  = 1,  − 1), we have the following problems: Unique solutions exist for the problems (12) on the interval 0 ≤  ≤ 1, and they form the fundamental solution system  0 (),  = 1,  − 1 for the following homogeneous equation: where  0 () is presented in the following form: Lemma 1.Let conditions (a) and (b) be satisfied.Then, the fundamental solution system   (, ),  = 1,  of the singularly perturbed equation (10) permits the following asymptotic representations: as  → 0.

Cauchy Function and Boundary Functions
Following previous work [17], let us introduce the Cauchy function.
Definition 2. Function (, , ), defined at 0 ≤  ≤  ≤ 1, is called the Cauchy function of (10), if it satisfies the homogeneous equation ( 10) according to  and within  =  initial conditions: The following theorem is valid.
Proof.By expanding  ()  (, , ),  = 0,  − 1 in entries of the th column, we obtain From ( 15), the minors W()  (, , ), W (, ) for sufficiently small  > 0 can be represented in the following form: where  0 () = () and This readily implies that From ( 26) with regard to (19) and (21), we obtain the desired estimates (22).Definition 5. Functions Φ  (, ),  = 1,  are referred to as boundary functions of the boundary value problems (1) and (2), if they satisfy the homogeneous equation (10) and the boundary conditions Consider the determinant where with (15), the entries (1, ) can be represented in the following form: This, together with the estimate exp((1/) ∫ 1 0 ()) = (  ) (where  is an arbitrary positive integer), implies that where Now, let us expand () in the entries of the last column: Here, from (15), the minors   () can be expressed in the following form: where  0 is the ( − 1)th-order determinant introduced in Section 1.
By substituting (30) and ( 33) into (32), we obtain This, together with condition (c), implies that the determinant (28) permits the asymptotic representation: The following theorem is valid.
where  > 0 and  > 0 are the constants independent of .
Proof.Formulae ( 22), (37), and (38) together with (5) imply the following estimates: By estimating solution (58) and taking into account (60), we obtain (59), which completes the proof of the theorem.Now, let us formulate the boundary conditions for the following unperturbed (degenerate) equation: obtained from (1) within  = 0. On the basis of Theorem 9, we must conclude that in (58), the coefficient of  1 approaches zero when  → 0 and the coefficients of   ,  = 2,  have the order (1).Therefore, the boundary conditions for the solution of the unperturbed (61) are defined with help of the boundary conditions (2), containing  2 , . . .,  +1 : Next, we show that (61) and boundary conditions (62) actually define the degenerate problem.