Analytical Aspects of the Theory of Tikhonov Systems

Abstract

It is proved that the solutions of Tikhonov systems, in addition to having the property of a limiting transition, are pseudoholomorphic under certain conditions additional to the conditions of A.N. Tikhonov’s theorem. At the same time, the number of fast and slow variables can be anything. Both initial and boundary value problems for systems of this type are considered.

1. Introduction

A.N. Tikhonov’s theorem on the limit transition [1] served as an impetus for the development of one of the most practically used methods for solving nonlinear singularly perturbed problems—the Vasilyeva–Butuzov–Nefedov boundary function method [1,2,3]. On the other hand, Poincaré’s theorems on the decomposition of [4], which are the basis of the analytical theory of differential equations, and the regularization method of S.A. Lomov [5,6] allowed us to formulate conditions for the usual (and not asymptotic) convergence of series by degrees of a small parameter representing solutions of singularly perturbed equations [7]. All those items should eventually lead to the creation of a rigorous mathematical theory of the boundary layer based on the ideas of algebra and functional analysis.

In [8], the conditions for the existence of pseudoholomorphic solutions of initial problems for Tikhonov systems with one slow variable were given. In this paper, the number of fast and slow variables can be arbitrary. In addition, boundary value problems are considered, which means that the use of the pseudoholomorphic continuation algorithm [9] is required.

The main advantage of the holomorphic regularization method is the fact that there is no need for the estimation of the reminder term. The usual convergence of series by degrees of a small parameter allows one to solve singularly perturbed boundary problems with great advantage, in comparison with the methods based only on the asymptotical convergence. The investigation of the theory of Tikhonov systems has not only a theoretical value but also a practical one: the solution methods for the systems of that type can be used in chemistry, biology, theoretical and applied mechanics.

The holomorphic regularization method allows us to create an analytical theory of singularly perturbed boundary problems’ solution, and that fact has a great theoretical and practical value.

2. Construction of the First Integrals for Tikhonov Systems

Let us investigate a Tikhonov system with k slow and m fast variables

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\mathit{z}),\hfill \\ \epsilon {\displaystyle \frac{d\mathit{z}}{dt}}=\mathit{g}(t,\mathit{y},\mathit{z})\hfill \end{array}\right.$$

(1)

on the time interval $[0,T]$. Here, $\mathit{y}=\{y_1,\dots ,y_k\}$, $\mathit{z}=\{z_1,\dots ,z_m\}$, $\mathit{f}=\{f_1,\dots ,f_k\}$, $\mathit{g}=\{g_1,\dots ,g_m\}$, $\epsilon$ is a small positive parameter.

Firstly, we study the Cauchy problem with initial conditions

$$\mathit{y}(0,\epsilon )={\mathit{y}}^0,\mathit{z}(0,\epsilon )={\mathit{z}}^0,$$

(2)

where ${\mathit{y}}^0=\{y_1^0,\dots ,y_k^0\}$, ${\mathit{z}}^0=\{z_1^0,\dots ,z_m^0\}$.

Let the conditions $(\alpha )$ be met:

(1_α)

The functions $\mathit{f}(t,\mathit{y},\mathit{z})$ and $\mathit{g}(t,\mathit{y},\mathit{z})$ are analytic in some closed domain ${\overline{\Omega }}_{t\mathit{y}\mathit{z}}\subset {\mathbb{R}}^{k+m+1}$ and admit an analytic continuation to some polycircle of ${\mathbb{C}}^{k+m+1}$.

(2_α)

Vector $\mathit{g}(t,\mathit{y},\mathit{z})$ for all $(t,\mathit{y},\mathit{z})\in {\Omega }_{t\mathit{y}\mathit{z}}$ has a nonzero value.

(3_α)

The segment $[0,T]$ of the axis of the variable t and the starting point $(0,{\mathit{y}}^0,{\mathit{z}}^0)$ belong to the domain ${\Omega }_{t\mathit{y}\mathit{z}}$.

We will also take into consideration the conditions $(\beta )$ for Tikhonov’s theorems on the limit transition:

(1_β)

If $\mathit{z}=\widehat{\mathit{Z}}(t,\mathit{y})=\{{\widehat{Z}}_1(t,\mathit{y}),\dots ,{\widehat{Z}}_m(t,\mathit{y})\}$ is an isolated root of the equation $\mathit{g}(t,\mathit{y},\mathit{z})=\mathbf{0}$ ($\mathbf{0}\in {\mathbb{R}}^m$), then the initial problem

$$\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\widehat{\mathit{Z}}(t,\mathit{y})),\hfill \\ \mathit{y}(0)={\mathit{y}}^0\hfill \end{array}$$

(3)

has the only analytical solution in the circle $\left|t\right|⩽T$ $\widehat{\mathit{Y}}(t)=\{{\widehat{Y}}_1(t),\dots ,{\widehat{Y}}_k(t)\}$.

(2_β)

The rest point $\mathit{z}=\widehat{\mathit{Z}}(t,\mathit{y})$ of a system

$$\frac{d\mathit{z}}{ds}=\mathit{g}(t,\mathit{y},\mathit{z}),s⩾0$$

(4)

is asymptotically stable by Lyapunov uniformly with respect to any compact of ${\overline{\omega }}_{t\mathit{y}}$ that is a projection of ${\overline{\Omega }}_{t\mathit{y}\mathit{z}}$ on the space ${\mathbb{R}}^{k+1}$. Equation (4) is called attached and in its $(t,\mathit{y})$ is considered as a vector parameter.

(3_β)

The solution $\mathit{z}=\check{\mathit{Z}}(s)$ of the Cauchy problem for the initial attached system

$$\frac{d\mathit{z}}{ds}=\mathit{g}(0,{\mathit{y}}^0,{\mathit{z}}^0),s⩾0$$

satisfies the limit condition

$$\check{\mathit{Z}}(s)\to \widehat{\mathit{Z}}(0,{\mathit{y}}^0),s\to +\infty .$$

As [1] is known, when these conditions are met, there is a limit transition

$$\begin{array}{c}\underset{\epsilon \to +0}{lim}\mathit{y}(t,\epsilon )=\widehat{\mathit{Y}}(t)\forall t\in [0,T],\hfill \\ \underset{\epsilon \to +0}{lim}\mathit{z}(t,\epsilon )=\widehat{\mathit{Z}}(t,\widehat{\mathit{Y}}(t))\forall t\in (0,T].\hfill \end{array}$$

To apply the holomorphic regularization method to the system (1), we introduce first-order partial differential operators:

$$\begin{array}{c}{L}^{\mathit{f}}={\displaystyle \frac{\partial }{\partial t}}+f_1{\displaystyle \frac{\partial }{\partial y_1}}+\dots +f_k{\displaystyle \frac{\partial }{\partial y_k}},\hfill \\ L^{\mathit{g}}=g_1{\displaystyle \frac{\partial }{\partial z_1}}+\dots +g_m{\displaystyle \frac{\partial }{\partial z_m}}.\hfill \end{array}$$

Then the first integral equation for this system takes the following form:

$$\epsilon L^{\mathit{f}}U+L^{\mathit{g}}U=0.$$

(5)

Considering the operator $L^{\mathit{f}}$ to be subordinate to the operator $L^{\mathit{g}}$, we look for a solution to Equation (5) in the form of a regular series in degrees $\epsilon$, i.e.,

$$U(t,\mathit{y},\mathit{z},\epsilon )=U_0(t,\mathit{y},\mathit{z})+\epsilon U_1(t,\mathit{y},\mathit{z})+\dots +{\epsilon }^n{U}_n(t,\mathit{y},\mathit{z})+\dots .$$

(6)

Its coefficients can be found as a result of solving a series of equations

$$\begin{array}{c}{L}^{\mathit{g}}{U}_0=0,\hfill \\ L^{\mathit{g}}{U}_1=-L^{\mathit{f}}{U}_0,\hfill \\ \dots \dots \dots \dots \dots \hfill \\ L^{\mathit{g}}{U}_n=-L^{\mathit{f}}{U}_{n-1},\hfill \\ \dots \dots \dots \dots \dots \hfill \end{array}$$

(7)

We further denote the projection ${\overline{\Omega }}_{t\mathit{y}\mathit{z}}$ onto the space ${\mathbb{R}}^m$ as ${\overline{\theta }}_{\mathit{z}}$ and so we formulate an additional condition to the conditions $(\alpha )$.

(4_α)

For each point $(t,\mathit{y})\in {\overline{\omega }}_{t\mathit{y}}$, only one phase trajectory of the autonomous system (4) passes through any point ${\theta }_{\mathit{z}}$.

In the following, we consider the solutions of the series (7) to be vectors.As a solution to the first equation of this series, we take the vector

$${\mathit{U}}_0=\{{\phi }_1(t),\dots ,{\phi }_m(t),y_1-{\widehat{Y}}_1(t),\dots ,y_k-{\widehat{Y}}_k(t)\},$$

where ${\phi }_1(t),\dots ,{\phi }_m(t)$ are analytic functions in the circle $\left|t\right|⩽T$.

To solve the remaining equations of the series, we use the integral method of solving first-order partial differential equations [10]. Let ${\Lambda }^m$ be an analytically smooth surface in ${\mathbb{R}}^m$ (${\mathbb{R}}^m$ is the space of fast variables $z_1,\dots ,z_m$ passing through the point ${\mathit{z}}_0$). It is set using m functions, i.e.,

$${\Lambda }^m=\{\mathit{z}\in {\mathbb{R}}^m:z_i={\lambda }_i(\tilde{\mathit{z}}),i=\overline{1,m}\},$$

where $\tilde{\mathit{z}}=\{{\tilde{z}}_1,\dots ,{\tilde{z}}_{m-1}\}$ is a coordinate vector of this surface, and ${\lambda }_i(\tilde{\mathit{z}})$ is analytic in some area of space ${\mathbb{R}}^{m-1}$ functions.

Let us investigate the Cauchy problem

$$\begin{array}{c}{\displaystyle \frac{d\mathit{z}}{ds}}=\mathit{g}(t,\mathit{y},\mathit{z}),s⩾0,\hfill \\ \mathit{z}{|}_{s=0}=\mathit{\lambda }(\tilde{\mathit{z}}),\mathit{\lambda }=\{{\lambda }_1,\dots ,{\lambda }_m\},\hfill \end{array}$$

(8)

in which t and $\mathit{y}$ act as parameters, and let $\mathit{z}=\overline{\mathit{Z}}(t,\tilde{\mathit{z}},s)$ be its solution. The condition $4_{\alpha }$ together with the existence and uniqueness theorem guarantee unambiguous solvability of the equation $\overline{\mathit{Z}}(t,\tilde{\mathit{z}},s)=\mathit{z}$ with respect to $\tilde{\mathit{z}}$ and s: $s=S(t,\mathit{z})$, $\tilde{\mathit{z}}=\tilde{\mathit{Z}}(t,\mathit{z})$. Let $R(t)$ be the notation for the variable replacement operator $(s,\tilde{\mathit{z}})$ on the variable $\mathit{z}$ and let $R^{-1}(t,s)$ be the notation for a reverse substitution operator. Then, the solution of the initial problem

$$L^{\mathit{g}}W=F,W{|}_{\mathit{z}\in {\Lambda }^m}=0,$$

is given by the following formula [10]:

$$W(t,\mathit{z})=\underset{0}{\overset{s}{\int }}F(t,\overline{\mathit{Z}}(t,\tilde{\mathit{z}},s_1))d{s}_1{|}_{\genfrac{}{}{0pt}{}{s=S(t,\mathit{z})}{\tilde{\mathit{z}}=\tilde{\mathit{Z}}(t,\mathit{z})}},$$

(9)

under the condition of transversality of the phase trajectories of the system (8) to the specified surface.

To construct a solution to the second equation of (7), we further use the notation ${\tilde{V}}^{\left[i\right]}(t,\mathit{y},\mathit{z})$, $i=\overline{1,m}$ for functionally independent solutions (for $(t,\mathit{y})\in {\omega }_{t\mathit{y}}$) of the equation $L^{\mathit{g}}V=1$, and consider the fact that ${\tilde{V}}^{\left[i\right]}(0,{\mathit{y}}_0,{\mathit{z}}_0)=0$. Then, it is easy to see that as the first m components of the vector ${\mathit{U}}_1$, we can take the functions

$$V^{\left[i\right]}=-{\phi }_i^{\prime }(t){\tilde{V}}^{\left[i\right]},i=\overline{1,m}.$$

The remaining k components of this vector are found using formula (9):

$${({\mathit{U}}_1)}^{\left[j\right]}=-R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_j-{\widehat{Y}}_j(t))d{s}_1,j=\overline{1,k}.$$

The solutions of the other equations of series (7) can also be found using this formula.

To prove the convergence of the series

$${\mathit{U}}_0+\epsilon {\mathit{U}}_1+\dots +{\epsilon }^n{\mathit{U}}_n+\dots ,$$

in the expression for the operator $L^{\mathit{g}}$, we replace the variables $z_1,\dots ,z_m$ with the variable z and apply the method of evaluating contour integrals described in [11]. As a result, $(m+k)$ independent integrals of system (1) are constructed, being analytical at the point $\epsilon =0$:

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1^{\prime }(t){\tilde{V}}^{\left[1\right]}(t,\mathit{y},\mathit{z})-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}{V}^{\left[1\right]}d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^{2}R(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}^{-1}(t,s_1)L^{\mathit{f}}R(t)\underset{0}{\overset{s_1}{\int }}{R}^{-1}(t,s_2)L^{\mathit{f}}{V}^{\left[1\right]}d{s}_2-\dots ={\phi }_1(t)/\epsilon ,}\hfill \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \\ {\displaystyle {\phi }_m^{\prime }(t){\tilde{V}}^{\left[m\right]}(t,\mathit{y},\mathit{z})-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}{V}^{\left[m\right]}d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^{2}R(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}^{-1}(t,s_1)L^{\mathit{f}}R(t)\underset{0}{\overset{s_1}{\int }}{R}^{-1}(t,s_2)L^{\mathit{f}}{V}^{\left[m\right]}d{s}_2-\dots ={\phi }_m(t)/\epsilon ,}\hfill \\ {\displaystyle y_1-{\widehat{Y}}_1(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t))d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^{2}R(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}^{-1}(t,s_1)L^{\mathit{f}}R(t)\underset{0}{\overset{s_1}{\int }}{R}^{-1}(t,s_2)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t))d{s}_2-\dots =0,}\hfill \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \\ {\displaystyle y_k-{\widehat{Y}}_k(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_k-{\widehat{Y}}_k(t))d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^{2}R(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}^{-1}(t,s_1)L^{\mathit{f}}R(t)\underset{0}{\overset{s_1}{\int }}{R}^{-1}(t,s_2)L^{\mathit{f}}(y_k-{\widehat{Y}}_k(t))d{s}_2-\dots =0.}\hfill \end{array}\right.$$

(10)

3. Pseudoholomorphic Solutions of Tikhonov Systems

The concept of a pseudoholomorphic (pseudoanalytical) solution of a singularly perturbed problem first arose within the framework of the regularization method of S.A. Lomov [5,6,7]. It is by no means an alternative to the concept of an asymptotic solution, but only emphasizes the fact that with an accurate description of the boundary layer, the regular part of the solution is an ordinary power series. This is achieved by isolating special manifolds described by functions for which the point $\epsilon =0$ is essentially special for almost all values of time [7]. In linear problems, the boundary layer is dictated by the spectrum of the limit operator; in weakly nonlinear ones, the method of normal forms by V.F. Safonov [12] has proven itself well. A much more complicated situation arises in highly nonlinear problems. There is no clear algorithm for choosing the so-called regularizing functions.

We proceed to the presentation of the concept of a pseudoholomorphic solution.

Definition 1.

The solution $\{y_1(t,\epsilon ),\dots ,y_k(t,\epsilon ),z_1(t,\epsilon ),\dots ,z_m(t,\epsilon )\}$ of systems (1) is called pseudoholomorphic at the point $\epsilon =0$ if there exists a vector function $\mathit{\phi }=\{{\phi }_1(t),\dots ,{\phi }_m(t)\}$, analytic in the circle $\left|t\right|⩽T$ such that for any arbitrarily small positive ε there is a segment $[0,T_{\epsilon }]$, which has the following representation:

$$\left\{\begin{array}{c}{\displaystyle \mathit{y}(t,\epsilon )=\sum _{n=0}^{\infty }{\epsilon }^n{\tilde{\mathit{Y}}}_n(t,\mathit{\phi }(t)/\epsilon ),}\hfill \\ {\displaystyle \mathit{z}(t,\epsilon )=\sum _{n=0}^{\infty }{\epsilon }^n{\tilde{\mathit{Z}}}_n(t,\mathit{\phi }(t)/\epsilon ),}\hfill \end{array}\right.$$

(11)

and the series in (11) converge uniformly on the specified segment.

We give sufficient conditions for the existence of such solutions, considering the conditions $(\alpha )$, $(\beta )$ as fulfilled.

Theorem 1.

Let the vector function $\mathit{\phi }=\{{\phi }_1(t),\dots ,{\phi }_m(t)\}$ be analytic in the circle $\left|t\right|⩽T$, ${\phi }_i(0)=0$, ${\phi }_i(t)$ strictly monotonically decreases on the segment $[0,T]$ $(i=\overline{1,m})$, and the system of equations

$$\left\{\begin{array}{c}{\phi }_1^{\prime }{\tilde{V}}^{\left[1\right]}(t,{\mathit{y}}^0,\mathit{z})={\phi }_1(t)/\epsilon ,\hfill \\ \dots \dots \dots \dots \dots \dots \hfill \\ {\phi }_m^{\prime }{\tilde{V}}^{\left[m\right]}(t,{\mathit{y}}^0,\mathit{z})={\phi }_m(t)/\epsilon \hfill \end{array}\right.$$

(12)

has a solution of the form

$$\mathit{z}={\mathit{Z}}_0(t,{\mathit{y}}^0,{\mathit{z}}^0,\mathsf{\Phi }(\mathit{\phi }(t)/\epsilon )),$$

where the vector is a function $\mathsf{\Phi }(\eta )=\{{\Phi }_1(\eta ),\dots ,{\Phi }_m(\eta )\}$ and $q_i={\mathsf{\Phi }}_i(\eta )$$(i=\overline{1,m})$ are integer functions with asymptotic values $a_i$ $(i=\overline{1,m})$, and that function is strictly monotonically increasing on the real axis.

Then if the function ${\mathit{Z}}_0(t,{\mathit{y}}^0,{\mathit{z}}^0,\mathit{q})$ is bounded on the set ${\mathit{T}}_{t\mathit{y}\mathit{z}}^0\times (a_1,{\Phi }_1(0)]\times \dots \times (a_m,{\Phi }_m(0)]$, in which ${\mathit{T}}_{t\mathit{y}\mathit{z}}^0$ is an arbitrary compact set of ${\Omega }_{t\mathit{y}\mathit{z}}$, then the solution is $\{\mathit{y}(t,\epsilon ),\mathit{z}(t,\epsilon )\}$. Cauchy problems (1) and (2) will be pseudoholomorphic at the point $\epsilon =0$.

Proof of Theorem 1.

Using the monotony of the functions ${\Phi }_i(\eta )$ $(i=\overline{1,m})$, we write a system equivalent to system (10):

$$\left\{\begin{array}{c}{\Phi }_1({\phi }_1^{\prime }{\tilde{V}}^{\left[1\right]}(t,\mathit{y},\mathit{z}))+\epsilon {\Psi }_1(t,\mathit{y},\mathit{z},{\mathit{y}}^0,{\mathit{z}}^0,\epsilon )=q_1,\hfill \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \\ {\Phi }_m({\phi }_m^{\prime }{\tilde{V}}^{\left[m\right]}(t,\mathit{y},\mathit{z}))+\epsilon {\Psi }_m(t,\mathit{y},\mathit{z},{\mathit{y}}^0,{\mathit{z}}^0,\epsilon )=q_m,\hfill \\ {\displaystyle \mathit{y}-\widehat{\mathit{Y}}(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(\mathit{y}-\widehat{\mathit{Y}}(t))d{s}_1+\dots =\mathbf{0}.}\hfill \end{array}\right.$$

(13)

Here, ${\Psi }_i$ are analytical functions of their arguments; $q_i={\Phi }_i({\phi }_i(t)/\epsilon )$, $q_i^{*}$ are further new variables $(i=\overline{1,m})$.

Take $q_i^{*}>a_i$ very close to $a_i$$(i=\overline{1,m})$ and construct a compact set

$${\mathit{T}}_{\mathit{a}}=[0,T]\times {\tilde{\mathit{T}}}_{\mathit{y}\mathit{z}}^0\times [q_1^{*},{\Phi }_1(0)]\times \dots [q_m^{*},{\Phi }_m(0)],$$

in which ${\tilde{\mathit{T}}}_{\mathit{y}\mathit{z}}^0$ are projections of ${\mathit{T}}_{t\mathit{y}\mathit{z}}^0$ on to ${\mathbb{R}}^{k+m}$, which is the space of variables $(\mathit{y},\mathit{z})$. If we put (13) at $\epsilon =0$ in the system, then the solution of the resulting system is put down as follows:

$$\left\{\begin{array}{c}\mathit{z}={\mathit{Z}}_0(t,{\mathit{y}}^0,{\mathit{z}}^0,\mathit{q}),\hfill \\ \mathit{y}=\widehat{\mathit{Y}}(t,{\mathit{y}}^0),\hfill \end{array}\right.$$

here, it is convenient to consider the function $\widehat{\mathit{Y}}$ dependent on ${\mathit{y}}^0$.

Since all the conditions of the implicit function theorem are met, then, in some neighborhood ${\sigma }_{t{\mathit{y}}^0{\mathit{z}}^0\mathit{q}}$ of each point ${\mathit{T}}_{\mathit{a}}$, there is a solution

$$\left\{\begin{array}{c}\mathit{y}=\mathit{Y}(t,{\mathit{y}}^0,{\mathit{z}}^0,\mathit{q},\epsilon ),\hfill \\ \mathit{z}=\mathit{Z}(t,{\mathit{y}}^0,{\mathit{z}}^0,\mathit{q},\epsilon )\hfill \end{array}\right.$$

(14)

of the specified system, analytical in some neighborhood of the point $\epsilon =0$. From the open cover $\{{\sigma }_{t{\mathit{y}}^0{\mathit{z}}^0\mathit{q}}\}$ of the compact set ${\mathit{T}}_{\mathit{a}}$, we choose a finite subcover, and then the functions in (14) are analytic by $\epsilon$ uniformly on this compact, in the smallest of the neighborhoods of the point $\epsilon =0$ corresponding to the specified subcover. The theorem is proved. □

4. Pseudoholomorphic Continuation of Solutions of Singularly Perturbed Problems

We assume, without limiting generality, that ${\Phi }_i(0)=1$, $a_i=0$ $\forall i=\overline{1,m}$. Let the vector functions in (14) be analytic for $\left|\epsilon \right|<{\epsilon }_0$ and let a fixed parameter in system (1) satisfy the inequality $0<\epsilon <{\epsilon }_0$. If for each $i=\overline{1,m}$ all curves ${\Gamma }_i=\{(t,q_i):q_i={\Phi }_i({\phi }_i(t)/\epsilon ),t\in [0,T]\}$ belong entirely to rectangles ${\pi }_i=[0,T]\times [q_i^{*},1]$, respectively, then system (1) has a pseudoholomorphic solution on the entire segment $[0,T]$, represented as rows

$$\left\{\begin{array}{c}\mathit{y}(t,\epsilon )={\displaystyle \sum _{n=0}^{\infty }}{\epsilon }^n{\mathit{Y}}_n(t,\mathsf{\Phi }(\mathit{\phi }(t)/\epsilon )),\hfill \\ \mathit{z}(t,\epsilon )={\displaystyle \sum _{n=0}^{\infty }}{\epsilon }^n{\mathit{Z}}_n(t,\mathsf{\Phi }(\mathit{\phi }(t)/\epsilon )),\hfill \end{array}\right.$$

(15)

converging uniformly over the entire segment $[0,T]$. If the curves ${\Gamma }_i$ do not completely belong to the rectangles ${\pi }_i$, then the rows in (15) converge uniformly on some segment $[0,t_1]\subset [0,T]$. As $t_1$, as it is easy to see, we can take the minimal root of the equations ${\Phi }_i({\phi }_i(t)/\epsilon )=q_i^{*}$, $i=\overline{1,m}$, due to the monotony of the left parts of these equations (all rectangles ${\pi }_i$ are strictly internal to the rectangle $[0,T]\times [0,1]$).

We denote the solution as: $\{\mathit{y}(t,\epsilon ),\mathit{z}(t,\epsilon )\}$ by $\{{\mathit{y}}_0(t,\epsilon ),{\mathit{z}}_0(t,\epsilon )\}$. To continue them to the right, we should take into consideration the following initial problem:

$$\begin{array}{c}\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\mathit{z}),\hfill \\ \epsilon {\displaystyle \frac{d\mathit{z}}{dt}}=\mathit{g}(t,\mathit{y},\mathit{z}),\hfill \end{array}\right.\hfill \\ \mathit{y}(t_1,\epsilon )={\mathit{y}}_0(t_1,\epsilon ),\mathit{z}(t_1,\epsilon )={\mathit{z}}_0(t_1,\epsilon ).\hfill \end{array}$$

(16)

Let us construct an analytically smooth surface ${\Lambda }_1^m$ in the space ${\mathbb{R}}^m$ through the point ${\mathit{z}}_0(t_1,\epsilon )$ and, according to the scheme outlined in the previous paragraph of this article, we find $(k+m)$ independent integrals of system (16) on the interval $[t_1,T]$. In this case, a system similar to system (10) takes the following form:

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1^{\prime }(t){\tilde{V}}_1^{\left[1\right]}(t,\mathit{y},\mathit{z})-\epsilon R_1(t)\underset{0}{\overset{s}{\int }}{R}_1^{-1}(t,s_1)L^{\mathit{f}}{V}_1^{\left[1\right]}d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^2{R}_1(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}_1^{-1}(t,s_1)L^{\mathit{f}}{R}_1(t)\underset{0}{\overset{s_1}{\int }}{R}_1^{-1}(t,s_2)L^{\mathit{f}}{V}_1^{\left[1\right]}d{s}_2-\dots =}\hfill \\ {\displaystyle =({\phi }_1(t)-{\phi }_1(t_1))/\epsilon ,}\hfill \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \\ {\displaystyle {\phi }_m^{\prime }(t){\tilde{V}}_1^{\left[m\right]}(t,\mathit{y},\mathit{z})-\epsilon R_1(t)\underset{0}{\overset{s}{\int }}{R}_1^{-1}(t,s_1)L^{\mathit{f}}{V}_1^{\left[m\right]}d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^2{R}_1(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}_1^{-1}(t,s_1)L^{\mathit{f}}{R}_1(t)\underset{0}{\overset{s_1}{\int }}{R}_1^{-1}(t,s_2)L^{\mathit{f}}{V}_1^{\left[m\right]}d{s}_2-\dots =}\hfill \\ {\displaystyle =({\phi }_m(t)-{\phi }_m(t_1))/\epsilon ,}\hfill \\ {\displaystyle y_1-{\widehat{Y}}_1(t)-\epsilon R_1(t)\underset{0}{\overset{s}{\int }}{R}_1^{-1}(t,s_1)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t))d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^2{R}_1(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}_1^{-1}(t,s_1)L^{\mathit{f}}{R}_1(t)\underset{0}{\overset{s_1}{\int }}{R}_1^{-1}(t,s_2)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t))d{s}_2-\dots =0,}\hfill \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \\ {\displaystyle y_k-{\widehat{Y}}_k(t)-\epsilon R_1(t)\underset{0}{\overset{s}{\int }}{R}_1^{-1}(t,s_1)L^{\mathit{f}}(y_k-{\widehat{Y}}_k(t))d{s}_1+}\hfill \\ {\displaystyle +{\epsilon }^2{R}_1(t)\underset{0}{\overset{s}{\int }}d{s}_1{R}_1^{-1}(t,s_1)L^{\mathit{f}}{R}_1(t)\underset{0}{\overset{s_1}{\int }}{R}_1^{-1}(t,s_2)L^{\mathit{f}}(y_k-{\widehat{Y}}_k(t))d{s}_2-\dots =0.}\hfill \end{array}\right.$$

(17)

Here, ${\tilde{V}}_1^{\left[i\right]}(t,\mathit{y},\mathit{z})$ are functionally independent solutions of the equation $L^{\mathit{g}}V=1$ such that ${\tilde{V}}_1^{\left[i\right]}(t_1,{\mathit{y}}_0(t_1,\epsilon ),{\mathit{z}}_0(t_1,\epsilon ))=0$. At the same time, $V_1^{\left[i\right]}=-{\phi }_i^{\prime }(t){\tilde{V}}_1^{\left[i\right]}$, $i=\overline{1,m}$, and the operators $R_1(t)$ and $R_1^{-1}(t,s)$ correspond to the surface ${\Lambda }_1^m$. The interval of existence of a pseudoholomorphic solution $\{{\mathit{y}}_1(t,\epsilon ),{\mathit{z}}_1(t,\epsilon )\}$ for the initial problem (16) is defined as the minimal root of the equations

$$\frac{{\phi }_i(t_1)}{\epsilon }=\frac{{\phi }_m(t)-{\phi }_m(t_1)}{\epsilon },i=\overline{1,m}.$$

(18)

If these equations have no solutions, it meas that system (16) has a pseudoholomorphic solution at the point $\epsilon =0$ on the entire segment $[t_1,T]$. Then, a pseudoholomorphic solution of the Cauchy problem (1) on the segment $[0,T]$ serves as a set of functions $(\{{\mathit{y}}_0(t,\epsilon ),{\mathit{z}}_0(t,\epsilon )\},\{{\mathit{y}}_1(t,\epsilon ),{\mathit{z}}_1(t,\epsilon )\})$.

On the contrary, let $t_2\in (t_1,T]$ be the smallest root of Equation (18). To find the interval of existence of a pseudoholomorphic solution $\{{\mathit{y}}_2(t,\epsilon ),{\mathit{z}}_2(t,\epsilon )\}$, for the system constructed in the next step, we take the minimal root $t_3$ of the equations

$$\frac{{\phi }_i(t)-{\phi }_i(t_2)}{\epsilon }=\frac{{\phi }_i(t_1)}{\epsilon },i=\overline{1,m}.$$

Such an interval will be the segment $[t_2,t_3]$, etc. Let us set an estimate for $t_{j+1}-t_j$. According to the Lagrange formula we have

$$|{\phi }_i(t_{j+1})-{\phi }_i(t_j)|=|{\phi }_i^{\prime }({\tilde{t}}_j)|(t_{j+1}-t_j),$$

where ${\tilde{t}}_j\in (t_j,t_{j+1})$.

From here,

$$t_{j+1}-t_j=\frac{|{\phi }_i(t_1)|}{|{\phi }_i^{\prime }({\tilde{t}}_j)|}⩾\frac{\underset{i}{min}\left|{\phi }_i(t_1)\right|}{\underset{i,t}{max}\left|{\phi }_i^{\prime }(t)\right|},$$

and, as there exists a positive constant l such that

$$\underset{i,t}{max}\left|{\phi }_i^{\prime }(t)\right|⩽l,$$

then, in a finite number of steps, it is possible to reach the point $t=T$.

Thus, it is possible to construct a pseudoholomorphic solution to the Cauchy problem (1) on the entire segment $[0,T]$, which is very important for solving the boundary value problems.

5. Holomorphic Regularization of the Boundary Value Problem for the Tikhonov System $(\mathit{k}=\mathit{m}=\mathbf{2})$

Consider the boundary value problem for the Tikhonov system in the case where $k=m=2$ (the case of one fast variable is studied in [8]). As a result, we have a system

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\mathit{z}),\hfill \\ \epsilon {\displaystyle \frac{d\mathit{z}}{dt}}=\mathit{g}(t,\mathit{y},\mathit{z}),\hfill \end{array}\right.$$

(19)

in which $\mathit{f}=\{f_1,f_2\}$, $\mathit{g}=\{g_1,g_2\}$, $\mathit{y}=\{y_1,y_2\}$ and $\mathit{z}=\{z_1,z_2\}$, with boundary conditions

$$\mathit{y}(0,\epsilon )={\mathit{y}}^0,\mathit{y}(1,\epsilon )={\mathit{y}}^1,$$

(20)

where ${\mathit{y}}^0=\{y_1^0,y_2^0\}$, ${\mathit{y}}^1=\{y_1^1,y_2^1\}$.

First, let us consider the Cauchy problem

$$\mathit{y}(0,\epsilon )={\mathit{y}}^0,\mathit{z}(0,\epsilon )={\mathit{z}}^0,$$

(21)

while ${\mathit{z}}^0=\{z_1^0,z_2^0\}$.

To solve the problems (19) and (20), we just need to define ${\mathit{z}}^0$. Let us put $\epsilon =0$ in system (19) and get a degenerate system

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\mathit{z}),\hfill \\ \mathbf{0}=\mathit{g}(t,\mathit{y},\mathit{z}).\hfill \end{array}\right.$$

(22)

Let $\widehat{\mathit{Z}}=\{{\widehat{Z}}_1(t,\mathit{y}),{\widehat{Z}}_2(t,\mathit{y})\}$ be the stable root [1] of the second equation of this system, and $\widehat{\mathit{Y}}=\{{\widehat{Y}}_1(t),{\widehat{Y}}_2(t)\}$ is the only analytical solution in the circle $\left|t\right|⩽T$

$$\frac{d\mathit{y}}{dt}=\mathit{f}(t,\mathit{y},\widehat{\mathit{Z}}(t,\mathit{y}))$$

with the initial condition $\mathit{y}(0)={\mathit{y}}^0$.

An analogue of the characteristics system (8) is a system

$$\frac{d\mathit{z}}{ds}=\mathit{g}(t,\mathit{y},\mathit{z})$$

(23)

with conditions on the starting line ${\Lambda }^2$

$$z_1{|}_{s=0}=z_1^0,z_2{|}_{s=0}=\tilde{x}.$$

(24)

In system (23), $(t,\mathit{y})$ acts as a parameter and s is an independent variable. Let us assume that $g_1(t,\mathit{y},\mathit{z})\ne 0$ (see condition $4_{\alpha }$), and therefore, the phase trajectories of this system are transversal (non-tangential) to ${\Lambda }^2$. Denote the solution of the problems (23) and (24) by $\overline{\mathit{Z}}=\{{\overline{Z}}_1(t,\mathit{y},\tilde{x},s,z_1^0),{\overline{Z}}_2(t,\mathit{y},\tilde{x},s,z_1^0)\}$ and let the functions $s=S(t,\mathit{y},\mathit{z},z_1^0)$, $\tilde{x}=\tilde{X}(t,\mathit{y},\mathit{z},z_1^0)$ be the solution of an algebraic system

$$\overline{\mathit{Z}}(t,\mathit{y},\tilde{x},s,z_1^0)=\mathit{z}.$$

In accordance with those statements, $R(t)$ is the variable replacement operator $(s,\tilde{x})$ on variables $(z_1,z_2)$ and $R^{-1}(t,s)$ is its inverse.

In the investigated case, $L^{\mathit{g}}=g_1{\partial }_{z_1}+g_2{\partial }_{z_2}$, and the solution of the equation $L^{\mathit{g}}W=F(t,\mathit{z})$ with the initial condition $W{|}_{\mathit{z}\in {\Lambda }^2}$ is given by the formula

$$W(t,\mathit{z})=\underset{0}{\overset{s}{\int }}F(t,\overline{\mathit{Z}}(t,\tilde{x},s_1))d{s}_1{|}_{\genfrac{}{}{0pt}{}{s=S(t,\mathit{z})}{\tilde{x}=\tilde{X}(t,\mathit{z})}}.$$

(25)

If ${\phi }_1(t)$, ${\phi }_2(t)$ are regularizing functions, and ${\tilde{V}}^{\left[1\right]}(t,\mathit{y},\mathit{z})$, ${\tilde{V}}^{\left[2\right]}(t,\mathit{y},\mathit{z})$ are functionally independent solutions of the equation $L^{\mathit{g}}\tilde{V}=1$, turning to zero at the point $(0,{\mathit{y}}^0,{\mathit{z}}^0)$, then the four independent first integrals of system (19), determining its solution with the initial conditions (21), has the following form:

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1^{\prime }(t){\tilde{V}}^{\left[1\right]}(t,\mathit{y},\mathit{z})-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}{V}_1^{\left[1\right]}d{s}_1+\dots ={\phi }_1(t)/\epsilon ,}\hfill \\ {\displaystyle {\phi }_2^{\prime }(t){\tilde{V}}^{\left[2\right]}(t,\mathit{y},\mathit{z})-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}{V}_1^{\left[2\right]}d{s}_1+\dots ={\phi }_2(t)/\epsilon ,}\hfill \\ {\displaystyle y_1-{\widehat{Y}}_1(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t))d{s}_1+\dots =0,}\hfill \\ {\displaystyle y_2-{\widehat{Y}}_2(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_2-{\widehat{Y}}_2(t))d{s}_1+\dots =0,}\hfill \end{array}\right.$$

(26)

where $L^{\mathit{f}}={\partial }_t+f_1{\partial }_{y_1}+f_2{\partial }_{y_2}$.

We formulate a statement about the existence of a pseudoholomorphic solution to the boundary value problem.

Theorem 2.

If the conditions of Theorem 1 are met for $m=2$ and the solution $\{{\check{z}}_1(\epsilon ),{\check{z}}_2(\epsilon ),{\check{z}}_1^0(\epsilon ),{\check{z}}_2^0(\epsilon )\}$ of the system

$$\left\{\begin{array}{c}{\displaystyle {\phi }_1^{\prime }(t){\tilde{V}}^{\left[1\right]}(1,y_1^1,y_2^1,{\check{z}}_1,{\check{z}}_2,{\check{z}}_1^0,{\check{z}}_2^0)={\phi }_1(1)/\epsilon ,}\hfill \\ {\displaystyle {\phi }_2^{\prime }(t){\tilde{V}}^{\left[2\right]}(1,y_1^1,y_2^1,{\check{z}}_1,{\check{z}}_2,{\check{z}}_1^0,{\check{z}}_2^0)={\phi }_2(1)/\epsilon ,}\hfill \\ {\displaystyle R(1)\underset{0}{\overset{s}{\int }}{R}^{-1}(1,s_1)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t)){|}_{t=1}d{s}_1=(y_1^1-{\widehat{Y}}_1(1))/\epsilon ,}\hfill \\ {\displaystyle R(1)\underset{0}{\overset{s}{\int }}{R}^{-1}(1,s_1)L^{\mathit{f}}(y_2-{\widehat{Y}}_2(t)){|}_{t=1}d{s}_1=(y_2^1-{\widehat{Y}}_2(1))/\epsilon }\hfill \end{array}\right.$$

is such that the functions $S(t,\mathit{y},\mathit{z},{\check{z}}_1^0(\epsilon ))$ and $\tilde{X}(t,\mathit{y},\mathit{z},{\check{z}}_1^0(\epsilon ))$ are limited at $\epsilon \to +0$ when $(t,\mathit{y},\mathit{z})\in {\Omega }_{t\mathit{y}\mathit{z}}$, then the solution of the boundary value problems (19) and (20), defined by system (26), is pseudoholomorphic at the point $\epsilon =0$.

Proof of Theorem 2.

It follows from the fact that from the limitation of the function $S(t,\mathit{y},\mathit{z},{\check{z}}_1^0(\epsilon ))$ follows the convergence of series (26), and from the limitation of the function $\tilde{X}(t,\mathit{y},\mathit{z},{\check{z}}_1^0(\epsilon ))$ follows the asymptotic stability of the solution of this boundary value problem obtained using the implicit function theorem. □

Example 1.

Let us investigate the boundary value problem $(m=k=2)$ on the segment $[0,1]$

$$\begin{array}{c}\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\mathit{z}),\hfill \\ \epsilon {\displaystyle \frac{d{z}_1}{dt}}=h_1^2(t,\mathit{y})-z_1^2,\hfill \\ \epsilon {\displaystyle \frac{d{z}_2}{dt}}=h_2^2(t,\mathit{y})-z_2^2,\hfill \end{array}\right.\hfill \\ \mathit{y}(0,\epsilon )={\mathit{y}}^0,\mathit{y}(1,\epsilon )={\mathit{y}}^1,\hfill \end{array}$$

(27)

where $\mathit{f}=\{f_1,f_2\}$, $\mathit{y}=\{y_1,y_2\}$, $\mathit{z}=\{z_1,z_2\}$, ${\mathit{y}}^0=\{y_1^0,y_2^0\}$, ${\mathit{y}}^1=\{y_1^1,y_2^1\}$, $\mathit{h}=\{h_1,h_2\}$.

We will assume that $f_1$, $f_2$, $h_1$, $h_2$ are integer functions of their arguments and let $h_1(t,\mathit{y})>0$, $h_2(t,\mathit{y})>0$ in some domain ${\omega }_{t\mathit{y}}\subset {\mathbb{R}}^3$ containing the starting point $(0,{\mathit{y}}^0)$. Then, as it is known, $\widehat{\mathit{Z}}=\{{\widehat{Z}}_1(t,\mathit{y}),{\widehat{Z}}_2(t,\mathit{y})\}$, where ${\widehat{Z}}_1(t,\mathit{y})=h_1(t,\mathit{y})$, ${\widehat{Z}}_2(t,\mathit{y})=h_2(t,\mathit{y})$, is a stable solution of the system of the last two equations (if we put $\epsilon =0$). Next, we denote by $\widehat{\mathit{Y}}(t)=\{{\widehat{Y}}_1(t),{\widehat{Y}}_2(t)\}$ the analytical solution of the initial problem in the circle $\left|t\right|\le 1$

$$\frac{d\mathit{y}}{dt}=\mathit{f}(t,\mathit{y},\mathit{h}),\mathit{y}(0,\epsilon )={\mathit{y}}^0.$$

(28)

First, let us set the Cauchy problem: find a solution to system (27) with initial conditions

$$\mathit{y}(0,\epsilon )={\mathit{y}}^0,\mathit{z}(0,\epsilon )={\mathit{z}}^0,$$

(29)

where ${\mathit{z}}^0=\{z_1^0,z_2^0\}$.

We compose a system of equations of characteristics corresponding to a linear partial differential operator of the first order:

$$L^{\mathit{h}}=(h_1^2(t,\mathit{y})-z_1^2){\partial }_{z_1}+(h_2^2(t,\mathit{y})-z_2^2){\partial }_{z_2}.$$

It has the following form:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{z}_1}{ds}}=h_1^2(t,\mathit{y})-z_1^2,\hfill \\ {\displaystyle \frac{d{z}_2}{ds}}=h_2^2(t,\mathit{y})-z_2^2.\hfill \end{array}\right.$$

(30)

As the “initial surface” ${\Lambda }^2$, we choose the straight line $z_1=z_1^0$, which is legitimate, since $h_1\ne 0$ and, therefore, the characteristics are transversal to it. Let us put the initial conditions on the line ${\Lambda }^2$:

$$z_1{|}_{s=0}=z_1^0,z_2{|}_{s=0}=\tilde{x},$$

(31)

where $\tilde{x}$ is the coordinate on the initial ”surface”. We have a solution to problems (30) and (31):

$$\left\{\begin{array}{c}{\overline{Z}}_1(t,\mathit{y},s,\tilde{x})={\displaystyle \frac{h_1((h_1+z_1^0)e^{s{h}_1}-(h_1-z_1^0)e^{-s{h}_1})}{(h_1+z_1^0)e^{s{h}_1}+(h_1-z_1^0)e^{-s{h}_1}}},\hfill \\ {\overline{Z}}_2(t,\mathit{y},s,\tilde{x})={\displaystyle \frac{h_2((h_2+\tilde{x})e^{s{h}_2}-(h_2-\tilde{x})e^{-s{h}_2})}{(h_2+\tilde{x})e^{s{h}_2}+(h_2-\tilde{x})e^{-s{h}_2}}}.\hfill \end{array}\right.$$

(32)

From the solution of (32), the initial problem of the system of equations of characteristics, we find

$$\left\{\begin{array}{c}S(t,\mathit{y},\mathit{z})={\displaystyle \frac{1}{2h_1}}ln\left({\displaystyle \frac{h_1+z_1}{h_1-z_1}}·{\displaystyle \frac{h_1-z_1^0}{h_1+z_1^0}}\right),\hfill \\ \tilde{X}(t,\mathit{y},\mathit{z})={\displaystyle \frac{h_2\left(\frac{h_2+z_2}{h_2-z_2}·\frac{h_1+z_1^0}{h_1+z_1}·\frac{h_1-z_1}{h_1-z_1^0}-1\right)}{1+\frac{h_2+z_2}{h_1+z_1}·\frac{h_1+z_1^0}{h_1+z_1}·\frac{h_1-z_1}{h_1-z_1^0}}}.\hfill \end{array}\right.$$

It is not difficult to see that the functions

$${\tilde{V}}^{\left[1\right]}=\frac{1}{2h_1(t,\mathit{y})}ln\left|\frac{h_1(t,\mathit{y})+z_1}{h_1(t,\mathit{y})-z_1}\right|-\frac{1}{2h_1(0,{\mathit{y}}^0)}ln\left|\frac{h_1(0,\mathit{y})+z_1^0}{h_2(0,\mathit{y})-z_1^0}\right|,$$

$${\tilde{V}}^{\left[2\right]}=\frac{1}{2h_2(t,\mathit{y})}ln\left|\frac{h_2(t,\mathit{y})+z_2}{h_2(t,\mathit{y})-z_2}\right|-\frac{1}{2h_2(0,{\mathit{y}}^0)}ln\left|\frac{h_2(0,\mathit{y})+z_2^0}{h_1(0,\mathit{y})-z_2^0}\right|$$

are independent solutions of the equation $L^{\mathit{h}}V=1$, turning to zero at the point ($t=0$, $y_1=y_1^0$, $y_2=y_2^0$, $z_1=z_1^0$, $z_2=z_2^0$).

From the form $V^{\left[1\right]}$ and $V^{\left[2\right]}$ it follows that we can take the following functions as regularizing functions

$${\phi }_1(t)=-2\underset{0}{\overset{t}{\int }}{h}_1(\tau ,{\widehat{Y}}_1(\tau ),{\widehat{Y}}_2(\tau ))d\tau ,{\phi }_2(t)=-2\underset{0}{\overset{t}{\int }}{h}_2(\tau ,{\widehat{Y}}_1(\tau ),{\widehat{Y}}_2(\tau ))d\tau .$$

So, to define ${\mathit{Z}}_0(t,e^{{\phi }_1(t)/\epsilon },e^{{\phi }_2(t)/\epsilon })$ we have a system of equations (relative to $z_1$, $z_2$)

$$\left\{\begin{array}{c}{\displaystyle ln\left|\frac{h_1(t,\widehat{\mathit{Y}}(t))+z_1}{h_1(t,\widehat{\mathit{Y}}(t))-z_1}\right|-\frac{h_1(t,\widehat{\mathit{Y}}(t))}{h_1(0,{\mathit{y}}^0)}ln\left|\frac{h_1(0,{\mathit{y}}^0)+z_1^0}{h_1(0,{\mathit{y}}^0)-z_1^0}\right|=}\hfill \\ {\displaystyle =(-2/\epsilon )\underset{0}{\overset{t}{\int }}{h}_1(\tau ,\widehat{\mathit{Y}}(\tau ))d\tau ,}\hfill \\ {\displaystyle ln\left|\frac{h_2(t,\widehat{\mathit{Y}}(t))+z_2}{h_2(t,\widehat{\mathit{Y}}(t))-z_2}\right|-\frac{h_2(t,\widehat{\mathit{Y}}(t))}{h_2(0,{\mathit{y}}^0)}ln\left|\frac{h_2(0,{\mathit{y}}^0)+z_2^0}{h_1(0,{\mathit{y}}^0)-z_2^0}\right|=}\hfill \\ {\displaystyle =(-2/\epsilon )\underset{0}{\overset{t}{\int }}{h}_2(\tau ,\widehat{\mathit{Y}}(\tau ))d\tau .}\hfill \end{array}\right.$$

(33)

From system (26), taking into account that

$$L^{\mathit{f}}={\partial }_t+f_1(t,\mathit{y},\mathit{z}){\partial }_{y_1}+f_2(t,\mathit{y},\mathit{z}){\partial }_{y_2},$$

it is possible to find a first-order approximation by ε for the slow variables $\{y_1,y_2\}$:

$$\begin{array}{c}{\displaystyle y_{1,1}(t,\epsilon )={\widehat{Y}}_1(t)+}\hfill \\ {\displaystyle +\epsilon \underset{0}{\overset{S(t,\mathit{y},\mathit{z})}{\int }}\left(-{\partial }_t{\widehat{Y}}_1+f_1(t,\mathit{y},\overline{\mathit{Z}}(t,\mathit{y},s_1,\tilde{x})){|}_{\tilde{x}=\tilde{X}(t,\mathit{y},\mathit{z})}\right)d{s}_1{|}_{\genfrac{}{}{0pt}{}{\mathit{y}=\widehat{\mathit{Y}}(t)}{\mathit{z}={\mathit{Z}}_0(t,e^{\mathit{\phi }(t)/\epsilon })}},}\hfill \\ {\displaystyle y_{2,1}(t,\epsilon )={\widehat{Y}}_2(t)+}\hfill \\ {\displaystyle +\epsilon \underset{0}{\overset{S(t,\mathit{y},\mathit{z})}{\int }}\left(-{\partial }_t{\widehat{Y}}_2+f_2(t,\mathit{y},\overline{\mathit{Z}}(t,\mathit{y},s_1,\tilde{x})){|}_{\tilde{x}=\tilde{X}(t,\mathit{y},\mathit{z})}\right)d{s}_1{|}_{\genfrac{}{}{0pt}{}{\mathit{y}=\widehat{\mathit{Y}}(t)}{\mathit{z}={\mathit{Z}}_0(t,e^{\mathit{\phi }(t)/\epsilon })}}.}\hfill \end{array}$$

(34)

To determine the zero approximations of $z_1^0$ and $z_2^0$, we set $t=1$ in Equation (33), and in the equalities in (34), we put $t=1$, $y_{1,1}=y_1^1$, $y_{2,1}=y_2^1$ and solve the resulting system of four equations with respect to $z_1^0$, $z_2^0$, $z_1$, $z_2$. We substitute the found $z_1^0$, $z_2^0$ into equality (34) and find the first (by ε) approximations for $y_1(t,\epsilon )$ and $y_2(t,\epsilon )$ for slow variables of the boundary value problem.

Example 2.

Let us consider on the segment $[0,1]$ the Cauchy problem for a Tikhonov system in which slow variables enter in a nonlinear way, and fast variables enter in a linear way:

$$\begin{array}{c}\left\{\begin{array}{c}{\displaystyle \frac{d{\mathit{y}}_1}{dt}}={\mathit{f}}_1(t,\mathit{y},\mathit{z}),\hfill \\ {\displaystyle \frac{d{\mathit{y}}_2}{dt}}={\mathit{f}}_2(t,\mathit{y},\mathit{z}),\hfill \\ \epsilon {\displaystyle \frac{d{z}_1}{dt}}=-(e^t+1)z_1+2e^{-t}{z}_2,\hfill \\ \epsilon {\displaystyle \frac{d{z}_2}{dt}}=e^{2t}{z}_1-3z_2,\hfill \end{array}\right.\hfill \\ y_1(0,\epsilon )=y_1^0,y_2(0,\epsilon )=0,z_1(0,\epsilon )=1,z_2(0,\epsilon )=0.\hfill \end{array}$$

(35)

In accordance with the approach for solving Tikhonov systems outlined in [1], we set $\epsilon =0$ and write a degenerate system

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{y}}{dt}}=\mathit{f}(t,\mathit{y},\mathit{z}),\hfill \\ \mathbf{0}=\mathit{A}(t)\mathit{z},\hfill \end{array}\right.$$

where ${\displaystyle \mathit{A}(t)=\left(\begin{array}{cc}-(e^t+1)& 2e^{-t}\\ e^{2t}& -3\end{array}\right)}$.

Since the eigenvalues ${\lambda }_1=-1$, ${\lambda }_2=-3-e^t$ of the matrix $\mathbf{A}(t)$ are negative for all $t\in \mathbb{R}$, then all the conditions of Tikhonov’s theorem on the limit transition are met. Further, suppose that $\{{\widehat{Y}}_1(t),{\widehat{Y}}_2(t)\}$ is an analytical solution of a system of differential equations in a circle $\left|t\right|\le 1$

$$\frac{d\mathit{y}}{dt}=\mathit{f}(t,\mathit{y},0),$$

with initial conditions $y_1(0)=y_1^0$, $y_2(0)=0$. In accordance with the method of holomorphic regularization, we make up the first integral equation

$$\epsilon L^{\mathit{f}}U+L^{\mathit{g}}U=0,$$

(36)

where

$$L^{\mathit{f}}={\partial }_t+f_1(t,\mathit{y},\mathit{z}){\partial }_{y_1}+f_2(t,\mathit{y},\mathit{z}){\partial }_{y_2},L^{\mathit{g}}=(-(e^t+1)z_1+2e^{-t}{z}_2){\partial }_{z_1}+(e^{2t}{z}_1-3z_2){\partial }_{z_2}$$

are first-order linear partial differential operators.

We will construct the solution of Equation (36) in the form of the regular series by degrees of a small parameter, considering the operator $L^{\mathit{f}}$ subordinate to the operator $L^{\mathit{g}}$:

$$U(t,\mathit{y},\mathit{z},\epsilon )=U_0(t,\mathit{y},\mathit{z})+\epsilon U_1(t,\mathit{y},\mathit{z})+\dots +U_n(t,\mathit{y},\mathit{z})+\dots .$$

(37)

We have the following series of problems for determining the coefficients of series (37):

$$\left\{\begin{array}{c}{L}^{\mathit{g}}{U}_0=0,\hfill \\ L^{\mathit{g}}{U}_1=-L^{\mathit{f}}{U}_0,\hfill \\ \dots \dots \dots \dots \dots \hfill \\ L^{\mathit{g}}{U}_n=-L^{\mathit{f}}{U}_{n-1},\hfill \\ \dots \dots \dots \dots \dots \hfill \end{array}\right.$$

(38)

It is easy to see that we can take the functions $U_0^{\left[1\right]}=-t$, $U_0^{\left[2\right]}=-3t-e^t+1$, $U_0^{\left[3\right]}=y_1-{\widehat{Y}}_1(t)$, $U_0^{\left[4\right]}=y_2-{\widehat{Y}}_2(t)$ as solutions to the first equation of this series.

To solve the remaining equations of series (38), we write down a system of equations of characteristics:

$$\frac{d\mathit{z}}{ds}=\mathbf{A}(t)\mathit{z},$$

(39)

and for the “initial surface” on the plane $(z_1,z_2)$, let us choose a straight line Λ: $z_1-2e^{-2t}-1=0$ (since t enters the system as a parameter, then at each moment of time it is its own straight line). Let $\tilde{x}=z_2$ be the coordinate on Λ. Then,

$$\left(\begin{array}{c}{\overline{Z}}_1(t,\tilde{x},s)\\ {\overline{Z}}_2(t,\tilde{x},s)\end{array}\right)=\frac{(2e^{-t}+1)\tilde{x}+e^t}{e^t+2}\left(\begin{array}{c}2e^{-t}\\ e^t\end{array}\right)e^{-s}+\frac{e^t}{e^t+2}\left(\begin{array}{c}-1\\ e^t\end{array}\right)e^{-(3+e^t)s}$$

(40)

is a solution of system (39), such that $\overline{Z}(t,\tilde{x},0)\in \Lambda$. From vector equality (40) we find s and $\tilde{x}$:

$$s=\frac{1}{e^t+3}ln(z_1-2z_2{e}^{-2t})\equiv S(t,\mathit{z}),\tilde{x}=\frac{z_1{e}^t+z_2-e^t}{2e^{-t}+1}\equiv \tilde{X}(t,\mathit{z}).$$

(41)

Note that the functions included in equalities (40) and (41) do not depend on $\mathit{y}$. This is due to the fact that the equations for fast variables in the original system contain neither $y_1$ nor $y_2$.

Let us return to the series of Equation (38). You can directly verify that with the previously selected $U_0^{\left[1\right]}$ and $U_0^{\left[2\right]}$, the functions $U_1^{\left[1\right]}=-ln(e^t{z}_1+z_2)$ and $U_1^{\left[2\right]}=-ln(e^{2t}{z}_1-2z_2)$ satisfy the second equation of this series.

We have four functionally independent first integrals of system (35):

$$\left\{\begin{array}{c}{\displaystyle ln(e^t{z}_1+z_2)-\epsilon U_2^{\left[1\right]}(t,\mathit{z})-\dots =-t/\epsilon ,}\hfill \\ {\displaystyle ln(e^{2t}{z}_1-2z_2)-\epsilon U_2^{\left[2\right]}(t,\mathit{z})-\dots =-(3t+e^t-1)/\epsilon ,}\hfill \\ {\displaystyle y_1-{\widehat{Y}}_1(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_1-{\widehat{Y}}_1(t))d{s}_1+\dots =0,}\hfill \\ {\displaystyle y_2-{\widehat{Y}}_2(t)-\epsilon R(t)\underset{0}{\overset{s}{\int }}{R}^{-1}(t,s_1)L^{\mathit{f}}(y_2-{\widehat{Y}}_2(t))d{s}_1+\dots =0,.}\hfill \end{array}\right.$$

(42)

Here, as usual, $R(t)$ is a transition operator from variables $(\tilde{x},s)$ to variables $(z_1,z_2)$, $R^{-1}(t,s)$ is a transition operator from variables $(z_1,z_2)$, to variables $(\tilde{x},s)$, and for $U_2^{\left[1\right]}$ and $U_2^{\left[2\right]}$ the following formulas hold:

$$\left\{\begin{array}{c}{\displaystyle U_2^{\left[1\right]}(t,\mathit{z})=e^t\underset{0}{\overset{S(t,\mathit{z})}{\int }}\frac{{\overline{Z}}_1(t,\tilde{X}(t,\mathit{z}),s_1)d{s}_1}{e^t{\overline{Z}}_1(t,\tilde{X}(t,\mathit{z}),s_1)+{\overline{Z}}_2(t,\tilde{X}(t,\mathit{z}),s_1)},}\hfill \\ {\displaystyle U_2^{\left[2\right]}(t,\mathit{z})=2e^{2t}\underset{0}{\overset{S(t,\mathit{z})}{\int }}\frac{{\overline{Z}}_1(t,\tilde{X}(t,\mathit{z}),s_1)d{s}_1}{e^{2t}{\overline{Z}}_1(t,\tilde{X}(t,\mathit{z}),s_1)-2{\overline{Z}}_2(t,\tilde{X}(t,\mathit{z}),s_1)}.}\hfill \end{array}\right.$$

The solution $(\mathit{y},\mathit{z})$ of the Cauchy problem is determined from system (42) using the implicit function theorem. In particular,

$${\displaystyle {\mathit{Z}}_0=\left(\begin{array}{c}{\displaystyle \frac{2e^{-t/\epsilon }+e^{(-3t-e^t+1)/\epsilon }}{2e^t+e^{2t}}}\\ {\displaystyle \frac{e^{t-t/\epsilon }-e^{(-3t-e^t+1)/\epsilon }}{e^t+2}}\end{array}\right),}$$

$${\mathit{y}}_{1,1}=\widehat{\mathit{Y}}(t)+\epsilon \underset{0}{\overset{S(t,\mathit{z})}{\int }}(-{\partial }_t\widehat{\mathit{Y}}(t)+\mathit{f}(t,\widehat{\mathit{Y}}(t),\mathit{Z}(t,\tilde{X}(t,{\mathit{Z}}_0),s_1))d{s}_1.$$

6. Conclusions

Thus, Tikhonov systems are a class of singularly perturbed problems that have pseudoholomorphic solutions. In our opinion, this is primarily due to the asymptotic stability of the attached systems. This is the determining factor in the emergence of both the boundary layer and the actual limit transition. The results presented in this article allow us to introduce the possibility of constructing an analytical theory of singular perturbations as a section of differential equations [13,14]. It should also be noted that the algebraic approach is actively used in the holomorphic regularization method [9].