AN EXPANSION FORMULA CONNECTED WITH A BOUNDARY-VALUE PROBLEM WITH A COMPLEX PARAMETER FOR SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS

G. B. MUSTAFAEV

The application of M. L. Rasulov’s residue method to the solution of mixed problems for a system of equations is connected with the question of expanding an arbitrary vector function of a certain class in a series in the residues of the solutions of the corresponding spectral problem (a boundary-value problem with a complex parameter) [1].

In the works of M. L. Rasulov [1], [2] this method was applied to the solution of mixed problems whose boundary conditions do not contain higher derivatives. Accordingly, the validity of the main expansion formula for an arbitrary vector function was proved by him for spectral problems that likewise do not contain higher derivatives.

The application of this method to the solution of mixed problems whose boundary conditions contain higher derivatives of the unknown functions requires an investigation of the validity of the main expansion formulas for arbitrary vector functions (from a certain class) in series in the residues of the solutions of spectral problems whose boundary conditions contain higher derivatives of the unknown vector function.

In this connection, in the present paper we prove the validity of the main expansion formula (30), connected with a spectral problem for a system of first-order equations whose boundary conditions contain derivatives of the unknown vector function*.

This question was investigated by A. I. Vagabov for the special case when the coefficients of the system depend linearly on the parameter, while the boundary conditions do not depend on it [3].

In addition, in the present paper a certain generalization is obtained of the concept of regularity of boundary conditions [1, 4, 3].

Consider the problem of finding a solution of the system of equations

\[ \frac{dy_i}{dx}-\sum_{j=1}^{n} a_{i,j}(x,\lambda)y_j=h_i(x),\quad (i=1,2,\ldots,n) \tag{1} \]

under the boundary conditions

\[ L_i(y)\equiv \sum_{j=1}^{n}\left\{\alpha_{i,j}(\lambda)\frac{dy_j(a,\lambda)}{dx} +\beta_{i,j}(\lambda)\frac{dy_j(b,\lambda)}{dx}+\right. \]

* Under certain smoothness conditions on the coefficients of the system and on the right-hand sides, formula (30) holds even if the boundary conditions (2) contain derivatives of \(y_i\) of arbitrary order.

\[ + \alpha_{i,j}^{(0)}(\lambda)y_j(a,\lambda)+\beta_{i,j}^{(0)}(\lambda)y_j(b,\lambda)\biggr\}=0, \tag{2} \]

where

\[ a_{i,j}(x,\lambda)=\lambda a_{i,j}(x)+\sum_{\nu=0}^{q}\lambda^{-\nu}a_{i,j,\nu}(x), \tag{3} \]

\(\alpha_{i,j}(\lambda)\), \(\beta_{i,j}(\lambda)\), \(\alpha_{i,j}^{(0)}(\lambda)\), \(\beta_{i,j}^{(0)}(\lambda)\) are polynomials in \(\lambda\). In what follows, matrices composed of \(a_{i,j}(x,\lambda)\), \(a_{i,j}(x)\), \(a_{i,j,\nu}(x)\), \(\alpha_{i,j}(\lambda)\), \(\beta_{i,j}(\lambda)\), \(\alpha_{i,j}^{(0)}(\lambda)\), \(\beta_{i,j}^{(0)}(\lambda)\) will be denoted by the corresponding letters without the indices \(i,j\), and the columns of the \(y_i\) by \(y\).

The following restrictions are imposed on the coefficients of equations (1) and the boundary conditions (2):

\(1^\circ\). On the interval \([a,b]\) the functions

\[ \frac{d^p a_{i,j}(x)}{dx^p},\quad \frac{d^{p-1}a_{i,j,0}(x)}{dx^{p-1}},\ldots,\ a_{i,j,p-1}(x) \]

(\(p>1\)) are continuously differentiable once, and the functions \(a_{i,j,\nu}(x)\) (\(\nu>p-1\)) are continuous.

\(2^\circ\). For \(x\in[a,b]\) the roots \(\vartheta_1(x),\ldots,\vartheta_n(x)\) of the characteristic equation

\[ \det(a(x)-\vartheta e)=0 \tag{4} \]

are distinct and different from zero; their arguments and the arguments of their differences do not depend on \(x\).

\(3^\circ\). For \(\lambda\in D(R)\) (where \(D(R)\) denotes the domain of values of \(\lambda\) satisfying the inequality \(|\lambda|>R\)) the rank of the matrix

\[ A(\lambda)= \begin{pmatrix} \alpha_{1,1}(\lambda)\ldots \alpha_{1,n}(\lambda)& \beta_{1,1}(\lambda)\ldots \beta_{1,n}(\lambda)& \alpha_{1,1}^{(0)}(\lambda)\ldots \alpha_{1,n}^{(0)}(\lambda)& \beta_{1,1}^{(0)}(\lambda)\ldots \beta_{1,n}^{(0)}(\lambda)\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \alpha_{n,1}(\lambda)\ldots \alpha_{n,n}(\lambda)& \beta_{n,1}(\lambda)\ldots \beta_{n,n}(\lambda)& \alpha_{n,1}^{(0)}(\lambda)\ldots \alpha_{n,n}^{(0)}(\lambda)& \beta_{n,1}^{(0)}(\lambda)\ldots \beta_{n,n}^{(0)}(\lambda) \end{pmatrix} \]

is equal to \(n\).

The solution of problem (1), (2) is found by the usual method in the form

\[ y_i(x,\lambda,h)=\int_a^b \sum_{j=1}^{n}G_{i,j}(x,\xi,\lambda)h_j(\xi)\,d\xi+ \frac{\Delta_i(x,\lambda)}{\Delta(\lambda)}, \tag{5} \]

where

\[ G_{i,j}(x,\xi,\lambda)=\frac{\Delta_{i,j}(x,\xi,\lambda)}{\Delta(\lambda)}, \tag{6} \]

\[ \Delta_{i,j}(x,\xi,\lambda)= \begin{vmatrix} g_{i,j}(x,\xi,\lambda) & \psi_{i,1}(x,\lambda)\ldots \psi_{i,n}(x,\lambda)\\ L_1(g_{i,j}(x,\xi,\lambda))_x & u_{1,1}(\lambda)\ldots u_{1,n}(\lambda)\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ L_n(g_{i,j}(x,\xi,\lambda))_x & u_{n,1}(\lambda)\ldots u_{n,n}(\lambda) \end{vmatrix}, \]

\[ \Delta_i(x,\lambda)= \begin{vmatrix} 0 & \psi_{i,1}(x,\lambda) & \cdots & \psi_{i,n}(x,\lambda)\\ \displaystyle \sum_{l=1}^{n}\bigl(\alpha_{1,l}(\lambda)h_l(a)+\beta_{1,l}(\lambda)h_l(b)\bigr) & u_{1,1}(\lambda) & \cdots & u_{1,n}(\lambda)\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \displaystyle \sum_{l=1}^{n}\bigl(\alpha_{n,l}(\lambda)h_l(a)+\beta_{n,l}(\lambda)h_l(b)\bigr) & u_{n,1}(\lambda) & \cdots & u_{n,n}(\lambda) \end{vmatrix}, \]

\[ \Delta(\lambda)= \left| \begin{array}{ccc} u_{1,1}(\lambda)&\ldots&u_{1,n}(\lambda)\\ \ldots&\ldots&\ldots\\ u_{n,1}(\lambda)&\ldots&u_{n,n}(\lambda) \end{array} \right|, \]

\[ u_{i,j}(\lambda)=L_i\bigl(y_j(x,\lambda)\bigr), \]

\[ g_{i,j}(x,\xi,\lambda)=\pm \frac12\sum_{s=1}^n y_{i,s}(x,\lambda)z_{j,s}(\xi,\lambda) \tag{7} \]

\[ +\ \text{for } a\le \xi \le x\le b,\qquad -\ \text{for } a\le x\le \xi\le b \]

\[ z_{j,s}(\xi,\lambda)=\frac{W_{j,s}(\xi,\lambda)}{W(\xi,\lambda)} . \]

Here \(y_{i,s}(x,\lambda)\) \((i,s=1,2,\ldots,n)\) is a fundamental system of particular solutions of the corresponding homogeneous system (1), \(W(\xi,\lambda)\) is the Wronskian determinant of these solutions, \(W_{j,s}(\xi,\lambda)\) is the algebraic complement of the element \((j,s)\) in \(W(\xi,\lambda)\), and \(L_i(g_{i,j}(x,\xi,\lambda))_x\) denotes the application of the operator \(L_i\) to the \(j\)-th column of the matrix \(g(x,\xi,\lambda)\) as to a function of \(x\).

To prove formula (30), it is necessary to obtain an asymptotic representation for the solution (5).

For this purpose, consider the straight lines defined by the equations

\[ \arg\lambda=\pm \frac{\pi}{2}-\psi_{k,s}, \]

where

\[ \psi_{k,s}=\arg(\vartheta_k(x)-\vartheta_s(x))\quad k\ne s\quad (k,s=1,2,\ldots,n). \]

These straight lines divide the \(\lambda\)-plane into a finite number of sectors \((\Sigma_j)\). Then, with a suitable numbering of the \(\vartheta(x)\) roots of the characteristic equation (4), in the sector \((\Sigma_j)\) the inequalities

\[ \operatorname{Re}\lambda\vartheta_{k_1}(x)\le \operatorname{Re}\lambda\vartheta_{k_2}(x)\le \ldots \le \operatorname{Re}\lambda\vartheta_{k_n}(x), \]

hold; or, denoting

\[ \vartheta_{k_p}(x)=\varphi_p(x)\quad (p=1,2,\ldots,n), \]

we shall have

\[ \operatorname{Re}\lambda\varphi_1(x)\le \operatorname{Re}\lambda\varphi_2(x)\le \ldots \le \operatorname{Re}\lambda\varphi_n(x). \tag{8} \]

Then, according to the theorem of Ya. D. Tamarkin (see [4]), there exists a fundamental matrix of solutions of the homogeneous system corresponding to system (1), whose elements in the sector \((\Sigma_j)\) have the asymptotic representation

\[ y_{i,s}(x,\lambda)= e^{\lambda\int_a^x \varphi_s(z)\,dz} \left\{ \sum_{\nu=0}^{p}\lambda^{-\nu}y_{i,s,\nu}(x) +\frac{E_{is}(x,\lambda)}{\lambda^{p+1}} \right\} \quad \text{for } |\lambda|\ge R, \tag{9} \]

where \(y_{i,s,\nu}(x)\) are \((p-1-\nu)\)-times continuously differentiable functions\(^*\);

\[ \text{* Here } p \text{ is a natural number. Its choice is determined by the regularity condition of the boundary conditions (2), given below.} \]

\(E_{i,s}(x,\lambda)\) are continuous in \(x\) on the interval \([a,b]\) and bounded in \(\lambda\) in the sector \((\Sigma_j)\) for \(|\lambda|\ge R\) and \(x\in [a,b]\).

Using the homogeneous system corresponding to system (1), we find

\[ \frac{dy_{i,s}(x,\lambda)}{dx} = \lambda e^{\lambda\int_a^x \varphi_s(z)\,dz} \left\{ \sum_{\nu=0}^{p}\lambda^{-\nu}\eta_{i,s,\nu}(x) + \frac{\bar E_{i,s}(x,\lambda)}{\lambda^{p+1}} \right\}, \tag{10} \]

where

\[ \eta_{i,s,\nu}(x) = \sum_{k+m=\nu}\sum_{l=1}^{n} a_{i,l,k-1}(x)y_{l,s,m}(x), \]

\[ a_{i,l,-1}(x)=a_{i,l}(x). \]

Substituting (9), (10) into (7), for \(u_{i,j}(\lambda)\) we obtain the following asymptotic representation:

\[ \begin{aligned} u_{i,j}(\lambda) ={}& \sum_{l=1}^{n} \left\{ \lambda\alpha_{i,l}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}\eta_{l,j,\nu}(a) + \alpha_{i,l}^{(0)}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}y_{l,j,\nu}(a) + \frac{E_{i,l}(a,\lambda)}{\lambda^{p+1}} \right\} \\ &+ \sum_{l=1}^{n} \left\{ \lambda\beta_{i,l}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}\eta_{l,j,\nu}(b) + \beta_{i,l}^{(0)}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}y_{l,j,\nu}(b) + \frac{F_{il}(b,\lambda)}{\lambda^{p+1}} \right\} e^{\lambda\int_a^b \varphi_j(z)\,dz}. \end{aligned} \tag{11} \]

We introduce the following notation:

\[ \widetilde P(\lambda)=P(\lambda)+\frac{E(\lambda)}{\lambda}, \tag{12} \]

where

\[ \left|\frac{E(\lambda)}{P(\lambda)}\right|\le C \quad\text{as }|\lambda|\to\infty, \]
where \(C\) is a constant,

\[ \int_a^b \varphi_j(z)\,dz=\omega_j, \]

\[ \sum_{l=1}^{n} \left\{ \lambda\alpha_{i,l}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}\eta_{l,j,\nu}(a) + \alpha_{i,l}^{(0)}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}y_{l,j,\nu}(a) \right\} = A_{i,j}(\lambda), \]

\[ \sum_{l=1}^{n} \left\{ \lambda\beta_{i,l}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}\eta_{l,j,\nu}(b) + \beta_{il}^{(0)}(\lambda)\sum_{\nu=0}^{p}\lambda^{-\nu}y_{l,j,\nu}(b) \right\} = B_{i,j}(\lambda). \tag{13} \]

Thus, we have:

\[ u_{i,j}(\lambda)=\widetilde A_{i,j}(\lambda)+\widetilde B_{i,j}(\lambda)e^{\lambda\omega_j}, \]

\[ \Delta(\lambda)= \left| \begin{array}{cccc} \widetilde A_{1,1}(\lambda)+\widetilde B_{1,1}(\lambda)e^{\lambda\omega_1} & \cdots & \widetilde A_{1,n}(\lambda)+\widetilde B_{1,n}(\lambda)e^{\lambda\omega_n} \\ \cdots & \cdots & \cdots \\ \widetilde A_{n,1}(\lambda)+\widetilde B_{n,1}(\lambda)e^{\lambda\omega_1} & \cdots & \widetilde A_{n,n}(\lambda)+\widetilde B_{n,n}(\lambda)e^{\lambda\omega_n} \end{array} \right|. \]

For the study of the asymptotic behavior of the determinant \(\Delta(\lambda)\) and of the asymptotic distribution of its zeros, one should simplify the asymptotic representation of the determinant \(\Delta(\lambda)\).

For this purpose, consider the set of points of the \(\lambda\)-plane satisfying the condition \(\operatorname{Re}\lambda\omega_k=0\) \((k=1,2,\ldots,n)\). According to condition \(2^0\), for \(\lambda\ne 0\) we obtain

\[ \arg\lambda=\pm\frac{\pi}{2}-\gamma_k,\quad \text{where } \gamma_k=\arg\omega_k\quad (k=1,2,\ldots,n). \]

The equations obtained determine straight lines passing through the origin. If each such straight line is considered as split into two rays with origin at the coordinate origin, then all these equations determine \(2\mu\) \((2\mu\le 2n)\) distinct rays \(d_1,d_2,\ldots,d_{2\mu}\). We denote by \(\frac{\pi}{2}-\gamma_j\) the argument of the ray \(d_j\); the rays \(d_j\) will be regarded as arranged so that

\[ 0\le \gamma_1<\gamma_2<\cdots<\gamma_{2\mu}<2\pi. \]

Suppose further that we have a second set of rays \(d'_j\) \((j=1,2,\ldots,2\mu)\), chosen arbitrarily, distinct from the rays \(d_j\) and going in the sequence

\[ d'_1,\ d_1,\ d'_2,\ d_2,\ d'_3,\ d_3,\ldots,\ d'_{2\mu},\ d_{2\mu},\ d'_1. \]

The rays \(d'_j\) \((j=1,2,\ldots,2\mu)\) divide the entire \(\lambda\)-plane into \(2\mu\) sectors \((T_1),(T_2),\ldots,(T_{2\mu})\). Then the boundaries of the sectors \((T_j)\) and \((\Sigma_s)\) divide the entire \(\lambda\)-plane into sectors, which we shall denote by \((R_j)\). Each sector \((R_j)\) lies simultaneously in one of the sectors \((T_j)\) and in one of the sectors \((\Sigma_s)\). Consider one of the sectors \((R_j)\) with fixed number \(j\).

Let the sector \((R_j)\) lie in some sector \((\Sigma_s)\) in which, under a suitable numbering of the roots of the characteristic equation, the inequalities (8) are satisfied.

Assume that \(\omega_{1j},\omega_{2j},\ldots,\omega_{\nu_j,j}\) are those of the numbers \(\omega_1,\omega_2,\ldots,\omega_n\) which lie on the straight line passing through the origin and making an angle \(\gamma_j^*\) with the real axis. By condition \(2^0\), the arguments of the numbers \(\omega_{kj}\) do not depend on \(x\). Then, obviously, \(\arg\omega_{kj}\) is equal either to \(\gamma_j\) or to \(\pi+\gamma_j\).

In the first case we have \(\omega_{kj}=|\omega_{kj}|e^{\sqrt{-1}\gamma_j}\), and in the second case \(\omega_{kj}=-|\omega_{kj}|e^{\sqrt{-1}\gamma_j}\). Then these numbers admit the following representations:

\[ \omega_{kj}=r_{kj}e^{\sqrt{-1}\gamma_j}\quad (k=1,2,\ldots,\nu_j), \]

where \(r_{kj}\) are real numbers, numbered in increasing order:

\[ r_{1j}<r_{2j}<\cdots<r_{s_j j}<0<r_{s_j+1,j}<\cdots<r_{\nu_j j}. \tag{14} \]

Here, if all numbers \(r_{kj}>0\), we put \(s_j=0\); if all numbers \(r_{kj}\) are negative, we put \(s_j=\nu_j\).

If from the set of numbers \(\omega_1,\omega_2,\ldots,\omega_n\) we exclude all the numbers \(\omega_{kj}\), then the remaining numbers can be divided into two groups \(\omega_k^{(1)}\) and \(\omega_k^{(2)}\).

* If there are no such numbers, then near the ray \(d_j\) the characteristic determinant \(\Delta(\lambda)\) may have no zeros for large \(\lambda\).

To the first group \(\omega_k^{(1)}\) we assign those of the numbers \(\omega_k\) for which, in the sector \((R_j)\),

\[ \operatorname{Re}\lambda\omega_k^{(1)}\to -\infty \quad \text{as } |\lambda|\to\infty . \]

To the second group \(\omega_k^{(2)}\) we assign the numbers \(\omega_k\) for which, in the sector \((R_j)\),

\[ \operatorname{Re}\lambda\omega_k^{(2)}\to +\infty \quad \text{as } |\lambda|\to\infty . \]

Thus, in the sector \((R_j)\) we shall have:

\[ \operatorname{Re}\lambda\omega_k^{(1)}\to -\infty,\quad k=1,2,\ldots,x_j; \]

as \(|\lambda|\to\infty\),

\[ \operatorname{Re}\lambda\omega_k^{(2)}\to +\infty,\quad k=x_j+\nu_j+1,\ldots,n \quad \text{as } |\lambda|\to\infty . \]

Obviously, under the substitution

\[ z=\lambda e^{\sqrt{-1}\gamma_j} \]

the ray \(d_j\) of the \(\lambda\)-plane is mapped into the positive part of the imaginary axis of the \(z\)-plane, and the sector \((R_j)\) into the sector \((R)\) of the \(z\)-plane containing the positive part of the imaginary axis. Indeed, from the equality \(\arg z=\arg\lambda+\gamma_j\), for \(\lambda\) lying on the ray \(d_j\), we obtain

\[ \arg z=\frac{\pi}{2}-\gamma_j+\gamma_j=\frac{\pi}{2}. \]

Thus the numbers \(\omega_k\) can be renumbered so that in the sector \((R_j)\) the inequalities

\[ \operatorname{Re}\lambda\omega_1^{(1)}\ll\ldots\ll \operatorname{Re}\lambda\omega_{x_j}^{(1)} \ll \operatorname{Re}\lambda\omega_{1j}\ll\ldots\ll \operatorname{Re}\lambda\omega_{s_jj}\ll 0\ll \]

\[ \ll \operatorname{Re}\lambda\omega_{s_j+1,j}\ll\ldots\ll \operatorname{Re}\lambda\omega_{\nu_jj}\ll \operatorname{Re}\lambda\omega_{x_j+\nu_j+1}^{(2)} \ll\ldots\ll \operatorname{Re}\lambda\omega_n^{(2)}. \tag{15} \]

Then the characteristic determinant \(\Delta(\lambda)\) can be written in the form

\[ \Delta(\lambda)= \left| \begin{array}{cccccc} \tilde A_{1,1}(\lambda)+\tilde B_{1,1}(\lambda)e^{\lambda\omega_1^{(1)}}& \ldots& \tilde A_{1,x_j}(\lambda)+\tilde B_{1,x_j}(\lambda)e^{\lambda\omega_{x_j}^{(1)}}& & & \\[2mm] & & & \tilde A_{1,x_j+1}(\lambda)+\tilde B_{1,x_j+1}(\lambda)e^{\lambda\omega_{1j}}& \ldots& \\[2mm] \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\[2mm] \tilde A_{n,1}(\lambda)+\tilde B_{n,1}(\lambda)e^{\lambda\omega_1^{(1)}}& \ldots& \tilde A_{n,x_j}(\lambda)+\tilde B_{n,x_j}(\lambda)e^{\lambda\omega_{x_j}^{(1)}}& & & \\[2mm] & & & \tilde A_{n,x_j+1}(\lambda)+\tilde B_{n,x_j+1}(\lambda)e^{\lambda\omega_{1j}}& \ldots& \\[2mm] \ldots& \tilde A_{1,x_j+s_j}(\lambda)+\tilde B_{1,x_j+s_j}(\lambda)e^{\lambda\omega_{s_jj}}& \tilde A_{1,x_j+s_j+1}(\lambda)+\tilde B_{1,x_j+s_j+1}(\lambda)e^{\lambda\omega_{s_j+1,j}}& \ldots& & \\[2mm] \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\[2mm] \ldots& \tilde A_{n,x_j+s_j}(\lambda)+\tilde B_{n,x_j+s_j}(\lambda)e^{\lambda\omega_{s_jj}}& \tilde A_{n,x_j+s_j+1}(\lambda)+\tilde B_{n,x_j+s_j+1}(\lambda)e^{\lambda\omega_{s_j+1,j}}& \ldots& & \\[2mm] \ldots& \tilde A_{1,x_j+\nu_j}(\lambda)+\tilde B_{1,x_j+\nu_j}(\lambda)e^{\lambda\omega_{\nu_jj}}& \tilde A_{1,x_j+\nu_j+1}(\lambda)+ \tilde B_{1,x_j+\nu_j+1}(\lambda)e^{\lambda\omega_{x_j+\nu_j+1}^{(2)}}& \ldots& \tilde A_{1,n}(\lambda)+\tilde B_{1,n}(\lambda)e^{\lambda\omega_n^{(2)}}\\[2mm] \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\[2mm] \ldots& \tilde A_{n,x_j+\nu_j}(\lambda)+\tilde B_{n,x_j+\nu_j}(\lambda)e^{\lambda\omega_{\nu_jj}}& \tilde A_{n,x_j+\nu_j+1}(\lambda)+ \tilde B_{n,x_j+\nu_j+1}(\lambda)e^{\lambda\omega_{x_j+\nu_j+1}^{(2)}}& \ldots& \tilde A_{n,n}(\lambda)+\tilde B_{n,n}(\lambda)e^{\lambda\omega_n^{(2)}} \end{array} \right|. \]

Taking \(\exp \lambda\omega_k^{(2)}\) outside the determinant sign and representing the resulting determinant as a sum of determinants, we obtain

\[ \Delta(\lambda)=e^{\lambda\omega} \left\{ \widetilde M_{1,j}(\lambda)e^{\lambda\sum_{k=1}^{s_j}\omega_{k,j}} +\cdots+ \widetilde M_{\sigma_j,j}(\lambda)e^{\lambda\sum_{k=s_j+1}^{\nu_j}\omega_{k,j}} \right\}, \tag{16} \]

where

\[ \omega=\sum_{k=\varkappa_j+\nu_j+1}^{n}\omega_k^{(2)}, \]

\[ M_{1,j}(\lambda)= \left| \begin{array}{cccccccc} A_{1,1}(\lambda)&\cdots&A_{1,\varkappa_j}(\lambda)&B_{1,\varkappa_j+1}(\lambda)&\cdots&B_{1,\varkappa_j+s_j}(\lambda)&A_{1,\varkappa_j+s_j+1}(\lambda)&\cdots\\ &&&\cdots&A_{1,\varkappa_j+\nu_j}(\lambda)&B_{1,\varkappa_j+\nu_j+1}(\lambda)&\cdots&B_{1,n}(\lambda)\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ A_{n,1}(\lambda)&\cdots&A_{n,\varkappa_j}(\lambda)&B_{n,\varkappa_j+1}(\lambda)&\cdots&B_{n,\varkappa_j+s_j}(\lambda)&A_{n,\varkappa_j+s_j+1}(\lambda)&\cdots\\ &&&\cdots&A_{n,\varkappa_j+\nu_j}(\lambda)&B_{n,\varkappa_j+\nu_j+1}(\lambda)&\cdots&B_{n,n}(\lambda) \end{array} \right|, \]

\[ M_{\sigma_j,j}(\lambda)= \left| \begin{array}{ccccc} A_{1,1}(\lambda)&\cdots&A_{1,\varkappa_j+s_j}(\lambda)&B_{1,\varkappa_j+s_j+1}(\lambda)&\cdots B_{1,n}(\lambda)\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ A_{n,1}(\lambda)&\cdots&A_{n,\varkappa_j+s_j}(\lambda)&B_{n,\varkappa_j+s_j+1}(\lambda)&\cdots B_{n,n}(\lambda) \end{array} \right|. \]

Consider an exponential polynomial of the form

\[ H(z)=M_1(z)e^{m_1z}+\cdots+M_\sigma(z)e^{m_\sigma z}, \tag{a} \]

where \(M_s(z)=z^{n_s}\left\{d_s+\dfrac{E_s(z)}{z}\right\}\), \(E_s(z)\) are bounded for large \(\lambda\), \(m_s\), \(n_s\), \(d_s\) are constants.

A term of this exponential polynomial will be called dominant if the real part of its exponent is not smaller than the real parts of the exponents of the other terms. If, however, the real part of its exponent is not greater than the real parts of the exponents of the other terms, then this term will be called subordinate.

We have obtained \(\Delta(\lambda)=e^{\lambda\omega}\Delta_0(\lambda)\), where

\[ \Delta_0(\lambda)= \widetilde M_{1,j}(\lambda)e^{\lambda\sum_{k=1}^{s_j}\omega_{k,j}} +\cdots+ \widetilde M_{\sigma_j,j}(\lambda)e^{\lambda\sum_{k=s_j+1}^{\nu_j}\omega_{k,j}} \tag{17} \]

and where

\[ M_{s,j}(\lambda)=\lambda^{n_{s,j}} \left\{ d_{s,j}+\frac{E_{s,j}(\lambda)}{\lambda} \right\}, \]

\(E_{s,j}(\lambda)\) are bounded for large \(\lambda\), and \(n_{s,j}\), \(d_{s,j}\) are constants.

As is seen, \(\Delta_0(\lambda)\) is an exponential polynomial of type \((a)\).

Definition. If, in the sectors \((R_j)\), in the expansion of the characteristic determinant \(\Delta(\lambda)\) into an exponential polynomial of the form \((a)\), when passing from one sector \((R_j)\) to another the exponential function of a subordinate term passes into the exponential function of a dominant term, and conversely, then the boundary conditions will be called regular\(^*\), if

\[ n_{1,j}=n_{\sigma_j,j},\qquad n_{s,j}<n_{1,j}\quad \text{for } s\ne 1,\sigma_j\ (j=1,2,\ldots,2\mu). \]

\(^*\) This will occur, for example, in the case when the ratio of the moduli of the roots of the characteristic equation does not depend on \(x\).

If, however, in passing from one sector \((R_j)\) to another in the expansion of the characteristic determinant in a sum of type \((a)\), the exponential function of the highest term passes into the exponential function of a term that is not the lowest, then the boundary conditions will be called regular if \(n_{s,j}=n_{1,j}\) for \(s=2,3,\ldots,\sigma_j\) \((j=1,2,\ldots,2\mu)\).

In what follows, as an additional restriction, we shall assume that the boundary conditions are regular in our sense.

Without dwelling here on a detailed investigation (see [1]), we indicate that in this case the determinant \(\Delta(\lambda)\) has a countable number of zeros, which can be distributed into \(2\mu\) groups; the values of the \(j\)-th group lie in a strip \((D_h^j)\) of finite width, parallel to the ray \(d_j\) and containing it.

If from the \(\lambda\)-plane one removes the interiors of small circles of radius \(\delta\) with centers at the zeros of \(\Delta(\lambda)\) and at the point \(\lambda=0\), then in the remaining part the inequality holds

\[ \left|\lambda^{-p} e^{-\lambda \omega_k^{(2)}} \Delta(\lambda)\right| \geqslant K_\delta, \tag{17'} \]

where \(K_\delta\) is a positive number depending on \(\delta\). Choose a sequence of closed contours \(T_\nu\) \((\nu=1,2,\ldots)\), situated outside some \(\delta\)-neighborhood of these zeros, such that the distance of the nearest point of the contour \(T_\nu\) \((\nu=1,2,\ldots)\) from the origin tends to \(\infty\).*)

It is now necessary to obtain asymptotic representations for

\[ z_{j,s}(\xi,\lambda)=\frac{W_{j,s}(\xi,\lambda)}{W(\xi,\lambda)} \]

in the sector \((R_j)\).

Taking into account the absence in \(W_{j,s}(\xi,\lambda)\) of the column with the multiplier \(\exp\left(\lambda\int_a^\xi \varphi_s(z)\,dz\right)\), it is easy to obtain the asymptotic representation

\[ z_{j,s}(\xi,\lambda) = e^{-\lambda\int_a^\xi \varphi_s(z)\,dz} \left| \zeta_{j,s,0}(\xi)+\frac{E_{j,s}(\xi,\lambda)}{\lambda} \right|, \tag{18} \]

where

\[ \zeta_{j,s,0}(\xi)=\frac{D_{j,s,0}(\xi)}{D_0(\xi)},\qquad D_0(\xi)= \left| \begin{array}{cccc} y_{1,1,0}(\xi) & \cdots & y_{1,n,0}(\xi)\\ \cdots & \cdots & \cdots & \cdots\\ y_{n,1,0}(\xi) & \cdots & y_{n,n,0}(\xi) \end{array} \right|, \]

and \(D_{j,s,0}(\xi)\) is the algebraic cofactor of the element \(y_{j,s,0}(\xi)\) in the determinant \(D_0(\xi)\); \(\zeta_{j,s,0}(\xi)\) is continuously differentiable once on the interval \([a,b]\).

In order to obtain a suitable asymptotic expression for the solution (5), form the determinant \(\Delta_{l,j}(x,\xi,\lambda)\), multiplying columns with numbers \(2,3,\ldots,n+1\), respectively, by

\[ \frac{1}{2}z_{j,1}(\xi,\lambda),\ldots, \frac{1}{2}z_{j,\tau_j}(\xi,\lambda),\, -\frac{1}{2}z_{j,\tau_j+1}(\xi,\lambda),\ldots \]

\[ \ldots,-\frac{1}{2}z_{j,n}(\xi,\lambda) \]

*) The possibility of choosing such contours \(T_\nu\) follows from Theorem 7 of [1]. From the choice of the contours \(T_\nu\) it is obvious that on these contours inequality (17) is satisfied uniformly with respect to \(\nu\).

and adding it to the first column, where \(\tau_j = \chi_j + s_j\) is the number determined in inequalities (15).

The determinant obtained in this way will be denoted by
\[ \Delta_{i,j,0}(x,\xi,\lambda). \]
The elements of the first column of this determinant will be denoted, respectively, by
\[ g_{i,j,0}(x,\xi,\lambda),\qquad L_{i,0}(g_{i,j}(x,\xi,\lambda))_x . \]
As is seen from (10), we have:
\[ g_{i,j,0}(x,\xi,\lambda)= \begin{cases} \displaystyle \sum_{s=1}^{\tau_j} y_{i,s}(x,\lambda)z_{j,s}(\xi,\lambda), & \text{for } a\le \xi \le x \le b,\\[1.2em] \displaystyle -\sum_{s=\tau_j+1}^{n} y_{i,s}(x,\lambda)z_{j,s}(\xi,\lambda), & \text{for } a\le x\le \xi \le b, \end{cases} \tag{19} \]
\[ \begin{aligned} L_{i,0}(g_{i,j}(x,\xi,\lambda))_x ={}&-\sum_{s=\tau_j+1}^{n} z_{j,s}(\xi,\lambda)\times\\ &\times\left\{\sum_{l=1}^{n}\left(\alpha_{i,l}(\lambda)\frac{dy_{l,s}(a,\lambda)}{dx} +\alpha_{i,l}^{(0)}(\lambda)y_{l,s}(a,\lambda)\right)\right\}\\ &+\sum_{s=1}^{\tau_j} z_{j,s}(\xi,\lambda) \left\{\sum_{l=1}^{n}\left(\beta_{i,l}(\lambda)\frac{dy_{l,s}(b,\lambda)}{dx} +\beta_{i,l}^{(0)}(\lambda)y_{l,s}(b,\lambda)\right)\right\}. \end{aligned} \tag{20} \]

Expanding the determinant \(\Delta_{i,j,0}(x,\xi,\lambda)\) by the elements of the first row, we obtain
\[ \Delta_{i,j,0}(x,\xi,\lambda) =g_{i,j,0}(x,\xi,\lambda)\Delta(\lambda) +\sum_{k=1}^{n}(-1)^k y_{i,k}(x,\lambda)\Delta_{i,j,0,1,k+1}(\xi,\lambda), \tag{21} \]
where \(\Delta_{i,j,0,1,k+1}(\xi,\lambda)\) is the cofactor of the element \((1,k+1)\) in the determinant \(\Delta_{i,j,0}(x,\xi,\lambda)\), which is obtained from \(\Delta(\lambda)\) if in it the column with number \((k)\) is replaced by the column
\[ \begin{pmatrix} L_{1,0}(g_{i,j}(x,\xi,\lambda))_x\\ \cdots\\ L_{n,0}(g_{i,j}(x,\xi,\lambda))_x \end{pmatrix}. \]

Further, substituting \(L_{i,0}(g_{i,j}(x,\xi,\lambda))_x\) from formula (20) into the expression for the determinant \(\Delta_{i,j,0,1,k+1}(\xi,\lambda)\) and expanding the determinant thus obtained into a sum of such determinants, outside the signs of which one may factor out \(z_{j,s}(\xi,\lambda)\) as a common factor of the elements of the column with number \((k)\), we obtain
\[ \Delta_{i,j,0,1,k+1}(\xi,\lambda) = \sum_{s=\tau_j+1}^{n} z_{j,s}(\xi,\lambda)a_{k,s}(\lambda) -\sum_{s=1}^{\tau_j} z_{j,s}(\xi,\lambda)b_{k,s}(\lambda), \tag{22} \]
where \(a_{k,s}(\lambda)\), \(b_{k,s}(\lambda)\) are determinants obtained from \(\Delta(\lambda)\) by deleting the column with number \((k)\) and inserting, respectively, columns consisting of the elements

$$ \sum_{l=1}^{n}\left\{\alpha_{i,l}(\lambda)\frac{dy_{l,s}(a,\lambda)}{dx} +\alpha_{i,l}^{(0)}(\lambda)y_{l,s}(a,\lambda)\right\}, $$

$$ \sum_{l=1}^{n}\left\{\beta_{i,l}(\lambda)\frac{dy_{l,s}(b,\lambda)}{dx} +\beta_{i,l}^{(0)}(\lambda)y_{l,s}(b,\lambda)\right\} $$

$$ (i=1,2,\ldots,n). $$

Taking into account (19), (21), and (22), for the first term of formula (5) we obtain

$$ \begin{aligned} \bar y_i(x,\lambda) &=\int_a^b \sum_{j=1}^n \frac{\Delta_{i,j,0}(x,\xi,\lambda)}{\Delta(\lambda)}h_j(\xi)\,d\xi \\ &=\int_a^x \sum_{j=1}^n\sum_{s=1}^{\tau_j} y_{i,s}(x,\lambda)z_{j,s}(\xi,\lambda)h_j(\xi)\,d\xi \\ &\quad-\int_x^b \sum_{j=1}^n\sum_{s=\tau_j+1}^{n} y_{i,s}(x,\lambda)z_{j,s}(\xi,\lambda)h_j(\xi)\,d\xi \\ &\quad+\int_a^b \sum_{j=1}^n\sum_{k=1}^n\sum_{s=\tau_j+1}^{n} (-1)^k y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda) \frac{a_{k,s}(\lambda)}{\Delta(\lambda)}h_j(\xi)\,d\xi \\ &\quad-\int_a^b \sum_{j=1}^n\sum_{k=1}^n\sum_{s=1}^{\tau_j} (-1)^k y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda) \frac{b_{k,s}(\lambda)}{\Delta(\lambda)}h_j(\xi)\,d\xi . \tag{23} \end{aligned} $$

Let us consider the growth of the integrand functions on the right-hand side of (23) in the sector \((R_j)\) outside the \(\delta\)-neighborhood of the spectrum.

According to the asymptotic formulas (9), (18) and the inequalities (15), the product \(y_{i,s}(x,\lambda)z_{j,s}(\xi,\lambda)\) is bounded in the sector \((R_j)\). As for the ratios

\[ \frac{a_{k,s}(\lambda)}{\Delta(\lambda)}, \qquad \frac{b_{k,s}(\lambda)}{\Delta(\lambda)}, \]

we shall proceed with them in the following way.

If one takes \(\exp \lambda\omega\) outside the determinant signs \(a_{k,s}(\lambda)\), \(b_{k,s}(\lambda)\), where

\[ \omega=\sum_{k=\tau_j+1}^{n}\omega_k, \]

uses the notation (12) and the formulas (9), (10), (11), then \(a_{k,s}(\lambda)\), \(b_{k,s}(\lambda)\) can be written in the following form:

$$ a_{k,s}(\lambda)= \begin{cases} a_{k,s,0}(\lambda)e^{\lambda\omega}, & \text{for } k=1,2,\ldots,\tau_j,\\ a_{k,s,0}(\lambda)e^{\lambda\omega-\lambda\omega_k}, & \text{for } k=\tau_j+1,\ldots,n, \end{cases} $$

$$ \begin{aligned} b_{k,s}(\lambda)&= \begin{cases} b_{k,s,0}(\lambda)e^{\lambda\omega+\lambda\omega_s}, & \text{for } k=1,2,\ldots,\tau_j,\\ b_{k,s,0}(\lambda)e^{\lambda\omega+\lambda\omega_s-\lambda\omega_k}, & \text{for } k=\tau_j+1,\ldots,n. \end{cases} \tag{24} \end{aligned} $$

Now consider the behavior of the integrand functions \(y_{i,k}(x,\lambda)\times z_{j,s}(\xi,\lambda)\dfrac{a_{k,s}(\lambda)}{\Delta(\lambda)}\), \(y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda)\dfrac{b_{k,s}(\lambda)}{\Delta(\lambda)}\) in the sector \((R_j)\) outside the \(\delta\)-neighborhood of the spectrum.

\[ \begin{aligned} 1.\quad y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda)\, \frac{a_{k,s}(\lambda)}{\Delta(\lambda)} &= e^{\lambda\int_a^x \varphi_k(z)\,dz-\lambda\int_a^\xi \varphi_s(z)\,dz} \bigl(\widetilde y_{i,k,0}(x)\times {}\\ &\qquad {}\times \widetilde \xi_{j,s,0}(\xi)\bigr)E_{k,s}^{(1)}(\lambda) \quad \text{for } k\leq \tau_j,\ s\geq \tau_j+1 . \end{aligned} \]

\[ \begin{aligned} 2.\quad y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda)\, \frac{a_{k,s}(\lambda)}{\Delta(\lambda)} &= e^{-\lambda\int_x^b \varphi_k(z)\,dz-\lambda\int_a^\xi \varphi_s(z)\,dz} \bigl(\widetilde y_{i,k,0}(x)\times {}\\ &\qquad {}\times \widetilde \xi_{j,s,0}(\xi)\bigr)E_{k,s}^{(1)}(\lambda) \quad \text{for } k\geq \tau_j+1,\ s\geq \tau_j+1 . \end{aligned} \]

\[ \begin{aligned} 3.\quad y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda)\, \frac{b_{k,s}(\lambda)}{\Delta(\lambda)} &= e^{\lambda\int_a^x \varphi_k(z)\,dz+\lambda\int_\xi^b \varphi_s(z)\,dz} \bigl(\widetilde y_{i,k,0}(x)\times {}\\ &\qquad {}\times \widetilde \xi_{j,s,0}(\xi)\bigr)E_{k,s}^{(2)}(\lambda) \quad \text{for } k\leq \tau_j,\ s\leq \tau_j . \end{aligned} \]

\[ \begin{aligned} 4.\quad y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda)\, \frac{b_{k,s}(\lambda)}{\Delta(\lambda)} &= e^{-\lambda\int_x^b \varphi_k(z)\,dz+\lambda\int_\xi^b \varphi_s(z)\,dz} \bigl(y_{i,k,0}(x)\times {}\\ &\qquad {}\times \xi_{j,s,0}(\xi)\bigr)E_{k,s}^{(2)}(\lambda) \end{aligned} \tag{25} \]

for \(k\geq \tau_j+1,\ s\leq \tau_j\), where \(E_{k,s}^{(i)}(\lambda)\) \((i=1,2)\) are bounded in the sector \((R_j)\) outside the \(\delta\)-neighborhood of the spectrum.

Now compute the limit of the integrals over the above-mentioned contours \(T_\nu\) as \(\nu\to\infty\) from the solution (23). As is seen from (19) and (23),

\[ \int_{T_\nu} \overline y_i(x,\lambda)\,d\lambda = J_{1,\nu}(x)+J_{2,\nu}(x), \]

where

\[ J_{1,\nu}(x) = \int_{T_\nu} d\lambda \int_a^b \sum_{j=1}^n g_{i,j,0}(x,\xi,\lambda)h_j(\xi)\,d\xi, \]

\[ J_{2,\nu}(x) = \int_{T_\nu} d\lambda \int_a^b \sum_{j=1}^n \Omega_{i,j}(x,\xi,\lambda)h_j(\xi)\,d\xi, \]

\[ \Omega_{i,j}(x,\xi,\lambda) = \sum_{k=1}^n (-1)^k \left\{ \sum_{s=\tau_j+1}^n y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda) \frac{a_{k,s}(\lambda)}{\Delta(\lambda)} - \sum_{s=1}^{\tau_j} y_{i,k}(x,\lambda)z_{j,s}(\xi,\lambda) \frac{b_{k,s}(\lambda)}{\Delta(\lambda)} \right\}. \tag{26} \]

Taking formulas (25), (26) into account, with the help of known lemmas (see [1], 6, 7), one can show that the limit of \(J_{2,\nu}(x)\) as \(\nu\to\infty\) is equal to zero.

According to condition \(2^\circ\), the roots of the characteristic equation may be represented in the form

\[ \varphi_k(x)=c_k q_k(x) \quad \text{for } x\in [a,b], \]

where \(c_k\) are, generally speaking, complex numbers, and \(q_k(x)>0\) for \(x\in [a,b]\).

For convenience we introduce the notation:

\[ \int_a^b \varphi_k(z)\,dz = \int_a^b c_k q_k(z)\,dz = c_k x_{0,k}, \]

\[ \int_a^x \varphi_k(z)\,dz = \int_a^x c_k q_k(z)\,dz = c_k x_k, \]

\[ \int_a^\xi \varphi_k(z)\,dz = \int_a^\xi c_k q_k(z)\,dz = c_k \xi_k . \]

Consider the integral

\[ J_{1,\nu}(x) = \int_{T_\nu} d\lambda \int_a^b \sum_{j=1}^n g_{i,j,0}(x,\xi,\lambda)h_j(\xi)\,d\xi = \]

\[ = \sum_{(R_j)} \int_{T_\nu\cap R_j} d\lambda \sum_{j=1}^n \left\{ \int_a^x \sum_{s=1}^{\tau_j} \widetilde{(y_{i,s,0}(x)\xi_{j,s,0}(\xi))} e^{\lambda c_s(x_s-\xi_s)} h_j(\xi)\,d\xi \right. \]

\[ \left. - \int_x^b \sum_{s=\tau_j+1}^{n} \widetilde{(y_{i,s,0}(x)\xi_{j,s,0}(\xi))} e^{\lambda c_s(x_s-\xi_s)} h_j(\xi)\,d\xi \right\}. \tag{27} \]

Denote by \(J_{1,\nu}^{(0)}(x)\) the expression obtained from (27) by omitting the tilde sign. Then one can prove that

\[ J_{1,\nu}(x)-J_{1,\nu}^{(0)}(x)\to 0 \]

uniformly as \(\nu\to\infty\).

Exactly as in [1], after simple transformations we obtain:

\[ J_{1,\nu}^{(0)}(x) = -2\sqrt{-1} \sum_{s=1}^n \sum_{j=1}^n y_{i,s,0}(x) \int_a^b \xi_{j,s,0}(\xi)\, \frac{\sin\bigl(r_\nu |c_k| (x_s-\xi_s)\bigr)} {\varphi_s(\xi)(x_s-\xi_s)} h_j(\xi)\,d\xi, \tag{28} \]

where \(r_\nu\) is the distance from the origin to the nearest point of intersection of \(T_\nu\) with the imaginary axis. According to a well-known formula from the theory of the Fourier transform, the integral on the right-hand side of (28), for every function

\[ h_j(\xi)\frac{\xi_{j,s,0}(\xi)}{\varphi_s(\xi)} \]

from \(L_2(a,b)\), tends to

\[ \pi\,\frac{\xi_{j,s,0}(x)}{\varphi_s(x)}h_j(x) \]

as \(\nu\to\infty\) in the sense of the \(L_2\) metric:

\[ J_{1,\nu}(x) \Longrightarrow -2\pi\sqrt{-1} \sum_{j=1}^n \sum_{s=1}^n y_{i,s,0}(x) \frac{\xi_{j,s,0}(x)}{\varphi_s(x)} h_j(x). \]

We rewrite these relations in matrix form:

\[ -\frac{1}{2\pi\sqrt{-1}}\lim_{\nu\to\infty}\int_{T_\nu} d\lambda \int_a^b G(x,\xi,\lambda)h(\xi)\,d\xi = \]

\[ = y_0(x) \begin{pmatrix} \dfrac{1}{\varphi_1(x)} & 0 & \cdots & 0\\ \cdot & \cdot & \cdots & \cdot\\ 0 & 0 & \cdots & \dfrac{1}{\varphi_n(x)} \end{pmatrix} \xi_0(x)h(x). \]

Since \(y_0(x)\xi_0(x)=E\), \((E\) is the identity matrix\()\),

\[ y_0(x) \begin{pmatrix} \varphi_1(x) & 0 & \cdots & 0\\ \cdot & \cdot & \cdots & \cdot\\ 0 & 0 & \cdots & \varphi_n(x) \end{pmatrix} = a(x)y_0(x), \]

we obtain

\[ -\frac{1}{2\pi\sqrt{-1}}\lim_{\nu\to\infty}\int_{T_\nu} d\lambda \int_a^b G(x,\xi,\lambda)h(\xi)\,d\xi = (a(x))^{-1}h(x). \]

Now let us compute the limit of the integral

\[ \frac{-1}{2\pi\sqrt{-1}}\int_{T_\nu} \frac{\Delta_i(x,\lambda)}{\Delta(\lambda)}\,d\lambda . \]

Expanding the determinant \(\Delta_i(x,\lambda)\) with respect to the elements of the first row, we shall have

\[ \Delta_i(x,\lambda)=\sum_{k=1}^n (-1)^k y_{i,k}(x,\lambda)\Delta_{1,k}(\lambda). \tag{29} \]

Substituting the asymptotic expression (11) into (29) and taking \(e^{\lambda\omega}\) outside the determinant sign (where

\[ \omega=\sum_{k=\tau_j+1}^n \omega_k \]

), we obtain:

1) for \(k\le \tau_j\),

\[ \Delta_{1,k}(\lambda)=e^{\lambda\omega}\Delta_{1,k,0}(\lambda), \]

2) for \(k\ge \tau_j+1\),

\[ \Delta_{1,k}(\lambda)=e^{\lambda\omega-\lambda\omega_k}\Delta_{1,k,0}(\lambda). \]

Expand the determinant \(\Delta_{1,k,0}(\lambda)\) with respect to the elements of the \(k\)-th column:

\[ \Delta_{1,k,0}(\lambda)= \sum_{r=1}^n \left( \sum_{l=1}^n \alpha_{r,l}(\lambda)h_l(a) + \sum_{l=1}^n \beta_{r,l}(\lambda)h_l(b) \right) \Delta_{1,r,k,0}(\lambda), \]

where \(\Delta_{1,r,k,0}(\lambda)\) are the algebraic complements of the elements \(u_{r,k}(\lambda)\) in \(\Delta(\lambda)\).

Taking out from all columns of the determinant \(\Delta_{1,r,k,0}(\lambda)\) the highest power of \(\lambda\), and taking into account the absence in it of the \(k\)-th column, we obtain

\[ \Delta_{1,r,k,0}(\lambda)=\lambda^{p-p_k}(\widetilde d_{r,k}). \]

Therefore

\[ \frac{\Delta_i(x,\lambda)}{\Delta(\lambda)} = \sum_{k=1}^{\tau_j} E_{i,k}^{(1)}(x,\lambda) e^{\lambda\int_a^x \tau_k(z)\,dz} + \sum_{k=\tau_j+1}^{n} E_{i,k}^{(2)}(x,\lambda) e^{-\lambda\int_x^b \varphi_k(z)\,dz}, \]

where \(E_{i,k}^{(1)}(x,\lambda), E_{i,k}^{(2)}(x,\lambda)\to 0\) as \(\nu\to\infty\) or \(|\lambda|\to\infty\).

According to a known lemma (see [4], p. 170),

\[ \lim_{\nu\to\infty} \int_{T_\nu} \frac{\Delta_i(x,\lambda)}{\Delta(\lambda)}\,d\lambda =0. \]

Thus we have proved

Theorem. Under conditions \(1^\circ, 2^\circ, 3^\circ\), and if the boundary conditions (2) are regular, there exists a sequence of expanding closed contours \(T_\nu\) \((\nu=1,2,\ldots)\) such that for every vector-function \(h(x)\) from \(L_2(a,b)\) the expansion formula holds

\[ -\frac{1}{2\pi\sqrt{-1}}\, \lim_{\nu\to\infty} \int_{T_\nu} Y(x,\lambda,h)\,d\lambda = \]

\[ = -\frac{1}{2\pi\sqrt{-1}}\, \lim_{\nu\to\infty} \int_{T_\nu} d\lambda \left\{ \int_a^b G(x,\xi,\lambda)h(\xi)\,d\xi + \frac{\Delta_i(x,\lambda)}{\Delta(\lambda)} \right\} = \]

\[ = (a(x))^{-1}h(x) \tag{30} \]

in the sense of \(L_2(a,b)^{*)}\).

References

1. Rasулов M. L. The method of the contour integral. Moscow, “Nauka” Publishing House, 1964.
2. Rasулов M. L. Mat. sb., 30 (72), 1952, pp. 509–528.
3. Vагabov A. I. Solution of one-dimensional mixed problems for a hyperbolic system of first order. Scientific Notes of the Azerbaijan State University, Series of Physical and Mathematical Sciences, No. 3, 1963.
4. Tamarkina Ya. D. On certain general problems in the theory of ordinary linear differential equations and expansions of arbitrary functions in series. Petrograd, 1917.

Received by the editors
March 15, 1965

Azerbaijan State University
named after S. M. Kirov

\({}^{*)}\) Under certain smoothness restrictions on \(h(x)\), it is easy to show that formula 30 holds in the pointwise sense.