UDC 517.946.42

ON AN ASYMPTOTIC REPRESENTATION OF THE SOLUTION OF A MIXED PROBLEM FOR A DIFFERENTIAL EQUATION OF HYPERBOLIC TYPE WITH A SMALL PARAMETER

I. I. MARKUSH

One of the most widely used methods for the approximate solution of differential equations is the method of a small parameter. The simplest scheme for applying this method consists in taking, as an approximate solution of a differential equation containing a small parameter, the solution of the equation in which the parameter is set equal to zero. Such an equation is, as a rule, simpler and in most cases can be integrated, whereas the full equation cannot be integrated. In this connection the question necessarily arises of the extent to which such an approximate solution reflects the properties of the true solution. This question, which reduces to the investigation of the dependence of the solution of the full equation on the parameter, contains considerable difficulties, since it must be answered without integrating the equation.

In the case where the small parameter occurs at the highest derivatives (the parameter enters the equation singularly), deep results have recently been obtained by Gradshtein, Vasil’eva, Vishik and Lyusternik (see [1] and the bibliography), M. Zlamal [16], and others.

The case of continuous dependence of an equation on a small parameter is characterized by the fact that, if this parameter is sufficiently small, then the solution of the full equation is represented with arbitrarily high accuracy by the solution of the simplified equation (in which the parameter is set equal to zero).

In this case (when the small parameter enters the equation continuously), outstanding results have been obtained by Poincaré, Krylov, Bogolyubov, Shtokalo, Mitropol’skii, Erugin, Feshchenko, and others. However, almost all investigations by the above-mentioned scholars concerned linear and nonlinear ordinary differential equations. Partial differential equations have been studied comparatively little.

§ 1. In the present work we shall consider the question of an asymptotic representation of the solution of a mixed problem for a linear partial differential equation of hyperbolic type whose coefficients depend on a small parameter.

Consider a differential equation of the form

\[ u_{tt}=A(\tau,x,\varepsilon)u_{xx}+B(\tau,x,\varepsilon)u_x- \]
\[ -C(\tau,x,\varepsilon)u+\varepsilon\sum_{j=1}^{N}F_j(\tau,x,\varepsilon)e^{i\theta_j(t,\varepsilon)} \tag{1} \]

with initial and boundary conditions of the form

\[ u(0,x)=\varphi(x), \qquad u_t(0,x)=\psi(x), \tag{2} \]

\[ a u_x(t,0)+b u(t,0)=0, \qquad c u_x(t,\pi)+d u(t,\pi)=0, \tag{3} \]

where \(\varepsilon\) is a real small parameter, with \(\tau=\varepsilon t\) (slow time), \(0\leq \tau \leq L\) (\(L\) is a constant), \(0\leq x\leq \pi\);

\[ \frac{d\theta_j}{dt}=k_j(\tau)>0 \]

(\(k_j(\tau)\) are real slowly varying functions). We shall assume that the functions \(A(\tau,x,\varepsilon)\), \(B(\tau,x,\varepsilon)\), \(C(\tau,x,\varepsilon)\), \(F_j(\tau,x,\varepsilon)\), differentiable with respect to \(\tau\), admit representations in the form of asymptotic series

\[ A(\tau,x,\varepsilon)=A_0(x)+\sum_{s=1}^{\infty}\varepsilon^s A_s(\tau,x), \]

\[ B(\tau,x,\varepsilon)=B_0(x)+\sum_{s=1}^{\infty}\varepsilon^s B_s(\tau,x), \]

\[ C(\tau,x,\varepsilon)=C_0(x)+\sum_{s=1}^{\infty}\varepsilon^s C_s(\tau,x), \qquad F_j(\tau,x,\varepsilon)=\sum_{s=1}^{\infty}\varepsilon^s F_j^{(s)}(\tau,x) \tag{4} \]

(let \(B_0(x)=A_0'(x)\)). The function \(A(\tau,x,\varepsilon)\) is twice differentiable with respect to \(x\), with \(A_0(x)>0\); \(C_0(x)\) is a continuous function; \(\varphi(x)\), \(\psi(x)\) are known functions satisfying all the necessary conditions required for application of the generalized Fourier method (these conditions will be specified later); \(a,b,c,d\) are constants (\(a^2+b^2>0,\ c^2+d^2>0\)). If in the boundary conditions \(a,b,c,d\) depend on \(t\), i.e., are variable, then by means of the substitution \(u(x,t)=v(x,t)\exp\{w(x,t)\}\), where

\[ w(x,t)=-x\left\{b(t)a^{-1}(t)+\frac{x}{2}\left[d(t)c^{-1}(t)-b(t)a^{-1}(t)\right]\right\}, \tag{5} \]

the variable boundary conditions of the form (3) are reduced to constant boundary conditions of the second type \(v_x(0,t)=v_x(1,t)=0\), where \(0\leq x\leq 1\).

An equation of hyperbolic type whose coefficients depend on a small parameter, with boundary conditions of the first type, \(u(0,t)=u(1,t)=0\), was considered in [2]. In that work an asymptotic construction was given under the assumption that the equation with variable coefficients degenerates, for \(\varepsilon=0\), into an equation with constant coefficients of the form \(u_{tt}=u_{xx}\). We assume that for \(\varepsilon=0\) equation (1) degenerates into an equation with variable coefficients (depending only on \(x\)) of the form

\[ u_{tt}=(A_0(x)u_x)_x-C_0(x)u, \tag{6} \]

and the boundary conditions have the form (3). In addition, we consider a generalized “resonance” in the sense that several of the functions \(k_j^2(\tau)\), for certain values of \(\tau\), take on a value equal to the number \(\lambda_1\) (\(\lambda_1\) is an eigenvalue of the right-hand side of equation (6)), and we give a justification of the asymptotic method in a more natural way.

We note that the most general results for the mixed problem for hyperbolic equations whose coefficients depend on \(x\) and \(t\) were obtained by O. Ladyzhenskaya by the method of finite differences. In particular, she proved the theorem on existence and uniqueness of the solution of the mixed-

the posed problem with boundary conditions of the form \(u|_s=\chi(t)\). We give an asymptotic representation of the solution of the mixed problem (1)—(3), assuming that the coefficients of the equation depend in a special way on \(t\) and on the small parameter \(\varepsilon\) \((\tau=\varepsilon t)\). This method makes it possible to find directly an approximate solution that tends to the exact one as \(\varepsilon\to0\), by taking a definite number of terms in the series.\(^{1)}\)

§ 2. We shall seek the solution of equation (1) under conditions (2), (3) in the form\(^{2)}\)

\[ u(x,t,\varepsilon)=\sum_{n=1}^{\infty} z_n(t,\varepsilon) W_n(x), \tag{7} \]

where \(W_n(x)\) are the eigenfunctions of the boundary-value problem

\[ L_0W\equiv (A_0(x)W')'-C_0(x)W=-\lambda W, \tag{8} \]

\[ aW'_x(0)+bW(0)=0,\qquad cW'_x(\pi)+dW(\pi)=0, \tag{9} \]

which is obtained by applying the Fourier method to equation (1), putting \(\varepsilon=0\) in it, under the boundary conditions (3). It is known that the spectrum of the boundary-value problem (8), (9) is discrete. Let \(\lambda_1,\lambda_2,\ldots\) be the eigenvalues, and \(W_1(x),W_2(x),\ldots\) the corresponding eigenfunctions of the boundary-value problem (8), (9). Assuming that (7) may be differentiated twice with respect to \(x\) and with respect to \(t\) (the conditions that must be imposed on the coefficients can be found in [3] and other works), we find the corresponding derivatives, substitute them into equation (1) and into conditions (2), (3).

First consider the following obvious equalities:

\[ L_0u=\sum_{n=1}^{\infty} z_n(t,\varepsilon)L_0W_n(x) =-\sum_{n=1}^{\infty}\lambda_n z_n(t,\varepsilon)W_n(x), \]

\[ u_{tt}=\sum_{n=1}^{\infty}z''_n(t,\varepsilon)W_n(x). \tag{10} \]

Now expanding the functions \(F_j(\tau,x,\varepsilon)\) and \(\widetilde L_1W_n(x)\) (\(\widetilde L_1\) is the differential operator whose coefficients depend on \(\tau\) and \(x\)) in Fourier series with respect to the eigenfunctions of the Sturm—Liouville operator \(L_0\) (8), and assuming that the double sums may be interchanged, and taking (10) into account, we obtain

\[ \sum_{n=1}^{\infty}\left[ z''_n+\lambda_n z_n -\varepsilon\sum_{\nu=1}^{\infty}Q_{n\nu}(\tau,\varepsilon)z_\nu -\varepsilon\sum_{j=1}^{N}G_{jn}(\tau,\varepsilon)e^{i\theta_j} \right]W_n(x)=0, \]

where

\[ Q_{n\nu}(\tau,\varepsilon) = \int_{0}^{\pi} \left\{ \sum_{s=0}^{\infty}\varepsilon^s \left[ A_{s+1}(\tau,x)\frac{d^2}{dx^2} + B_{s+1}(\tau,x)\frac{d}{dx} - C_{s+1}(\tau,x) \right]W_n(x) \right\} W_\nu(x)\,dx, \]

\[ G_{jn}(\tau,\varepsilon) = \int_{0}^{\pi}F_j(\tau,x,\varepsilon)W_n(x)\,dx \qquad (j=1,2,\ldots,N). \tag{11} \]

\(^{1)}\) Certain equations and systems with a small parameter were considered by the author in papers [8—13].

\(^{2)}\) A justification of the generalized Fourier method is given in papers [5, 14, 15].

From (11) we obtain an infinite system of differential equations with slowly varying coefficients of the form

\[ z_n''+\lambda_n z_n = \varepsilon \sum_{\nu=1}^{\infty} Q_{n\nu}(\tau,\varepsilon)z_\nu + \varepsilon \sum_{j=1}^{N} G_{jn}(\tau,\varepsilon)e^{i\theta_j}. \tag{12} \]

Expanding the functions \(\varphi(x)\) and \(\psi(x)\) in series in the eigenfunctions \(W_n(x)\),

\[ \varphi(x)=\sum_{n=1}^{\infty}\alpha_n W_n(x); \qquad \psi(x)=\sum_{n=1}^{\infty}\beta_n W_n(x), \tag{13} \]

from the initial conditions (2) we obtain

\[ z_n(0,\varepsilon)=\alpha_n,\qquad z_n'(0,\varepsilon)=\beta_n. \tag{14} \]

Thus, the mixed problem for equation (1) with the initial and boundary conditions (2), (3) is formally reduced to the boundary-value problem (8), (9) and to the Cauchy problem for the infinite system of ordinary linear differential equations of the second order (12) with initial conditions (14)\({}^{1)}\). We shall not here study in detail the questions under what conditions a solution of problem (1), (2), (3) exists in the classical or in the generalized sense (these questions were investigated in [3]); rather, applying the generalized Fourier method, we find an asymptotic representation of the solution of the infinite system of differential equations (12) thereby obtained, give a justification of the asymptotic method, and show the asymptotic character of the solution obtained for the mixed problem. Concerning the coefficients of system (12) we shall assume that they are differentiable with respect to \(\tau\) and that the series

\[ \sum_{n=1}^{\infty}\sum_{\nu=1}^{\infty} \frac{1}{\lambda_\nu^{2}} \left| \frac{d^{k}Q_{n\nu}(\tau,\varepsilon)}{d\tau^{k}} \right|^{2}; \qquad \sum_{n=1}^{\infty} \frac{1}{\lambda_n^{2}} \left| \frac{d^{k}G_{jn}(\tau,\varepsilon)}{d\tau^{k}} \right|^{2} \tag{15} \]

converge absolutely and uniformly. These conditions are needed for the justification of the asymptotic method in constructing the required approximations. It is clear that all of them will follow from the general conditions imposed on the coefficients, but they are also satisfied when all \(B_s(\tau,x)\), \(C_s(\tau,x)\), \(F_i^{(s)}(\tau,x)\) and their derivatives with respect to \(\tau\) are square-integrable on the interval \([0,\pi]\). In solving the Cauchy problem (12), (14) we shall consider two cases:

1) the “resonant” case, when for certain values of \(\tau\) one or several of the functions \(k_l^2(\tau)\) \((l=1,\ldots,r;\ 1\le r\le N)\) become equal to one of the numbers \(\lambda_n\) (we shall assume that \(k_l^2(\tau)\) may be equal to the number \(\lambda_1\));

2) the “nonresonant” case, when for all values of \(\tau\) none of the functions \(k_j^2(\tau)\) assumes the values \(\lambda_n\) \((n=1,2,\ldots)\)\({}^{2)}\).

\({}^{1)}\) Results concerning the boundary-value problem (8), (9) may be found, for example, in the book [4] and in other monographs. The Cauchy problem (12), (14) reduces to an infinite system of integral equations of Volterra type. Existence and uniqueness of the solution of the latter may be shown by the method of successive approximations. In this direction, substantial results were obtained in [5, 14, 15].

\({}^{2)}\) We note that methods of asymptotic integration of nonlinear ordinary equations with slowly varying coefficients and passage through resonance in nonstationary processes were investigated by Yu. A. Mitropol’skii [6]. For linear equations in Hilbert space, these questions were studied in [7].

§ 3. We proceed to the construction of an asymptotic solution of system (12) for the “resonance” case. For this case the following holds.

Theorem 1. If the functions \(Q_{nv}(\tau,\varepsilon)\), \(G_{jn}(\tau,\varepsilon)\), \(k_j(\tau)\) are infinitely differentiable with respect to \(\tau\) and the conditions (15) are satisfied, then a formal solution of system (12) can be represented in the form

\[ z_n(t,\varepsilon)= \sum_{l=1}^{r}\left\{\left[\delta_{n,1}+\Pi_n(\tau,\varepsilon)\right]\zeta_l+ +P_{nl}(\tau,\varepsilon)\right\}e^{i\theta_l} +\varepsilon\sum_{j=r+1}^{N}R_{jn}(\tau,\varepsilon)e^{i\theta_j}, \tag{16} \]

where \(\delta_{n,1}\) is the Kronecker symbol \((n=1,2,\ldots)\), and \(\zeta_l(t)=\xi_l(t)+i\eta(t)\) are determined from the system of first-order differential equations

\[ \zeta_l'(t)=\{D(\tau,\varepsilon)+i[\Omega(\tau,\varepsilon)-k_l(\tau)]\}\zeta_l(t)+T_l(\tau,\varepsilon), \tag{17} \]

where it is assumed that the following expansions hold:

\[ \Pi_n(\tau,\varepsilon)=\sum_{s=1}^{\infty}\varepsilon^s\Pi_n^{(s)}(\tau), \qquad P_{nl}(\tau,\varepsilon)=\sum_{s=1}^{\infty}\varepsilon^s P_{nl}^{(s)}(\tau), \]

\[ R_{jn}(\tau,\varepsilon)=\sum_{s=0}^{\infty}\varepsilon^s R_{jn}^{(s)}(\tau), \]

\[ D(\tau,\varepsilon)=\sum_{s=1}^{\infty}\varepsilon^sD_s(\tau), \qquad \Omega(\tau,\varepsilon)=\sum_{s=0}^{\infty}\varepsilon^s\Omega_s(\tau), \tag{18} \]

\[ T_l(\tau,\varepsilon)=\sum_{s=1}^{\infty}\varepsilon^sT_l^{(s)}(\tau). \]

The proof of Theorem 1 consists in indicating an algorithm for constructing the functions entering into the asymptotic expansion (18), in such a way that the sequence \(z_n\) from (16) formally satisfies system (12).

We find the corresponding derivatives of expression (16), taking (17) into account, and, comparing the coefficients of \(\zeta_l\exp\{i\theta_l\}\), \(\exp\{i\theta_l\}\), \(\exp\{i\theta_j\}\) \((l=1,2,\ldots,r;\ 1\le r\le N;\ j=1,2,\ldots,N)\), obtain the following relations\(^1\):

\[ (\delta_{n,1}+\Pi_n)(D+i\Omega)^2+2\varepsilon \Pi_n^{\tau'}(D+i\Omega)+ \]

\[ +\varepsilon(\delta_{n,1}+\Pi_n)(D'+i\Omega')+\varepsilon^2\Pi_n^{\tau''}+ \]

\[ +\lambda_n(\delta_{n,1}+\Pi_n) = \varepsilon\sum_{\nu=1}^{\infty}Q_{n\nu}(\delta_{\nu,1}+\Pi_\nu); \tag{19} \]

\[ (\delta_{n,1}+\Pi_n)\left[(D+i\Omega)T_l+ik_lT_l+\varepsilon T_l^{\tau'}\right]+ \]

\[ +2i\varepsilon k_l P_{nl}^{\tau'}+i\varepsilon k_l^{\tau'}P_{nl}+2\varepsilon T_l\Pi_n^{\tau'}+ \]

\(^1\) Here \(\Pi_n^{\tau'}\), \(D'\), \(\Omega'\), \(k_{jn}^{\tau'}\), \(T_l^{\tau'}\) denote derivatives with respect to \(\tau\).

\[ + \varepsilon^2 \ddot P_{nl}^{\tau^2} - k_l^2 P_{nl} + \lambda_n P_{nl} = \varepsilon \sum_{\nu=1}^{\infty} Q_{n\nu} P_{\nu l} + \varepsilon G_{ln}; \tag{20} \]

\[ i\varepsilon k_j \dot R_{jn}^{\tau} + 2 i\varepsilon k_j \dot R_{jn}^{\tau} + \varepsilon^2 \ddot R_{jn}^{\tau^2} - \]

\[ - k_j^2 R_{jn} + \lambda_n R_{jn} = \varepsilon \sum_{\nu=1}^{\infty} Q_{n\nu} R_{j\nu} + G_{jn} \tag{21} \]

\[ (n=1,2,\ldots;\quad l=1,2,\ldots,r;\quad 1\le r\le N;\quad j=r+1,\ldots,N). \]

§ 4. If now in these relations we separate the terms standing at each power of \(\varepsilon\), then we obtain an infinite system of equations for the successive determination of the desired terms of the expansion (18).

Comparing the coefficients at \(\varepsilon^0\) in relation (19), we have
\[ \delta_{n,1}(\lambda_n-\Omega_0^2)=0. \]
Hence we obtain
\[ \Omega_0=\pm\sqrt{\lambda_1}. \tag{22} \]

From relation (19), comparing the coefficients at \(\varepsilon^1\), we have
\[ 2i\Omega_0 \times \]

\[ \times (D_1+i\Omega_1)\delta_{n,1} = (\Omega_0^2-\lambda_n)\Pi_n' + \sum_{\nu=1}^{\infty} Q_{n\nu}^{(0)}\delta_{\nu,1}. \]
Hence, for \(n=1\), we obtain
\[ D_1+i\Omega_1=(2i\Omega_0)^{-1}Q_{11}^{(0)}(\tau), \tag{23} \]
and for \(n\ge 2\) we have
\[ \Pi_n^{(1)}(\tau)=(\lambda_n-\Omega_0^2)^{-1}Q_{n,1}^{(0)}(\tau). \tag{24} \]

We note that the function \(\Pi_1^{(1)}(\tau)\) is not determined; it may be arbitrary (we put it equal to zero). From formulas (23), (24) it is seen that the functions \(D_1(\tau)+i\Omega_1(\tau)\) and \(\Pi_n^{(1)}(\tau)\) are differentiable with respect to \(\tau\). We shall show further that the series
\[ \sum_{n=2}^{\infty}\lambda_n^2|\Pi_n^{(1)}(\tau)|^2 \]
converges absolutely and uniformly. Indeed, squaring (24), multiplying by \(\lambda_n^2\), and summing over \(n\) \((n\ge 2)\), we have
\[ \sum_{n=2}^{\infty}\lambda_n^2|\Pi_n^{(1)}(\tau)|^2. \]

Introduce the notation:
\[ \Delta_{np}=\sum_{\nu=n+1}^{n+p}\lambda_\nu^2[\Pi_\nu^{(1)}(\tau)]^2. \]
It is obvious that
\[ |\Delta_{np}| \le \sum_{\nu=n+1}^{n+p}\lambda_\nu^2|\Pi_\nu^{(1)}(\tau)|^2 = \sum_{\nu=n+1}^{n+p} \frac{|Q_{\nu,1}^{(0)}|^2}{(\lambda_\nu-\Omega_0^2)^2}\lambda_\nu^2 = \]

\[ = \sum_{\nu=n+1}^{n+p} |Q_{\nu,1}^{(0)}|^2 \left(1-\frac{\Omega_0^2}{\lambda_\nu}\right)^{-2} \le \sum_{\nu=n+1}^{n+p}|Q_{\nu,1}^{(0)}|^2. \tag{25} \]

The series \(\sum_{n=1}^{\infty} |Q_{n,1}^{(0)}|^2\) converges absolutely and uniformly by virtue of conditions (15). The quantity \(\left(1-\dfrac{\Omega_0^2}{\lambda_\nu}\right)\to 1\) as \(\nu\to\infty\), since \(\lambda_\nu\to\infty\).

Thus, the series \(\sum_{n=2}^{\infty} \lambda_n^2 |\Pi_n^{(1)}|^2\) converges absolutely and uniformly.

Let us prove that the series \(\sum_{n=2}^{\infty} \lambda_n^2 \left|\dfrac{d^k\Pi_n^{(1)}(\tau)}{d\tau^k}\right|^2\) also converges absolutely and uniformly. For this, we find the derivative of order \(k\) with respect to \(\tau\) of the function \(\Pi_n^{(1)}(\tau)\) from (24), square it, multiply by \(\lambda_n^2\), and, summing over \(n\) \((n\geqslant 2)\), obtain

\[ \sum_{n=2}^{\infty}\lambda_n^2\left|\frac{d^k\Pi_n^{(1)}(\tau)}{d\tau^k}\right|^2 = \sum_{n=2}^{\infty}\left|\frac{d^k Q_{n,1}^{(0)}}{d\tau^k}\right|^2 \left(1-\frac{\Omega_0^2}{\lambda_n}\right)^{-2}. \tag{26} \]

It follows from this (taking into account conditions (15) and the condition \(\left(1-\dfrac{\Omega_0^2}{\lambda_n}\right)\to 1\)) that the indicated series converges absolutely and uniformly\(^1\). Comparing the coefficients of \(\varepsilon^2\) in relation (19), we obtain

\[ D_2(\tau)+i\Omega_2(\tau) = \frac{1}{2i\Omega_0} \left\{ \sum_{\nu=2}^{\infty} Q_{1,\nu}^{(0)}\Pi_\nu^{(1)} + Q_{11}^{(1)} - (D_1+i\Omega_1)^2 - \left(D_1^{\tau}+i\Omega_1^{\tau}\right) \right\} \quad (n=1), \tag{27} \]

\[ \Pi_n^{(2)}(\tau) = \frac{1}{\lambda_n-\Omega_0^2} \left[ \sum_{\nu=2}^{\infty} Q_{n\nu}^{(0)}\Pi_\nu^{(1)} + Q_{n1}^{(1)} - 2i\Omega_0(D_1+i\Omega_1)\Pi_n^{(1)} - 2i\Omega_0\Pi_n^{\tau_1(1)} \right] \quad (n\geqslant 2). \tag{28} \]

The function \(\Pi_1^{(2)}(\tau)\) is not determined; it may be arbitrary. To prove the differentiability of the function \(D_2(\tau)+i\Omega_2(\tau)\), we shall show that the series

\[ \sum_{\nu=2}^{\infty} Q_{1,\nu}^{(0)}(\tau)\Pi_\nu^{(1)}(\tau) \]

converges absolutely and uniformly, and that the series obtained by termwise differentiation of it with respect to \(\tau\) converge uniformly \((0\leqslant \tau\leqslant L)\).

Indeed, on the basis of Bunyakovsky’s inequality we have

\[ \sum_{\nu=2}^{\infty}\left|Q_{1,\nu}^{(0)}(\tau)\Pi_\nu^{(1)}(\tau)\right| = \sum_{\nu=2}^{\infty}\frac{1}{\lambda_\nu} \left|Q_{1,\nu}^{(0)}(\tau)\right|\cdot \Pi_\nu^{(1)}(\tau)\lambda_\nu \leqslant \sqrt{ \sum_{\nu=2}^{\infty}\frac{1}{\lambda_\nu^2} \left|Q_{1,\nu}^{(0)}(\tau)\right|^2 } \sqrt{ \sum_{\nu=2}^{\infty}\lambda_\nu^2 \left|\Pi_\nu^{(1)}(\tau)\right|^2 }. \tag{29} \]

\(^1\) The uniform convergence of the indicated series is necessary for proving the differentiability of higher approximations of the functions \(D_s(\tau)+i\Omega_s(\tau)\) and \(\Pi_n^{(s)}(\tau)\).

From inequality (29), taking into account (15) and (25), it is seen that the indicated series converges absolutely and uniformly. The uniform convergence of the series obtained as a result of termwise differentiation with respect to \(\tau\) of the series indicated above follows from conditions (15), (25), (26). Then, on the basis of (27), taking into account (23) and the conditions of Theorem 1, we conclude that the function \(D_2(\tau)+i\Omega_2(\tau)\) is differentiable with respect to \(\tau\). Similarly, the differentiability with respect to \(\tau\) of the functions \(\Pi_\nu^{(2)}(\tau)\) \((\nu \geqslant 2)\) and the uniform convergence of the series
\[ \sum_{\nu=2}^{\infty} |\Pi_\nu^{(2)}|^2 \lambda_\nu^2, \]
\[ \sum_{\nu=2}^{\infty} \left| \frac{d^k \Pi_\nu^{(2)}}{d\tau^k} \right|^2 \lambda_\nu^2 \]
are proved. We shall show that, by the indicated method, one can find any approximation of the functions \(D(\tau,\varepsilon)+i\Omega(\tau,\varepsilon)\) and \(\Pi_n(\tau,\varepsilon)\). Comparing the coefficients of \(\varepsilon^m\) in relation (19), we obtain
\[ D_m+i\Omega_m= \frac{1}{2i\Omega_0} \left\{ \sum_{\nu=2}^{\infty}\sum_{s=0}^{m-2} \left[ Q_{1\nu}^{(s)}\Pi_\nu^{(m-s-1)} +Q_{11}^{(m-1)} - \right.\right. \]
\[ \left.\left. -\sum_{s=1}^{m-1}(D_s+i\Omega_s)(D_{m-s}+i\Omega_{m-s}) -(D_{m-1}^{\tau\prime}+i\Omega_{m-1}^{\tau\prime}) \right] \right\}; \tag{27″} \]
\[ \Pi_n^{(m)}(\tau)= \frac{1}{\lambda_n-\Omega_0^2} \left\{ \sum_{\nu=2}^{\infty}\sum_{s=0}^{m-2} Q_{n\nu}^{(s)}\Pi_\nu^{(m-s-1)} +Q_{n1}^{(n-1)} -\sum_{j=1}^{m-1}\sigma_j(\tau)\Pi_n^{(j)} +\right. \]
\[ \left. +2\sum_{s=1}^{m-2}(D_s+i\Omega_s)\Pi_n^{\tau_1(m-s-1)} +2i\Omega_0\Pi_n^{\tau_1(m-1)} + \right. \]
\[ \left. +\sum_{s=1}^{m-2}(D_s^{\tau\prime}+i\Omega_s^{\tau\prime})\Pi_n^{(m-s-1)}(\tau) +\Pi_n^{\tau^2\prime\prime(m-2)}(\tau) \right\} \quad (m=3,4,5,\ldots), \tag{28″} \]
where
\[ \sigma_1(\tau)= \sum_{s=1}^{m-2} \left[ (D_s+i\Omega_s)(D_{m-s-1}+i\Omega_{m-s-1}) + \right. \]
\[ \left. +2i\Omega_0(D_{m-1}+i\Omega_{m-1}) \right]; \]
\[ \sigma_2(\tau)= \sum_{s=1}^{m-3} \left[ (D_s+i\Omega_s)(D_{m-s-2}+i\Omega_{m-s-2})+ \right. \]
\[ \left. +2i\Omega_0(D_{m-2}+i\Omega_{m-2}) \right]; \]
\[ \ldots\ldots\ldots\ldots\ldots \]
\[ \sigma_{m-1}(\tau)=2i\Omega_0(D_1+i\Omega_1) \quad (m=3,4,\ldots). \]

The differentiability of the indicated functions is proved analogously to how it was proved for the preceding approximations. This process of finding successive approximations \(D_s(\tau)+i\Omega_s(\tau)\) and \(\Pi_n^{(s)}(\tau)\) can be continued indefinitely.

§ 5. We proceed to finding the coefficients in the expansions of the functions \(P_{nl}(\tau,\varepsilon)\) and \(T_l(\tau,\varepsilon)\) \((n=1,2,\ldots;\ l=1,2,\ldots,r;\ 1\leq r\leq N)\). For this we use relation (20). Comparing the coefficients of \(\varepsilon^1\) in this relation, we obtain

\[ (\lambda_n-k_l^2(\tau))P_{ln}^{(1)}(\tau) = G_{ln}^{(0)}(\tau)-i(\Omega_0+k_l(\tau))T_l^{(1)}(\tau)\delta_{n,1}. \tag{30} \]

We note that in the present case the functions \(k_l^2(\tau)\), for some values of \(\tau\) \((0\leq \tau\leq L)\), may be equal to the number \(\lambda_1\), but are different from the numbers \(\lambda_n\) \((n=2,3,\ldots)\). Taking this into account, we choose the functions \(T_l^{(1)}(\tau)\) so that the right-hand side of relation (30) for \(n=1\) is equal to zero. Then we obtain that \(P_{1,l}^{(1)}(\tau)\) are arbitrary functions (we set them equal to zero), and for the remaining approximations we have

\[ T_l^{(1)}(\tau)=i(\Omega_0+k_l)^{-1}G_{l1}^{(0)}(\tau); \qquad P_{nl}^{(1)}(\tau)=(\lambda_n-k_l^2(\tau))^{-1}G_{ln}^{(0)}(\tau). \tag{31} \]

From formulas (31) and the conditions of Theorem 1 it follows that \(T_l^{(1)}(\tau)\) and \(P_{nl}^{(1)}(\tau)\) are differentiable with respect to \(\tau\). Next we shall show that the series

\[ \sum_{n=2}^{\infty}|P_{nl}^{(1)}(\tau)|^2; \qquad \sum_{n=2}^{\infty}\left|\frac{d^kP_{nl}^{(1)}(\tau)}{d\tau^k}\right|^2 \quad (k=1,2,\ldots) \tag{32} \]

converge uniformly for all values of \(\tau\) \((0\leq \tau\leq L)\). Indeed, squaring the second of formulas (31) and summing over \(n\) \((n\geq 2)\), we have

\[ \sum_{n=2}^{\infty}|P_{nl}^{(1)}(\tau)|^2 = \sum_{n=2}^{\infty} \frac{1}{\lambda_n^2}|G_{ln}^{(0)}(\tau)|^2 \left(1-\frac{k_l^2(\tau)}{\lambda_n}\right)^{-2}. \tag{33} \]

Since the functions \(k_l^2(\tau)\) are bounded on the segment \([0,L]\), and \(\lambda_n\to\infty\) as \(n\to\infty\), it follows that

\[ \left(1-\frac{k_l^2(\tau)}{\lambda_n}\right)\to 1. \]

Thus, on the basis of condition (15), series (33) converges uniformly.

Now we find the \(k\)-th derivative with respect to \(\tau\) of \(P_{nl}^{(1)}(\tau)\) from (31), square the result obtained and, summing over \(n\), obtain

\[ \sum_{n=2}^{\infty}\left|\frac{d^kP_{nl}^{(1)}(\tau)}{d\tau^k}\right|^2 = \]

\[ = \sum_{n=2}^{\infty} \left| \sum_{\mu=0}^{k} C_k^\mu \frac{d^{k-\mu}}{d\tau^{k-\mu}} \left(1-\frac{k_l^2(\tau)}{\lambda_n}\right)^{-1} \frac{1}{\lambda_n} \frac{d^\mu}{d\tau^\mu}G_{ln}^{(0)}(\tau) \right|^2. \tag{34} \]

We note that for any \(\mu,k\) and for all \(n\geq 2\) the functions

\[ \frac{d^{k-\mu}}{d\tau^{k-\mu}} \left(1-\frac{k_l^2}{\lambda_n}\right)^{-1} \]

are bounded. This follows from the fact that the functions \(k_l^2(\tau)\) and their derivatives with respect to \(\tau\) are bounded on the segment \([0,L]\) and \(k_l^2(\tau)\neq \lambda_n,\ n\geq 2\). Thus, taking into account condition (15), series (34) converges uniformly, i.e. the second of the series (32) is uniformly convergent.

We now turn to the determination of the functions \(T_l(\tau,\varepsilon)\), \(P_{nl}(\tau,\varepsilon)\) in the second approximation. To this end we equate the coefficients of \(\varepsilon^2\) in relation (20). Having done this, we obtain the following relation:

\[ (\lambda_n-k_l^2(\tau))P_{nl}^{(2)} = \sum_{\nu=1}^{\infty} Q_{n\nu}^{(0)}P_{\nu l}^{(1)}+G_{ln}^{(1)} - \]

\[ -i(\Omega_0+k_l)T_l^{(2)}\delta_{n,1} -(D_1+i\Omega_1)T_l^{(1)}\delta_{n,1} - \]

\[ -i(\Omega_0+k_l)T_l^{(1)}\Pi_n^{(1)} -\stackrel{\tau_1}{T_l^{(1)}}\delta_{n,1} -2ik_l\stackrel{\tau_1}{P_{nl}^{(1)}}-ik_l^2P_{nl}^{(1)}. \tag{35} \]

We now choose the functions \(T_l^{(2)}(\tau)\) so that the right-hand side of relation (35) for \(n=1\) becomes zero. Then from this relation, taking into account that \(P_{1,l}^{(1)}(\tau)\) and \(\Pi_1^{(1)}(\tau)\) have been set equal to zero, we obtain

\[ T_l^{(2)}(\tau) = i(\Omega_0+k_l)^{-1} \left\{ \sum_{\nu=2}^{\infty} Q_{1,\nu}^{(0)}P_{\nu l}^{(1)} +G_{l,1}^{(1)} -\right. \]

\[ \left. -(D_1+i\Omega_1)T_l^{(1)} -\stackrel{\tau_1}{T_l^{(1)}}(\tau) \right\}; \]

\[ P_{nl}^{(2)}(\tau) = (\lambda_n-k_l^2(\tau))^{-1} \left\{ \sum_{\nu=2}^{\infty} Q_{n\nu}^{(0)}P_{\nu l}^{(1)} +G_{ln}^{(1)} -\right. \]

\[ \left. -i(\Omega_0+k_l)T_l^{(1)}\Pi_n^{(1)} -2ik_l\stackrel{\tau_1}{P_{nl}^{(1)}} -ik_l^2P_{nl}^{(1)} \right\}. \tag{36} \]

Let us note that \(P_{1,l}^{(2)}(\tau)\) may be arbitrary. We set them equal to zero.

We now show that the series
\[ \sum_{\nu=2}^{\infty} Q_{1,\nu}^{(0)}(\tau)P_{\nu,l}^{(1)}(\tau), \]
as well as the series obtained by termwise differentiation of it with respect to \(\tau\), converge uniformly. Applying Bunyakovsky’s inequality, we have

\[ \sum_{\nu=2}^{\infty} \left|Q_{1,\nu}^{(0)}(\tau)P_{\nu,l}^{(1)}(\tau)\right| \le \sqrt{\sum_{\nu=2}^{\infty}\left|Q_{1,\nu}^{(0)}(\tau)\right|^2} \sqrt{\sum_{\nu=2}^{\infty}\left|P_{\nu,l}^{(1)}(\tau)\right|^2}. \tag{37} \]

On the basis of conditions (15) and (33), the series (37) converge absolutely and uniformly for all \(l\). The uniform convergence of the series obtained as a result of termwise differentiation of the series (37) follows from conditions (15) and from the uniform convergence of the series (34). Thus, we conclude that the functions \(T_l^{(2)}(\tau)\) are differentiable with respect to \(\tau\).

To prove the differentiability of the functions \(P_{nl}^{(2)}(\tau)\) with respect to \(\tau\), it is necessary to show that the series

\[ \sum_{n,\nu=2}^{\infty}(\lambda_\nu-k_l^2)^{-1}Q_{n\nu}^{(0)}P_{\nu l}^{(1)} \]

and the series obtained by termwise differentiation with respect to \(\tau\) converge uniformly. Applying Bunyakovsky’s inequality to these series and taking into account conditions (15), (33), (34), we prove the uniform convergence of the indicated series, i.e. the functions \(P_{nl}^{(2)}(\tau)\) are differentiable with respect to \(\tau\).

We now show that the series \(\sum\limits_{n=2}^{\infty}|P_{nl}^{(2)}(\tau)|^2\) converges uniformly for any \(l\) (\(l=1,2,\ldots,r;\ 1\le r<N\)). To prove the uniform convergence of this series it suffices to prove, as follows from the second formula (36), the uniform convergence of the following series:

\[ \sum_{n=2}^{\infty}\left|\sum_{\nu=2}^{\infty} (\lambda_\nu-k_l^2(\tau))^{-1}Q_{n\nu}^{(0)}P_{\nu l}^{(1)}\right|^2; \]

\[ \sum_{n=2}^{\infty}\left|(\lambda_n-k_l^2)^{-1}(\Omega_0+k_l)T_l^{(1)}\Pi_n^{(1)}(\tau)\right|^2; \tag{38} \]

\[ \sum_{n=2}^{\infty}\left|\frac{d}{d\tau}P_{nl}^{(1)}(\tau)(\lambda_n-k_l^2(\tau))^{-1}\right|^2; \]

\[ \sum_{n=2}^{\infty}\left|(\lambda_n-k_l^2)^{-1}P_{nl}^{(1)}(\tau)\right|^2. \]

We show the uniform convergence of the first of the series (38). Using Bunyakovsky’s inequality, we obtain

\[ \sum_{n=2}^{\infty}\left|\sum_{\nu=2}^{\infty} (\lambda_\nu-k_l^2)^{-1}Q_{n\nu}^{(0)}P_{\nu l}^{(1)}\right|^2 \le \]

\[ \le \sum_{n,\nu=2}^{\infty}\frac{1}{\lambda_\nu^2}|Q_{n\nu}^{(0)}|^2 \left(1-\frac{k_l^2}{\lambda_\nu}\right)^{-2} \sum_{\nu=2}^{\infty}|P_{\nu l}^{(1)}|^2 . \tag{38′} \]

On the basis of conditions (15) and (33), the series on the right-hand side of inequality (38′) converge absolutely and uniformly; hence the given series also converges absolutely and uniformly. To prove the uniform convergence of the second of the series (38), it suffices to show that the series \(\sum\limits_{n=2}^{\infty}|(\lambda_n-k_l^2)^{-1}\Pi_n^{(1)}G_{l1}^{(0)}|\) converges absolutely and uniformly. The last series can be written as follows:

\[ \sum_{n=2}^{\infty}|(\lambda_n-k_l^2)^{-1}\Pi_n^{(1)}G_{l1}^{(0)}|^2 \le \sum_{n=2}^{\infty}\frac{1}{\lambda_n^2}|G_{l1}^{(0)}|^2|\Pi_n^{(1)}|^2 \left(1-\frac{k_l^2}{\lambda_n}\right)^{-2} \le \]

\[ \le \sum_{n=2}^{\infty}\frac{1}{\lambda_n^2}|G_{l1}^{(0)}|^2 \left(1-\frac{k_l^2}{\lambda_n}\right)^{-2} \sum_{n=2}^{\infty}|\Pi_n^{(1)}|^2 . \tag{38″} \]

It follows from this (on the basis of the assertions proved above) that the indicated series converges absolutely and uniformly. The uniform convergence of the third and fourth of the series (38) follows from the assertions proved earlier.

Similarly, as was proved above, for the series \(\sum\limits_{n=2}^{\infty}\left|\dfrac{d^kP_{nl}^{(1)}}{d\tau^k}\right|^2\) it is suffi-

is the uniform convergence of the series \(\sum_{n=2}^{\infty}\left|\dfrac{d^k P_{nl}^{(2)}}{d\tau^k}\right|^2\) for the second approximation of the functions \(P_{nl}^{(2)}(\tau)\). Thus, \(P_{nl}^{(2)}(\tau)\) are differentiable with respect to \(\tau\). In the indicated way, any approximation of the functions \(T_l(\tau,\varepsilon)\), \(P_{nl}(\tau,\varepsilon)\) can be found. For example, comparing the coefficients of \(\varepsilon^m\) in relation (20) and carrying out analogous reasoning, we obtain

\[ T_l^{(m)}(\tau)=i(\Omega_0+k_l)^{-1}\left\{\sum_{\nu=2}^{\infty}\sum_{s=0}^{m-2}Q_{n\nu}^{(s)}P_{\nu l}^{(m-s-1)}(\tau)+G_{ln}^{(m-1)}-\right. \]

\[ \left. -\sum_{s=1}^{m-1}(D_{m-s}+i\Omega_{m-s})T_l^{(s)}+T_l^{\tau_1(m-1)}(\tau)\right\}; \]

\[ P_{nl}^{(m)}(\tau)=(\lambda_n-k_l^2)^{-1}\left\{\sum_{\nu=2}^{\infty}\sum_{s=0}^{m-2}Q_{n\nu}^{(s)}P_{\nu l}^{(m-s-1)}(\tau)+G_{ln}^{(m-1)}(\tau)-\right. \]

\[ -\sum_{\mu=1}^{m-2}\omega_\mu(\tau)\Pi_n^{(\mu)}-i(\Omega_0+k_l)\sum_{s=1}^{m-1}\Pi_n^{(m-s)}T_l^{(s)}- \]

\[ -i(\Omega_0+k_l)T_n^{(m-1)}-2\sum_{s=1}^{m-1}T_l^{(s)}\Pi_n^{(m-s-1)}- \]

\[ -\sum_{s=1}^{m-2}T_l^{(m-s-1)}(\tau)\Pi_n^{(s)}-\frac{\tau^2}{2}P_{nl}^{\prime\prime(m-2)}(\tau)- \]

\[ \left.-2ik_lP_{nl}^{\tau_1(m-1)}-ik_lP_{nl}^{\tau_1'(m-1)}(\tau)\right\}; \]

where

\[ \omega_\mu=\sum_{s=1}^{m-\mu-1}(D_s+i\Omega_s)T_l^{(m-s-\mu)}(\tau) \qquad (\mu=1,2,\ldots,m-1); \]

\[ \omega_{m-2}=(D_1+i\Omega_1)T_l^{(1)}. \]

The differentiability of these functions is proved analogously to the preceding approximations. The process indicated above for finding approximations to the functions \(T_l(\tau,\varepsilon)\) and \(P_{nl}(\tau,\varepsilon)\) can be continued indefinitely.

§ 6. We now proceed to the determination of the functions \(R_{jn}^{(s)}(\tau)\). For this purpose we use relation (21). Comparing the coefficients of \(\varepsilon^0\) in this relation, we obtain

\[ R_{jn}^{(0)}(\tau)=\left(\lambda_n-k_j^2(\tau)\right)^{-1}G_{jn}^{(0)}(\tau) \qquad (n=1,2,\ldots; \]

\[ j=r+1,\ldots,N;\quad 1\leq r\leq N). \tag{39} \]

From formulas (39) and the conditions of Theorem 1 it is seen that the functions \(R_{jn}^{(0)}(\tau)\) are diffe-

differentiable with respect to \(\tau\). We shall also show that the series \(\sum_{n=1}^{\infty}|R_{jn}^{(0)}(\tau)|^2\) converges uniformly. Indeed, squaring equality (39) and summing over \(n\), we have

\[ \sum_{n=1}^{\infty}|R_{jn}^{(0)}(\tau)|^2 = \sum_{n=1}^{\infty} \frac{1}{\lambda_n^2}\, |G_{jn}^{(0)}(\tau)|^2 \left(1-\frac{k_l^2(\tau)}{\lambda_n}\right)^{-2}. \tag{40} \]

On the basis of (15), the series (40) converges uniformly, which was to be proved. The series obtained by termwise differentiation with respect to \(\tau\) of expression (39) also converges uniformly. Indeed, finding the \(k\)-th derivative with respect to \(\tau\) of \(R_{jn}^{(0)}(\tau)\) from (39), squaring the result obtained and summing over \(n\), we obtain

\[ \sum_{n=1}^{\infty} \left| \frac{d^k R_{jn}^{(0)}}{d\tau^k} \right|^2 = \sum_{n=1}^{\infty} \left| \sum_{\mu=0}^{k} C_k^\mu \frac{d^{k-\mu}}{d\tau^{k-\mu}} \times \right. \]

\[ \left. \times \left(1-\frac{k_l^2}{\lambda_n}\right)^{-1} \frac{1}{\lambda_n}\, \frac{d^\mu}{d\tau^\mu} G_{jn}^{(0)}(\tau) \right|^2 . \tag{41} \]

Applying Bunyakovsky’s inequality to the right-hand side of relation (41) and taking into account the conditions of Theorem 1 and condition (15), we see that the series (41) converges uniformly. Comparing the coefficients of \(\varepsilon^1\) in relation (21), we obtain

\[ R_{jn}^{(1)}(\tau) = (\lambda_n-k_j^2)^{-1} \left\{ \sum_{\nu=1}^{\infty} Q_{n\nu}^{(0)}R_{j\nu}^{(0)} + G_{jn}^{(1)} - \right. \]

\[ \left. - i\dot{k}_j R_{jn}^{(0)} - 2ik_j \dot{R}_{jn}^{(0)} \right\}. \tag{42} \]

To prove the differentiability of the functions \(R_{jn}^{(1)}(\tau)\), it is enough to show that the following series will converge absolutely and uniformly:

\[ \sum_{n,\nu=1}^{\infty} (\lambda_\nu-k_j^2)^{-1}Q_{n\nu}^{(0)}R_{j\nu}^{(0)}; \qquad \sum_{n,\nu=1}^{\infty} \frac{d^k}{d\tau^k} Q_{n\nu}^{(0)}R_{j\nu}^{(0)} . \tag{43} \]

Applying Bunyakovsky’s inequality to the first of the series (43), we have

\[ \sum_{n,\nu=1}^{\infty} \left| (\lambda_\nu-k_j^2)^{-1}Q_{n\nu}^{(0)}R_{j\nu}^{(0)} \right| \le \]

\[ \le \left\{ \sum_{n,\nu=1}^{\infty} \frac{1}{\lambda_\nu^2} |Q_{n\nu}^{(0)}|^2 \left(1-\frac{k_j^2}{\lambda_\nu}\right)^{-2} \right\}^{\frac12} \left\{ \sum_{\nu=1}^{\infty}|R_{j\nu}^{(0)}|^2 \right\}^{\frac12}. \tag{44} \]

On the basis of conditions (15) and (40), the series (44) converges absolutely and uniformly. The uniform convergence of the second of the series (43) follows from conditions (15) and the uniform convergence of the series (41). Thus, the functions \(R_{jn}^{(1)}(\tau)\) are differentiable with respect to \(\tau\).

We shall further show that the following series converge absolutely and uniformly:

\[ \sum_{n=1}^{\infty}|R_{jn}^{(1)}(\tau)|^2, \qquad \sum_{n=1}^{\infty} \left| \frac{d^k R_{jn}^{(1)}(\tau)}{d\tau^k} \right|^2 . \tag{45} \]

Taking into account formulas (40), (41) and the conditions of Theorem 1, we see that in order to prove the uniform convergence of the first of the series (45) it is sufficient to prove the uniform convergence of the following series:

\[ \sum_{n=1}^{\infty}\left|\sum_{\nu=1}^{\infty} (\lambda_\nu-k_j^2(\tau))^{-1}Q_{n\nu}^{(0)}R_{j\nu}^{(0)} \right|^2 \]

\[ (j=r+1,\ldots,N;\quad 1\le r\le N). \tag{46} \]

Applying Bunyakovsky’s inequality to the series (46), we obtain

\[ \sum_{n=1}^{\infty}\left|\sum_{\nu=1}^{\infty} (\lambda_\nu-k_j^2)^{-1}Q_{n\nu}^{(0)}R_{j\nu}^{(0)} \right|^2\le \]

\[ \le \sum_{n,\nu=1}^{\infty}\frac{1}{\lambda_\nu^2}|Q_{n\nu}^{(0)}|^2 \left(1-\frac{k_j^2}{\lambda_\nu}\right)^{-2} \sum_{\nu=1}^{\infty}|R_{j\nu}^{(0)}|^2 . \tag{47} \]

The series (47) converges uniformly on the basis of conditions (15) and (40).

Similarly, as was shown in formula (41) for \(R_{jn}^{(0)}(\tau)\), the uniform convergence of the second series (45) for \(R_{jn}^{(1)}(\tau)\) is proved.

We show that by the method indicated above one can find any approximation. For example, comparing the coefficients of \(\varepsilon^m\) in relation (21), we obtain

\[ R_{jn}^{(m)}(\tau)= \frac{1}{\lambda_n-k_j^2(\tau)} \left\{ \sum_{\nu=1}^{\infty}\sum_{s=0}^{m-1} Q_{n\nu}^{(0)}R_{j\nu}^{(m-s-1)} +\right. \]

\[ \left. +G_{jn}^{(m)}-ik_j\frac{\tau}{R}_{jn}^{(m-1)} -2ik_j\frac{\tau}{R}_{jn}^{(m-1)} -\frac{\tau^2}{R}_{jn}^{\prime\prime(m-2)} \right\}. \tag{48} \]

The differentiability of the functions \(R_{jn}^{(m)}(\tau)\) is shown in the same way as was shown above for the preceding approximations. The indicated process of finding successive approximations for the unknown functions \(R_{jn}(\tau,\varepsilon)\) can be continued indefinitely. Thus, by indicating a method for finding the coefficients of the expansion (18), we have proved Theorem 1.

For the “nonresonant” case the following holds.

Theorem 2. If the conditions of Theorem 1 are satisfied, then formal asymptotic solutions of system (12) can be represented as

\[ z_n(t,\varepsilon)=\delta_{n,1}\zeta+\varepsilon \sum_{j=1}^{N}\tilde R_{jn}(\tau,\varepsilon)e^{i\theta_j}, \tag{49} \]

where \(\delta_{n,1}\) is the Kronecker symbol, and \(\zeta(t)=\xi(t)+i\eta(t)\) is determined by the first-order differential equation

\[ \zeta'(t)=\bigl[\tilde D(\tau,\varepsilon)+i\tilde\Omega(\tau,\varepsilon)\bigr]\zeta(t). \tag{50} \]

The functions \(\tilde R_{jn}, \tilde D, \tilde\Omega\) have a representation in the form of asymptotic series in powers of the small parameter.

The proof of Theorem 2 is analogous to the proof of Theorem 1. We find the corresponding derivatives with respect to \(t\) of expression (49), substitute them into system (12), taking (50) into account, and, comparing the coefficients of \(\zeta(t)\) and \(e^{i\theta_j}\), obtain two relations. Separating in these relations the terms that stand

with equal powers of \(\varepsilon\), we obtain an infinite system of equations which makes it possible to determine the terms of the asymptotic expansions.

§ 7. In the preceding sections we formally reduced the mixed problem (1)—(3) to the boundary-value problem (8), (9) and the Cauchy problem for the infinite system (12) with initial conditions (14), and also indicated an algorithm for constructing a formal solution of this system. The aim of the present section is to justify the indicated asymptotic method1.

We shall first prove the existence and uniqueness of the solution of problem (12), (14), and then prove that the difference between the exact solutions \(z_n\) and the \(m\)-th approximations (the \(m\)-th approximations are obtained by cutting off the asymptotic series (16), (17) after the \(m\)-th terms) has order \(O(\varepsilon^m)\), and, finally, prove the asymptotic character of the solution of the mixed problem (1)—(3). For this we introduce some concepts of functional analysis. The set of real numbers \((\xi_1,\xi_2,\ldots)\) satisfying the condition

\[ \sum_{n=1}^{\infty} |\xi_n| < \infty \]

will be called, as usual, the space \(l_1\). We shall call this sequence a point of the space \(l_1\), or a vector, and shall denote it as follows:
\[ \vec{\xi}=(\xi_1,\xi_2,\ldots)\equiv(\xi_n). \]
If as the norm of any vector \(\vec{\xi}\) we choose the number
\[ \|\vec{\xi}\|=\sum_{n=1}^{\infty}|\xi_n|, \]
then \(l_1\) is a Banach space. Let us also introduce the space \(L_1(0,L)\), which is the set of sequences of continuous functions \((f_1(\tau,\varepsilon), f_2(\tau,\varepsilon),\ldots)\equiv(f_n(\tau,\varepsilon))\) satisfying the condition:
\[ \sum_{n=1}^{\infty}\max_{\tau}|f_n(\tau,\varepsilon)|<\infty. \]
If by the norm of the vector-function \((f_n)\) we mean the number
\[ \|f\|=\sum_{n=1}^{\infty}\max_{\tau}|f_n(\tau,\varepsilon)|, \]
then \(L_1(0,L)\) is a Banach space. By means of the substitution

\[ z_n(t,\varepsilon)=\frac{1}{\sqrt{\lambda_n}}\int_0^t \sin\sqrt{\lambda_n}(t-s)y_n(s,\varepsilon)\,ds+ \]

\[ +\alpha_n\cos\sqrt{\lambda_n}\,t+ \frac{\beta_n}{\sqrt{\lambda_n}}\sin\sqrt{\lambda_n}\,t \tag{51} \]

we reduce system (12) with initial conditions (14) to an infinite system of integral equations. Indeed, we have

\[ z'_n(t,\varepsilon)=\int_0^t \cos\sqrt{\lambda_n}(t-s)y_n(s,\varepsilon)\,ds- \]

\[ -\sqrt{\lambda_n}\alpha_n\sin\sqrt{\lambda_n}\,t+ \beta_n\cos\sqrt{\lambda_n}\,t, \tag{51'} \]

\[ z''_n(t,\varepsilon)=-\lambda_n z_n(t,\varepsilon)+y_n(t,\varepsilon). \]

Substituting the values from (51) into system (12), we obtain

\[ y_n(t,\varepsilon)=\sum_{\nu=1}^{\infty}\int_0^t K_{n\nu}(\tau,s,\varepsilon)y_\nu(s,\varepsilon)\,ds+\dot f_n(\tau,\varepsilon), \tag{52} \]

where

\[ K_{n\nu}(\tau,s,\varepsilon)=-\frac{\varepsilon}{\sqrt{\lambda_\nu}}Q_{n\nu}(\tau,\varepsilon)\sin\sqrt{\lambda_\nu}(t-s), \]

\[ f_n(\tau,\varepsilon)=\varepsilon\sum_{\nu=1}^{\infty}Q_{n\nu}(\tau,\varepsilon) \left[\alpha_\nu\cos\sqrt{\lambda_\nu}t+ \frac{\beta_\nu}{\sqrt{\lambda_\nu}}\sin\sqrt{\lambda_\nu}t\right] +\varepsilon\sum_{j=1}^{N}G_{jn}(\tau,\varepsilon)e^{i\theta_j}. \tag{53} \]

By a solution of system (52) we shall mean a sequence of functions
\((y_n(t,\varepsilon))\in L_1(0,L)\), for which the series
\(\sum_{\nu=1}^{\infty}K_{n\nu}(\tau,s,\varepsilon)y_\nu(s,\varepsilon)\), for every \(n\), converges uniformly, represents a continuous function, and satisfies this system. The following holds.

Theorem 3. If
\[ \sum_{n=1}^{\infty}\max_{\tau}|f_n(\tau,\varepsilon)|<\infty,\qquad |K_{n\nu}(\tau,s,\varepsilon)|<\varepsilon a_{n\nu}, \]
where the matrix \(a_{n\nu}\) is bounded in \(l_1\), i.e. for every \(\vec{\xi}\in l_1\),

\[ \|(a_{n\nu})\vec{\xi}\|=\sum_{n=1}^{\infty}\left|\sum_{\nu=1}^{\infty}a_{n\nu}\xi_\nu\right| \le C\sum_{n=1}^{\infty}|\xi_n|=C\|\vec{\xi}\|, \]

where \(C\) is some constant independent of \(\vec{\xi}\), then system (52) has a solution, moreover a unique one, which satisfies the condition
\[ \sum_{n=1}^{\infty}\max_t |y_n(t,\varepsilon)|<\infty. \]

The theorem is proved by the method of successive approximations. Let

\[ y_n^{(0)}(t,\varepsilon)=f_n(\tau,\varepsilon)\quad (n=1,2,\ldots),\qquad \left(0\le \tau\le L,\quad 0\le t\le \frac{L}{\varepsilon}\right) \]
be the zeroth approximation, and let the subsequent approximations be defined from the equalities:

\[ y_n^{(k)}(t,\varepsilon)= \sum_{\nu=1}^{\infty}\int_0^t K_{n\nu}(\tau,s,\varepsilon)y_\nu^{(k-1)}(s,\varepsilon)\,ds +f_n(\tau,\varepsilon). \tag{54} \]

Let us estimate the difference

\[ |y_n^{(1)}-y_n^{(0)}|= \left|\sum_{\nu=1}^{\infty}\int_0^t K_{n\nu}(\tau,s,\varepsilon)y_\nu^{(0)}(s,\varepsilon)\,ds\right| \le \]

\[ \le \sum_{\nu=1}^{\infty}\varepsilon t a_{n\nu}\max_\tau |f_n(\tau,\varepsilon)|. \]

It follows from this that

\[ \begin{aligned} \left|y_n^{(2)}-y_n^{(1)}\right| &= \left| \sum_{\nu=1}^{\infty}\int_{0}^{t} K_{n\nu}\left[y_n^{(1)}-y_n^{(0)}\right]\,ds \right| \leqslant \\[4pt] &\leqslant \sum_{\nu=1}^{\infty}\int_{0}^{t} \varepsilon a_{n\nu}\left|y_n^{(1)}-y_n^{(0)}\right|\,ds \leqslant \\[4pt] &\leqslant \sum_{\nu=1}^{\infty}\int_{0}^{t} \varepsilon a_{n\nu} \sum_{\nu_1=1}^{\infty} \varepsilon s a_{\nu\nu_1}\max_{\tau}|f_{\nu_1}|\,ds \leqslant \\[4pt] &\leqslant \frac{(\varepsilon t)^2}{2} \sum_{\nu=1}^{\infty} a_{n\nu} \sum_{\nu_1=1}^{\infty} a_{\nu\nu_1}\max_{\tau}|f_{\nu_1}|. \end{aligned} \]

Applying the method of complete mathematical induction, we prove the inequality

\[ \left|y_n^{(k)}-y_n^{(k-1)}\right| \leqslant \frac{(\varepsilon t)^k}{k!} \sum_{\nu=1}^{\infty} a_{n\nu} \sum_{\nu_1=1}^{\infty} a_{\nu\nu_1}\ldots \]

\[ \ldots \sum_{\nu_{k-1}=1}^{\infty} a_{\nu_{k-2},\nu_{k-1}}\max_{\tau}|f_{\nu_{k-1}}|. \]

Then

\[ \sum_{n=1}^{\infty}\max_t \left|y_n^{(k)}-y_n^{(k-1)}\right| \leqslant \frac{(LC)^k}{k!}\,\|f\|, \tag{55} \]

i.e.

\[ \left\|y^{(k)}-y^{(k-1)}\right\| \leqslant \frac{(LC)^k}{k!}\,\|f\|, \]

where

\[ y=(y_1,y_2,\ldots)\equiv(y_n); \qquad f=(f_1,f_2,\ldots)\equiv(f_n). \]

Continuing further, as usual, we obtain the following inequality:

\[ \left\|y^{(\mu+k)}-y^{(k)}\right\| \leqslant \sum_{i=k}^{\mu+k-1} \left\|y^{(i+1)}-y^{(i)}\right\| \leqslant \sum_{i=k}^{\mu+k-1} \frac{(LC)^{i+1}}{(i+1)!}\,\|f\|, \]

whence follows the Cauchy property of the sequence \((y_n^{(k)})\). Since the space \(L_1(0,L)\) is complete, there exists the limit \(\lim_{k\to\infty}(y_n^{(k)})=(y_n)\). By virtue of the continuity of the operator (52), we obtain that \((y_n)\) is a solution of the system of integral equations (52), and therefore, on the basis of the substitution (51), we obtain a solution of system (12). The uniqueness of the solution of the system follows from the inequality

\[ \|\Phi\|\leqslant \frac{(LC)^k}{k!}\,\|\Phi\|, \tag{56} \]

where \((\Phi_n)\) is a solution of the corresponding homogeneous system

\[ \Phi_n(t,\varepsilon)= \sum_{\nu=1}^{\infty}\int_{0}^{t} K_{n\nu}(\tau,s,\varepsilon)\Phi_n(s,\varepsilon)\,ds. \tag{56'} \]

Indeed, let \((\Phi_n^*)\) and \((\Phi_n^{**})\) be solutions of system (52), with \((\Phi_n^*)-(\Phi_n^{**})\ne 0\). Then \((\Phi_n)=(\Phi_n^*)-(\Phi_n^{**})\) is a solution of system (56), different from zero: \((\Phi_n)\ne 0\). This leads to a contradiction on the basis of (56).

Lemma 1. Under the condition of the preceding theorem the inequality holds
\[ \|y\|\le e^{LC}\|f\|, \tag{57} \]
where \((y_n)\) is a solution of system (52).

Indeed, on the basis of inequality (55) we have
\[ \begin{aligned} \|y\|&=\sum_{n=1}^{\infty}\max_t |y_n| \le \sum_{n=1}^{\infty}\max_t |y_n^{(0)}|+\\ &\quad+\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\max_t |y_n^{(k)}-y_n^{(k-1)}|\le\\ &\le \|f\|+\sum_{k=1}^{\infty}\frac{(LC)^k}{k!}\|f\| =\sum_{k=0}^{\infty}\frac{(LC)^k}{k!}\|f\|=e^{LC}\|f\|. \end{aligned} \]

Lemma 2. If for \(0<\varepsilon\le \varepsilon_0\) the coefficients of system (12) have a sufficient number of derivatives with respect to \(\tau\), then the \(m\)-th approximations \(z_n^m\) of this system satisfy it with accuracy up to quantities of order \(O(\varepsilon^{m+1})\).

The proof of this lemma follows from the very construction of the approximate solutions. For the “resonant” case the following theorem holds.

Theorem 4. If the exact solutions \(z_n\) of system (12) and the \(m\)-th approximations \(z_n^m\) are taken under the same initial conditions and the conditions of Lemma 2 are satisfied, then in the case of “resonance” there exists a positive constant \(C_m\), independent of \(\varepsilon\), such that the inequality holds1
\[ \|z-z^m\|\le C_m\varepsilon^m . \]

For convenience in proving this theorem we introduce the notation
\[ x_n^m=z_n-z_n^m . \tag{58} \]

Using this notation and Lemma 2, system (12) in the case of “resonance” can be written in the form
\[ \frac{d^2x_n^m}{dt^2}+\lambda_n x_n^m =\varepsilon\sum_{\nu=1}^{\infty} Q_{n\nu}(\tau,\varepsilon)x_\nu^m +\varepsilon^{m+1}H_n^{(m)}(\tau,\varepsilon). \tag{59} \]

By means of the substitution
\[ x_n^m=\frac{1}{\lambda_n}\int_0^t \sin\sqrt{\lambda_n}\,(t-s)\,y_n^m(s,\varepsilon)\,ds \tag{60} \]
system (59) is reduced to the infinite system of integral equations
\[ y_n^m(t,\varepsilon)= \sum_{\nu=1}^{\infty}\int_0^t \widetilde K_{n\nu}(\tau,s,\varepsilon)y_\nu^m(s,\varepsilon)\,ds +\widetilde f_n(\tau,\varepsilon), \tag{61} \]

where \(\widetilde K_{n\nu}(\tau,s,\varepsilon)\) and \(\widetilde f_n(\tau,\varepsilon)\) have the form

\[ \widetilde K_{n\nu}(\tau,s,\varepsilon) = \varepsilon\,\frac{\sqrt{\lambda_n}}{\lambda_\nu}\, Q_{n\nu}(\tau,\varepsilon)\sin\sqrt{\lambda_\nu}(t-s); \]

\[ \widetilde f(\tau,\varepsilon) = \varepsilon^{m+1}\sqrt{\lambda_n}\,H_n^{(m)}(\tau,\varepsilon). \tag{62} \]

From formula (60) we have

\[ |x_n^m(t,\varepsilon)| \leq \frac{1}{\lambda_n} \int_0^t \left|\sin\sqrt{\lambda_n}(t-s)\right|\times \]

\[ {}\times |y_n^m(s,\varepsilon)|\,ds \leq \frac{1}{\lambda_n} \int_0^t \max_s |y_n^m(s,\varepsilon)|\,ds \leq \]

\[ \leq \frac{1}{\lambda}\max_t |y_n^m(t,\varepsilon)|\,\frac{L}{\varepsilon} \qquad (\lambda=\min_n\lambda_n). \]

Then we shall have

\[ \sum_{n=1}^{\infty}\max_t |x_n^m| \leq \frac{L}{\varepsilon\lambda} \sum_{n=1}^{\infty}\max_t |y_n^m(t,\varepsilon)|. \]

Hence, on the basis of the condition of the theorem and formula (57), we obtain

\[ \|x^m\| \leq \frac{L}{\varepsilon\lambda}\|y^m\| \leq \frac{L}{\varepsilon\lambda}e^{LC}\|H^{(m)}\|\varepsilon^{m+1} = C_m\varepsilon^m, \]

where \(C_m=\lambda^{-1}\exp\{LC\}\|H^{(m)}\|\). Returning now to the notation of formula (58), we have

\[ \|z-z^m\|\leq C_m\varepsilon^m. \tag{63} \]

Inequality (63) proves the asymptotic character of the approximate solutions of system (12).

§ 8. We now return to the mixed problem (1)—(3). By a classical solution of the indicated problem we shall mean a function \(u(x,t,\varepsilon)\), continuous together with the partial derivatives \(u_t, u_{tt}, u_x, u_{xx}\) in the region \(0\leq x\leq\pi;\ 0\leq t\leq \dfrac{L}{\varepsilon}\). Taking into account the form of the sought solution given by formula (7), it is not difficult to see that, in order to justify the Fourier method, it is sufficient that the series

\[ \sum_{n=1}^{\infty}\lambda_n\max_t |z_n(t,\varepsilon)| \]

converge, since the convergence of the other series for \(u, u_{tt}, u_x, u_{xx}\) will follow from the convergence of this series.

We indicate the conditions that must be imposed on problem (1)—(3) in order that series (7) converge uniformly and give a solution of the indicated mixed problem. Since the eigenfunctions \(W_n(x)\) of the principal part of the equation are bounded in the aggregate, it is sufficient for the convergence of series (7) that

the series \(\sum_{n=1}^{\infty}\max_t |z_n|\) converges, i.e., that the function \(z_n\) belongs to the space \(L_1(0,L)\). For the existence of \(u_{tt}\) it is sufficient that the series \(\sum_{n=1}^{\infty}\max_t |z'_n|\) converge; and for this, as is seen from the system (12) itself, the convergence of the above-mentioned series \(\sum_{n=1}^{\infty}\lambda_n \max_t |z_n|\) is sufficient. For the existence of the derivative \(u_x\) it is sufficient that the series \(\sum_{n=1}^{\infty}\max_t |z_n|\max_x |W'_n(x)|\) converge. Since \(W'_n(x)=O(n)\), the convergence of the preceding series ensures the convergence of the latter. As for the derivative \(u_{xx}\) of the series (7), its existence follows from equation (1) itself. It is not difficult to verify that, in order for there to exist a classical solution of the mixed problem (1)—(3), representable in the form (7), it is sufficient to impose the following conditions.

1) The coefficients \(A_0(x)\) are continuous together with their derivatives up to and including the second order; \(C_0(x)\) is a continuous function.

2) The coefficients \(A_s(\tau,x)\), \(B_s(\tau,x)\), \(C_s(\tau,x)\) are continuous together with their derivatives with respect to \(x\) up to the second order and vanish at the endpoints of the interval \([0,\pi]\).

3) The free term \(f=\sum_{j=1}^{N} F_j e^{i\theta_j}\), \(L_0 f\) are continuous and satisfy the boundary conditions (3).

4) The initial function \(\varphi(x)\) has on the interval \([0,\pi]\) a continuous derivative of third order, and \(\varphi(x)\), \(L_0\varphi(x)\) satisfy the boundary conditions (9); while the function \(\psi(x)\) has on this interval a continuous derivative of second order, and \(\psi(x)\), \(L_0\psi(x)\) satisfy the boundary conditions (9).

Finally, let us show that the difference between the exact solution of the mixed problem
\[ u(x,t,\varepsilon)=\sum_{n=1}^{\infty} z_n(t,\varepsilon)W_n(x) \]
and the \(m\)-th approximation
\[ u_m(x,t,\varepsilon)=\sum_{n=1}^{\infty} z_n^m(t,\varepsilon)W_n(x), \]
where \(z_n^m(t,\varepsilon)\) is obtained by truncating the series (16) and (17) at the \(m\)-th terms, has order \(O(\varepsilon^m)\), i.e.,
\[ u(x,t,\varepsilon)=u_m(x,t,\varepsilon)+O(\varepsilon^m). \]
Consider the difference:
\[ u(x,t,\varepsilon)-u_m(x,t,\varepsilon) =\sum_{n=1}^{\infty}\left[z_n(t,\varepsilon)-z_n^m(t,\varepsilon)\right]W_n(x), \]
\[ \max_{x,t}|u-u_m|\leq \sum_{n=1}^{\infty}\max_t |z_n-z_n^m|\,\max_x |W_n(x)|\leq \]
\[ \leq C_1\sum_{n=1}^{\infty}\max_t |z_n-z_n^m| \leq C_1\|z-z^m\| \leq C_1 C_m\varepsilon^m=\widetilde C_m\varepsilon^m. \]

Thus, we see that the solution of the mixed problem (1)—(3) has the asymptotic character
\[ u=u_m+O(\varepsilon^m). \]

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Received by the editors
January 22, 1965

Uzhgorod State University

1. In the “nonresonant” case we obtain the estimate \(\|z_n-z_n^m\|\le C_m\varepsilon^{m+1}\). ↩↩