UDC 517.94; 517.948.35

ON THE THEORY OF OPERATORS OF THE FORM \(\dfrac{d^m}{dt^m}-A\)

V. K. ROMANKO

Introduction. In this paper we study the equation

\[ Lu \equiv \frac{d^m u}{dt^m} - Au = f \tag{L} \]

under “nonlocal” boundary conditions of the form

\[ B_j u \equiv a_j \left. \frac{d^{j-1}u}{dt^{j-1}}\right|_{t=0} + b_j \left. \frac{d^{j-1}u}{dt^{j-1}}\right|_{t=1} = 0, \tag{\(\Gamma\)} \]

\[ j = 1, 2, \ldots, m. \]

(Problem \((L)-(\Gamma)\).) Here \(t \in [0,1]\); \(a_j\) and \(b_j\) are complex numbers such that \(a_j+b_j \ne 0\) for all \(j=1,2,\ldots,m\); \(A\) is a differential polynomial in \(x\) with constant complex coefficients on the \(n\)-dimensional torus \(T^n\), or, in other words, considered on the set of functions periodic in \(x=(x_1,\ldots,x_n)\).

A special case of the conditions \((\Gamma)\) is formed by the so-called splitting boundary conditions, i.e., the conditions obtained from \((\Gamma)\) when \(a_j=0\), \(j \le \nu+1\), and \(b_j=0\), \(\nu+1<j\le m\). These conditions have the following form:

\[ \left. \frac{d^k u}{dt^k}\right|_{t=1} = \left. \frac{d^l u}{dt^l}\right|_{t=0} =0, \tag{\(\Gamma_0\)} \]

\[ k=0,1,\ldots,\nu-1;\qquad l=0,1,\ldots,m-\nu-1. \]

Without loss of generality one may assume that

\[ 0 \le \nu \le \left[\frac{m}{2}\right]. \]

The usefulness of considering equations of the form \((L)\) on a manifold that is the direct product of the torus \(T^n\) and the interval \([0,1]\) was first shown in [1], where certain “pathological” operators were considered, and then in [2], where the problem \((L)-(\Gamma_0)\) was studied in detail in the case \(m=1\), and in [3], where the problem \((L)-(\Gamma)\) with operators \(A\) on \(T^1\) was considered.

We are interested in the question of under what necessary and sufficient conditions on \(A\), \((\Gamma)\), and \((\Gamma_0)\) there exists and is bounded an inverse operator to \(L\). It is clear that the choice of the problem \((\Gamma)\) or \((\Gamma_0)\) depends essentially on the location of the points of the spectrum of the operator \(A\) in the complex plane \(C\), since the location of the spectrum of \(A\) in \(C\), in turn, determines the location of the roots of the characteristic equation for \((L)\) (to each point \(\lambda_s\) of the point spectrum of \(A\) there corresponds its own characteristic equation; see § 2).

This paper studies the character of this dependence. If the operator \(A\) is such that, for a given \(m\), the number of characteristic roots with nonnegative real part is the same for all \(s\), then boundary conditions \((\Gamma_0)\) are taken, and under them the operator \(L^{-1}\) exists and is bounded if and only if the number \(\nu\) in \((\Gamma_0)\) is chosen “correctly.” Otherwise the operator \(L^{-1}\), even if it exists, is unbounded. If, however, the number of characteristic roots with nonnegative real part varies depending on the location of the points of the spectrum of \(A\) in \(C\), then the boundary conditions \((\Gamma_0)\) are no longer sufficient, and one must take conditions of the form \((\Gamma)\). If the conditions \((\Gamma)\) are “weakly regular” (see § 2) and there exists a \(\delta>0\) such that the points of the spectrum of \(A\) do not fall into the \(\delta\)-neighborhood of the points of the discrete spectrum of the operator

\[ \mathcal L \equiv \frac{d^m}{dt^m} \]

under \((\Gamma)\), then the operator \(L^{-1}\) exists and is bounded.

If \((\Gamma)\) are “irregular,” then the operator \(L^{-1}\), even if it exists, is unbounded. Examples of operators \(A\) entering into our considerations are hypoelliptic differential polynomials with constant complex coefficients.

Finally, if the spectrum of \(A\) is such that it is not possible to choose conditions \((\Gamma)\) ensuring the existence of the operator \(L^{-1}\), in particular if the spectrum of \(A\) fills the whole plane \(C\), then we indicate boundary conditions with certain projection operators specially constructed from the operator \(A\), under which there always exists a bounded inverse to \(L\).

The plan of the article is as follows. In § 1 some definitions are given and the operators \(A\) considered by us are described. In § 2 the solutions of the problems \((L)\)—\((\Gamma)\) and \((L)\)—\((\Gamma_0)\) are written out under the assumption that \(A\) is a numerical parameter; estimates of these solutions are established; lemmas on the roots of the characteristic equation are proved; and formulas, taken from [4], for the discrete spectrum of

\[ \mathcal L \equiv \frac{d^m}{dt^m} \]

under the conditions \((\Gamma)\) and \((\Gamma_0)\) are written out. In § 3 the main results are formulated and proved.

§ 1. Basic definitions

Let the torus \(T^n\) be obtained by identifying the opposite faces of the cube
\(0 \leq x_k \leq 2\pi,\ k=1,2,\ldots,n\). If \(A(s)\) is a polynomial with constant complex coefficients,

\[ A(s)=\sum_{|\alpha|\leq r} A_\alpha s^\alpha,\qquad s^\alpha=s_1^{\alpha_1}\cdots s_n^{\alpha_n},\qquad |\alpha|=\alpha_1+\cdots+\alpha_n, \]

then by \(A=A(-iD)\) we denote the corresponding differential operator, for which

\[ A(-iD)\exp i(s,x)=A(s)\exp i(s,x);\qquad (s,x)=s_1x_1+\cdots+s_nx_n . \]

On \(T^n\) define the complex Hilbert space \(H_x\), generated by the scalar product and norm

\[ (u,v)_x=\int_{T^n} u\bar v\,dx,\qquad |u,H_x|^2=(u,u)_x . \]

By the operator \(A:H_x\to H_x\) we mean the closure of the operator \(A(-iD)\), initially defined on functions from \(C^\infty(T^n)\). Denote by \(S\) the set of integer vectors \(s=(s_1,\ldots,s_n)\), where each \(s_i=0,\pm1,\pm2,\ldots\). By \(A(S)\) denote the set belonging to the complex plane \(C\) ...

values of the polynomial \(A(s)\), as \(s\) ranges over all of \(S\). For the operator \(A\) the following assertions are valid (see [2]):

Lemma 1. The operator \(A\) is normal, i.e. \(AA^*=A^*A\).

Lemma 2. The spectrum \(\sigma_A\) of the operator \(A\) consists of the closure in \(C\) of the set \(A(S)\), which forms the discrete spectrum \(\sigma_d A\) of the operator \(A\).

Let now \(H\) be the Hilbert space of functions obtained by completing the set \(u(t,x)\in C^\infty,\ t\in[0,1],\ x\in T^n\), with respect to the norm

\[ |u,H|^2=\int_0^1 |u(t,x),H_x|^2\,dt. \tag{1} \]

Lemma 3. For any \(u\in H\), in the sense of equality in \(H\), the representation

\[ u(t,x)=\sum_s u_s(t)\exp i(s,x);\qquad |u,H|^2=(2\pi)^n\sum_s |u_s(t),H_t|^2, \tag{2} \]

holds, where

\[ |u_s,H_t|^2=\int_0^1 |u_s(t)|^2\,dt,\qquad s\in S. \]

(For \(u\in C^\infty\), \(u_s(t)\in C^\infty\).)

The validity of this assertion is well known.

Definition. We shall say that \(u\in H\) belongs to the domain of definition of the operator \(L\), if there exists a sequence of smooth functions \(\{u_i\}\), satisfying the conditions \((\Gamma)\) (or \((\Gamma_0)\)), and \(f\in H\) such that

\[ |u_i-u,H|\to 0,\qquad |Lu_i-f,H|\to 0,\qquad i\to\infty. \]

The corresponding solution of problem \((L)-(\Gamma)\) (problem \((L)-(\Gamma_0)\)) will be called a strong solution.

§ 2. Some results for ordinary differential equations

Let \(A\) be a numerical complex parameter. In this case we write down the explicit formulas for the solution of problem \((L)-(\Gamma)\) that will be used below.

1. Let \(A=0\). Then the general solution of \((L)\) has the form

\[ u(t)=\frac{1}{(m-1)!}\int_0^t (t-\tau)^{m-1} f(\tau)\,d\tau+\sum_{k=0}^{m-1} c_k t^k. \tag{3} \]

Since the determinant \(\Delta\) of the system of linear equations for the \(c_k\) in (3) under the conditions \((\Gamma)\), under our assumptions on \(a_j\) and \(b_j\), is always different from zero, the solution of problem \((L)-(\Gamma)\) for \(A=0\) is given by the formula

\[ u(t)=\frac{1}{(m-1)!}\int_0^t (t-\tau)^{m-1} f(\tau)\,d\tau -\frac{1}{\Delta}\sum_{k=0}^{m-1}\Delta_k t^k, \tag{4} \]

where

\[ \Delta=1!\,2!\,\ldots\,(m-1)!\prod_{j=1}^{m}(a_j+b_j), \]

and \(\Delta_k\) are obtained from \(\Delta\) by replacing in it the \(k\)-th column by the column consisting of

\[ \varkappa_r=\int_0^1 \frac{(1-\tau)^{m-1-r}}{(m-1-r)!}\,f(\tau)\,d\tau,\qquad r=0,1,\ldots,m-1. \]

In particular, the solution of problem \((L)-(\Gamma_0)\) for \(A=0\) has the form

\[ u(t)=\frac{1}{(m-1)!}\int_0^t (t-\tau)^{m-1} f(\tau)\,d\tau -\frac{1}{\Delta}\sum_{k=m-\nu}^{m-1}\Delta_k t^k, \tag{5} \]

where

\[ \Delta=\prod_{\nu\ge i>k\ge 1}(i-k), \]

and the \(\Delta_k\) are obtained from \(\Delta\) by replacing in it the \(k\)-th column by the column consisting of

\[ \varkappa_r=\int_0^1 \frac{(1-\tau)^{m-1-r}}{(m-1-r)!} f(\tau)\,d\tau, \qquad r=m-\nu,\ldots,m-1. \]

II. Let now \(|A|\ge \delta>0\), and let \(\omega_1,\omega_2,\ldots,\omega_m\) be the roots of the \(m\)-th degree of unity.

We shall call the following equation the characteristic equation for \((L)\):

\[ \lambda^m-A=0. \tag{6} \]

We shall study the location of the roots \(\lambda_j,\ j=1,2,\ldots,m,\) of this equation. Introduce the following notation. By \(l_+=l_+(m,A)\) denote the number of roots of (6) with \(\operatorname{Re}\lambda>0\), by \(l_-=l_-(m,A)\) the number of roots of (6) with \(\operatorname{Re}\lambda<0\), and by \(l_0=l_0(m,A)\) the number of roots of (6) with \(\operatorname{Re}\lambda=0\). Obviously, we have \(l_+ + l_- + l_0=m\).

Lemma 4. Let \(m=2\varkappa-1\). If \(\operatorname{Re} A>0\), then for \(\varkappa\) odd \(l_+=\varkappa,\ l_-=\varkappa-1\), and for \(\varkappa\) even \(l_+=\varkappa-1,\ l_-=\varkappa\). If \(\operatorname{Re} A<0\), then for \(\varkappa\) odd \(l_+=\varkappa-1,\ l_-=\varkappa\), and for \(\varkappa\) even \(l_+=\varkappa,\ l_-=\varkappa-1\). If \(\operatorname{Re} A=0\), then \(l_+=l_-=\varkappa-1\).

Proof. The proof is exactly the same in all cases; therefore we restrict ourselves to the case \(\operatorname{Re} A>0\).

If \(\arg A\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\), put \(\alpha=A^{1/m}\), \(\arg\alpha\in\left(-\dfrac{\pi}{2m},\dfrac{\pi}{2m}\right)\), and \(\lambda_j=\alpha\omega_j\). Number the numbers \(\omega_j\) in the following way:

\[ \omega_j=\exp\left(i\frac{2\pi j}{m}\right), \qquad j=0,\mp1,\ldots,\mp(\varkappa-1). \]

It is necessary to count the number of \(\omega_j\) with \(\operatorname{Re}(\alpha\omega_j)>0\). Such \(\omega_j\) are found from the condition

\[ \arg\alpha+\arg\omega_j\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right). \]

Hence we have that

\[ j\in\left(-\frac{\varkappa}{2},\frac{\varkappa}{2}\right). \]

If \(\varkappa=2\varkappa_1\), then the number of such \(j\) is

\[ 2(\varkappa_1-1)+1=2\varkappa_1-1=\varkappa-1, \]

whereas if \(\varkappa=2\varkappa_1+1\), then the number of such \(j\) is \(2\varkappa_1+1=\varkappa\).

Lemma 4 is proved.

Lemma 5. Let \(m=2\varkappa\). If \(\varkappa\) is odd, then for \(\arg A=\pi\), \(l_+=l_-=\varkappa-1\), and for \(\arg A\ne\pi\), \(l_+=l_-=\varkappa\). If \(\varkappa\) is even, then for \(\arg A=0\), \(l_+=l_-=\varkappa-1\), and for \(\arg A\ne0\), \(l_+=l_-=\varkappa\).

Proof. Let \(\arg A\in[0,2\pi)\), and let, for example, \(\varkappa\) be even. Put \(\lambda_j=\alpha\omega_j,\ \arg\alpha\in\left[0,\dfrac{2\pi}{m}\right)\). If \(\arg\alpha=0\), then \(l_+\) is equal to

number of \(\operatorname{Re}\omega_j>0\), which in turn is equal to \((\chi-1)\), and there are two purely imaginary roots. Let \(\arg\alpha\in\left(0,\dfrac{2\pi}{m}\right)\).

Let us number the numbers \(\omega_j\) in the following way:

\[ \omega_j=\exp\left(i\,\frac{2\pi j}{m}\right),\qquad j=0,\mp1,\ldots,\mp(\chi-1),\chi . \]

From the condition that \(\arg\alpha+\arg\omega_j\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\), we find that \(j\in\left(-\dfrac{\chi}{2},\dfrac{\chi+1}{2}\right)\). If \(\chi=2\chi_1\), then \(j\in\left(-\chi_1,\chi_1+\dfrac{1}{2}\right)\). The number of integer \(j\)’s from this interval is \(\chi\). The case of odd \(\chi\) is considered analogously. Lemma 5 is proved.

Definition. The decomposing conditions \((\Gamma_0)\) are called regular if

\[ \nu=l_+ \quad \text{or} \quad \nu=l_+ + l_0 . \]

In the opposite case \((\Gamma_0)\) are called irregular.

We now write out the formulas for the solution of the problems \((L)-(\Gamma)\) and \((L)-(\Gamma_0)\) in the case \(A\ne0\). If \(\arg A\in[-\pi,\pi)\), put \(\alpha=A^{1/m}\), \(\arg\alpha\in\left[-\dfrac{\pi}{m},\dfrac{\pi}{m}\right)\).

Remark. In what follows we shall assume that, for a given \(\alpha\), the roots \(\omega_j\) are numbered so that
\[ \operatorname{Re}(\alpha\omega_1)<\cdots<\operatorname{Re}(\alpha\omega_m). \]
This can always be done (see [4]).

The solution of problem \((L)-(\Gamma)\), under the assumption that \(\Delta(\alpha,\omega)\ne0\), is written in the form

\[ u(t)= \frac{(-1)^m}{\alpha^{m-1}W(\omega)} \sum_{p=1}^{m}\exp\alpha\omega_p t \times \]

\[ \times \left[ (-1)^p W_p(\omega)\int_0^t \exp(-\alpha\omega_p\tau)f(\tau)\,d\tau - \frac{1}{\Delta(\alpha,\omega)} \times \right. \]

\[ \left. \times \sum_{q=1}^{m}(-1)^q W_q(\omega)\Delta_{pq}(\alpha,\omega) \int_0^1 \exp(-\alpha\omega_q\tau)f(\tau)\,d\tau \right]. \tag{7} \]

Here \(W(\omega)\) is the Vandermonde determinant of the roots \(\omega\); \(W_p(\omega)\) is the determinant obtained from \(W(\omega)\) by deleting the last row and the \(p\)-th column, while the determinant \(\Delta(\alpha,\omega)\) has the form

\[ \Delta(\alpha,\omega)= \sum_{p=1}^{m}\sum_l \Theta_l \exp\alpha|\omega_l|+\Theta, \tag{8} \]

where \(l=(l_1,\ldots,l_p)\); \(l_1<l_2<\cdots<l_p\); \(|\omega_l|=\omega_{l_1}+\cdots+\omega_{l_p}\); \(l_j=1,2,\ldots,m\). The expansion of \(\Delta(\alpha,\omega)\) contains \(2^m\) terms \(\Theta_l\exp\alpha|\omega_l|\), where each \(\Theta\) is a determinant of a special form. We shall be interested in the coefficients of the exponentials with the largest exponents in (8). Since for any given \(\alpha\) (see Lemmas 4, 5) when \(m=2\mu-1\), \(\operatorname{Re}(\alpha\omega_j)<0\),

\(j \leqslant \mu-1\), and \(\operatorname{Re}(\alpha\omega_j)>0,\ j>\mu+1\), while for \(m=2\mu\), \(\operatorname{Re}(\alpha\omega_j)<0,\ j\leqslant\mu-1\), and \(\operatorname{Re}(\alpha\omega_j)>0,\ j\geqslant\mu+2\), then it is not hard to see that, for \(m=2\mu-1\), such coefficients will be the numbers \(\Theta_{(\mu,\ldots,m)}\), \(\Theta_{(\mu+1,\ldots,m)}\), and, for \(m=2\mu\), the numbers \(\Theta_{(\mu+2,\ldots,m)}\), \(\Theta_{(\mu,\ldots,m)}\), \(\Theta_{(\mu+1,\ldots,m)}\), and \(\Theta_{(\mu,\mu+2,\ldots,m)}\). All these coefficients depend on \(a_j,\ b_j,\ \omega\). In [4], for other purposes, the principal part of (8) as \(\alpha\to\infty\) was studied. This is not sufficient for us; however, the expressions for the above-written coefficients \(\Theta\) coincide with the coefficients of the principal part (8) written out in [4]. In the notation adopted in [4], for \(m=2\mu-1\),
\(\Theta_{(\mu,\ldots,m)}=\Theta_1,\ \Theta_{(\mu+1,\ldots,m)}=\Theta_0\), while for \(m=2\mu\),
\(\Theta_1=\Theta_{(\mu,\mu+2,\ldots,m)}\), \(\Theta_{-1}=\Theta_{(\mu+1,\ldots,m)}\), and \(\Theta_0=\Theta_{(\mu+2,\ldots,m)}+\Theta_{(\mu,\ldots,m)}\).

Definition. The conditions \((\Gamma)\) are called weakly regular if, for \(m=2\chi-1\), the numbers \(\Theta_1\) and \(\Theta_0\), and, for \(m=2\chi\), the number \(\Theta_{-1}\), are different from zero. If the conditions \((\Gamma)\) are regular (see [4]), then they are weakly regular. The \(\Delta_{pq}(\alpha,\omega)\) written in formula (7) is obtained from \(\Delta(\alpha,\omega)\) by replacing the \(p\)-th column by the column of \(b_j\omega_q\exp\alpha\omega_q\). As is easily seen, for \(p\ne q\), \(\Delta_{pq}(\alpha,\omega)\) is equal to \(\Delta(\alpha,\omega)\), in which the \(p\)-th column contains \(b_j\omega_q^{j-1}\exp\alpha\omega_q\), while the \(q\)-th column contains \(a_j\omega_q^{j-1}\), \(j=1,2,\ldots,m\).

The solution of problem \((L)-(\Gamma_0)\), under the condition that \(\Delta(\alpha,\omega)\ne0\), also has the form (7), with the only difference that the determinant

\[ \Delta(\alpha,\omega)=\sum_r(-1)^{\varphi_r}W(\omega_r)W_r(\omega_r)\exp\alpha|\omega_r|, \tag{9} \]

where

\[ r=(r_1,\ldots,r_\nu);\qquad \varphi_r=m\nu-\frac{1}{2}\nu(\nu-1)+|r|;\qquad r=r_1+\cdots+r_\nu; \]

\[ r_j=1,2,\ldots,m;\qquad r_1<r_2<\cdots<r_\nu;\qquad \omega_r=(\omega_{r_1},\ldots,\omega_{r_\nu}); \]

\[ |\omega_r|=\omega_{r_1}+\cdots+\omega_{r_\nu}, \]

and \(\Delta_{pq}(\alpha,\omega)\) is obtained from \(\Delta(\alpha,\omega)\) by replacing the \(p\)-th column by a column in which the first \(\nu\) elements are respectively equal to \(\omega_q^j\exp\alpha\omega_q,\ j=0,1,\ldots,\nu-1\), and the remaining elements are equal to zero. In other words,

\[ \Delta_{pq}(\alpha,\omega)= \sum_{r',\,r'\not\ni p} (-1)^{\varphi_{r'}}W(\omega_{r'})W_{r'}(\omega_q,\omega_{r'}) \exp\alpha(\omega_q+|\omega_{r'}|), \]

where

\[ r'=(r_1,\ldots,r_{\nu-1});\qquad \omega_{r'}=(\omega_{r_1},\ldots,\omega_{r_{\nu-1}});\qquad |\omega_{r'}|=\omega_{r_1}+\cdots+\omega_{r_{\nu-1}}. \]

As is known (see [4]), the asymptotics of the eigenvalues \(\lambda\) of the operator \(\mathcal L\equiv\dfrac{d^m}{dt^m},\ t\in[0,1]\), under regular \((\Gamma)\), is given by the formulas:

1) \(m=2\chi-1\)

\[ \lambda_k=(2k\pi i)^m \left[ 1+\frac{m\ln_0\xi}{2k\pi i} +O\left(\frac{1}{k^2}\right) \right], \]

where

\[ \xi=-\frac{\Theta_0}{\Theta_1};\qquad \ln_0\xi=\ln|\xi|+i\arg\xi;\qquad k=\pm1,\ \pm2,\ldots; \tag{10} \]

2) \(m=2\chi\)

\[ \lambda_k=(2k\pi i)^m \left[ 1+\frac{m\ln_0\xi}{2|k|\pi i} +O\left(\frac{1}{k^2}\right) \right], \]

where \(k=\pm 1,\ \pm 2,\ldots;\ \xi=\begin{cases}\xi',& k>0,\\ \xi'',& k<0,\end{cases}\ \xi',\xi''\) are the roots of the equation \(\Theta_1 \xi^2+\Theta_0\xi+\Theta_{-1}=0\). The asymptotics of the eigenvalues \(\lambda\) of the operator \(\mathcal L=\dfrac{d^m}{dt^m}\), \(t\in[0,1]\), under decomposing conditions \((\Gamma_0)\) is given by the formulas (see [4]):

\[ \lambda_k=\rho_k^m;\quad \rho_k\in\varphi=\left\{\rho,\ \arg\rho\in\left[-\frac{\pi}{m},\frac{\pi}{m}\right)\right\},\quad k=\pm 1,\pm 2,\pm 3,\ldots, \]

\[ \rho_k=\pi\, \frac{ k+\dfrac12-\dfrac{1}{2m}\,[\nu(\nu+1)+(m-\nu)(m-\nu-1)] }{ \sin\dfrac{\nu\pi}{m} } + \]

\[ +O\left(\frac1k\right),\quad \nu\ \text{even}, \tag{11} \]

\[ \rho_k=\pi\exp\left(-\frac{i\pi}{m}\right)\, \frac{ k+\dfrac12-\dfrac{1}{2m}\,[\nu(\nu+1)+(m-\nu)(m-\nu-1)] }{ \sin\dfrac{\nu\pi}{m} } + \]

\[ +O\left(\frac1k\right),\quad \nu\ \text{odd}. \]

If \(A=0\), then for any smooth solution of the problem \((L)—(\Gamma)\) or \((L)—(\Gamma_0)\) the estimate is obvious:

\[ |u(t)|^2\le C\,|f,H_t|^2,\quad C>0. \tag{12} \]

Under the assumption that \(A\ne0\) and \(\Delta(\alpha,\omega)\ne0\), we shall now establish estimates for smooth solutions of the problem \((L)—(\Gamma)\) under weakly regular conditions \((\Gamma)\) and of the problem \((L)—(\Gamma_0)\) under proper conditions \((\Gamma_0)\).

Lemma 6. If \(|\Delta(\alpha,\omega)|\ge \delta>0\), then for smooth solutions of the problem \((L)—(\Gamma)\) under weakly regular conditions \((\Gamma)\) and of the problem \((L)—(\Gamma_0)\) under proper conditions \((\Gamma_0)\), the estimate

\[ |u(t)|\le \frac{C}{|\alpha|^{m-1}}\,|f,H_t|,\quad C=C(\delta)>0 \tag{13} \]

holds.

Proof. We shall use the solution formula (7), putting \(\alpha\omega_p=r_p+iq_p\), where \(r_p\) and \(q_p\) are real. If, for brevity, we set

\[ u(t)=-\frac{(-1)^m}{\alpha^{m-1}W(\omega)}\sum_{p=1}^{m}v_p(t), \]

then

\[ |u(t)|^2\le \frac{C}{|\alpha|^{2m-2}}\sum_{p=1}^{m}|v_p(t)|^2,\quad C>0, \]

and to prove (13) it is necessary to obtain an estimate of each term \(v_p(t)\) in formula (7). For this it suffices to consider separately the case \(r_p>0\),

\(r_p<0\) and \(r_p=0\). For \(|\alpha|\leqslant N\) estimate (13) is obvious. For \(|\alpha|\to\infty\), by virtue of the conditions of Lemma 6, from formulas (8), (9) we have

\[ |\Delta(\alpha,\omega)|=O[\exp(r_{m-\nu+1}+\cdots+r_m)]. \tag{14} \]

In formula (14) all \(r_j>0,\ j=m-\nu+1,\ldots,m\), and for each prescribed value \(A\ne0\) the number \(\nu\) is chosen equal to \(l_+\) or \(l_+ + l_0\).

1) Let \(r_p>0\). Then, using the formula
\[ \int_0^t=\int_0^1-\int_t^1, \]
we write \(v_p(t)\) in the following form:

\[ \begin{aligned} |v_p(t)|^2 = \exp 2r_p t \bigg| &(-1)^{p+1} W_p(\omega)\int_t^1 \exp(-\alpha\omega_p\tau) f(\tau)\,d\tau \\ &+(-1)^p W_p(\omega)\left[1-\frac{\Delta_{pp}(\alpha,\omega)}{\Delta(\alpha,\omega)}\right] \int_0^1 \exp(-\alpha\omega_p\tau) f(\tau)\,d\tau \\ &-\frac{1}{\Delta(\alpha,\omega)} \sum_{\substack{q=1\\ q\ne p}}^m (-1)^q W_q(\omega)\Delta_{pq}(\alpha,\omega) \int_0^1 \exp(-\alpha\omega_q\tau) f(\tau)\,d\tau \bigg|^2 . \end{aligned} \]

The expression \([\Delta(\alpha,\omega)-\Delta_{pp}(\alpha,\omega)]\) has exactly the form of \(\Delta(\alpha,\omega)\), except that, instead of the \(p\)-th column, it contains the column consisting of \(a_j\omega_p^{\,j-1}\), \(j=1,2,\ldots,m\), if the conditions \((\Gamma)\) are taken, and the column in which the first \(\nu\) elements are zero and the remaining elements are respectively \(\omega_p^{\,j}\), \(j=0,1,\ldots,m-\nu-1\), if the conditions \((\Gamma_0)\) are taken.

Since each term in the expansion of the determinant \([\Delta(\alpha,\omega)-\Delta_{pp}(\alpha,\omega)]\) may have in the exponent no more than \((\nu-1)\) numbers \(r_j>0\), it follows from (14) that

\[ \left|\frac{\Delta-\Delta_{pp}}{\Delta}\right|^2 \exp 2r_p t \leqslant M_1 . \]

For the same reason, for \(q\ne p;\ p=1,2,\ldots,m\),

\[ \left|\frac{\Delta_{pq}(\alpha,\omega)}{\Delta(\alpha,\omega)}\right|^2 \exp 2r_p t \leqslant M_2 . \]

Using these estimates, as well as the Cauchy—Bunyakovsky inequality, we obtain the following:

\[ \begin{aligned} |v_p(t)|^2 \leqslant C'\exp 2r_p t \bigg\{& \left|\int_t^1 \exp(-\alpha\omega_p\tau) f(\tau)\,d\tau\right|^2 +\left|\frac{\Delta-\Delta_{pp}}{\Delta}\right|^2 \\ &\times \left|\int_0^1 \exp(-\alpha\omega_p\tau) f(\tau)\,d\tau\right|^2 \\ &+\sum_{\substack{q=1\\ q\ne p}}^m \left|\frac{\Delta_{pq}(\alpha,\omega)}{\Delta(\alpha,\omega)}\right|^2 \left|\int_0^1 \exp(-\alpha\omega_q\tau) f(\tau)\,d\tau\right|^2 \bigg\} \\ &\leqslant C''\|f,H_t\|^2 \left\{\frac{1}{r_p}+\sum_{q\ne p}\frac{1}{|r_q|}+C_0\right\}, \end{aligned} \]

where \(C_0=0\), if there is no \(r_q=0,\ q\ne p\). Hence
\[ |v_p(t)|^2 \leqslant C_1^2\|f,H_t\|^2 . \]

2) Let \(r_p < 0\). Arguing in the same way as above, we have

\[ \left|\frac{\Delta_{pq}(\alpha,\omega)}{\Delta(\alpha,\omega)} \exp \alpha\omega_q \right|\leq M,\qquad q=1,2,\ldots,m. \]

Then from formula (7) we at once obtain in this case the estimate for \(v_p(t)\)

\[ |v_p(t)|^2 \leq C'\,|f,H_t|^2 \left\{\frac{1}{|r_p|}+\sum_q \frac{1}{|r_q|}+C_0\right\}. \]

Hence

\[ |v_p(t)|^2 \leq C_2^2\,|f,H_t|^2. \]

The case \(r_p=0\) is considered analogously to 2).

Let us note that all constants in the estimates depend only on \(m,\ a_j,\ b_j,\ \omega_j\). From the estimates obtained, (13) follows.

Lemma 6 is proved.

§ 3. Main theorems

Definition. The operator \(A\), for a given \(m\), is called stable if there exists an \(N\) such that \(l_+(m,A(s))=\mathrm{const}\) for every \(s\in S\), \(|s|>N\). From Lemmas 4, 5 it follows at once that \(A\) will be stable if and only if the polynomial \(A(s)\) satisfies one of the following conditions:

1) if \(m=2\chi-1\), then either \(\operatorname{Re} A(s)\geq 0\) for all \(s\in S\), or \(\operatorname{Re} A(s)\leq 0\) for all \(s\in S\), \(|s|>N\);

2) if \(m=2\chi\), then for \(\chi\) even either \(\arg A=0\) for all \(s\in S\), or \(\arg A\ne 0\) for all \(s\in S\), while for \(\chi\) odd either \(\arg A(s)=\pi\) for all \(s\in S\), or \(\arg A(s)\ne\pi\) for all \(s\in S\), \(|s|>N\).

A stable operator \(A\) for all \(m\) is given, for example, by the polynomial \(A(s)=s^2\), while an example of an operator \(A\) that is stable for no \(m\) is the operator generated by the polynomial \(A(s)=s\). The eigenvalues of the operator \(L\) under regular conditions \((\Gamma)\) or under decomposing conditions \((\Gamma_0)\), as is not difficult to see, are given by the formula

\[ \lambda_{sk}=-A(s)+\lambda_k, \tag{15} \]

where \(s\in S\), and the values \(\lambda_k\) are the eigenvalues of the operator

\[ \mathcal{L}=\frac{d^m}{dt^m},\qquad t\in[0,1] \]

under the conditions \((\Gamma)\) or \((\Gamma_0)\), and their asymptotics are given respectively by formula (10) or (11).

Denote by \(\Lambda(m,\Gamma)\) the set of eigenvalues of the operator \(\mathcal{L}\) under the conditions \((\Gamma)\), and by \(d[z,\Lambda(m,\Gamma)]\) the distance from the point \(z\in C\) to the set \(\Lambda(m,\Gamma)\). By \(R_\delta(\Gamma)\), for a given \(\delta>0\), we denote the set of points \(\{z\in C,\ d[z,\Lambda(m,\Gamma)]\geq\delta\}\).

Analogously, for the operator \(\mathcal{L}\) under the conditions \((\Gamma_0)\), the sets \(\Lambda(m,\Gamma_0)\), \(d[z,\Lambda(m,\Gamma_0)]\), and \(R_\delta(\Gamma_0)\) are defined. If \(A\) is stable and \((\Gamma_0)\) are regular, then from (11) it follows that, adding, if necessary, a constant to \(A(S)\), one can always find a \(\delta>0\) such that

\[ \sigma A=\overline{A(S)}\subseteq R_\delta(\Gamma_0). \tag{16} \]

(In what follows it is always assumed that, for the stable operator \(A\), condition (16) is fulfilled.)

Theorem 1. If the operator \(A\) is stable, then problem \((L)-(\Gamma_0)\) under regular conditions \((\Gamma_0)\) has a unique strong solution for every \(f\in H\). Under irregular conditions \((\Gamma_0)\), whenever not all

the roots of (6) are purely imaginary, the operator \(L^{-1}\), if it exists, is unbounded.

Remark. The case of purely imaginary roots of (6) is possible only for \(m=1,2\).

Proof. Let the conditions \((\Gamma_0)\) be proper. In this case, to prove the theorem it is enough to establish the existence of a smooth solution of the problem \((L)—(\Gamma_0)\), for which the energy inequality

\[ |u,H|\leq C|f,H|,\quad C>0. \tag{\(\Phi\)} \]

holds.

By virtue of (16), zero is not an eigenvalue of \(L\). Therefore the Fourier coefficients \(u_s(t)\) of the smooth solution \(u(t,x)\) of the problem \((L)—(\Gamma_0)\) can be uniquely determined in the representation (2) as solutions of the problem

\[ \frac{d^m u_s(t)}{dt^m}-A(s)u_s(t)=f_s(t), \]

\[ \left.\frac{d^k u_s}{dt^k}\right|_{t=1} = \left.\frac{d^l u_s}{dt^l}\right|_{t=0} =0,\quad k=0,1,\ldots,\nu-1;\quad l=0,1,\ldots,m-\nu-1, \tag{17} \]

where \(f_s(t)\) are the Fourier coefficients in the expansion (2) for \(f\). Therefore a smooth solution of \((L)—(\Gamma_0)\) exists, and to obtain \((\Phi)\), as follows from Lemma 3, it is enough to establish the inequality

\[ |u_s,H_t|\leq C|f_s,H_t|, \tag{\(\Phi_s\)} \]

where the constant \(C\) does not depend on \(s\).

We shall use the solution formulas (5), (7), (9). In these formulas

\[ A=A(s),\quad \alpha=\alpha(s). \]

If \(s\in S\) is such that \(|A(s)|\leq M\), then \((\Phi_s)\) is obvious. If, however, \(s\in S\) is such that \(|A(s)|\geq \delta>0\), then \((\Phi_s)\) follows from Lemma 6. Indeed, by virtue of (16) there exists \(\delta_1>0\) such that \(|\Delta(\alpha_s,\omega)|\geq \delta_1\) uniformly in \(s\in S\), and hence, by virtue of (13),

\[ |u_s(t)|\leq \frac{C_1}{|\alpha_s|^{m-1}}\,|f_s,H_t|,\quad C_1=C_1(\delta_1)>0. \]

Since \(|A(s)|\geq \delta>0\), for the \(s\in S\) under consideration

\[ |u_s(t)|\leq C|f_s,H_t|,\quad C=\operatorname{const}>0. \]

This implies the inequality \((\Phi_s)\) and, consequently, the operator \(L^{-1}\) exists and is bounded.

It remains to show that \(L^{-1}\) is defined on all of \(H\). For any \(f\in H\) there exists a sequence of smooth functions \(\{f_i\}\), which are finite sums of the form (2) with smooth coefficients \(f_{s,i}(t)\), such that \(|f_i-f,H|\to 0\) as \(i\to\infty\). Under the assumptions of the theorem, as shown above, the solution of the problem \((L)—(\Gamma_0)\) for \(f=f_i\) always exists and is a finite sum of the form (2), in which \(u_{s,i}\) are determined from \(f_{s,i}\) on the basis of (17). By the inequality \((\Phi)\), the \(u_i\) form a convergent sequence in \(H\), whose limit is the solution of \((L)—(\Gamma_0)\).

The first part of Theorem 1 is proved. Let us prove its second part.

Let the conditions \((\Gamma_0)\) be improper and not all roots of (16) purely imaginary. Consider a sequence \(\{s^r\}_{r=1}^{\infty}\subset S\), \(A_r=A(s^r)\ne0\), \(A_r\subset R_\delta(\Gamma_0)\) for

for all \(s\), \(\alpha_r=\alpha(s^r)\to\infty\), \(r\to\infty\), and the corresponding sequence of right-hand sides \(f^r=\exp i(s^r,x)\). Then \(\lvert f^r,H\rvert^2=(2\pi)^n\), and the solution of problem \((L)-(\Gamma_0)\) for the chosen right-hand side will be the function
\[ u^r(t,x)=u^r(t)\times \exp i(s^r,x), \]
where \(u^r(t)\) is computed by formula (7), (9) with \(\alpha=\alpha_r\) and \(f=1\). We have
\[ \lvert u^r(t,x),H\rvert^2=(2\pi)^n\lvert u^r(t),H_t\rvert^2, \tag{18} \]
\[ u^r(t)=\frac{(-1)^m}{A_rW(\omega)} \sum_{p=1}^{m}\exp\alpha_r\omega_p t \times \]
\[ {}\times\left[ \frac{(-1)^p W_p(\omega)}{\omega_p} \left(1-\exp(-\alpha_r\omega_p t)\right) -\frac{1}{\Delta(\alpha_r,\omega)}\times \right. \]
\[ \left. {}\times \sum_{q=1}^{m} \frac{(-1)^q W_q(\omega)}{\omega_q} \Delta'_{pq}(\alpha_r,\omega) \left(\exp\alpha_r\omega_q-1\right) \right], \]
where \(\Delta(\alpha_r,\omega)\) is given by (9), and \(\Delta'_{pq}\) are obtained from \(\Delta_{pq}\) after factoring out \(\exp\alpha_r\omega_q\) from the \(p\)-th column.

Let \(\operatorname{Re}(\alpha_r\omega_p)>0\). Denoting the coefficient of \(\exp\alpha_r\omega_p t\) by \(v_p(t)\), we have
\[ v_p(t)=\frac{(-1)^m}{A_rW(\omega)} \frac{(-1)^p W_p(\omega)}{\omega_p}\times \]
\[ {}\times\left\{ \left[\exp\alpha_r\omega_p(t-1)-1\right] -\exp\alpha_r\omega_p t\left[\exp(-\alpha_r\omega_p)-1\right]\times \right. \]
\[ {}\times\left[ 1+\frac{\Delta'_{pp}(\alpha_r,\omega)}{\Delta(\alpha_r,\omega)} \right] + \left\{ \frac{(-1)^{m+1}}{A_rW(\omega)} \frac{\exp\alpha_r\omega_p t}{\Delta(\alpha_r,\omega)} \times \right. \]
\[ \left. \left. {}\times \sum_{\substack{q=1\\ q\ne p}}^{m} \frac{(-1)^q W_q(\omega)}{\omega_q} \Delta'_{pq}(\alpha_r,\omega) \left[\exp\alpha_r\omega_q-1\right] \right\} \right\}. \]

Since, as \(\alpha_r\to\infty\),
\[ \lvert\Delta'_{pq}\rvert = O\left(\exp\{\operatorname{Re}\alpha_r(\omega_{p_1}+\cdots+\omega_{p_{\nu-1}})\}\right) \]
and
\[ \lvert\Delta\rvert = O\left(\exp\{\operatorname{Re}\alpha_r(\omega_{p_1}+\cdots+\omega_{p_\nu})\}\right), \]
where \(\operatorname{Re}\alpha_r(\omega_{p_1}+\cdots+\omega_{p_\nu})>0\), the expression in the first brace is bounded.

Consider the expression in the second brace. If \(\operatorname{Re}\alpha_r\omega_q>0\), then the principal part of this expression is the ratio
\[ \frac{\exp\alpha_r\bigl(\lvert\omega_r'\rvert+\omega_q+\omega_p t\bigr)} {\exp\alpha_r\lvert\omega_r'\rvert}. \]
It follows from this that in the case when \(0<t<l_+-1\) and when not all roots of equation (6) are purely imaginary,
\[ \operatorname{Re}\alpha_r\bigl(\lvert\omega_r'\rvert+\omega_q+\omega_p t\bigr)>0, \]
and the ratio under consideration increases without bound as \(\alpha_r\to\infty\). Hence we obtain that \(\lvert u^r(t)\rvert\to\infty\) as \(\alpha_r\to\infty\).

Theorem 1 is proved.

We now consider operators \(A\) that are not stable. To describe correct boundary-value problems for \(L\) with such operators \(A\), it is already necessary to bring in the conditions \((\Gamma)\).

Theorem 2. If the non-stable operator \(A\) and the weakly regular conditions \((\Gamma)\) are such that there exists \(\delta>0\) for which \(\sigma A\subseteq R_\delta(\Gamma)\), then there exists a unique strong solution of problem \((L)-(\Gamma)\) for every \(f\in H\). If, however, the conditions \((\Gamma)\) are not weakly regular, then the operator \(L^{-1}\), if it exists, is unbounded.

Proof. The scheme of the proof of Theorem 2 is the same as that of Theorem 1; therefore we shall note only the main points of the proof.

As in Theorem 1, in this case it suffices, for weakly regular \((\Gamma)\), to establish the existence of a smooth solution of problem \((L)-(\Gamma)\) for which the energy inequality \((\Phi)\) is valid. By the assumptions of Theorem 2, the Fourier coefficients \(u_s(t)\) of the smooth solution \(u(t,x)\) of problem \((L)-(\Gamma)\) are uniquely determined in representation (2) as solutions of the problem

\[ \frac{d^m u_s(t)}{dt^m}-A(s)u_s(t)=f_s(t), \]

\[ B_j u_s \equiv a_j\frac{d^{j-1}u_s}{dt^{j-1}}\bigg|_{t=0} +b_j\frac{d^{j-1}u_s}{dt^{j-1}}\bigg|_{t=1}=0,\qquad j=1,2,\ldots,m, \tag{19} \]

and, in order to obtain \((\Phi)\), it is sufficient to establish \((\Phi_s)\). If one uses the solution formulas (4), (7), (8), then the inequality \((\Phi_s)\) follows from (13), since from the conditions of Theorem 2 we have that \(|\Delta(\alpha_s,\omega)|\ge \delta>0\) uniformly in \(s\in S\). The proof that \(L^{-1}\) is defined on all of \(H\) is carried out exactly as in Theorem 1.

We pass to the second part of Theorem 2. Suppose that the conditions \((\Gamma)\) are not weakly regular. Consider a sequence \(\{s^r\}_{r=1}^{\infty}\subset S\), \(A_r=A(s^r)\ne0\), \(\alpha_r=\alpha(s^r)\to\infty\), \(r\to\infty\), such that \(|\Delta(\alpha_r,\omega)|\ge\delta>0\) for all \(r=1,2,\ldots\), where \(\Delta\) is given by formula (8), and a sequence of right-hand sides
\[ f^r=\exp i(s^r,x). \]
Then \(|f^r,H|^2=(2\pi)^n\), and the solution of problem \((L)-(\Gamma)\) with such a right-hand side will be
\[ u^r(t,x)=u^r(t)\exp i(s^r,x), \]
where \(u^r(t)\) is computed by formula (18), in which \(\Delta\) is given by formula (8).

The further course of the argument is the same as in Theorem 1.

It remains to indicate how specifically to choose \(\{\alpha_r\}_{r=1}^{\infty}\), depending on which of the numbers \(\Theta\) tends to zero.

a) \(m=2\varkappa-1\). If \(\Theta_0=\Theta_1=0\), then as \(\{\alpha_r\}\) one may take an arbitrary sequence. If \(\Theta_1=0\), then for odd \(\varkappa\) we take
\[ \arg\alpha_r\in\left(-\frac{\pi}{2m},\frac{\pi}{2m}\right), \]
and for even \(\varkappa\)
\[ \arg\alpha_r\in\left(\frac{\pi}{2m},\frac{3\pi}{2m}\right). \]
If \(\Theta_0=0\), then for odd \(\varkappa\) we take
\[ \arg\alpha_r\in\left(\frac{\pi}{2m},\frac{3\pi}{2m}\right), \]
and for even \(\varkappa\)
\[ \arg\alpha_r\in\left(-\frac{\pi}{2m},\frac{\pi}{2m}\right). \]

b) \(m=2\varkappa\). If \(\Theta_{-1}=\Theta_1=\Theta_0=0\), then we take an arbitrary sequence \(\{\alpha_r\}\). If only \(\Theta_{-1}=0\), then for odd \(\varkappa\) we take \(\arg\alpha_r\ne\pi\), and for even \(\varkappa\) we take \(\arg\alpha_r\ne0\). This completes the proof of Theorem 2.

Remark 1. As shown in [2], for \(m=1\) the condition \(\sigma A\subseteq R_\delta(\Gamma)\) is also necessary. In the general case we have not been able to prove this.

Remark 2. Under nonregular conditions \((\Gamma)\) cases are possible in which the operator \(L^{-1}\) does not exist at all for any given operator \(A\) of the type under consideration, or \(L^{-1}\), although it always exists, is always unbounded. We shall show this by an example.

Let \(m=2\). In this case the only boundary conditions \((\Gamma)\) of the type considered by us are the following conditions:

\[ \left. \begin{aligned} u\big|_{t=0}+\mu u\big|_{t=1}&=0,\\[4pt] \frac{du}{dt}\bigg|_{t=0}-\mu \frac{du}{dt}\bigg|_{t=1}&=0 \end{aligned} \right\} \tag{$\Gamma$} \]

If \(\mu=\pm 1\), then the eigenvalues \(\lambda\) of the operator \(\mathscr L u \equiv \dfrac{d^2u}{dt^2}\) under the conditions \((\Gamma)\) fill the entire complex plane \(C\). Indeed, let \(\lambda=\rho^2\). Then in the case \(\mu=+1\), for \(\rho=0\) an eigenfunction is
\[ u(t)=t-\frac12, \]
while for \(\rho\ne0\) an eigenfunction is
\[ u(t)=\exp(-\rho t)-\exp(-\rho+\rho t), \]
and in the case \(\mu=-1\), for \(\rho=0\) an eigenfunction is \(u(t)=1\), while for \(\rho\ne0\) an eigenfunction is
\[ u(t)=\exp \rho t+\exp(-\rho t)\,\frac{1+\exp(-\rho)}{1+\exp \rho}. \]

Consequently, the eigenvalues of the operator \(\mathscr L\) under the conditions \((\Gamma)\) fill the entire plane \(C\). Therefore, as is seen from (15), the operator \(L^{-1}\) does not exist.

Now let \(\mu\ne\pm1\). Then \(\Delta=2(\mu^2-1)\ne0\), and hence, for any prescribed \(A\), the operator \(L^{-1}\) exists. The solution of problem \((L)-(\Gamma)\) with right-hand side \(f^r=\exp i(s^r,x)\), \(r=1,2,\ldots\), will be
\[ u^r(t,x)=u^r(t)\exp i(s^r,x), \]
where
\[ u^r(t)=\frac{1}{2A_r}\left[ \frac{1}{1-\mu}\exp \alpha_r t +\frac{\mu}{\mu-1}\exp \alpha_r(t-1) +\frac{1}{\mu+1}\exp(-\alpha_r t) +\frac{\mu}{\mu+1}\exp \alpha_r(1-t)-2 \right], \]
\[ A_r=A(s^r),\qquad \alpha_r=\sqrt{A_r},\qquad \arg\alpha_r\ne\pm\frac{\pi}{2}. \]

From this we have \(\lvert u^r(t)\rvert\to\infty\) as \(\lvert\alpha_r\rvert\to\infty\).

Remark 3. In considering boundary conditions \((\Gamma)\) of a more general form than ours, the regularity requirement for \((\Gamma)\) may prove insufficient for describing the asymptotics \(\Lambda(\alpha,\omega)\) used in our constructions. An example is provided by the conditions \((\Gamma)\), considered in [1] for the ultrahyperbolic operator \(L\). Although the conditions \((\Gamma)\) considered there are not weakly regular, nevertheless they give a permissible extension of \(L\).

We pass to the consideration of the case when \(\sigma A\) does not belong to \(R_\delta(\Gamma)\) for any \(\delta>0\), in particular, when \(\sigma A=C\). Let \(m=2\chi\) \((m=2\chi-1)\) and let \(A\) be fixed. We divide the set \(S\) into two subsets \(S^+\) and \(S^-\), assuming that \(s\in S^+\) if in equation (6), where \(A=A(s)\), \(l_+=\chi\), and \(s\in S^-\) if \(l_+=\chi-1\). The partition of \(S\) induces the partition of \(H_\chi\) into the sum of orthogonal subspaces
\[ H_\chi=H_\chi^+\oplus H_\chi^-, \]
where \(H_\chi^+\) \((H_\chi^-)\) is the closed linear span of the set of vectors of the form \(\exp i(s,x)\), \(S\in S^+\) \((s\in S^-)\).

Denote by \(\mu^+\), \(\mu^-\) the projection operators in \(H_\chi^+\), \(H_\chi^-\), respectively, and consider boundary conditions, for \(m=2\chi-1\), of the form
\[ \mu^-\frac{d^k u}{dt^k}\bigg|_{t=0} +\mu^+\frac{d^k u}{dt^k}\bigg|_{t=1}=0, \tag{$\Gamma_1$} \]

\[ \mu^{-}\left.\frac{d^{l}u}{dt^{l}}\right|_{t=1} +\mu^{+}\left.\frac{d^{l}u}{dt^{l}}\right|_{t=0}=0, \tag{\(\Gamma_{1}\)} \]

\[ k=0,\,1,\ldots,\varkappa-1;\qquad l=0,\,1,\ldots,\varkappa-2, \]

and for \(m=2\varkappa\), conditions of the form

\[ \mu^{-}\left.\frac{d^{k}u}{dt^{k}}\right|_{t=0} +\mu^{+}\left.\frac{d^{k}u}{dt^{k}}\right|_{t=1}=0, \]

\[ \mu^{-}\left.\frac{d^{l}u}{dt^{l}}\right|_{t=1} +\mu^{+}\left.\frac{d^{l}u}{dt^{l}}\right|_{t=0}=0, \tag{\(\Gamma_{2}\)} \]

\[ \mu^{-}\left.\frac{d^{\varkappa}u}{dt^{\varkappa}}\right|_{t=0} +\mu^{+}\left.\frac{d^{\varkappa-1}u}{dt^{\varkappa-1}}\right|_{t=1}=0, \]

\[ k=0,\,1,\ldots,\varkappa-1;\qquad l=0,\,1,\ldots,\varkappa-2. \]

Theorem 3. For any operator \(A\), the problem \((L)—(\Gamma_{1})\) for \(m=2\varkappa-1\) and the problem \((L)—(\Gamma_{2})\) for \(m=2\varkappa\) has a unique strong solution for every \(f\in H\).

The scheme of proof of this theorem is the same as in Theorem 1. Let us first note that the Fourier coefficients of the expansion
\[ u(t,x)=\sum_{s\in S}u_s(t)\exp i(s,x) \]
of a smooth solution of \((L)—(\Gamma_{1})\) and \((L)—(\Gamma_{2})\) are uniquely determined as solutions of problem (17), where \(\nu=\varkappa\) if \(s\in S^{+}\), and \(\nu=\varkappa-1\) if \(s\in S^{-}\). Therefore, for \(s\in S^{+}\) and for \(s\in S^{-}\) separately, Lemma 6 gives the estimate \((\Phi_s)\). Hence follows the energy inequality \((\Phi)\), from which (see Theorem 1) we obtain the assertion of Theorem 3.

References

1. Dezin A. A. Dokl. Akad. Nauk SSSR, 148, No. 5, 1013—1016, 1963.
2. Dezin A. A. Dokl. Akad. Nauk SSSR, 164, No. 5, 963—966, 1965.
3. Romanko V. K. Dokl. Akad. Nauk SSSR, 159, No. 2, 269—272, 1964.
4. Naimark M. A. Linear Differential Operators. Gostekhizdat, 1954.

Received by the editors
October 19, 1966

Moscow Institute of Physics and Technology