MATHEMATICS

M. A. RUTMAN

A BOUNDEDNESS CRITERION FOR SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS HAVING A HIGHEST TERM

(Presented by Academician I. G. Petrovskii on 16 VI 1962)

The present work adjoins (1–6) and, in a certain sense, is the concluding one in this series of papers.

We shall consider the differential equation

\[ \frac{\partial^{p_1+p_2+\cdots+p_n}y} {\partial t_1^{p_1}\partial t_2^{p_2}\cdots\partial t_n^{p_n}} - \sum_{(q_1q_2\ldots q_n)} A_{q_1q_2\ldots q_n} \frac{\partial^{q_1+q_2+\cdots+q_n}y} {\partial t_1^{q_1}\partial t_2^{q_2}\cdots\partial t_n^{q_n}} =x \tag{1} \]

in the domain \(0 \leqslant t_1, t_2, \ldots, t_n < \infty\).

Here \(y=y(t_1,t_2,\ldots,t_n)\), \(x=x(t_1,t_2,\ldots,t_n)\) are vector-functions whose values belong to a Banach (complex) space \(E\); \(A_{q_1q_2\ldots q_n}=A_{q_1q_2\ldots q_n}(t_1,t_2,\ldots,t_n)\) are families of linear operators acting in \(E\). The first term on the left-hand side is assumed to be the “highest” one: this means that \(p_j \geqslant q_j\),

\[ \sum_1^n p_j > \sum_1^n q_j \]

for every term of

\[ \sum_{(q_1q_2\ldots q_n)} . \]

The Cauchy–Goursat boundary conditions natural for equation (1) shall, for simplicity and without loss of generality, be considered homogeneous:

\[ y\big|_{t_j=0} = \frac{\partial y}{\partial t_j}\bigg|_{t_j=0} = \cdots = \frac{\partial^{p_j-1}y}{\partial t_j^{p_j-1}}\bigg|_{t_j=0} =0 \quad (j=1,2,\ldots,n). \tag{2} \]

Suppose further that the coefficients \(A_{q_1\ldots q_n}(t_1,\ldots,t_n)\) of equation (1) satisfy the following conditions:

\(1^\circ\). All families \(A_{q_1\ldots q_n}(t_1,\ldots,t_n)\) are compact. (For finite-dimensional \(E\) this means the uniform boundedness \(\|A_{q_1\ldots q_n}(t_1,\ldots,t_n)\|\).)

\(2^\circ\). All \(A_{q_1\ldots q_n}(t_1,\ldots,t_n)\) have weak variation at infinity (see, for example, (7)). This means that for every \(\varepsilon>0\) there exists \(T=T(\varepsilon)>0\) such that, whenever

\[ \sum_1^n t_j' \geqslant T;\qquad \sum_1^n t_j'' \geqslant T;\qquad \sum_1^n |t_j'-t_j''|<1 \]

necessarily

\[ \|A(t_1',\ldots,t_n')-A(t_1'',\ldots,t_n'')\|<\varepsilon . \]

Let us introduce the following additional notation. Let \(A^{(\omega)}\) be some limit operator generated by the family \(A(t_1,\ldots,t_n)\) as

\[ \sum_1^n t_j \to \infty . \]

Under

By \(\{A_{q_1\ldots q_n}^{(\omega)}\}\) we shall mean the totality (corresponding to \(\sum_{(q_1\ldots q_n)}\)) of limit operators \(A_{q_1\ldots q_n}^{(\omega)}\), generated by the families \(A_{q_1\ldots q_n}(t_1,\ldots,t_n)\) and the common sequences \(t_j\).

Finally, using equation (1) and a certain limiting totality \(\{A_{q_1\ldots q_n}^{(\omega)}\}\), we construct the following operator-function, depending on the complex parameters \(\lambda_1,\lambda_2,\ldots,\lambda_n\):

\[ \Gamma^{(\omega)}=\Gamma^{(\omega)}(\lambda_1,\ldots,\lambda_n) =\left(\lambda_1^{p_1}\ldots \lambda_n^{p_n} I -\sum_{(q_1\ldots q_n)} \lambda_1^{q_1}\ldots \lambda_n^{q_n} A_{q_1\ldots q_n}^{(\omega)}\right)^{-1}. \tag{3} \]

A point \((\lambda_1,\ldots,\lambda_n)\) at which \(\Gamma^{(\omega)}(\lambda_1,\ldots,\lambda_n)\) exists and is bounded will be called regular for \(\Gamma^{(\omega)}\). Any other point we shall call singular.

In the particular case when (1) is the ordinary differential equation of first order \(\dfrac{dy}{dt}-Ay=x\), formula (3) gives \(\Gamma^{(\omega)}(\lambda)=(\lambda I-A^{(\omega)})^{-1}\), and the singular points of \(\Gamma^{(\omega)}\) coincide with the points of the spectrum of \(A^{(\omega)}\).

The main result of the present work is the following

Theorem. In order that, in the boundary-value problem (1)—(2), to every bounded right-hand side \(\sup \|x(t_1,\ldots,t_n)\|<\infty\) there correspond a bounded solution \(\sup \|y(t_1,\ldots,t_n)\|<\infty\), it is necessary and sufficient that every (i.e., constructed from any limiting totality \(\{A_{q_1\ldots q_n}^{(\omega)}\}\)) operator-function \(\Gamma^{(\omega)}(\lambda_1,\ldots,\lambda_n)\) have no singular points in the domain \(\operatorname{Re}\lambda_j\ge 0,\ j=1,2,\ldots,n;\) in other words, that every singular point of \(\Gamma^{(\omega)}\) have at least one “coordinate” lying in the left (open) half-plane.

Not being able to present here a detailed proof of the theorem, we give only some of the most essential supporting points.

First of all it is necessary to consider the case when the coefficients \(A_{q_1\ldots q_n}\) are constant (i.e., do not depend on \(t_1,\ldots,t_n\)). The passage from this case to an equation with variable coefficients satisfying conditions \(1^\circ\) and \(2^\circ\) has, however, been set out by us, for an equation of a more special form than (1), in the papers \(({}^3,{}^6)\).

For constant operator coefficients the boundary-value problem (1)—(2) is equivalent to the operator (integral) equation

\[ y-\sum_{(q_1\ldots q_n)} S_1^{p_1-q_1}\ldots S_n^{p_n-q_n} A_{q_1\ldots q_n}y = S_1^{p_1}\ldots S_n^{p_n}x, \tag{4} \]

where

\[ S_jx(t_1,\ldots,t_n)=\int_0^{t_j} x(t_1,\ldots,t_{j-1},s,t_{j+1},\ldots,t_n)\,ds. \]

The solution of equation (4) is given by the following easily verified formula:

\[ \begin{aligned} y={}&\left(\frac{1}{2\pi i}\right)^n \oint_{\gamma_1}\oint_{\gamma_2}\cdots\oint_{\gamma_n} (I-\lambda_1 S_1)^{-1}(I-\lambda_2 S_2)^{-1}\cdots (I-\lambda_n S_n)^{-1} \\ &\times \left(I-\sum_{(q_1\ldots q_n)} \frac{A_{q_1\ldots q_n}} {\lambda_1^{p_1-q_1}\ldots \lambda_n^{p_n-q_n}} \right)^{-1} S_1^{p_1}\ldots S_n^{p_n}x\, \frac{d\lambda_1}{\lambda_1}\cdots \frac{d\lambda_n}{\lambda_n}. \end{aligned} \tag{5} \]

The contours \(\gamma_1,\gamma_2,\ldots,\gamma_n\) here must be chosen so that every point \((\lambda_1,\lambda_2,\ldots,\lambda_n)\), with \(\lambda_j\) lying on \(\gamma_j\) or outside it \((j=1,2,\ldots,n)\), is regular for the operator-function (3).

If one takes into account that

\[ (I-\lambda_j S_j)^{-1}S_j^{p_j} =(I-\lambda_j S_j)^{-1}\frac{1}{\lambda_j^{p_j}} -\frac{I+\lambda_j S_j+\cdots+\lambda_j^{p_j-1}S_j^{p_j-1}}{\lambda_j^{p_j}}, \]

then from (5) it is not difficult to obtain the proof of sufficiency.

A considerably more delicate fact, in our opinion, is necessity. The proof of necessity can also be obtained by means of formula (5); however, in doing so one must use certain nontrivial properties of the operator-function (3), or, equivalently, of the operator-function

\[ \Delta(\lambda_1,\ldots,\lambda_n) = \left( I-\sum_{(q_1\ldots q_n)} \frac{A_{q_1\ldots q_n}} {\lambda_1^{p_1-q_1}\cdots \lambda_n^{p_n-q_n}} \right)^{-1}. \]

We give one of them—the one most essential for the proof of the theorem and, as it seems to us, of independent interest.

Let \((\lambda_1^0,\ldots,\lambda_n^0)\) be a singular point of \(\Delta(\lambda_1,\ldots,\lambda_n)\) that is a limit point for the set of regular points. Then one of the following two assertions holds.

\(\alpha)\) There exists an element \(x_0\in E,\ \|x_0\|=1\), such that

\[ \Delta(\lambda_1,\ldots,\lambda_n)x_0 = \frac{\lambda_1\cdots\lambda_n} {(\lambda_1-\lambda_1^0)\cdots(\lambda_n-\lambda_n^0)}x_0 - \]

\[ -\Delta(\lambda_1,\ldots,\lambda_n) \sum_{(k_1\ldots k_m)} \frac{P_{k_1\ldots k_m}(\lambda_1,\ldots,\lambda_n)} {(\lambda_{k_1}-\lambda_{k_1}^0)\cdots(\lambda_{k_m}-\lambda_{k_m}^0)}x_0. \]

Here \(P_{k_1\ldots k_m}(\lambda_1,\ldots,\lambda_n)\) are polynomials in \(\lambda_1,\ldots,\lambda_n\), and each of the denominators \((\lambda_{k_1}-\lambda_{k_1}^0)\cdots(\lambda_{k_m}-\lambda_{k_m}^0)\) consists of factors \(\lambda_1-\lambda_1^0,\ldots,\lambda_n-\lambda_n^0\) taken in an incomplete set (i.e., with \(m<n\)).

\(\beta)\) There exists a sequence of elements \(x_j\in E,\ \|x_j\|=1\), such that (with the same notation)

\[ \Delta(\lambda_1,\ldots,\lambda_n)x_j = \frac{\lambda_1\cdots\lambda_n} {(\lambda_1-\lambda_1^0)\cdots(\lambda_n-\lambda_n^0)}x_j - \]

\[ -\Delta(\lambda_1,\ldots,\lambda_n) \sum_{(k_1\ldots k_m)} \frac{P_{k_1\ldots k_m}(\lambda_1,\ldots,\lambda_n)} {(\lambda_{k_1}-\lambda_{k_1}^0)\cdots(\lambda_{k_m}-\lambda_{k_m}^0)}x_j+z_j, \]

where \(z_j\to\theta\). (In fact, \(\beta)\) contains \(\alpha)\) as a special case.)

The structure of

\[ \sum_{(k_1\ldots k_m)} \]

is cumbersome in the general case and plays no essential role in the proof. For the case when (1) is a hyperbolic equation of the 2nd order:

\[ \frac{\partial^2 y}{\partial t_1\,\partial t_2} -A\frac{\partial y}{\partial t_1} -B\frac{\partial y}{\partial t_2} -Cy=x, \]

relation \(\alpha)\) is written as follows:

\[ \left(I-\frac{A}{\lambda}-\frac{B}{\mu}-\frac{C}{\lambda\mu}\right)^{-1}x_0 = \frac{\lambda\mu}{(\lambda-\lambda_0)(\mu-\mu_0)}x_0 - \]

\[ -\left(I-\frac{A}{\lambda}-\frac{B}{\mu}-\frac{C}{\lambda\mu}\right)^{-1} \left\{ \frac{A}{\lambda_0}\frac{\mu}{\mu-\mu_0} + \frac{B}{\mu_0}\frac{\lambda_0}{\lambda-\lambda_0} + \frac{C}{\lambda_0\mu_0} \left( \frac{\mu_0}{\mu-\mu_0} + \frac{\lambda_0}{\lambda-\lambda_0} \right) \right\}x_0. \]

In conclusion we note that the main theorem presented in this paper is readily extended to the case when the boundedness of \(x\) and \(y\) is understood in the following sense:

\[ \sup e^{-\alpha (t_1+\cdots+t_n)} \|x(t_1,\ldots,t_n)\|<\infty;\qquad \sup e^{-\alpha (t_1+\cdots+t_n)} \|y(t_1,\ldots,t_n)\|<\infty, \]

where \(\alpha\) is a given real number. This makes it possible to obtain from it not only criteria for the boundedness of solutions, but also certain estimates of exponential growth, given in \((2^{-6})\) for an equation of a more particular form.

Odessa Hydrometeorological
Institute

Received
12 V 1962

CITED LITERATURE

\(^{1}\) M. G. Krein, UMN, 3, no. 3 (25), 166 (1948).
\(^{2}\) M. A. Rutman, DAN, 101, No. 6, 993 (1955).
\(^{3}\) M. A. Rutman, DAN, 108, No. 5, 770 (1956).
\(^{4}\) M. A. Rutman, UMN, 12, no. 1 (73), 234 (1957).
\(^{5}\) M. A. Rutman, DAN, 124, No. 4, 764 (1959).
\(^{6}\) M. A. Rutman, Tr. Odessa Hydrometeorol. Inst., vol. XXVII, 11 (1962).
\(^{7}\) K. P. Persidskii, Izv. Phys.-Math. Soc. at Kazan Univ., part 1, 8, ser. 3, 47 (1936).