UDC 517.949.2

MATHEMATICS

I. I. MARMERSHTEIN

ON THE BOUNDEDNESS OF SOLUTIONS OF CERTAIN SYSTEMS OF LINEAR EQUATIONS IN PARTIAL DIFFERENCES

(Presented by Academician I. G. Petrovskii, 22 IV 1969)

We consider the boundary-value problem:

\[ \Delta_1 y(t_1,t_2)-\sum_{j=0}^{k} P_j \Delta_2^{\,j} y(t_1,t_2)=f(t_1,t_2), \]

\[ y\big|_{0\le t_1<\delta_1}=g(t_1,t_2). \tag{1} \]

Here \(y(t_1,t_2)\), \(f(t_1,t_2)\), \(g(t_1,t_2)\) are vector functions with values belonging to some complex Banach space \(E\), given in the domain \(0\le t_1,t_2<\infty\) and bounded in every domain \(0\le t_i\le a_i<\infty\), \(i=1,2\). The coefficients of the equation \(P_j\) \((j=0,1,2,\ldots,k)\) are linear operators acting in \(E\). In contrast to \((^5,^6)\), we do not require pairwise commutativity of these operators.

\(\Delta_1,\Delta_2\) are difference operators:

\[ \Delta_1 y(t_1,t_2)=\bigl[y(t_1+\delta_1,t_2)-y(t_1,t_2)\bigr]/\delta_1, \]

\[ \Delta_2 y(t_1,t_2)=\bigl[y(t_1,t_2+\delta_2)-y(t_1,t_2)\bigr]/\delta_2. \]

\[ (\delta_1,\delta_2>0) \]

The system (1) does not belong to the class of equations considered by M. A. Rutman in \((^3)\), since it has no leading term.

In the present paper, necessary and sufficient conditions are obtained under which to every pair of vector functions \(f(t_1,t_2)\) and \(g(t_1,t_2)\), bounded in the domain \(0\le t_1,t_2<\infty\), there corresponds a bounded solution \(y(t_1,t_2)\) of problem (1). Such a problem will be called stable.

Consider the family of operators \(\sum_{j=0}^{k} P_j \alpha^j\), where \(\alpha\) ranges over the closed disk \(|z+1/\delta_2|\le 1/\delta_2\). Let \(\Pi_\alpha\) be the set of spectral points of this family. (By the spectrum of an operator \(A\) we mean the set of points \(\lambda\) at which \(A-\lambda I\) has no bounded inverse.)

Theorem 1. In order that problem (1) be stable, it is necessary and sufficient that the set \(\Pi_\alpha\) lie inside the disk \(|z+1/\delta_1|<1/\delta_1\).

Without loss of generality, we shall consider problem (1) with zero initial conditions. We transform system (1) into the form

\[ \Delta_1 y(t_1,t_2)-\sum_{j=0}^{k} Q_j y(t_1,t_2+j\delta_2)=f(t_1,t_2), \]

\[ y\big|_{0\le t_1<\delta_1}=0, \tag{2} \]

where the coefficients \(Q_j\) are related to \(P_j\) by certain linear relations. Taking these relations into account, we obtain the following equivalent formulation of the theorem.

Problem (2) is stable if and only if the set of spectral points of the family of operators \(\sum_{j=0}^{k} Q_j \alpha^j\), where \(\alpha\) runs over the closed unit circle, lies inside the circle \(|z+1/\delta_1|<1/\delta_1\).

It is easy to show that system (2) is equivalent to the equation

\[ y(t_1,t_2)-\sum_{j=0}^{k} S_1Q_jy(t_1,t_2+j\delta_2)=S_1f(t_1,t_2), \]

or

\[ y(t_1,t_2)-S_1Ay(t_1,t_2)=S_1f(t_1,t_2). \tag{3} \]

The vector-functions \(y(t_1,t_2)\) and \(f(t_1,t_2)\) on the grid \(\{t_1=n\delta_1;\ t_2=m\delta_2;\ n,m=0,1,2,\ldots\}\) define infinite matrices whose entries are vectors from \(E\).

The operator \(A\) on the indicated grid is realized by the infinite matrix:

\[ A= \begin{pmatrix} Q_0 & Q_1 & \cdot & \cdot & \cdot & Q_k & 0 & 0 & \cdot & \cdot & \cdot\\ 0 & Q_0 & Q_1 & \cdot & \cdot & \cdot & Q_{k-1} & Q_k & 0 & \cdot & \cdot & \cdot\\ 0 & 0 & Q_0 & Q_1 & \cdot & \cdot & \cdot & Q_{k-1} & Q_k & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \end{pmatrix}, \]

where \(Q_j\) \((j=0,1,2,\ldots,k)\) are operators acting in \(E\).

\(S_1\) is the operator inverse to the difference operator \(\Delta_1\). On the grid under consideration it is realized by multiplying the matrix \(\{f(n\delta_1,m\delta_2)\}_{n,m=0}^{\infty}\) on the right by the infinite matrix

\[ \begin{pmatrix} 0 & \delta_1 I & \delta_1 I & \cdots\\ 0 & 0 & \delta_1 I & \cdots\\ 0 & 0 & 0 & \cdots \end{pmatrix} \]

(\(0\) and \(I\) are, respectively, the zero and identity operators acting in the space \(E\)).

The solution of equation (3) can be represented in the form:

\[ y=-(2\pi i)^{-1}\oint_{\gamma}(I-\lambda S_1)^{-1}(A-\lambda I)^{-1}S_1f\,d\lambda. \tag{4} \]

(see (1)). Here \(\gamma\) is an arbitrary contour surrounding the spectrum of the operator \(A\).

In what follows the following lemma is used.

Lemma. Every spectral point of the operator \(A\) belongs to the spectrum of the operator

\[ \sum_{j=0}^{k} Q_j\alpha^j \]

for some \(|\alpha|\leqslant 1\).

We give the scheme of the proof of the theorem.

Sufficiency. The condition of the theorem makes it possible to choose the contour \(\gamma\) so that it lies entirely inside the circle \(|z+1/\delta_1|<1/\delta_1\). The operator \((I-\lambda S_1)^{-1}S_1\) is represented in the form

\[ (I-\lambda S_1)^{-1}S_1= \begin{pmatrix} 0 & \delta_1 I & \delta_1(1+\lambda\delta_1)I & \delta_1(1+\lambda\delta_1)^2I & \delta_1(1+\lambda\delta_1)^3I & \cdots\\ 0 & 0 & \delta_1 I & \delta_1(1+\lambda\delta_1)I & \delta_1(1+\lambda\delta_1)^2I & \cdots\\ 0 & 0 & 0 & \delta_1 I & \delta_1(1+\lambda\delta_1)I & \cdots \end{pmatrix}. \]

It follows from this that this operator is uniformly bounded on the contour \(\gamma\). Sufficiency is easily obtained from formula (4).

Necessity. Suppose first that for some \(\alpha_0\) \((|\alpha_0|\leqslant 1)\) the operator

\[ \sum_{j=0}^{k} Q_j\alpha_0^j \]

has a spectral point \(\lambda_0\) lying outside

of the circle \(|z+1/\delta_1|<1/\delta_1\). For simplicity we restrict ourselves to the case when \(\lambda_0\) is an eigenvalue and \(g_0\) is the corresponding eigenvector:

\[ \sum_{j=0}^{k} Q_j \alpha_0^j g_0=\lambda_0 g_0 . \]

As the right-hand side of equation (2) we take the vector-function which, on the grid \(\{t_1=n\delta_1,\ t_2=m\delta_2,\ n,m=0,1,2,\ldots\}\), defines the matrix

\[ \begin{pmatrix} g_0 & g_0 & g_0 & \cdots\\ \alpha_0 g_0 & \alpha_0 g_0 & \alpha_0 g_0 & \cdots\\ \alpha_0^2 g_0 & \alpha_0^2 g_0 & \alpha_0^2 g_0 & \cdots \end{pmatrix}. \]

Then \((\lambda_0 I-A)f_0=0\) and

\[ (\lambda I-A)f_0=(\lambda-\lambda_0)f_0 . \]

Hence

\[ (\lambda I-A)^{-1}f_0=\frac{f_0}{\lambda-\lambda_0}. \]

Substituting this expression into the solution formula (4), we obtain

\[ y=(I-\lambda_0 S_1)^{-1}S_1 f_0 . \]

From this the unboundedness of the solution \(y\) follows immediately.

If \(\lambda_0\) lies on the boundary of the circle \(|z+1/\delta_1|<1/\delta_1\), then the right-hand side must be given by the matrix

\[ \begin{pmatrix} g_0 & g_0 e^{i\varphi} & g_0 e^{2i\varphi} & \cdots\\ \alpha_0 g_0 & \alpha_0 g_0 e^{i\varphi} & \alpha_0 g_0 e^{2i\varphi} & \cdots\\ \alpha_0^2 g_0 & \alpha_0^2 g_0 e^{i\varphi} & \alpha_0^2 g_0 e^{2i\varphi} & \cdots \end{pmatrix}, \]

where \(e^{i\varphi}=1+\lambda_0\delta_1\).

Using the methods developed in (2–4), one can extend the result obtained to equations with variable coefficients. Consider the boundary-value problem

\[ \Delta_1 y(t_1,t_2)-\sum_{j=0}^{k} P_j(t_1,t_2)\Delta_2^{\,j}y(t_1,t_2)=f(t_1,t_2), \tag{5} \]

\[ y\big|_{0\le t_1<\delta_1}=g(t_1,t_2). \]

We shall assume that the operator-coefficients \(P_j(t_1,t_2)\) satisfy the following conditions:

1. The operator-functions \(P_j(t_1,t_2)\) \((j=0,1,2,\ldots,k)\) are continuous throughout the entire domain of variation of the independent variables.
2. All families \(P_j(t_1,t_2)\) are compact.
3. The \(P_j(t_1,t_2)\) have weak variation at \(t_1\)-infinity.

This means that for every \(\varepsilon>0\) there exists \(T=T(\varepsilon)\) such that from the conditions

\[ t_1'>T,\qquad t_1''>T,\qquad |t_1'-t_1''|+|t_2'-t_2''|\le 1 \]

it follows that

\[ \|P_j(t_1',t_2')-P_j(t_1'',t_2'')\|<\varepsilon . \]

Consider all possible \(\omega_{t_1}\)-limit operators \(P_j^{(\omega_{t_1})}\), generated by sequences of points \((t_1^{(n)},t_2^{(n)})\), common for all \(P_j(t_1,t_2)\) \((j=0,1,2,\ldots,k)\), with \(t_1^{(n)}\to\infty\). Form the family of operators

\[ \sum_{j=0}^{k} P_j^{(\omega_{t_1})}\alpha^j, \]

where \(\alpha\) ranges over the closed circle \(|z+1/\delta_2|\le 1/\delta_2\). Let \(\Omega_\alpha\) be the set of points of the spectrum of this family.

Theorem 2. In order that, in problem (5), every bounded vector-functions \(f(t_1,t_2)\) and \(g(t_1,t_2)\) correspond to a bounded solution \(y(t_1,t_2)\), it is necessary and sufficient that the set \(\Omega_\alpha\) lie inside the circle \(|z+1/\delta_1|<1/\delta_1\).

The results presented are applicable, in particular, to the problems

\[ \Delta_1 y(t_1,t_2)-a\Delta_2 y(t_1,t_2)-A(t_1,t_2)y(t_1,t_2)=f(t_1,t_2), \]

\[ y\big|_{0\le t_1<\delta_1}=g(t_1,t_2); \]

\[ \Delta_1 y(t_1,t_2)-a^2\Delta_2^2 y(t_1,t_2)-A(t_1,t_2)y(t_1,t_2)=f(t_1,t_2), \]

\[ y\big|_{0\le t_1<\delta_1}=g(t_1,t_2) \]

(\(a\) is a real number), which are difference analogues of the differential problems

\[ \partial u/\partial t-a\,\partial u/\partial x-A(x,t)u=f(x,t), \]

\[ u(x,0)=\varphi(x); \]

\[ \partial u/\partial t-a^2\partial^2 u/\partial x^2-A(x,t)u=f(x,t), \]

\[ u(x,0)=\varphi(x), \]

considered by E. Ya. Melamed in (4). The stability criteria established by him for these problems can be obtained from the last theorem as limiting cases.

Odessa Hydrometeorological Institute

Received
19 IV 1969

REFERENCES

1. M. A. Rutman, DAN, 101, No. 6, 993 (1955).
2. M. A. Rutman, DAN, 108, No. 5, 770 (1956).
3. M. A. Rutman, DAN, 159, No. 2, 273 (1964).
4. E. Ya. Melamed, DAN, 120, No. 6, 1194 (1958).
5. I. I. Marmerstein, Izv. vyssh. uchebn. zaved., Mat., No. 3, 84 (1966).
6. I. I. Marmerstein, Ibid., No. 4, 60 (1967).