Z. I. Rekhlitskii

ON ESTIMATES OF THE GROWTH OF SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATIONS IN PARTIAL DERIVATIVES OF HYPERBOLIC TYPE

(Presented by Academician I. G. Petrovskii, 11 XII 1964)

In notes \((^{1-5})\), stability criteria (boundedness on a half-axis) of solutions were obtained for a broad class of ordinary differential-difference equations. In the present note we have succeeded in obtaining a necessary and sufficient criterion for the growth of solutions for differential-difference equations of the form

\[ \frac{\partial^{p_1+p_2+\cdots+p_n}Y} {\partial t_1^{p_1}\partial t_2^{p_2}\cdots \partial t_n^{p_n}} - A(t_1,t_2,\ldots,t_n)Y(t_1-a_1,t_2-a_2,\ldots,t_n-a_n) = \]

\[ = f(t_1,t_2,\ldots,t_n), \]

\[ p_1,p_2,\ldots,p_n \geqslant 1;\qquad a_k=a_k(t_1,t_2,\ldots,t_n)\geqslant 0 \quad (0\leqslant t_k<\infty), \]

where \(A(t_1,t_2,\ldots,t_n)\) is a linear operator-function acting in a complex Banach space \(\mathfrak{E}\), and \(f(t_1,t_2,\ldots,t_n)\), \(Y(t_1,t_2,\ldots,t_n)\) are continuous vector-functions with range belonging to \(\mathfrak{E}\). The main result may be formulated in the form of the following theorem.

Theorem. Consider the boundary-value problem

\[ -\frac{\partial^{p_1+p_2+\cdots+p_n}Y} {\partial t_1^{p_1}\partial t_2^{p_2}\cdots \partial t_n^{p_n}} - A(t_1,t_2,\ldots,t_n)Y(t_1-a_1,t_2-a_2,\ldots,t_n-a_n) = \]

\[ = f(t_1,t_2,\ldots,t_n), \]

\[ p_1,p_2,\ldots,p_n \geqslant 1;\qquad a_k=a_k(t_1,t_2,\ldots,t_n)\geqslant 0 \]

\[ (0\leqslant t_1,t_2,\ldots,t_n<\infty), \]

\[ \frac{\partial^{q_k}Y}{\partial t_k^{q_k}} = \varphi_{q_k}(t_1,t_2,\ldots,t_n) \]

\[ (t_k\leqslant 0;\ q_k=0,1,\ldots,p_k-1;\ k=1,2,\ldots,n). \tag{1} \]

Let the continuous operator-function \(A(t_1,t_2,\ldots,t_n)\) satisfy the conditions:

1) the family of operators is compact;

2) \(A(t_1,t_2,\ldots,t_n)\) has weak variation at infinity, i.e., for any \(\varepsilon>0\) there exists a \(T=T(\varepsilon)>0\) such that for all \(t_k'\), \(t_k''>T\) and \(|t_k'-t_k''|<1\),

\[ \|A(t_1',t_2',\ldots,t_n')-A(t_1'',t_2'',\ldots,t_n'')\|<\varepsilon. \]

The functions \(a_k(t_1,t_2,\ldots,t_n)\geqslant 0\) will be assumed continuous, bounded, and satisfying condition 2).

Consider all possible limit operators \(A_\omega\) and the corresponding limit values of the functions \(a_k^{(\omega)}\), to which the families of operators and functions converge on common sequences:

\[ A\bigl(t_1^{(m)},\, t_2^{(m)},\, \ldots,\, t_n^{(m)}\bigr)\to A_\omega;\qquad a_k\bigl(t_1^{(m)},\, t_2^{(m)},\, \ldots,\, t_n^{(m)}\bigr)\to a_k^{(\omega)} \]

\[ \bigl(t_k^{(m)}\to\infty\bigr). \]

Then, in order that the solution \(Y\) of the boundary-value problem (1) satisfy the condition \(\|Y\|\le C\exp[\alpha(t_1+t_2+\cdots+t_n)]\) for all \(\|f\|\le C\exp[\alpha(t_1+t_2+\cdots+t_n)]\), it is necessary and sufficient that, for all \(\Lambda_\omega\in \operatorname{sp.} A_\omega\) and the corresponding \(a_k^{(\omega)}\), all roots \(z(z_1,z_2,\ldots,z_n)\) of the equation

\[ z_1^{p_1}z_2^{p_2}\cdots z_n^{p_n} -\Lambda_\omega \exp\left[-\sum_{k=1}^{n} a_k^{(\omega)} z_k\right]=0 \tag{2} \]

satisfy the condition

\[ \sup_{(z)}\left(\min_k \operatorname{Re} z_k\right)<\alpha,\qquad k=1,2,\ldots,n. \tag{3} \]

Proof of the theorem will be carried out according to the following plan:

I. First we prove the theorem for the case when \(A(t_1,t_2,\ldots,t_n)\equiv A\); \(a_k(t_1,t_2,\ldots,t_n)\equiv a_k\) are constants. By replacing the unknown function, we reduce the boundary-value problem (1) to a problem with zero initial functions: \(\varphi_{q_k}(t_1,t_2,\ldots,t_n)\equiv 0\), which is equivalent to the operator equation

\[ Y-AS_1^{p_1}S_2^{p_2}\cdots S_n^{p_n}K_1K_2\cdots K_nY = S_1^{p_1}S_2^{p_2}\cdots S_n^{p_n}f, \tag{4} \]

where

\[ S_i f=\int_0^{t_i} f(t_1,\ldots,\tau,\ldots,t_n)\,d\tau \]

is the integration operator, and

\[ K_i f=f(t_1,\ldots,t_i-a_i,\ldots,t_n) \]

is the “delay” operator. It is convenient to consider equation (4) in the space of continuous functions \(\widetilde E_0\) such that \(f(t_1,\ldots,t_i,\ldots,t_n)=0\) \((t_i<0)\). In the space \(\widetilde E_0\) we approximate the operators \(K_i\) and \(S_i\) by the operators

\[ K_i(m)f=f(t_1,\ldots,t_i-m_i/m,\ldots,t_n), \]

\[ \sigma_i(m)f=\frac1m\sum_{k=0}^{\infty} f(t_1,\ldots,t_i-k/m,\ldots,t_n),\qquad m_i/m\to a_i \]

\[ (m_i,m\to\infty;\ m_i,m\text{ are natural numbers}). \]

We approximate equation (4) by the operator-difference equation:

\[ Y_m-A\prod_{i=1}^{n}\sigma_i^{p_i}(m)K_i(m)Y_m = \prod_{i=1}^{n}\sigma_i^{p_i}(m)f. \tag{5} \]

It is easy to show that \(\|Y_m-Y\|<\varepsilon\) for \(m\ge N\), uniformly on every finite interval.

Consider the auxiliary function

\[ \Phi_m=\Phi_m(z_1,z_2,\ldots,z_n) = \sum_{k_1,k_2,\ldots,k_n=0}^{\infty} Y_m\left(\frac{k_1}{m},\,\frac{k_2}{m},\,\ldots,\,\frac{k_n}{m}\right) z_1^{k_1}z_2^{k_2}\cdots z_n^{k_n}. \]

Apply the operators \(K_i(m)\) and \(\sigma_i(m)\) to \(\Phi_m\), taking into account that \(Y_m(t_1,t_2,\ldots,t_n)\) belongs to \(\widetilde E_0\).

We shall have:

\[ K_i(m)\Phi_m=z_i^{m_i}\Phi_m;\qquad \sigma_i(m)\Phi_m=\frac{1}{m(1-z_i)}\Phi_m. \]

The function \(\Phi_m\) obviously satisfies equation (5) if, in place of \(f\), we substitute the function

\[ F_m(z_1,z_2,\ldots,z_n)= \sum_{k_1,k_2,\ldots,k_n=0}^{\infty} f\left(\frac{k_1}{m},\frac{k_2}{m},\ldots,\frac{k_n}{m}\right) z_1^{k_1}z_2^{k_2}\cdots z_n^{k_n}. \]

From equation (5) we find the unknown function \(\Phi_m\) and obtain the formula

\[ \Phi_m(z_1,z_2,\ldots,z_n) = \left( \left(\frac{1-z_1}{1/m}\right)^{p_1} \left(\frac{1-z_2}{1/m}\right)^{p_2} \cdots \left(\frac{1-z_n}{1/m}\right)^{p_n} - Az_1^{m_1}z_2^{m_2}\cdots z_n^{m_n} \right)^{-1} F_m(z_1,z_2,\ldots,z_n). \tag{6} \]

The proof of the theorem is based on formula (6). The growth of the solution \(Y_m(t_1,t_2,\ldots,t_n)\) depends on the position of the singular points of the operator

\[ B_m(z_1,z_2,\ldots,z_n) = \left( \left(\frac{1-z_1}{1/m}\right)^{p_1} \left(\frac{1-z_2}{1/m}\right)^{p_2} \cdots \left(\frac{1-z_n}{1/m}\right)^{p_n} - Az_1^{m_1}z_2^{m_2}\cdots z_n^{m_n} \right)^{-1} \tag{7} \]

relative to the points \(z_k=0\). If we put \(z_k=\xi_k^{1/m}\), then

\[ B_m(\xi_1,\xi_2,\ldots,\xi_n)\to B(\xi_1,\xi_2,\ldots,\xi_n)\qquad (m\to\infty), \]

where

\[ B(\xi_1,\xi_2,\ldots,\xi_n) = \left( (-\ln \xi_1)^{p_1}(-\ln \xi_2)^{p_2}\cdots(-\ln \xi_n)^{p_n} - A\xi_1^{a_1}\xi_2^{a_2}\cdots \xi_n^{a_n} \right)^{-1}. \]

It is convenient to replace \(-\ln \xi_k=z_k\), \(\xi_k=e^{-z_k}\). We shall have

\[ B_1(z_1,z_2,\ldots,z_n) = \left( z_1^{p_1}z_2^{p_2}\cdots z_n^{p_n} - A\exp\left[-\sum_{k=1}^{n}a_kz_k\right] \right)^{-1}. \tag{8} \]

Suppose that for all \(\lambda\in\mathrm{sp.}\,A\) all roots of the equation

\[ z_1^{p_1}z_2^{p_2}\cdots z_n^{p_n} - \lambda\exp\left[-\sum_{k=1}^{n}a_kz_k\right]=0 \tag{2'} \]

satisfy condition (3). Then the operator (8) will be holomorphic for all \(z_k\) with \(\operatorname{Re} z_k\geq \alpha_1\) \((\alpha_1<\alpha)\), and the operators (7), obviously, are holomorphic for all \(|z_k|\leq \exp[-\alpha_1/m]\), beginning with \(m\geq N\). This makes it possible to estimate the Taylor coefficients \(b_{k_1k_2\ldots k_n}\) of the function \(B_m(z_1,z_2,\ldots,z_n)\), and consequently also the function \(\Phi_m(z_1,z_2,\ldots,z_n)\) from formula (6), on the basis of Cauchy’s inequality in strengthened form,

\[ \|b_{k_1k_2\ldots k_n}\| \leq \exp\left[\frac{\alpha_1}{m}(k_1+k_2+\cdots+k_n)\right] \times \]

\[ \times \oint\oint\cdots\oint_{|z_k|=e^{-\alpha_1/m}} \|B_m(z_1,z_2,\ldots,z_n)\|\,|dz_1|\,|dz_2|\cdots|dz_n| \qquad(\alpha_1<\alpha). \]

Carrying out sufficiently precise estimates, we shall have:

\[ \|Y_m(k_1/m,k_2/m,\ldots,k_n/m)\| \leq C\exp[\alpha(k_1/m+k_2/m+\cdots+k_n/m)] \]

\[ (m\geq N); \]

\(C\) does not depend on \(m\). Hence follows the estimate for \(Y(t_1,t_2,\ldots,t_n)\)

\[ \|Y(t_1,t_2,\ldots,t_n)\| \leq C\exp[\alpha(t_1+t_2+\cdots+t_n)] \qquad(\alpha>0). \]

The sufficiency of the theorem is proved.

II. Suppose that, for some \(\lambda_0\in\mathrm{sp.}\,A\), the roots \(z(z_1,z_2,\ldots,z_n)\) of equation (2′) satisfy the condition

\[ \sup_{(z)}\left(\min_k \operatorname{Re} z_k\right)=\alpha_0\geq \alpha. \]

Consider two

cases: \(a_0>a\) and \(a_0=a\). In the case \(a_0>a\) there exist roots \(z_k\) with \(\operatorname{Re} z_k\ge a_1>a\). One can choose such a \(\lambda_0\in\operatorname{sp.} A\) and such roots \(z_k\) with \(\operatorname{Re} z_k\ge a_1\) for which there exists a sequence of regular points \(\lambda_m\) of the operator \(A\), converging to \(\lambda_0\in\operatorname{sp.} A\) \((\lambda_m\to\lambda_0,\ m\to\infty)\). Then one can choose sequences of vectors \(f_m\) and \(e_m\) such that

\[ Af_m=\lambda_m f_m+e_m;\quad \|f_m\|=1;\quad \|e_m\|\to 0. \]

If in formula (6) we put \(f(t_1,t_2,\ldots,t_n)=f_m\) and estimate from below the Taylor coefficients \(Y_m(k_1/m,k_2/m,\ldots,k_n/m)\), then one can obtain an estimate for \(Y(t_1,t_2,\ldots,t_n)\):

\[ \|Y(t_1,t_2,\ldots,t_n)\|\ge C\exp\,[a_1(t_1+t_2+\cdots+t_n)],\quad C>0\ (a_1>a). \]

In this case the necessity of the theorem is proved.

In the case when

\[ \sup_{(z)}\left(\min_k \operatorname{Re} z_k\right)=a_0=a, \]

one can perturb the operator \(A\) by an arbitrarily small amount in norm so that, for some \(\lambda_0\in\operatorname{sp.} A\), \(a_0>a\), and obtain the preceding case. The theorem is proved.

III. In the case when the operator-function \(A(t_1,t_2,\ldots,t_n)\) and the functions \(a_k(t_1,t_2,\ldots,t_n)\) are not constant, the proof of the theorem relies on formula (6) and conditions 1), 2) of the theorem.

Remark 1. It should be noted that the boundary-value problem (1) for bounded functions \(f(t_1,t_2,\ldots,t_n)\) always has an unbounded solution. This follows from the fact that the roots of equation (2), for \(n>1\), always satisfy the condition \(\sup_{(z)}(\min_k \operatorname{Re} z_k)\ge 0\), whereas for boundedness it is necessary that \(\sup_{(z)}(\min_k \operatorname{Re} z_k)<0\).

Remark 2. For \(n=1\) and \(a=0\) one obtains the necessary and sufficient boundedness criterion for the solution that was obtained by us in note \({}^{1}\).

Remark 3. In the case when all \(a_k(t_1,t_2,\ldots,t_n)\equiv 0\), the growth criterion can be formulated as follows: in order that the solution of equation (1) satisfy the inequality
\(\|Y(t_1,t_2,\ldots,t_n)\|\le C\exp[\alpha(t_1+t_2+\cdots+t_n)]\) for all
\(\|f\|\le C\exp[\alpha(t_1+t_2+\cdots+t_n)]\), it is necessary and sufficient that, for all limiting operators \(A_\omega\), the condition hold:

\[ \lambda_\omega\in\operatorname{sp.} A_\omega;\quad \max \operatorname{Re}\sqrt[p]{\lambda_\omega}<\alpha \quad (p=p_1+p_2+\cdots+p_n). \tag{9} \]

Condition (9), as is easy to see, is equivalent to the fact that all roots
\(z(z_1,z_2,\ldots,z_n)\) of the equation
\(z_1^{p_1}z_2^{p_2}\cdots z_n^{p_n}-\lambda_\omega=0\) satisfy condition (3). Criterion (9) was obtained by M. A. Rutman in note \({}^{6}\).

Odessa Hydrometeorological Institute

Received
8 XII 1964

CITED LITERATURE

\({}^{1}\) Z. I. Rekhlitskii, DAN 111, No. 1 (1956).
\({}^{2}\) Z. I. Rekhlitskii, DAN, 118, No. 3 (1958).
\({}^{3}\) Z. I. Rekhlitskii, DAN, 125, No. 1 (1959).
\({}^{4}\) Z. I. Rekhlitskii, DAN, 127, No. 5 (1959).
\({}^{5}\) Z. I. Rekhlitskii, DAN, 149, No. 2 (1963).
\({}^{6}\) M. A. Rutman, UMN, 12, 1 (73), 234 (1957).