UDC 517.949.22

On the Operational Method for Solving Linear Differential-Difference Equations with Constant Coefficients

O. A. Dyshin

§ 1. Formulation of the Basic Initial Problem and Existence of Its Solution

For the differential-difference equation

\[ \sum_{i=0}^{m}\sum_{j=0}^{n} a_{ij} u^{(j)}(t-\omega_i)=f(t), \qquad t>0, \tag{1} \]

\[ 0=\omega_0<\omega_1<\cdots<\omega_m \]

we pose the initial problem: to find a solution \(u(t)\) of equation (1) for \(t>0\), continuous together with its \(n\) derivatives, satisfying the initial conditions

\[ u^{(j)}(t)=\varphi^{(j)}(t), \qquad t\in E_0=[-\omega_m,0]\quad (j=0,1,\ldots,n-1). \tag{2} \]

The substitution

\[ u^{(k-1)}(t)=y_k(t)\qquad (k=1,\ldots,n) \tag{3} \]

reduces equation (1) to a system of linear differential-difference equations of the form

\[ \sum_{i=0}^{m}\left[A_i y'(t-\omega_i)+B_i y(t-\omega_i)\right]=f^{*}(t), \qquad t>0, \tag{4} \]

\[ 0=\omega_0<\omega_1<\cdots<\omega_m, \]

where \(f^{*}(t)\) and \(y(t)\) are \(n\)-dimensional vector functions, namely

\[ f^{*}(t)= \begin{pmatrix} f(t)\\ 0\\ \cdot\\ \cdot\\ 0 \end{pmatrix}, \qquad y(t)= \begin{pmatrix} y_1(t)\\ \cdot\\ \cdot\\ y_n(t) \end{pmatrix}, \tag{5} \]

and \(A_i, B_i\) are square matrices of order \((n,n)\), with \(\det A_0=a_{0n}\).

The initial conditions (2) take the form

\[ y(t)=\varphi^{*}(t), \qquad t\in E_0=[-\omega_m,0], \tag{6} \]

where \(\varphi^{*}(t)\) is an \(n\)-dimensional vector function of the form (5) with components \(\varphi(t), \varphi'(t), \ldots, \varphi^{(n-1)}(t)\).

In [1], for the system of equations (4) with initial conditions on the interval \(E_0=[0,\omega_m]\), an existence and uniqueness theorem for the solution was proved, which, under conditions (6), is reformulated as follows.

Theorem 1. Let a system of equations (4) with initial conditions (6) be given, and let \(\det A_0 \ne 0\).

Denote by \(S\) the set of points
\[ t=\sum_{i=0}^{m} j_i\omega_i,\qquad j_i \text{ are nonnegative integers}, \]
and let \(S_1\) and \(S_2\) be the sets of points
\[ S_1=S\cap[0,\infty),\qquad S_2=S\cap(0,\infty). \]

Suppose that \(f^{*}(t)\in C^0[0,\infty)\,^{*}\), except for a finite number of jump discontinuities at points of the set \(S_1\), and \(\varphi^{*}(t)\in C^1[-\omega_m,0]\). Then there exists one and only one vector function \(y(t)\in C^0[0,\infty)\) satisfying equation (4) for \(t>0\), \(t\notin S_2\), and the initial condition (6). Moreover, \(y(t)\in C^1[0,\infty)\), except for possible jumps of \(y'(t)\) at points of the set \(S_1\).

If \(f^{*}(t)\) has no discontinuities on \([0,\infty)\), then \(y(t)\in C^1[0,\infty)\) and satisfies equation (4) for \(t>0\) with the initial condition (6) if and only if
\[ \sum_{i=0}^{m}\left[A_i\varphi^{*\,\prime}(-\omega_i)+B_i\varphi^{*}(-\omega_i)\right]=f^{*}(0). \tag{7} \]

To prove the theorem, it is enough to write equation (4) in the form
\[ \frac{d}{dt}\left(e^{\frac{B_0}{A_0}t}y(t)\right) = \frac{1}{A_0}e^{\frac{B_0}{A_0}t}v(t), \tag{8} \]
where
\[ v(t)=f^{*}(t)-\sum_{i=1}^{m}\left[A_i y'(t-\omega_i)+B_i y(t-\omega_i)\right], \]
and to apply to equation (8) the method of successive integration (the method of steps) with step \(\omega_1\).

From Theorem 1 we conclude that if in equation (1)
1) \(a_{0n}\ne 0\),
2) \(f(t)\in C^0[0,\infty)\), \(\varphi(t)\in C^n[-\omega_m,0]\), and
\[ \sum_{i=0}^{m}\sum_{j=0}^{n} a_{ij}\varphi^{(j)}(-\omega_i)=f(0), \]
then the solution \(u(t)\) of the basic initial-value problem exists and is uniquely determined. It can, for example, be obtained by the method of successive integration with step \(h\le \omega_1\). But the application of such a method for approximate computations becomes very difficult if \(\omega_1\) is small in comparison with the interval on which the solution is required to be determined; in this case the computations will, generally speaking, be nonuniform and cumbersome.

Remark. If in equation (1) \(a_{0n}=0\), \(a_{mn}\ne 0\), then by the substitution \(t-\omega_m=-t'\) we obtain, with respect to the function \(\tilde u(t')=u(-t')=u(t-\omega_m)\), an equation of type (1) with \(\tilde a_{0n}\ne 0\), \(\tilde\omega_i=\omega_m-\omega_{m-i}\). Existence and uniqueness—

\(*\) In saying that a vector or matrix belongs to the class \(C^0[0,\infty)\), we shall mean that all its components belong to this class.

the existence of the solution \(\tilde u(t')\) of the initial problem, to which the basic initial problem for equation (1) is thereby reduced, is proved with the aid of a theorem similar to the one given above, if \(S\) is taken to be the set of points of the form \(t'=\)

\[ = \sum_{i=0}^{m} j_i\omega_i \quad (j_i=+1,0,-1,-2,\ldots), \]

and \(S_1, S_2\) are the sets of points:

\[ S_1=S\cap(-\infty,\omega_m], \qquad S_2=S\cap(-\infty,\omega_m). \]

§ 2. SOLUTION OF THE BASIC INITIAL PROBLEM BY MEANS OF THE LAPLACE TRANSFORM

In applying the Laplace integral to the solution of the basic initial problem for equation (1), it is necessary first of all to justify the Laplace-transformability of equation (1) with the initial conditions (2), assuming a priori that the function \(f(t)\) has a convergent Laplace integral.

For this purpose we shall prove the following theorem.

Theorem 2. Suppose that, for equation (1), conditions 1), 2) of § 1 are satisfied. If, moreover, for all \(t\geqslant 0\)

\[ |f(t)|\leqslant c_1 e^{c_2 t}, \qquad c_1\geqslant 0,\quad c_2\geqslant 0, \tag{9} \]

then there exist nonnegative constants \(c_3,\ c_4\), whose values are determined by the relations (27), such that the solution of the basic initial problem satisfies the inequalities

\[ \bigl|u^{(j)}(t)\bigr|\leqslant c_3 e^{c_4 t}, \qquad t\geqslant 0 \quad (j=0,1,\ldots,n). \tag{10} \]

Proof. Write equation (1) in the form

\[ \sum_{l=0}^{n} a_l u^{(l)}(t)=\psi(t), \qquad t>0, \tag{11} \]

where

\[ a_l=a_{0l}a_{0n}^{-1}\ (l=0,1,\ldots,n), \qquad \psi(t)=a_{0n}^{-1}\left[f(t)-\sum_{i=1}^{m}\sum_{j=0}^{n} a_{ij}u^{(j)}(t-\omega_i)\right]. \]

We shall integrate equation (11) successively on the intervals \(\Delta_k=((k-1)h,\ kh]\) \((k=1,2,\ldots)\), \(0<h\leqslant \omega_1\). To this end, at each \(k\)-th step one has to solve the ordinary differential equation

\[ \sum_{l=0}^{n} a_l u^{(l)}(t)=\psi(t), \qquad t\in\Delta_k \tag{12} \]

with the initial conditions

\[ u^{(j)}((k-1)h)=\eta_{kj}\qquad (j=0,1,\ldots,n-1). \tag{13} \]

On the basis of (2), \(\eta_{1j}=\varphi^{(j)}(0)\) \((j=0,1,\ldots,n-1)\), while for subsequent \(k\) the quantities \(\eta_{kj}\) \((j=0,1,\ldots,n-1)\) are determined as the values of the functions \(u^{(j)}(t)\) at the right endpoint of the interval \(\Delta_{k-1}\).

By virtue of Theorem 1, the solution \(u(t)\) of equation (11) with the initial data (2) obtained in this way is continuous, together with its \(n\) derivatives, for all \(t\geqslant 0\).

Introduce a new variable \(t'\) by the substitution

\[ t'=\alpha t \quad (\alpha>0), \tag{14} \]

the parameter \(\alpha\) will subsequently be subject to a more stringent restriction. Setting
\(\tilde a_l=\alpha^{\,l-n}a_l\ (l=0,1,\ldots,n)\), \(\tilde\omega_i=\alpha\omega_i\ (i=0,1,\ldots,m)\), \(\delta=\alpha h\), \(\tilde f(t',\alpha)=f(t'/\alpha)\ (t'>0)\), \(\tilde u(t'\alpha)=u(t'/\alpha)\ (t'>-\alpha\omega_m)\), \(\tilde\varphi(t',\alpha)=\varphi(t'/\alpha)\ (-\alpha\omega_m\le t'\le0)\), let us consider the infinite sequence of initial problems:
\[ \sum_{l=0}^{n}\tilde a_l \tilde u_{10}^{(l)}(t',\alpha) =(a_{0n}\alpha^n)^{-1} \left[ \tilde f(t',\alpha)-\sum_{i=1}^{m}\sum_{j=0}^{n} \alpha^j a_{ij}\tilde y_0^{(j)}(t'-\tilde\omega_i,\alpha) \right],\quad t'>0, \]
\[ \tilde u_{10}^{(j)}(0,\alpha)=\tilde\varphi^{(j)}(0,\alpha)\quad (j=0,1,\ldots,n-1), \]
\[ \sum_{l=0}^{n}\tilde a_l \tilde u_{20}^{(l)}(t',\alpha) =(a_{0n}\alpha^n)^{-1} \left[ \tilde f(t',\alpha)-\sum_{i=1}^{m}\sum_{j=0}^{n} \alpha^j a_{ij}\tilde y_1^{(j)}(t'-\tilde\omega_i,\alpha) \right],\quad t'>\delta, \tag{15} \]
\[ \tilde u_{20}^{(j)}(\delta,\alpha)=\tilde u_{10}^{(j)}(\delta,\alpha)\quad (j=0,1,\ldots,n-1) \]
\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]
\[ \sum_{l=0}^{n}\tilde a_l \tilde u_{k0}^{(l)}(t',\alpha) =(a_{0n}\alpha^n)^{-1} \left[ \tilde f(t',\alpha)-\sum_{i=1}^{m}\sum_{j=0}^{n} \alpha^j a_{ij}\tilde y_{k-1}^{(j)}(t'-\tilde\omega_i,\alpha) \right], \]
\[ t'>(k-1)\delta, \]
\[ \tilde u_{k0}^{(j)}((k-1)\delta,\alpha) = \tilde u_{k-1,0}^{(j)}((k-1)\delta,\alpha) \quad (j=0,1,\ldots,n-1), \]
where the functions \(\tilde y_k(t',\alpha)\) in the right-hand sides of equations (15) are defined as follows:
\[ \tilde y_0(t',\alpha)= \begin{cases} \tilde\varphi(t',\alpha), & -\alpha\omega_m\le t'\le0,\\ \tilde\varphi_1(t',\alpha), & t'>0, \end{cases} \]
\[ \tilde y_k(t',\alpha)= \begin{cases} \tilde y_{k-1}(t',\alpha), & -\alpha\omega_m\le t'\le(k-1)\delta,\\ \tilde u_{k0}(t',\alpha), & t'>(k-1)\delta \end{cases} \tag{16} \]
\[ (k=1,2,\ldots), \]
\(\tilde\varphi_1(t',\alpha)=\varphi_1(t'/\alpha)\ (t'>0)\), where \(\varphi_1(t)\) is some function of the class \(C_n[0,\infty)\) with exponential growth at infinity, for which
\(\varphi_1^{(j)}(0)=\varphi^{(j)}(0)\ (j=0,1,\ldots,n)\). Then, for any \(\alpha>0\) and \(t\in A_k\),
\[ u(t)=u_{k0}(t)=\tilde u_{k0}(t\alpha,\alpha) \]
and the functions \(\tilde y_k(t',\alpha)\ (k=1,2,\ldots)\) are continuous in \(t'\) for all \(t'\ge-\alpha\omega_m\). Integrating successively equations (15) and “gluing” the functions \(u_{k0}(t)\ (k=1,2,\ldots)\) at the points \(t_k=k\delta/\alpha\ (k=1,2,\ldots)\), we obtain the same solution \(u(t)\) of the original initial problem for \(t>0\) as when integrating equations (12), (13).

Along with each \(k\)-problem (15), we shall consider the system of initial problems
\[ \sum_{l=0}^{n-p}\tilde a_{l+p}\tilde u_{kp}^{(l)}(t',\alpha) = \tilde\psi_k(t',\alpha) - \sum_{l=0}^{p-1}\tilde a_l\tilde u_{kl}^{(l)}(t',\alpha), \quad t'>(k-1)\delta, \]
\[ \tilde u_{kp}^{(j)}((k-1)\delta,\alpha) = \tilde u_{k-1,p}^{(j)}((k-1)\delta,\alpha) \quad (j=0,1,\ldots,n-p-1) \]
\[ (p=1,2,\ldots,n-1), \tag{17} \]

\[ \widetilde u_{kn}(t',\alpha)=\widetilde\psi_k(t',\alpha)-\sum_{l=0}^{n-1}\widetilde a_l\,u_{kl}(t',\alpha),\qquad t'>(k-1)\delta, \]

where \(\psi_k(t',\alpha)\) denotes the right-hand side of the \(k\)-th equation (15). From (14)—(17) it follows that

\[ u^{(j)}(t)=u_{kj}(t)=\alpha^j\widetilde u_{kj}(t',\alpha),\qquad t\in \Delta_k\quad (j=0,1,\ldots,n). \tag{18} \]

Let us prove, for all functions \(\widetilde u_{kj}(t',\alpha)\) \((j=0,1,\ldots,n)\), on the interval \(\widetilde\Delta_k=((k-1)\delta,k\delta]\), the validity of estimates of type (10) with constants independent of \(k\).

We shall use the Laplace transform to solve the initial-value problems (15), (17). By the general property of ordinary differential equations with constant coefficients whose right-hand sides are Laplace-transformable, the first of equations (15) and equations (17) for \(k=1\) are Laplace-transformable.

Let

\[ D_p(s,\alpha)\equiv \sum_{l=0}^{n-p}\widetilde a_{l+p}s^l=0 \qquad (p=0,1,\ldots,n-1) \tag{19} \]

be the characteristic equations associated with equations (15) and (17) for \(k=1\). Since \(a_{0n}\ne 0\), \(D_p(s,\alpha)\) is a polynomial in the variable \(s\) of degree \(n-p\) for fixed \(\alpha\). Therefore each of the fractions \(D_p^{-1}(s,\alpha)\) is decomposed into partial fractions:

\[ D_p^{-1}(s,\alpha)= \sum_{i=1}^{r^{(p)}(\alpha)} \sum_{\lambda=1}^{\lambda_i^{(p)}(\alpha)} \frac{A_{i\lambda}^{(p)}(\alpha)} {\bigl(s-s_i^{(p)}(\alpha)\bigr)^\lambda}. \]

Here \(s_i^{(p)}(\alpha)\) \((1\le i\le r^{(p)}(\alpha)\le n-p)\) are the distinct roots of the \(p\)-th equation (19); \(\lambda_i^{(p)}(\alpha)\) are the multiplicities of the roots \(s_i^{(p)}(\alpha)\).

For each \(p=0,1,\ldots,n-1\) we shall distinguish two cases:

\[ \text{a) }\sum_{j=p}^{n-1}|a_{0j}|>0 \qquad\text{and}\qquad \text{b) }\sum_{j=p}^{n-1}|a_{0j}|=0. \]

Then each of the functions

\[ \widetilde D_p(\widetilde s,\alpha)= \sum_{l=0}^{n-p}\widetilde a_{n-l}s^l \qquad (p=0,1,\ldots,n-1) \]

is, in case a), a polynomial in the variable \(\widetilde s=s^{-1}\) for fixed \(\alpha\). The degree of the \(p\)-th such polynomial does not exceed \(n-p\), and, consequently,

\[ \widetilde D_p^{-1}(\widetilde s,\alpha)= \sum_{i=1}^{\widetilde r^{(p)}(\alpha)} \sum_{\lambda=1}^{\lambda_i^{(p)}(\alpha)} \frac{\widetilde A_{i\lambda}^{(p)}(\alpha)} {\bigl(\widetilde s-\widetilde s_i^{(p)}(\alpha)\bigr)^\lambda}, \]

where \(\widetilde s_i^{(p)}(\alpha)\) \((1\le i\le \widetilde r^{(p)}(\alpha)\le r^{(p)}(\alpha))\) are the distinct roots of the equation \(\widetilde D_p(\widetilde s,\alpha)=0\), which simultaneously serve as nonzero roots of the \(p\)-th equation (19), which we shall denote by \(s_i^{\prime(p)}(\alpha)\).

It is known that the original of the fraction \((s-s_i)^{-\lambda}\), where \(\lambda\) is a positive integer, is the function \(t^{\lambda-1}/(\lambda-1)!\,e^{s_i t}\). Denoting by \(\widetilde\psi_{kp}(t',\alpha)\) the right-

parts of equations (15)—(17), which define the functions \(\tilde u_{kp}(t',\alpha)\), for the functions \(\tilde u_{1p}(t',\alpha)\) \((p=0,1,\ldots,n-1)\) we shall have, in case a), the integral representations for \(t'>0\):

\[ \begin{aligned} \tilde u_{1p}(t',\alpha)={}& \sum_{i=1}^{r^{(p)}(\alpha)} \sum_{\lambda=1}^{\lambda_i^{(p)}(\alpha)} A_{i\lambda}^{(p)}(\alpha)e^{s_i^{(p)}(\alpha)t'} \int_0^{t'} \frac{(t'-\tau)^{\lambda-1}}{(\lambda-1)!} e^{-s_i^{(p)}(\alpha)\tau}\tilde\psi_{1p}(\tau,\alpha)\,d\tau \\ &+ \sum_{l=1}^{n-p}\sum_{j=0}^{l-1} \sum_{i=1}^{\tilde r^{(p)}(\alpha)} \sum_{\lambda=1}^{\lambda_i^{(p)}(\alpha)} (-1)^\lambda \tilde a_{l+p}A_{i\lambda}^{(p)}(\alpha) \left\{\sum_{\mu=0}^{\lambda-1} C_\lambda^\mu\bigl(s_i^{\prime(p)}(\alpha)\bigr)^{\lambda-\mu} e^{s_i^{\prime(p)}(\alpha)t'} \right. \\ &\qquad\qquad\qquad\qquad\qquad\times \int_0^{t'} \frac{(t'-\tau)^{\lambda-\mu-1}}{(\lambda-\mu-1)!} \frac{\tau^{\,n-p-l+j}}{(n-p-l+j)!} e^{-s_i^{\prime(p)}(\alpha)\tau}\,d\tau + \frac{(t')^{\,n-p-l+j}}{(n-p-l+j)!} \\ &\qquad\qquad\qquad\qquad\qquad\left.\times \bigl(s_i^{\prime(p)}(\alpha)\bigr)^\lambda \right\}\eta_{1,j+p}(\alpha), \end{aligned} \tag{20} \]

where \(\eta_{1,j+p}(\alpha)=\alpha^{-j-p}\varphi^{(j+p)}(0)\).

If, however, for some \(0\le p\le n-1\) we have case b), then for the function \(\tilde u_{1p}(t',\alpha)\), instead of the corresponding formula from (20), the following formula is valid for \(t'>0\):

\[ \tilde u_{1p}(t',\alpha)= \int_0^{t'} dt_{n-p}\int_0^{t_{n-p}}dt_{n-p-1}\cdots \int_0^{t_2}\tilde\psi_{1p}(t_1,\alpha)\,dt_1 + \sum_{j=0}^{n-p-1}\eta_{1,j+p}(\alpha)\frac{(t')^j}{j!}. \tag{20_1} \]

Let us agree on the notation:

\[ x_i^{(p)}(\alpha)=\operatorname{Re}s_i^{(p)}(\alpha) \qquad (i=1,\ldots,r^{(p)}(\alpha);\ p=0,1,\ldots,n-1), \]

\[ x_0(\alpha)= \max_{1\le i\le r^{(p)}(\alpha),\ 0\le p\le n-1} x_i^{(p)}(\alpha), \qquad x_p(\alpha)= \min_{1\le i\le r^{(p)}(\alpha)} \bigl(\gamma-x_i^{(p)}(\alpha)\bigr), \]

\[ K=\max_{t\in E_0,\ 0\le j\le n}|\varphi^{(j)}(t)|, \qquad M=\sum_{i=1}^{m}\sum_{j=0}^{n}|a_{ij}|, \]

\[ c=\max(c_1,KM) \]

and define \(\gamma=\gamma(\alpha)\) by the inequality

\[ \gamma>\max\left(\frac{c_2}{\alpha},\ x_0(\alpha)\right). \tag{21} \]

Define the constants \(A_p\) and \(C_p\) \((p=0,1,\ldots,n-1)\), depending on \(\alpha,\gamma,\delta\), in case a) as

\[ A_p= \frac{1}{|a_{0n}|\alpha^n x_p(\alpha)} \sum_{i=1}^{r^{(p)}(\alpha)} \sum_{\lambda=1}^{\lambda_i^{(p)}(\alpha)} |A_{i\lambda}^{(p)}(\alpha)| \frac{\delta^{\lambda-1}}{(\lambda-1)!}, \]

\[ C_p=\sum_{l=1}^{n-p}\sum_{j=0}^{l-1}\sum_{i=1}^{\tilde r^{(p)}(\alpha)} \sum_{\lambda=1}^{\lambda_i^{(p)}(\alpha)} \alpha^{l+p-n}\left|\tilde A_{i\lambda}^{(p)}(\alpha) -\frac{a_{0,l+p}}{a_{0n}}\right| \left\{\frac{1}{\chi_p(\alpha)}\times \right. \]

\[ \left. \times \sum_{\mu=0}^{\lambda-1} C_\lambda^\mu \left|s_i^{(p)}(\alpha)\right|^{\lambda-\mu} \frac{\delta^{\lambda-\mu-1}}{(\lambda-\mu-1)!}+1 \right\} \left|s_i^{(p)}(\alpha)\right|^\lambda \frac{\delta^{\,n-p-l+j}}{(n-p-l+j)!} \]

and in case b)

\[ A_p=\frac{\delta^{\,n-p-1}}{|a_{0n}|\alpha^n\gamma(n-p-1)!}, \qquad C_p=\sum_{j=0}^{n-p-1}\frac{\delta^j}{j!}. \]

When choosing the parameter \(\alpha\), we shall proceed from the condition

\[ \frac{1}{|a_{0n}|\alpha^n}\left(2+q\sum_{l=0}^{n-1}\alpha^l|a_{0l}|\right)<\frac{q}{2}, \tag{22} \]

assuming \(q\) to be given, \(q>0\). If, further, \(\gamma\) is such that, along with (21), the conditions

\[ 2A_0+C_0<\frac{q}{2},\qquad A_p\left(2+q\sum_{l=0}^{p-1}\alpha^l|a_{0l}|\right)+C_p<\frac{q}{2} \quad (p=1,\ldots,n-1), \tag{23} \]

are satisfied, then from (20), \((20_1)\) we obtain the inequalities

\[ \left|\tilde u_{1j}(t',\alpha)\right|<cq e^{\gamma t'}, \qquad 0<t'\leq\delta \quad (j=0,1,\ldots,n). \tag{24} \]

We now carry out induction with respect to \(k\). Suppose that

\[ \left|\tilde u_{\nu j}(t',\alpha)\right|<cq e^{\gamma t'}, \qquad t'\in\tilde A_\nu \quad (j=0,1,\ldots,n;\ \nu=1,2,\ldots,k-1) \tag{25} \]

with constants \(\alpha\) and \(\gamma\) satisfying (21)—(23). In order to obtain estimates of the functions \(\tilde u_{kj}(t',\alpha)\) on the interval \(\tilde\Delta_k\), it is necessary first to make the substitution \(t''=t'-(k-1)\delta\). Put, for all \(t''\geq0\) \((t'\geq(k-1)\delta)\),

\[ \tilde u_{kp}(t'',\alpha)=\tilde u_{kp}(t''+(k-1)\delta,\alpha) \]

and

\[ \tilde\psi_{kp}(t'',\alpha)=\tilde\psi_{kp}(t''+(k-1)\delta,\alpha) \qquad (p=0,1,\ldots,n). \]

The Laplace transform gives, for the functions \(\tilde u_{kp}(t'',\alpha)\) \((p=0,1,\ldots,n-1)\), for \(t''>0\), formulas of the form (20), in whose right-hand sides \(t'\), \(\tilde\psi_{1p}(\tau,\alpha)\), \(\eta_{1,j+p}(\alpha)\) are replaced respectively by \(t''\), \(\tilde\psi_{kp}(\tau,\alpha)\), \(\eta_{k,j+p}(\alpha)\), and moreover

\[ \eta_{k,j+p}(\alpha)=\tilde u_{k-1,j+p}(0,\alpha) =\tilde u_{k-1,j+p}((k-1)\delta,\alpha). \]

Let

\[ d(\alpha)=\sum_{i=1}^{n}\sum_{j=0}^{n}\alpha^j|a_{ij}|. \]

We impose on the parameters \(\alpha\) and \(\gamma\) entering the inequalities (25), in addition to (21)—(23), another \(n+1\) conditions:

\[ 2d(\alpha)<|a_{0n}|\alpha^n e^{\gamma\delta},\qquad 2d(\alpha)A_p<e^{\gamma\delta} \quad p=0,1,\ldots,n-1). \tag{26} \]

Then from (25) and the integral representations for \(\tilde u_{kp}(t'',\alpha)\) \((p=0,1,\ldots,n-1)\) it is clear that

\[ \left|\tilde u_{kj}(t'',\alpha)\right| <cq e^{\gamma(t''+(k-1)\delta)}, \qquad 0<t''\leq\delta \quad (j=0,1,\ldots,n) \]

and, consequently,

\[ \left|\tilde u_{kj}(t',\alpha)\right|<cq e^{\gamma t'},\qquad t'\in \tilde\Delta_k \quad (j=0,1,\ldots,n). \]

Thus, if the relations (21)—(23), (26) are satisfied for \(\alpha\) and \(\gamma\), then for any \(k=1,2,\ldots\) the inequalities

\[ \left|\tilde u_{kj}(t',\alpha)\right|<cq e^{\gamma t'},\qquad t'\in \tilde\Delta_k \quad (j=0,1,\ldots,n) \]

hold. Hence, taking (24) and (18) into account,

\[ \left|u^{(j)}(t)\right|<cq\,\alpha^j e^{\alpha\gamma t}\le c_3 e^{c_4 t}, \qquad t\ge 0 \quad (j=0,1,\ldots,n), \]

where

\[ c_3=\max(cq,K)\max_{0\le j\le n}\alpha^j \quad\text{and}\quad c_4=\alpha\gamma . \tag{27} \]

The assertion of the theorem is proved, since the constants \(c_3\), \(c_4\), defined by the equalities (27), do not depend on \(t\).

Theorem 2, in addition to substantiating the applicability of the operational method to the solution of the principal initial problem for equation (1), gives effective estimates for the growth of the solution \(u(t)\) and of its \(n\) derivatives as \(t\to\infty\), which makes it possible to find the solution directly by the Riemann—Mellin formula:

\[ u(t)=\frac{1}{2\pi i}\lim_{\omega\to\infty} \int_{\sigma-i\omega}^{\sigma+i\omega} h^{-1}(s)\,[F(s)+Q(s)]\,e^{st}\,ds, \qquad \sigma>c_4, \]

where \(F(s)\) is the Laplace integral of the function \(f(t)\);

\[ h(s)=\sum_{i=0}^{m}\sum_{j=0}^{n} a_{ij}s^j e^{-\omega_i s}; \]

\[ Q(s)= \sum_{i=0}^{m}\sum_{j=1}^{n}\sum_{k=0}^{j-1} a_{ij}\varphi^{(k)}(0)s^{j-k-1}e^{-\omega_i s} - \sum_{i=1}^{m}\sum_{j=0}^{n} a_{ij}\int_{-\omega_i}^{0}\varphi^{(j)}(t)e^{-s(t+\omega_i)}\,dt. \]

Remark. The application of the Laplace integral to the solution of equation (1) with coefficients \(a_{0n}\ne0\), \(a_{in}=0\) \((i=1,\ldots,m)\) and with zero initial conditions was considered in [2].

References

1. Bellman R., Cooke K. Differential-Difference Equations. New York, Acad. Press, 1963.

2. Ditkin V. A., Kuznetsov P. I. Handbook of Operational Calculus. Moscow—Leningrad, 1951.

Received by the editors
November 9, 1965

Computing Center
Academy of Sciences of the USSR