MATHEMATICS

Z. I. Rekhlitskii

CRITERIA FOR BOUNDEDNESS OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE DELAY OF THE ARGUMENT

(Presented by Academician I. G. Petrovskii on 22 VII 1957)

In our note \((^1)\), criteria were established for the stability of solutions of differential equations of the form

\[ \frac{dy}{dt}-A(t)y(t-a)=x(t)\qquad (0\leq t<\infty;\ a>0). \]

In the present article we give necessary and sufficient criteria for boundedness of solutions on the half-axis \((0,\infty)\) of differential equations of the form:

\[ \frac{dy}{dt}-A(t)y(t-\alpha(t))=x(t)\qquad (\alpha(t)\geq 0); \]

\[ \frac{dy}{dt}-A(t)y(t-a)-B(t)y(t)=x(t)\qquad (a>0). \]

Consider the space \(\widetilde{E}\) of continuous functions \(x=x(t)\) \((-\infty<t<\infty)\) with range belonging to the complex Banach space \(\widetilde{\mathfrak{C}}\). Let \(\widetilde{E}_0\) be the subspace of continuous functions \(\{x(t)\}\) that vanish for \(t<0\). Using some results of the works \((^1\text{–}^4)\), one can show that in the space \(\widetilde{E}\) the following theorems hold.

Theorem 1. Consider the boundary-value problem:

\[ \begin{aligned} \frac{dy}{dt}-\lambda y(t-\alpha(t))&=x(t)\qquad (0\leq t<\infty),\\ y(t)&=\varphi(t)\qquad (t\leq 0;\ \alpha(t)\geq 0). \end{aligned} \tag{1} \]

Suppose that the continuous, bounded function \(\alpha(t)\) admits a representation \(\alpha(t)=\alpha_1(t)+\alpha_2(t)\), which satisfies the conditions:

1) the derivative \(\alpha_1'(t)\) exists;
2) \(\lim \alpha_1'(t)=0\) as \(t\to+\infty\);
3) \(\lim \dfrac{\alpha_2(t)}{t}=0\) as \(t\to+\infty\).

Let \(\lim \alpha(t)=a>0\) as \(t\to+\infty\).
In order that the boundary-value problem (1) have a bounded solution \(y(t)\) for all bounded \(x(t)\) and \(\varphi(t)\), it is necessary and sufficient that all roots \(z\) of the equation \(1-ze^{\lambda a z}=0\) lie outside the unit circle.

If, however, \(a=0\), then for boundedness of \(y(t)\) it is necessary and sufficient that \(\lambda\) lie in the open left half-plane.

Theorem 2. Consider the boundary-value problem

\[ \begin{aligned} \frac{dy}{dt}-Ay(t-\alpha(t))&=x(t)\qquad (0\leq t<\infty),\\ y(t)&=\varphi(t)\qquad (t\leq 0;\ \alpha(t)\geq 0). \end{aligned} \tag{2} \]

Let \(A\) be a linear bounded operator acting in \(\mathfrak{E}\); let \(\alpha(t)\) satisfy the same conditions as in boundary-value problem (1).

If \(a>0\), then in order that boundary-value problem (2) have a bounded solution \(y(t)\) for all bounded \(x(t)\) and \(\varphi(t)\), it is necessary and sufficient that, for every \(\lambda\) in the spectrum of the operator \(A\), all roots \(z\) of the equation \(1 - z e^{\lambda a z}=0\) lie outside the unit circle.

If, however, \(a=0\), then for boundedness of \(y(t)\) it is necessary and sufficient that the spectrum of the operator \(A\) lie in the open left half-plane.

Theorem 3. Consider the boundary-value problem

\[ \frac{dy}{dt} - A(t)y(t-\alpha(t)) = x(t)\qquad (0 \leq t < \infty), \]

\[ y(t)=\varphi(t)\qquad (t \leq 0;\ \alpha(t)\geq 0). \tag{3} \]

Let \(A(t)\) be an operator-valued function admitting a representation
\(A(t)=A_1(t)+A_2(t)\), which satisfies the conditions:

1) for each fixed \(t\), the operators \(A_1(t)\) and \(A_2(t)\) are linear, bounded, and act in \(\mathfrak{E}\);

2) the family of operators \(\{A_1(t)\}\) is compact: from every sequence \(\{A_1(t_n)\}\) one can extract a part convergent in norm;

3) there exists a strong derivative \(A_1'(t)\);

4) \(\lim \|A_1'(t)\|=0\) as \(t\to+\infty\);

5) \(\lim \|A_2(t)\|=0\) as \(t\to+\infty\).

Let \(\alpha(t)\) satisfy the conditions of Theorem 1.

Consider the family \(\{a_\omega\}\) of all possible limiting values of the function \(\alpha(t)\) as \(t\to+\infty\). To each \(a_\omega\), evidently, there corresponds some law according to which \(t\) tends to \(+\infty\). Assign to each \(a_\omega\) the family \(\{A_\omega\}\) of those limiting operators which are generated by the family \(\{A(t)\}\) under the same law of tendency of \(t\) to \(+\infty\).

In order that boundary-value problem (3) have a bounded solution \(y(t)\) for all bounded \(x(t)\) and \(\varphi(t)\), it is necessary and sufficient that, for every \(\lambda\) from the spectrum of at least one limiting operator \(A_\omega\) corresponding to \(a_\omega>0\), all roots \(z\) of the equation \(1-z e^{\lambda a_\omega z}=0\) lie outside the unit circle.

If, however, \(a_\omega=0\), then all \(\lambda\) must lie in the open left half-plane. In particular, when \(a=0\), for boundedness of the solution \(y(t)\) it is necessary and sufficient that all points of the spectra of all limiting operators of the family \(\{A(t)\}\) lie in the open left half-plane.

Theorem 4. Consider the boundary-value problem:

\[ \frac{dy}{dt}-\alpha(t)y(t-a)-\beta(t)y(t)=x(t)\qquad (0\leq t<\infty), \]

\[ y(t)=\varphi(t)\qquad (t\leq 0;\ a>0). \tag{4} \]

Let \(\alpha(t)\) and \(\beta(t)\) be complex-valued functions admitting representations
\(\alpha(t)=\alpha_1(t)+\alpha_2(t)\), \(\beta(t)=\beta_1(t)+\beta_2(t)\), which satisfy the conditions:

1) \(\alpha_1(t), \alpha_2(t), \beta_1(t), \beta_2(t)\) are continuous and bounded;

2) there exist continuous derivatives \(\alpha_1'(t)\) and \(\beta_1'(t)\);

3) \(\displaystyle \lim_{t\to+\infty}|\alpha_1'(t)|=0,\quad \lim_{t\to+\infty}|\beta_1'(t)|=0;\)

4) \(\displaystyle \lim_{t\to+\infty}\alpha_2(t)=0,\quad \lim_{t\to+\infty}\beta_2(t)=0.\)

Let \(\{\alpha_\omega\}\) and \(\{\beta_\omega\}\) be all possible limiting values of the functions \(\alpha(t)\) and \(\beta(t)\), generated by an arbitrary sequence \(t_n\to+\infty\):

\[ \alpha_\omega=\lim_{t_n\to+\infty}\alpha(t_n), \]

\[ \beta_\omega=\lim_{t_n\to+\infty}\beta(t_n). \]

Then, in order that the boundary-value problem (4) have a bounded solution \(y(t)\) for all bounded \(x(t)\) and \(\varphi(t)\), it is necessary and sufficient that all roots \(z\) of the equation

\[ 1 - z e^{(\alpha_\omega z + \beta_\omega)a} = 0 \tag{5} \]

lie outside the unit circle.

Consider a boundary-value problem of the form

\[ \begin{gathered} \frac{dy}{dt} - A(t)y(t-a) - B(t)y(t) = x(t) \qquad (0 \leqslant t < \infty),\\ y(t)=\varphi(t) \qquad (t \leqslant 0;\ a>0), \end{gathered} \tag{6} \]

where \(A(t)\), \(B(t)\) are compact operator-functions, commuting with each other, acting in a Banach space \(\mathfrak{S}\).

One can formulate a sufficient criterion for boundedness of solutions of the boundary-value problem (6), completely corresponding to Theorem 4; \(\alpha_\omega\) and \(\beta_\omega\) in the characteristic equation (5) should be replaced by points of the spectra \(\lambda_\omega\) and \(\mu_\omega\) of the limiting operators \(A_\omega\) and \(B_\omega\), generated by one and the same sequence \(t_n \to +\infty\).

Odessa Pedagogical Institute
named after K. D. Ushinsky

Received
15 I 1957

REFERENCES CITED

1. Z. I. Rekhlitskii, DAN, 111, No. 1 (1956).
2. M. A. Rutman, DAN, 101, No. 2 (1955).
3. M. A. Rutman, DAN, 101, No. 6 (1955).
4. A. A. Rutman, DAN, 108, No. 5 (1956).