Mathematics

E. Ya. Melamed

On the Stability of Solutions of Some Differential Boundary-Value Problems with Partial Derivatives in a Banach Space

(Presented by Academician I. G. Petrovskii, 19 II 1958)

M. G. Krein \((^1)\) was the first to consider the question of boundedness of solutions of differential equations in a Banach space. M. A. Rutman, in \((^2)\), proposed a method that makes it possible to study the question of stability of solutions of certain operator equations, and applied it in \((^{3-5})\) to the qualitative investigation of solutions of certain linear differential equations with partial derivatives. All of them belong to equations of the form

\[ \frac{\partial^{p_1+p_2+\cdots+p_n}u} {\partial t_1^{p_1}\partial t_2^{p_2}\cdots \partial t_n^{p_n}} - \sum A_{q_1q_2\ldots q_n} \frac{\partial^{q_1+q_2+\cdots+q_n}u} {\partial t_1^{q_1}\partial t_2^{q_2}\cdots \partial t_n^{q_n}} = v, \]

where \(p_i \geqslant g_i\), \(\sum_{i=1}^{n} p_i > \sum_{i=1}^{n} q_i\). By applying to both sides of such an equation a product of Volterra integration operators, it is reduced to an equation with continuous operators, considered in \((^2)\).

We have considered several boundary-value problems that do not belong to the indicated type.

Let us consider in the half-plane \(Q:\ -\infty < x < \infty,\ t \geqslant 0\) the differential boundary-value problems:

\[ \frac{\partial u}{\partial t} - a\frac{\partial u}{\partial x} - A(x,t)u = f(x,t), \qquad u(x,0)=\varphi(x); \tag{1} \]

\[ \frac{\partial u}{\partial t} - a^2\frac{\partial^2 u}{\partial x^2} - A(x,t)u = f(x,t), \qquad u(x,0)=\varphi(x) \quad (a\ne 0); \tag{2} \]

\[ \frac{\partial^2 u}{\partial t^2} + 2\alpha \frac{\partial u}{\partial t} - a^2\frac{\partial^2 u}{\partial x^2} - A(x,t)u = f(x,t), \qquad u(x,0)=\varphi(x), \qquad u_t'(x,0)=\psi(x), \tag{3} \]

where \(u(x,t)\) is the unknown function; \(f(x,t)\), \(\varphi(x)\), \(\psi(x)\) are given functions with values in a complex Banach space \(\widetilde{E}\), defined and continuous in the half-plane \(Q\); \(\alpha\) and \(a\) are real numbers \((\alpha>0)\); \(A(x,t)\) is a continuous operator-valued function with values in the normed ring \(R\) of all linear continuous operators acting in \(\widetilde{E}\).

Regarding the operator-function \(A(x,t)\) we shall assume the following:

\(1^\circ\). The family \(\{A(x,t)\}\) is compact.

\(2^\circ\). The operator-function \(A(x,t)\) has “small variation at \(t\)-infinity”: for a sufficiently small number \(\varepsilon>0\) there exists a \(T>0\) such that for any \(x_1\) and \(x_2\), from the conditions \(t_1>T,\ t_2>T,\ |t_1-t_2|\leqslant 1\) it follows that
\[ \|A(x_1,t_1)-A(x_2,t_2)\|<\varepsilon . \]

We shall call an operator \(A_\omega\in R\) an \(\omega_t\)-limit operator for the operator-function \(A(x,t)\) if there exists a sequence of points \((x_n,t_n)\in Q,\ t_n\to\infty\), such that
\[ \lim_{n\to\infty} A(x_n,t_n)=A_\omega . \]

We shall call the boundary-value problems (1), (2) stable if, to any functions \(f(x,t)\) and \(\varphi(x)\) uniformly bounded in \(Q\), there corresponds a solution uniformly bounded in \(Q\).

We shall call the boundary-value problem (3) stable if, to any functions \(f(x,t)\), \(\varphi(x)\), \(\varphi'(x)\), and \(\psi(x)\) uniformly bounded in \(Q\), there corresponds a solution uniformly bounded in \(Q\).

Theorem 1. In order that the boundary-value problems (1) and (2) be stable, it is necessary and sufficient that the spectra of all \(\omega_t\)-limit operators for the operator-function \(A(x,t)\) lie inside the left half-plane.

Theorem 2. In order that the boundary-value problem (3) be stable, it is necessary and sufficient that the spectra of all \(\omega_t\)-limit operators for the operator-function \(A(x,t)\) lie inside the region bounded by the parabola \(\eta^2=-4\alpha^2\xi\).

The boundary-value problem (3) leads to the equation of electrical oscillations in an infinite conductor.

Let us note that, from Theorem 1, for the boundary-value problem (1) with \(a=0\), one obtains the well-known result of M. G. Krein (see \((^1)\), Theorem 3), refined by M. A. Rutman in \((^4)\). In the finite-dimensional case this refinement was made by N. Ya. Lyashchenko \((^6)\).

Odessa Pedagogical Institute
named after K. D. Ushinsky

Received
27 I 1958

REFERENCES

\(^1\) M. G. Krein, Uspekhi Mat. Nauk, 3, 25, 166 (1948).
\(^2\) M. A. Rutman, DAN, 101, No. 2 (1955).
\(^3\) M. A. Rutman, DAN, 101, No. 6 (1955).
\(^4\) M. A. Rutman, DAN, 108, No. 5 (1956).
\(^5\) M. A. Rutman, Uspekhi Mat. Nauk, 12, issue 1 (1957).
\(^6\) N. Ya. Lyashchenko, DAN, 96, No. 2 (1954).