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).