Portal de Periódicos da CAPES

On the Uniquenes of Bounded Solutions to $u'(t) = A(t)u(t)$ and $u''(t) = A(t)u(t)$ in Hilbert Space

1973; Society for Industrial and Applied Mathematics; Volume: 4; Issue: 2 Linguagem: Inglês

10.1137/0504024

1095-7154

Howard A. Levine,

Differential Equations and Boundary Problems

Let $A:D_A \to H$ be a symmetric linear operator. If 0 is not an eigenvalue of A, then every solution A to $u'(t) = Au(t)$, $ - \infty < t < \infty $, is either identically zero or satisfies $\sup _{ - \infty < t < \infty } \| {u(t)} \| = + \infty $. This result is proved via an elementary argument and then extended in two directions: (i) $A = A(t)$, $t \in ( - \infty ,\infty )$, and (ii) $A = A_ + + A_ - $, where $A_ + $ is symmetric, zero is not an eigenvalue of $A_ + $, $A_ - $ is skew symmetric and $\operatorname{Re} (A_ + x,A_ - x) > - \| {A_ + x} \|^2 $ for all $x \in D_A $, $x \ne 0$. This inequality is sharp. A similar analysis is carried out for $u''(t) = A(t)u(t)$. A number of examples from partial differential equations are given.