Stability theorems for some functional equations

Abstract:

Functional-differential equations of the form \[ \dot u(t) = - \int _0^t {A(t - \tau )g(u(\tau ))d\tau + f(t,u(t))} \] are considered. Here $u(t)$ is to be an element of a Hilbert space $\mathcal {H},A(t)$ a family of bounded symmetric operators on $\mathcal {H}$ and g an operator with domain in $\mathcal {H}$. g may be unbounded. A is called strongly positive if there exists a semigroup exp St, where S is symmetric and $(S\xi ,\xi ) \leqq - m{\left \| \xi \right \|^2},m > 0$, such that ${A^ \ast } = A - \exp$ St is positive, that is, \[ \int \nolimits _0^T \left ( {v(t),\int _0^t {{A^\ast }(t - \tau )v(\tau )} } \right )d\tau \geqq 0,\] for all smooth $v(t)$. It is shown that if A is strongly positive, and g and f are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.