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Stability theorems for some functional equations


Authors: R. C. MacCamy and J. S. W. Wong
Journal: Trans. Amer. Math. Soc. 164 (1972), 1-37
MSC: Primary 45M05
DOI: https://doi.org/10.1090/S0002-9947-1972-0293355-X
MathSciNet review: 0293355
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Abstract: Functional-differential equations of the form

$\displaystyle \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\Vert \xi \right\Vert^2},m > 0$, such that $ {A^ \ast } = A - \exp $ St is positive, that is,

$\displaystyle \mathop \int \nolimits_0^T \left( {v(t),\int_0^t {{A^\ast}(t - \tau )v(\tau )} } \right)d\tau \geqq 0,$

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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0293355-X
Keywords: Functional-differential equation, Gårding's inequality, Laplace transform, partial differential-functional equation, positivity, strong ellipticity, strong positivity, symmetric, weak boundedness, weak convergence, weak stability, weak uniform continuity
Article copyright: © Copyright 1972 American Mathematical Society

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