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A singular perturbation problem in integrodifferential equations. (English) Zbl 0809.45008
Summary: The solution \(w\) to the integro differential equation \(w' = Aw + K^* Aw + f\) is approximated by solutions \(w\) to the perturbed equation \(\varepsilon^ 2 u'' + u' = Au + K^* Au + f_ \varepsilon\) with initial conditions \(w(0) = w_ 0\), \(w'(0) = w_ 1\), \(u(0) = u_ 0\), \(u'(0) = u_ 1\). The functions \(u,w\) are in a Banach space \(X\), the operator \(A\) is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and the kernel \(K\) is such that the convolution is a bounded linear operator for \(t \geq 0\).
With some convergence conditions on the data \(u,f\), and smoothness conditions on \(K\), we prove that as \(\varepsilon\) approaches 0, \(u\) approaches \(w(t)\) and \(u'\) approaches \(w'\) in \(X\) uniformly on \([0,T]\) for any fixed \(T>0\). An application to viscoelasticity is given.

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45L05 Theoretical approximation of solutions to integral equations
34K30 Functional-differential equations in abstract spaces
47N20 Applications of operator theory to differential and integral equations
74Hxx Dynamical problems in solid mechanics
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