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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.

##### MSC:
 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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