## Positivity and regularity of hyperbolic Volterra equations in Banach spaces.(English)Zbl 0608.45007

We derive necessary and sufficient conditions for two classes of linear Volterra equations of the form (*) $$u=f+a*Au$$ to admit finite wave speed as well as continuity across the wave front. These conditions are based on the positivity of the fundamental solution. One of these classes serves as a model in linear viscoelasticity. These results are then used to obtain several general theorems on existence, positivity, regularity and asymptotic behavior of the resolvent for (*).

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45M05 Asymptotics of solutions to integral equations 74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
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### References:

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