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On stability and robust stability of positive linear Volterra equations in Banach lattices. (English) Zbl 1218.45013

The authors first introduce the notion of positive linear Volterra integro-differential equations in Banach lattices. Then they give a characterization of positive linear Volterra equations in terms of positivity of the co-semigroup generated by \(A\), where \(A\) is the infinitesimal generator of a co-semigroup, and positivity of the kernel function. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, they deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given. An example is given to illustrate the obtained results. Their analysis is based on the theory of positive co-semigroups on Banach lattices.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45M10 Stability theory for integral equations
45J05 Integro-ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
45D05 Volterra integral equations
45M20 Positive solutions of integral equations
46B42 Banach lattices
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