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Uniformly continuous \(L^1\) solutions of Volterra equations and global asymptotic stability. (English) Zbl 1190.45010

This paper concerns the following Volterra integro-differential equation \[ x'(t)= -a(t)x(t)+ \int^t_0 b(t,s) x(s)\,ds, \] where \(a\) and \(b\) are continuous functions. Lyapunov functions are constructed and they allow to obtain sufficient conditions so that the solutions of the given equation aer absolutely Riemann integrable on \([0,+\infty)\) and have bounded derivatives. By Barbalat’s lemma it has been proved that under certain hypotheses the zero solution is stable and that all solutions approach zero as \(t\to+\infty\).

MSC:

45M10 Stability theory for integral equations
45J05 Integro-ordinary differential equations
45D05 Volterra integral equations
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