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Convergence of solutions for two delays Volterra integral equations in the critical case. (English) Zbl 1197.45006
Consider the delay Volterra equation
$y(t)=f(t)+\int_{t-\tau}^{t-\delta}k(t,s)g(y(s))ds,\quad t\geq\tau$
(with initial value $$y|_{[0,\tau]}$$), where $$g$$ satisfies a Lipschitz condition with constant $$L$$. Assuming that $$L|k(t,s)|\leq\beta(s+\delta)$$ with $$\beta(t)\leq(\tau-\delta)^{-1}-c/t$$ it is shown that any two solutions $$y,y_*$$ of the equation satisfy $$|y(t)-y_*(t)|\to0$$ as $$t\to\infty$$, and actually the difference can be estimated by $$z(t)-z(t-(\tau-\delta))\to0$$ where $$z$$ is a solution of the auxiliary linear delay differential equation $$z'(t)=\beta(t)(z(t-\delta)-z(t-\tau))$$. The case is critical in the sense that if, roughly speaking, all inequalities in the assumptions are inverted (in particular, if $$L$$ denotes the lower Lipschitz constant of $$g$$) there holds a similar lower bound for $$|y(t)-y_*(t)|$$.

##### MSC:
 45G10 Other nonlinear integral equations 45M10 Stability theory for integral equations 45D05 Volterra integral equations 45M05 Asymptotics of solutions to integral equations
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##### References:
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