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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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