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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 $$y(t)=f(t)+\int_{t-\tau}^{t-\delta}k(t,s)g(y(s))ds,\quad t\ge\tau$$ (with initial value $y|_{[0,\tau]}$), where $g$ satisfies a Lipschitz condition with constant $L$. Assuming that $L|k(t,s)|\le\beta(s+\delta)$ with $\beta(t)\le(\tau-\delta)^{-1}-c/t$ 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 Nonsingular nonlinear integral equations 45M10 Stability theory of integral equations 45D05 Volterra integral equations 45M05 Asymptotic theory of integral equations
Full Text:
References:
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