Convergence of solutions for two delays Volterra integral equations in the critical case. (English) Zbl 1197.45006

Messina, Eleonora; Muroya, Yoshiaki; Russo, Elvira; Vecchio, Antonia

Appl. Math. Lett. 23, No. 10, 1162-1165 (2010).

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 \(|y(t)-y_*(t)|\).

Reviewer: Martin Väth (Berlin)

MSC:

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

Keywords:

Volterra delay equation; asymptotic behavior; stability; delay differential equation; convergence of solution; critical case; unbounded solution

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