zbMATH — the first resource for mathematics

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

45G10 Other nonlinear integral equations
45M10 Stability theory for integral equations
45D05 Volterra integral equations
45M05 Asymptotics of solutions to integral equations
Full Text: DOI
[1] Iannelli, M.; Kostova, T.; Milner, F.A., A method for numerical integration of age- and size-structured population models, Numer. methods partial differential equations, 25, 918-930, (2009) · Zbl 1172.92026
[2] Breda, D.; Cusulin, C.; Iannelli, M.; Maset, S.; Vermiglio, R., Stability analysis of age-structured population equations by pseudospectral differencing methods, J. math. biol., 54, 701-720, (2007) · Zbl 1114.92054
[3] Messina, E.; Russo, E.; Vecchio, A., A stable numerical method for Volterra integral equations with discontinuous kernel, J. math. anal. appl., 337, 1383-1393, (2008) · Zbl 1142.65110
[4] Messina, E.; Russo, E.; Vecchio, A., A convolution test equation for double delay integral equations, J. comput. appl. math., 228, 589-599, (2009) · Zbl 1169.65123
[5] Corduneanu, C., Integral equations and applications, (1991), Cambridge University Press · Zbl 0714.45002
[6] Linz, P., Analytical and numerical methods for Volterra equations, (1985), SIAM Philadelphia · Zbl 0566.65094
[7] Diblík, J.; Růžičková, M., Exponential solutions of equation \(\dot{y}(t) = \beta(t) [y(t - \delta) - y(t - \tau)]\), J. math. anal. appl., 294, 273-287, (2004) · Zbl 1058.34099
[8] Diblík, J.; Růžičková, M., Convergence of the solutions of the equation \(\dot{y}(t) = \beta(t) [y(t - \delta) - y(t - \tau)]\) in the critical case, J. math. anal. appl., 331, 1361-1370, (2007) · Zbl 1125.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.