Asymptotic stability of solutions to Volterra-renewal integral equations with space maps (Q448278)

The following linear Volterra-renewal integral equations are considered NEWLINE\[NEWLINE u(x,t)=f(x,t)+\int\limits_{0}^{t}k(t-\eta)u(g(x,t,\eta),\eta)d\eta, NEWLINE\]NEWLINE with \(t>0,\:x\in\Omega:=[a,b]\), where \(k(t)\geq 0,\:f(x,t)\geq 0\) and \(g(x,t,\eta)\) are known continuous functions. Their solutions depend on the space variable, via a map transformation.NEWLINENEWLINEThe authors investigate the asymptotic behavior of these solutions, and study the asymptotic stability of a numerical method based on direct quadrature in time and interpolation in space. There are also some numerical experiments showing that this method behaves according to the theoretical results.