Asymptotic of solutions of Volterra-Hammerstein equations (Q1398491)

The authors study the asymptotic of solutions of Volterra-Hammerstein equations of the form \[ x(t)=\int_0^tK(t-s)[x(s)+\varphi(x(s))] ds+f(t),\tag{1} \] where \(K\in L_1[0,+\infty)\), \(\varphi\in C(\mathbb{R})\) and \(f\in C[0,+\infty)\). This topic is related to the zeros \(1-\widehat{K}(z)\), where \(\widehat{K}(z)=\int_0^{+\infty}e^{-zt}K(t) dt\) is the Laplace transformation of the kernel \(K(t)\). The authors formulate a few theorems, in which the main assumptions concern the number and location of roots of the equation \(1-\widehat{K}(z)=0\). To obtain more information about the behaviour of solutions of (1) it is assumed that \(f\) has the Taylor decomposition on infinity of order \(m\).