A note on the uniqueness and attractive behavior of solutions for nonlinear Volterra equations (Q1607584)

The authors consider the nonlinear Volterra integral equations of convolution type \[ u(x)= \int_0^\infty k(x-s)g(u(s)) ds,\tag{1} \] with positive kernel \(k(x)\) and positive nonlinearity \(g(x)\). They study the nontrivial solutions of equation (1) in the sense that \(u(x) \not\equiv 0\) on \([0, \delta)\) for any positive \(\delta\). Under assumptions that \(g(x)\) and \(k(x)\) are continuous and \(g(x)\) and \(K(x)=\int_0^x k(t) dt\) are strictly increasing functions they show that positive solutions of the equation (1) must be locally bounded and global attractors of positive functions.