On nonlinear Volterra equations of convolution type (Q1048387)

The author proves some results on existence and uniqueness and some results on nonexistence of nontrivial nonnegative solutions of the two nonlinear Volterra integral equations \[ u^\alpha(x) = a(x) \int_0^x k(x-t) b(t) u(t) dt + f(x), \quad 0 < x \leq \ell, \leqno(1) \] and \[ u^\alpha(x) = \int_0^x u(x-t) u(t) dt. \leqno(2) \] All the functions appearing in these equations are real-valued and strictly positive on \((0,\ell)\), where \(0 < \ell \leq \infty\). The analysis of (1) is based on a generalized monotonicity notion, and several different sets of sufficient conditions are given for the existence of a unique positive (and sometimes nondecreasing) solution \(u\) of (1). About equation (2) it is shown that the condition \(\alpha > 2\) is necessary for the existence of a locally bounded nontrivial solution of (2) and sufficient for the existence of a locally integrable nontrivial solution of (2), and that (2) does not have any nontrivial solution which belongs locally to \(L^s(0,\infty)\) for \(s \geq 1/\alpha\) if \(0 \leq \alpha \leq 1\) and for \(s = 1\) if \(1 < \alpha \leq 2\).