Picard and Adomian methods for quadratic integral equation (Q619678)

The authors investigate the Volterra integral equation \(x(t) = a(t) + g(t, x(t)) \int_{0}^{t} f(s, x(s))ds\) under standard continuity and Lipschitz conditions on the given functions. They prove that the integral equation has a positive continuous solution if \(a\), \(f\) and \(g\) map to \((0,\infty)\). A Picard iteration is used to provide a sequence of approximate solutions that converges to the exact solution. The analysis of the Adomian method seems to be somewhat unprecise as the authors use the condition \(h<1\) which does not appear to be justified by the assumptions.