The author studies Volterra integral equations of the form \(u(x)=\int^ x_ 0k(x-s)g(u(s))ds\) for \(x\geq 0\), where \(k\geq 0\) is an integrable function and \(g\) is an increasing absolutely continuous function with \(g(0)=0\). Sufficient conditions and necessary conditions for the existence of a positive nontrivial solution are obtained. An example is given.