The authors establish the existence of positive \(L^p\)-solution of the Hammerstein integral equation \[ y(t)=h(t)+\int_0^T k(t,s)f(s,y(s)) ds \quad\text{for a.e. } t\in[0,T) \] where \([0,T)\) is either bounded or unbounded interval contained in \(R\). First, for simplicity, the case when \(h\equiv 0\) is considered. The function \(f\) is a.e. positive, nondecreasing in view of the second variable, satisfies the classical Caratheódory's assumptions and some kind of growth condition. The kernel \(k\) is measurable and satisfies some regularity conditions. The main tool in the proofs is Krasnoselskii's fixed point theorem for completely continuous operators acting on a cone contained in a Banach space. To illustrate the obtained results the authors consider the case \(f(t,y)=y^n\), \(n\geq 0\).