The existence theorems of the random solutions for random Hammerstein equation with random kernel. II (Q1861751)

[For part I see \textit{G. Li} and \textit{L. Debnath}, ibid. 13, 111-115 (2000; Zbl 0957.45022).] Two theorems are proved establishing the existence of nontrivial, nonnegative, random continuous solutions of the random Hammerstein integral equation \[ u(\omega,x)= \int_Gk(\omega, x,y)f\bigl( y,u(\omega,y) \bigr)dy \] where \(G\) is a bounded closed domain in \(\mathbb{R}^n\) and \(f(x,u)=\sum^m_{i=1} a_i(x)u^{\alpha_i}\), \(\alpha_i>0\). The first theorem also guarantees the existence of at least two nontrivial solutions.