The existence of nonnegative solutions of the system of \(n\) nonlinear equations \(x=\lambda AF(x)\) is shown, where the parameter \(\lambda>0\), \(A\) is real \(n\times n\) matrix, \(AF(x)\geq 0\), \(x\in \mathbb R_{+}^{n}\), \(F(x)\) maps \(\mathbb R_{+}^{n}\) to \(\mathbb R^{n}\). The negative elements of \(A\) also can be used which improve other results. Sharp bounds of \(\lambda\) are received.