Bifurcation theorems for Hammerstein nonlinear integral equations

In this paper, we establish two results assuring that $\lambda=0$ is a bifurcation point in $L^\infty(\Omega)$ for the Hammerstein integral equation \begin{center} $u(x)=\lambda\int_\Omega k(x,y)f(y,u(y))dy.$ \end{center} We also present an application to the two-point boundary value problem \[ \left\{ \begin{array}{ll} -u''=\lambda f(x,u) & \mbox{a.e. in $[0,1]$ } \\ u(0)=u(1)=0 \end{array} \right. \]