I will present the results on the existence of solutions to semi-linear equation

    Lx+N(x)=0,

where L is a linear and N a nonlinear operator defined on Hilbert space. I concentrate on the case when 0 is in an essential spectrum of L. The two main methods which I use are: topological degree in infinite-dimensional spaces and the spectral theory for linear operators in Hilbert spaces. This results are part of my Ph.D. project.