The authors consider the operator equation in the form:

$\left(1\right)\phantom{\rule{1.em}{0ex}}Lx=Nx$ , where

$L$ is a Fredholm mapping of index zero and

$N$ is

$L$-completely continuous. By using Brouwer degree theory and a continuation theorem based on Mawhin’s coincidence degree theory there is developed an abstract existence theorem at resonance for the equation (1). As application of this result sufficient conditions are proved for the existence of

$2\pi $-periodic solutions to semilinear equations at resonance, where the kernel of the linear part has dimension

$\ge 2$. Finally, some ilustration examples on the theory are given.