Periodic solutions of a class of integral equations

Based on the fixed point index theory for a Banach space, nontrivial periodic solutions are found for a class of integral equation of the form $$ \phi (x)=\int_{[x,x+\omega ]\cap \Omega }K(x,y)f(y,\phi (y-\tau (y)))dy, \quad x\in \Omega , $$ where $\Omega $ is a closed subset of $\mathbb R^{N}$ with perioidc structure