SOLVABILITY OF SOME FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED DIFFUSION IN A SQUARE

. We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems in a square in two dimensions with periodic boundary conditions. They contain the normal diffusion in one direction and the superdiffusion in the other direction. We work in a constrained subspace of H2 using the fixed point technique. The elliptic equation involves a second order differential operator satisfying the Fredholm property. It is established that, under reasonable technical assumptions, the convergence in the appropriate function spaces of the integral kernels yields the existence and convergence in H02 of the solutions. We generalize the results obtained in our preceding work [11] for the analogous equation considered in the whole R2 which contained a non-Fredholm operator.