Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion

We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion in one direction and the anomalous diffusion in the other direction in H 2 ( R 2 ) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under the reasonable technical conditions, the convergence in L 1 ( R 2 ) of the integral kernels implies the existence and convergence in H 2 ( R 2 ) of the solutions.