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

Abstract

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 H2(R2) 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 L1(R2) of the integral kernels implies the existence and convergence in H2(R2) of the solutions.