Solvability of some systems of non-fredholm integro-differential equations with mixed diffusion

Abstract

We prove the existence in the sense of sequences of solutions for some system of integro-differential type equations in two dimensions containing the normal diffusion in one direction and the anomalous diffusion in the other direction in H-2 (R-2, R-N) using the fixed point technique. The system of elliptic equations contains second order differential operators without the Fredholm property. It is established that, under the reasonable technical assumptions, the convergence in L-1 (R-2) of the integral kernels yields the existence and convergence in H-2 (R-2, R-N) of the solutions. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities.