Messoud Efendiev, Vitali Vougalter
Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion
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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 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.