Messoud Efendiev, Vitali Vougalter
On the solvability of some systems of Fredholm integro-differential equations with mixed diffusion in a square
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ABSTRACT. We establish the existence in the sense of sequences of solutions for a
certain system of integro-differential equations in a square in two dimensions with periodic boundary conditions involving the normal diffusion in one direction and the superdiffusion in the other direction
in a constrained subspace of H^2 for the vector functions via the fixed point technique. The system of elliptic equations contains a second order
differential operator, which satisfies the Fredholm property. It is demonstrated that, under the reasonable technical conditions, the convergence in the appropriate function spaces of the integral kernels
implies the existence and convergence in H_{c}^{2}(\Omega, {\mathbb R}^{N}) of the solutions. We generalize the results derived in our previous article [18] for the analogical system studied in the whole
R^2 which involved non-Fredholm operators. Let us emphasize that the study
of the systems is more complicated than of the scalar case and requires to
overcome more cumbersome technicalities.