Solvability of some integro-differential equations with the bi-Laplacian and transport

We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H⁴ spaces. Our argument is based on the fixed point technique when the elliptic equations contain fourth order differential operators with and without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in L¹ of the integral kernels yields the existence and convergence in H⁴ of the solutions.