On solvability of some boundary value problems for the non-local polyharmonic equation with boundary operators of the Hadamard Type

In this paper the solvability problems of some boundary value problems for a non-local polyharmonic equation are studied. A non-local polyharmonic equation is represented by using some orthogonal matrix. The properties and examples of such matrices are given. In the current boundary value problem, which being considered in the paper, the fractional order differentiation operators are used as boundary operators. These operators are defined as derivatives of the Hadamard-Caputo type. Note that in particular cases of the parameters of the boundary conditions we obtain well known conditions of the Dirichlet, Neumann, and Robin type problems. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. The exact solvability conditions for the problem under study are found. In addition, we obtained representation for the solution of the fractional boundary problem for polyharmonic operator. (paper)