ON THE SOLVABILITY OF CERTAIN BOUNDARY VALUE PROBLEMS FOR A NONLOCAL POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER

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Abstract

This paper investigates the solvability of certain boundary value problems for a nonlocal polyharmonic equation. The nonlocal polyharmonic operator is introduced through a new class of nonlocal polyharmonic operators using involution-type mappings. Boundary value problems with Dirichlet conditions and boundary operators of fractional order are considered. The solvability of a boundary value problem with Dirichlet conditions for the corresponding nonlocal polyharmonic equation is studied. First, the existence and uniqueness of the solution to Problem D are established, and the corresponding theorem is formulated. Based on this theorem, an explicit form of the Green's function and an integral representation of the solution to Problem D are obtained. The second problem under consideration involves boundary operators of fractional order, which are defined using modified Hadamard derivatives. The properties of mutual invertibility of the Hadamard-type integro-differential operators   and  are presented, and the smoothness of the functions  and  in Hölder classes is established. Based on these properties of the integro-differential operators, the existence and uniqueness of the solution to the corresponding problem  are proved under the condition , and in the case , a necessary and sufficient condition for the existence of a solution is determined. As a result, theorems are proved that confirm the existence and uniqueness of solutions for the problems under consideration.