Solvability of an elliptic partial differential equation with boundary condition involving fractional derivatives

Modified Riemann - Liouville integro-differential operators in the class of harmonic functions and their applications

“…We note that in the case = = 0, i.e., as = ≡ I is the identity mapping, in the one-dimensional case similar problems were studied in work [16], while the case 2 was considered in works [17], [18], [19]. We note also that similar nonlocal problems for boundary operators of fractional order were studied in works [2], [20], [21], [22].…”

“…Then it is easy to show that by condition (21), kernel ( , ) is continuous on Ω × Ω, and by condition (22) it is symmetric, i.e., ( , ) = ( , ). Therefore, Fredholm theorems [23] are applicable to integral equation (20). It implies easily the statement of the theorem.…”

“…If conditions (13) and (16) are satisfied, then a constant solves homogeneous problem (15). Therefore, in this case the homogeneous integral equation corresponding to (20) has one linearly independent solution and this solution is the function ( ) = , where is an arbitrary constant. Since * ( , ) = ( , ) = ( , ), the dual equation corresponding to (20) also has the solution * ( ) = .…”

On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

“…We mention that in the case N D Ň D 0, that is, when B N N D B Ň N Á I is the unit operator, analogous problems have been studied in [18][19][20]. We also mention that analogous problem with boundary operators of fractional order have been studied in [15,21].…”

Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials

“…Many classical operators of harmonic analysis were studied, in Morreytype spaces and similar classes of functions, during the last decades (see e.g. [3] [9], [28], [24], [29], [30], [36]). Definition 1.2.…”