Existence of solutions to fractional boundary value problems at resonance in Hilbert spaces

• Main Authors: Phan Dinh Phung, Ha Binh Minh
• Format: Article
• Language: English
• Published: SpringerOpen 2017-07-01
• Series: Boundary Value Problems
• Subjects: coincidence degree; three-point boundary value problem; fractional differential equation; resonance
• Online Access: http://link.springer.com/article/10.1186/s13661-017-0836-3

Abstract We study the existence of solutions to a nonlinear fractional differential equation in Hilbert spaces associated with three-point boundary conditions at resonance x ( 0 ) = θ , D α − 1 x ( 1 ) = A D α − 1 x ( η ) $$x(0) = \theta,\qquad D^{\alpha -1}x(1) = AD^{\alpha -1}x(\eta) $$ by using Mawhin’s continuation theorem. We propose a new technique to improve the conditions on A which have been used previously. In addition, a necessary and sufficient condition for that the fractional differential operator is Fredholm with zero-index is established, especially for the first time when the fractional differential operator takes values in an arbitrary Hilbert space.