Existence of solutions to fractional differential equations with three-point boundary conditions at resonance with general conditions

Abstract

This paper presents a new technique to investigate the existence of solutions to fractional three-point boundary value problems at resonance in a Hilbert space. Based on the proposed method, the restricted conditions A 2 ξ 2α−2 = Aξ α−1 and A 2 ξ 2α−2 = I on the operator A, which have been used in [18], are removed. It is shown that the system under consideration admits at least one solution by applying coincidence degree theory. Finally, an illustrative example is presented.where x l 2 = ∞ i=1 |x i | 2 is the norm of x = (x 1 , x 2 , . . . , •) ⊤ in l 2 .