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

Abstract

We study the existence of solutions to a nonlinear fractional differential equation in Hilbert spaces associated with three-point boundary conditions at resonanceby 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.

Existence of Solutions for a Fractional Boundary Value Problem at Resonance

“…Thus, if c 1 > 1, c 1 f t, c 1 t − 1(H3) is verified. It follows from Theorem 3 that fractional boundary value problem(14) has at least one solution.…”

Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in ℝn

“…In [14], P.D. Phung removed the restriction on matrix A and studied the solvability of the same problem as in [13].…”

“…Then, P.D. Phung [15] used similar methods to study the following three-point boundary conditions in the fractional differential equations at resonance:…”

“…Recently, the solvability of integer or fractional differential equations with a wide range of boundary value conditions at resonance in R n has been researched. We direct readers to [13][14][15][16][17][18][19][20][21] for details.…”

“…Therefore, in this study we consider the characterization of different constraints on different state variables, in other words, we introduce matrices to control the constraints on state variables so that the expression of the equation can be richer. However, other works [13][14][15][16]20,21] under the condition of zero boundary value (similar to u(0) = 0) studied n equations of the problem. Gupta in [10] proposed that many multi-point boundary value problems can be transformed into four-point boundary value problems under certain conditions, so studying four-point BVPs is more meaningful.…”