Abstract
In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable x∈Rn and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained.
Highlights
We show the solvability of boundary value problems (BVPs) (1), (2) when B 6= I, C 6= I, |( I − ηC )( I − B) +
We show the solvability of BVP (1), (2) when B = I, | I − C | = 0
We study the BVP (1) and (30) using Theorem 1
Summary
Motivated by [9], Bai [6] researched a four-point boundary value problem, and proved the existence and multiplicity results by making use of the method of upper and lower solutions established by the coincidence degree theorem. P.D. Phung [15] used similar methods to study the following three-point boundary conditions in the fractional differential equations at resonance:. Motivated by the above ideas, we consider the following fractional-order equations with a new boundary value condition in Rn : c. It is worth mentioning that, inspired by [14], in Section 4, we remove the restriction on the matrix C, and give the existence theorem of the solution of the problem only under the most basic resonance conditions (refer to case (2))
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