Solvability For Fractional Order Multi-point Boundary Value Problems At Resonance

The research on the theory and application of fractional multi-point boundary value problems have been paid widely attention and have been made great progress and development. Fractional differential equation is a valuable tool, widely used in many fields. The fractional differential equation boundary value problems at resonance attracted the attention of the majority of scholars. In this paper, the Mawhin theorem of coincidence degree theory is used to solve the existence of solutions for fractional multi-point boundary value problems at resonance.Firstly, describes the development of fractional order differential equation and the Caputo fractional derivative theory. Lists the previous studies on fractional differential equations and introduces the fractional differential equation boundary value problems to be discussed in this article.Secondly, on the basis of reference, changes the fractional differential equation and make it universal. The basic hypothesis and relative lemma of the boundary value problem for fractional differential equation are given, establishes proper operators,the existence of solutions for fractional boundary value problems at resonance is researched under the premise of Caputo differential order number 1 < ? ? 2,0 < ? ? 1 based on the coincidence degree theory. In addition, the dimensions of nuclear is 1, the existence conditions on solutions are obtained.Thirdly, a new class of boundary condition is studied based on this, which makes it more meaningful. The basic hypothesis and relative lemma of the boundary value problem for fractional differential equation are given, changes boundary value conditions and establishes proper operators, the existence of solutions for fractional multi-point boundary value problems at resonance is considered under the premise of Caputo differential order number 1 < ? ? 2, 0 < ? ? 1 based on the coincidence degree theory. In addition, the dimensions of the nuclear is 2, the existence conditions on solutions are established.Finally, summarizes the main results of the full text, and proposes the future research direction.