Solvability For Some Conformable Fractional Boundary Value Problem At Resonance

The study of boundary value problems of fractional differential equations is closely related to other branches of mathematics.Although many scholars have studied these types of equations,the theories to establish them have only just begun.When studying fractional calculus and its applications in science and engineering,several fractional derivatives are mainly involved.The most common are fractional derivatives such as Riemann-Liouville,Caputo,Hadamard and Grunwald-Letnikov,which have been successfully applied to complex dynamics that appear in physics,biology,engineering and many other fields.As far as we know,in the past ten years,there were many research results in the discussion of boundary value problems of Riemann-Liouville fractional and Caputo fractional differential equations.However,the research on boundary value problems of newly defined Conformable fractional and New Conformable fractional differential equations has just started.Therefore,the boundary value problems of Conformable fractional and New Conformable fractional differential equations need to be further explored.On the other hand,differential equation boundary value problems are divided into resonant differential equation boundary value problems and non-resonant differential equation boundary value problems.The differential operator of the non-resonant boundary value problem is reversible,and the differential operator of the resonance boundary value problem is irreversible.This paper will study the following three types of resonance boundary value problems for fractional differential equations:1)The three-point resonance boundary value problem of New Conformable fractional differential equation is studied.Firstly,a new normed linear space is defined,and the completeness of the space is proved,and then a projection operator is given,and the existence of understanding is studied using Mawhin's coincidence theory.2)Using Mawhin's coincidence degree theory,under the three-point boundary condition,the boundary value problem of the Conformable fractional differential equation is studied,and the existence of the solution is obtained.3)The multi-point resonance boundary value problem of New Conformable fractional differential equation is studied.First,a suitable Banach space and its norm are defined,and then a projection operator is given.Using Mawhin's coincidence degree theory,the existence of solution for the multi-point boundary value problem is obtained,and an example is given to verify the existence of solution for the equation.