Existence Of Solutions For Some Boundary Value Problems Of Fractional Differential Equations

Compared with integer calculus,fractional calculus can solve practical problems more effectively.As we all know,Riemann-Liouville and Caputo type fractional integrals are defined on the basis of n iterations of integer order integrals,and the derivative definitions of both depend on fractional integrals.Later,the Conformable fractional derivative defined by T.Abdeljawad only depends on the limit definition of the derivative.Immediately afterwards,F.Jarad obtained the New Conformable fractional integral based on n iterations of the Conformable fractional integral.Studies have shown that,the New Conformable fractional calculus can help us better understand the complexity of the dynamics of phenomena in porous media.In order to have a further study on differential equations,by using degree theory and fixed point theorem,the existence of solutions for boundary value problems of fractional differential equations is obtained.The main researches are as follows:1)The existence of nontrivial solutions for the boundary value problems of Caputo type nonlinear fractional differential equations is studied.By using Laplace transformation,the equation satisfied by eigenvalues ? in the boundary value problem of linear differential equation is obtained.And then,combined with degree theory,the existence of nontrivial solutions for the boundary value problem of nonlinear fractional differential equation is obtained.Finally,three different examples are given to estimate the existence range of eigenvalues by using Matlab,which is the prerequisite for using Leray-Schauder degree theory when solving practical problems.2)By using the Banach contraction mapping principle and Leggett Williams fixed point theorem,the existence of solution for two-point boundary value problem of New Conformable fractional differential equations is given.Last,an example is given to illustrate our results,respectively.3)By using the fixed point theory on mixed monotone operators,we obtain the existence of solutions for three-point boundary value problems of New Conformable fractional differential equations.4)Using the fixed point theorem on compression and expansion of cones,we obtain that there is at least one positive solution for the multi-point boundary value problem of New Conformable fractional differential equations.