Existence Of Solutions For Some Boundary Value Problems Of Hilfer Fractional Differential Equations At Resonance

The boundary value of fractional order differential equations is an analytical problem of solving equations under given conditions.It is closely related to many disciplines and has a wide application background.The boundary value problems of differential equations are divided into resonance boundary value problems and non-resonant boundary value problems.The most important difference between them is whether the differential operator corresponding to the boundary value problem is reversible.Previously,most scholars studied the resonance boundary value problem of differential equations on the basis of Riemann-Liouville and Caputo derivatives,and solved many practical problems.Later,the Riemann-Liouville and Caputo derivatives were generalized to the Hilfer fractional derivative,and the differential equations obtained by the Hilfer derivative were used to solve a wider range of practical problems.Therefore,it is particularly important to study the resonance boundary value problems of Hilfer fractional order differential equations,and it is of certain practical significance.In this paper,by defining several appropriate Banach Spaces,constructing appropriate projection operators P and Q,and using Mawhin’s coincidence degree theory,we obtain the existence of solutions of four kinds of boundary value problems at resonance: the Hilfer fractional integral boundary value problem,the mixed Hilfer fractional three-point boundary value problem,the Hilfer fractional impulse problem with dim Ker L=2 and the double-order Hilfer fractional integral boundary value problem,and give examples to explain the conclusions.The above are the research results for linear differential operators,and for nonlinear operators,by defining two appropriate Banach spaces,constructing appropriate operators,and using the extension for the continuous theorem of Ge and Ren to obtain the existence of two types of problem solutions: the Hilfer fractional integral boundary value problem with p-Laplacian and the Hilfer fractional functional boundary value problem with p-Laplacian,and the conclusions are illustrated with examples respectively.