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In order to extend the basic theory of boundary value problems, the existence of solutions for a class of Hilfer fractional impulsive differential equations with finite impulsive points was studied. Firstly, the integral equation equivalent to the differential equation was obtained; Secondly, appropriate Banach spaces and norms were defined, and appropriate operators were constructed. When the nonlinear term satisfies different conditions, sufficient conditions for the existence of solutions of such boundary value problems were obtained by using Krasnoselskii fixed point theorem; Finally, two examples were used to illustrate the universality of the research results. It is shown that the solution of the boundary value problem of impulsive differential equations with Hilfer fractional derivative exists. By using the Krasnoselskii fixed-point theorem, the existence of solutions for impulsive differential equation boundary value problems with Hilfer fractional order can be effectively solved, which provides some reference for solving other types of impulsive fractional differential equation boundary value problems.

Abstract This article investigates the existence of solutions of mixed Hilfer fractional differential equations with p-Laplacian under the functional boundary conditions at resonance. By defining Banach spaces with appropriate norms, constructing suitable operators, and using the extension of the continuity theorem, some of the current results are extended to the nonlinear situation, and some new existence results of the problem are obtained. Finally, an example is given to verify our main results.

In order to expand the basic theory of boundary value problems of fractional differential equations,the existence of solutions of double-order Hilfer fractional differential equations under Riemann-Stieltjes integral boundary conditions under resonance conditions was studied.Firstly,two suitable Banach spaces were constructed.Secondly,appropriate operators in Banach spaces were defined and Mawhin′s coincidence theory was used to prove the existence of the solution to the double-order Hilfer fractional resonance boundary value problem.Finally,an example was given to illustrate the correctness of the results.The results show that the solution of the double-order Hilfer fractional resonant boundary value problem exists in a suitable Banach space.Using Mawhin′s coincidence degree theory to study the existence of solutions to the double-order Hilfer fractional resonance boundary value problem expands the range of orders in the differential operator,enriches the theory of solvability of fractional differential equations and provides an important theoretical basis for the study of fractional differential equations in aerodynamics,economics,control theory and other fields.