Existence of solutions for double-order Hilfer fractional boundary value problems at resonance

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.