double-order Hilfer分数阶共振边值问题解的存在性

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

为了拓展分数阶微分方程边值问题的基本理论,研究了共振情形下double-order Hilfer分数阶微分方程在Riemann-Stieltjes积分边界条件下解的存在性。首先,构造2个合适的Banach空间;然后,在Banach空间中定义恰当的算子并使用Mawhin重合度理论,获得double-order Hilfer分数阶共振边值问题解的存在性;最后,通过例子验证结果的正确性。结果表明,在合适的Banach空间中,double-order Hilfer分数阶共振边值问题的解具有存在性。采用Mawhin重合度理论方法研究double-order Hilfer分数阶共振边值问题解的存在性,扩展了微分算子阶数的取值范围,丰富了分数阶微分方程的可解性理论,为微分方程在空气动力学、经济学、控制理论等领域的应用提供了理论参考。

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.