一类具p-Laplacian算子的分数阶边值问题正解的存在性

Existence of Solutions to a Class of Fractional Order Boundary Value Problems with p-Laplacian Operators

含p-Laplacian算子的微分方程被广泛地应用于物理学和自然现象等各个领域。在含p-Laplacian算子的基础上,讨论一类新的具有任意阶Caputo导数的微分方程边值问题正解存在性问题。通过求解与微分方程等价的积分方程得到积分方程的格林函数及其相应性质,再定义一个Banach空间中的算子和最大模范数,并利用Arzela-Ascoli定理证明定义的算子为全连续算子,最后利用Kranoselskii不动点定理证明所研究的分数阶微分方程边值问题的正解存在。

Differential equations with p-Laplacian operator are widely applied in different fields of physics and natural phenomena.On the basis of p-Laplacian operator,the existence of positive solutions to a class of new boundary value problems of differential equations with arbitrary order Caputo derivative is discussed.Green’s functions of integral equation and properties are obtained by solving the integral equation which is equivalent to differential equation,an operator and maximum norm on a Banach space are defined and Arzela-Ascoli theorem is used to prove that the defined operator is continuous.Finally,the existence of positive solutions to boundary value problems is proved by Kranoselskii’s fixed point theorem.