带有 $p$-Laplacian 算子的分数阶非线性积分边值问题的唯一正解与和算子方法

摘要/Abstract

摘要：

该文研究了一类带有 $p$-Laplacian 算子并且非线性项 $f$ 中含有分数阶导数项以及边界条件中含有非线性积分项的分数阶边值问题唯一正解的存在性. 通过构造适当的辅助边值问题和锥上的等价类, 利用锥理论与和算子方法获得了该边值问题唯一正解存在的充分条件, 建立了一致收敛于唯一正解的单调迭代格式, 最后给出了一个具体的例子作为所获结论的应用.

关键词: 分数阶微分方程, 边值问题, $p$-Laplacian 算子, 正解, 锥

Abstract:

This paper investigates the existence of unique positive solution for a class of fractional boundary value problems involving the $p$-Laplacian operator, a fractional derivative term in the nonlinearity $f$ and nonlinear integral terms in the boundary conditions. By constructing appropriate auxiliary boundary value problems and equivalence classes on cone, and using the theory of cone and the method of sum operators, some sufficient conditions for the existence of unique positive solution are obtained, in addition, a monotone iterative sequence uniformly converging to the unique positive solution is constructed. Finally, an example is given to illustrate the main result.

Key words: Fractional differential equation, Boundary value problem, $p$-Laplacian operator, Positive solution, Cone

中图分类号:

O175.8

王文霞. 带有 $p$-Laplacian 算子的分数阶非线性积分边值问题的唯一正解与和算子方法[J]. 数学物理学报, 2023, 43(6): 1731-1743.

Wang Wenxia. The Method of Sum Operator and Unique Positive Solution for Fractional Nonlinear Integral Boundary Value Problems with $p$-Laplacian Operator[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1731-1743.

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