Eigenvalue problem for fractional differential equations with nonlinear integral and disturbance parameter in boundary conditions

Boundary value problems

Wenxia Wang¹  Xiaotong Guo² 

[1] Department of Mathematics, Taiyuan Normal University, Taiyuan, P.R. China;School of Software, North University of China, Taiyuan, P.R. China

关键词: integral boundary value problem;    eigenvalue problem;    disturbance parameter;    Caputo fractional derivative;    positive solution;    cone;    34B18;    34B15;   

DOI  :  10.1186/s13661-016-0548-0

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

This paper is concerned with the existence, nonexistence, uniqueness, and multiplicity of positive solutions for a class of eigenvalue problems of nonlinear fractional differential equations with a nonlinear integral term and a disturbance parameter in the boundary conditions. By using fixed point index theory we give the critical curve of eigenvalue λ and disturbance parameter μ that divides the range of λ and μ for the existence of at least two, one, and no positive solutions for the eigenvalue problem. Furthermore, by using fixed point theorem for a sum operator with a parameter we establish the maximum eigenvalue interval for the existence of the unique positive solution for the eigenvalue problem and show that such a positive solution depends continuously on the parameter λ for given μ. In particular, we give estimates for the critical value of parameters. Two examples are given to illustrate our main results.

【 授权许可】

CC BY