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

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.