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. and disturbance parameter mu that divides the range of lambda and mu 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. for given mu. In particular, we give estimates for the critical value of parameters. Two examples are given to illustrate our main results.