Abstract
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.
Highlights
1 Introduction and preliminaries Fractional differential equations have been extensively investigated in recent years, due to a wide range of applications in various fields of sciences and engineering such as control, porous media, electromagnetic, and so forth; see [ – ] and the references therein
The eigenvalue problems are one of the most active fields in differential equation theories, and the eigenvalue problems of nonlinear fractional differential equations have been concerned by some authors; see [ – ]
The purpose of this paper is to find the critical curve of parameters λ and μ dividing the range of λ and μ for the existence of at least two, one, and no positive solutions and to establish the maximum eigenvalue interval for the existence of the unique positive solution for the eigenvalue problem
Summary
For any given μ > , there exists λ∗(μ) > such that FEP ( ) has a unique positive solution xλ for λ ∈ [ , λ∗(μ)) and has no positive solution for λ ≥ λ∗(μ) Such a solution xλ satisfies the following properties:. For given μ > , FEP ( ) has a unique positive solution xλ for λ ∈ [ , +∞) (ii) xλ is nondecreasing in λ for λ ∈ [ , +∞); (iii) xλ is continuous with respect to λ for λ ∈ [ , +∞); (iv) if f (t, ) ≡ , limλ→ + xλ – xμ = and limλ→+∞ xλ = +∞, where xμ is the unique fixed point of Tμ in Pe. Applying Theorem . If F∞ > , for given μ > : there exists λ∗(μ) ≥
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