The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives

• DOI: 10.1016/j.amc.2012.02.014
• Publication Date: 2012-03-06T09:14:50Z
• Authors: Xinguang Zhang; Lishan Liu; Yonghong Wu
• Keywords: Derivative (finance); Singular solution

Abstract In this paper, we study the following singular eigenvalue problem for a higher order fractional differential equation - D α x ( t ) = λ f ( x ( t ) , D μ 1 x ( t ) , D μ 2 x ( t ) , … , D μ n - 1 x ( t ) ) , 0 t 1 , x ( 0 ) = 0 , D μ i x ( 0 ) = 0 , D μ x ( 1 ) = ∑ j = 1 p - 2 a j D μ x ( ξ j ) , 1 ⩽ i ⩽ n - 1 , where n ≥ 3 , n ∈ N , n - 1 α ⩽ n , n - l - 1 α - μ l n - l , for l = 1 , 2 , … , n - 2 , and μ - μ n - 1 > 0 , α - μ n - 1 ≤ 2 , α - μ > 1 , a j ∈ [ 0 , + ∞ ) , 0 ξ 1 ξ 2 ⋯ ξ p - 2 1 , 0 ∑ j = 1 p - 2 a j ξ j α - μ - 1 1 , D α is the standard Riemann–Liouville derivative, and f : ( 0 , + ∞ ) n → [ 0 , + ∞ ) is continuous. Firstly, we give the Green function and its properties. Then we established an eigenvalue interval for the existence of positive solutions from Schauder’s fixed point theorem and the upper and lower solutions method. The interesting point of this paper is that f may be singular at x i = 0 , for i = 1 , 2 , … , n .