Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

and Applied Analysis 3 Lemma 2.1 see 8, 9 . 1 If x ∈ L1 0, 1 , ρ > σ > 0, and n ∈ N, then IIx t I x t , DtIx t Iρ−σx t , 2.4 DtIx t x t , d dtn Dtx t Dt x t . 2.5 2 If ν > 0, σ > 0, then Dttσ−1 Γ σ Γ σ − ν t σ−ν−1. 2.6 Lemma 2.2 see 8 . Assume that x ∈ L1 0, 1 and μ > 0. Then IDtx t x t c1tμ−1 c2tμ−2 · · · cntμ−n, 2.7 where ci ∈ R i 1, 2, . . . , n , n is the smallest integer greater than or equal to μ. Let x t In−1v t , and consider the following modified integro-differential equation: −Dtn−1v t λf ( t, In−1v t , Iμn−1−μ1v t , . . . , Iμn−1−μn−2v t , v t ) , v 0 v′ 0 0 v 1 m−2 ∑ j 1 ajv ( ξj ) . 2.8 The following Lemmas 2.3–2.5 are obtained by Zhang et al. 10 . Lemma 2.3. The higher order multipoint boundary value problem 1.1 has a positive solution if and only if nonlinear integro-differential equation 2.8 has a positive solution. Moreover, if v is a positive solution of 2.8 , then x t In−1v t is positive solution of the higher order multipoint boundary value problem 1.1 . Lemma 2.4. If 2 < μ − μn−1 < 3 and α ∈ L1 0, 1 , then the boundary value problem Dtn−1w t λα t 0, w 0 w′ 0 0, w 1 m−2 ∑ j 1 ajw ( ξj ) 2.9 has the unique solution w t λ ∫1 0 K t, s α s ds, 2.10 4 Abstract and Applied Analysis where K t, s k t, s tμ−μn−1−1 1 −m−2 j 1 ajξ μ−μn−1−1 j m−2 ∑ j 1 ajk ( ξj , s ) , 2.11 is the Green function of the boundary value problem 2.9 , and k t, s ⎧ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪⎩ t 1 − s μ−μn−1−1 − t − s μ−μn−1−1 Γ ( μ − μn−1 ) , 0 ≤ s ≤ t ≤ 1, t 1 − s μ−μn−1−1 Γ ( μ − μn−1 ) , 0 ≤ t ≤ s ≤ 1. 2.12 Lemma 2.5. The Green function of the boundary value problem 2.9 satisfies