Nontrivial Solutions for a Class of Fractional Differential Equations with Integral Boundary Conditions and a Parameter in a Banach Space with Lattice

and Applied Analysis 3 Krein-Rutman theorem see 34 , we infer that if r B / 0, then there exist φ ∈ P \ {θ} and g∗ ∈ P ∗ \ {θ} such that Bφ r B φ, B∗g∗ r B g∗. 2.1 For a given constant δ > 0, set P ( g∗, δ ) { x ∈ P, g∗ x ≥ δ‖x‖, 2.2 then P g∗, δ is also a cone in E. Definition 2.1 see 30 . Let D ⊂ E and F : D → E a nonlinear operator. F is said to be quasiadditive on lattice, if there exists y0 ∈ E such that Fx Fx Fx− y0, ∀x ∈ D. 2.3 Definition 2.2 see 30 . Let B be a positive linear operator. The operator B is said to satisfy H condition, if there exist φ ∈ P \ {θ}, g∗ ∈ P ∗ \ {θ} such that 2.1 holds, and B maps P into P g∗, δ . Definition 2.3 see 4 . The Riemann-Liouville fractional integral of order α > 0 of a function y : 0,∞ → R is given by I 0 y t 1 Γ α ∫ t 0 t − s α−1y s ds 2.4 provided the right-hand side is pointwise defined on 0,∞ . Definition 2.4 see 4 . The Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : 0,∞ → R is given by D 0 y t 1 Γ n − α ( d dt )n ∫ t 0 y s t − s α−n 1 ds, 2.5 where n α 1, α denotes the integer part of the number α, provided that the right-hand side is pointwise defined on 0,∞ . Lemma 2.5 see 29 . Let P be a normal solid cone in E and A : E → E completely continuous and quasiadditive on lattice. Suppose that the following conditions are satisfied: i there exist a positive bounded linear operator B1, u∗ ∈ P and u1 ∈ P , such that −u∗ ≤ Ax ≤ B1x u1, ∀x ∈ P ; 2.6 ii there exist a positive bounded linear operator B2 and u2 ∈ P , such that Ax ≥ B2x − u2, ∀x ∈ −P ; 2.7 4 Abstract and Applied Analysis iii r B1 < 1, r B2 < 1, where r Bi is the spectral radius of Bi i 1, 2 . Then there exists R0 > 0 such that for R > R0, the topological degree deg I −A,BR, θ 1. Lemma 2.6 see 29 . Let P be a normal cone of E, and A : E → E a completely continuous operator. Suppose that there exist positive bounded linear operator B0 and u0 ∈ P , such that |Ax| ≤ B0|x| u0, ∀x ∈ E. 2.8 If r B0 < 1, then there existsR0 > 0 such that forR > R0 the topological degree deg I−A,BR, θ 1. Lemma 2.7 see 30 . Let P be a solid cone in E and A : E → E a completely continuous operator withA BF, where F is quasiadditive on lattice, and B is a positive bounded linear operator satisfying H condition. Suppose that i there exist a1 > r−1 B and y1 ∈ P such that Fx ≥ a1x − y1, ∀x ∈ P ; 2.9 ii there exist 0 < a2 < r−1 B and y2 ∈ P such that Fx ≥ a2x − y2, ∀x ∈ −P . 2.10 Then there exists R0 > 0 such that for R > R0 the topological degree deg I −A,BR, θ 0. Lemma 2.8 see 30 . LetΩ ⊂ E be a bounded open set which contains θ. Suppose thatA : Ω → E is a completely continuous operator which has no fixed point on ∂Ω. If i there exists a positive bounded linear operator B such that |Ax| ≤ B0|x|, ∀x ∈ ∂Ω; 2.11 ii r B0 ≤ 1, then the topological degree deg I −A,Ω, θ 1. Lemma 2.9 see 4 . Let α > 0. If one assumes u ∈ C 0, 1 ∩L 0, 1 , then the fractional differential equation D 0 u t 0, 2.12 has u t C1tα−1 C2tα−2 · · · CNtα−N , Ci ∈ R, i 1, 2, . . . ,N, as unique solution, where N is the smallest integer greater than or equal to α. Lemma 2.10 see 4 . Assume that u ∈ C 0, 1 ∩ L 0, 1 with a fractional derivative of order α > 0 that belongs to C 0, 1 ∩ L 0, 1 . Then I 0 D α 0 u t u t C1t α−1 C2tα−2 · · · CNtα−N, 2.13 for some Ci ∈ R, i 1, 2, . . . ,N, where N is the smallest integer greater than or equal to α. Abstract and Applied Analysis 5 In the following, we present Green’s function of the fractional differential equation boundary value problem. Lemma 2.11. Given y ∈ C 0, 1 , the problem D 0 u t y t 0, u 0 u′ 0 u′′ 0 0, u 1 λ ∫ηand Applied Analysis 5 In the following, we present Green’s function of the fractional differential equation boundary value problem. Lemma 2.11. Given y ∈ C 0, 1 , the problem D 0 u t y t 0, u 0 u′ 0 u′′ 0 0, u 1 λ ∫η