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


Introduction
Fractional differential equations have been of great interest for many researchers recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of sciences and engineering such as control, porous media, electromagnetic, and other fields. For an extensive collection of such results, we refer the readers to the monographs by Samko et al. 1 , Podlubny 2 , and Kilbas et al. 3 . Recently, there are some papers dealing with the existence of solutions or positive solutions for nonlinear fractional differential equation by means of techniques of nonlinear analysis fixed point theorems, Leray-Schauder theory, adomian decomposition method, lower and upper solution method, etc. ; see 4-16 . As is well known, the first eigenvalue is a character of great significance for the linear operator. For some integer order differential equations, many authors have investigated the existence of positive and nontrivial solutions concerning the first eigenvalue corresponding to the relevant linear operators when the nonlinearities are sublinear, see [17][18][19][20][21][22] for reference.
On the other hand, papers 23-26 obtained similar results to the sublinear case. The main discussion is based on the concepts of dual space, dual cone, and a constructed cone on them.
Recently, Xu et al. 27 and Bai 28 obtained the existence results of positive solutions for some fractional differential equations under the conditions with respect to the first eigenvalue corresponding to the relevant linear operators.
In two recent papers 29, 30 , Sun and Liu established some results about the computation of the topological degree for nonlinear operators which are not cone mappings using the lattice structure.
Motivated by the above papers, by using results for the computation of topological degree under the lattice structure, we investigate the existence of nontrivial solutions for the following nonlinear fractional differential equations with integral boundary conditions: is the standard Riemann-Liouville derivative. In this paper, it is not required that nonlinearity f t, u ≥ 0, for all u ≥ 0. To the author's knowledge, few papers are available in the literature to study the existence of solutions for fractional differential equations with integral boundary conditions under the lattice structure. The method used in this paper is different from those in previous works. This paper is organized as follows. In Section 2 corresponding Green's function for BVP 1.1 is derived and its properties are also discussed. The main results and their proof are presented in Section 3.

Background Materials and Green's Function
Let E be a Banach space with a cone P . Then E becomes an ordered Banach space under the partial ordering ≤ which is induced by P . P is said to be normal if there exists a positive constant N such that θ ≤ x ≤ y implies x ≤ N y . P is called solid if it contains interior points, that is, int P / ∅. P is called total if E P − P . If P is solid, then P is total. For the concepts and the properties about the cone we refer to 31, 32 .
We call E a lattice under the partial ordering ≤ if sup{x, y} and inf{x, y} exist for arbitrary x, y ∈ E. For x ∈ E, let x sup{x, θ}, x − sup{−x, θ}, x and x − are called the positive part and the negative part of x, respectively, and obviously x x − x − . Take |x| x x − , then |x| ∈ P . One can refer to 33 for the definition and the properties of the lattice. Let x x , x − −x − as denoted in 29,30 . Then x ∈ P , x − ∈ −P and x x x − . Let B : E → E be a bounded linear operator. B is said to be positive if B P → P . In this case, B is an increase operator, namely, for x, y ∈ E, x ≤ y implies Bx ≤ By. Let B : E → E be a positively completely continuous operator, r B a spectral radius of B, B * the conjugated operator of B, P * the conjugated cone of P . Since P ⊂ E is a total cone, according to the famous Krein-Rutman theorem see 34 , we infer that if r B / 0, then there exist ϕ ∈ P \ {θ} and g * ∈ P * \ {θ} such that For a given constant δ > 0, set then P g * , δ is also a cone in E.
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 B 1 , u * ∈ P and u 1 ∈ P , such that ii there exist a positive bounded linear operator B 2 and u 2 ∈ P , such that 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 B 0 and u 0 ∈ P , such that  ii r B 0 ≤ 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 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 for some C i ∈ R, i 1, 2, . . . , N, where N is the smallest integer greater than or equal to α.

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
Proof. We may apply Lemma 2.10 to reduce 2.14 to an equivalent integral equation Consequently, the general solution of 2.14 is 2.18 6 Abstract and Applied Analysis By u 0 u 0 u 0 0, we get that C 2 C 3 C 4 0. On the other hand, boundary condition u 1 λ η 0 u s ds combining with Therefore, the unique solution of the problem 2.14 is Γ α y s ds.

2.21
For t ≤ η, one has G t, s y s ds.

2.22
Abstract and Applied Analysis 7 For t ≥ η, one has G t, s y s ds.

3.21
Therefore, for u ∈ P , we get by Lemma 2.12, 3.20 and 3.21 that

3.22
This means that g * Bu ≥ δ Bu , where δ m 1 /M 1 1 0 t α−1 g * t dt. So, B P ⊂ P g * , δ . Thus, we have shown that B satisfies H condition. By Lemma 2.7, we know that there exists R > R 0 such that deg I − A, T R , θ 0. 3.23