Existence of Nontrivial Solutions for Unilaterally Asymptotically Linear Three-Point Boundary Value Problems

and Applied Analysis 3 (H 2 ) lim φ→+∞ (g(t, φ)/φ) = ξ uniformly on [0, 1]. (H 3 ) lim φ→−∞ (g(t, φ)/φ) = ρ uniformly on [0, 1]. (H 4 ) g(t, 0) ≡ 0, lim φ→0 (g(t, φ)/φ) = η uniformly on [0, 1]. By [4], it is well known that BVP (1) can be converted to the following nonlinear Hammerstein equation: u (t) = ∫ 1 0 Z (t, s) g (s, u (s)) ds, t ∈ [0, 1] , (11) where Z (t, s) = k (t, s) + αt 1 − αβ k (β, s) , (12) k (t, s) = { t (1 − s) , 0 ≤ t ≤ s ≤ 1, s (1 − t) , 0 ≤ s ≤ t ≤ 1. (13) Define the operators (Φu) (t) = ∫ 1 0 Z (t, s) g (s, u (s)) ds, t ∈ [0, 1] , (14) (Lu) (t) = ∫ 1 0 Z (t, s) u (s) ds, t ∈ [0, 1] , (15) (Gu) (t) = g (t, u (t)) , t ∈ [0, 1] , (16) where Z(t, s) is defined by (12), and obviously Φ = LG. It is obvious that fixed points of Φ are solutions of BVP (1) (see [4]). Lemma 8 (see [13]). Let δ be a positive number. The eigenvalues of the linear operator δL are

Many problems of different areas of physics and applied mathematics can be changed into multipoint boundary value problems for ordinary differential equations (see [1]).In [2], Gupta firstly studied three-point boundary value problems for nonlinear second order ordinary differential equations in 1992.Since then, many authors have been concerned with second order multipoint boundary value problems (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein).For example, some authors have studied the existence and multiplicity of positive solutions for nonlinear multipoint boundary value problems under the condition that the nonlinear term may be nonnegative by applying Krasnosel'skii's fixed point theorem, theory of fixed point index, and so on (see [3][4][5][6][7][8]).Meanwhile, some authors considered the existence of nontrivial solutions when the nonlinear term can be negative; for example, see [9][10][11] and references therein.For instance, in [10], under the assumption of non-well-ordered upper and lower solutions, some multiplicity results for solutions of three-point boundary value problems (1) have been obtained using the fixed point index theory.On the other hand, some authors have considered the existence of sign-changing solutions to some boundary value problems (see [12][13][14][15][16]18] and references therein).For example, in [13], by using the fixed point index method, Xu and Sun have considered the existence of signchanging solutions for the following three-point boundary value problem: In [18], Rynne has considered the following second order -point boundary value problem: where  : (−∞, +∞) → (−∞, +∞) is continuous.The author has used global bifurcation theorem to obtain signchanging solutions of the boundary value problem (3) under some conditions on the asymptotic behavior of .
The organization of this paper is as follows.In Section 2, some preliminaries and lemmas are given including some properties of the lattice and some lemmas that will be used to prove the main results.In Section 3, we shall give the main results.Finally, in Section 4, concrete examples are given to illustrate applications of obtained main results.

Preliminaries and Some Lemmas
Let  be an ordered Banach space in which the partial ordering ≤ is induced by a cone  ⊆ . is called normal if there exists  > 0 such that  ≤  ≤  implies ‖‖ ≤ ‖‖ (see [26]).
Definition 2 (see [16,[21][22][23]).Let  ⊂  and  :  →  be a nonlinear operator. is said to be quasi-additive on lattice, if there exists V * ∈  such that Definition 3 (see [21]).Let  be a Banach space with a cone  and let  :  →  be a nonlinear operator.We say that  is a unilaterally asymptotically linear operator along , if there exists a bounded linear operator  such that lim where  is said to be the derived operator of  along  and will be denoted by    .
Remark 4. The operator  in Definition 3 is not assumed to be a cone mapping.
Let  be a cone of Banach space . is said to be a positive fixed point of  if  ∈ ( \ {}) is a fixed point of ;  is said to be a negative fixed point of  if  ∈ ((−) \ {}) is a fixed point of ;  is said to be a sign-changing fixed point of  if  ∉ ( ∪ (−)) is a fixed point of  (see [21][22][23]).
Lemma 5 (see [21]).Let  be a Banach space with a lattice structure, let  be a normal cone of , and let  :  →  be completely continuous and quasi-additive on lattice.Suppose that there exist ] 1 , ] 2 ∈  and a positive bounded linear operator  :  →  with () < 1, such that In addition, assume that  = ; the Fréchet derivative    of  at  exists, 1 is not an eigenvalue of    , the sum of the algebraic multiplicities for all eigenvalues of    , lying in the interval (1, ∞), is an odd number, and    exists, (   ) < 1.Then  has at least one nontrivial fixed point.
Then  has at least one nontrivial fixed point.
Lemma 7 (see [21]).Suppose that  is an ordered Banach space with a lattice structure,  is a normal cone of , and  is quasi-additive on the lattice.Assume that Then  has at least three nontrivial fixed points containing one sign-changing fixed point.
For convenience, we list the following conditions.
(H 1 ) The sequence of positive solutions of the equation Abstract and Applied Analysis By [4], it is well known that BVP (1) can be converted to the following nonlinear Hammerstein equation: where Define the operators where (, ) is defined by (12), and obviously Φ = .It is obvious that fixed points of Φ are solutions of BVP (1) (see [4]).
Lemma 8 (see [13]).Let  be a positive number.The eigenvalues of the linear operator  are and the algebraic multiplicity of each positive eigenvalue /  of the linear operator  is equal to 1, where   is defined by (10).
Proof.By [4], we know that (i) holds.The proof of (ii) is similar to that of [16,[21][22][23][24][25], so we omit it.We easily know that Lemma 10.Let Φ and  be defined as (14) so by Definition 3, we have Φ   = .(ii) Similar to the proof of (i), we can prove that conclusion (ii) holds.
Then BVP (1) has at least one nontrivial solution.
By 0 <  <  1 and Lemma 9, ( So the conditions of Lemma 5 are satisfied.Lemma 5 assures that Φ has at least one nontrivial fixed point.So BVP (1) has at least one nontrivial solution.
Then BVP (1) has at least one nontrivial solution.
Then BVP (1) has at least three nontrivial solutions, containing a sign-changing solution.
From ( 45) and ( 15), for any  ∈  \ {}, we have that By (48) and the condition (i), we know that Φ is strongly increasing; so condition (i) of Lemma 7 is satisfied.
In the following, we prove that Φ  ∞ = .In fact, by (H 2 ) (H 3 ) and  =  = , for any  > 0, there exists R > 0 such that so we have Φ  ∞ = .By condition (ii) and Lemma 8, we know that 1 is not the eigenvalue of Φ  ∞ , and the sum of the algebraic multiplicities for all eigenvalues of Φ  ∞ , lying in the interval (1, ∞), is an even number ñ.So condition (iv) of Lemma 7 is satisfied.Therefore, Lemma 7 guarantees that Theorem 13 is valid.Remark 14.In [13][14][15], the authors considered the boundary value problem (2).In this paper, it is obvious that we generalize and improve the nonlinear term , and we also obtain that BVP (1) has at least a sign-changing solution.The methods we use are different from those of [13][14][15].In [18], the author has used global bifurcation theorem to obtain signchanging solutions of the boundary value problem (3), so the methods we use are different from those of [18].

Applications
In this section, some examples are given to illustrate our main results obtained in Section 3.