Existence of multiple solutions for nonlinear multi-point boundary value problems

In this paper, we study some nonlinear second order multi-point boundary value problems. We first give a lemma about the characteristic values of the corresponding linear operator. Then, by fixed point theorems in the recent existing literature, we obtain the existence of multiple solutions for these nonlinear second order multi-point boundary value problems, including two positive solutions, two negative solutions, and one sign-changing solution.

In [5], Zhang and Sun studied the following nonlinear multi-point boundary value problem: is continuous, and h is singular at t = 0 and t = 1; 0 < ξ 1 < ξ 2 < · · · < ξ m-2 < 1; a i > 0 (i = 1, 2, . . . , m -2). The authors proved the existence of the first eigenvalue of the relevant linear operator, and they considered the existence of positive solutions for BVP (1.3). The method they used is the fixed point index theory.
In [15], Li considered the following second order three-point boundary value problem:  [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], we consider boundary value problem (1.1) in this paper. By the existing fixed point theorems due to Sang et al. [19], we obtain the existence results of multiple nontrivial solutions for BVP (1.1) for asymptotically linear case, including two positive solutions, one sign-changing, and two negative solutions. Characteristic value is an important index of the linear operator. One of the features of this paper is that we give a lemma about the characteristic values of the relevant linear operator about BVP (1.1). The other feature of this paper is that the obtained main theorems are new results for BVP (1.1), which improve and generalize BVP (1.4).
The rest of the paper is organized as follows. We introduce the definition of e-continuous operator and the used fixed point theorems due to Sang et al. [19] in Sect. 2. We give the main lemma about the characteristic values of the relevant linear operator, prove some auxiliary lemmas that we need, and obtain the main result of the existence of multiple solutions for BVP (1.1) in Sect. 3. We provide an example to illustrate our main result in Sect. 4.

Preliminaries
In the following, we mainly introduce the e-continuous operator and list the used fixed point theorems due to Sang et al. [19]. For detailed concepts and properties about the cone, we can refer to [21][22][23].
Let E be a Banach space, P be a cone of E. Let A be an operator. If Au = u with u / ∈ P ∪ (-P), then u is said to be a sign-changing fixed point of A. The linear operator B is said to be positive if B(P) ⊂ P. Definition 2.1 (see [20]) Let A : D → E be an operator, e ∈ P\{θ }, and u 0 ∈ D. If, for any given > 0, there exists δ > 0 such that then A is said to be e-continuous at u 0 . If A is e-continuous at every point u ∈ D, then A is said to be e-continuous on D.
Lemma 2.1 (see [19]) Let E be a Banach space, P be a normal and total cone of E; B : E → E be a positive linear completely continuous operator and be also e-continuous on E; F : E → E be a continuous and bounded increasing operator and A = BF. Assume that Then A has at least two positive fixed points, one sign-changing solution, fixed point, and two negative fixed points.

Main results
It is obvious that P is a normal and total cone of E (see [21][22][23]).
For convenience, we first give the following assumptions to be used in the rest of this paper.
(H 1 ) The sequence of positive solutions of the equation Lemma 3.1 (see [5]) For y(t) ∈ E, the following boundary value problem has a unique solution Define the following operators: where T = LG, G(t, s) is defined by (3.2).

Lemma 3.2
Assume that (H 1 ) holds. Then the sequence of positive characteristic values of the linear operator L defined by (3.6) is λ 1 < λ 2 < · · · < λ n < · · · and the positive characteristic values λ n have algebraic multiplicity one.
Proof Let ξ be a positive characteristic value and u(t) be a characteristic function corresponding to the characteristic value ξ .
From Lemma 3.1, we obtain (3.8) Then the form of the general solution for the differential equation (3.8) is Since u (0) = 0, we know that C 1 = 0. Then (3.9) can be reduced to By (H 1 ), we know that ξ is one of the values λ 1 < λ 2 < · · · < λ n < · · · , and the corresponding characteristic function is where C is a nonzero constant.
By ordinary method, we can know that dim Ker(Iλ n L) = 1. It is obvious that we only need to prove that Ker(Iλ n L) 2 ⊂ Ker(Iλ n L). (3.12) Take any u ∈ Ker(Iλ n L) 2 . If (Iλ n L)u = θ , then (Iλ n L)u is a characteristic function of the linear operator L corresponding to the characteristic value λ n . So we have where b is a nonzero constant.
Then the form of the general solution for the differential equation (3.13) is (3.14) Since u (0) = 0, we know that C 1 = 0. By (3.14) and cos From (3.15), (3.16) and Applying the Schwarz inequality and (3.17), we have Combining Since m-2 i=1 α i < 1, we know that (3.18) is a contradiction. So (3.12) holds. By (3.10) and (3.11), we know that the algebraic multiplicity of characteristic value λ n is 1.

Lemma 3.3 The linear operator L is e(t)-continuous on E.
Proof Take u 0 ∈ E. For any given > 0, we where e(t) = 1. Hence L is e(t)-continuous on u 0 ∈ E. By the arbitrariness of u 0 , L is e(t)continuous on E.
Then condition (ii) of Lemma 2.1 is satisfied.
(iii) From Lemma 3.5, T ∞ = γ L. So T ∞ is increasing and λ n γ is the characteristic value of T ∞ , where λ n is defined by (H 1 ). By (H 4 ), since r(T ∞ ) = γ λ 1 , γ > λ 1 , and γ = λ n , we have that r(T ∞ ) > 1 and 1 is not a characteristic value of T ∞ . Hence condition (iii) of Lemma 2.1 holds. From the above proof, Theorem 3.1 holds by Lemma 2.1.

Application
The following nonlinear four-point boundary value problem is studied: ⎧ ⎨ ⎩ -u (t) = g(t, u(t)), 0 ≤ t ≤ 1, So, by Theorem 3.1, BVP (4.1) has at least two positive solutions, one sign-changing solution, and two negative solutions.