Study on the Existence of Sign-Changing Solutions of Case Theory Based a Class of Differential Equations Boundary-Value Problems

By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.


Introduction
The existence of nonlinear three-point boundary-value problems has been studied [1]- [6], and the existence of sign-changing solutions is obtained.In the past, most studies were focused on the cone fixed point index theory [7] [8] [9], just a few took use of case theory to study the topological degree of non-cone mapping and the calculation of fixed point index, and the case theory was combined with the topological degree theory to study the sign-changing solutions.Recent study Ref.
[10] [11] have given the method of calculating the topological degree under the case structure, and taken use of the fixed point theorem of non-cone mapping to study the existence of nontrivial solutions for the nonlinear Sturm-Liouville problems.
Inspired by the Ref. [8]- [13] and by using the new fixed point theorem under the case structure, this paper studies three-point boundary-value problems for A Existence of the sign-changing solution, constant 0 1, 0 Boundary-value problem (1) is equivalent to Hammerstein nonlinear integral equation hereunder Of which ( ) It's easy to get: is the solution of boundary-value problem (1), and is the solution of operator equation u Au = .
We note that, in Ref. [9] [10], an abstract result on the existence of signchanging solutions can be directly applied to problem (1).After the necessary preparation, when the non-linear term f is under certain assumptions, we get the existence of sign-changing solution of such boundary-value problems.
Compared with the Ref. [8], we can see that we generalize and improve the nonlinear term f , and remove the conditions of strictly increasing function, and the method is different from Ref. [8].
For convenience, we give the following conditions.

Knowledge
Provided P is the cone of E in Banach space, the semi order in E is exported by cone P.
For convenience, we use the following signs: , and E's norm as ⋅ , i.e.
( ) , then P is the normal cone of E, and E becomes a case under semi order ≤.

Now we give the definitions and theorems
Def 1 [10] provided , :  ( ) , then x is a sign-changing fixed point.
Lemma 1 [6] ( ) is completely continuous operator, and Lemma 3 A is a quasi additive operator under case structure.
Proof: Similar to the proofs in Lemma 4.3.1 in Ref. [10], get Lemma 3 works.
Lemma 4 [6] the eigenvalues of the linear operator K are , , , , And the sum of algebraic multiplicity of all eigenvalues is 1, of which n λ is defined by (H 2 ).
The lemmas hereunder are the main study bases.
Lemma 5 [10] provided E is Banach space, P is the normal cone in E, is completely continuous operator, and quasi additive operator under case structure.Provided that 1) There exists positive bounded linear operator 1 B , and 1 B 's ( ) 2) There exists positive bounded linear operator 2 B , 2 B 's ( ) 2 1 r B < , and 3) Aθ θ = , there exists Frechet derivative A θ ′ of A at θ , 1 is not the eigenvalue of A θ ′ , and the sum µ of algebraic multiplicity of A θ ′ 's all eigenvalues in the range ( ) Then A exists three nonzero fixed points at least: one positive fixed point, one negative fixed point and a sign-changing fixed point.

Results
Theorem provided (H 1 ) (H 2 ) (H 3 ) works, boundary-value problem (1) exists a sign-changing solution at least, and also a positive solution and a negative solution.
Proof provided linear operator is a positive bounded linear operator.Lemma 4 gets K's ( ) , d u t m G t s s = ∫ , obviously, 0 u P ∈ .Such that, for any ( ) Consequently (1) (2) in lemma 5 works.
contains internal point, i.e. int P ≠ ∅ , then P is a solid cone.E becomes a case when semi order ≤, i.e. any , DOI: 10.4236/apm.2017.712042688 Advances in Pure Mathematics cone; if P is an operator (generally a nonlinear).