The existence of nonlinear three-point boundary-value problems has been studied  -  , and the existence of sign-changing solutions is obtained. In the past, most studies were focused on the cone fixed point index theory    , 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.   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. Relevant studies as    .
Inspired by the Ref.  -  and by using the new fixed point theorem under the case structure, this paper studies three-point boundary-value problems for A class of nonlinear second-order equations
Existence of the sign-changing solution, constant , .
Boundary-value problem (1) is equivalent to Hammerstein nonlinear integral equation hereunder
Of which is the Green function hereunder
Defining linear operator K as follow
Let , , obviously composition operator , i.e.
It’s easy to get: is the solution of boundary-value problem (1), and is the solution of operator equation .
We note that, in Ref.   , an abstract result on the existence of sign- changing solutions can be directly applied to problem (1). After the necessary preparation, when the non-linear term is under certain assumptions, we get the existence of sign-changing solution of such boundary-value problems. Compared with the Ref.  , we can see that we generalize and improve the nonlinear term , and remove the conditions of strictly increasing function, and the method is different from Ref.  .
For convenience, we give the following conditions.
(H1) continues, , , and .
(H2) , and , make , of which is the positive sequence of .
(H3) exists , make .
Provided P is the cone of E in Banach space, the semi order in E is exported by cone P. If the constant , and , then P is a normal cone; if P contains internal point, i.e. , then P is a solid cone.
E becomes a case when semi order £, i.e. any , and is existed, for , , , we call positive and negative of x respectively, call as the modulus of x. Obviously, , , , .
For convenience, we use the following signs: , . Such that , .
Provided Banach space , and E’s norm as , i.e.
. Let , 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  provided is an operator (generally a nonlinear). If , then A is an additive operator under case structure; if , and , then A is a quasi additive operator.
Def 2 provided x is a fixed point of A, if , then x is a positive fixed point; if , then x is a negative fixed point; if , then x is a sign-changing fixed point.
Lemma 1  is a nonnegative continuous function of ,
and when , , of which .
Lemma 2 is completely continuous operator, and is completely continuous operator.
Lemma 3 A is a quasi additive operator under case structure.
Proof: Similar to the proofs in Lemma 4.3.1 in Ref.  , get Lemma 3 works.
Lemma 4  the eigenvalues of the linear operator K are
. And the sum of algebraic multiplicity of all eigenvalues is
1, of which is defined by (H2).
The lemmas hereunder are the main study bases.
Lemma 5  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 , and ’s , and , get
2) There exists positive bounded linear operator , ’s , and , get
3) , there exists Frechet derivative of A at , 1 is not the eigenvalue of , and the sum of algebraic multiplicity of ’s all eigenvalues in the range is a nonzero even number,
Then A exists three nonzero fixed points at least: one positive fixed point, one negative fixed point and a sign-changing fixed point.
Theorem provided (H1) (H2) (H3) 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 , Lemma 2 knows is a positive bounded linear operator. Lemma 4 gets K’s , so .
(H3) knows and gets
Let , obviously, . Such that, for any ,
And for any , from (H1), obviously gets .
For any , there
Consequently (1) (2) in lemma 5 works.
We note that can get , from (H2), we know , and gets
i.e. , from lemma 4 we get linear operator K’s eigenvalue is , then ’s eigenvalue is . Because , let be the sum of
algebraic multiplicity of ’s all eigenvalues in the range , then is an even number.
From (H1) , , there
Easy to get
Lemma (1) for any , ,
consequently . Such that
Such that (3) in lemma 5 works. According to lemma 5, A exists three nonzero fixed points at least: one positive fixed point, one negative fixed point and one sign-changing fixed point. Which states that boundary-value problem (1) has three nonzero solutions at least: one positive solution, one negative solution and one sign-changing solution.
Provided that all conditions of the theorem are satisfied, and is an odd function, then boundary-value problem (1) has four nonzero solutions at least: one positive solution, one negative solution and two sign-changing solutions.
By using case theory to study the topological degree of non-cone mapping and the calculation of fixed point index, it’s an attempt to combine case theory and topological degree theory, the author thinks it’s an up-and-coming topic and expects to have further progress on that.
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