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.

• Case Theory
• Boundary-Value Problems
• Fixed Point Theorem
• Sign-Changing Solutions

1. 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. Relevant studies as [12] [13] [14] .

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 class of nonlinear second-order equations

{ u ″ ( t ) + f ( u ( t ) ) = 0 ,   0 ≤ t ≤ 1 ; u ′ ( 0 ) = 0 ,   u ( 1 ) = α u ( η ) , (1)

Existence of the sign-changing solution, constant 0 < α < 1 , 0 < η < 1 , f ∈ C ( R , R ) .

Boundary-value problem (1) is equivalent to Hammerstein nonlinear integral equation hereunder

u ( t ) = ∫ 0 1 G ( t , s ) f ( u ( s ) ) d s ,   0 ≤ t ≤ 1 (2)

Of which G ( t , s ) is the Green function hereunder

G ( t , s ) = 1 1 − α { ( 1 − s ) − α ( η − s ) , 0 ≤ s ≤ η , 0 ≤ t ≤ s ; ( 1 − s ) , η ≤ s ≤ 1 , 0 ≤ t ≤ s ; ( 1 − α η ) − t ( 1 − α ) , 0 ≤ s ≤ η , s ≤ t ≤ 1 ; ( 1 − α y ) − t ( 1 − α ) , η ≤ s ≤ 1 , s ≤ t ≤ 1.

Defining linear operator K as follow

( K u ) ( t ) = ∫ 0 1 G ( t , s ) u ( s ) d s ,   u ∈ C [ 0 , 1 ] . (3)

Let F u ( t ) = f ( u ( t ) ) , t ∈ [ 0 , 1 ] , obviously composition operator A = K F , i.e.

( A u ) ( t ) = ∫ 0 1 G ( t , s ) f ( u ( s ) ) d s ,   0 ≤ t ≤ 1 (4)

It’s easy to get: u ∈ C 2 [ 0 , 1 ] is the solution of boundary-value problem (1), and u ∈ C [ 0 , 1 ] is the solution of operator equation u = A u .

We note that, in Ref. [9] [10] , 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 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.

(H₁) f ( u ) : R → R continues, f ( u ) u > 0 , ∀ u ∈ R , u ≠ 0 , and f ( 0 ) = 0 .

(H₂) lim u → 0 f ( u ) u = β , and n 0 ∈ N , make λ 2 n 0 < β < λ 2 n 0 + 1 , of which 0 < λ 1 < λ 2 < ⋯ < λ n < λ n + 1 < ⋯ is the positive sequence of cos x = α cos η x .

(H₃) exists ε > 0 , make lim | u | → + ∞ sup f ( u ) u ≤ λ 1 − ε .

2. Knowledge

Provided P is the cone of E in Banach space, the semi order in E is exported by cone P. If the constant N > 0 , and θ ≤ x ≤ y ⇒ ‖ x ‖ ≤ N ‖ y ‖ , then P is a normal cone; if P contains internal point, i.e. int P ≠ ∅ , then P is a solid cone.

E becomes a case when semi order £, i.e. any x , y ∈ E , sup { x , y } and inf { x , y } is existed, for x ∈ E , x + = sup { x , θ } , x − = sup { − x , θ } , we call positive and negative of x respectively, call | x | = x + + x − as the modulus of x. Obviously, x + ∈ P , x − ∈ ( − P ) , | x | ∈ P , x = x + − x − .

For convenience, we use the following signs: x + = x + , x − = − x − . Such that x = x + + x − , | x | = x + − x − .

Provided Banach space E = C [ 0 , 1 ] , and E’s norm as ‖   ⋅   ‖ , i.e.

‖ u ‖ = max 0 ≤ t ≤ 1 | u ( t ) | . Let P = { u ∈ E | u ( t ) ≥ 0 , t ∈ [ 0 , 1 ] } , 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 D ⊂ E , A : D → E is an operator (generally a nonlinear). If A x = A x + + A x − , ∀ x ∈ E , then A is an additive operator under case structure; if v ∗ ∈ E , and A x = A x + + A x − + v ∗ , ∀ x ∈ E , then A is a quasi additive operator.

Def 2 provided x is a fixed point of A, if x ∈ ( P \ { θ } ) , then x is a positive fixed point; if x ∈ ( ( − P ) \ { θ } ) , then x is a negative fixed point; if x ∉ ( P ∪ ( − P ) ) , then x is a sign-changing fixed point.

Lemma 1 [6] G ( t , s ) is a nonnegative continuous function of [ 0 , 1 ] × [ 0 , 1 ] ,

and when t , s ∈ [ 0 , 1 ] , G ( t , s ) ≥ γ G ( 0 , s ) , of which γ = α ( 1 − η ) 1 − α η .

Lemma 2 K : P → P is completely continuous operator, and A : E → E 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. [10] , get Lemma 3 works.

Lemma 4 [6] the eigenvalues of the linear operator K are

1 λ 1 , 1 λ 2 , ⋯ , 1 λ n , 1 λ n + 1 , ⋯ . And the sum of algebraic multiplicity of all eigenvalues is

1, of which λ n is defined by (H₂).

The lemmas hereunder are the main study bases.

Lemma 5 [10] provided E is Banach space, P is the normal cone in E, A : E → E is completely continuous operator, and quasi additive operator under case structure. Provided that

1) There exists positive bounded linear operator B 1 , and B 1 ’s r ( B 1 ) < 1 , and u ∗ ∈ P , u 1 ∈ P , get

− u ∗ ≤ A x ≤ B 1 x + u 1 , ∀ x ∈ P ;

2) There exists positive bounded linear operator B 2 , B 2 ’s r ( B 2 ) < 1 , and u 2 ∈ P , get

A x ≥ B 2 x − u 2 , ∀ x ∈ ( − P ) ;

3) A θ = 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 ( 1 , ∞ ) is a nonzero even number,

A ( P \ { θ } ) ⊂ P ° ,     A ( ( − P ) \ { θ } ) ⊂ − P °

Then A exists three nonzero fixed points at least: one positive fixed point, one negative fixed point and a sign-changing fixed point.

3. Results

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 B = ( λ 1 − ε 2 ) K , Lemma 2 knows B : C [ 0 , 1 ] → C [ 0 , 1 ] is a positive bounded linear operator. Lemma 4 gets K’s r ( K ) = 1 λ 1 , so r ( B ) = ( λ 1 − ε 2 ) r ( K ) = 1 − ε 2 λ 1 < 1 .

(H₃) knows m > 0 and gets

f ( u ) ≤ ( λ 1 − ε 2 ) u + m ,   ∀ t ∈ [ 0 , 1 ] ,   u ≥ 0 (5)

f ( u ) ≥ ( λ 1 − ε 2 ) u − m ,   ∀ t ∈ [ 0 , 1 ] ,   u ≤ 0 (6)

Let u 0 ( t ) = m ∫ 0 1 G ( t , s ) d s , obviously, u 0 ∈ P . Such that, for any u ( t ) ∈ P ,

there

( A u ) ( t ) = ∫ 0 1 G ( t , s ) f ( u ( s ) ) d s ≤ ∫ 0 1 G ( t , s ) ( ( λ 1 − ε 2 ) u + m ) d s ≤ ( λ 1 − ε 2 ) ∫ 0 1 G ( t , s ) u ( s ) d s + m ∫ 0 1 G ( t , s ) d s = ( λ 1 − ε 2 ) K u ( t ) + u 0 ( t ) = B u ( t ) + u 0 ( t )

And for any u ∗ ∈ P , from (H₁), obviously gets ( A u ) ( t ) ≥ − u ∗ ( t ) .

For any u ( t ) ∈ − P , there

( A u ) ( t ) = ∫ 0 1 G ( t , s ) f ( u ( s ) ) d s ≥ ∫ 0 1 G ( t , s ) ( ( λ 1 − ε 2 ) u − m ) d s ≥ ( λ 1 − ε 2 ) ∫ 0 1 G ( t , s ) u ( s ) d s − m ∫ 0 1 G ( t , s ) d s = ( λ 1 − ε 2 ) K u ( t ) − u 0 ( t ) = B u ( t ) − u 0 ( t )

Consequently (1) (2) in lemma 5 works.

We note that f ( 0 ) = 0 can get A θ = θ , from (H₂), we know ∀ ε > 0 , and ∃ r > 0 gets

| f ( u ) − β u | ≤ ε u ,   | u | ≤ r

Then

| ( F u ) ( t ) − λ u ( t ) | = | f ( u ( t ) ) − β u ( t ) | ≤ ε ‖ u ‖ ,   ∀ ‖ u ‖ ≤ r

‖ A u − A θ − β K u ‖ = ‖ K ( F u ) − β K u ‖ ≤ ε ‖ K ‖ ‖ u ‖ ,   ∀ ‖ u ‖ ≤ r

Such that

lim ‖ u ‖ → 0 ‖ A u − A θ − β K u ‖ ‖ u ‖ = 0

i.e. A ′ θ = β K , from lemma 4 we get linear operator K’s eigenvalue is 1 λ n , then A ′ θ ’s eigenvalue is β λ n . Because λ 2 n 0 < β < λ 2 n 0 + 1 , let μ be the sum of

algebraic multiplicity of A ′ θ ’s all eigenvalues in the range ( 1 , ∞ ) , then μ = 2 n 0 is an even number.

From (H₁) f ( u ) u > 0 , u ∈ R \ { 0 } , there

f ( u ( t ) ) > 0 ,   ∀ t ∈ [ 0 , 1 ] ,   u ( t ) > 0 ,

f ( u ( t ) ) < 0 ,   ∀ t ∈ [ 0 , 1 ] ,   u ( t ) < 0.

Easy to get

F ( P \ { θ } ) ⊂ P \ { θ } ,   F ( ( − P ) \ { θ } ) ⊂ ( − P ) \ { θ } ,

Lemma (1) for any u ( t ) ∈ P , ( K u ) ( t ) = ∫ 0 1 G ( t , s ) u ( s ) d s ≥ γ ∫ 0 1 G ( 0 , s ) u ( s ) d s ,

consequently K ( P \ { θ } ) ⊂ P ° . Such that

A ( P \ { θ } ) ⊂ P ° ,   A ( ( − P ) \ { θ } ) ⊂ − P ° ,

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.

4. Conclusion

Provided that all conditions of the theorem are satisfied, and f ( u ) 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.

Note

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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