Sign-Changing Solutions for Nonlinear Operator Equations

In recent years, motivated by some ecological problems, much attention has been attached to the existence of signchanging solutions for nonlinear partial differential equations (see [1–4] and the references therein). We note that the proofs of main results in [1–4] depend upon critical point theory. However, some concrete nonlinear problems have no variational structures [5]. To overcome this difficulty, in [6], Zhang studied the existence of sign-changing solution for nonlinear operator equations by using the cone theory and combining uniformly positive condition. Xu [7] studied multiple sign-changing solutions to the followingm-point boundary value problems:


Introduction
In recent years, motivated by some ecological problems, much attention has been attached to the existence of signchanging solutions for nonlinear partial differential equations (see [1][2][3][4] and the references therein).We note that the proofs of main results in [1][2][3][4] depend upon critical point theory.However, some concrete nonlinear problems have no variational structures [5].To overcome this difficulty, in [6], Zhang studied the existence of sign-changing solution for nonlinear operator equations by using the cone theory and combining uniformly positive condition.
We list some assumptions as follows.
Theorem 1 (see [7]).Suppose that conditions ( 1 )-( 4 ) are satisfied.Then the problem (1) has at least two sign-changing solutions.Moreover, the problem (1) also has at least two positive solutions and two negative solutions.
Based on [7], many authors studied the sign-changing solutions of differential and difference equations.For example, Yang [8] considered the existence of multiple signchanging solutions for the problem (1).Compared with Theorem 1, Yang employed the following assumption which is different from (A 4 ).

(A 󸀠
4 ) There exists  > 0 such that Pang et al. [9] investigated multiple sign-changing solutions of fourth-order differential equation boundary value problems.Moreover, Wei and Pang [10] established the existence theorem of multiple sign-changing solutions for fourth-order boundary value problems.Y. Li and F. Li [11] studied two sign-changing solutions of a class of second-order integral boundary value problems by computing the eigenvalues and the algebraic multiplicities of the corresponding linear problems.He et al. [12] discussed the existence of signchanging solutions for a class of discrete boundary value problems, and a concrete example was also given.Very recently, Yang [13] investigated the following discrete fourth Neumann boundary value problems The author employed similar conditions with ( 1 )-( 4 ) and obtained a similar result to Theorem 1 (see Theorem 5.1 in [13]).
The main purpose of this paper is to abstract more general conditions from ( 1 )-( 4 ) of Theorem 1, obtain the existence theorem of sign-changing solutions for general operator equations, and, then, apply the abstract result obtained in this paper to nonlinear elliptic partial differential equations.

Preliminaries and Some Lemmas
For the discussion of the following sections, we state here preliminary definitions and known results on cones, partial orderings, and topological degree theory, which can be found in [14][15][16][17][18].
Let  be a real Banach space.Given a cone  ⊂ , we define a partial ordering ≤ with respect to  by  ≤  if and only if − ∈ .A cone  is said to be normal if there exists a constant  > 0 such that  ≤  ≤  implies ‖‖ ≤ ‖‖; the smallest  is called the normal constant of . is called solid if it contains interior, that is, int  ̸ = 0.If  ≤  and  ̸ = , we write  < ; if cone  is solid and  −  ∈ int , we write  ≪ . is reproducing if  −  =  and total if  −  = .Let  :  →  be a bounded linear operator. is said to be positive if () ⊂ .An operator  is strongly increasing; that is,  <  implies  ≪ .If  is a linear operator,  is strongly increasing which implies  is strongly positive.A fixed point  of operator  is said to be a sign-changing fixed point if  ∉ ∪(−).If  0 ∈ \{} satisfies  0 =  0 , where  is some real number, then  is called an eigenvalue of  and  0 is called an eigenfunction belonging to the eigenvalue .
Definition 2 (see [16]).Let  1 ,  2 be real Banach spaces and let  ⊂  1 contain the outside of a ball { : ‖‖ ≤ }, and  :  →  2 .The operator  is called asymptotically linear if there is a bounded linear operator  : The operator  involved in the definition of an asymptotically linear operator  is uniquely determined.It is called the derivative of  at infinity and is denoted by   ∞ .
Definition 3 (see [16,18]).Let  be a retract of , and let  ⊂  be a relatively bounded open set of .Suppose that  :  →  is completely continuous and has no fixed point on .Let the positive integer (, , ) be defined by where  :  →  is an arbitrary retraction, and  is a large enough positive number such that  ⊂   = { |  ∈ , ‖‖ < }.Then (, , ) is called the fixed point index of  on  with respect to .
Lemma 6 (see [19]).Let  be a normal and total cone in , and let  :  →  be a completely continuous increasing operator.Then the following assertions hold Lemma 7 (see [18]).Let  be an open set of ,  :  →  be completely continuous,  0 ∈ , and  0 =  0 .Assume that  is Fréchet differentiable at  0 and 1 is not an eigenvalue of    0 , then  0 is an isolated fixed point, and where  is the sum of algebraic multiplicities of the real eigenvalues of    0 in (0, 1).
Lemma 8 (see [18]).Suppose that  :  →  is a completely continuous and asymptotically linear operator.If 1 is not an eigenvalue of the linear operator   ∞ , then there exists  0 > 0 such that for all  ≥  0 , where  is the sum of the algebraic multiplicities of the real eigenvalues of   ∞ in (0, 1).
Then  has at least two sign-changing fixed points, two positive fixed points, and two negative fixed points.
According to the theory of elliptic partial differential equations (see [20,21]), we know that for each  ∈ (Ω), the linear boundary value problem has a unique solution   ∈  2 (Ω).Define the operator  by Then  : (Ω) →  2 (Ω) is a linear completely continuous operator and has an unbounded sequence of eigenvalues: In order to obtain multiple sign-changing solutions of (35), we give the following assumptions.
where  1 is the first normalized eigenfunction of  corresponding to its first eigenvalue  1 .It follows from (E 2 ) and (E 3 ) that conditions (H 1 ) and (H 2 ) of Theorem 11 hold.
In the following, we prove that (H The proof is completed. Remark 13.It follows from conditions (E 2 ) and (E 4 ) that  0 ‖‖ < 1.We should point out that the initial ideas of condition (E 4 ) and the general one (H 3 ) are motivatedby [24].