The following results will be used throughout the paper.
Let , . Then E is a real Banach space with the norm . Let , and P is a normal solid cone of E, .
Let the operators K, F, A be defined by
and , respectively, where
Remark 1 (1) are completely continuous. (2) , . In fact, since it is obvious in the other case, we only need to prove the case . Now we suppose that . Then
Definition 2.1 
We call E a lattice under the partial ordering ≤, if and exist for arbitrary .
Remark 2 is a lattice under the partial ordering ≤ that is deduced by the cone of E.
Definition 2.2 
Let E be a Banach space with a cone be a nonlinear operator. We call that A is a unilaterally asymptotically linear operator along , if there exists a bounded linear operator L such that
L is said to be the derived operator of A along and will be denoted by . Similarly, we can also define a unilaterally asymptotically linear operator along . Specially, if , We call that A is a unilaterally asymptotically linear operator along P and −P.
Definition 2.3 
Let and be a nonlinear operator. A is said to be quasi-additive on lattice, if there exists such that
where , .
Remark 3 It is easy to see that the operators F and defined by (2.1) are both quasi-additive on the lattice .
Let us list some conditions and preliminary lemmas to be used in this paper.
(H1) is strictly increasing in u, and for all , .
(H2) uniformly on . There exists a positive integer such that
where are the positive solutions of the equation
(H3) uniformly on , and .
Lemma 2.1 For any , is a solution of the following problem:
if and only if
is a solution of the integral equation
where is defined by (2.2).
Proof On the one hand, integrating the equation
over for three times, we have
Combining them with boundary condition , we conclude that
On the other hand, since
Moreover, we get . □
Remark 4 Considering Lemma 2.1, we find that u is a solution of the problem (1.1) if and only if u is a fixed point of the operator .
From the following lemma, we can obtain the eigenvalues and the algebraic multiplicity of the linear operator K.
The eigenvalues of the linear operator
and the algebraic multiplicity of each positive eigenvalue is equal to 1, where are the positive solutions of (2.3).
Proof Let be a positive eigenvalue of the linear operator K, and be an eigenfunction corresponding to eigenvalue . By Lemma 2.1, we have
The auxiliary equation of the differential equation (2.4) has roots −μ, , , where . Thus the general solution of (2.4) is
Applying the condition , we obtain , , where .
Applying the second condition , that is,
Considering (2.3), we see that μ is one of , that is,
are eigenvalues of the linear operator K and the eigenfunction corresponding to the eigenvalue is
where C is a nonzero constant.
Next we prove that the algebraic multiplicity of the eigenvalue is 1. From (2.6), any two eigenfunctions corresponding to the same eigenvalue are merely nonzero constant multiples of each other, that is,
Now we show that
Obviously, we only need to show that
In fact, for any , is an eigenfunction of linear operator K corresponding to the eigenvalue if . Considering (2.6), there exists a nonzero constant σ such that
By direct computation, we have
It is easy to see that the solution for the corresponding homogeneous equation of (2.7) is of the form
Then, by an ordinary differential equation method, we see that the general solution of (2.7) is of the form
is the special solution of the equation
is the special solution of the equation
Applying the condition , we obtain , . From the condition , we obtain
From (2.5), we have . Thus it follows from (2.8) that
which implies that
which is a contradiction of
Therefore, the algebraic multiplicity of the eigenvalue is 1. □
Lemma 2.3 Suppose that (H1) holds and is a solution of the (1.1), then . Similarly, if is a solution of the (1.1), then .
Proof The proof is obvious. □
Lemma 2.4 Suppose that (H1)-(H3) hold. Then the operator A is Fréchet differentiable at θ and ∞, and , .
Proof Since (H3): uniformly on . That is, for any , there exists such that
From (H1), it is easy to see that , . Then, for any with , we have
which means .
Since (H2): uniformly on . That is, for any , there exists such that
Let . Then
Therefore, . □
Remark 5 Suppose (H2) holds. Similar to Lemma 2.4, we have
Lemma 2.5 
Suppose that E is an ordered Banach space with a lattice structure, P is a normal solid cone in E, and the nonlinear operator A is quasi-additive on the lattice. Assume that
A is strongly increasing on P and −P;
both and exist with and ; 1 is not an eigenvalue of or corresponding a positive eigenvector;
; the Fréchet derivative of A at θ is strongly positive, and ;
the Fréchet derivative of A at ∞ exists; 1 is not an eigenvalue of ; the sum β of the algebraic multiplicities for all eigenvalues of lying in the interval is an even number.
Then A has at least three nontrivial fixed points containing one sign-changing fixed point.