In the sequel we always take E = C[0,1] with the norm ||u|| = max0≤t≤1|u(t)| and P = {u ∈ C[0, 1] | u(t) ≥ 0, 0 ≤ t ≤ 1}. Then P is a solid cone in E and E is a lattice under the partial ordering ≤ induced by P.
A solution of BVP (P) is a fourth differentiable function u : [0,1] → R such that u satisfies (P). u is said to be a positive solution of BVP (P) if u(t) > 0, 0 < t < 1. Let G(t, s) be Green's function of homogeneous linear problem u(4)(t) = 0, u(0) = u(1) = u'(0) = u'(1) = 0. From Yao [11] we have
and
(G1) G(t, s) ≥ 0, 0 ≤ t, s ≤ 1;
(G2) G(t, s) = G(s, t);
(G3) G(t, s) ≥ p(t)G(τ; s), 0 ≤ t, s, τ ≤ 1, where .
Lemma 3.1 Let and . Then
(3.1)
Proof. Since G(0, s) = G(1, s) = 0, 0 ≤ s ≤ 1, H(0) = H(1) = 0, then q(s)H(t) = G(t, s) = H(t) holds for t = 0 and t = 1. If 0 < t ≤ s ≤ 1 and t < 1, then
and
Similarly, (3.1) holds for 0 ≤ s ≤ t < 1 and t > 0. The proof is complete. □
It is well known that the problem (P) is equivalent to the integral equation
Let
(3.2)
(3.3)
Lemma 3.2 Let B be defined by (3.3). Then B is a u0- bounded linear operator.
Proof. Let , t ∈ [0,1]. For any u ∈ P\{θ}, by Lemma 3.1
we have
Take arbitrarily , then
Let , η = ||u|| > 0. Then
This indicates that B : E → E is a u0- bounded linear operator. □
From Lemma 2.1 we have r(B) ≠ 0 and r-1(B) is the only eigenvalue of B. Denote λ1 = r-1(B).
Now let us list the following conditions which will be used in this article:
(H1) there exist constants α and β with α > β ≥ 0 satisfying
(3.4)
(H2) there exists a constant γ ≥ 0 satisfying
(3.5)
(H3) .
Theorem 3.1 Suppose that (H1) and (H2) hold. Then for any , BVP (P) has at least one nontrivial solution, where λ1 = r-1(B) is the only eigenvalue of B, B is denoted by (3.3), ι = max{β, γ}.
Proof. Let (Fu)(t) = f(t, u(t)). Then A = BF, where A is denoted by (3.2). By Remark 2.2, F is quasi-additive on lattice. Applying the Arzela-Ascoli theorem and a standard argument, we can prove that A : E → E is a completely continuous operator.
Now we show that λA = λBF has at least one nontrivial fixed point, which is the nontrivial solution of BVP (P).
On account of (G3) we have that such that
Notice that , where G(t, s) ≥ 0, G(t, s) ∈ C([0,1] × [0,1]). From Lemma 3.2 B is a u0- bounded linear operator. By Lemma 2.1 we have r(B) ≠ 0 and λ1 = r-1(B) is the only eigenvalue of B. Then there exist and g* ∈ P*\{θ} such that (2.3) holds. Notice that λ > 0, from Remark 2.3, λB satisfies H condition.
By (3.4) and (3.5), there exist r > 0, M > 0 and such that
(3.6)
(3.7)
(3.8)
By (3.6) and (3.7), we have (2.4) and (2.5) hold, where a1 = α - ε, a2 = β + ε.
Let B1 = λB. Then . Obviously, for any , a1> r-1(B1), a2< r-1(B1). From Lemma 2.2 there exists R0> 0 such that for any R > max{R0, r},
(3.9)
Let B2 = λ(γ + ε)B. From (3.8) we have |λAu| ≤ B2|u|, also . Without loss of generality we assume that λA has no fixed point on ∂T
r
, where T
r
= {u ∈ C[0,1] | ||u|| < r}. By Lemma 2.3 we have
(3.10)
It is easy to see from (3.9) and (3.10) that λA has at least one nontrivial fixed point. Thus problem (P) has at least one nontrivial solution. □
Remark 3.1 If α = +∞, β = γ = 0, then for any λ > 0 problem (P) has at least one nontrivial solution.
Theorem 3.2 Suppose that (H1) holds. Assume f(t, 0) ≡ 0, ∀t ∈ [0,1] and
(3.11)
Then for any and , BVP (P) has at least one nontrivial solution.
Proof. Since f(t, 0) ≡ 0, ∀t ∈ [0,1], then Aθ = θ. By (3.11) we have that the Frechet derivative of A at θ exists and
Notice that , then 1 is not an eigenvalue of . By the famous Leray-Schauder theorem there exists r > 0 such that
(3.12)
where κ is the sum of algebraic multiplicities for all eigenvalues of lying in the interval (0, 1). From the proof of Theorem 3.1 we have that (3.9) holds for any . By (3.9) and (3.12), λA has at least one nontrivial fixed point. Thus problem (P) has at least one nontrivial solution. □