MULTIPLE POSITIVE SOLUTIONS OF STRUM-LIOUVILLE EQUATIONS WITH SINGULARITIES

The existence of multiple positive solutions for Strum-Liouville
boundary value problems with singularities is investigated. By
applying a fixed point theorem of cone map, some existence and
multiplicity results of positive solutions are derived. Our
results improve and generalize those in some well-known results.

Recently the authors [10] investigated the Strum-Liouville equation which has singularities at 0 and 1, the existence of one positive solution is established by applying the fixed point index theory.
However, the authors [1,4,6,8,10] only investigated the existence of positive solution for BVP(1.1).Liu and Li [5] get the existence of multiple positive solutions in the special case as follows with p(t) ≡ 1 and g(t) ≡ 1 by using the fixed pointed theorem.When we choose g(t) ≡ 1, Wang et al. [11] established some nonexistence, existence, and multiplicity results for the BVP(1.1)which are based on the Schauder fixed point theorem, the method of upper and lower solutions, and the Leray-Schauder degree theory.Furthermore, Ma and Thompson [7] studied the existence of multiple positive solutions by applying bifurcation techniques.The purpose of this paper is to consider BVP(1.1) in which g(t) ∈ L 1 (0,1) and F(t,u) satisfies weaker conditions than those in [1,[4][5][6][7][8][9][10][11], the existence of multiple positive solutions for BVP(1.1) is obtained by using a fixed point theorem.Our method is different from the ones in those papers and our results are often new even when p(t) ≡ for all t ∈ (0,1) and x(t) satisfies BVP(1.1).
Lemma 2.1.Let K be a cone in a Banach space X.Assume that Ω is a bounded open subset of X with θ ∈ Ω and let A : Lemma 2.2.Assume that A : K ∩ (Ω) → K is completely continuous, and there exists B : We denote by G(t,s) Green's function for the homogeneous boundary value problem (2.1) Zenggui Wang et al. 3 We know that G(t,s) is nonnegative on [0,1] × [0,1] and is expressed by where ρ = αδ + αγ 1 0 (dr/ p(r)) + βγ.Lemma 2.3 [2].Green's function G(t,s) has the following properties: (s,s), where . (2. 3) Define an operator A by Proof.Similar to the proof of [10, Lemma 2.1], we can prove that A is a completely continuous operator.
Proof.Without loss of generality, we assume that there exists > 0 such that x = Ax for x ∈ K with 0 < x < , otherwise there is a fixed point in K and this would complete the proof.
In view of (H 2 ), there exists η > 0 and N > 0 such that F(t,u) ≤ ηu q , for u ≥ C 0 , then where Choose sufficiently large R > 0 such that Zenggui Wang et al. 5 We will prove that In fact, if not, there exist we have which yields which is a contradiction to (3.9), so (3.10) holds.By Lemma 2.1, we have (3.7) and (3.14) together imply Consequently, according to [2, Theorem 2.3.2],A has a fixed point u * ∈ K ∩ (Ω R \Ω r ), so BVP(1.1)has at least one positive solution u * .This completes the proof.
Proof.By virtue of (H 3 ), there exist ξ > 0 and . (3.16) Without loss of generality, we assume that u = Au for u ∈ K ∩ ∂Ω R , otherwise the conclusion holds.Define B : where Then it is easy to verify B : K ∩ ∂Ω → K is completely continuous and inf K∩∂ΩR Bu > 0, φ ∈ K\θ with ψ = C, C is a const.
We now prove that In fact, if not, there are λ 1 ≥ 0 and which contradicts the definition of λ * , hence (3.18) holds, by Lemma 2.2 By virtue of (H 4 ), there are μ > 0 and > 0 such that 0 ≤ F(t,u) ≤ μu l for 0 ≤ u ≤ .Take then we now prove that In fact, if it is not true, there exist u 0 ∈ K ∩ ∂Ω r and λ 0 ≥ 1 such that Au 0 = λ 0 u 0 .Then if then BVP(1.1) has at least two positive solutions.
Proof.By the proof of Theorems 3.1 and 3.2, there exist 0 < r < T 1 < R such that (3.7) and (3.20) hold, respectively.We now prove that Otherwise, there are that is, then BVP(1.1) has at least two positive solutions.
Proof.By the proof of Theorems 3.1 and 3.2, there exist 0 < r < T 2 < R such that (3.14) and (3.25) hold, respectively.Define B : Then it is easy to verify B : K ∩ ∂Ω → K is completely continuous and inf K∩∂Ωr Bu > 0.
We now prove that Otherwise, there are . This completes the proof.Similar to the proof of Theorem 3.5, we can get the following theorem.
Remark 3.7.In fact, if T 1 = T 2 = T, the conditions of Theorems 3.5 and 3.6 both cannot ensure that BVP(1.1) has at least three positive solutions or even one positive solution.
The reason is that the conditions (H 5 ) and (H 6 ) imply that the inequalities hold, respectively, however, the latter two contradict each other.
Remark 3.8.In this paper, if p(t) = 1, F(t,u) = f (u), all the theorems above still hold and the results are new.Here the function f (u) and the boundary conditions are more general than in [1,6,7] where f (u) only satisfies lim u→0+ f (u)/u = 0 (or ∞), lim u→∞ f (u)/u = ∞ (or 0) and only the cases β = 0, δ = 0 are considered.In addition, our method is different from those methods in [1,6,7].
Remark 3.9.In the proof of theorems, one of the key steps is to find the operator B.
We note that it is more general than the ones in [6,[8][9][10][11].We think not only about the superlinear, sublinear cases but also the general cases.Hence, our results improve and generalize those in some well-known papers.

Examples
In this section, we provide some examples to illustrate the validity of the results established in Section 2.