Positive Solutions for Singular Sturm-Liouville Boundary Value Problems with Integral Boundary Conditions ∗

In this paper, we study the second-order nonlinear singular Sturm-Liouville boundary value problems with Riemann-Stieltjes integral boundary conditions     −(p(t)u ' (t)) ' +q(t)u(t) = f(t,u(t)), 0 < t < 1, �1u(0) −�1u ' (0) = R 1 0 u(�)d�(�), where f(t,u) is allowed to be singular at t = 0,1 and u = 0. Some new results for the existence of positive solutions of the boundary value problems are obtained. Our results extend some known results from the nonsingular case to the singular case, and we also improve and extend some results of the singular cases.


Introduction
We investigate the existence of positive solutions for the second-order nonlinear singular Sturm-
In this paper, the integral BVP in (1.1) has a more general form where the nonlinear term f (t, u) is allowed to be singular at t = 0, 1 and u = 0. We obtain the existence criteria of at least one positive solution for BVP (1.1) in the two cases which are β 1 , β 2 > 0 and β 1 = 0 or β 2 = 0.
Boundary value problems (BVPs) arise from applied mathematics, biology, engineering and so on.The existence of positive solutions to nonlocal BVPs has been extensively studied in recent years.There are many results on the existence of positive solutions for three-point BVPs [1,2], m-point BVPs [3,4].
It is well known that BVPs with Riemann-Stieltjes integral boundary conditions include twopoint, three-point, multi-point and the Riemann integral BVPs as special cases.Such BVPs have attracted the attention of researchers such as [5]- [16].In [5] and [6], the existence and uniqueness of a solution of BVPs were studied.In [7]- [16], the sufficient conditions for the existence of positive solutions of BVPs were given and many optimal results were obtained.In addition, many papers investigated the existence of solutions for the singular BVPs, for example, [1,2,4,5,6], [11]- [16].
In [12], Webb used the methodology of [13] to study the existence of multiple positive solutions of nonlocal BVP of the form where g, f are non-negative functions, typically f is continuous and g ∈ L 1 may have pointwise singularities.The case when f has no singularity at u = 0 is covered in [12] for the more general case when the BCs allow Riemann-Stieltjes integrals with sign changing measures.Using the same general method, other nonlocal problems of arbitrary order are studied in [14].
In [15], by means of the fixed point theorem, Jiang, Liu and Wu concerned with the second-order where h is allowed to be singular at t = 0, 1 and f (t, u) may be singular at u = 0. BVP (1.3) is the spacial case of BVP (1.1), when p ≡ 1 and q ≡ 0. In [15], [1] and [2], Liu, Jiang and co-author used the same condition to control the singularity of f at u = 0 for those BVPs (see (H2) in [1] and [15], (H3) in [2]).In this paper, our condition is less restrictive than that one (see (3.4)), and the conditions of the existence of solutions is simpler than the one in [15] when β 1 , β 2 > 0.
In [16], by using some results from the mixed monotone operator theory, Kong concerned with positive solutions of the second order singular BVP where f (t, u) may be singular at t = 0, 1 and u = 0.When β 1 , β 2 = 0, (1.1) becomes BVP (1.4).
Kong [16] studied the existence and uniqueness of positive solutions of (1.4).In this paper, we use different methods from [16] to control the singularity of f at u = 0. We improve and extend the results in [16] (see Remark 3.5).
The rest of this paper is organized as follows.In section 2, we present some lemmas that are used to prove our main results.In section 3, the existence of positive solutions for BVP (1.1) is stated and proved when β 1 , β 2 > 0 and β 1 = 0 or β 2 = 0, respectively.EJQTDE, 2010 No. 77, p. 3 Lemma 2.1 (See [3]) Suppose φ and ψ be the solutions of the linear problems respectively.Then (i) φ is strictly increasing on [0, 1], and φ(t) > 0 on (0, 1]; (ii) ψ is strictly decreasing on [0, 1], and ψ(t) > 0 on [0, 1); ) is a constant and w > 0, φ and ψ are linearly independent. Let if and only if u can be expressed by Let Then a(t) and b(t) are solutions of respectively. Denote We will use the following hypothesis: if and only if u can be expressed by The equation (2.2) is proved in [12] using the methods of [13] with a different notation from the one here. and Lemma 2.5 Suppose (H1) holds.Then (1) A(s) and B(s) are nonnegative and bounded on [0, 1]; (2) a(t) is strictly decreasing on [0, 1], and a(t) > 0 on [0, 1); we can easily obtain the following Lemma 2.6 from Lemma 2.4 and Lemma 2.5.
Let E be a Banach space, K ⊂ E a cone.K is said to be reproducing if E = K − K, and is a total cone if E = K − K. (See [18] and [19]).Lemma 2.9 (See [18] Page 219 Proposition 19.1)Let E be a Banach space and K ⊂ E a cone.
Then we can get that K = Ø ⇒ K is reproducing.The converse fails.
Lemma 2.10 Suppose that (H1) holds.Then T : K → K is a completely continuous, positive, linear operator and the spectral radius r(T ) > 0.
Proof Since (H1) holds, by Lemma 2.4, 2.5 and Lemma 2.6, it is easy to show T : K → K is a completely continuous, positive, linear operator.Noticing Lemma 2.6, we can get the spectral radius r(T ) > 0 from Lemma 2.5 in [17].
Lemma 2.11 (Krein-Rutman theorem.See [18] Page 226 Theorem 19.2) Let E be a Banach space, K ⊂ E a total cone and T ∈ L(E) a compact, linear, operator with T (K) ⊂ K (positive) and spectral radius r(T ) > 0. Then r(T ) is an eigenvalue with an eigenvector in K.
According to Lemmas above, we can let u 0 denote the eigenfunction in K corresponding to its eigenvalue r(T ) such that r(T )u 0 = T u 0 .We write λ = (r(T )) −1 .
(2.4) Lemma 2.12 (See [20]) Let K be a cone of a real Banach space E, Ω be a bounded open set in E. Suppose A : K ∩ Ω → K is a completely continuous operator.If there exists u 0 ∈ K\{θ} such that u − Au = ρu 0 for all u ∈ ∂Ω ∩ K and all ρ ≥ 0, then i(A, Ω ∩ K, K) = 0. Proof.This is the same as the first part of Theorem 3.1.
Proof.We denote If (3.5) is not true, there exist u * ∈ ∂B R ∩ K and ρ 1 ≥ 1 such that We have Let We have γr 0 ≤ u * * ≤ R 0 .
We can get A n has a fixed point It is easy to see that Ω is uniformly bounded.And we have Hence, It is similar to the proof above, we can show That is, for each u n ∈ Ω, we have where M 1 = max γr0≤u≤R g(u) 1 0 h(s)ds + M 0 .In the following, we prove that Ω is equicontinuous.
It follows that the {u n } n>n0 has a subsequence which is uniformly convergent on [0, 1] from Ascoli-Arzela theorem.Without loss of generality, we can assume that {u n } n>n0 itself converges uniformly to u on [0, 1], then r 0 ≤ u ≤ R and u ∈ K.
By means of Theorem 3.3, we can obtain that the BVP (3.16) has at least one positive solution.