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Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. The Sturm-Liouville BVP in Banach space Advances in Difference Equations 2011, 2011:65 doi:10.1186/1687-1847-2011-65 Su Hua H. Su (jnsuhua@163.com) Lishan Liu L. Liu (lls@mail.qfnu.edu.cn) Xinjun Wang X. Wang (wangxj566@sina.com) ISSN 1687-1847 Article type Research Submission date 12 October 2011 Acceptance date 21 December 2011 Publication date 21 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/65 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Hua et al. ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Sturm–Liouville BVP in Banach space Hua Su∗ 1 , Lishan Liu2 and Xinjun Wang3 1 School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan Shandong 250014, China 2 School of Mathematical Sciences, Qufu Normal University, Qufu Shandong 273165, China 3 School of Economics, Shandong University, Jinan Shandong 250014, China ∗ Corresponding author: jnsuhua@163.com Email addresses: LL: lls@mail.qfnu.edu.cn XW: wangxj566@sina.com Abstract We consider the existence of single and multiple positive solutions for fourth-order Sturm–Liouville boundary value problem in Banach space. The suﬃcient condition for the existence of single and multiple positive solutions is obtained by ﬁxed theorem of strict set contraction operator in the frame of the ODE technique. Our results signiﬁcantly extend and improve many known results including singular and nonsingular cases. 1 Introduction The boundary value problems (BVPs) for ordinary diﬀerential equations play a very important role in both theory and application. They are used to describe a large number of physical, biological, and chemical phenomena. In this article, we will study the existence of positive solutions for the following fourth-order 1
nonlinear Sturm–Liouville BVP in a real Banach space E  1 ′′′ ′  p(t) (p(t)u (t)) = f (u(t)), 0 < t < 1,          α1 u(0) − β1 u′ (0) = 0, γ1 u(1) + δ1 u′ (1) = 0,     (1.1)   α u′′ (0) − β lim p(t)u′′′ (t) = 0,  2 2  t→0+         γ2 u′′ (1) + δ2 lim p(t)u′′′ (t) = 0,   t→1− s dτ where αi , βi , δi , γi ≥ 0 (i = 1, 2) are constants such that ρ1 = β1 γ1 + α1 γ1 + α1 δ1 > 0, B (t, s) =, ρ2 = p(τ ) t β2 γ2 + α2 γ2 B (0, 1) + α2 δ2 > 0, and p ∈ C 1 ((0, 1), (0, +∞)). Moreover p may be singular at t = 0 and/or 1. BVP (1.1) is often referred to as the deformation of an elastic beam under a variety of boundary conditions, for detail, see [1–17]. For example, BVP (1.1) subject to Lidstone boundary value conditions u(0) = u(1) = u′′ (0) = u′′ (1) = 0 are used to model such phenomena as the deﬂection of elastic beam simply supported at the endpoints, see [1, 3, 5, 7–11, 13–14]. We notice that the above articles use the completely continuous operator and require f satisﬁes some growth condition or assumptions of monotonicity which are essential for the technique used. The aim of this article is to consider the existence of positive solutions for the more general Sturm– Liouville BVP by using the properties of strict set contraction operator. Here, we allow p have singularity at t = 0, 1, as far as we know, there were fewer works to be done. This article attempts to ﬁll part of this gap in the literature. This article is organized as follows. In Section 2, we ﬁrst present some properties of the Green functions that are used to deﬁne a positive operator. Next we approximate the singular fourth-order BVP to singular second-order BVP by constructing an integral operator. In Section 3, the suﬃcient condition for the existence of single and multiple positive solutions for BVP (1.1) will be established. In Section 4, we give one example as the application. 2 Preliminaries and lemmas In this article, we all suppose that (E, · 1) is a real Banach space. A nonempty closed convex subset P in E is said to be a cone if λP ∈ P for λ ≥ 0 and P {−P } = {θ}, where θ denotes the zero element of E . The cone P deﬁnes a partial ordering in E by x ≤ y iﬀ y − x ∈ P . Recall the cone P is said to be normal 2
if there exists a positive constant N such that 0 ≤ x ≤ y implies x ≤ N y 1 . The cone P is normal if 1 every order interval [x, y ] = {z ∈ E |x ≤ z ≤ y } is bounded in norm. In this article, we assume that P ⊆ E is normal, and without loss of generality, we may assume that the normality of P is 1. Let J = [0, 1], and C (J, E ) = {u : J → E | u(t) continuous}, C i (J, E ) = {u : J → E | u(t) is i-order continuously diﬀerentiable}, i = 1, 2, . . . . For u = u(t) ∈ C (J, E ), let u = max u(t) 1 , then C (J, E ) is a Banach space with the norm ·. t∈J A function u(t) is said to be a positive solution of the BVP (1.1), if u ∈ Deﬁnition 2.1 C 2 ([0, 1], E ) C 3 ((0, 1), E ) satisﬁes u(t) ≥ 0, t ∈ (0, 1], pu′′′ ∈ C 1 ((0, 1), E ) and the BVP (1.1), i.e., u ∈ C 2 ([0, 1], P ) C 3 ((0, 1), P ) and u(t) ≡ θ, t ∈ J . We notice that if u(t) is a positive solution of the BVP (1.1) and p ∈ C 1 (0, 1), then u ∈ C 4 (0, 1). Now we denote that H (t, s) and G(t, s) are the Green functions for the following boundary value problem   −u′′ = 0, 0 < t < 1,      α u(0) − β u′ (0) = 0, γ u(1) + δ u′ (1) = 0  1 1 1 1 and  1 ′ ′  p(t) (p(t)v (t)) = 0, 0 < t < 1,        ′  α2 v (0) − t→0+ β2 p(t)v (t) = 0, lim        γ2 v (1) + lim δ2 p(t)v ′ (t) = 0,  t→1− respectively. It is well known that H (t, s) and G(t, s) can be written by    (β + α1 s)(δ1 + γ1 (1 − t)), 0 ≤ s ≤ t ≤ 1,  1 1 H (t, s) = (2.1) ρ1    (β + α t)(δ + γ (1 − s)), 0 ≤ t ≤ s ≤ 1 1 1 1 1 and    (β + α B (0, s)) (δ + γ B (t, 1)) , 0 ≤ s ≤ t ≤ 1, 1 2 2 2 2  G(t, s) = (2.2) ρ2    (β + α B (0, t)) (δ + γ B (s, 1)) , 0 ≤ t ≤ s ≤ 1, 2 2 2 2 s dτ where ρ1 = γ1 β1 + α1 γ1 + α1 δ1 > 0, B (t, s) = , ρ2 = α2 δ2 + α2 γ2 B (0, 1) + β2 γ2 > 0. p(τ ) t 3
It is easy to verify the following properties of H (t, s) and G(t, s) (I) G(t, s) ≤ G(s, s) < +∞, H (t, s) ≤ H (s, s) < +∞; (II) G(t, s) ≥ ρG(s, s), H (t, s) ≥ ξH (s, s), for any t ∈ [a, b] ⊂ (0, 1), s ∈ [0, 1], where δ2 + γ2 B (b, 1) β2 + α2 B (0, a) ρ = min , , δ2 + γ2 B (0, 1) β2 + α2 B (0, 1) δ1 + γ1 (1 − b) β1 + α1 a ξ = min , . δ 1 + γ1 β1 + α1 Throughout this article, we adopt the following assumptions (H1 ) p ∈ C 1 ((0, 1), (0, +∞)) and satisﬁes 1 1 ds 0< < +∞, 0 < λ = G(s, s)p(s)ds < +∞. p(s) 0 0 (H2 ) f (u) ∈ C (P \ {θ}, P ) and there exists M > 0 such that for any bounded set B ⊂ C (J, E ), we have α(f (B (t))) ≤ M α(B (t)), 2M λ < 1. (2.3) where α(·) denote the Kuratowski measure of noncompactness in C (J, E ). The following lemmas play an important role in this article. Lemma 2.1 [17]. Let B ⊂ C [J, E ] be bounded and equicontinuous on J , then α(B ) = sup α(B (t)). t∈J Lemma 2.2 [16]. Let B ⊂ C (J, E ) be bounded and equicontinuous on J , let α(B ) is continuous on J and     u(t)dt : u ∈ B  ≤ α(B (t))dt. α   J J Lemma 2.3 [16]. Let B ⊂ C (J, E ) be a bounded set on J . Then α(B (t)) ≤ 2α(B ). Now we deﬁne an integral operator S : C (J, E ) → C (J, E ) by 1 Sv (t) = H (t, τ )v (τ )d τ. (2.4) 0 Then, S is linear continuous operator and by the expressed of H (t, s), we have   (Sv )′′ (t) = −v (t), 0 < t < 1,         (2.5) ′  α1 (Sv )(0) − β1 (Sv ) (0) = 0,       γ (Sv )(1) + δ (Sv )′ (1) = 0.  1 1 4
Lemma 2.4. The Sturm–Liouville BVP (1.1) has a positive solution if and only if the following integral- diﬀerential boundary value problem has a positive solution of  1 ′ ′  p(t) (p(t)v (t)) + f (Sv (t)) = 0, 0 < t < 1,        (2.6) ′  α2 v (0) − t→0+ β2 p(t)v (t) = 0, lim        γ2 v (1) + lim δ2 p(t)v ′ (t) = 0,  t→1− where S is given in (2.4). Proof In fact, if u is a positive solution of (1.1), let u = Sv , then v = −u′′ . This implies u′′ = −v is a solution of (2.6). Conversely, if v is a positive solution of (2.6). Let u = Sv , by (2.5), u′′ = (Sv )′′ = −v . Thus, u = Sv is a positive solution of (1.1). This completes the proof of Lemma 2.1. So, we only need to concentrate our study on (2.6). Now, for the given [a, b] ⊂ (0, 1), ρ as above in (II), we introduce K = {u ∈ C (J, P ) : u(t) ≥ ρu(s), t ∈ [a, b], s ∈ [0, 1]}. It is easy to check that K is a cone in C [0, 1]. Further, for u(t) ∈ K, t ∈ [a, b], we have by normality of cone P with normal constant 1 that u(t) ≥ρ u . 1 Next, we deﬁne an operator T given by 1 T v (t) = G(t, s)p(s)f (Sv (s))ds, t ∈ [0, 1], (2.7) 0 Clearly, v is a solution of the BVP (2.6) if and only if v is a ﬁxed point of the operator T . Through direct calculation, by (II) and for v ∈ K, t ∈ [a, b], s ∈ J , we have 1 T v (t) = G(t, s)p(s)f (Sv (s))ds 0 1 ≥ρ G(s, s)p(s)f (Sv (s))ds = ρT v (s). 0 So, this implies that T K ⊂ K . Lemma 2.5. Assume that (H1 ), (H2 ) hold. Then T : K → K is strict set contraction. Proof Firstly, The continuity of T is easily obtained. In fact, if vn , v ∈ K and vn → v in the sup norm, then for any t ∈ J , we get 1 T vn (t) − T v (t) ≤ f (Svn (t)) − f (Sv (t)) G(s, s)p(s)ds, 1 1 0 5
so, by the continuity of f, S , we have T vn − T v = sup T vn (t) − T v (t) → 0. 1 t∈J This implies that T vn → T v in the sup norm, i.e., T is continuous. Now, let B ⊂ K is a bounded set. It follows from the the continuity of S and (H2 ) that there exists a positive number L such that f (Sv (t)) ≤ L for any v ∈ B . Then, we can get 1 T v (t) ≤ Lλ < ı, ∀ t ∈ J, v ∈ B. 1 So, T (B ) ⊂ K is a bounded set in K . For any ε > 0, by (H1 ), there exists a δ ′ > 0 such that δ′ 1 ε ε G(s, s)p(s) ≤ G(s, s)p(s) ≤ , . 6L 6L 0 1−δ ′ Let P = max p(t). It follows from the continuity of G(t, s) on [0, 1] × [0, 1] that there exists δ > 0 such t∈[δ ′ ,1−δ ′ ] that ε |G(t, s) − G(t′ , s)| ≤ , |t − t′ | < δ, t, t′ ∈ [0, 1]. 3P L Consequently, when |t − t′ | < δ, t, t′ ∈ [0, 1], v ∈ B , we have 1 (G(t, s) − G(t′ , s))p(s)f (Sv (s))ds ′ T v (t) − T v (t ) = 1 0 1 δ′ |G(t, s) − G(t′ , s)|p(s) f (Sv (s)) 1 ds ≤ 0 1−δ ′ |G(t, s) − G(t′ , s)|p(s) f (Sv (s)) 1 ds + δ′ 1 |G(t, s) − G(t′ , s)|p(s) f (Sv (s)) 1 ds + 1−δ ′ δ′ 1 ≤ 2L G(s, s)p(s)ds + 2L G(s, s)p(s)ds 0 1−δ ′ 1 |G(t, s) − G(t′ , s)|ds +P L 0 ≤ ε. 6
This implies that T (B ) is equicontinuous set on J . Therefore, by Lemma 2.1, we have α(T (B )) = sup α(T (B )(t)). t∈J Without loss of generality, by condition (H1 ), we may assume that p(t) is singular at t = 0, 1. So, There exists {ani }, {bni } ⊂ (0, 1), {ni } ⊂ N with {ni } is a strict increasing sequence and lim ni = +ı such that i→+ı 0 < · · · < ani < · · · < an1 < bn1 < · · · < bni < · · · < 1; p(t) ≥ ni , t ∈ (0, ani ] ∪ [bni , 1), p(ani ) = p(bni ) = ni ; lim ani = 0, lim bni = 1. (2.9) i→+ı i→+ı Next, we let   t ∈ (0, ani ] ∪ [bni , 1);  n, i  pni (t) =    p(t), t ∈ [ani , bni ].  Then, from the above discussion we know that (p)ni is continuous on J for every i ∈ N and pni (t) ≤ p(t); pni (t) → p(t), ∀ t ∈ (0, 1), as i → +ı. For any ε > 0, by (2.9) and (H1 ), there exists a i0 such that for any i > i0 , we have that an i 1 ε ε 2L G(s, s)p(s)ds < , 2L G(s, s)p(s)ds < (2.10) . 2 2 0 bni Therefore, for any bounded set B ⊂ C [J, E ], by (2.4), we have S (B ) ⊂ B . In fact, if v ∈ B , there exists D > 0 such that v ≤ D, t ∈ J . Then by the properties of H (t, s), we can have 1 1 S v (t) ≤ H (t, τ ) v (τ ) 1 dτ ≤ D H (t, τ )dτ ≤ D, 1 0 0 i.e., S (B ) ⊂ B . Then, by Lemmas 2.2 and 2.3, (H2 ), the above discussion and note that pni (t) ≤ p(t), t ∈ (0, 1), as 7
t ∈ J, i > i0 , we know that   1   α(T (B )(t)) = α G(t, s)p(s)f (Sv (s))ds ∈ B    0   1   ≤ G(t, s)[p(s) − pni (s)]f (Sv (s))ds ∈ B  α   0   1   +α  G(t, s)pni (s)f (Sv (s))ds ∈ B    0 an i 1 ≤ 2L G(s, s)p(s)ds + 2L G(s, s)p(s)ds 0 bni 1 + α (G(t, s)pni (s)f (Sv (s)) ∈ B ) ds 0 1 ≤ ε+ G(s, s)p(s)α (f (Sv (s)) ∈ B ) ds 0 ≤ ε + 2M λα(B ). Since the randomness of ε, we get α(T (B )(t)) ≤ 2M λα(B ), t ∈ J. (2.11) So, it follows from (2.8) (2.11) that for any bounded set B ⊂ C [J, E ], we have α(T (B )) ≤ 2M λα(B ). And note that 2M λ < 1, we have T : K → K is a strict set contraction. The proof is completed. Remark 1. When E = R, (2.3) naturally hold. In this case, we may take M as 0, consequently, T : K → K is a completely continuous operator. So, our condition (H1 ) is weaker than those of the above mention articles. Our main tool of this article is the following ﬁxed point theorem of cone. Theorem 2.1 [16]. Suppose that E is a Banach space, K ⊂ E is a cone, let Ω1 , Ω2 be two bounded open sets of E such that θ ∈ Ω1 , Ω1 ⊂ Ω2 . Let operator T : K ∩ (Ω2 \ Ω1 ) −→ K be strict set contraction. Suppose that one of the following two conditions hold, (i) T x ≤ x , ∀ x ∈ K ∩ ∂ Ω1 , Tx ≥ x , ∀ x ∈ K ∩ ∂ Ω2 ; 8
(ii) T x ≥ x , ∀ x ∈ K ∩ ∂ Ω1 , T x ≤ x , ∀ x ∈ K ∩ ∂ Ω2 . Then, T has at least one ﬁxed point in K ∩ (Ω2 \ Ω1 ). Theorem 2.2 [16]. Suppose E is a real Banach space, K ⊂ E is a cone, let Ωr = {u ∈ K : u ≤ r}. Let operator T : Ωr −→ K be completely continuous and satisfy T x = x, ∀ x ∈ ∂ Ωr . Then (i) If T x ≤ x , ∀ x ∈ ∂ Ωr , then i(T, Ωr , K ) = 1; (ii) If T x ≥ x , ∀ x ∈ ∂ Ωr , then i(T, Ωr , K ) = 0. 3 The main results Denote f (x) 1 f (x) 1 f0 = lim , f∞ = lim . x1 x1 + 1 →∞ 1 →0 x x In this section, we will give our main results. Theorem 3.1. Suppose that conditions (H1 ), (H2 ) hold. Assume that f also satisfy 1 ∗ (A1 ): f (x) ≥ ru , ξ H (τ, τ )x(τ )d τ ≤x ≤ r; 1 0 1 (A2 ): f (x) ≤ Ru∗ , 0 ≤ x ≤ R, 1 where u∗ and u∗ satisfy 1 b ∗ G(s, s)p(s)u (s)ds ≥ 1, u∗ (s) G(s, s)p(s)ds ≤ 1. ρ 1 0 a 1 Then, the boundary value problem (1.1) has a positive solution. Proof of Theorem 3.1. Without loss of generality, we suppose that r < R. For any u ∈ K , we have u(t) ≥ρ u , t ∈ [a, b]. (3.1) 1 we deﬁne two open subsets Ω1 and Ω2 of E Ω1 = { u ∈ K : u < r } , Ω2 = { u ∈ K : u < R } For u ∈ ∂ Ω1 , by (3.1), we have r = u ≥ u(s) ≥ ρ u = ρr, s ∈ [a, b]. (3.2) 1 9
Then, for u ∈ K ∩ ∂ Ω1 , by (2.4), (3.2), (II), for any s ∈ [a, b], u ∈ K ∩ ∂ Ω1 , we have 1 1 ≥ H (τ, τ ) u(τ ) 1 dτ ≥ H (τ, τ )u(τ )dτ ≥ S u(s) r 1 0 0 1 1 1 = H (s, τ )u(τ )dτ ≥ξ H (τ, τ )u(τ )dτ . 0 0 1 1 So, for u ∈ K ∩ ∂ Ω1 , if (A1 ) holds, we have 1 b G(s, s)p(s)u∗ (s)rds T u(t) = G(t, s)p(s)f (Su(s))ds ≥ rρ ≥r= u . 1 0 a 1 1 Therefore, we have Tu ≥ u , ∀ u ∈ ∂ Ω1 . (3.3) On the other hand, as u ∈ K ∩ ∂ Ω2 , by (2.4), (3.2), (II), for any s ∈ [a, b], u ∈ K ∩ ∂ Ω2 , we have 1 1 R≥ H (τ, τ ) u(τ ) 1 dτ ≥ H (τ, τ )u(τ )dτ ≥ S u(s) ≥ 0. 0 0 1 For u ∈ K ∩ ∂ Ω2 , if (A2 ) holds, we know 1 1 T u(t) = G(t, s)p(s)f (Su(s))ds ≤ G(t, s)p(s)u∗ (s)ds R 1 0 0 1 1 1 1 ≤ G(t, s)p(s) u∗ (s) dsR ≤ G(s, s)p(s)ds u∗ (s) 1 R ≤ R = u . 1 0 0 Thus T (u) ≤ u , ∀ u ∈ ∂ Ω2 . (3.4) Therefore, by (3.2), (3.3), Lemma 2.5 and r < R, we have that T has a ﬁxed point v ∈ (Ω2 \ Ω1 ). Obviously, v is positive solution of problem (2.6). Now, by Lemma 2.4 we see that u = Sv is a position solution of BVP (1.1). The proof of Theorem 3.1 is complete. Theorem 3.2. Suppose that conditions (H1 ), (H2 ) and (A1 ) in Theorem 3.1 hold. Assume that f also satisfy (A3 ): f0 = 0; (A4 ): f∞ = 0. Then, the boundary value problem (1.1) have at least two solutions. 10
Proof of Theorem 3.2. First, by condition (A3 ), (2.4) and the property of limits, we can have lim + 1 →0 u b |f (Su) = 0. Then, for any m > 0 such that m G(s, s)p(s)ds ≤ 1, there exists a constant u 1 1 a ρ∗ ∈ (0, r) such that f (Su) ≤ m u 1, 0< u ≤ ρ∗ , u = 0. (3.5) 1 1 Set Ωρ∗ = {u ∈ K : u < ρ∗ }, for any u ∈ K ∩ ∂ Ωρ∗ , by (3.4), we have f (Su) ≤m u ≤ mρ∗ . 1 1 For u ∈ K ∩ ∂ Ωρ∗ , we have 1 1 T u(t) = G(t, s)p(s)f (Su(s))ds ≤ G(t, s)p(s) f (Su(s)) ds 1 1 0 0 1 b b ≤ G(t, s)p(s)mρ∗ ds ≤ ρ∗ m G(s, s)p(s)ds ≤ ρ∗ = u . a a Therefore, we can have Tu ≤ u , ∀ u ∈ ∂ Ωρ∗ . Then by Theorem 2.2, we have i(T, Ωρ∗ , K ) = 1. (3.6) Next, by condition (A4 ), (2.4) and the property of limits, we can have lim f (Su) = 0. Then, u 1 1 1 →ı u b for any m > 0 such that m G(s, s)p(s)ds ≤ 1, there exists a constant ρ0 > 0 such that a f (Su) ≤ m u 1, (3.7) u > ρ0 . 1 1 We choose a constant ρ∗ > max {r, ρ0 }, obviously, ρ∗ < r < ρ∗ . Set Ωρ∗ = {u ∈ K : u < ρ∗ }, for any u ∈ K ∩ ∂ Ωρ∗ , by (3.6), we have ≤ mρ∗ . f (Su) ≤m u 1 1 For u ∈ K ∩ ∂ Ωρ∗ , we have 1 1 T u(t) = G(t, s)p(s)g (s)f (Su(s))ds ≤ G(t, s)p(s)g (s) f (Su(s)) 1 ds 1 0 0 1 b b G(t, s)p(s)mρ∗ ds ≤ ρ∗ m G(s, s)p(s)ds ≤ ρ∗ = u . ≤ a a 11
Therefore, we can have Tu ≤ u , ∀ u ∈ ∂ Ωρ∗ . Then by Theorem 2.2, we have i(T, Ωρ∗ , K ) = 1. (3.8) Finally, set Ωr = {u ∈ K : u < r}, For any u ∈ ∂ Ωr , by (A2 ), Lemma 2.2 and also similar to the latter proof of Theorem 3.1, we can also have Tu ≥ u , ∀ u ∈ ∂ Ωr . Then by Theorem 2.2, we have i(T, Ωr , K ) = 0. (3.9) Therefore, by (3.5), (3.7), (3.8), and ρ∗ < r < ρ∗ , we have i(T, Ωr \ Ωρ∗ , k ) = −1, i(T, Ωρ∗ \ Ωr , k ) = 1. Then T have ﬁxed point v1 ∈ Ωr \ Ωρ∗ , and ﬁxed point v2 ∈ Ωρ∗ \ Ωr . Obviously, v1 , v2 are all positive solutions of problem (2.6). Now, by Lemma 2.4 we see that u1 = Sv1 , u2 = Sv2 are two position solutions of BVP (1.1). The proof of Theorem 3.2 is complete. 4 Application In this section, in order to illustrate our results, we consider some examples. Now, we consider the following concrete second-order singular BVP (SBVP) Example 4.1. Consider the following SBVP ′ 1 √ ′′′  3 1 1 3 √ tu (t) + 160 u 2 + u 3 = θ, 0 < t < 1,  t3 3      (4.1) ′ ′  u(0) − 3u (0) = θ, u(1) + 2u (1) = θ,    √ √   3u′′ (0) − lim 1 3 tu′′′ (t) = θ, u′′ (1) + lim 1 3 tu′′′ (t) = θ,    +3 −3 t→0 t→1 where α1 = γ1 = 1, β1 = 3, δ1 = 2, β2 = γ2 = δ2 = 1, α2 = 3, 12
1√ 1 1 3 p(t) = t, f (u) = 160(u 2 + u 3 ). 3 Then obviously, 1 1 3 dt = , f∞ = 0, f0 = 0, p(t) 2 0 By computing, we know that the Green’s function are    (3 + s) (3 − t) , 0 ≤ s ≤ t ≤ 1,  1 H (t, s) = 6   (3 + t) (3 − s) , 0 ≤ t ≤ s ≤ 1.     (1 + 3s)(2 − t), 0 ≤ s ≤ t ≤ 1,  1 G(t, s) = 7   (1 + 3t)(2 − s), 0 ≤ t ≤ s ≤ 1.  It is easy to note that 0 ≤ G(s, s) ≤ 1 and conditions (H1 ), (H2 ), (A3 ), (A4 ) hold. Next, by computing, we know that ρ = 0.44, ξ = 0.8. We choose r = 3, u∗ = 104, as 1.05 = ρξr ≤ b G(s, s)p(s)u∗ (s)ds = 1.3 > 1, because of the monotone increasing u = max{u(t), t ∈ J } ≤ 3 and ρ a of f (u) on [0, ∞), then f (u) ≥ f (1.05) = 326.4, 1.05 ≤ u ≤ 3. Therefore, as 1 1 ρξr = ρξ H (τ, τ ) u dτ ≤ ξ H (τ, τ )u(τ )dτ , 0 0 so we have 1 ∗ f (u) ≥ ru , ξ H (τ, τ )u(τ )dτ ≤ u ≤ r, 0 then conditions (A1 ) holds. Then by Theorem 3.2, SBVP (4.1) has at least two positive solutions u1 , u2 and 0 < u1 < 3 < u2 . Competing interests We declare that we have no signiﬁcant competing ﬁnancial, professional, or personal interests that might have inﬂuenced the performance or presentation of the work described in this manuscript. 13
Authors’ contributions All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript. Acknowledgments The authors would like to thank the reviewers for their valuable comments and constructive suggestions. HS and LL were supported ﬁnancially by the Shandong Province Natural Science Foundation (ZR2009AQ004), NSFC (11026108, 11071141) and XW was supported by the Shandong Province planning Foundation of Social Science (09BJGJ14), Shandong Province Natural Science Foundation (Z2007A04). References 1. Ma, RY, Wang, HY: On the existence of positive solutions of fourth order ordinary diﬀerential equation. Appl. Anal. 59, 225–231 (1995) 2. Schroder, J: Fourth-order two-point boundary value problems: estimates by two side bounds. Nonlinear Anal. 8, 107–144 (1984) 3. Agarwal, RP, Chow, MY: Iterative methods for a fourth order boundary value problem. J. Comput. Appl. Math. 10, 203–217 (1984) 4. Agarwal, RP: On the fourth-order boundary value problems arising in beam analysis. Diﬀ. Integral Equ. 2, 91–110 (1989) 5. Ma, RY: Positive solutions of fourth-order two point boundary value problem. Ann. Diﬀ. Equ. 15, 305–313 (1999) 6. O’Regan, D: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 161, 78–116 (1991) 7. Gupta, CP: Existence and uniqueness theorems for a bending of an elastic beam equation at resonance. J. Math. Anal. Appl. 135, 208–225 (1988) 8. Gupta, CP: Existence and uniqueness theorems for a bending of an elastic beam equation. Appl. Anal. 26, 289–304 (1988) 9. Yang, YS: Fourth order two-point boundary value problem. Proc. Am. Math. Soc. 104, 175–180 (1988) 14
10. Li, H, Sun, J: Positive solutions of sublinear Sturm–Liouville problems with changing sign nonlinearity. Comput. Math. Appl. 58, 1808–1815 (2009) 11. Yang, J, Wei, Z, Liu, K: Existence of symmetric positive solutions for a class of Sturm–Liouville-like boundary value problems. Appl. Math. Comput. 214, 424–432 (2009) 12. Xu, F, Su, H, Positive solutions of fourth-order nonlinear singular boundary value problems. Nonlinear Anal. 68(5), 1284–1297 (2008) 13. Liu, B: Positive solutions of fourth-order two point boundary value problem. Appl. Math. Comput. 148, 407–420 (2004) 14. Wei, Z: Positive solutions of singular dirichlet boundary value problems. Chin. Ann. Math. 20(A), 543–552 (2007) (in Chinese) 15. Wong, PJY, Agarwal, RP: Eigenvalue of lidstone boundary value problems. Appl. Math. Comput. 104, 15–31 (1999) 16. Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cone. Academic Press, Inc., New York (1988) 17. Zhang, X: Positive solutions for three-point semipositone boundary value problems with convex nonlinearity. J. Appl. Math. Comput. 30, 349–367 (2009) 15