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Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 594128, 21 pages doi:10.1155/2011/594128 Research Article Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations Huiqin Lu School of Mathematical Sciences, Shandong Normal University, Jinan, 250014 Shandong, China Correspondence should be addressed to Huiqin Lu, lhy@sdu.edu.cn Received 16 October 2010; Revised 22 December 2010; Accepted 27 January 2011 Academic Editor: Kanishka Perera Copyright q 2011 Huiqin Lu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from 0, θ Our nonlinearity f t, u, v, w may be singular at u,v, t and/or t 1 Introduction Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on Therefore, singular boundary value problems have been investigated extensively in recent years see 1–4 and references therein This paper investigates the following fourth-order nonlinear singular eigenvalue problem: u4 t λf t, u t , u t , u t , t ∈ 0, , 1.1 u0 u u u 0, where λ ∈ 0, ∞ is a parameter and f satisfies the following hypothesis: H f ∈ C 0, × 0, ∞ × 0, ∞ × −∞, , 0, ∞ , and there exist constants αi , βi , Ni , i 1, 2, −∞ < α1 ≤ ≤ β1 < ∞, −∞ < α2 ≤ ≤ β2 < ∞, ≤ α3 ≤ β3 < 1, Boundary Value Problems i βi < 1; < Ni ≤ 1, i w ∈ −∞, , f satisfies 1, 2, such that for any t ∈ 0, , u, v ∈ 0, ∞ , cβ1 f t, u, v, w ≤ f t, cu, v, w ≤ cα1 f t, u, v, w , ∀0 < c ≤ N1 , c f t, u, v, w ≤ f t, u, cv, w ≤ c f t, u, v, w , ∀0 < c ≤ N2 , c f t, u, v, w ≤ f t, u, v, cw ≤ c f t, u, v, w , ∀0 < c ≤ N3 β2 α2 β3 α3 1.2 Typical functions that satisfy the above sublinear hypothesis H are those taking the form m1 m2 m3 f t, u, v, w pi,j,k t uri vsj wσk , 1.3 i j 1k where pi,j,k t ∈ C 0, , 0, ∞ , ri , sj ∈ R, ≤ σk < 1, max{ri , 0} max{sj } σk < 1, i 1, 2, , m1 , j 1, 2, , m2 , k 1, 2, , m3 The hypothesis H is similar to that in 5, Because of the extensive applications in mechanics and engineering, nonlinear fourthorder two-point boundary value problems have received wide attentions see 7–12 and references therein In mechanics, the boundary value problem 1.1 BVP 1.1 for short describes the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps The term u in f represents bending effect which is useful for the stability analysis of the beam BVP 1.1 has two special features The first one is that the nonlinearity f may depend on the first-order derivative of the unknown function u, and the second one is that the nonlinearity f t, u, v, w may be singular at u, v, t and/or t In this paper, we study the existence of positive solutions and the structure of positive solution set for the BVP 1.1 Firstly, we construct a special cone and present a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from 0, θ Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory By singularity of f, we mean that the function f in 1.1 is allowed to be unbounded at the points u 0, v 0, t 0, and/or t A function u t ∈ C2 0, ∩ C4 0, is called a positive solution of the BVP 1.1 if it satisfies the BVP 1.1 u t > 0, −u t > for t ∈ 0, and u t > for t ∈ 0, For some λ ∈ 0, ∞ , if the BVP 1.1 has a positive solution u, then λ is called an eigenvalue and u is called corresponding eigenfunction of the BVP 1.1 The existence of positive solutions of BVPs has been studied by several authors in the literature; for example, see 7–20 and the references therein Yao 15, 18 studied the following BVP: u4 t f t, u t , u t , t ∈ 0, \ E, 1.4 u0 u u u 0, where E ⊂ 0, is a closed subset and mesE 0, f ∈ C 0, \E × 0, ∞ × 0, ∞ , 0, ∞ In 15 , he obtained a sufficient condition for the existence of positive solutions of BVP 1.4 Boundary Value Problems by using the monotonically iterative technique In 13, 18 , he applied Guo-Krasnosel’skii’s fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP 1.4 and the following BVP: u4 t f t, u t , t ∈ 0, , 1.5 u u u1 u These differ from our problem because f t, u, v in 1.4 cannot be singular at u 0, v and the nonlinearity f in 1.5 does not depend on the derivatives of the unknown functions In this paper, we first establish a necessary and sufficient condition for the existence of positive solutions of BVP 1.1 for any λ > by using the following Lemma 1.1 Efforts to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs by the lower and upper solution method can be found, for example, in 5, 6, 21–23 In 5, 6, 22, 23 they considered the case that f depends on even order derivatives of u Although the nonlinearity f in 21 depends on the first-order derivative, where the nonlinearity f is increasing with respect to the unknown function u Papers 24, 25 derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity f t, u does not depend on the derivatives of the unknown functions, and f t, u is decreasing with respect to u Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated see 26–28 and references therein Ma and An 26 and Ma and Xu 27 discussed the global structure of positive solutions for the nonlinear eigenvalue problems and obtained the existence of an unbounded connected branch of positive solution set by using global bifurcation theorems see 29, 30 The terms f u in 26 and f t, u, u in 27 are not singular at t 0, 1, u 0, u Yao 14 obtained one or two positive solutions to a singular elastic beam equation rigidly fixed at both ends by using Guo-Krasnosel’skii’s fixed point theorem, but the global structure of positive solutions was not considered Since the nonlinearity f t, u, v, w in BVP 1.1 may be singular at u, v, t and/or t 1, the global bifurcation theorems in 29, 30 not apply to our problem here In Section 4, we also investigate the global structure of positive solutions for BVP 1.1 by applying the following Lemma 1.2 The paper is organized as follows: in the rest of this section, two known results are stated In Section 2, some lemmas are stated and proved In Section 3, we establish a necessary and sufficient condition for the existence of positive solutions In Section 4, we prove that the closure of positive solution set possesses an unbounded connected branch which comes from 0, θ Finally we state the following results which will be used in Sections and 4, respectively Lemma 1.1 see 31 Let X be a real Banach space, let K be a cone in X, and let Ω1 , Ω2 be bounded open sets of E, θ ∈ Ω1 ⊂ Ω1 ⊂ Ω2 Suppose that T : K ∩ Ω2 \ Ω1 → K is completely continuous such that one of the following two conditions is satisfied: T x ≤ x , x ∈ K ∩ ∂Ω1 ; T x ≥ x , x ∈ K ∩ ∂Ω2 T x ≥ x , x ∈ K ∩ ∂Ω1 ; T x ≤ x , x ∈ K ∩ ∂Ω2 Then, T has a fixed point in K ∩ Ω2 \ Ω1 Boundary Value Problems Lemma 1.2 see 32 Let M be a metric space and a, b ⊂ R1 Let {an }∞ and {bn }∞ satisfy n n a < · · · < an < · · · < a1 < b1 < · · · < bn < · · · < b, lim an n→ ∞ {Cn : n Suppose also that following conditions: a, lim bn n→ ∞ 1.6 b 1, 2, } is a family of connected subsets of R1 × M, satisfying the Cn ∩ {an } × M / ∅ and Cn ∩ {bn } × M / ∅ for each n ∞ n For any two given numbers α and β with a < α < β < b, relatively compact set of R1 × M Then there exists a connected branch C of lim supn → ∞ Cn C ∩ {λ} × M / ∅, where lim supn → ∞ Cn Cn ∩ α, β × M is a such that ∀λ ∈ a, b , 1.7 {x ∈ M : there exists a sequence xni ∈ Cni such that xni → x, i → ∞ } Some Preliminaries and Lemmas Let E E, · P {u ∈ C2 0, : u 0, u 0, u 0}, u is a Banach space, where u maxt∈ 0,1 |u t | Define u∈E:u t ≥ t− t2 u , u t ≥ 1−t max{ u , u , u }, then u , −u t ≥ t u , t ∈ 0, 2.1 It is easy to conclude that P is a cone of E Denote Pr {u ∈ P : u < r}; ∂Pr {u ∈ P : u r} 2.2 Let G0 t, s ⎧ ⎨s, ≤ s ≤ t ≤ 1, ⎩t, ≤ t ≤ s ≤ 1, G t, s G0 t, r G0 r, s dr 2.3 Boundary Value Problems Then G t, s is the Green function of homogeneous boundary value problem u4 t u0 u ⎧ ⎪s ⎪ ⎪ ⎨3 ⎪ t3 ⎪ ⎪ ⎩ G t, s G1 t, s 0, t ∈ 0, , u s t2 − s 2 u 0, st − t , ≤ s ≤ t ≤ 1, t s − t2 ts − s , ⎧ ⎪s − t , ⎪ ⎨ ≤ t ≤ s ≤ 1, 2.4 ≤ s ≤ t ≤ 1, : Gt t, s ⎪ s t2 ⎪ ⎩ − s − s , ≤ t ≤ s ≤ 1, 2 ⎧ ⎨s, ≤ s ≤ t ≤ 1, : −Gt t, s ⎩t, ≤ t ≤ s ≤ G2 t, s Lemma 2.1 G t, s , G1 t, s , and G2 t, s have the following properties: 1, 2, for all t, s ∈ 0, G t, s > 0, Gi t, s > 0, i G t, s ≤ s t − t /2 , G1 t, s ≤ s − t , G2 t, s ≤ t or s , for all t, s ∈ 0, maxt∈ 0,1 G t, s ≤ 1/2 s, maxt∈ 0,1 Gi t, s ≤ s, i 1, 2, for all s ∈ 0, G t, s ≥ s/2 t − t2 /2 , G1 t, s ≥ s/2 − t , G2 t, s ≥ st, for all t, s ∈ 0, Proof From 2.4 , it is easy to obtain the property 2.18 We now prove that property is true For ≤ s ≤ t ≤ 1, by 2.4 , we have s3 G t, s st2 s3 − 2 st − st2 ≤ st − s 1−t , G1 t, s st2 s t− t2 , 2.5 G2 t, s ≤ t or s For ≤ t ≤ s ≤ 1, by 2.4 , we have G t, s G1 t, s s− t3 t3 − 2 ts − s t − ≤ s − ts 2 ts2 st2 ≤ st − 2 s 1−t , s t− t2 , 2.6 G2 t, s ≤ t or s Consequently, property holds From property , it is easy to obtain property We next show that property is true From 2.4 , we know that property holds for s 6 Boundary Value Problems For < s ≤ 1, if s ≤ t ≤ 1, then G t, s s t− t2 s − t2 t− 2 t2 s − t− G1 t, s s 1−t ≥ t2 ts G t, s ≥t− − s t2 t− t2 t− 2 ≥ 1−t , t− t2 t2 − > t− t2 G2 t, s ≥ st; 2.7 if ≤ t ≤ s, then ≥ t − ts G1 t, s t s ≥ 1− − ≥ 1−t , s 2 t2 t− ≥ t− t2 , 2.8 G2 t, s ≥ st Therefore, property holds Lemma 2.2 Assume that u ∈ P \ {θ}, then u u t− t2 u u ≤ u ≤ u , ≤u t ≤ t− t u t2 and u ≤ u 1−t u u 2, ≤ −u t ≤ u , ≤ u 2.9 ≤ u t ≤ − t u 2, 2.10 ∀t ∈ 0, Proof Assume that u ∈ P \ {θ}, then u t ≥ 0, −u t ≥ 0, t ∈ 0, , so t u t∈ 0,1 t u max u t∈ 0,1 u max t∈ 0,1 u s ds ≥ u s ds ≤ u , u s ds max t∈ 0,1 u s ds max −u s ds t −u s ds t 1 u 1 u , − s ds −u s ds ≤ u 2.11 , −u s ds ≥ s u ds u Therefore, 2.9 holds From 2.9 , we get u max u , u , u u 2.12 , Boundary Value Problems By 2.9 and the definition of P , we can obtain that ut t G0 t, s −u s ds ≤ sds 0 t− u tds t t2 u t− t2 u 2, ∀t ∈ 0, , ut ≥ t2 t− u t u ≥ t− −u s ds ≤ − t t2 − t u 2, u t 1−t u t ≥ t u ≥ 1 − t u 2, ≤ −u t ≤ u t u u ∀t ∈ 0, , u 2, u 2, ∀t ∈ 0, , ∀t ∈ 0, 2.13 Thus, 2.10 holds For any fixed λ ∈ 0, ∞ , define an operator Tλ by Tλ u t :λ G t, s f s, u s , u s , u s ds, ∀u ∈ P \ {θ} 2.14 ∀u ∈ P \ {θ}, 2.15 Then, it is easy to know that Tλ u t G1 t, s f s, u s , u s , u s ds, λ Tλ u t −λ G2 t, s f s, u s , u s , u s ds, ∀u ∈ P \ {θ} 2.16 Lemma 2.3 Suppose that (H) and 0< hold Then Tλ P \ {θ} ⊂ P sf s, s − s2 , − s, −1 ds < ∞ 2.17 Boundary Value Problems Proof From H , for any t ∈ 0, , u, v ∈ 0, ∞ , w ∈ −∞, , we easily obtain the following inequalities: cα1 f t, u, v, w ≤ f t, cu, v, w ≤ cβ1 f t, u, v, w , −1 ∀c ≥ N1 , cα2 f t, u, v, w ≤ f t, u, cv, w ≤ cβ2 f t, u, v, w , −1 ∀c ≥ N2 , cα3 f t, u, v, w ≤ f t, u, v, cw ≤ cβ3 f t, u, v, w , −1 ∀c ≥ N3 2.18 For every u ∈ P \ {θ}, t ∈ 0, , choose positive numbers c1 ≤ min{N1 , 1/8 N1 u }, c2 ≤ −1 −1 min{N2 , 1/4 N2 u }, c3 ≥ max{N3 , N3 u } It follows from H , 2.10 , Lemma 2.1, and 2.17 that Tλ u t G t, s f s, u s , u s , u s ds λ ≤ ≤ ≤ ≤ λ λ λ sf s, c1 1 α sc1 α sc1 us c1 s − s2 /2 us c1 s − s2 /2 u c1 β1 α1 −β1 α2 −β2 β3 −α3 c c2 c3 u α c2 s2 s− β1 α c2 u c2 β1 β2 α3 λ , c2 u s c2 − s β2 β2 u s −c3 β c3 α3 u c3 β c3 u s u s − s , −1 c3 c2 − s −c3 sf s, s − f α3 s, s − f s, s − ds s2 , − s, −1 ds s2 , − s, −1 ds s2 , − s, −1 ds < ∞ 2.19 Similar to 2.19 , from H , 2.10 , Lemma 2.1, and 2.17 , for every u ∈ P \ {θ}, t ∈ 0, , we have Tλ u t G1 t, s f s, u s , u s , u s ds λ ≤λ sf s, c1 us c1 s − s2 /2 α −β1 α2 −β2 β3 −α3 c2 c3 ≤ c1 u β1 β2 α3 λ s− s2 , c2 u s u s − s , −1 c3 c2 − s −c3 sf s, s − s2 , − s, −1 ds < ∞ ds Boundary Value Problems − Tλ u t G2 t, s f s, u s , u s , u s ds λ ≤λ sf s, c1 us c1 s − s2 /2 α −β1 α2 −β2 β3 −α3 c2 c3 ≤ c1 s− β1 β2 α3 λ u s2 , c2 u s u s − s , −1 c3 c2 − s −c3 sf s, s − ds s2 , − s, −1 ds < ∞ 2.20 Thus, Tλ is well defined on P \ {θ} From 2.4 and 2.14 – 2.16 , it is easy to know that Tλ u 0, Tλ u 0, Tλ u 0, Tλ u t λ G t, s f s, u s , u s , u s ds t2 λ t− t2 λ t− ≥ t2 t− ≥ sf s, u s , u s , u s ds max G τ, s f s, u s , u s , u s ds τ∈ 0,1 Tλ u , ∀t ∈ 0, , u ∈ P \ {θ}, Tλ u t G1 t, s f s, u s , u s , u s ds λ ≥ ≥ − Tλ u 1−t λ sf s, u s , u s , u s ds 1 max G1 τ, s f s, u s , u s , u s ds 1−t λ τ∈ 0,1 1 − t Tλ u , ∀t ∈ 0, , u ∈ P \ {θ}, t G2 t, s f s, u s , u s , u s ds λ ≥ tλ sf s, u s , u s , u s ds ≥ tλ t max G2 τ, s f s, u s , u s , u s ds τ∈ 0,1 Tλ u , ∀t ∈ 0, , u ∈ P \ {θ} Therefore, T P \ {θ} ⊂ P follows from 2.21 2.21 10 Boundary Value Problems Obviously, u∗ is a positive solution of BVP 1.1 if and only if u∗ is a positive fixed point of the integral operator Tλ in P Lemma 2.4 Suppose that (H) and 2.17 hold Then for any R > r > 0, Tλ : PR \ Pr → P is completely continuous Proof First of all, notice that Tλ maps PR \ Pr into P by Lemma 2.3 Next, we show that Tλ is bounded In fact, for any u ∈ PR \ Pr , by 2.10 we can get r t− t2 ≤u t ≤ t− t2 R, r − t ≤ u t ≤ − t R, rt ≤ −u t ≤ R, ∀t ∈ 0, 2.22 −1 Choose positive numbers c1 ≤ min{N1 , r/8 N1 }, c2 ≤ min{N2 , r/4 N2 }, c3 ≥ max{N3 , −1 N3 R} This, together with H , 2.22 , 2.16 , and Lemma 2.1 yields that Tλ u t G2 t, s f s, u s , u s , u s ds λ ≤λ sf s, c1 ≤λ α sc1 us c1 s − s2 /2 us c1 s − s2 /2 β1 s− α −β1 α2 −β2 β3 −α3 β1 β2 α3 c2 c3 R λ < ∞, , c2 u s c2 − s α c2 ≤ c1 s2 sf s, s − u s u s − s , −1 c3 c2 − s −c3 β2 β c3 u s −c3 α3 f s, s − ds s2 , − s, −1 ds s2 , − s, −1 ds ∀t ∈ 0, , u ∈ PR \ Pr 2.23 Thus, Tλ is bounded on PR \ Pr Now we show that Tλ is a compact operator on PR \ Pr By 2.23 and Ascoli-Arzela theorem, it suffices to show that Tλ V is equicontinuous for arbitrary bounded subset V ⊂ PR \ Pr Since for each u ∈ V , 2.22 holds, we may choose still positive numbers c1 ≤ min{N1 , −1 −1 r/8 N1 }, c2 ≤ min{N2 , r/4 N2 }, c3 ≥ max{N3 , N3 R} Then Tλ u t λ f s, u s , u s , u s ds t ≤ C0 f s, s − t :H t , s2 , − s, −1 ds t ∈ 0, , 2.24 Boundary Value Problems where C0 11 α −β1 α2 −β2 β3 −α3 β1 β2 α3 c2 c3 R λc1 1 Notice that H t dt f 0 s2 , − s, −1 ds dt s, s − s2 , − s, −1 dt ds t s, s − f C0 s C0 0 sf C0 s, s − 2.25 s2 , − s, −1 ds < ∞ Thus for any given t1 , t2 ∈ 0, with t1 ≤ t2 and for any u ∈ V , we get t2 − Tλ u Tλ u t1 t2 ≤ t dt ≤ Tλ u t1 t2 2.26 H t dt t1 From 2.25 , 2.26 , and the absolute continuity of integral function, it follows that Tλ V is equicontinuous Therefore, Tλ V is relatively compact, that is, Tλ is a compact operator on PR \ Pr Finally, we show that Tλ is continuous on PR \ Pr Suppose un , u ∈ PR \ Pr , n 1, 2, and un − u → 0, n → ∞ Then un t → u t , un t → u t and un t → u t as n → ∞ uniformly, with respect to t ∈ 0, From H , choose still positive numbers c1 ≤ −1 −1 min{N1 , r/8 N1 }, c2 ≤ min{N2 , r/4 N2 }, c3 ≥ max{N3 , N3 R} Then ≤ f t, un t , un t , un t ≤ G2 t, s f s, un s , un s , un s t, t − t2 , − t, −1 , t ∈ 0, , ≤ C0 sf s, s − s2 , − s, −1 , t ∈ 0, , s ∈ 0, ≤ C0 f 2.27 By 2.17 , we know that sf s, s − s2 /2, − s, −1 is integrable on 0, Thus, from the Lebesgue dominated convergence theorem, it follows that lim n→ ∞ Tλ un − Tλ u ≤ lim λ n→ ∞ lim n→ ∞ Tλ un − Tλ u s f s, un s , un s , un s λ s lim f s, un s , un s , un s n→ ∞ − f s, u s , u s , u s ds 2.28 − f s, u s , u s , un s ds Thus, Tλ is continuous on PR \ Pr Therefore, Tλ : PR \ Pr → P is completely continuous 12 Boundary Value Problems A Necessary and Sufficient Condition for Existence of Positive Solutions In this section, by using the fixed point theorem of cone, we establish the following necessary and sufficient condition for the existence of positive solutions for BVP 1.1 Theorem 3.1 Suppose (H) holds, then BVP 1.1 has at least one positive solution for any λ > if and only if the integral inequality 2.17 holds Proof Suppose first that u t be a positive solution of BVP 1.1 for any fixed λ > Then there exist constants Ii i 1, 2, 3, with < Ii < < Ii , i 1, such that I1 t − t2 ≤ u t ≤ I2 t − t2 , I3 − t ≤ u t ≤ I4 − t , t ∈ 0, 3.1 u u u 0, that u t ≤ In fact, it follows from u t ≥ 0, t ∈ 0, and u for t ∈ 0, and u t ≤ 0, u t ≥ for t ∈ 0, By the concavity of u t and u t , we have u t ≥ tu 1−t u u t ≥ tu t u ≥ 1−t u 1−t t− t2 u , 3.2 ∀t ∈ 0, u , On the other hand, ut t G0 t, s −u s ds ≤ t2 u u t s −u s ds t 1−t −u s ds ≤ − t u , t −u s ds t t− u t2 u , 3.3 ∀t ∈ 0, t Let I1 holds min{ u , 1/2}, let I2 I4 max{ u , 2}, and let I3 min{ u , 1/2}, then 3.1 Boundary Value Problems 13 −1 −1 −1 −1 Choose positive numbers c1 ≤ N1 I2 , c2 ≤ N2 I4 , c3 ≥ max{N3 , N3 u } This, together with H , 1.2 , and 2.18 yields that f t, t − t2 , − t, −1 f ≤ t, c1 α c1 α ≤ c1 t − t2 /2 1−t c3 u t , c2 u t, u t c1 u t c2 u t c3 −u t β1 t − t2 /2 c1 u t c1 I1 C∗ −u t α c2 β1 α c2 −β3 1−t c2 u t c2 I3 β2 β2 c3 α3 c3 α3 − c3 −u t c3 u t f t, u t , u t , u t , β3 f t, u t , u t , u t β3 f t, u t , u t , u t t ∈ 0, , 3.4 where C∗ α −β1 α2 −β2 β3 −α3 −β1 −β2 c2 c3 I1 I3 c1 1 −u s λ β3 Hence, integrating 3.4 from t to 1, we obtain f s, s − t s2 , − s, −1 ds ≤ C∗ −u t , t ∈ 0, 3.5 t ∈ 0, , 3.6 Since −u t increases on 0, , we get β3 −u t λ f s, s − t s2 , − s, −1 ds ≤ C∗ −u t , that is, λ f s, s − t −u t s2 , − s, −1 ds ≤ C∗ −u t β3 , t ∈ 0, 3.7 Notice that β3 < 1, integrating 3.7 from to 1, we have 1 t s λ f s, s − s2 , − s, −1 ds dt ≤ C∗ − β3 −1 −u 1−β3 3.8 s, s − s2 , − s, −1 dt ds ≤ C∗ − β3 −1 −u 1−β3 3.9 That is, λ f 0 Thus, sf s, s − s2 , − s, −1 ds < ∞ 3.10 14 Boundary Value Problems By an argument similar to the one used in deriving 3.5 , we can obtain λ α3 −u s f s, s − t β −α1 β2 −α2 α3 −β3 −α1 −α2 c2 c3 I2 I4 c1 where C∗ s, s − f λ t s2 , − s, −1 ds ≥ C∗ −u t , t ∈ 0, , 3.11 So, s2 , − s, −1 ds ≥ C∗ u −α3 t ∈ 0, −u t , 3.12 Integrating 3.12 from to 1, we have 1 t s λ f s, s − s2 , − s, −1 ds dt ≥ C∗ u −α3 −u 3.13 s, s − s2 , − s, −1 dt ds ≥ C∗ u −α3 −u 3.14 That is, λ f 0 So, sf s, s − s2 , − s, −1 ds > 3.15 This and 3.10 imply that 2.17 holds Now assume that 2.17 holds, we will show that BVP 1.1 has at least one positive solution for any λ > By 2.17 , there exists a sufficient small δ > such that 1−δ sf s, s − δ s2 , − s, −1 ds > 3.16 For any fixed λ > 0, first of all, we prove Tλ u where < r ≤ min{N1 , N2 , N3 , λδ1 Let u ∈ ∂Pr , then r t− t2 ≤u t ≤r t− t2 2 ≥ u 2, β3 −3 β1 β2 ≤ N1 t − t2 ∀u ∈ ∂Pr , 1−δ δ , δr ≤ rt ≤ −u t ≤ r ≤ N3 , sf s, s − s2 /2, − s, −1 ds 3.17 1/ 1− β1 β2 β3 } r − t ≤ u t ≤ r − t ≤ N2 − t , ∀t ∈ δ, − δ 3.18 Boundary Value Problems 15 From Lemma 2.1, 3.18 , and H , we get Tλ u sf s, δ 1−δ ≥ δλ s δ ≥δ ≥ δ1 β1 r G2 t, s f s, u s , u s , u s ds t∈ δ,1−δ 1−δ ≥ δλ ≥ λ max Tλ u us s − s2 /2 β1 us s − s2 /2 r β2 δr β3 s2 s− β2 u s 1−s 1−δ λ , u s − s , −1 −u s 1−s −u s s, s − sf δ β3 −3 β1 β2 r β1 β2 β3 1−δ λ ≥r u 2, f s2 , − s, −1 ds s, s − 3.19 s2 , − s, −1 ds sf s, s − δ β3 ds s2 , − s, −1 ds u ∈ ∂Pr Thus, 3.17 holds Next, we claim that Tλ u α −β ≤ u 2, ∀u ∈ ∂PR , 3.20 −1 −1 where R ≥ max{8N1 , 4N2 , λN3 3 sf s, s − s2 /2, − s, −1 ds Let c N3 /R, then for u ∈ ∂PR , we get −1 N1 t − t2 ≤ R t2 t− ≤u t ≤ R t− −cu t ≤ c u t2 , −1 N2 − t ≤ cR N3 , 1/ 1− β1 β2 β3 } R 1−t ≤u t ≤R 1−t , ∀t ∈ 0, 3.21 Therefore, by Lemma 2.1 and H , it follows that Tλ u t λ G2 t, s f s, u s , u s , u s ds ≤λ sf s, us s − s2 /2 s− s2 , u s − s , −1 1−s c −cu s ds 16 Boundary Value Problems ≤λ u s s − s2 /2 ≤ Rβ1 Rβ1 β2 α3 −β3 N3 R β2 β3 β1 β2 u s 1−s Rα3 λ sf s, s − α3 −β3 N3 λ sf s, s − ≤R β3 c α3 −cu s f s, s − s2 , − s, −1 ds s2 , − s, −1 ds s2 , − s, −1 ds u ∈ ∂PR u 2, 3.22 This implies that 3.20 holds By Lemmas 1.1 and 2.4, 3.17 , and 3.20 , we obtain that Tλ has a fixed point in PR \ Pr Therefore, BVP 1.1 has a positive solution in PR \ Pr for any λ > Unbounded Connected Branch of Positive Solutions In this section, we study the global continua results under the hypotheses H and 2.17 Let L { λ, u ∈ 0, ∞ × P \ {θ} : λ, u satisfies BVP 1.1 }, 4.1 then, by Theorem 3.1, L ∩ {λ} × P / ∅ for any λ > Theorem 4.1 Suppose (H) and 2.17 hold, then the closure L of positive solution set possesses an unbounded connected branch C which comes from 0, θ such that i for any λ > 0, C ∩ {λ} × P / ∅, and ii lim λ,uλ ∈C,λ → uλ 0, lim λ,uλ ∈C,λ → ∞ uλ ∞ Proof We now prove our conclusion by the following several steps First, we prove that for arbitrarily given < λ1 < λ2 < ∞, L∩ λ1 , λ2 ×P is bounded In fact, let R ⎧ ⎨ −1 −1 max 8N1 , 4N2 , ⎩ then for u ∈ P \ {θ} and u α −β3 λ2 N3 −1 N1 t − s2 sf s, s − , − s, −1 ds 1/ 1− β1 β2 β3 ⎫ ⎬ ⎭ , 4.2 ≥ R, we get t2 −1 N2 − t ≤ ≤ R t− t2 ≤u t ≤ t− R − t ≤ u t ≤ − t u 2, t2 u 2, ∀t ∈ 0, 4.3 Boundary Value Problems 17 Therefore, by Lemma 2.1 and H , it follows that Tλ u ≤ Tλ2 u ≤ λ2 ≤ λ2 sf s, u s , u s , u s ds sf s, us s − s2 /2 β1 β2 N3 u ≤ λ2 u λ2 u β1 β2 β3 < u 2, N3 s− α3 −β3 u α3 −β3 s2 u N3 u s −u s − s , −1 1−s N3 u , α3 sf s, s − sf s, s − ds s2 , − s, −1 ds 4.4 s2 , − s, −1 ds ∀λ ∈ λ1 , λ2 Let r ⎧ ⎨ N1 , N2 , N3 , ⎩ λ1 δ 1−δ β3 −3 β1 β2 sf δ s2 s, s − , − s, −1 ds 1/ 1− β1 β2 β3 ⎫ ⎬ , ⎭ 4.5 where δ is given by 3.16 Then for u ∈ P \ {θ} and u u t− t2 ≤u t ≤ r t − δ u t2 ≤ N1 t − ≤t u t2 ; u ≤ r, we get − t ≤ u t ≤ r − t ≤ N2 − t , ≤ −u t ≤ r ≤ N3 , ∀t ∈ δ, − δ 4.6 Therefore, by Lemma 2.1 and H , it follows that Tλ u ≥ Tλ1 u ≥ λ1 max t∈ δ,1−δ ≥ δλ1 1−δ sf s, δ ≥ δλ1 1−δ s δ G2 t, s f s, u s , u s , u s ds us s − s2 /2 us s − s2 /2 β1 s− s2 u s 1−s , β2 u s − s , −1 −u s 1−s −u s β3 f s, s − ds s2 , − s, −1 ds 18 Boundary Value Problems ≥δ ≥ δ1 β1 u β3 −3 β1 β2 u β2 δ u β3 1−δ β1 β2 β3 λ1 u 1−δ λ1 sf s, s − δ sf s, s − δ s2 , − s, −1 ds s2 , − s, −1 ds u ∈ ∂Pr > u 2, 4.7 Therefore, u Tλ u has no positive solution in λ1 , λ2 × P \ PR ∪ consequence, L ∩ λ1 , λ2 × P is bounded By the complete continuity of Tλ , L ∩ λ1 , λ2 × P is compact Second, we choose sequences {an }∞ and {bn }∞ satisfy n n λ1 , λ2 × Pr As a < · · · < an < · · · < a1 < b1 < · · · < bn < · · ·, lim an n→ ∞ 0, lim bn n→ ∞ 4.8 ∞ We are to prove that for any positive integer n, there exists a connected branch Cn of L satisfying Cn ∩ {an } × P / ∅, Cn ∩ {bn } × P / ∅ 4.9 Let n be fixed, suppose that for any bn , u ∈ L ∩ {bn } × P , the connected branch Cu of ∅ Since Cu is compact, L ∩ an , bn × P , passing through bn , u , leads to Cu ∩ {an } × P there exists a bounded open subset Ω1 of an , bn × P such that Cu ⊂ Ω1 , Ω1 ∩ {an } × P ∅, ∅, where Ω1 and later ∂Ω1 denote the closure and boundary of Ω1 and Ω1 ∩ an , bn × {θ} with respect to an , bn × P If L ∩ ∂Ω1 / ∅, then Cu and L ∩ ∂Ω1 are two disjoint closed subsets of L ∩ Ω1 Since L ∩ Ω1 is a compact metric space, there are two disjoint compact subsets M1 and M2 of L ∩ Ω1 such that L ∩ Ω1 M1 ∪ M2 , Cu ⊂ M1 , and L ∩ ∂Ω1 ⊂ M2 Evidently, γ : dist M1 , M2 > Denoting by V the γ /3-neighborhood of M1 and letting Ωu Ω1 ∩ V , then it follows that Cu ⊂ Ωu , Ωu ∩ {an } × P ∪ an , bn × {θ} ∅, L ∩ ∂Ωu ∅ 4.10 If L ∩ ∂Ω1 ∅, then taking Ωu Ω1 It is obvious that in {bn } × P , the family of {Ωu ∩ {bn } × P : bn , u ∈ L} makes up an open covering of L ∩ {bn } × P Since L ∩ {bn } × P is a compact set, there exists a finite k subfamily {Ωui ∩ {bn } × P : bn , ui ∈ L}k which also covers L ∩ {bn } × P Let Ω i i Ωui , then L ∩ {bn } × P ⊂ Ω, Ω∩ {an } × P ∪ an , bn × {θ} ∅, L ∩ ∂Ω ∅ 4.11 Boundary Value Problems 19 Hence, by the homotopy invariance of the fixed point index, we obtain i Tbn , Ω ∩ {bn } × P , P i Tan , Ω ∩ {an } × P , P 4.12 By the first step of this proof, the construction of Ω, 4.4 , and 4.7 , it follows easily that there exist < rn < Rn such that ∩ {bn } × Prn Ω ∩ {bn } × P ⊂ {bn } × PRn , Ω ∩ {bn } × P ∅, 4.13 i Tbn , Prn , P 0, 4.14 i Tbn , PRn , P 4.15 However, by the excision property and additivity of the fixed point index, we have from 4.12 and 4.14 that i Tbn , PRn , P 0, which contradicts 4.15 Hence, there exists some bn , u ∈ L ∩ {bn } × P such that the connected branch Cu of L ∩ an , bn × P containing bn , u satisfies that Cu ∩ {an } × P / ∅ Let Cn be the connected branch of L including Cu , then this Cn satisfies 4.9 By Lemma 1.2, there exists a connected branch C∗ of lim supn → ∞ Cn such that C∗ ∩ {λ} × P / ∅ for any λ > Noticing lim supn → ∞ Cn ⊂ L, we have C∗ ⊂ L Let C be the connected branch of L including C∗ , then C ∩ {λ} × P / ∅ for any λ > Similar to 4.4 and 4.7 , for any λ > 0, λ, uλ ∈ C, we have, by H , 4.2 , 4.3 , 4.5 , 4.6 , and Lemma 2.1, uλ Tλ uλ ≤λ α3 −β3 N3 uλ β1 β2 ≤ λ uλ sf s, uλ s , uλ s , uλ s ds α3 uλ sf s, s − s2 , − s, −1 ds 4.16 β1 β2 β3 λ uλ ≤ λRβ1 β2 β3 α3 −β3 N3 N3 α3 −β3 s2 sf s, s − , − s, −1 ds sf s, s − uλ Tλ uλ ≥ λδ ≥ λδ1 ≥ λδ1 uλ t∈ δ,1−δ β1 β3 −3 β1 β2 β3 −3 β1 β2 ≥ λ max uλ G2 t, s f s, uλ s , uλ s , uλ s ds β2 δ uλ β β β uλ 2 1−δ r β1 s2 , − s, −1 ds, β2 β3 δ 1−δ δ 1−δ β3 δ sf s, s − s2 , − s, −1 ds s2 sf s, s − , − s, −1 ds sf s, s − s2 , − s, −1 ds, 4.17 20 Boundary Value Problems where δ is given by 3.16 Let λ → in 4.16 and λ → lim λ,uλ ∈C,λ → uλ 0, lim ∞ in 4.17 , we have λ,uλ ∈C,λ → ∞ uλ ∞ 4.18 Therefore, Theorem 4.1 holds and the proof is complete Acknowledgments This work is carried out while the author is visiting the University of New England The author thanks Professor Yihong Du for his valuable advices and the Department of Mathematics for providing research facilities The author also thanks the anonymous referees for their careful reading of the first 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