We investigate in this paper the following second-order multipoint boundary value problem: -(Lφ)(t)=λf(t,φ(t)), 0≤t≤1, φ′0=0, φ1=∑i=1m-2βiφηi. Under some conditions, we obtain global structure of positive solution set of this boundary value problem and the behavior of positive solutions with respect to parameter λ by using global bifurcation method. We also obtain the infinite interval of parameter λ about the existence of positive solution.

National Natural Science Foundation of China115712071. Introduction

In this paper, we shall study the following second-order multipoint boundary value problem: (1)-Lφt=λft,φt,0≤t≤1,φ′0=0,φ1=∑i=1m-2βiφηi,where (Lφ)(t)=ptφ′t′+q(t)φ(t), ηi∈(0,1),0<η1<η2<⋯<ηm-2<1, βi∈[0+∞), and λ is a positive parameter.

The multipoint boundary value problems for ordinary differential equations play an important role in physics and applied mathematics, and so on. The existence and multiplicity of nontrivial solutions for multipoint boundary value problems have been extensively considered (including positive solutions, negative solutions, or sign-changing solutions) by using the fixed point theorem with lattice, fixed point index theory, coincidence degree theory, Leray-Schauder continuation theorems, upper and lower solution method, and so on (see [1–25] and references therein). On the other hand, some scholars have studied the global structure of nontrivial solutions for second-order multipoint boundary value problems (see [26–32] and references therein).

There are few papers about the global structure of nontrivial solutions for the boundary value problem (1). Motivated by [1, 26–32], we shall investigate the global structure of positive solutions of the boundary value problem (1). In [1], the authors only have studied the existence of positive solutions, but in this paper, we prove that the set of nontrivial positive solutions of the boundary value problem (1) possesses an unbounded connected component.

This paper is arranged as follows. In Section 2, some notation and lemmas are presented. In Section 3, we prove the main results of the boundary value problem (1). Finally, in Section 4, two examples are given to illustrate the main results obtained in Section 3.

2. Preliminaries

Let E be a Banach space, P⊂E be a cone, and A:P→P be a completely continuous operator.

Definition 1 (see [<xref ref-type="bibr" rid="B33">33</xref>]).

Let Ω⊂P be an open set, A:Ω→P, and λ0∈(0,+∞). If, for any ϵ>0, there exists the solution (λ,x)∈R+×Ω of the equation x=λAx, satisfying (2)λ-λ0<ϵ,0<x<ϵ, then λ0 is called a bifurcation point of the cone operator A.

Definition 2 (see [<xref ref-type="bibr" rid="B33">33</xref>]).

Let Ω⊂P be an open set, A:Ω→P, and λ0∈(0,+∞). If, for any ϵ>0, there exists the solution (λ,x)∈R+×Ω of the equation x=λAx, satisfying (3)λ-λ0<ϵ,x>1ϵ,then λ0 is called an asymptotic bifurcation point of the cone operator A.

Definition 3 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

Let T:E→E be a linear operator and T map P into P. The linear operator T is u0-positive if there exists u0∈P∖{θ} such that, for any x∈P∖{θ}, we can find an integer n and real numbers α0>0,β0>0 such that α0u0≤Tnx≤β0u0.

Lemma 4 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

Let Ω(P) be an open set of P. Assume that the operator A has no fixed points on ∂Ω(P). If there exists a linear operator B and u∗∈P∖{θ} such that

Ax≥Bx,x∈∂Ω(P);

for some n,Bnu∗≥u∗,

then i(A,Ω(P),P)=0.Lemma 5 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

Let A:P→P be completely continuous and T be a completely continuous u0-bounded linear operator. If, for any x∈P, Ax≥Tx, λAx=x, then λ≤1/r(T), where 1/r(T) is unique eigenvalue of T corresponding to positive eigenfunction.

Lemma 6 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

Let M be a compact metric space and A and B be disjoint, compact subsets of M. If there does not exist connected subset C of M such that C∩A≠∅ and C∩B≠∅, then there exist disjoint compact subsets MA and MB such that A⊂MA,B⊂MB and M=MA∪MB.

3. Main Results

Let E=C[0,1] with the norm φ(t)=maxt∈[0,1]φt; then E is a Banach space. Let P={φ∈E∣φ(t)≥0,t∈[0,1]}. Obviously, P is a normal cone of E.

In this paper, we always assume that

p(t)∈C1[0,1],p(t)>0,q(t)∈C[0,1],q(t)≤0.

Lemma 7 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Suppose that (H1) holds. Let Φ1 and Φ2 be the solutions of (4)Lφt=0,0<t<1,φ′0=0,φ1=1,(5)Lφt=0,0<t<1,φ0=1,φ1=0,respectively. Then

Φ1 is increasing on [0,1] and Φ1>0,t∈[0,1];

Φ2 is decreasing on [0,1] and Φ2>0,t∈[0,1).

Let (6)Gt,s=1ρΦ1tΦ2s,0≤t≤s≤1,1ρΦ1sΦ2t,0≤s≤t≤1,where ρ=-Φ1(0)Φ2′(0)>0 by [1]. Let (7)Kt,s=Gt,s+D-1Φ1t∑i=1m-2βiGηi,s,0≤t,s≤1,where D=1-∑i=1m-2βiΦ1(ηi).

Define the operators A, B, and F: (8)Aφt=∫01Kt,sp~sfs,φsds,(9)Bφt=∫01Kt,sp~sφsds,(10)Fφt=ft,φt, where p~(s)=1/psexp(∫0sp′(x)/p(x)dx) and K(t,s) is defined by (7).

Obviously, A=BF. It is easy to know that the solutions of the boundary value problem (1) are equivalent to the solutions of the equation (11)φ=λAφ.

Let L={(λ,φ)∈(0,+∞)×P∣φ=λAφ,φ≠θ}¯ be the closure of nontrivial positive solution set of (11). Then L is also the closure of the nontrivial positive solution set of the boundary value problem (1).

We give the following assumptions:

∑i=1m-2βiΦ1(ηi)<1, where Φ1(t) is the solution of (4).

f:[0,1]×R+→R+ is continuous, f(t,0)=0,uniformlyont∈[0,1].

liminfu→0+(f(t,u)/u)≥α>0,uniformlyont∈[0,1].

limsupu→+∞(f(t,u)/u)=0,uniformlyont∈[0,1].

Lemma 8 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Suppose that (H1)–(H3) are satisfied. Then, for the operator B defined by (9), the spectral radius r(B)≠0 and B has a positive eigenfunction corresponding to its first eigenvalue λ1=(r(B))-1.

Theorem 9.

Suppose that (H1)–(H4) are satisfied. Then

the operator A defined by (8) has at least a bifurcation point λ∗∈[0,λ1/α] corresponding to positive solution; the operator A has no bifurcation points in (λ1/α,+∞) corresponding to positive solution, where λ1 is defined by Lemma 8;

L possesses an unbounded connected component C⊂(0,+∞)×P passing through (λ∗,θ), and C∩((λ1/α,+∞)×{θ})=∅, where λ1 is defined by Lemma 8.

Proof.

By (H1)–(H3), it is easy to know that A:P→P is completely continuous and B:P→P is completely continuous; and Aθ=θ. By Lemma 8, we have r(B)=1/λ1.

By (H3), for any ϵ>0, there exists rϵ>0 such that (12)ft,uu≥α-ϵ,∀t∈0,1,0≤u≤rϵ,that is, (13)ft,u≥α-ϵu,∀t∈0,1,0≤u≤rϵ.

Let Nrϵ(P)={φ∈P∣φ<rϵ}. From (8) and (13), for any φ∈N¯rϵ(P), we have (14)Aφt≥α-ϵ∫01Kt,sp~sφsds=α-ϵBφt=Tφt,where T=(α-ϵ)B. Clearly, T:P→P is completely continuous and r(T)=(α-ϵ)r(B)=(α-ϵ)/λ1.

By Lemma 7 and (6) and (7), it follows that (15)Φ1tD-1∑i=1m-2βiGηi,s≤Kt,s≤1ρΦ2s+D-1∑i=1m-2βiGηi,sΦ1t,∀t,s∈0,1.

For any φ∈P, by (14) and (15), we have (16)Tφt≥α-ϵΦ1t∫01D-1∑i=1m-2βiGηi,sp~sφsds,Tφt≤α-ϵΦ1t∫011ρΦ2s+D-1∑i=1m-2βiGηi,sp~sφsds.

Let u0=Φ1(t). It follows from (16) that T is a u0-bounded operator by Definition 3. By Krein-Rutman theorem, there exists φ∗∈P∖{θ} such that (17)Tφ∗=rTφ∗.

By (14) and (17), we have (18)λAφ≥λTφ,∀φ∈∂NrϵP,λ≥λ1α-ϵ,λTφ∗=λrTφ∗≥φ∗,∀λ≥λ1α-ϵ.

So, by (18) and Lemma 4, we have (19)iλA,NrϵP,P=0,∀λ≥λ1α-ϵ.

In the following, we prove that the operator A has at least one bifurcation point on [0,λ1/(α-ϵ)] and has no bifurcation points on (λ1/(α-ϵ),+∞).

We shall prove that, for any ϵ¯∈(0,rϵ), there must exist λϵ¯∈[0,λ1/(α-ϵ)] and φϵ¯∈∂Nϵ¯(P) such that (20)φϵ¯=λϵ¯Aφϵ¯,where Nϵ¯(P)={φ∈P∣φ<ϵ¯}.

Without loss of generality, we may assume that the equation (λ1/α-ϵ)Aφ=φ has no solutions on ∂Nϵ¯(P). By (19), we get (21)iλ1α-ϵA,Nϵ¯P,P=0.

Obviously, (22)i0,Nϵ¯P,P=1.

Set (23)Ht,φ=φ-tλ1α-ϵAφ,t∈0,1.

By (21) and (22) and the homotopy invariance of fixed point index, there exists t∗∈[0,1] such that H(t∗,φ)=θ has a solution φϵ¯∗∈∂Nϵ¯(P). Namely, φϵ¯∗=λϵ¯∗Aφϵ¯∗, where λϵ¯∗=λ1t∗/(α-ϵ)∈[0,λ1/(α-ϵ)].

Choose 1/n<rϵ. Then there exist λn∈[0,λ1/(α-ϵ)] and φn∈P with φn=1/n such that φn=λnAφn. And φn≠θ,φn→θ(n→∞). Assume that λn→λ∗(n→∞). Then λ∗∈[0,λ1/(α-ϵ)] is a bifurcation point of the cone operator A.

By (14) and Lemma 5, for any 0<r<rϵ, the equation φ=λAφ has no solutions in (λ1/(α-ϵ),+∞)×∂Nr(P), where Nr(P)={φ∈P∣φ<r}. Hence, A has no bifurcation points in (λ1/(α-ϵ),+∞) corresponding to positive solution, and LP∩((λ1/(α-ϵ),+∞)×{θ})=∅.

Let G={(λ,θ)∣λ∈[0,λ1/(α-ϵ)],λis a bifurcation point of the cone operator A}. By the above proof, we know that G≠∅. If, for any λ∈[0,λ1/(α-ϵ)], the connected component Cλ of LP is bounded, which passes through (λ,θ), then Cλ is a compact set.

Let Qλ be an open neighborhood of Cλ in [0,+∞)×P. If ∂Qλ∩LP≠∅, then Z=Q¯λ∩Lp is a compact metric space, and ∂Qλ∩LP and Cλ are disjoint, compact subsets of Z. Since Cλ has the property of maximal connectivity, there exists no connected subset C~ of Z such that C~∩Cλ=∅ and C~∩(∂Qλ∩LP)=∅. By Lemma 6, we know that there exist two compact subsets Z1,Z2 of Z such that (24)Z=Z1∪Z2,Z1∩Z2=∅,Cλ⊂Z1,∂Qλ∩LP⊂Z2.

Obviously, the distance ρ=d(Z1,Z2)>0. Let Qλ′={u∈[0,+∞)×P∣d(u,Z1)<ρ/3}. Then Qλ′ is an open neighborhood of Z1. Let Qλ′′=Qλ∩Qλ′. Then ∂Qλ′′∩LP=∅. Let (25)Qλ∗=Qλ,if∂Qλ∩LP=∅,Qλ′′,if∂Qλ∩LP≠∅.

Clearly, Qλ∗ is a bounded open set of [0,+∞)×P, and ∂Qλ∗∩LP=∅. Hence {Qλ∗∣(λ,θ)∈G} is an open covering of G. Since G is compact, there exist (λi,θ)∈G(i=1,2,…,n) such that {Qλi∗∣i=1,2,…,n} is also an open covering of G. Let Q∗=⋃i=1nQλi∗. Then Q∗ is a bounded open set of [0,+∞)×P, and G⊂Q∗,∂Q∗∩LP=∅.

Take sufficiently large λ~>λ1/(α-ϵ) such that Q¯∗⊂[0,λ~]×P. For 0<r<rϵ, let Ur=[0,λ~]×Nr(P), where Nr(P)={φ∈P∣φ<r}. Evidently, Ur is an open set of [0,λ~]×P, and ∂Ur=[0,λ~]×∂Nr(P). And λAφ=φ has no nontrivial solutions on ∂Ur∖Q∗ when r is sufficiently small.

Let X=Q∗∪Ur. Then ∂X⊂∂Q∗∪∂(Ur∖Q∗). Since [0,λ~]×{θ}⊂X and ∂Q∗∩LP=∅, we know that φ=λAφ has no solutions on ∂X. By the general homotopy invariance of topological degree, we get (26)iλ~A,Xλ~,P=i0A,X0,P=1.

Since Q∗⊂[0,λ~]×P, Q∗(λ~)=({λ~}×P)∩Q∗=∅, so (27)Xλ~=λ~×P∩Ur=Urλ~=λ~×P∩NrP=Nrλ~,P. Therefore, by (20), we have (28)iλ~A,Xλ~,P=iλ~A,Nrλ~,P,P=iλ~A,NrP,P=0,which contradicts (26).

Hence, LP possesses an unbounded connected component Cλ⊂(0,+∞)×P passing through (λ,θ).

By the above proof and the arbitrariness of ϵ, we obtain that (i) the cone operator A has at least a bifurcation point λ∗∈[0,λ1/α] (the cone operator A has no bifurcation point in (λ1/α,+∞)) and (ii) LP possesses an unbounded connected component C⊂(0,+∞)×P passing through (λ∗,θ), and C∩((λ1/α,+∞)×{θ})=∅.

Theorem 10.

Suppose that (H1)–(H3) and (H5) are satisfied. Then the operator A has no asymptotic bifurcation points in [0,+∞).

Proof.

For any λ0∈[0,+∞), there exists sufficiently small ϵ0>0 such that (29)λ0ϵ0<λ1,where λ1 is defined by Lemma 8.

By (H5), for the above ϵ0>0, there exists R>0 such that (30)ft,uu≤ϵ0,∀t∈0,1,u≥R,that is, (31)ft,u≤uϵ0,∀t∈0,1,u≥R.Set M=maxt∈[0,1],0≤u≤Rf(t,u); then (32)ft,u≤ϵ0u+M,∀t∈0,1,u≥0.

Let G(λ0)={φ∈P∣φ=λAφ,0≤λ≤λ0}. For any φ¯∈G(λ0), there exists λ¯∈[0,λ0] such that φ¯=λ¯Aφ¯. By (32), we have (33)φ¯t=λ¯Aφ¯t=λ¯∫01Kt,sp~sfs,φ¯sds≤λ0ϵ0∫01Kt,sp~sφ¯sds+M∫01Kt,sp~sds=T¯φ¯t+v0,where T¯=λ0ϵ0B,v0=M∫01K(t,s)p~(s)ds. It follows from (33) that r(T¯)=λ0ϵ0r(B)<1.

By (33), we get that φ¯(t)≤(I-T¯)-1v0. So φ¯≤(I-T¯)-1v0. Therefore, G(λ0) is bounded. By the arbitrariness of λ0, the operator A has no asymptotic bifurcation point in (0,+∞).

By Theorems 9 and 10, we have the following theorem.

Theorem 11.

Suppose that (H1)–(H5) are satisfied. Then, for any λ∈(λ1/α,+∞), the boundary value problem (1) has at least one positive solution.

Furthermore, we take α=+∞ in (H4), that is, the following condition (H4′).

liminfu→0+(ft,u/u)=+∞,uniformlyont∈[0,1].

Theorem 12.

Suppose that (H1)–(H3)(H5) and (H4′) are satisfied. Then

the operator A has no asymptotic bifurcation points in [0,+∞);

LP possesses an unbounded connected component C⊂(0,+∞)×P passing through (0,θ), and C∩((0,+∞)×{θ})=∅.

Proof.

Since (H1)–(H3) and (H5) are satisfied, it follows from Theorem 10 that (i) holds.

By (H4′), for sufficiently large M>0, there exists rM>0 such that (34)ft,uu≥M,∀t∈0,1,0≤u≤rM, that is,(35)ft,u≥Mu,∀t∈0,1,0≤u≤rM.

Let N~(P)={φ∈P∣φ<rM}. From (7) and (35), for any φ∈N~(P), we have (36)Aφt≥M∫01Kt,sp~sφsds=MBφt=T~φt, where T~=MB. Clearly, T~:P→P is completely continuous and r(T~)=Mr(B)=M/λ1.

Similar to the proof of Theorem 9, we obtain that the operator A has a bifurcation point λ∗∈[0,λ1/M] corresponding to positive solution and LP possesses an unbounded connected component C⊂(0,+∞)×P passing through (λ∗,θ), and C∩((λ1/M,+∞)×{θ})=∅. Since M can take sufficiently large value, we know that (i) and (ii) hold. The proof is completed.

It follows from Theorem 12 that we have the following theorem.

Theorem 13.

Suppose that (H1)–(H3)(H4′) and (H5) are satisfied. Then, for any λ∈(0,+∞), the boundary value problem (1) has at least one positive solution.

4. Applications

In this section, two examples are given to illustrate our main results.

Example 14.

Consider the following boundary value problem: (37)-φ′′t=λft,φt,0≤t≤1,φ′0=0,φ1=12φ12,where(38)ft,u=2u+u2t,t∈0,1,u∈0,10,100t+10u+10,t∈0,1,u∈10,+∞.

By simple calculations, λ1≈6.9497. The nonlinear term f satisfies the conditions of Theorem 11. Thus, for any λ>3.4749, the boundary value problem (37) has at least one positive solution by Theorem 11.

Example 15.

Consider the following boundary value problem: (39)-φ′′t=λft,φt,0≤t≤1,φ′0=0,φ1=12φ12,where(40)ft,u=tu+u1/3,t∈0,1,u∈0,1,t+u,t∈0,1,u∈1,+∞.

The nonlinear term f satisfies the conditions of Theorem 13. Thus, for any λ>0, the boundary value problem (39) has at least one positive solution by Theorem 13.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The project is supported by the National Science Foundation of China (11571207).

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