Global Behavior of the Components for the Second Order m-Point Boundary Value Problems

Volume 2008, Article ID 254593,10pages doi:10.1155/2008/254593

Research Article

Global Behavior of the Components for the Second Order m-Point Boundary Value Problems

Yulian An^{1, 2}and Ruyun Ma¹

1Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

2Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China

Correspondence should be addressed to Yulian An,an [email protected] Received 9 October 2007; Accepted 16 December 2007

Recommended by Kanishka Perera

We consider the nonlinear eigenvalue problemsu rfu 0, 0 < t < 1,u0 0,u1 _m−2

i1 αiuη_i, wherem ≥ 3,η_i ∈0,1, andαi > 0 fori 1, . . . , m−2, with_m−2

i1αi <1;r ∈R;

f ∈ C¹R,R. There exist two constantss2 < 0 < s1such thatfs1 fs2 f0 0 and f0:lim_u→0fu/u∈0,∞,f_∞:lim_|u|→∞fu/u∈0,∞. Using the global bifurcation tech- niques, we study the global behavior of the components of nodal solutions of the above problems.

Copyrightq2008 Y. An and R. Ma. 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.

1. Introduction

In1, Ma and Thompson were concerned with determining values of real parameterr, for which there exist nodal solutions of the boundary value problems:

uratfu 0, 0< t <1,

u0 u1 0, 1.1

whereaandfsatisfy the following assumptions:

H1 f ∈CR,Rwithsfs>0 fors /0;

H2 there existf0, f∞∈0,∞such that f0lim

|s|→0

fs

s , f∞ lim

|s|→∞

fs

s ; 1.2

H3 a:0,1→0,∞is continuous andat/≡0 on any subinterval of0,1.

Using Rabinowitz global bifurcation theorem, Ma and Thompson established the following theorem.

Theorem 1.1. Let (H1), (H2), and (H3) hold. Assume that for somek∈N, either λk

f∞ < r < λk

f0 1.3

or

λ_k

f₀ < r < λ_k

f∞. 1.4

Then1.1have two solutionsu_kandu⁻_ksuch thatu_k has exactlyk−1 zeros in0,1and is positive near 0, andu⁻_khas exactlyk−1 zeros in0,1and is negative near 0. In1.3and1.4,λ_kis thekth eigenvalue of

ϕλatϕ0, 0< t <1, ϕ0 ϕ1 0. 1.5

Recently, Ma2extended this result and studied the global behavior of the components of nodal solutions of1.1under the following conditions:

H1 f ∈CR,Rand there exist two constantss2<0< s1, such thatfs1 fs2 f0 0 andsfs>0 fors∈R\ {0, s1, s2};

H4 fsatisfies Lipschitz condition ins2, s1.

Using Rabinowitz global bifurcation theorem, Ma established the following theorem.

Theorem 1.2. LetH1, (H2), (H3), and (H4) hold. Assume that for somek∈N, λk

f∞ < λk

f0. 1.6

Then

iifr∈λk/f∞, λk/f0, then1.1have at least two solutionsu^±_k,∞, such thatu_k,∞has exactly k−1 zeros in0,1and is positive near 0, andu⁻_k,∞has exactlyk−1 zeros in0,1and is negative near 0,

iiifr∈λk/f0,∞, then1.1have at least four solutionsu^±_k,∞andu^±_k,0, such thatu_k,∞(resp., u_k,0) has exactlyk−1 zeros in0,1and is positive near 0;u⁻_k,∞(resp.,u⁻_k,0) has exactlyk−1 zeros in 0,1and is negative near 0.

Remark 1.3. LetH1,H2,H3, andH4hold. Assume that for somek∈N,λ_k/f₀< λ_k/f_∞. Similar results toTheorem 1.2have also been obtained.

Making a comparison between the above two theorems, we see that asfhas two zeros s1, s2 : s2 < 0 < s1, the bifurcation structure of the nodal solutions of 1.1 becomes more complicated: two new nodal solutions are obtained whenr >max{λk/f0, λk/f∞}.

In 3, Ma and O’Regan established some existence results which are similar to Theorem 1.1of the nodal solutions of them-point boundary value problems

ufu 0, 0< t <1,

u0 0, u1 ^m−2

i1

αiu

ηi 1.7

under the following condition:

H1 f ∈C¹R,Rwithsfs>0 fors /0.

Remark 1.4. For other results about the existence of nodal solution of multipoint boundary value problems, we can see4–7.

Of course an interesting question is, as form-point boundary value problems, when f possesses zeros inR\ {0}, whether we can obtain some new results which are similar to Theorem 1.2.

We consider the eigenvalue problems

urfu 0, 0< t <1, 1.8

u0 0, u1 ^m−2

i1

αiu ηi

, 1.9

where m ≥ 3,ηi ∈ 0,1,andαi > 0 for i 1, . . . , m−2. Also using the global bifurcation techniques, we study the global behavior of the components of nodal solutions of1.8,1.9 and give a positive answer to the above question. However, whenm-point boundary value condition1.9is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate.

In the following paper, we assume that H0 α_i>0 fori1, . . . , m−2,with 0<_m−2

i1 α_i<1;

H1 f ∈ C¹R,Rand there exist two constantss₂ < 0 < s₁, such thatfs1 fs2

f0 0;

H2 there existf₀, f∞∈0,∞such that f0lim

|s|→0

fs

s , f∞ lim

|s|→∞

fs

s . 1.10

The rest of the paper is organized as follows.Section 2contains preliminary definitions and some eigenvalue results of corresponding linear problems of 1.8, 1.9. In Section 3, we give two Rabinowize-type global bifurcation theorems. Finally, inSection 4, we consider two bifurcation problems related to1.8,1.9, and use the global bifurcation theorems from Section 3to analyze the global behavior of the components of nodal solutions of1.8,1.9.

2. Preliminary definitions and eigenvalues of corresponding linear problems LetY C0,1with the norm

u∞max

t∈0,1ut. 2.1

Let

X

u∈C¹0,1|u0 0, u1 ^m−2

i1

αiu ηi

,

E

u∈C²0,1|u0 0, u1 ^m−2

i1

αiu ηi

2.2

with the norm

uXmax

u∞,u∞}, uEmax{u∞,u∞,u∞

, 2.3

respectively. DefineL:E→Y by setting

Lu:−u, u∈E. 2.4

ThenLhas a bounded inverseL⁻¹:Y →Eand the restriction ofL⁻¹toX, that is,L⁻¹:X →X is a compact and continuous operator, see3,4,8.

LetER×Eunder the product topology. As in9, we add the points{λ,∞|λ∈R}

to our spaceE. For anyC¹functionu, ifux0 0,thenx0is a simple zero ofuifux0/0.

For any integerk ≥ 1 and anyν ∈ {±}, define setsS^ν_k, T_k^ν ⊂ C²0,1consisting of functions u∈C²0,1satisfying the following conditions:

S^ν_k:

iu0 0, νu0>0;

iiuhas only simple zeros in0,1and has exactlyk−1 zeros in0,1;

T_k^ν:

iu0 0,νu0>0,andu1/0;

iiuhas only simple zeros in0,1and has exactlykzeros in0,1;

iiiuhas a zero strictly between each two consecutive zeros ofu.

Remark 2.1. Obviously, ifu∈T_k^ν, thenu∈S^ν_koru∈S^ν_k1. The setsT_k^νare open inEand disjoint.

Remark 2.2. The nodal properties of solutions of nonlinear Sturm-Liouville problems with sep- arated boundary conditions are usually described in terms of sets similar toS^ν_k, see1,2,5,9–

11. However, Rynne4 stated that T_k^ν are more appropriate than S^ν_k when the multipoint boundary condition1.9is considered.

Next, we consider the eigenvalues of the linear problem

Luλu, u∈E. 2.5

We call the set of eigenvalues of2.5the spectrum ofL, and denote it byσL. The following lemmas can be found in3,4,12.

Lemma 2.3. Let (H0) hold. The spectrumσLconsists of a strictly increasing positive sequence of eigenvaluesλk,k1,2, . . . ,with corresponding eigenfunctionsϕkx sin

λkx. In addition, ilimk→∞λk∞;

iiϕk∈T_k, for eachk≥1,andϕ1is strictly positive on0,1.

We can regard the inverse operator L⁻¹ : Y → Eas an operatorL⁻¹ : Y → Y. In this setting, eachλ_k,k1,2, . . . ,is a characteristic value ofL⁻¹, with algebraic multiplicity defined to be dim_∞

j1NI−λ_kL^−1j, whereNdenotes null-space andIis the identity onY.

Lemma 2.4. Let (H0) hold. For eachk ≥ 1, the algebraic multiplicity of the characteristic valueλ_k, k1,2, . . . ,ofL⁻¹:Y →Y is equal to 1.

3. Global bifurcation

Letg∈C¹R,Rand satisfy

g0 g0 0. 3.1

Consider the following bifurcation problem:

Luμugu, μ, u∈R×X. 3.2

Obviously,u≡0 is a trivial solution of3.2for anyμ∈R. About nontrivial solutions of3.2, we have the following.

Lemma 3.1see4, Proposition 4.1. Let (H0) hold. Ifμ, u∈Eis a nontrivial solution of 3.2, thenu∈T_k^νfor somek, ν.

Remark 3.2. From Lemmas2.3and3.1, we can see thatT_k^ν are more effectual than the setS^ν_k when the multipoint boundary condition1.9is considered. In fact, eigenfunctionsϕkx sin

λ_kx,k 1,2, . . . , of2.5do not necessarily belong to S_k. In3,4, there were some special examples to show this problem.

Also, in4, Rynne obtained the following Rabinowitz-type global bifurcation result for 3.2.

Lemma 3.3see4, Theorem 4.2. Let (H0) hold. For eachk ≥1 andν, there exists a continuum C^ν_k⊂Eof solution of3.2with the following properties:

1^o λk,0∈ C^ν_k;

2^o C^ν_k\ {λk,0} ⊂R×T_k^ν; 3^o C^ν_kis unbounded inE.

Now, we consider another bifurcation problem

Luμuhu, μ, u∈R×X, 3.3

where we suppose thath∈C¹R,Rand satisfy

|x|→∞lim hx

x 0. 3.4

TakeΛ⊂Ras an interval such thatΛ∩ {λj|j ∈N}{λk}andMas a neighborhood of λk,∞whose projection onRlies inΛand whose projection onEis bounded away from 0.

Lemma 3.4. Let (H0) and3.4hold. For eachk≥1 andν, there exists a continuumD_k^ν⊂Eof solution of3.3which meetsλk,∞and either

1^o D^ν_k\ Mis bounded inEin which caseD^ν_k\ Mmeets{λ,0|λ∈R}or 2^o D^ν_k\ Mis unbounded inE.

Moreover, if2^ooccurs andD^ν_k\ Mhas a bounded projection onR, thenD^ν_k\ Mmeetsμ,∞, whereμ∈ {λj|j∈N}withμ /λk.

In every case, there exists a neighborhoodO ⊂ M ofλk,∞such thatμ, u ∈ D_k^ν∩ O and μ, u/ λk,∞impliesμ, u∈R×T_k^ν.

Remark 3.5. A continuumD^ν_k ⊂ Eof solution of 3.3meetsλk,∞which means that there exists a sequence{λn, un} ⊂ D^ν_ksuch thatunE→ ∞andλn→λk.

Proof. Obviously,3.3is equivalent to the problem

uμL⁻¹uL⁻¹hu, μ, u∈R×X. 3.5

Note thatL⁻¹ :X →Xis a compact and continuous linear operator. In addition, the mapping u→L⁻¹huis continuous and compact, and satisfiesL⁻¹hu ouXatu∞; moreover, u²_XL⁻¹hu/u²_Xis compactsimilar proofs can be found in9. Hence, the problem3.3is of the form considered in9, and satisfies the general hypotheses imposed in that paper. Then by 9, Theorem 1.6 and Corollary 1.8together with Lemmas 2.3 and2.4 inSection 2, there exists a continuumD_k^ν⊂R×Xof solutions of3.3which meetsλk,∞and either

1^o D^ν_k\ Mis bounded inR×Xin which caseD^ν_k\ Mmeets{λ,0|λ∈R}or 2^o D^ν_k\ Mis unbounded inR×X.

Moreover, if (2^o) occurs andD_k^ν\ Mhas a bounded projection onR, thenD^ν_k\ Mmeets μ,∞whereμ∈ {λj|j∈N}withμ /λk.

In every case, there exists a neighborhoodO ⊂ Mofλk,∞such thatμ, u ∈ D^ν_k∩ O andμ, u/ λk,∞impliesμ, u∈R×T_k^ν.

On the other hand, by3.5and the continuity of the operatorL⁻¹:Y →E, the setD^ν_klies inEand the injectionD^ν_k→Eis continuous. Thus,D^ν_kis also a continuum inEand the above properties hold inE.

Now, we assume that

h0 0. 3.6

Lemma 3.6. Let (H0) and3.6hold. Ifμ, u ∈Eis a nontrivial solution of 3.3, thenu∈T_k^νfor somek, ν.

Proof. The proof ofLemma 3.6is similar to the proof ofLemma 3.14, Proposition 4.1; we omit it.

Remark 3.7. If3.6holds,Lemma 3.6guarantees thatD_k^ν inLemma 3.4is a component of so- lutions of3.3in T_k^ν which meets λk,∞. Otherwise, if there existη1, y1 ∈ D_k^ν ∩T_k^ν and η2, y₂∈ D^ν_k∩T_h^νfor somek /h∈N, then by the connectivity ofD^ν_k, there existsη∗, y_∗∈ D^ν_k such thaty_∗ has a multiple zero point in0,1. However, this contradictsLemma 3.6. Hence, if3.6holds andD^ν_kinLemma 3.4is unbounded inR×E, thenD_k^νhas unbounded projection onR.

4. Statement of main results

We return to the problem1.8,1.9. LetH1, H2hold and letζ, ξ∈C¹R,Rbe such that

fu f0uζu, fu f∞uξu. 4.1

Clearly

ζ0 0, ξ0 0,

|u|→0lim ζu

u ζ0 0, lim

|u|→∞

ξu

u 0. 4.2

Let us consider

Lu−rf₀urζu 4.3

as a bifurcation problem from the trivial solutionu≡0, and

Lu−rf∞urξu 4.4

as a bifurcation problem from infinity. We note that4.3and4.4are the same, and each of them is equivalent to1.8,1.9.

The results ofLemma 3.3for4.3can be stated as follows: for each integerk ≥1,ν ∈ {,−}, there exists a continuumC^ν_k,0of solutions of4.3joiningλk/f0,0to infinity, andC^ν_k,0\ {λk/f0,0} ⊂R×T_k^ν.

The results ofLemma 3.4for4.4can be stated as follows: for each integerk ≥1,ν ∈ {,−}, there exists a continuumD^ν_k,∞of solutions of4.4meetingλk/f∞,∞.

Theorem 4.1. Let (H0),H1, and (H2) hold. Then iforr, u∈ C_k,0∪ C⁻_k,0,

s2< ut< s1, t∈0,1; 4.5

iiforr, u∈ D_k,∞∪ D_k,∞⁻ ,

t∈0,1maxut> s1, or min

t∈0,1ut< s2. 4.6

Proof ofTheorem 4.1. Suppose on the contrary that there existsr, u∈ C_k,0∪ C⁻_k,0∪ D_k,∞∪ D⁻_k,∞

such that either

max

ut|t∈0,1

s₁ 4.7 or

min

ut|t∈0,1

s2. 4.8 Sinceu∈T_k^ν, byRemark 2.1,u∈S^ν_koru∈S^ν_k1. We assumeu∈S^ν_k. Whenu∈S^ν_k1, we can prove all the following results with small modifications. Let

0τ₀< τ₁<· · ·< τk−1<1 4.9 denote the zeros ofu. We divide the proof into two cases.

Case 1max{ut|t∈0,1}s1. In this case, there existsj∈ {0, . . . , k−2}such that max

ut|t∈

τj, τj1

s1 or max{ut|t∈τk−1,1 s1, 0≤ut≤s₁, t∈

τ_j, τ_j1

, or t∈

τ_k−1,1 .

4.10

Sinceu1 _m−2

i1 α_iuηiandH0, we claimu1< s₁.

Lett0∈τj, τj1 ort0∈τk−1,1such thatut0 s1, thenut0 0. Note that f

ut0

fs1 0. 4.11 By the uniqueness of solutions of1.8subject to initial conditions, we see thatut ≡ s₁ on 0,1. This contradicts1.9andH0.

Therefore,

max

ut|t∈0,1

/s1. 4.12

Case 2min{ut|t∈0,1}s2. In this case, the proof is similar toCase 1, we omit it.

Consequently, we obtain the resultsiandii.

Theorem 4.2. Let (H0),H1, and (H2) hold. Assume that for somek∈N, λ_k

f_∞ < λ_k f₀

resp., λ_k f₀ < λ_k

f_∞

. 4.13

Then

iif r ∈ λk/f∞, λ_k/f₀ (resp., r ∈ λk/f₀, λ_k/f∞, then1.8, 1.9have at least two solutionsu^±_k,∞(resp.,u^±_k,0), such thatu_k,∞∈T_kandu⁻_k,∞∈T_k⁻ (resp.,u_k,0∈T_kandu⁻_k,0 ∈ T_k⁻),

iiifr∈λk/f0,∞(resp.,r ∈λk/f∞,∞, then1.8,1.9have at least four solutionsu^±_k,∞

andu^±_k,0, such thatu_k,∞,u_k,0∈T_k, andu⁻_k,∞,u⁻_k,0∈T_k⁻.

Remark 4.3. Making a comparison between results in3and the above theorem, we see that asf has two zeros s₁, s₂ : s₂ < 0 < s₁, the bifurcation structure of the nodal solutions of 1.8,1.9becomes more complicated: the component of the solutions of1.8,1.9from the trivial solution atλk/f0,0and the component of the solutions of1.8,1.9from infinity at λk/f_∞,∞are disjoint; two new nodal solutions are born whenr >max{λk/f₀, λ_k/f_∞}.

Proof ofTheorem 4.2. Since1.8,1.9have a unique solutionu≡0, we get C_k,0∪ C⁻_k,0∪ D_k,∞∪ D_k,∞⁻

⊂

μ, z∈E|μ≥0

. 4.14

TakeΛ⊂Ras an interval such thatΛ∩ {λj/f_∞ |j ∈N} {λk/f_∞}andMas a neigh- borhood ofλk/f∞,∞whose projection onRlies inΛand whose projection onEis bounded away from 0. Then byLemma 3.4,Remark 3.7, andLemma 3.6we have that eachν ∈ {,−},

D_k,∞^ν \ Msatisfies one of the following:

1^o D^ν_k,∞\ Mis bounded inEin which caseD^ν_k,∞\ Mmeets{λ,0|λ∈R};

2^o D^ν_k,∞\ Mis unbounded inEin which case Proj_RD_k,∞\ Mis unbounded.

Obviously,Theorem 4.1iiimplies that1^odoes not occur. SoD_k,∞\ Mis unbounded inE. Thus

Proj_R D_k,∞

⊃ λk

f∞,∞ ,

Proj_R D⁻_k,∞

⊃ λ_k

f_∞,∞

.

4.15

ByTheorem 4.1, for anyr, u∈C_k,0∪ C⁻_k,0, u∞<max

s₁,s₂:s^∗. 4.16

Equations4.16,1.8, and1.9imply that uE<max

rmax

|s|≤s^∗fs, s^∗

, 4.17

which means that the sets{μ, z∈ C_k,0 | μ∈0, d}and{μ, z∈ C⁻_k,0 | μ∈0, d}are bound- ed for any fixedd∈0,∞. This, together with the fact thatC_k,0resp.,C⁻_k,0joinsλk/f₀,0to infinity, yields that

Proj_R C_k,0

⊃ λ_k

f₀,∞

,

Proj_R C⁻_k,0

⊃ λk

f0,∞

.

4.18

Combining4.15with4.18, we conclude the desired results.

Acknowledgments

This paper is supported by the NSFCno. 10671158, the NSF of Gansu Provinceno. 3ZS051- A25-016, NWNU-KJCXGC-03-17, the Spring-sun programno. Z2004-1-62033, SRFDPno.

20060736001, the SRF for ROCS, SEM2006311, and LZJTU-ZXKT-40728.

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