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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 354513, 13 pages
doi:10.5402/2012/354513
Research Article

Bifurcation of Sign-Changing Solutions for π‘š -Point Boundary Value Problems

Yulian An1,2Β and Maoan Han1

1Department of Mathematics, Shanghai Normal University, Shanghai 200235, China
2Department of Mathematics, Shanghai Institute of Technology, Shanghai 201418, China

Received 13 February 2012; Accepted 21 March 2012

Academic Editors: R.Β Avery and A. I.Β Hai

Copyright Β© 2012 Yulian An and Maoan Han. 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.

Abstract

With the help of bifurcation techniques, some multiplicity results and global structure for sign-changing solutions of some π‘š -point boundary value problems are obtained when the nonlinear term is sublinear at 0.

1. Introduction and Main Results

In this paper, we consider the π‘š -point boundary value problems: 𝑒 ξ…ž ξ…ž + πœ† 𝑓 ( 𝑒 ) = 0 , 𝑑 ∈ ( 0 , 1 ) , ( 1 . 1 ) 𝑒 ( 0 ) = 0 , 𝑒 ( 1 ) = π‘š βˆ’ 2  𝑖 = 1 𝛼 𝑖 𝑒 ξ€· πœ‚ 𝑖 ξ€Έ , ( 1 . 2 ) where integer π‘š β‰₯ 3 , πœ‚ 𝑖 ∈ ( 0 , 1 ) , πœ† is a positive parameter, and 𝑓 ∈ 𝐢 1 ( ℝ , ℝ ) . We study the multiplicity and global structure of sign-changing solutions of (1.1) and (1.2) under the assumptions:(A1) 𝛼 𝑖 > 0 for 𝑖 = 1 , … , π‘š βˆ’ 2 with βˆ‘ 0 < π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 < 1 ;(A2) 𝑓 ∈ 𝐢 1 ( ℝ , ℝ ) satisfies 𝑠 𝑓 ( 𝑠 ) > 0 for 𝑠 β‰  0 ;(A3) 𝑓 0 ∢ = l i m | 𝑠 | β†’ 0 𝑓 ( 𝑠 ) / 𝑠 = 0 ;(A4) 𝑓 ∞ ∢ = l i m | 𝑠 | β†’ ∞ 𝑓 ( 𝑠 ) / 𝑠 = 0 .

Multipoint boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics. The existence of solutions of second-order multipoint boundary value problems has been extensively studied in the literature, see [1–4] and the references therein. Particularly, many authors have studied the existence of sign-changing solutions for various nonlinear boundary value problems, see for example [5–10].

Recently, the global structure of solutions of nonlinear multipoint boundary value problems has also been investigated by several authors using bifurcation methods, see [7–10]. These papers dealt with the case 𝑓 0 ∈ ( 0 , ∞ ) , and relatively little is known about the global structure of solutions when 𝑓 satisfying 𝑓 0 = 0 . The main reason is that the global bifurcation techniques cannot be used directly in this case. Very recently, [11] investigated the global structure of positive solutions for a class of boundary value problems with 𝑓 0 = 0 . However, to our knowledge there is no paper studying the global structure of sign-changing solutions for nonlinear multipoint boundary value problems under the assumption 𝑓 0 = 0 . The purpose of present paper is to fill this gap.

In this paper, we consider the global structure of nodal solutions of (1.1) and (1.2), a kind of sign-changing having a given number of zeros, when 𝑓 0 = 𝑓 ∞ = 0 . We find that the discussion is more complicated, when sign-changing solutions are concerned. Eigenvalue theory and Sturm’s comparison theorem play important roles in our discussion.

Now, we introduce some notations as follows.

Let π‘Œ = 𝐢 [ 0 , 1 ] with the norm β€– 𝑒 β€– ∞ = m a x 𝑑 ∈ [ 0 , 1 ] | 𝑒 ( 𝑑 ) | . Let 𝑋 = { 𝑒 ∈ 𝐢 1 [ 0 , 1 ] ∣ 𝑒 ( 0 ) = 0 , 𝑒 ( 1 ) = βˆ‘ π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 𝑒 ( πœ‚ 𝑖 ) } , and 𝐸 = { 𝑒 ∈ 𝐢 2 βˆ‘ [ 0 , 1 ] ∣ 𝑒 ( 0 ) = 0 , 𝑒 ( 1 ) = π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 𝑒 ( πœ‚ 𝑖 ) } equipped with the norm: β€– 𝑒 β€– 𝑋 ξ€½ = m a x β€– 𝑒 β€– ∞ , β€– β€– 𝑒 ξ…ž β€– β€– ∞ ξ€Ύ ξ€½ , β€– 𝑒 β€– = m a x β€– 𝑒 β€– ∞ , β€– β€– 𝑒 ξ…ž β€– β€– ∞ , β€– β€– 𝑒 ξ…ž ξ…ž β€– β€– ∞ ξ€Ύ . ( 1 . 3 )

For any 𝐢 1 function 𝑒 , if 𝑒 ( π‘₯ 0 ) = 0 and 𝑒 β€² ( π‘₯ 0 ) β‰  0 , then π‘₯ 0 is called a simple zero of 𝑒 . For any integer π‘˜ β‰₯ 1 and any 𝜈 ∈ { + , βˆ’ } , let 𝑆 𝜈 π‘˜ , 𝑇 𝜈 π‘˜ βŠ‚ 𝐢 2 [ 0 , 1 ] be sets consisting of functions 𝑒 ∈ 𝐢 2 [ 0 , 1 ] satisfying the following conditions: 𝑆 𝜈 π‘˜ : (i) 𝑒 ( 0 ) = 0 , 𝜈 𝑒 β€² ( 0 ) > 0 ; (ii) 𝑒 has only simple zeros in [ 0 , 1 ] and has exactly π‘˜ βˆ’ 1 zeros in ( 0 , 1 ) . 𝑇 𝜈 π‘˜ : (i) 𝑒 ( 0 ) = 0 , 𝜈 𝑒 β€² ( 0 ) > 0 and 𝑒 β€² ( 1 ) β‰  0 ;(ii) 𝑒 β€² has only simple zeros in ( 0 , 1 ) and has exactly π‘˜ zeros in ( 0 , 1 ) ;(iii) 𝑒 has a zero strictly between each two consecutive zeros of 𝑒 β€² .

Remark 1.1. If 𝑒 ∈ 𝑇 𝜈 π‘˜ , then 𝑒 ∈ 𝑆 𝜈 π‘˜ or 𝑒 ∈ 𝑆 𝜈 π‘˜ + 1 . The sets 𝑇 𝜈 π‘˜ ( π‘˜ = 1 , 2 , … ) are open in 𝐸 and disjoint [8].

Lemma 1.2 (See [8]). Let (A1) and (A2) hold. If ( πœ‡ , 𝑒 ) is a nontrivial solution of (1.1) and (1.2). Then, 𝑒 ∈ 𝑇 𝜈 π‘˜ for some π‘˜ , 𝜈 .
Let 𝕏 = ℝ Γ— 𝑋 with the product topology. As in [12], we add the point { ( πœ† , ∞ ) ∣ πœ† ∈ ℝ } to the space 𝕏 . Denote πœƒ ∈ 𝑋 , πœƒ ( 𝑑 ) ≑ 0 , 𝑑 ∈ [ 0 , 1 ] .

The main results of this paper are as follows.

Theorem 1.3. Let (A1)–(A4) hold. Then, there exists a component π’ž 𝜈 π‘˜ βŠ‚ ( 0 , ∞ ) Γ— 𝑇 𝜈 π‘˜ of solutions of (1.1) and (1.2), which joins ( ∞ , πœƒ ) to ( ∞ , ∞ ) (see Figure 1(a)) such that P r o j ℝ π’ž 𝜈 π‘˜ = [ 𝜌 𝜈 π‘˜ , ∞ ) for some 𝜌 𝜈 k > 0 . Here, π’ž 𝜈 π‘˜ joins ( ∞ , πœƒ ) to ( ∞ , ∞ ) meaning that: l i m ( πœ† , 𝑒 ) ∈ π’ž 𝜈 π‘˜ , β€– 𝑒 β€– ≀ 1 , πœ† β†’ + ∞ β€– 𝑒 β€– = 0 , l i m ( πœ† , 𝑒 ) ∈ π’ž 𝜈 π‘˜ , β€– 𝑒 β€– > 1 , πœ† β†’ + ∞ β€– 𝑒 β€– = + ∞ . ( 1 . 4 )

fig1
Figure 1

Corollary 1.4. Let (A1)–(A4) hold. Then, there exists πœ† 𝜈 π‘˜ β‰₯ 𝜌 𝜈 π‘˜ > 0 such that (1.1) and (1.2) have at least two solutions in 𝑇 𝜈 π‘˜ for πœ† ∈ ( πœ† 𝜈 π‘˜ , ∞ ) .

Remark 1.5. Theorem 1.3 extends the result stated in [11]. Meanwhile, Theorem 1.3 and Corollary 1.4 do not only obtain the multiplicity of nodal solutions of (1.1) and (1.2), but also describe the global structure of these solutions.

2. Preliminary Lemmas

The following definition and lemmas about superior limit and component are important to prove Theorem 1.3.

Definition 2.1 (See [13]). Let π‘Š be a Banach space, and { 𝐢 𝑛 ∣ 𝑛 = 1 , 2 , … } be a family of subsets of π‘Š . Then, the superior limit π’Ÿ of { 𝐢 𝑛 } is defined by: π’Ÿ ∢ = l i m s u p 𝑛 β†’ ∞ 𝐢 𝑛 = ξ€½ ξ€½ 𝑛 π‘₯ ∈ π‘Š ∣ βˆƒ 𝑖 ξ€Ύ βŠ‚ β„• , π‘₯ 𝑛 𝑖 ∈ 𝐢 𝑛 𝑖 , s u c h t h a t π‘₯ 𝑛 𝑖 ξ€Ύ . ⟢ π‘₯ ( 2 . 1 )

Lemma 2.2 (See [13]). Each connected subset of metric space π‘Š is contained in a component, and each component of π‘Š is closed.

Lemma 2.3 (See [11]). Let π‘Š be a Banach space and 𝐢 𝑛 a family of closed connected subsets of π‘Š . Assume that:
(i) there exist 𝑧 𝑛 ∈ 𝐢 𝑛 , 𝑛 = 1 , 2 , … , and 𝑧 βˆ— ∈ π‘Š such that 𝑧 𝑛 β†’ 𝑧 βˆ— ;
(ii) π‘Ÿ 𝑛 = ∞ , where π‘Ÿ 𝑛 = s u p { β€– π‘₯ β€– ∣ π‘₯ ∈ 𝐢 𝑛 } ;
(iii) for all 𝑅 > 0 , ( βˆͺ ∞ 𝑛 = 1 𝐢 𝑛 ) ∩ 𝐡 𝑅 is a relative compact set of W , where 𝐡 𝑅 = { π‘₯ ∈ π‘Š ∣ β€– π‘₯ β€– ≀ 𝑅 } . ( 2 . 2 ) Then, there exists an unbounded connected component π’ž in π’Ÿ such that 𝑧 βˆ— ∈ π’ž .

Define a linear operator 𝐿 ∢ 𝐸 β†’ π‘Œ by: L u ∢ = βˆ’ 𝑒 ξ…ž ξ…ž , 𝑒 ∈ 𝐸 . ( 2 . 3 ) We consider the linear eigenvalues problem: 𝐿 𝑒 = πœ† 𝑒 , 𝑒 ∈ 𝐸 . ( 2 . 4 ) Let πœ† π‘˜ be the π‘˜ th eigenvalue of (2.4), and πœ‘ π‘˜ an eigenfunction corresponding to πœ† π‘˜ . The following lemma or similar result can be found in [7–9].

Lemma 2.4. Let (A1) hold. Then, 0 < πœ† 1 < πœ† 2 < β‹― < πœ† π‘˜ < πœ† π‘˜ + 1 < β‹― , l i m π‘˜ β†’ ∞ πœ† π‘˜ = ∞ . ( 2 . 5 ) For each π‘˜ ∈ 𝑁 , algebraic multiplicity of πœ† π‘˜ is equal to 1, and the corresponding eigenfunction πœ‘ π‘˜ ∈ 𝑇 + π‘˜ and is strictly positive on (0,1).

Define a map 𝑇 πœ† ∢ π‘Œ β†’ 𝐸 by: 𝑇 πœ† ξ€œ 𝑒 ( 𝑑 ) = πœ† 1 0 𝐻 ( 𝑑 , 𝑠 ) 𝑓 ( 𝑒 ( 𝑠 ) ) 𝑑 𝑠 , ( 2 . 6 ) where βˆ‘ 𝐻 ( 𝑑 , 𝑠 ) = 𝐺 ( 𝑑 , 𝑠 ) + π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 𝐺 ξ€· πœ‚ 𝑖 ξ€Έ , 𝑠 βˆ‘ 1 βˆ’ π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 πœ‚ 𝑖 ξ‚» 𝑑 , 𝐺 ( 𝑑 , 𝑠 ) = ( 1 βˆ’ 𝑑 ) 𝑠 , 0 ≀ 𝑠 ≀ 𝑑 ≀ 1 , 𝑑 ( 1 βˆ’ 𝑠 ) , 0 ≀ 𝑑 ≀ 𝑠 ≀ 1 . ( 2 . 7 )

It is clear that 𝑇 πœ† ∢ π‘Œ β†’ 𝑋 is completely continuous provided that (A1) and (A2) hold.

Lemma 2.5. Let (A1) and (A2)  hold, and { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } βŠ‚ ( 0 , ∞ ) Γ— 𝑇 𝜈 π‘˜ be a sequence of solutions of (1.1) and (1.2). Assume that πœ‡ 𝑙 ≀ 𝐢 0 for some constant 𝐢 0 > 0 , and l i m 𝑙 β†’ ∞ β€– 𝑦 𝑙 β€– = ∞ . Then, l i m 𝑙 β†’ ∞ β€– β€– 𝑦 𝑙 β€– β€– ∞ = ∞ . ( 2 . 8 )

Proof. From the relation 𝑦 𝑙 ( 𝑑 ) = πœ‡ 𝑙 ∫ 1 0 𝐻 ( 𝑑 , 𝑠 ) 𝑓 ( 𝑦 𝑙 ( 𝑠 ) ) 𝑑 𝑠 , we conclude that 𝑦 ξ…ž 𝑙 ( 𝑑 ) = πœ‡ 𝑙 ∫ 1 0 𝐻 𝑑 ( 𝑑 , 𝑠 ) 𝑓 ( 𝑦 𝑙 ( 𝑠 ) ) 𝑑 𝑠 . Then, β€– β€– 𝑦 ξ…ž 𝑙 β€– β€– ∞ ≀ 𝐢 0  βˆ‘ 1 + π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 βˆ‘ 1 βˆ’ π‘š βˆ’ 2 𝑖 = 1 𝛼 𝑖 πœ‚ 𝑖 ξƒͺ ξ€œ 1 0 | | 𝑓 ξ€· 𝑦 𝑙 ξ€Έ | | ( 𝑠 ) 𝑑 𝑠 . ( 2 . 9 ) Equations (2.9) and (1.1) imply that { β€– 𝑦 ξ…ž 𝑙 β€– ∞ } , { β€– 𝑦 𝑙 ξ…ž ξ…ž β€– ∞ } are bounded, whenever { β€– 𝑦 𝑙 β€– ∞ } is bounded.

3. Proof of the Main Results

We will construct a sequence of functions { 𝑓 [ 𝑛 ] } which is asymptotic linear at 0 and satisfies l i m 𝑛 β†’ ∞ s u p 𝑠 ∈ R | | 𝑓 [ 𝑛 ] | | ( 𝑠 ) βˆ’ 𝑓 ( 𝑠 ) = 0 , l i m 𝑛 β†’ ∞ ξ€· 𝑓 [ 𝑛 ] ξ€Έ 0 ∢ = l i m 𝑛 β†’ ∞  l i m | 𝑠 | β†’ 0 𝑓 [ 𝑛 ] ( 𝑠 ) 𝑠 ξƒͺ = 0 . ( 3 . 1 ) By means of some corresponding auxiliary equations, we can obtain a sequence of unbounded components { 𝐢 π‘˜ 𝜈 [ 𝑛 ] } via Rabinowitz’s global bifurcation theorem [14]. Based on the sequence, we can find an unbounded component 𝐢 𝜈 π‘˜ satisfying: 𝐢 𝜈 π‘˜ βŠ‚ l i m s u p 𝑛 β†’ ∞ 𝐢 π‘˜ 𝜈 [ 𝑛 ] , ( 3 . 2 ) and joining ( ∞ , πœƒ ) with ( ∞ , ∞ ) . We do it as follows.

For each 𝑛 ∈ β„• , define 𝑓 [ 𝑛 ] ( 𝑠 ) ∢ ℝ β†’ ℝ by: 𝑓 [ 𝑛 ] ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ‚€ 1 ( 𝑠 ) = 𝑓 ( 𝑠 ) , 𝑠 ∈ 𝑛  βˆͺ ξ‚€ 1 , ∞ βˆ’ ∞ , βˆ’ 𝑛  , ξ‚€ 1 𝑛 𝑓 𝑛   βˆ’ 1 𝑠 , 𝑠 ∈ 𝑛 , 1 𝑛 ξ‚„ . ( 3 . 3 ) Then, 𝑓 [ 𝑛 ] β‹‚ 𝐢 ∈ 𝐢 ( ℝ , ℝ ) 1 ( ℝ ⧡ { Β± 1 / 𝑛 } , ℝ ) with 𝑠 𝑓 [ 𝑛 ] ξ€· 𝑓 ( 𝑠 ) > 0 , βˆ€ 𝑠 β‰  0 , [ 𝑛 ] ξ€Έ 0 ξ‚€ 1 = 𝑛 𝑓 𝑛  . ( 3 . 4 ) By (A3), it follows that l i m 𝑛 β†’ ∞ ξ€· 𝑓 [ 𝑛 ] ξ€Έ 0 = 0 . ( 3 . 5 ) Now let us consider the auxiliary family of problems: 𝑒 ξ…ž ξ…ž + πœ† 𝑓 [ 𝑛 ] ( 𝑒 ) = 0 , 𝑑 ∈ ( 0 , 1 ) , 𝑒 ( 0 ) = 0 , 𝑒 ( 1 ) = π‘š βˆ’ 2  𝑖 = 1 𝛼 𝑖 𝑒 ξ€· πœ‚ 𝑖 ξ€Έ . ( 3 . 6 ) From Proposition  4.1 in [8], we obtain the following.

Lemma 3.1. Let (A1) and (A2) hold. If ( πœ‡ , 𝑒 ) is a nontrivial solution of (3.6). Then, 𝑒 ∈ 𝑇 𝜈 π‘˜ for some π‘˜ , 𝜈 .

Let 𝑔 [ 𝑛 ] ∈ 𝐢 ( ℝ , ℝ ) such that: 𝑓 [ 𝑛 ] ξ€· 𝑓 ( 𝑒 ) = [ 𝑛 ] ξ€Έ 0 𝑒 + 𝑔 [ 𝑛 ] ξ‚€ 1 ( 𝑒 ) = 𝑛 𝑓 𝑛  𝑒 + 𝑔 [ 𝑛 ] ( 𝑒 ) . ( 3 . 7 ) Note that l i m | 𝑠 | β†’ 0 𝑔 [ 𝑛 ] ( 𝑠 ) 𝑠 = 0 . ( 3 . 8 ) Let us consider ξ€· 𝑓 𝐿 𝑒 βˆ’ πœ† [ 𝑛 ] ξ€Έ 0 𝑒 = πœ† 𝑔 [ 𝑛 ] ( 𝑒 ) , ( 3 . 9 ) as a bifurcation problem from the trivial solution 𝑒 ≑ πœƒ .

Equation (3.9) can be converted to the equivalent form: ξ€œ 𝑒 ( 𝑑 ) = 1 0 ξ€Ί πœ† ξ€· 𝑓 𝐻 ( 𝑑 , 𝑠 ) [ 𝑛 ] ξ€Έ 0 𝑒 ( 𝑠 ) + πœ† 𝑔 [ 𝑛 ] ξ€» ( 𝑒 ( 𝑠 ) ) 𝑑 𝑠 ∢ = πœ† 𝐿 βˆ’ 1 𝑓 ξ€Ί ξ€· [ 𝑛 ] ξ€Έ 0 ξ€» 𝑒 ( β‹… ) ( 𝑑 ) + πœ† 𝐿 βˆ’ 1 ξ€Ί 𝑔 [ 𝑛 ] ξ€» ( 𝑒 ( β‹… ) ) ( 𝑑 ) . ( 3 . 1 0 ) Note that β€– 𝐿 βˆ’ 1 [ 𝑔 [ 𝑛 ] ( 𝑒 ) ] β€– = π‘œ ( β€– 𝑒 β€– ) for 𝑒 near πœƒ in 𝑋 . Applying Lemma 2.4, the global bifurcation result of Rabinowitz [14] for (3.9) can be stated as follows: for each integer π‘˜ β‰₯ 1 , 𝜈 ∈ { + , βˆ’ } , there exists a continuum 𝐢 π‘˜ 𝜈 [ 𝑛 ] of solutions of (3.9) joining ( πœ† π‘˜ / ( 𝑓 [ 𝑛 ] ) 0 , πœƒ ) to infinity in 𝕏 . Moreover, 𝐢 π‘˜ 𝜈 [ 𝑛 ] ⧡ ( πœ† π‘˜ / ( 𝑓 [ 𝑛 ] ) 0 , πœƒ ) βŠ‚ ( 0 , ∞ ) Γ— 𝑇 𝜈 π‘˜ .

For properties of 𝐢 π‘˜ 𝜈 [ 𝑛 ] , we give the following lemmas.

Lemma 3.2. Let (A1)–(A4) hold. Then for each fixed 𝑛 , 𝐢 π‘˜ 𝜈 [ 𝑛 ] joins ( πœ† π‘˜ / ( 𝑓 [ 𝑛 ] ) 0 , πœƒ ) to ( ∞ , ∞ ) in 𝕏 (see Figure 1(b)).

Proof. We divide the proof into two steps.
Step 1. We show that s u p { πœ† ∣ ( πœ† , 𝑒 ) ∈ 𝐢 π‘˜ 𝜈 [ 𝑛 ] } = ∞ . Assume on the contrary that s u p { πœ† ∣ ( πœ† , 𝑒 ) ∈ 𝐢 π‘˜ 𝜈 [ 𝑛 ] } = ∢ 𝐢 0 < ∞ . Let { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } βŠ‚ 𝐢 π‘˜ 𝜈 [ 𝑛 ] be such that: | | πœ‡ 𝑙 | | + β€– β€– 𝑦 𝑙 β€– β€– β†’ ∞ . ( 3 . 1 1 ) Similar to the argument of Lemma 2.5, we conclude that β€– 𝑦 𝑙 β€– ∞ β†’ ∞ .
Since ( πœ‡ 𝑙 , 𝑦 𝑙 ) ∈ π’ž π‘˜ 𝜈 [ 𝑛 ] , we have 𝑦 𝑙 ξ…ž ξ…ž ( 𝑑 ) + πœ‡ 𝑙 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ 𝑦 ( 𝑑 ) = 0 , 𝑑 ∈ ( 0 , 1 ) , 𝑙 ( 0 ) = 0 , 𝑦 𝑙 ( 1 ) = π‘š βˆ’ 2  𝑖 = 1 𝛼 𝑖 𝑦 𝑙 ξ€· πœ‚ 𝑖 ξ€Έ . ( 3 . 1 2 ) Set 𝑣 𝑙 ( 𝑑 ) = 𝑦 𝑙 ( 𝑑 ) / β€– 𝑦 𝑙 β€– ∞ . Then, β€– 𝑣 𝑙 β€– ∞ = 1 , and 𝑣 𝑙 ξ…ž ξ…ž ( 𝑑 ) + πœ‡ 𝑙 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ ( 𝑑 ) β€– β€– 𝑦 𝑙 β€– β€– ∞ = 0 , 𝑑 ∈ ( 0 , 1 ) . ( 3 . 1 3 ) Using l i m | 𝑒 | β†’ 0 𝑓 ( 𝑒 ) / 𝑒 = 0 , we can show that l i m 𝑙 β†’ ∞ | | 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ | | ( 𝑑 ) β€– β€– 𝑦 𝑙 β€– β€– ∞ = 0 . ( 3 . 1 4 ) The proof is similar to that of Theorem  1 in [12], and therefore we omit it. Equations (3.13) and (3.14) imply that β€– 𝑣 𝑙 ξ…ž ξ…ž β€– ∞ ≀ 𝑀 for some constant 𝑀 > 0 , independent of 𝑙 . Hence, { 𝑣 𝑙 } has a convergent subsequence in 𝑋 . Without loss of generality, we assume that there exists ( πœ‡ βˆ— , 𝑣 βˆ— ) ∈ [ 0 , 𝐢 0 ] Γ— 𝑋 with: β€– β€– 𝑣 βˆ— β€– β€– ∞ = 1 , ( 3 . 1 5 ) such that l i m 𝑙 β†’ ∞ ξ€· πœ‡ 𝑙 , 𝑣 𝑙 ξ€Έ = ξ€· πœ‡ βˆ— , 𝑣 βˆ— ξ€Έ , i n ℝ Γ— π‘Œ . ( 3 . 1 6 ) Note that (3.12) is equivalent to 𝑣 𝑙 ( 𝑑 ) = πœ‡ 𝑙 ξ€œ 1 0 𝑓 𝐻 ( 𝑑 , 𝑠 ) [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ ( 𝑠 ) β€– β€– 𝑦 𝑙 β€– β€– ∞ 𝑑 𝑠 , 𝑑 ∈ ( 0 , 1 ) . ( 3 . 1 7 ) Combining this with (3.16) and using (3.14) and the Lebesgue dominated convergence theorem, we have 𝑣 βˆ— ( 𝑑 ) = πœ‡ βˆ— ξ€œ 1 0 𝐻 ( 𝑑 , 𝑠 ) 0 𝑑 𝑠 = 0 , 𝑑 ∈ ( 0 , 1 ) . ( 3 . 1 8 ) This contradicts (3.15). Therefore, ξ€½ s u p πœ† ∣ ( πœ† , 𝑦 ) ∈ π’ž π‘˜ 𝜈 [ 𝑛 ] ξ€Ύ = ∞ . ( 3 . 1 9 )
Step 2. We show that s u p { β€– 𝑒 β€– ∞ ∣ ( πœ† , 𝑒 ) ∈ 𝐢 π‘˜ 𝜈 [ 𝑛 ] } = ∞ . On the contrary, assume that s u p { β€– 𝑒 β€– ∞ ∣ ( πœ† , 𝑒 ) ∈ 𝐢 π‘˜ 𝜈 [ 𝑛 ] } = 𝑀 0 < ∞ . Then, there exists a sequence { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } βŠ‚ 𝐢 π‘˜ 𝜈 [ 𝑛 ] such that πœ‡ 𝑙 β€– β€– 𝑦 ⟢ ∞ , 𝑙 β€– β€– ∞ ≀ 𝑀 0 . ( 3 . 2 0 ) From Remark 1.1, we can take a subsequences of { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } , still denoted by { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } , such that { 𝑦 𝑙 } βŠ‚ 𝑇 𝜈 π‘˜ β‹‚ 𝑆 𝜈 π‘˜ or { 𝑦 𝑙 } βŠ‚ 𝑇 𝜈 π‘˜ β‹‚ 𝑆 𝜈 π‘˜ + 1 . Without loss of generality, we suppose that { 𝑦 𝑙 } βŠ‚ 𝑇 𝜈 π‘˜ β‹‚ 𝑆 𝜈 π‘˜ . When { 𝑦 𝑙 } βŠ‚ 𝑇 𝜈 π‘˜ β‹‚ 𝑆 𝜈 π‘˜ + 1 is considered, the proof is similar. We omit it.
Note that ( πœ‡ 𝑙 , 𝑦 𝑙 ) satisfies the autonomous equation: 𝑦 𝑙 ξ…ž ξ…ž + πœ‡ 𝑙 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ = 0 , 𝑑 ∈ ( 0 , 1 ) . ( 3 . 2 1 ) Therefore, the graph of 𝑦 𝑙 consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval [ 0 , 1 ] , with the following properties (ignoring the truncated bump) (see [8]): all the positive (respectively, negative) bumps (i) have the same shape (the shapes of the positive and negative bumps may be different); (ii) attain the same maximum (minimum) value.
Let 0 = 𝜏 0 𝑙 < 𝜏 1 𝑙 < β‹― < 𝜏 𝑙 π‘˜ βˆ’ 1 ( 3 . 2 2 ) denote the zeros of 𝑦 𝑙 in [ 0 , 1 ] . Then, after taking a subsequence if necessary, l i m 𝑙 β†’ ∞ 𝜏 𝑗 𝑙 ∢ = 𝜏 𝑗 ∞ , 𝑗 ∈ { 0 , 1 , … , π‘˜ βˆ’ 1 } . Clearly, 𝜏 0 ∞ = 0 . Set 𝜏 π‘˜ ∞ = 1 . We can choose at least one subinterval ( 𝜏 𝑗 ∞ , 𝜏 ∞ 𝑗 + 1 ) β‰œ 𝐼 𝑗 ∞ which is of length at least 1 / π‘˜ for some 𝑗 ∈ { 0 , 1 , … , π‘˜ βˆ’ 1 } . Then, for this 𝑗 , 𝜏 𝑙 𝑗 + 1 βˆ’ 𝜏 𝑗 𝑙 > 3 / 4 π‘˜ if 𝑙 is large enough. Put ( 𝜏 𝑗 𝑙 , 𝜏 𝑙 𝑗 + 1 ) β‰œ 𝐼 𝑗 𝑙 .
Obviously, for the above given π‘˜ , 𝜈 , and 𝑗 , 𝑦 𝑙 ( 𝑑 ) have the same sign on 𝐼 𝑗 𝑙 for all 𝑙 . Without loss of generality, we assume 𝑦 𝑙 ( 𝑑 ) > 0 , 𝑑 ∈ 𝐼 𝑗 𝑙 . ( 3 . 2 3 ) Armed with the information on the shape of 𝑦 𝑙 , it is easy to show that for the above given 𝐼 𝑗 𝑙 , β€– 𝑦 𝑙 β€– 𝐼 𝑗 𝑙 , ∞ ∢ = m a x 𝐼 𝑗 𝑙 𝑦 𝑙 ( 𝑑 ) ≀ 𝑀 0 , 𝑙 = 1 , 2 , … .
Let 𝜎 be a constant with 0 < 𝜎 < 3 / 8 π‘˜ . Since 𝑦 𝑙 is concave on 𝐼 𝑗 𝑙 , we have 𝑦 𝑙 β€– β€– 𝑦 ( 𝑑 ) β‰₯ 𝜎 𝑙 β€– β€– 𝐼 𝑗 𝑙 , ∞  𝜏 , βˆ€ 𝑑 ∈ 𝑗 𝑙 + 𝜎 , 𝜏 𝑙 𝑗 + 1 ξ‚„ . βˆ’ 𝜎 ( 3 . 2 4 ) Then, there must exist constants 𝛼 , 𝛽 with [ 𝛼 , 𝛽 ] βŠ‚ 𝐼 𝑗 ∞ and 𝑙 0 such that 𝑦 𝑙 β€– β€– 𝑦 ( 𝑑 ) β‰₯ 𝜎 𝑙 β€– β€– 𝐼 𝑗 𝑙 , ∞ > 0 , u n i f o r m l y f o r [ ] 𝑑 ∈ 𝛼 , 𝛽 a n d 𝑙 > 𝑙 0 . ( 3 . 2 5 ) On the other hand, note that 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ ( 𝑑 ) 𝑦 𝑙 ξ‚» 𝑓 ( 𝑑 ) β‰₯ i n f [ 𝑛 ] ( 𝑠 ) 𝑠 ∣ 0 < 𝑠 ≀ 𝑀 0 ξ‚Ό ξ‚€ 𝜏 > 0 , 𝑑 ∈ 𝑗 𝑙 , 𝜏 𝑙 𝑗 + 1  . ( 3 . 2 6 ) Using the relation: 𝑦 𝑙 ξ…ž ξ…ž ( 𝑑 ) + πœ‡ 𝑙 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ ( 𝑑 ) 𝑦 𝑙 𝑦 ( 𝑑 ) 𝑙 ξ‚€ 𝜏 ( 𝑑 ) = 0 , 𝑑 ∈ 𝑗 𝑙 , 𝜏 𝑙 𝑗 + 1  , ( 3 . 2 7 ) and Sturm’s comparison theorem, we deduce that 𝑦 𝑙 must change its sign on ( 𝛼 , 𝛽 ) if 𝑙 is sufficiently large, contradicting (3.25). Therefore, l i m 𝑙 β†’ ∞ β€– β€– 𝑦 𝑙 β€– β€– ∞ = ∞ . ( 3 . 2 8 ) Hence, 𝐢 π‘˜ 𝜈 [ 𝑛 ] joins ( πœ† π‘˜ / ( 𝑓 [ 𝑛 ] ) 0 , πœƒ ) to ( ∞ , ∞ ) in 𝕏 .

Lemma 3.3. Let (A1)–(A4)  hold. Then, there exists 𝜌 𝜈 π‘˜ > 0 such that  ∞  𝑛 = 1 𝐢 π‘˜ 𝜈 [ 𝑛 ] ξƒͺ  ξ€· ξ€· 0 , 𝜌 𝜈 π‘˜ ξ€Έ ξ€Έ Γ— 𝑋 = βˆ… . ( 3 . 2 9 )

Proof. The proof is similar to that of Lemma  4.3 in [11]. We omit it.

Lemma 3.4. Let (A1)–(A4) hold, and let 𝜌 𝜈 π‘˜ be as in Lemma 3.3. Then, there exist 𝑛 0 ∈ β„• and  πœ† 𝜈 π‘˜ β‰₯ 𝜌 𝜈 π‘˜ > 0 such that for any  πœ† πœ† > 𝜈 π‘˜ and 𝑒 ∈ 𝐢 π‘˜ 𝜈 [ 𝑛 ] : 𝐢 π‘˜ 𝜈 [ 𝑛 ] ∩   πœ† ( πœ† , 𝑒 ) ∣ πœ† β‰₯ 𝜈 π‘˜ ; β€– 𝑒 β€– ∞  = 1 = βˆ… , βˆ€ 𝑛 > 𝑛 0 . ( 3 . 3 0 )

Proof. Suppose on the contrary that there exists { ( πœ‡ 𝑙 , 𝑦 𝑙 ⋃ ) } βŠ‚ ( ∞ 𝑛 = 1 𝐢 π‘˜ 𝜈 [ 𝑛 ] ) β‹‚ ( ( 0 , ∞ ) Γ— 𝑋 ) such that l i m 𝑙 β†’ ∞ πœ‡ 𝑙 β€– β€– 𝑦 = ∞ , 𝑙 β€– β€– ∞ = 1 . ( 3 . 3 1 )
Now, the method used in the proof of Lemma 3.2, Step 2, is still valid. Let 𝜎 be a constant with 0 < 𝜎 < 3 / 8 π‘˜ . Taking subsequences again if necessary, still denoted by { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } , such that { 𝑦 𝑙 } βŠ‚ 𝑇 𝜈 π‘˜ β‹‚ 𝑆 𝜈 π‘˜ . Without loss of generality, we can also derive an interval [ 𝛼 , 𝛽 ] βŠ‚ 𝐼 𝑗 ∞ and 𝑙 0 such that 1 β‰₯ 𝑦 𝑙 β€– β€– 𝑦 ( 𝑑 ) β‰₯ 𝜎 𝑙 β€– β€– 𝐼 𝑗 𝑙 , ∞ = 𝜎 , u n i f o r m l y f o r [ ] 𝑑 ∈ 𝛼 , 𝛽 a n d 𝑙 > 𝑙 0 . ( 3 . 3 2 ) It is easy to find an integer 𝑛 0 ∈ β„• such that 1 / 𝑛 0 < 𝜎 . This implies that ξ‚» 𝑓 i n f [ 𝑛 ] ( 𝑠 ) 𝑠 ξ‚Ό ξ‚» ∣ 𝜎 < 𝑠 ≀ 1 = i n f 𝑓 ( 𝑠 ) 𝑠 ξ‚Ό ∣ 𝜎 < 𝑠 ≀ 1 , βˆ€ 𝑛 > 𝑛 0 . ( 3 . 3 3 ) Note that for 𝑛 > 𝑛 0 , 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ ( 𝑑 ) 𝑦 𝑙 ξ‚» ( 𝑑 ) β‰₯ i n f 𝑓 ( 𝑠 ) 𝑠 ξ‚Ό ∣ 𝜎 < 𝑠 ≀ 1 > 0 , u n i f o r m l y f o r 𝑑 ∈ ( 𝛼 , 𝛽 ) a n d 𝑙 > 𝑙 0 . ( 3 . 3 4 ) Combining these facts and the relation: 𝑦 𝑙 ξ…ž ξ…ž ( 𝑑 ) + πœ‡ 𝑙 𝑓 [ 𝑛 ] ξ€· 𝑦 𝑙 ξ€Έ ( 𝑑 ) 𝑦 𝑙 𝑦 ( 𝑑 ) 𝑙 ( 𝑑 ) = 0 , 𝑑 ∈ ( 𝛼 , 𝛽 ) , ( 3 . 3 5 ) and Sturm’s comparison theorem, we conclude that 𝑦 𝑙 must change its sign on ( 𝛼 , 𝛽 ) if 𝑙 is large enough. This contradicts (3.32), and the proof is done.

Lemma 3.5. Let (A1)–(A4) hold, and let 𝑛 0 be as in Lemma 3.4. Then, there exist πœ† 𝜈 π‘˜ β‰₯ 𝜌 𝜈 π‘˜ > 0 and πœ– ∈ ( 0 , 1 / 2 ) such that for any πœ† > πœ† 𝜈 π‘˜ and 𝑒 ∈ 𝐢 π‘˜ 𝜈 [ 𝑛 ] : 𝐢 π‘˜ 𝜈 [ 𝑛 ] ∩ ξ€½ ( πœ† , 𝑒 ) ∣ πœ† β‰₯ πœ† 𝜈 π‘˜ ; 1 βˆ’ 2 πœ– ≀ β€– 𝑒 β€– ∞ ξ€Ύ ≀ 1 + 2 πœ– = βˆ… , βˆ€ 𝑛 > 𝑛 0 . ( 3 . 3 6 )

Proof. Similar to the proof of Lemma 3.4, we can find a constant  πœ† 𝜈 π‘˜ > 0 such that β€– 𝑒 β€– ∞ β‰  1 provided that  πœ† ( πœ† , 𝑒 ) ∈ ( 𝜈 π‘˜ , ∞ ) Γ— 𝑇 𝜈 π‘˜ being a solution of (1.1) and (1.2).
Let πœ† 𝜈 π‘˜  πœ† = m a x { 𝜈 π‘˜ ,  πœ† 𝜈 π‘˜ } + 1 . We claim that there exists πœ– ∈ ( 0 , 1 / 2 ) such that (3.36) holds. Suppose on the contrary that there exists πœ‡ ξ€½ ξ€· 𝑙 , 𝑦 𝑙 βŠ‚  ξ€Έ ξ€Ύ ∞  𝑛 = 1 𝐢 π‘˜ 𝜈 [ 𝑛 ] ξƒͺ  πœ† ξ€· ξ€· 𝜈 π‘˜ ξ€Έ ξ€Έ , , ∞ Γ— 𝑋 ( 3 . 3 7 ) satisfying l i m 𝑙 β†’ ∞ πœ‡ 𝑙 = πœ‡ βˆ— β‰₯ πœ† 𝜈 π‘˜ l i m 𝑙 β†’ ∞ β€– β€– 𝑦 𝑙 β€– β€– ∞ = 1 . ( 3 . 3 8 ) We can discuss two cases.
Case 1. If πœ‡ βˆ— < ∞ . { 𝑦 𝑙 } is compact in 𝑋 implies that there exists a subsequence, still denoted by { 𝑦 𝑙 } , such that l i m 𝑙 β†’ ∞ 𝑦 𝑙 = 𝑦 βˆ— ∈ 𝑇 𝜈 π‘˜ , β€– 𝑦 βˆ— β€– ∞ = 1 . ( 3 . 3 9 ) Obviously, ( πœ‡ βˆ— , 𝑦 βˆ— ) is a solution of (1.1) and (1.2). It is impossible.Case 2. If πœ‡ βˆ— = ∞ . Taking subsequences again if necessary, still denoted by { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } , such that 1 / 2 ≀ β€– 𝑦 𝑙 β€– ∞ ≀ 3 / 2 . Using the same argument as Lemma 3.4, we can find a contradiction.

Proof of Theorem 1.3. We will prove that the superior limit of 𝐢 π‘˜ 𝜈 [ 𝑛 ] contains an unbounded component π’ž 𝜈 π‘˜ βŠ‚ ( 0 , ∞ ) Γ— 𝑇 𝜈 π‘˜ of solutions of (1.1) and (1.2), which joins ( ∞ , πœƒ ) to ( ∞ , ∞ ) . For π‘Ÿ > 0 , introduce Ξ© π‘Ÿ = ξ€½ 𝑒 ∈ π‘Œ ∣ β€– 𝑒 β€– ∞ ξ€Ύ . < π‘Ÿ ( 3 . 4 0 ) Set Ξ“ 𝜈 π‘˜ ξ€· [ ∢ = 0 , ∞ ) Γ— 𝑇 𝜈 π‘˜ ξ€Έ ⧡ ξ€½ ( πœ‚ , 𝑒 ) ∣ πœ‚ β‰₯ πœ† 𝜈 π‘˜ ; 𝑒 ∈ 𝑇 𝜈 π‘˜ , β€– 𝑒 β€– ∞ ξ€Ύ , Ξ£ ≀ 1 + πœ– 𝜈 π‘˜ ξ€½ ( ∢ = πœ‚ , 𝑒 ) ∣ πœ‚ β‰₯ πœ† 𝜈 π‘˜ ; 𝑒 ∈ 𝑇 𝜈 π‘˜ , β€– 𝑒 β€– ∞ ξ€Ύ . ≀ 1 βˆ’ πœ– ( 3 . 4 1 ) Let 𝑛 0 and πœ– be as in Lemma 3.5. Firstly, for each given nonnegative integer 𝑝 = 0 , 1 , 2 , … , and 𝑛 β‰₯ 𝑛 0 with ( πœ† π‘˜ / ( 𝑓 [ 𝑛 ] ) 0 ) β‰₯ πœ† 𝜈 π‘˜ + 𝑝 , we define the connected subset, ( 𝜁 π‘˜ 𝜈 [ 𝑛 ] ) 𝑝 , in 𝐢 π‘˜ 𝜈 [ 𝑛 ] satisfying (see Figure 2(a)):
(i) ( 𝜁 π‘˜ 𝜈 [ 𝑛 ] ) 𝑝 βŠ‚ ( 𝐢 π‘˜ 𝜈 [ 𝑛 ] ⧡ ( πœ† 𝜈 π‘˜ + 𝑝 , ∞ ) Γ— Ξ© 1 βˆ’ πœ– ) ;
(ii) ( 𝜁 π‘˜ 𝜈 [ 𝑛 ] ) 𝑝 joins { πœ† 𝜈 π‘˜ + 𝑝 } Γ— Ξ© 1 βˆ’ πœ– with infinity in Ξ“ 𝜈 π‘˜ .By Lemmas 2.2 and 2.3, l i m s u p 𝑛 β†’ ∞ ( 𝜁 π‘˜ 𝜈 [ 𝑛 ] ) 𝑝 contains a component ( 𝜁 𝜈 π‘˜ ) 𝑝 joining { πœ† 𝜈 π‘˜ + 𝑝 } Γ— Ξ© 1 βˆ’ πœ– with infinity in Ξ“ 𝜈 π‘˜ (see Figure 2(b)).
It is easy to verify that if ( πœ† , 𝑒 ) ∈ ( 𝜁 𝜈 π‘˜ ) 𝑝 ( 𝑝 = 0 , 1 , 2 , … ) , then ( πœ† , 𝑒 ) is a solution of (1.1) and (1.2), and 𝑒 ∈ 𝑇 𝜈 π‘˜ .
Next, by using Lemma 2.3 and the method in [11] (see (4.22)–(4.30) in [11]), we can find a component π’ž 𝜈 π‘˜ in l i m s u p 𝑝 β†’ ∞ ( 𝜁 𝜈 π‘˜ ) 𝑝 , which is unbounded both in Ξ“ 𝜈 π‘˜ and Ξ£ 𝜈 π‘˜ .
Finally, we show that π’ž 𝜈 π‘˜ joins ( ∞ , πœƒ ) with ( ∞ , ∞ ) . This will be done by the following three steps.
Step 1. We show that l i m πœ† β†’ + ∞ β€– 𝑒 β€– ∞ = 0 for ( πœ† , 𝑒 ) ∈ ( π’ž 𝜈 π‘˜ ∩ Ξ£ 𝜈 π‘˜ ) .
Suppose on the contrary that there exists { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } βŠ‚ π’ž 𝜈 π‘˜ with β€– 𝑦 𝑙 β€– ∞ ≀ 1 βˆ’ πœ– , and πœ‡ 𝑙 β€– β€– 𝑦 β†’ + ∞ , 𝑙 β€– β€– ∞ β‰₯ π‘Ž , ( 3 . 4 2 ) for some constant π‘Ž > 0 . Applying the method of proving Lemma 3.2, we can deduce a contradiction.Step 2. We show that s u p { πœ† ∣ ( πœ† , 𝑒 ) ∈ ( π’ž 𝜈 π‘˜ ∩ Ξ“ 𝜈 π‘˜ ) } = ∞ . By a similar argument as Lemma 3.2, we can get the conclusion.Step 3. We show that l i m πœ† β†’ + ∞ β€– 𝑒 β€– ∞ = + ∞ for ( πœ† , 𝑒 ) ∈ ( π’ž 𝜈 π‘˜ ∩ Ξ“ 𝜈 π‘˜ ) .
On the contrary, suppose that there exists { ( πœ‡ 𝑙 , 𝑦 𝑙 ) } βŠ‚ ( π’ž 𝜈 π‘˜ ∩ Ξ“ 𝜈 π‘˜ ) with πœ‡ 𝑙 β€– β€– 𝑦 β†’ + ∞ , 1 < 𝑙 β€– β€– ∞ ≀ 𝑀 , ( 3 . 4 3 ) for some constant 𝑀 > 0 . The proof can be done by the same argument as Lemma 3.2.
This completes the proof of Theorem 1.3.

fig2
Figure 2

Proof of Corollary 1.4. The result can be directly obtained by Theorem 1.3.

Acknowledgments

This paper is supported by NSFC (no. 10971139); China Postdoctoral Fund (no. 2011M500615); Scientific Innovation Projection of Shanghai Education Department (no. 11YZ225); SIT-YJ2009-16.

References

  1. J. R. Graef and L. Kong, β€œNecessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems,” Nonlinear Analysis, vol. 68, no. 6, pp. 1529–1552, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. M. K. Kwong and J. S. W. Wong, β€œThe shooting method and nonhomogeneous multipoint BVPs of second-order ODE,” Boundary Value Problems, vol. 2007, Article ID 64012, 16 pages, 2007. View at Zentralblatt MATH
  3. S. Liang and J. Zhang, β€œExistence of countably many positive solutions of nth-order m-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 527–537, 2009. View at Publisher Β· View at Google Scholar
  4. Z. Bai, β€œSolvability for a class of fractional m-point boundary value problem at resonance,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1292–1302, 2011.
  5. X. Xian and J. Sun, β€œOn sign-changing solution for some three-point boundary value problems,” Nonlinear Analysis, vol. 59, no. 4, pp. 491–505, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. X. Xian and D. O'Regan, β€œMultiplicity of sign-changing solutions for some four-point boundary value problem,” Nonlinear Analysis, vol. 69, no. 2, pp. 434–447, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. B. P. Rynne, β€œSecond-order, three-point boundary value problems with jumping non-linearities,” Nonlinear Analysis, vol. 68, no. 11, pp. 3294–3306, 2008. View at Publisher Β· View at Google Scholar
  8. B. P. Rynne, β€œSpectral properties and nodal solutions for second-order, m-point, boundary value problems,” Nonlinear Analysis, vol. 67, no. 12, pp. 3318–3327, 2007. View at Publisher Β· View at Google Scholar
  9. R. Ma and D. O'Regan, β€œNodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues,” Nonlinear Analysis, vol. 64, no. 7, pp. 1562–1577, 2006. View at Publisher Β· View at Google Scholar
  10. S. Jingxian, X. Xian, and D. O'Regan, β€œNodal solutions for m-point boundary value problems using bifurcation methods,” Nonlinear Analysis, vol. 68, no. 10, pp. 3034–3046, 2008. View at Publisher Β· View at Google Scholar
  11. R. Ma and Y. An, β€œGlobal structure of positive solutions for nonlocal boundary value problems involving integral conditions,” Nonlinear Analysis, vol. 71, no. 10, pp. 4364–4376, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. R. Ma and B. Thompson, β€œNodal solutions for nonlinear eigenvalue problems,” Nonlinear Analysis, vol. 59, no. 5, pp. 707–718, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. G. T. Whyburn, Topological Analysis, vol. 23 of Princeton Mathematical Series, Princeton University Press, Princeton, N J, USA, 1958.
  14. P. H. Rabinowitz, β€œSome global results for nonlinear eigenvalue problems,” vol. 7, pp. 487–513, 1971. View at Zentralblatt MATH