ISRN Mathematical Analysis

VolumeΒ 2012Β (2012), Article IDΒ 354513, 13 pages

http://dx.doi.org/10.5402/2012/354513

## Bifurcation of Sign-Changing Solutions for -Point Boundary Value Problems

^{1}Department of Mathematics, Shanghai Normal University, Shanghai 200235, China^{2}Department 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 -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 -point boundary value problems: where integer , is a positive parameter, and . We study the multiplicity and global structure of sign-changing solutions of (1.1) and (1.2) under the assumptions:(A1) for with ;(A2) satisfies for ;(A3);(A4).

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 [1β4] and the references therein. Particularly, many authors have studied the existence of sign-changing solutions for various nonlinear boundary value problems, see for example [5β10].

Recently, the global structure of solutions of nonlinear multipoint boundary value problems has also been investigated by several authors using bifurcation methods, see [7β10]. These papers dealt with the case , and relatively little is known about the global structure of solutions when satisfying . 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 . 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 . 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 . We find that the discussion is more complicated, when sign-changing solutions are concerned. Eigenvalue theory and Sturmβs comparison theorem play important roles in our discussion.

Now, we introduce some notations as follows.

Let with the norm . Let , and equipped with the norm:

For any function , if and , then is called a simple zero of . For any integer and any , let be sets consisting of functions satisfying the following conditions:: (i) ; (ii) has only simple zeros in and has exactly zeros in .: (i) and ;(ii) has only simple zeros in and has exactly zeros in ;(iii) has a zero strictly between each two consecutive zeros of .

*Remark 1.1. *If , then or . The sets are open in and disjoint [8].

Lemma 1.2 (See [8]). *Let (A1) and (A2) hold. If is a nontrivial solution of (1.1) and (1.2). Then, for some .**Let with the product topology. As in [12], we add the point to the space . Denote .*

The main results of this paper are as follows.

Theorem 1.3. *Let (A1)β(A4) hold. Then, there exists a component of solutions of (1.1) and (1.2), which joins to (see Figure 1(a)) such that for some . Here, joins to meaning that:
*

Corollary 1.4. *Let (A1)β(A4) hold. Then, there exists such that (1.1) and (1.2) have at least two solutions in for .*

*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 be a Banach space, and be a family of subsets of *. *Then, the superior limit of is defined by:

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

Lemma 2.3 (See [11]). *Let be a Banach space and a family of closed connected subsets of . Assume that:**(i) there exist , , and such that ;**(ii) , where ;**(iii) for all is a relative compact set of , where
**
Then, there exists an unbounded connected component in such that .*

Define a linear operator by: We consider the linear eigenvalues problem: Let be the th eigenvalue of (2.4), and an eigenfunction corresponding to . The following lemma or similar result can be found in [7β9].

Lemma 2.4. *Let (A1) hold. Then,
**
For each , algebraic multiplicity of is equal to 1, and the corresponding eigenfunction and is strictly positive on (0,1).*

Define a map by: where

It is clear that is completely continuous provided that (A1) and (A2) hold.

Lemma 2.5. *Let (A1) and (A2)ββhold, and be a sequence of solutions of (1.1) and (1.2). Assume that for some constant , and . Then,
*

*Proof. *From the relation we conclude that . Then,
Equations (2.9) and (1.1) imply that are bounded, whenever is bounded.

#### 3. Proof of the Main Results

We will construct a sequence of functions which is asymptotic linear at 0 and satisfies By means of some corresponding auxiliary equations, we can obtain a sequence of unbounded components via Rabinowitzβs global bifurcation theorem [14]. Based on the sequence, we can find an unbounded component satisfying: and joining with . We do it as follows.

For each , define by: Then, with By (A3), it follows that Now let us consider the auxiliary family of problems: From Propositionββ4.1 in [8], we obtain the following.

Lemma 3.1. *Let (A1) and (A2) hold. If is a nontrivial solution of (3.6). Then, for some .*

Let such that: Note that Let us consider as a bifurcation problem from the trivial solution .

Equation (3.9) can be converted to the equivalent form: Note that for near in . Applying Lemma 2.4, the global bifurcation result of Rabinowitz [14] for (3.9) can be stated as follows: for each integer , , there exists a continuum of solutions of (3.9) joining to infinity in . Moreover, .

For properties of , we give the following lemmas.

Lemma 3.2. *Let (A1)β(A4) hold. Then for each fixed , joins to in (see Figure 1(b)).*

*Proof. *We divide the proof into two steps.*Step 1. *We show that . Assume on the contrary that . Let be such that:
Similar to the argument of Lemma 2.5, we conclude that .

Since , we have
Set . Then, , and
Using , we can show that
The proof is similar to that of Theoremββ1 in [12], and therefore we omit it. Equations (3.13) and (3.14) imply that for some constant , independent of . Hence, has a convergent subsequence in . Without loss of generality, we assume that there exists with:
such that
Note that (3.12) is equivalent to
Combining this with (3.16) and using (3.14) and the Lebesgue dominated convergence theorem, we have
This contradicts (3.15). Therefore,
*Step 2. *We show that . On the contrary, assume that . Then, there exists a sequence such that
From Remark 1.1, we can take a subsequences of , still denoted by , such that or . Without loss of generality, we suppose that . When is considered, the proof is similar. We omit it.

Note that satisfies the autonomous equation:
Therefore, the graph of consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval , 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
denote the zeros of in . Then, after taking a subsequence if necessary, . Clearly, . Set . We can choose at least one subinterval which is of length at least for some . Then, for this , if is large enough. Put .

Obviously, for the above given , and , have the same sign on for all . Without loss of generality, we assume
Armed with the information on the shape of , it is easy to show that for the above given , .

Let be a constant with . Since is concave on , we have
Then, there must exist constants with and such that
On the other hand, note that
Using the relation:
and Sturmβs comparison theorem, we deduce that must change its sign on if is sufficiently large, contradicting (3.25). Therefore,
Hence, joins to in .

Lemma 3.3. *Let (A1)β(A4)ββhold. Then, there exists such that
*

*Proof. *The proof is similar to that of Lemmaββ4.3 in [11]. We omit it.

Lemma 3.4. *Let (A1)β(A4) hold, and let be as in Lemma 3.3. Then, there exist and such that for any and :
*

*Proof. *Suppose on the contrary that there exists such that

Now, the method used in the proof of Lemma 3.2, Step 2, is still valid. Let be a constant with . Taking subsequences again if necessary, still denoted by , such that . Without loss of generality, we can also derive an interval and such that
It is easy to find an integer such that . This implies that
Note that for ,
Combining these facts and the relation:
and Sturmβs comparison theorem, we conclude that must change its sign on if is large enough. This contradicts (3.32), and the proof is done.

Lemma 3.5. *Let (A1)β(A4) hold, and let be as in Lemma 3.4. Then, there exist and such that for any and :
*

*Proof. *Similar to the proof of Lemma 3.4, we can find a constant such that provided that being a solution of (1.1) and (1.2).

Let . We claim that there exists such that (3.36) holds. Suppose on the contrary that there exists
satisfying
We can discuss two cases.*Case 1. *If . is compact in implies that there exists a subsequence, still denoted by , such that
Obviously, is a solution of (1.1) and (1.2). It is impossible.*Case 2. *If . Taking subsequences again if necessary, still denoted by , such that . 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 contains an unbounded component of solutions of (1.1) and (1.2), which joins to . For , introduce
Set
Let and be as in Lemma 3.5. Firstly, for each given nonnegative integer , and with , we define the connected subset, , in satisfying (see Figure 2(a)):

(i);

(ii) joins with infinity in .By Lemmas 2.2 and 2.3, contains a component joining with infinity in (see Figure 2(b)).

It is easy to verify that if , then is a solution of (1.1) and (1.2), and .

Next, by using Lemma 2.3 and the method in [11] (see (4.22)β(4.30) in [11]), we can find a component in , which is unbounded both in and .

Finally, we show that joins with . This will be done by the following three steps.*Step 1. *We show that for .

Suppose on the contrary that there exists with , and
for some constant . Applying the method of proving Lemma 3.2, we can deduce a contradiction.*Step 2. *We show that . By a similar argument as Lemma 3.2, we can get the conclusion.*Step 3. *We show that for .

On the contrary, suppose that there exists with
for some constant . The proof can be done by the same argument as Lemma 3.2.

This completes the proof of Theorem 1.3.

*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

- J. R. Graef and L. Kong, βNecessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems,β
*Nonlinear Analysis*, vol. 68, no. 6, pp. 1529β1552, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. K. Kwong and J. S. W. Wong, βThe shooting method and nonhomogeneous multipoint BVPs of second-order ODE,β
*Boundary Value Problems*, vol. 2007, Article ID 64012, 16 pages, 2007. View at Google Scholar Β· View at Zentralblatt MATH - S. Liang and J. Zhang, βExistence of countably many positive solutions of $n$th-order $m$-point boundary value problems,β
*Journal of Computational and Applied Mathematics*, vol. 224, no. 2, pp. 527β537, 2009. View at Publisher Β· View at Google Scholar - Z. Bai, βSolvability for a class of fractional
*m*-point boundary value problem at resonance,β*Computers and Mathematics with Applications*, vol. 62, no. 3, pp. 1292β1302, 2011. View at Google Scholar - X. Xian and J. Sun, βOn sign-changing solution for some three-point boundary value problems,β
*Nonlinear Analysis*, vol. 59, no. 4, pp. 491β505, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. Xian and D. O'Regan, βMultiplicity of sign-changing solutions for some four-point boundary value problem,β
*Nonlinear Analysis*, vol. 69, no. 2, pp. 434β447, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. P. Rynne, βSecond-order, three-point boundary value problems with jumping non-linearities,β
*Nonlinear Analysis*, vol. 68, no. 11, pp. 3294β3306, 2008. View at Publisher Β· View at Google Scholar - B. P. Rynne, βSpectral properties and nodal solutions for second-order, $m$-point, boundary value problems,β
*Nonlinear Analysis*, vol. 67, no. 12, pp. 3318β3327, 2007. View at Publisher Β· View at Google Scholar - R. Ma and D. O'Regan, βNodal solutions for second-order $m$-point boundary value problems with nonlinearities across several eigenvalues,β
*Nonlinear Analysis*, vol. 64, no. 7, pp. 1562β1577, 2006. View at Publisher Β· View at Google Scholar - S. Jingxian, X. Xian, and D. O'Regan, βNodal solutions for $m$-point boundary value problems using bifurcation methods,β
*Nonlinear Analysis*, vol. 68, no. 10, pp. 3034β3046, 2008. View at Publisher Β· View at Google Scholar - R. Ma and Y. An, βGlobal structure of positive solutions for nonlocal boundary value problems involving integral conditions,β
*Nonlinear Analysis*, vol. 71, no. 10, pp. 4364β4376, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Ma and B. Thompson, βNodal solutions for nonlinear eigenvalue problems,β
*Nonlinear Analysis*, vol. 59, no. 5, pp. 707β718, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. T. Whyburn,
*Topological Analysis*, vol. 23 of*Princeton Mathematical Series*, Princeton University Press, Princeton, N J, USA, 1958. - P. H. Rabinowitz, βSome global results for nonlinear eigenvalue problems,β vol. 7, pp. 487β513, 1971. View at Google Scholar Β· View at Zentralblatt MATH