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Advances in Difference Equations

, 2009:671625 | Cite as

A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems

  • Ruyun Ma
  • Youji Xu
  • Chenghua Gao
Open Access
Research Article

Abstract

Let Open image in new window be an integer with Open image in new window , Open image in new window , Open image in new window . We consider boundary value problems of nonlinear second-order difference equations of the form Open image in new window , Open image in new window , Open image in new window , where Open image in new window , Open image in new window and, Open image in new window for Open image in new window , and Open image in new window , Open image in new window , Open image in new window . We investigate the global structure of positive solutions by using the Rabinowitz's global bifurcation theorem.

Keywords

Banach Space Open Covering Global Structure Connected Subset Global Bifurcation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be an integer with Open image in new window , Open image in new window , Open image in new window . We study the global structure of positive solutions of the problem

Here Open image in new window is a positive parameter, Open image in new window and Open image in new window are continuous. Denote Open image in new window and Open image in new window .

There are many literature dealing with similar difference equations subject to various boundary value conditions. We refer to Agarwal and Henderson [1], Agarwal and O'Regan [2], Agarwal and Wong [3], Rachunkova and Tisdell [4], Rodriguez [5], Cheng and Yen [6], Zhang and Feng [7], R. Ma and H. Ma [8], Ma [9], and the references therein. These results were usually obtained by analytic techniques, various fixed point theorems, and global bifurcation techniques. For example, in [8], the authors investigated the global structure of sign-changing solutions of some discrete boundary value problems in the case that Open image in new window . However, relatively little result is known about the global structure of solutions in the case that Open image in new window , and no global results were found in the available literature when Open image in new window . The likely reason is that the Rabinowitz's global bifurcation theorem [10] cannot be used directly in this case.

In the present work, we obtain a direct and complete description of the global structure of positive solutions of (1.1) under the assumptions:
Let Open image in new window denote the Banach space defined by
equipped with the norm

Let Open image in new window denote the Banach space defined by

equipped with the norm
To state our main results, we need the spectrum theory of the linear eigenvalue problem

Lemma 1.1 ([5, 11]).

Let (A1) hold. Then there exists a sequence Open image in new window satisfying that
  1. (i)

    Open image in new window is the set of eigenvalues of (1.7);

     
  2. (ii)
     
  3. (iii)
     
  4. (iv)
     

Let Open image in new window denote the closure of set of positive solutions of (1.1) in Open image in new window .

Let Open image in new window be a subset of Open image in new window . A component of Open image in new window is meant a maximal connected subset of Open image in new window , that is, a connected subset of Open image in new window which is not contained in any other connected subset of Open image in new window .

The main results of this paper are the following theorem.

Theorem 1.2.

Let (A1)–(A4) hold. Then there exists a component Open image in new window in Open image in new window which joins Open image in new window with Open image in new window , and

for some Open image in new window . Moreover, there exists Open image in new window such that (1.1) has at least two positive solutions for Open image in new window .

We will develop a bifurcation approach to treat the case Open image in new window directly. Crucial to this approach is to construct a sequence of functions Open image in new window which is asymptotic linear at Open image in new window and satisfies
By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components Open image in new window via Rabinnowitz's global bifurcation theorem [10], and this enables us to find an unbounded component Open image in new window satisfying

2. Some Preliminaries

In this section, we give some notations and preliminary results which will be used in the proof of our main results.

Definition 2.1 (see [12]).

Let Open image in new window be a Banach space, and let Open image in new window be a family of subsets of Open image in new window . Then the superior limit Open image in new window of Open image in new window is defined by

Definition 2.2 (see [12]).

A component of a set Open image in new window is meant a maximal connected subset of Open image in new window .

Lemma 2.3 ([12, Whyburn]).

Suppose that Open image in new window is a compact metric space, Open image in new window and Open image in new window are nonintersecting closed subsets of Open image in new window , and no component of Open image in new window interests both Open image in new window and Open image in new window . Then there exist two disjoint compact subsets Open image in new window and Open image in new window , such that Open image in new window , Open image in new window , Open image in new window .

Using the above Whyburn's lemma, Ma and An [13] proved the following lemma.

Lemma 2.4 ([13, Lemma Open image in new window ]).

Let Open image in new window be a Banach space, and let Open image in new window be a family of connected subsets of Open image in new window . Assume that

Then there exists an unbounded component Open image in new window in Open image in new window and Open image in new window .

It is easy to see that

Then the operator Open image in new window satisfies Open image in new window .

Using the standard arguments, we may prove the following lemma.

Lemma 2.5.

Assume that (A1)–(A2) hold. Then Open image in new window and Open image in new window is completely continuous.

Lemma 2.6.

Assume that (A1)–(A2) hold. If Open image in new window , then

Proof.

3. Proof of the Main Results

By (A3), it follows that
To apply the global bifurcation theorem, we extend Open image in new window to be an odd function Open image in new window by

Similarly we may extend Open image in new window to be an odd function Open image in new window for each Open image in new window .

Now let us consider the auxiliary family of the equations
Let us consider

as a bifurcation problem from the trivial solution Open image in new window .

Equation (3.8) can be converted to the equivalent equation

Further we note that Open image in new window for Open image in new window near Open image in new window in Open image in new window .

The results of Rabinowitz [10] for (3.8) can be stated as follows. For each integer Open image in new window , Open image in new window , there exists a continuum Open image in new window of solutions of (3.8) joining Open image in new window to infinity in Open image in new window . Moreover, Open image in new window

Lemma 3.1.

Let (A1)–(A4) hold. Then, for each fixed Open image in new window , Open image in new window joins Open image in new window to Open image in new window in Open image in new window .

Proof.

We divide the proof into two steps.

Step 1.

We show that Open image in new window .

Assume on the contrary that Open image in new window . Let Open image in new window be such that
Then Open image in new window . This together with the fact
implies that
Now, choosing a subsequence and relabelling if necessary, it follows that there exists Open image in new window with
such that
Moreover, using (3.13), (3.12), and the assumption Open image in new window , it follows that
and consequently, Open image in new window for Open image in new window . This contradicts (3.15). Therefore

Step 2.

We show that Open image in new window .

Assume on the contrary that Open image in new window . Let Open image in new window be such that
Since Open image in new window , for any Open image in new window , we have from (2.6) that

(where Open image in new window ), which yields that Open image in new window is bounded. However, this contradicts (3.19).

Therefore, Open image in new window joins Open image in new window to Open image in new window in Open image in new window .

Lemma 3.2.

Let (A1)–(A4) hold and let Open image in new window be a closed and bounded interval. Then there exists a positive constant Open image in new window , such that

Proof.

Assume on the contrary that there exists a sequence Open image in new window such that
Then, (3.11), (3.12), and (3.13) hold. Set Open image in new window , then
Now, choosing a subsequence and relabeling if necessary, it follows that there exists Open image in new window with
such that
Moreover, from (3.13), (3.12), and the assumption Open image in new window , it follows that
and consequently, Open image in new window for Open image in new window . This contradicts (3.24). Therefore

Lemma 3.3.

Let (A1)–(A4) hold. Then there exits Open image in new window such that

Proof.

Assume on the contrary that there exists Open image in new window such that Open image in new window . Then
which contradicts (3.30). Therefore, there exists Open image in new window , such that

Proof of Theorem 1.2.

Take Open image in new window . Let Open image in new window be as in Lemma 3.3, and let Open image in new window be a fixed constant satisfying Open image in new window and
It is easy to see that there exists Open image in new window , such that
This implies that
(see (2.10) for the definition of Open image in new window ), and accordingly, we may choose Open image in new window which is independent of Open image in new window . From Lemma 2.6 and (3.35), it follows that for Open image in new window ,
This together with the compactness of Open image in new window implies that there exists Open image in new window , such that

Notice that Open image in new window satisfies all conditions in Lemma 2.4, and consequently, Open image in new window contains a component Open image in new window which is unbounded. However, we do not know whether Open image in new window joins Open image in new window with Open image in new window or not. To answer this question, we have to use the following truncation method.

We claim that Open image in new window satisfies all of the conditions of Lemma 2.4.

we have from Lemmas 3.1–3.3 and (3.40) that for Open image in new window and Open image in new window ,
Thus, there exists Open image in new window , such that Open image in new window , and accordingly, condition (i) in Lemma 2.4 is satisfied. Obviously,

that is, condition (ii) in Lemma 2.4 holds. Condition (iii) in Lemma 2.4 can be deduced directly from the Arzelà -Ascoli theorem and the definition of Open image in new window . Therefore, the superior limit of Open image in new window contains a component Open image in new window joining Open image in new window with infinity in Open image in new window .

Similarly, for each Open image in new window , we may define a connected subset, Open image in new window , in Open image in new window satisfying

and the superior limit of Open image in new window contains a component Open image in new window joining Open image in new window with infinity in Open image in new window .

It is easy to verify that
Now, for each Open image in new window , let Open image in new window be a connected component containing Open image in new window . Let

From Lemma 2.4, it follows that Open image in new window is closed in Open image in new window , and furthermore, Open image in new window is compact in Open image in new window .

Let

If for some Open image in new window , Open image in new window , then Theorem 1.2 holds.

Assume on the contrary that Open image in new window for all Open image in new window .

For every Open image in new window , let Open image in new window be the component in Open image in new window which contains Open image in new window . Using the standard method, we can find a bounded open set Open image in new window in Open image in new window , such that

where Open image in new window and Open image in new window are the boundary and closure of Open image in new window in Open image in new window , respectively.

Evidently, the following family of the open sets of Open image in new window :
is an open covering of Open image in new window . Since Open image in new window is compact set in Open image in new window , there exist Open image in new window such that Open image in new window , and the family of open sets in Open image in new window :
is a finite open covering of Open image in new window . There is
and by (3.52), we have

where Open image in new window and Open image in new window are the boundary and closure of Open image in new window in Open image in new window , respectively.

Equation (3.58) together with (3.55) and (3.57) implies that

However, this contradicts (3.50).

Therefore, there exists Open image in new window such that Open image in new window which is unbounded in both Open image in new window and Open image in new window .

Finally, we show that Open image in new window joins Open image in new window with Open image in new window . This will be done by the following three steps.

Step 1.

We show that Open image in new window .

Suppose on the contrary that there exists Open image in new window with
which implies

This is impossible by (A3) and the assumption Open image in new window .

Step 2.

We show that Open image in new window .

Suppose on the contrary that there exists Open image in new window with Open image in new window and
for some constant Open image in new window , then

where Open image in new window . By (A2), it follows that Open image in new window . Obviously, (3.65) implies that Open image in new window is bounded. This is a contradiction.

Step 3.

We show that Open image in new window .

Suppose on the contrary that there exists Open image in new window with
for some constant Open image in new window , then

where Open image in new window . By (A2), it follows that Open image in new window . Obviously, (3.68) implies that Open image in new window is bounded. This is a contradiction.

To sum up, there exits a component Open image in new window which joins Open image in new window and Open image in new window .

Notes

Acknowledgments

This work was supported by the NSFC (no. 10671158), the NSF of Gansu Province (no. 3ZS051-A25-016), NWNU-KJCXGC-03-17, the Spring-Sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006 Open image in new window ).

References

  1. 1.
    Agarwal RP, Henderson J: Positive solutions and nonlinear eigenvalue problems for third-order difference equations. Computers & Mathematics with Applications 1998,36(10–12):347-355.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agarwal RP, O'Regan D: Boundary value problems for discrete equations. Applied Mathematics Letters 1997,10(4):83-89. 10.1016/S0893-9659(97)00064-5MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Agarwal RP, Wong F-H: Existence of positive solutions for nonpositive difference equations. Mathematical and Computer Modelling 1997,26(7):77-85. 10.1016/S0895-7177(97)00186-6MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Rachunkova I, Tisdell CC: Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions. Nonlinear Analysis. Theory, Methods & Applications 2007,67(4):1236-1245. 10.1016/j.na.2006.07.010MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Rodriguez J: Nonlinear discrete Sturm-Liouville problems. Journal of Mathematical Analysis and Applications 2005,308(1):380-391. 10.1016/j.jmaa.2005.01.032MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cheng SS, Yen H-T: On a discrete nonlinear boundary value problem. Linear Algebra and Its Applications 2000,313(1–3):193-201.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Zhang G, Feng W: On the number of positive solutions of a nonlinear algebraic system. Linear Algebra and Its Applications 2007,422(2-3):404-421. 10.1016/j.laa.2006.10.026MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ma R, Ma H: Existence of sign-changing periodic solutions of second order difference equations. Applied Mathematics and Computation 2008,203(2):463-470. 10.1016/j.amc.2008.05.125MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ma R: Nonlinear discrete Sturm-Liouville problems at resonance. Nonlinear Analysis. Theory, Methods & Applications 2007,67(11):3050-3057. 10.1016/j.na.2006.09.058MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rabinowitz PH: Some global results for nonlinear eigenvalue problems. Journal of Functional Analysis 1971, 7: 487-513. 10.1016/0022-1236(71)90030-9MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jirari A: Second-order Sturm-Liouville difference equations and orthogonal polynomials. Memoirs of the American Mathematical Society 1995,113(542):138.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Whyburn GT: Topological Analysis. Princeton University Press, Princeton, NJ, USA; 1958:119.MATHGoogle Scholar
  13. 13.
    Ma R, An Y:Global structure of positive solutions for superlinear seconde order Open image in new window-point boundary value problems. Topological Methods in Nonlinear Analysis. In pressGoogle Scholar

Copyright information

© Ruyun Ma et al. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouChina

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