- Research Article
- Open Access
A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems
© Ruyun Ma et al. 2009
- Received: 12 February 2009
- Accepted: 20 August 2009
- Published: 7 October 2009
- Banach Space
- Open Covering
- Global Structure
- Connected Subset
- Global Bifurcation
There are many literature dealing with similar difference equations subject to various boundary value conditions. We refer to Agarwal and Henderson , Agarwal and O'Regan , Agarwal and Wong , Rachunkova and Tisdell , Rodriguez , Cheng and Yen , Zhang and Feng , R. Ma and H. Ma , Ma , and the references therein. These results were usually obtained by analytic techniques, various fixed point theorems, and global bifurcation techniques. For example, in , the authors investigated the global structure of sign-changing solutions of some discrete boundary value problems in the case that . However, relatively little result is known about the global structure of solutions in the case that , and no global results were found in the available literature when . The likely reason is that the Rabinowitz's global bifurcation theorem  cannot be used directly in this case.
The main results of this paper are the following theorem.
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 ).
Definition 2.2 (see ).
Lemma 2.3 ([12, Whyburn]).
Using the above Whyburn's lemma, Ma and An  proved the following lemma.
Lemma 2.4 ([13, Lemma ]).
Using the standard arguments, we may prove the following lemma.
Equation (3.8) can be converted to the equivalent equation
The results of Rabinowitz  for (3.8) can be stated as follows. For each integer , , there exists a continuum of solutions of (3.8) joining to infinity in . Moreover,
We divide the proof into two steps.
Proof of Theorem 1.2.
Notice that satisfies all conditions in Lemma 2.4, and consequently, contains a component which is unbounded. However, we do not know whether joins with or not. To answer this question, we have to use the following truncation method.
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 . Therefore, the superior limit of contains a component joining with infinity in .
However, this contradicts (3.50).
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