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A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems
Advances in Difference Equations volume 2009, Article number: 671625 (2009)
Abstract
Let be an integer with
,
,
. We consider boundary value problems of nonlinear second-order difference equations of the form
,
,
, where
,
and,
for
, and
,
,
. We investigate the global structure of positive solutions by using the Rabinowitz's global bifurcation theorem.
1. Introduction
Let be an integer with
,
,
. We study the global structure of positive solutions of the problem

Here is a positive parameter,
and
are continuous. Denote
and
.
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 . 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 [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:
-
(A1)
;
-
(A2)
is continuous and
for
;
-
(A3)
, where
;
-
(A4)
, where
.
Let denote the Banach space defined by

equipped with the norm

Let denote the Banach space defined by

equipped with the norm

Define an operator by

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

Let (A1) hold. Then there exists a sequence satisfying that
-
(i)
is the set of eigenvalues of (1.7);
-
(ii)
for
;
-
(iii)
for
,
is one-dimensional subspace of
;
-
(iv)
for each
, if
, then
has exactly
simple generalized zeros in
.
Let denote the closure of set of positive solutions of (1.1) in
.
Let be a subset of
. A component of
is meant a maximal connected subset of
, that is, a connected subset of
which is not contained in any other connected subset of
.
The main results of this paper are the following theorem.
Theorem 1.2.
Let (A1)–(A4) hold. Then there exists a component in
which joins
with
, and

for some . Moreover, there exists
such that (1.1) has at least two positive solutions for
.
We will develop a bifurcation approach to treat the case directly. Crucial to this approach is to construct a sequence of functions
which is asymptotic linear at
and satisfies

By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components via Rabinnowitz's global bifurcation theorem [10], and this enables us to find an unbounded component
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 be a Banach space, and let
be a family of subsets of
. Then the superior limit
of
is defined by

Definition 2.2 (see [12]).
A component of a set is meant a maximal connected subset of
.
Lemma 2.3 ([12, Whyburn]).
Suppose that is a compact metric space,
and
are nonintersecting closed subsets of
, and no component of
interests both
and
. Then there exist two disjoint compact subsets
and
, such that
,
,
.
Using the above Whyburn's lemma, Ma and An [13] proved the following lemma.
Lemma 2.4 ([13, Lemma ]).
Let be a Banach space, and let
be a family of connected subsets of
. Assume that
-
(i)
there exist
,
, and
, such that
;
-
(ii)
, where
;
-
(iii)
for every
,
is a relatively compact set of
, where
(2.2)
Then there exists an unbounded component in
and
.
Let

It is easy to see that

Denote the cone in
by

Now we define a map by

Define an operator by

Then the operator satisfies
.
For , let

Using the standard arguments, we may prove the following lemma.
Lemma 2.5.
Assume that (A1)–(A2) hold. Then and
is completely continuous.
Lemma 2.6.
Assume that (A1)–(A2) hold. If , then

where

Proof.
Since for
, it follows that

3. Proof of the Main Results
Define by

Then with

By (A3), it follows that

To apply the global bifurcation theorem, we extend to be an odd function
by

Similarly we may extend to be an odd function
for each
.
Now let us consider the auxiliary family of the equations

Let be such that

Then

Let us consider

as a bifurcation problem from the trivial solution .
Equation (3.8) can be converted to the equivalent equation

Further we note that for
near
in
.
The results of Rabinowitz [10] 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,
Lemma 3.1.
Let (A1)–(A4) hold. Then, for each fixed ,
joins
to
in
.
Proof.
We divide the proof into two steps.
Step 1.
We show that .
Assume on the contrary that . Let
be such that

Then . This together with the fact

implies that

Since , we have that

Set . Then

Now, choosing a subsequence and relabelling if necessary, it follows that there exists with

such that

Moreover, using (3.13), (3.12), and the assumption , it follows that

and consequently, for
. This contradicts (3.15). Therefore

Step 2.
We show that .
Assume on the contrary that . Let
be such that

Since , for any
, we have from (2.6) that

(where ), which yields that
is bounded. However, this contradicts (3.19).
Therefore, joins
to
in
.
Lemma 3.2.
Let (A1)–(A4) hold and let be a closed and bounded interval. Then there exists a positive constant
, such that

Proof.
Assume on the contrary that there exists a sequence such that

Then, (3.11), (3.12), and (3.13) hold. Set , then

Now, choosing a subsequence and relabeling if necessary, it follows that there exists with

such that

Moreover, from (3.13), (3.12), and the assumption , it follows that

and consequently, for
. This contradicts (3.24). Therefore

Lemma 3.3.
Let (A1)–(A4) hold. Then there exits such that

Proof.
Assume on the contrary that there exists such that
. Then

Set , then

and for all ,

where ). Let

Then , and

which contradicts (3.30). Therefore, there exists , such that

Proof of Theorem 1.2.
Take . Let
be as in Lemma 3.3, and let
be a fixed constant satisfying
and

where

It is easy to see that there exists , such that

This implies that

(see (2.10) for the definition of ), and accordingly, we may choose
which is independent of
. From Lemma 2.6 and (3.35), it follows that for
,

This together with the compactness of implies that there exists
, such that

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.
Set

For with
, we define
a connected subset in
satisfying
-
(1)
;
-
(2)
joins
with infinity in
.
We claim that satisfies all of the conditions of Lemma 2.4.
Since

we have from Lemmas 3.1–3.3 and (3.40) that for and
,

Thus, there exists , such that
, 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 . Therefore, the superior limit of
contains a component
joining
with infinity in
.
Similarly, for each , we may define a connected subset,
, in
satisfying
-
(1)
;
-
(2)
joins
with infinity in
,
and the superior limit of contains a component
joining
with infinity in
.
It is easy to verify that

Now, for each , let
be a connected component containing
. Let

Set

then since

From Lemma 2.4, it follows that is closed in
, and furthermore,
is compact in
.
Let

then

If for some ,
, then Theorem 1.2 holds.
Assume on the contrary that for all
.
For every , let
be the component in
which contains
. Using the standard method, we can find a bounded open set
in
, such that


where and
are the boundary and closure of
in
, respectively.
Evidently, the following family of the open sets of :

is an open covering of . Since
is compact set in
, there exist
such that
, and the family of open sets in
:

is a finite open covering of . There is

Let

Then is a bounded open set in
,

and by (3.52), we have

where and
are the boundary and closure of
in
, respectively.
Equation (3.58) together with (3.55) and (3.57) implies that

However, this contradicts (3.50).
Therefore, there exists such that
which is unbounded in both
and
.
Finally, we show that joins
with
. This will be done by the following three steps.
Step 1.
We show that .
Suppose on the contrary that there exists with

Then

which implies

This is impossible by (A3) and the assumption .
Step 2.
We show that .
Suppose on the contrary that there exists with
and

for some constant , then

Thus

where . By (A2), it follows that
. Obviously, (3.65) implies that
is bounded. This is a contradiction.
Step 3.
We show that .
Suppose on the contrary that there exists with

for some constant , then

Thus

where . By (A2), it follows that
. Obviously, (3.68) implies that
is bounded. This is a contradiction.
To sum up, there exits a component which joins
and
.
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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 ).
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Ma, R., Xu, Y. & Gao, C. A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems. Adv Differ Equ 2009, 671625 (2009). https://doi.org/10.1155/2009/671625
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DOI: https://doi.org/10.1155/2009/671625
Keywords
- Banach Space
- Open Covering
- Global Structure
- Connected Subset
- Global Bifurcation