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Solutions and Green's Functions for Boundary Value Problems of Second-Order Four-Point Functional Difference Equations
Boundary Value Problems volume 2010, Article number: 973731 (2010)
Abstract
We consider the Green's functions and the existence of positive solutions for a second-order functional difference equation with four-point boundary conditions.
1. Introduction
In recent years, boundary value problems (BVPs) of differential and difference equations have been studied widely and there are many excellent results (see Gai et al. [1], Guo and Tian [2], Henderson and Peterson [3], and Yang et al. [4]). By using the critical point theory, Deng and Shi [5] studied the existence and multiplicity of the boundary value problems to a class of second-order functional difference equations

with boundary value conditions

where the operator is the Jacobi operator

Ntouyas et al. [6] and Wong [7] investigated the existence of solutions of a BVP for functional differential equations

where is a continuous function,
, and
.
Weng and Guo [8] considered the following two-point BVP for a nonlinear functional difference equation with -Laplacian operator

where ,
,
,
,
,
,
is continuous,
.
Yang et al. [9] considered two-point BVP of the following functional difference equation with -Laplacian operator:

where ,
, and
,
,
, and
are nonnegative real constants.
For and
, let

Then and
are both Banach spaces endowed with the max-norm

For any real function defined on the interval
and any
with
, we denote by
an element of
defined by
,
.
In this paper, we consider the following second-order four-point BVP of a nonlinear functional difference equation:

where and
,
,
,
,
is a continuous function,
and
for
,
,
, and
are nonnegative real constants, and
for
.
At this point, it is necessary to make some remarks on the first boundary condition in (1.9). This condition is a generalization of the classical condition

from ordinary difference equations. Here this condition connects the history with the single
. This is suggested by the well-posedness of BVP (1.9), since the function
depends on the term
(i.e., past values of
).
As usual, a sequence is said to be a positive solution of BVP (1.9) if it satisfies BVP (1.9) and
for
with
for
.
2. The Green's Function of (1.9)
First we consider the nonexistence of positive solutions of (1.9). We have the following result.
Lemma 2.1.
Assume that

or

Then (1.9) has no positive solution.
Proof.
From , we know that
is convex for
.
Assume that is a positive solution of (1.9) and (2.1) holds.
-
(1)
Consider that
.
If , then
. It follows that

which is a contradiction to the convexity of .
If , then
. If
, then we have

Hence

which is a contradiction to the convexity of . If
for
, then
is a trivial solution. So there exists a
such that
.
We assume that . Then

Hence

which is a contradiction to the convexity of .
If , similar to the above proof, we can also get a contradiction.
-
(2)
Consider that
.
Now we have

which is a contradiction to the convexity of .
Assume that is a positive solution of (1.9) and (2.2) holds.
-
(1)
Consider that
.
If , then we obtain

which is a contradiction to the convexity of .
If , similar to the above proof, we can also get a contradiction.
If , and so
, then there exists a
such that
. Otherwise,
is a trivial solution. Assume that
, then

which implies that

A contradiction to the convexity of follows.
If , we can also get a contradiction.
-
(2)
Consider that
.
Now we obtain

which is a contradiction to the convexity of .
Next, we consider the existence of the Green's function of equation

We always assume that
()
,
and
.
Motivated by Zhao [10], we have the following conclusions.
Theorem 2.2.
The Green's function for second-order four-point linear BVP (2.13) is given by

where

Proof.
Consider the second-order two-point BVP

It is easy to find that the solution of BVP (2.16) is given by


The three-point BVP

can be obtained from replacing by
in (2.16). Thus we suppose that the solution of (2.19) can be expressed by

where and
are constants that will be determined.
From (2.18) and (2.20), we have

Putting the above equations into (2.19) yields

By (H), we obtain
and
by solving the above equation:

By (2.19) and (2.20), we have

The four-point BVP (2.13) can be obtained from replacing by
in (2.19). Thus we suppose that the solution of (2.13) can be expressed by

where and
are constants that will be determined.
From (2.24) and (2.25), we get

Putting the above equations into (2.13) yields

By (H), we can easily obtain

Then by (2.17), (2.20), (2.23), (2.25), and (2.28), the solution of BVP (2.13) can be expressed by

where is defined in (2.14). That is,
is the Green's function of BVP (2.13).
Remark 2.3.
By (H), we can see that
for
. Let

Then .
Lemma 2.4.
Assume that () holds. Then the second-order four-point BVP (2.13) has a unique solution which is given in (2.29).
Proof.
We need only to show the uniqueness.
Obviously, in (2.29) is a solution of BVP (2.13). Assume that
is another solution of BVP (2.13). Let

Then by (2.13), we have


From (2.32) we have, for ,

which implies that

Combining (2.33) with (2.35), we obtain

Condition (H) implies that (2.36) has a unique solution
. Therefore
for
. This completes the proof of the uniqueness of the solution.
3. Existence of Positive Solutions
In this section, we discuss the BVP (1.9).
Assume that ,
.
We rewrite BVP (1.9) as

with .
Suppose that is a solution of the BVP (3.1). Then it can be expressed as

Lemma 3.1 (see Guo et al. [11]).
Assume that is a Banach space and
is a cone in
. Let
. Furthermore, assume that
is a completely continuous operator and
for
. Thus, one has the following conclusions:
-
(1)
if
 for
, then
;
-
(2)
if
 for
, then
.
Assume that . Then (3.1) may be rewritten as

Let be a solution of (3.3). Then by (3.2) and
, it can be expressed as

Let be a solution of BVP (3.1) and
. Then for
we have
and

Let

Then is a Banach space endowed with norm
and
is a cone in
.
For , we have by (H
) and the definition of
,

For every ,
, by the definition of
and (3.5), if
, we have

If , we have, by (3.4),

hence by the definition of , we obtain for

Lemma 3.2.
For every , there is
, such that

Proof.
For , and
, by the definitions of
and
, we have

Obviously, there is a , such that (3.11) holds.
Define an operator by

Then we may transform our existence problem of positive solutions of BVP (3.1) into a fixed point problem of operator (3.13).
Lemma 3.3.
Consider that .
Proof.
If and
,
and
, respectively. Thus, (H
) yields

It follows from the definition of that

which implies that .
Lemma 3.4.
Suppose that () holds. Then
is completely continuous.
We assume that
(H)
(H)
.
We have the following main results.
Theorem 3.5.
Assume that ()–(
) hold. Then BVP (3.1) has at least one positive solution if the following conditions are satisfied:
(H) there exists a
such that, for
, if
, then
;
(H) there exists a
such that, for
, if
, then
or
(H)
;
(H) there exists a
such that, for
, if
, then
;
(H) there exists an
, such that, for
, if
, then
where

Proof.
Assume that () and (
) hold. For every
, we have
, thus

which implies by Lemma 3.1 that

For every , by (3.8)–(3.10) and Lemma 3.2, we have, for
,
. Then by (3.13) and (
), we have

which implies by Lemma 3.1 that

So by (3.18) and (3.20), there exists one positive fixed point of operator
with
.
Assume that ()–(
) hold, for every
and
,
, by (
), we have

Thus we have from Lemma 3.1 that

For every , by (3.8)–(3.10), we have
,

Thus we have from Lemma 3.1 that

So by (3.22) and (3.24), there exists one positive fixed point of operator
with
.
Consequently, or
is a positive solution of BVP (3.1).
Theorem 3.6.
Assume that ()–(
) hold. Then BVP (3.1) has at least one positive solution if (
) and (
) or (
) and (
) hold.
Theorem 3.7.
Assume that ()–(
) hold. Then BVP (3.1) has at least two positive solutions if (H
), (H
), and (H
)  or (H
), (H
), and (H
) hold.
Theorem 3.8.
Assume that (H)–(H
) hold. Then BVP (3.1) has at least three positive solutions if (H
)–(H
) hold.
Assume that ,
, and
()
Define as follows:

which satisfies
H
.
Obviously, exists.
Assume that is a solution of (1.9). Let

where

By (1.9), (3.26), (3.27), (H), (H
), and the definition of
, we have


and, for ,

Let

Then by (3.27), (H), (H
), and the definition of
, we have
for
Thus, the BVP (1.9) can be changed into the following BVP:

with and
.
Similar to the above proof, we can show that (1.9) has at least one positive solution. Consequently, (1.9) has at least one positive solution.
Example 3.9.
Consider the following BVP:

That is,

Then we obtain

Let

where
By calculation, we can see that (H)–(H
) hold, then by Theorem 3.8, the BVP (3.33) has at least three positive solutions.
4. Eigenvalue Intervals
In this section, we consider the following BVP with parameter :

with .
The BVP (4.1) is equivalent to the equation

Let be the solution of (3.3),
. Then we have

Let and
be defined as the above. Define
by

Then solving the BVP (4.1) is equivalent to finding fixed points in . Obviously
is completely continuous and keeps the
invariant for
.
Define

respectively. We have the following results.
Theorem 4.1.
Assume that (H), (H
), (H
),
(H)  
,
(H)   
hold, where , then BVP (4.1) has at least one positive solution, where
is a positive constant.
Proof.
Assume that condition (H) holds. If
and
, there exists an
sufficiently small, such that

By the definition of , there is an
, such that for
,

It follows that, for and
,

For every and
, by (3.9), we have

Therefore by (3.13) and Lemma 3.2, we have

If then for a sufficiently small
, we have
Similar to the above, for every
, we obtain by (3.10)

If , choose
sufficiently large, such that

By the definition of , there is an
such that, for
and

For every , by (3.8)–(3.10) and (3.13), we have

which implies that

Finally, we consider the assumption . By the definition of
, there is
, such that, for
and
,

We now show that there is   , such that, for
,
In fact, for
 
and every
,
hence in a similar way, we have

which implies that

Theorem 4.2.
Assume that (H),(H
), and (H
) hold. If
or
, then there is a
such that for
, BVP (4.1) has at least one positive solution.
Proof.
Let be given. Define

Then , where
.
For every , we know that
. By the definition of operator
, we obtain

It follows that we can take such that, for all
and all
,

Fix . If
, for
, we obtain a sufficiently large
such that, for
,

It follows that, for and
,

For every , by the definition of
,
and the definition of Lemma 3.2, there exists a
such that
and
, thus
. Hence

If there is
, such that, for
and
,

where .
For every , by(3.8)–(3.10) and Lemma 3.2,

which by combining with (4.21) completes the proof.
Example 4.3.
Consider the BVP(3.33) in Example 3.9 with

where ,
is some positive constant,
.
By calculation, ,
, and
; let
. Then by Theorem(4.1), for
, the above equation has at least one positive solution.
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Acknowledgments
The authors would like to thank the editor and the reviewers for their valuable comments and suggestions which helped to significantly improve the paper. This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.
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Shujie, Y., Bao, S. Solutions and Green's Functions for Boundary Value Problems of Second-Order Four-Point Functional Difference Equations. Bound Value Probl 2010, 973731 (2010). https://doi.org/10.1155/2010/973731
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DOI: https://doi.org/10.1155/2010/973731