- Research Article
- Open access
- Published:
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ1_HTML.gif)
with boundary value conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ2_HTML.gif)
where the operator is the Jacobi operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ3_HTML.gif)
Ntouyas et al. [6] and Wong [7] investigated the existence of solutions of a BVP for functional differential equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ5_HTML.gif)
where ,
,
,
,
,
,
is continuous,
.
Yang et al. [9] considered two-point BVP of the following functional difference equation with -Laplacian operator:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ6_HTML.gif)
where ,
, and
,
,
, and
are nonnegative real constants.
For and
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ7_HTML.gif)
Then and
are both Banach spaces endowed with the max-norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ8_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ11_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ13_HTML.gif)
which is a contradiction to the convexity of .
If , then
. If
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ14_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ16_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ20_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ21_HTML.gif)
A contradiction to the convexity of follows.
If , we can also get a contradiction.
-
(2)
Consider that
.
Now we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ22_HTML.gif)
which is a contradiction to the convexity of .
Next, we consider the existence of the Green's function of equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ24_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ25_HTML.gif)
Proof.
Consider the second-order two-point BVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ26_HTML.gif)
It is easy to find that the solution of BVP (2.16) is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ28_HTML.gif)
The three-point BVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ29_HTML.gif)
can be obtained from replacing by
in (2.16). Thus we suppose that the solution of (2.19) can be expressed by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ30_HTML.gif)
where and
are constants that will be determined.
From (2.18) and (2.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ31_HTML.gif)
Putting the above equations into (2.19) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ32_HTML.gif)
By (H), we obtain
and
by solving the above equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ33_HTML.gif)
By (2.19) and (2.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ35_HTML.gif)
where and
are constants that will be determined.
From (2.24) and (2.25), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ36_HTML.gif)
Putting the above equations into (2.13) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ37_HTML.gif)
By (H), we can easily obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ38_HTML.gif)
Then by (2.17), (2.20), (2.23), (2.25), and (2.28), the solution of BVP (2.13) can be expressed by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ41_HTML.gif)
Then by (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ43_HTML.gif)
From (2.32) we have, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ44_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ45_HTML.gif)
Combining (2.33) with (2.35), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ47_HTML.gif)
with .
Suppose that is a solution of the BVP (3.1). Then it can be expressed as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ49_HTML.gif)
Let be a solution of (3.3). Then by (3.2) and
, it can be expressed as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ50_HTML.gif)
Let be a solution of BVP (3.1) and
. Then for
we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ51_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ52_HTML.gif)
Then is a Banach space endowed with norm
and
is a cone in
.
For , we have by (H
) and the definition of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ53_HTML.gif)
For every ,
, by the definition of
and (3.5), if
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ54_HTML.gif)
If , we have, by (3.4),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ55_HTML.gif)
hence by the definition of , we obtain for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ56_HTML.gif)
Lemma 3.2.
For every , there is
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ57_HTML.gif)
Proof.
For , and
, by the definitions of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ58_HTML.gif)
Obviously, there is a , such that (3.11) holds.
Define an operator by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ60_HTML.gif)
It follows from the definition of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ62_HTML.gif)
Proof.
Assume that () and (
) hold. For every
, we have
, thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ63_HTML.gif)
which implies by Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ64_HTML.gif)
For every , by (3.8)–(3.10) and Lemma 3.2, we have, for
,
. Then by (3.13) and (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ65_HTML.gif)
which implies by Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ66_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ67_HTML.gif)
Thus we have from Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ68_HTML.gif)
For every , by (3.8)–(3.10), we have
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ69_HTML.gif)
Thus we have from Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ70_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ71_HTML.gif)
which satisfies
H
.
Obviously, exists.
Assume that is a solution of (1.9). Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ72_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ73_HTML.gif)
By (1.9), (3.26), (3.27), (H), (H
), and the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ74_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ75_HTML.gif)
and, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ76_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ77_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ78_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ79_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ80_HTML.gif)
Then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ81_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ82_HTML.gif)
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 :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ83_HTML.gif)
with .
The BVP (4.1) is equivalent to the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ84_HTML.gif)
Let be the solution of (3.3),
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ85_HTML.gif)
Let and
be defined as the above. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ86_HTML.gif)
Then solving the BVP (4.1) is equivalent to finding fixed points in . Obviously
is completely continuous and keeps the
invariant for
.
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ87_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ88_HTML.gif)
By the definition of , there is an
, such that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ89_HTML.gif)
It follows that, for and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ90_HTML.gif)
For every and
, by (3.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ91_HTML.gif)
Therefore by (3.13) and Lemma 3.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ92_HTML.gif)
If then for a sufficiently small
, we have
Similar to the above, for every
, we obtain by (3.10)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ93_HTML.gif)
If , choose
sufficiently large, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ94_HTML.gif)
By the definition of , there is an
such that, for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ95_HTML.gif)
For every , by (3.8)–(3.10) and (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ96_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ97_HTML.gif)
Finally, we consider the assumption . By the definition of
, there is
, such that, for
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ98_HTML.gif)
We now show that there is   , such that, for
,
In fact, for
 
and every
,
hence in a similar way, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ99_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ100_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ101_HTML.gif)
Then , where
.
For every , we know that
. By the definition of operator
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ102_HTML.gif)
It follows that we can take such that, for all
and all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ103_HTML.gif)
Fix . If
, for
, we obtain a sufficiently large
such that, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ104_HTML.gif)
It follows that, for and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ105_HTML.gif)
For every , by the definition of
,
and the definition of Lemma 3.2, there exists a
such that
and
, thus
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ106_HTML.gif)
If there is
, such that, for
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ107_HTML.gif)
where .
For every , by(3.8)–(3.10) and Lemma 3.2,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ108_HTML.gif)
which by combining with (4.21) completes the proof.
Example 4.3.
Consider the BVP(3.33) in Example 3.9 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ109_HTML.gif)
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