We consider the Green's functions and the existence of positive solutions for a secondorder functional difference equation with fourpoint 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 secondorder 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 twopoint BVP for a nonlinear functional difference equation with Laplacian operator
where , , , , , , is continuous, .
Yang et al. [9] considered twopoint 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 maxnorm
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 secondorder fourpoint 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 wellposedness 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 secondorder fourpoint linear BVP (2.13) is given by
where
Proof.
Consider the secondorder twopoint BVP
It is easy to find that the solution of BVP (2.16) is given by
The threepoint 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 fourpoint 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 secondorder fourpoint 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.
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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