Solutions of 2th-Order Boundary Value Problem for Difference Equation via Variational Method
Advances in Difference Equations volume 2009, Article number: 730484 (2009)
The variational method and critical point theory are employed to investigate the existence of solutions for 2th-order difference equation for with boundary value condition by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.
Difference equations have been applied as models in vast areas such as finance insurance, biological populations, disease control, genetic study, physical field, and computer application technology. Because of their importance, many literature deals with its existence and uniqueness problems. For example, see [1–10].
We notice that the existing results are usually obtained by various analytical techniques, for example, the conical shell fixed point theorem [1, 6], Banach contraction map method , Leray-Schauder fixed point theorem [2, 10], and the upper and lower solution method . It seems that the variational technique combining with the critical point theory  developed in the recent decades is one of the effective ways to study the boundary value problems of difference equations. However because the variational method requires a "symmetrical'' functional, it is hard for the odd-order difference equations to create a functional satisfying the "symmetrical'' property. Therefore, the even-order difference equations have been investigated in most references.
Let , be integers, and , be a discrete interval in . Inspired by [5, 8], in this paper, we try to investigate the following th-order boundary value problem (BVP) of difference equation via variational method combining with some traditional analytical skills:
where is the forward difference operator; for and A variational functional for BVP (1.1)-(1.2) is constructed which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points of this functional. In order to prove the existence criteria of critical points of the functional, some lemmas are given in Section 2. Two criteria for the existence of at least one solution and two solutions for BVP (1.1)-(1.2) are established in Section 3 which is the generalization for BVP of the even-order difference equations. The existence results obtained in this paper are not found in the references, to the best of our knowledge.
For convenience, we will use the following notations in the following sections:
2. Variational Structure and Preliminaries
Let be a real reflexive Banach space with a norm , and let be a weakly lower (upper) semicontinuous functional, such that
then there exists such that
Furthermore, if has bounded linear Gâteaux derivative, then
Lemma 2.2 (mountain-pass lemma).
Let be a real Banach space, and let be continuously differential, satisfying the P-S condition. Assume that and is an open neighborhood of , but If then is the critical value of where
This means that there exists , s.t. ,
The following lemma will be used in the proof of Lemma 2.4.
If is a symmetric and positive-defined real matrix, is a real matrix, is the transposed matrix of Then is positive defined if and only if
Since is positive defined, then
Let be a Hilbert space defined by
with the norm
Hence is an -dimensional Hilbert space. For any , let then one can show that there exist constants s.t. ; that is, is an equivalent norm of (see [9, page 68]).
Since , , By using the inequality , , we have
Repeating the above process, we obtain
On the other hand, define , where is the combination number, then we can rewrite in a vector form, that is, where , , and
where Hence Note that
and by Lemma 2.3 with , we know that is positive defined. Hence all eigenvalues of are real and positive. Let be the minimal eigenvalue of these eigenvalues, then Therefore that is,
However, how to find the in Lemma 2.4 is a skillful and challenging task. The following lemma from  offers some help for the estimation of .
Lemma 2.5 (Brualdi ).
If is weak irreducible, then each eigenvalue is contained in the set
In Lemma 2.5, is the complex number set, and the denotations , , , can be found in . Since is positive defined, all eigenvalues are positive real numbers. Therefore, by Lemma 2.5, let
where . is a subset of and can be calculated directly from Define If , we can use this as in Lemma 2.4. If , then one needs to calculate the eigenvalues directly.
Define the functional on by
Then is with
where and is the inner product in . In fact, we have
The continuity of and the right-hand side of the inequality in Lemma 2.4 lead to (2.15).
Furthermore, for any we have , By using the following formula (e.g., see [14, page 28]):
Repeating the above process, we obtain
Let , that is,
Since is arbitrary, we know that the solution of BVP (1.1)-(1.2) corresponds to the critical point of
3. Main Results
Now we present our main results of this paper.
If there exist , , , and s.t.
then BVP (1.1)-(1.2) has at least one solution.
In fact, we can choose a suitable such that
Since there exists with we have
Then by Lemma 2.4, we obtain
Noticing that we have From Lemma 2.1, the conclusion of this lemma follows.
If there exists s.t. for all , and
where , satisfy either , or , then BVP (1.1)-(1.2) has at least one solution.
Assume that , Then for there exists such that as . We have from the continuity of that there is a such that for all , . When , one has for then
when , one has for then
Let , then we have for Therefore, we have
which implies and by Lemma 2.1, the conclusion of this lemma follows.
Assume that , Then for there exists such that as . We have from the continuity of that there is a such that for all , . When , one has , then we have
when , one has , then we have
Let , then we have for Therefore, by Theorem 3.1, the conclusion of this lemma follows.
Assume that , and
, is defined in Lemma 2.4;
satisfies (3.1) in Theorem 3.1 or satisfies the assumptions in Corollary 3.2.
Then BVP (1.1)-(1.2) has at least two solutions.
We first show that satisfies the P-S condition. Let satisfy that is bounded and If is unbounded, it possesses a divergent subseries, say as . However from (ii), we get (3.4) or (3.8), hence as , which is contradictory to the the fact that is bounded.
Next we use the mountain-pass lemma to finish the proof. By (i), for , there exists such that for . Then for . Now together with Lemma 2.3 we have
which implies that
where is the zero element in , and Since we have from (3.4) or (3.8) that there exists with that is, but Using Lemma 2.2, we have shown that is the critical value of with defined as
We denote as its corresponding critical point.
On the other hand, by Theorem 3.1 or Corollary 3.2, we know that there exists s.t. If the theorem is proved. If on the contrary, , that is, that implies for any , Taking in with , by the continuity of there exist s.t. , Hence , are two different critical points of that is, BVP (1.1)-(1.2) has at least two different solutions.
4. An Example
Consider the 6th-order boundary value problem for difference equation
Let , we have , Hence satisfies the conditions in Theorem 3.3, the boundary value problem (4.1) has at least two solutions.
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This research is partially supported by the NSF of China and NSF of Guangdong Province.
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Zou, Q., Weng, P. Solutions of 2th-Order Boundary Value Problem for Difference Equation via Variational Method. Adv Differ Equ 2009, 730484 (2009). https://doi.org/10.1155/2009/730484