Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem
This paper studies the existence of multiple solutions of the second-order difference boundary value problem , , . By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalue of linear difference problem , , near infinity and the trivial solution of the first equation is a local minimizer under some assumptions on .
Let , , and be the sets of real numbers, natural numbers, and integers, respectively. For any , , define .
Consider the second-order difference boundary value problem (BVP)
where and denotes the forward difference operator defined by , .
By a solution of the BVP (1.1), we mean a real sequence satisfying the BVP (1.1). For with , we say that if there exists at least one such that . We say that is positive (and write ) if for all , , and similarly, is negative () if for all , . The aim of this paper is to obtain the existence of multiple solutions of the BVP (1.1) and analyse the sign of solutions.
Recently, a few authors applied the minimax methods to examine the difference boundary value problems. For example, in , Agarwal et al. employed the Mountain Pass Lemma to study the following BVP:
and obtained the existence of multiple positive solutions, where may be singular at . In , Jiang and Zhou employed the Mountain Pass Lemma together with strongly monotone operator principle, to study the following difference BVP:
and obtained existence and uniqueness results, where is continuous. In , Cai and Yu employed the Linking Theorem and the Mountain Pass Lemma to study the following difference BVP:
and obtained the existence of multiple solutions, where is the ratio of odd positive integers, and are real sequences, for all , and , are two given constants, is continuous.
Although applications of the minimax methods in the field of the difference BVP have attracted some scholarly attention in the recent years, efforts in applying Morse theory to the difference BVP are scarce. The main purpose of this paper is to develop a new approach to the BVP (1.1) by using Morse theory. To this end, we first consider the following linear difference eigenvalue problem:
On the above eigenvalue problem, the following results hold; see .
Proposition 1.1. The eigenvalues of (1.5) are and the corresponding eigenfunction with is , .
Remark 1.2. (1) The set of functions is orthogonal on with respect to the weight function , that is,
Moreover, for each ,
(2) It is easy to see that is positive and changes sign for each , that is, and .
For (1.1), we assume that
Let By (1.9) we have
Assume that the following conditions on hold.If such that , then there exist and such that
where , , .
The main result of this paper is as follows.
Theorem 1.3. Let (1.8), (1.9) hold and (1) for all , (2)hold. Then the BVP (1.1) has at least three nontrivial solutions, with one positive solution and one negative solution, in each of the following cases: (i)() and ; (ii)() and .
To the author’s best knowledge, only Bin et al.  deal with the existence and multiplicity of nontrivial periodic solutions for asymptotically linear resonant difference problem by the aid of Su . In , satisfies
where , , , . In , the authors obtained the existence of one nontrivial periodic solution. Notice that (1.13) implies that (1.11) holds; however, () is not covered by (1.14). In fact, conditions (1.13) and (1.14) are borrowed from . The conditions in Theorem 1.3 coincide with the assumptions of Theorem in . The aim of this paper is to develop a new approach to study the discrete systems by using Morse theory, minimax theorems, and some analysis technique. We wish to have some breakthrough points with the aid of the method of discretization.
The remaining part of this paper proceeds as follows. In the next section, we establish the variational framework of the BVP (1.1) and collect some results which will be used in the proof of Theorem 1.3. In Section 3, we give the proof of Theorem 1.3. Finally, in Section 4, we give an example to illustrate our main result and summarize conclusions and future directions.
2. Variational Framework and Auxiliary Results
can be equipped with the norm and the inner product as follows:
where denotes the Euclidean norm in and denotes the usual scalar product in . It is easy to see that is a Hilbert space. Consider the functional defined on by
We claim that if is a critical point of , then is precisely a solution of the BVP (1.1). Indeed, for every , we have
So, if , then we have
Since is arbitrary, we obtain
Therefore, we reduce the problem of finding solutions of the BVP (1.1) to that of seeking critical points of the functional in .
see  for details.
Now we collect some results on Morse theory and the minimax methods.
Let be a real Hilbert space and . Denote
for . The following is the definition of the Palais-Smale condition ((PS) condition).
Definition 2.1. The functional satisfies the (PS) condition if any sequence such that is bounded and as has a convergent subsequence.
In , Cerami introduced a weak version of the (PS) condition as follows.
Definition 2.2. The functional satisfies the Cerami condition ( condition) if any sequence such that is bounded and as has a convergent subsequence.
Definition 2.3. The functional satisfies the condition at the level if for any and any neighborhood of , there are and a continuous deformation such that(1) for all ;(2) for all ;(3) is nonincreasing in for any ;(4). satisfies the condition if satisfies the ) condition for all .
Let be an isolated critical point of with , and let be a neighborhood of , the group
is called the th critical group of at , where denotes the th singular relative homology group of the pair over a field , which is defined to be quotient , where is the th singular relative closed chain group and is the th singular relative boundary chain group.
Let . If is bounded from below by and satisfies the ) condition for all , then the group
is called the th critical group of at infinity .
Assume that and satisfies the () condition. The Morse-type numbers of the pair are defined by
and the Betti numbers of the pair are
Thus, if , for some , then there must exist a critical point of with , which can be rephrased as follows.
Proposition 2.4. Let be a real Hilbert space and . Assume that and that satisfies the condition. If there exists some such that , then must have a critical point with .
In order to prove our main result, we need the following result about the critical group on .
Proposition 2.5. Let the functional be of the form where is a self-adjoint linear operator such that is isolated in , the spectrum of . Assume that satisfies Denote , , where () is the subspace of on which is positive (negative) definite. Assume that , are finite and that satisfies the condition. Then provided that satisfies the angle conditions at infinity. (): there exist and such that where , , , and .
Remark 2.6. Conditions (2.16) and (2.17) imply that is asymptotically quadratic. Bartsch and Li  introduced the notion of critical groups at infinity and proved that if satisfied some angle properties at infinity, the critical groups can be completely figured out. Proposition 2.5 is a slight improvement of [11, Proposition ] by Su and Zhao . There are many other papers considering concrete problems by computing the critical groups at infinity with different methods, for example, see [14–17].
Let denote the open ball in about of radius and let denote its boundary.
Theorem 2.7 (mountain pass lemma). Let be a real Banach space and satisfying the (PS) condition. Suppose and that (1) there are constants such that , (2) there is a such that then possesses a critical value . Moreover can be characterized as where
Definition 2.8 (mountain pass point). An isolated critical point of is called a mountain pass point, if .
Theorem 2.9. Let be a real Hilbert space. Suppose that has a mountain pass point , and that is a Fredholm operator with finite Morse index, satisfying then
3. Proof of Theorem 1.3
We give the proof of Theorem 1.3 in this section. Firstly, we prove that the functional satisfies the condition (Lemma 3.1) and compute the critical group (Lemma 3.2). Then, we employ the cut-off technique and the Mountain Pass Lemma to obtain two critical points of and compute the critical groups and (Lemmas 3.3 and 3.4). Finally, we prove Theorem 1.3.
Rewrite the functional as
Proof. We only prove the case where holds. Let such that
Then for all , we have
Denote with , and . Since is a finite-dimensional Hilbert space, it suffices to show that is bounded. Suppose that is unbounded. Passing to a subsequence we may assume that as .
By (1.11), for any , there exists such that Let in (3.3). Then by (2.7), (2.9), and (3.4), we have where And since is arbitrary, we have Similarly, let in (3.3), by (2.8) and (3.4) we get where And hence we also have By (3.7) and (3.10), we have By , there exist and such that This implies that and hence which is a contradiction to (3.2). Thus is bounded. The proof is complete.
Proof. We only prove the case (1). Define a bilinear function
Then by (2.7) we have
And hence there exists a unique continuous bounded linear operator such that
Since for all , we can conclude that is a self-adjoint operator and
Then has the form (2.16) with
and (1.11) implies that (2.17) holds. Let . Then . Next we show that implies that the angle condition at infinity holds.
If not, then for any and each , there exists with , such that On the other hand, (3.20) implies Thus, by there exist and such that Therefore, which is a contradiction to (3.21). Consequently holds and by Lemma 3.1 and Proposition 2.5, . Similarly, we can prove that (2) holds.
In order to obtain a mountain pass point, we need the following lemmas.
Lemma 3.3. Let and . If then the functional satisfies the (PS) condition.
Proof. We only prove the case . Let such that
as . Since is a finite-dimensional space, it suffices to show that is bounded in . Suppose that is unbounded. Passing to a subsequence we may assume that and for each , either or is bounded.
Noticing that for all ,
Denote , for a subsequence, converges to some with . By (3.29), we have If , then where with . If is bounded, then Since , there is an for which . So passing to the limit in (3.30), we have This implies that satisfies
Now, we claim that
In fact, let We only need to prove . If not, assume that . Then by (3.34), we have and hence . By induction, it is easy to get for all which is a contradiction to and hence (3.35) holds.
On the other hand, by Proposition 1.1 and Remark 1.2, we see that only the eigenfunction corresponding to the eigenvalue is positive, which is a contradiction to . The proof is complete.
Lemma 3.4. Under the conditions of Theorem 1.3, the functional has a critical point and ; the functional has a critical point and .
Proof. We only prove the case of . Firstly, we prove that satisfies the Mountain Pass Lemma and hence has a nonzero critical point . In fact, and by Lemma 3.3 we see that satisfies the (PS) condition. Clearly . Thus we need to show that satisfies and . To verify , by (1.8) and , there exist and with such that
for . So, for all , if , then for each , and
where . Let
Then and hence holds. For , by , we claim that there exist , such that
In fact, by assumption (1.9), there exist and such that for . Meanwhile, there exists such that for by virtue of the continuity of . Let , we get the conclusion.
Thus, if we choose with and , then as . Thus, we can choose a constant large enough with and such that . holds.
Therefore, by Theorem 2.7, has a critical point and similar to the proof of Lemma 3.3, we can prove that . So is also a critical point of . In the following we compute the critical group by using Theorem 2.9.
Assume that and that there exists such that This implies that satisfies Hence the eigenvalue problem has an eigenvalue . implies that must be a simple eigenvalue; see . So, . Since is a finite-dimensional Hilbert space, the Morse index of must be finite and must be a Fredholm operator. By Theorem 2.9, The proof is complete.
Remark 3.5. We can choose the neighborhood of such that for all . Therefore, Similarly,
Now, we give the proof of Theorem 1.3.
Proof of Theorem 1.3. We only prove the case (i). By Lemma 3.2, Hence by Proposition 2.4 the functional has a critical point satisfying Since by and , we see that is a local minimum of . Hence By Remark 3.5, (3.49), (3.51), and we get that , and are three nonzero critical points of with and . The proof is complete.
4. An Example and Future Directions
To illustrate the use of Theorem 1.3, we offer the following example.
Example 4.1. Consider the BVP
where is defined as follows:
It is easy to verify that satisfies (1.8), (1.9), (1.11), , and with . To verify the condition , note that for , we claim that
which implies that holds.
To this end, for any constant , we introduce another norm in as follows: Since is finite dimensional, there exist two constants such that
Now, by , for any small enough, it is easy to see that holds for large enough.
Set Since , , for large enough. And for large enough, we have Here and below we denote by various positive constants. Since Hence Since is small enough, we get (4.3) holds by the above and (4.5). Hence, by Theorem 1.3, BVP (4.1) has at least three nontrivial solutions.
Morse theory has been proved very useful in proving the existence and multiplicity of solutions of operator equations with variational frameworks. However, it is well known that the minimax methods is also a useful tool for the same purpose. The advantage of the minimax methods is that it provides an estimate of the critical value. But it is hard to distinguish critical points obtained by this methods with those by other methods, if the local behavior of the critical points is not very well known. However, critical groups serve as a topological tool in distinguishing isolated critical points. Hence, in order to obtain multiple solutions by using Morse theory, it is crucial to describe critical groups clearly.
A natural question is: can we use the same methods in this paper to other BVPs? Noticing that the key conditions which guarantee the multiplicity of solutions of the BVP (1.1) are as follows:(1)the BVP has a variational framework;(2)the eigenvalues of the corresponding linear BVP are nonzero and there is a one-sign eigenfunction,
hence, if the difference equation
subject to some other boundary value conditions satisfying (1) and (2), then we can obtain similar results to Theorem 1.3.
Example 4.2. Consider the BVP
Then is a -dimensional Hilbert space with the inner product
Define the functional on by
It is easy to see that is a critical point of in if and only if is a solution of the BVP (4.12).
The eigenvalues of the linear BVP are and the corresponding eigenfunctions are Hence, for all and for all . Therefore, the BVP (4.12) satisfies (1) and (2) and hence we can obtain similar results as in Theorem 1.3.
However, consider the following difference BVP:
It is easy to verify that the variational functional of the BVP (4.19) is
But, is an eigenvalue of the linear BVP:
The authors would like to express their thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project (2009).
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