Review Article | Open Access
Zhenguo Wang, Zhan Zhou, "Boundary Value Problem for a Second-Order Difference Equation with Resonance", Complexity, vol. 2020, Article ID 7527030, 10 pages, 2020. https://doi.org/10.1155/2020/7527030
Boundary Value Problem for a Second-Order Difference Equation with Resonance
In this paper, we study the existence and multiplicity of nontrivial solutions of a second-order discrete boundary value problem with resonance and sublinear or superlinear nonlinearity. The main methods are based on the Morse theory and the minimax methods. In addition, some examples are given to illustrate our results.
Let and be the sets of integers and real numbers, respectively. For , denotes the discrete interval if .
In this paper, we consider the existence and multiplicity of nontrivial solutions for the following discrete Dirichlet boundary value problem:where the forward difference operator is defined by ; and are real valued on , and is nonzero; and satisfies for each . Clearly, problem (1) has the trivial solution .
In the recent years, the existence of solutions for nonlinear difference equations has been widely studied by many authors. We note that these results were usually obtained by means of critical point theory, for example, the existence of ground state solutions , homoclinic orbits [2–5], the boundary value problem, and periodic solutions [6–8].
The boundary value problem (1) may be regarded as a discrete analogue of the following boundary value problem:which has been successfully applied to the modeling of astrophysics, gas dynamics, and chemically reacting system, see [9–11].
By various methods and techniques, many authors studied the similar second-order difference equation under various boundary value conditions. For example, Agarwal in  considered the existence of solutions for the boundary value problem (1) by the contraction mapping principle and Brouwer fixed point theorems in Euclidean space. Yu and Guo in  first used the critical point theory to study the following discrete boundary value problem:where and are the constants, and they proved the existence of solutions of problem (3) when the nonlinear term is sublinear or superlinear. We refer the readers to [13–16] and the reference therein for more information.
It is well known that the resonance exists in many real-world applications, and the equations with resonance have been extensively studied in various fields [17–21]. In fact, it is more difficult to study the boundary value problem under the case of resonance because the resonance case can change the local geometric properties of the critical points . In this paper, we study the existence and multiplicity of nontrivial solutions for problem (1) at resonance by means of the interaction between the nonlinearity and the spectrum of the symmetric matrix , where denotes a matrix whose elements are given by and for .
Note that in , the authors obtained the existence of one solution for (3) via the variational methods or the saddle point theorem. In the case where , the solution could be trivial, which was not considered in . In this paper, we will show that the boundary value problem (1) has at least one nontrivial solution. In fact, our Theorem 1 and Corollary 1 extend and complement the existence results given in  and establish the existence of multiple solutions.
Our results are obtained by combining the Morse theory [22–24], critical group computation, and the minimax methods including the local linking [17, 25]. We prove the existence of nontrivial solutions for problem (1) at resonance by the relationship between these nonlinear techniques and methods. In order to apply these ideas to identify the unknown critical point, we compute the corresponding critical groups of functional . Obviously, when we consider the existence of nontrivial solutions of problem (1) at resonance, we need to overcome much more difficulties than those in the literature .
We consider the -dimensional Banach space:
The space is a Hilbert space with the inner productendowed with the norm
Now, we define the -functional on as follows:for every , where , . We can compute the Frećhet derivative asfor all . It is clear that the critical points of are the solutions of problem (1).
For convenience, we identify with . Thus, we rewrite and asfurther,where denotes the transpose of and and are the symmetric matrices given by
Let be a real Banach space and . For , we write and .
In order to obtain the critical points of the functional on , we recall some basic concepts and results of the Morse theory.
Definition 1. (see ). is said to satisfy the Palais–Smale condition (PS condition for short), if any sequence for which is bounded and as possesses a convergent subsequence in .
In the Morse theory, functional is always required to satisfy the deformation condition (D), which was introduced by Bartsch and Li . Bartolo et al.  proved that satisfies the deformation condition (D) if satisfies the PS condition.
Assume that , and let be an isolated critical point of with , and is a neighborhood of , containing the unique critical point, then we callthe -th critical group of at , where stands for the -th singular relative homology group with integer coefficients [22, 23].
We say that is a homological nontrivial critical point of , if at least one of its critical groups is nontrivial.
Let satisfy the PS condition and all critical values of be greater than some , then the groupis called the -th critical group of at infinity .
Assume and satisfies the PS condition. The Morse-type numbers of the pair are defined by , and the Betti numbers of the pair are . By the Morse theory [22, 23], the following relations hold:By (14), it follows that for all . Hence, when for some and satisfies the PS condition, then must have a critical point satisfying . If has only one critical point , then , . If for some , then must have another critical point . Further, if are two critical points of and for some , then .
We need the following lemmas to prove the main results.
Lemma 1. (see ). Let be a real Banach space and satisfy the (D) condition and be bounded from below. If has a critical point that is homological nontrivial and is not the minimizer of , then has at least three critical points.
Obviously, we need to show that has a homological nontrivial critical point, which is not the global minimizer in this lemma.
Then, . Moreover, if or , then , where and denote the Morse index and the nullity of at 0, respectively.
From Lemma 2, if is a trivial critical point and has a local linking structure at 0, then 0 is a homological nontrivial critical point of .
Lemma 3 (see ). Let be a Banach space and satisfy the PS condition. Suppose splits as such that is bounded from below on and for as . Then, for .
Lemma 4. (see ). Let be tophological spaces and . Ifthen
2. Sublinear Case
In this section, we will consider the case where is sublinear in for each . Let the spectrum of the symmetric matrix consist of isolated eigenvalues of finite multiplicity denoted by
It follows thatfor every .
We assume the following conditions and are satisfied:(i) There exist such that(ii) There exist , for some , and such that
Theorem 1. Assume that the matrix is positive definite and satisfies and . Then, problem (1) has at least two nontrivial solutions in .
To prove Theorem 1, we need a series of lemmas.
Lemma 5. If the matrix is positive definite and holds. Then, satisfies the PS condition.
Proof. For any sequence , with is bounded and as , there exists a positive constant such that . By , we havewhere . Then, for any ,Since , thus is bounded in and the Bolzano–Weierstrass theorem implies that has a convergent subsequence.
Lemma 6. If the matrix is positive definite and holds. Then, is coercive on , that is, as .
Proof. If the matrix is positive definite, then , . Since is satisfied, it implies that there exist constants and such thatthenSincewe haveThe proof is complete.
Let denote the eigenspace of for some , denote the subspace of spanned by the eigenfunctions corresponding to the eigenvalues , and denote the subspace of spanned by the eigenfunctions corresponding to the eigenvalues , then we are given an orthogonal decomposition
Lemma 7. Assume that holds. Then, has a local linking at the origin with respect to and .
Proof. Let , for each and implies . By , we haveFor each , implies . We have the following estimates:This completes the proof.
Proof. Proof of Theorem 1. By Lemmas 5 and 6, satisfies the PS condition and is coercive, hence is bounded from below. Combining Lemma 7 with Lemma 2, the trivial solution is homological nontrivial and is not a minimizer. We apply Lemma 1 to conclude that problem (1) has at least two nontrivial solutions in .
Remark 1. (i) The conclusion of Theorem 1 holds, if the condition is replaced by the following condition: There exist , for some , and such that(ii)The condition means that problem (1) is resonant near 0 at the eigenvalue from the right side, and contains the situation .(iii)If for each , then problem (1) admits the trivial solution . In Theorem 1, we find two nontrivial solutions for (1). From and , we see that the existence of nontrivial solutions for problem (1) depends on the behaviors of the term or at 0 and at infinity .
Example 1. Let , and we consider the boundary value problem (1) withfor all . Then,Fix , , and , then the matrix ,is positive definite and admits two distinct eigenvalues given by .
For each , we havethus the condition holds.
Let , we see thatthen, the condition holds. Clearly, the conditions of Theorem 1 are satisfied, and problem (1) admits at least two nontrivial solutions in .
In fact, if for , then problem (1) isWe can show that and are the only two nontrivial solutions of problem (38) in .
Theorem 2. Assume that the matrix is negative definite, and the conditions and hold. Then, problem (1) possesses at least one nontrivial solution.
Proof. We now show that we can find at least one nontrivial critical point of functional .
Since the matrix is negative definite, in this case, for every and the inequality (20) holds. If satisfies , by , we haveWe notice that if is continuity and coercive, then has at least a local maximum by the above inequality. This means that . If , then , noticing Lemma 7, this contradicts Lemma 2. Hence, 0 is different from .
Theorem 3. Assume that matrix is negative semidefinite, and
Then, problem (1) has no nontrivial solution.
Proof. If we assume that (1) has a nontrivial solution, then has a nonzero critical point . Since and the matrix is negative semidefinite, one hasThis contradicts with . The proof is complete.
If the matrix is just nonsingular, we may suppose that its first positive eigenvalue is . We denote its eigenvalues as follows:Then, we have
Theorem 4. If the matrix is just nonsingular and , we assume that and the following conditions hold:(i) (ii) There exists such that for , Then, problem (1) has at least one nontrivial solution.
Proof. We show that has a local linking at 0. By , let , then there exists such thatSince holds, then we obtainFor every and , one has . Then,For every , implies . Thus, we getWe can apply Lemma 2 to conclude that has a local linking at 0, then , , and 0 is not local maximum in .
Let be the element of , and owing to , we havetherefore, and .
We see that satisfies the PS condition from , and we havefor every .
On the other hand, we havefor every . Hence, is bounded from below on .
It can be easily seen that the functional satisfies all the assumptions of Lemma 3; hence, we haveWe note that , and then has a critical point such that . Thus, has at least one nontrivial solution .
3. Superlinear Case
In this section, we will study the case where is superlinear in at infinity for each . We give the following assumption:(i) There exist such that
We apply a similar method in  to look for nontrivial critical points of in .
In order to prove the main results of this section, we need to make use of the following lemmas.
Lemma 8. If holds, then satisfies the PS condition.
Proof. Suppose that the eigenvalues of the matrix are , let . Using , there exist constants such thatFor any sequence , for some and as , then we haveSincethen,We haveFor any ,Note that , thus is bounded in , and the Bolzano–Weierstrass theorem implies that has a convergent subsequence.
Lemma 9. Assume that holds. Then, there exists a constant D > 0 such thatwhere is the unit sphere in .
Proof. From , we obtainLet , thenwe have as .
Let for . In view of , we haveLet , , andWe can compute the derivative of with respect to and obtainBy the implicit function theorem, there exists a unique such that and for . If , let , then and , for . Further, if , then .
Now, we define a functionObviously, is a continuous function.
We set a map as since is a strong deformation retract of , that is, .
Theorem 5. Assume that the conditions and hold. Then, problem (1) has at least one nontrivial solution.
Proof. Since the conditions and hold, it follows from Lemma 2, Lemma 7, and (, Theorem 4.2) that there exists a such thatwhere .
By Lemma 9, we haveUsing Lemma 4,Hence, has a critical point for whichThe proof is complete.
Note that the conclusion of Theorem 5 holds if the symmetric matrix is positive definite or negative definite or just nonsingular.
Example 2. Let , and we study the boundary value problem (1) withfor .
Then,Let , , and for each . The matrix ,admits three distinct eigenvalues given byClearly, we have