Abstract
The fourth-order discrete Dirichlet boundary value problem is also a discrete elastic beam problem. In this paper, the existence of infinitely many solutions to this problem is investigated through the critical point theory. By an important inequality we established and the oscillatory behavior of f either near the origin or at infinity, we obtain the existence of infinitely many solutions, which either converge to zero or unbounded. In the end, two examples are presented to illustrate our results.
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1 Introduction
Let \(\mathbb{Z}\) and \(\mathbb{R}\) denote the sets of integers and real numbers, respectively. Define \(\mathbb{Z}(a) = \{a, a + 1,\ldots \}\) and \(\mathbb{Z}(a, b)=\{a, a+1, \ldots , b\}\) for any \(a, b\in \mathbb{Z}\) with \(a\leq b\).
In this paper, we consider the following nonlinear fourth-order difference equation:
with the Dirichlet boundary value conditions
where T is a given positive integer, △ is the forward difference operator defined by \(\triangle u_{k} = u_{k+1} - u_{k}\), \(\triangle ^{2} u_{k} = \triangle (\triangle u_{k})\), \(p_{k}>0\) for all \(k\in \mathbb{Z}(-1, T)\), \(f: \mathbb{Z}(1, T)\times \mathbb{R} \to \mathbb{R} \) is continuous in the second variable.
Boundary value problem (1.1) with (1.2) can be regarded as a discrete analogue of the following fourth-order boundary value problem:
This problem gives the equilibrium state of a beam under simple bearing forces at both ends [1, 2]. In the mechanics of materials, the deformation of an elastic beam is usually modeled by the fourth-order problem (1.3) and some of its variants. For such issues, Agarwal [3] and Aftabizadeh [4] discussed the existence and uniqueness of solutions, Bonanno studied the multiplicity of solutions [5], and Graef et al. explored the existence of positive solutions [6].
In recent years, due to the wide applications of difference equations [7–9], the discrete elastic beam problems have attracted extensive attention of scholars. The methods include the fixed point theorem [10], invariant sets of descending flow [11], bifurcation techniques [12], etc. In 2003, the critical point theory was first used to prove the existence of periodic and subharmonic solutions of second-order difference equations [13]. Since then this method has been widely used to discuss periodic solutions [14], homoclinic solutions [15–17], and boundary value problems [18–23] for difference equations. In particular, the critical point theory is also used for boundary value problems of fourth-order difference equations [14, 23, 24]. Among them, Cai et al. obtained some sufficient conditions for the existence of at least two nontrivial solutions of the boundary value problem (1.1) with (1.2) for \(\lambda =1\) in [14].
In addition, He and Yu discussed the fourth-order difference equation
with the following boundary value conditions:
where \(a_{k}>0 \) for any \(k\in \mathbb{Z}(2,T+2)\) in [20]. It is clear that (1.4) is a special case of (1.1) when \(p_{k}\equiv 1\) for \(k\in \mathbb{Z}(-1,T)\) and f with the form \(f(k,u)=a_{k}g(u)\). By using the fixed point theorem, the existence of positive solutions to the boundary value problem (1.4) with (1.5) is obtained.
This paper aims to establish the existence results of infinite solutions to the boundary value problem (1.1) with (1.2) by the critical point theorem. To this end, we first construct a function space E and establish an important inequality between two norms in E, then, through the oscillation of nonlinear function f at the origin and at infinity, we obtain sufficient conditions for the existence of infinitely many solutions to the elastic beam problem (1.1) with (1.2).
The rest of this article is organized as follows. In Sect. 2, we establish a variational functional \(J_{\lambda}\) corresponding to the elastic beam problem (1.1) with (1.2) on the function space E. And we find that the critical points of \(J_{\lambda}\) are actually solutions to problem (1.1) with (1.2). Furthermore, we construct an inequality that plays an important role in proving our main results. The sufficient conditions for the existence of infinite solutions to problem (1.1) with (1.2) are established and proved in Sect. 3. In Sect. 4, we give two examples to illustrate the rationality and applicability of our conclusions.
2 Preliminaries
In this section, we first establish the variational framework associated with problem (1.1) with (1.2). We consider the T-dimensional Banach space
endowed with the norm
For each \(u\in E\), define
where
Define the functional \(J_{\lambda}\) on E as \(J_{\lambda}(u)=\Phi (u)-\lambda \Psi (u)\) for any \(u\in E\). Clearly, \(\Phi ,~\Psi \in C^{1}(E,\mathbb{R})\), and we have
for any \(u,v\in E\). This shows that critical points of functional \(J_{\lambda}\) are solutions to the boundary value problem (1.1) with (1.2).
Now we present the following result obtained by Ricceri in [25], which will be used to find the critical points of the problem (1.1) with (1.2).
Lemma 2.1
Let E be a real reflexive Banach space. For any \(x\in E\), \(J_{\lambda}(x)=\Phi (x)-\lambda \Psi (x)\), where \(\lambda \in \mathbb{R}^{+}\) and \(\Psi ,~\Phi \in C^{1}(E,\mathbb{R})\) with Φ coercive, that is, \(\lim_{\|x\|\rightarrow +\infty}\Phi (x)=+\infty \).
Assume that \(\inf_{E}\Phi < r\), let
where
When \(\alpha =0\) (or \(\beta =0\)), in the sequel, we agree to read \(1/\alpha \) (or \(1/\beta \)) as +∞.
- (I):
-
If \(\alpha <+\infty \), then for each \(\lambda \in (0,\frac{1}{\alpha} )\) the following alternatives hold: either
- (\(I_{1}\)):
-
\(J_{\lambda}\) possesses a global minimum or
- (\(I_{2}\)):
-
there is a sequence \(\{u_{n}\}\) of critical points of \(J_{\lambda}\) such that \(\lim_{n\rightarrow +\infty} \Phi (u_{n})=+\infty \).
- (H):
-
If \(\beta <+\infty \), then for each \(\lambda \in (0,\frac{1}{\beta} )\) the following alternatives hold: either
- (\(H_{1}\)):
-
there is a global minimum of Φ, which is a local minimum of \(J_{\lambda}\), or
- (\(H_{2}\)):
-
there is a sequence \(\{u_{n}\}\) of pairwise distinct critical points of \(J_{\lambda}\) with \(\lim_{n\rightarrow +\infty}\Phi (u_{n})=\inf_{E}\Phi \), which weakly converges to a global minimum of Φ.
Now we give the following inequality, which plays an important role in the proof of our main results.
Lemma 2.2
For any \(u\in E\), we have
Proof
Let \(\tau \in \mathbb{Z}(1,T)\) be such that
Noticing \(u_{-1}=u_{0}=0\), we have
By the Cauchy–Schwarz inequality, we have
Similarly, by the fact that \(u_{T+1}=u_{T+2}=0\), we have
If
then Lemma 2.1 holds. Otherwise,
Then
and
By (2.4), we have
We now show that
In fact, we consider the function \(v:[1,T]\rightarrow \mathbb{R}\) given by
Since
is increasing in \([1,T]\), and we see that there exists unique \(s=\frac{T+1}{2}\) such that
Therefore, v attains its minimum at \(s=\frac{T+1}{2}\), that is,
for \(s\in [1,T]\). Since \(\tau \in \mathbb{Z}(1,T)\), we have
which is the same as (2.2). □
3 Main results
In this section, we give our main results. Let
and
We have the following result.
Theorem 3.1
Suppose that there are two real sequences \(\{\omega _{n}\}\), \(\{c_{n}\}\) with \(\omega _{n}>0\) and \(\lim_{n\rightarrow +\infty}\omega _{n}=+\infty \) such that
and
where
Then, for each \(\lambda \in (\frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2\mu}, \frac{1}{\rho} )\), problem (1.1) with (1.2) admits an unbounded sequence of solutions.
Proof
It is obvious that
which means that \(\Phi (u)\) is coercive.
Define
If \(u\in E\) and \(\Phi (u)< r_{n}\), then we have the following inequality:
Considering Lemma 2.2, for any \(k\in \mathbb{Z}(1,T)\), we have
Furthermore, according to the definition of ϕ, we have
For any \(n\in \mathbb{Z}(1)\), take \((q_{n})_{k}=c_{n}\) for \(k\in \mathbb{Z}(1,T)\) and \((q_{n})_{-1}=(q_{n})_{0}=(q_{n})_{T}=(q_{n})_{T+1}=0\), then \(q_{n}\in E\) and
by exploiting (3.2). Therefore, from (3.4), we have
Moreover, combining (3.3), it is clear that \(\alpha \leq \liminf_{n\rightarrow +\infty}\phi (r_{n})\leq \rho <+ \infty \).
We assert that \(J_{\lambda}\) is unbounded from below. In fact, when \(\mu <+\infty \), since
there exists \(\varepsilon _{0}>0\) such that
From (3.1), we know that there exists a positive sequence \(\{a_{n}\} \) with \(\lim_{n\rightarrow +\infty}a_{n}=+\infty \) such that
For each \(n\in \mathbb{Z}(1)\), define \(\upsilon _{n}\in E\) with \((\upsilon _{n})_{k}=a_{n}\) for \(k\in \mathbb{Z}(1,T)\), then we have the following inequality:
The above inequality implies \(\lim_{n\rightarrow +\infty}J_{\lambda}(\upsilon _{n})=-\infty \). If \(\mu =+\infty \), it can be seen that there is a sequence of positive number \(\{\bar{a}_{n}\}\) with \(\lim_{n\rightarrow +\infty}\bar{a}_{n}=+\infty \) such that
from the definition of μ. Define \(\bar{\upsilon}_{n}\in E\) as \((\bar{\upsilon}_{n})_{k}=\bar{a}_{n}\) for \(k\in \mathbb{Z}(1,T)\), then
By combining (3.5) with (3.6), we can conclude that condition (\(I_{1}\)) of Lemma 2.1 does not hold. Therefore, the functional \(J_{\lambda}\) has a sequence of critical points with \(\lim_{n\rightarrow +\infty}\Phi (u_{n})=+\infty \), which means that the problem (1.1) with (1.2) admits an unbounded sequence of solutions. □
Corollary 3.2
If there is a sequence of positive numbers \(\{\tilde{\omega}_{n}\}\) with \(\tilde{\omega}_{n}\rightarrow +\infty \) as \(n\rightarrow +\infty \) such that
where
then, for each \(\lambda \in (\frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2\mu}, \frac{1}{\tilde{\rho}} )\), problem (1.1) with (1.2) admits an unbounded sequence of nontrivial solutions.
Proof
Taking \(c_{n}=0\) for all \(n\in \mathbb{Z}(1)\), it can be easily proved by Theorem 3.1. □
In particular, if the nonlinear function f in (1.1) with the form \(f(k,u)=a_{k}g(u)\), where \(a_{k}>0\) for \(k\in \mathbb{Z}(1,T)\), and \(p_{k}\equiv 1\) for \(k\in \mathbb{Z}(-1,T)\). Then (1.1) reads
Define
where
Then we have the following.
Corollary 3.3
Suppose that there are two real sequences \(\{\bar{\omega}_{n}\}\), \(\{\bar{c}_{n}\}\) with \(\bar{\omega}_{n}>0\) and \(\lim_{n\rightarrow +\infty}\bar{\omega}_{n}=+\infty \) such that
and
where
Then, for each \(\lambda \in \frac{1}{\sum_{k=1}^{T}a_{k}} (\frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2\bar{\mu}} ,\frac{1}{\bar{\rho}})\), problem (3.8) with (1.2) admits an unbounded sequence of nontrivial solutions.
Now, we discuss the existence of infinitely many solutions to the boundary value problem (1.1) with (1.2) by using the oscillatory behavior of the nonlinear function at the origin.
Theorem 3.4
Suppose that there are two real sequences \(\{z_{n}\}\) and \(\{\bar{z}_{n}\}\), where \(\bar{z}_{n}>0\) and \(\lim_{n\rightarrow +\infty}\bar{z}_{n}=0\), such that
and
where
Then, for each \(\lambda \in ( \frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2\mu},\frac{1}{\varrho}) \), problem (1.1) with (1.2) has a sequence of nontrivial solutions that converges to 0.
The proof of Theorem 3.4 is similar to that of Theorem 3.1, so we omit it.
Corollary 3.5
Suppose that there is a sequence \(\{\tilde{z}_{n}\}\) where \(\tilde{z}_{n}>0\) and \(\lim_{n\rightarrow +\infty}\tilde{z}_{n}=0\) such that
where
Then, for each \(\lambda \in (\frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2\mu}, \frac{1}{\bar{\varrho}} )\), problem (1.1) with (1.2) has a sequence of nontrivial solutions that converges to 0.
Considering the boundary value problem (3.8) with (1.2), we have the following result when the nonlinear function g oscillates at the origin.
Corollary 3.6
Suppose there are two real sequences \(\{b_{n}\}\), \(\{\bar{b}_{n}\}\) with \(\bar{b}_{n}>0\) and \(\lim_{n\rightarrow +\infty}\bar{b}_{n}=0\) such that
and
where
Then, for each \(\lambda \in \frac{1}{\sum_{k=1}^{T}a_{k}} ( \frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2\bar{\mu}} ,\frac{1}{{\sigma}} )\), problem (3.8) with (1.2) admits a sequence of nontrivial solutions that converges to 0.
4 Examples
Example 4.1
Consider (1.1) with (1.2) when
for any \(k\in \mathbb{Z}(1,T)\) and \(\epsilon >0\). Then, for \(u\geq 0\), it can be obtained by direct calculation
Obviously, \(f(u)\geq 0\) for \(u\in \mathbb{R}\), and \(F(u)\) is increasing at \((-\infty ,+\infty )\). Take
Then we have \(\lim_{n\rightarrow +\infty}\nu _{n} =\lim_{n \rightarrow +\infty}\tilde{\omega}_{n}=+\infty \) and
In addition,
Let ϵ be sufficiently small such that
which implies that (3.7) of Corollary 3.2 holds. Therefore, by Corollary 3.2, for any \(\lambda \in \frac{1}{T} ( \frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2+\epsilon}, \frac{16p_{*}}{(T+1)^{2}(T+3)\epsilon} )\), the boundary value problem (1.1) with (1.2) has an unbounded sequence of solutions.
Example 4.2
Consider (1.1) with (1.2) when
for any \(k\in \mathbb{Z}(1,T)\) and \(\epsilon >0\). Then, for \(u\neq 0\), we have
It can be seen that \(f(u)\geq 0\) for \(u\geq 0\), \(F(u)\) is increasing at \([0,+\infty )\) and \(F(-u)=F(u)\). It is easy to get that
Let \(\zeta _{n}=e^{-\frac{1}{\epsilon ^{2}} (\frac{\pi}{2}+2n\pi )}\), then \(\lim_{n\rightarrow +\infty}{\zeta}_{n}=0\), \({\zeta}_{n}>0\) for \(n\in \mathbb{Z}(1)\). After a simple calculation, we have
Take ϵ be small enough such that
which means that (3.13) of Corollary 3.5 holds. Hence, from Corollary 3.5, for any \(\lambda \in \frac{1}{T} ( \frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2+\epsilon}, \frac{16p_{*}}{(T+1)^{2}(T+3)\epsilon} )\), the boundary value problem (1.1) with (1.2) admits a sequence of solutions which converges to 0.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11971126) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT 16R16).
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 11971126) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT 16R16).
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Chen, Y., Zhou, Z. Existence of infinitely many solutions of nonlinear fourth-order discrete boundary value problems. Bound Value Probl 2022, 58 (2022). https://doi.org/10.1186/s13661-022-01640-y
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DOI: https://doi.org/10.1186/s13661-022-01640-y