 Research
 Open Access
Existence and multiple solutions to a discrete fourth order boundary value problem
 Xia Liu^{1, 2}Email author,
 Tao Zhou^{3} and
 Haiping Shi^{4, 5}
https://doi.org/10.1186/s1366201818791
© The Author(s) 2018
 Received: 15 June 2018
 Accepted: 8 November 2018
 Published: 20 November 2018
Abstract
In this article, we study a discrete fourth order boundary value problem. By making use of variational methods and critical point theory, we obtain some criteria for the existence and multiple solutions. Moreover, two examples are included to illustrate the applicability of the main results.
Keywords
 Boundary value problems
 Existence and multiple solutions
 Fourth order
 Critical point theory
 Discrete
MSC
 39A10
 34B05
 58E05
 65L10
1 Introduction and statement of the main results
Difference equations [1–10, 15–20, 22, 26–28, 30, 31] appear in numerous settings and forms, both as a fundamental tool in the discrete analogue of a differential equation and as a useful model for several economical and population problems.
In the last few years, variational methods and critical point theory have been used to study the existence and multiple solutions of discrete boundary value problems. In this article, we utilize this approach to obtain some sufficient conditions for the existence and multiple solutions to the boundary value problem (BVP for short) (1.1) with (1.2). What is more, two examples are included to illustrate the applicability of the main results.
Our main results are the following theorems.
Theorem 1.1
 \((F_{1})\) :

There exist two constants \(\delta_{1}>0\) and \(a_{1}\in ( 0,\frac{\lambda_{\min }}{2} ) \) such that$$ F(s,u)\leq a_{1}u^{2},\quad \forall s\in \mathbb{R}^{2}, \vert u \vert \leq \delta _{1}. $$
 \((F_{2})\) :
Remark 1.1
 \((H1)\) :

\(p_{n}>0\) for \(n\in \mathbb{Z}(0,N+1)\) and \(q_{n}>0\) for \(n\in \mathbb{Z}(0,N)\);
 \((H2)\) :

\(\lim_{ \vert n \vert \rightarrow \infty } \frac{ \vert F(n,u) \vert }{ \vert u \vert ^{2}}=0\) for \(n\in \mathbb{Z}(0,N)\);
 \((H3)\) :

\(\limsup_{ \vert n \vert \rightarrow \infty } \frac{ \vert F(n,u) \vert }{ \vert u \vert ^{2}} \leq 0\) for \(n\in \mathbb{Z}(0,N)\);
 \((H4)\) :

there exists \(\omega \in U\) such that \(\sum_{n=1}^{N} F(n, \omega_{n})>0\).
Theorem 1.2
 \((F_{3})\) :

\(\lim_{ \vert u \vert \rightarrow 0}\frac{F(s,u)}{u^{2}}=0\), \(\forall (s,u)\in \mathbb{R}^{2}\).
 \((F_{4})\) :

There exist three constants \(a_{4}>0\), \(\gamma >2\), and \(a_{5}>0\) such that$$ F(s,u)\geq a_{4} \vert u \vert ^{\gamma }a_{5}, \quad \forall (s,u)\in \mathbb{R}^{2}. $$
Theorem 1.3
Remark 1.2
Define \(f_{0}=\liminf_{u\rightarrow 0}\min_{n\in \mathbb{Z}(1,N)}\frac{f(n,u)}{u}\) and \(f^{\infty }= \limsup_{u\rightarrow 0}\max_{n\in \mathbb{Z}(1,N)} \frac{f(n,u)}{u}\).
 \((H_{1})\) :

\(p_{n}\geq 0\) and \(q_{n}\geq 0\) for \(n\in \mathbb{Z}(1,N)\) and there exists η with \(\eta <\underline{q}\) such that \(f^{\infty }\leq \eta \), where \(\underline{q}=\min_{n\in \mathbb{Z}(1,N)}q _{n}\);
 \((H_{2})\) :

\(f(n,u)\) is odd in u, i.e., \(f(n,u)=f(n,u)\) for \((n,u)\in \mathbb{Z}(1,N)\times \mathbb{R}\);
 \((H_{3})\) :

there exists \(m\in \{1,\ldots, N\}\) such that \(f_{0}> \lambda_{m}\).
Theorem 1.4
 \((\varPsi_{1})\) :

There exists a function \(\varPsi (u)\in C^{1}(\mathbb{R}, \mathbb{R})\) with \(\varPsi (u)\geq 0\) such that$$ \varPsi '(u)=\psi (u). $$
 \((\varPsi_{2})\) :

There exist two constants \(\delta_{2}>0\) and \(a_{6} \in (0,\frac{\lambda_{\min }}{2})\) such that$$ \varPsi (u)\leq a_{6}u^{2}, \quad \forall s\in \mathbb{R}^{2}, \vert u \vert \leq \delta _{2}. $$
 \((\varPsi_{3})\) :
Corollary 1.1
2 Variational framework
In this section, we shall establish the corresponding variational framework for BVP (1.1) with (1.2) which will be of fundamental importance in proving our main results.
Remark 2.1
Consequently, we reduce the problem of finding a solution of BVP (1.1) with (1.2) to that of seeking a critical point of the functional J on U. Denote the \(k\times k\) matrices S and R.
For \(k=1\), let \(S=R=(0)\).
It is easy to see that 0 is an eigenvalue of M, \((1,1,\ldots,1)^{*}\) is an eigenvector associated with 0. M is semipositive definite. Let \(\lambda_{1},\lambda_{2},\ldots,\lambda_{k}\) be the eigenvalues of M.
3 Some basic lemmas
Assume that U is a real Banach space and \(J\in C^{1}(U,\mathbb{R})\). As usual, J is said to satisfy the Palais–Smale condition if every sequence \(\{ u^{(j)} \} \subset U\) such that \(\{ J ( u ^{(j)} ) \} \) is bounded and \(J^{\prime } ( u^{(j)} ) \rightarrow 0\) (\(j\rightarrow \infty \)) has a convergent subsequence. The sequence \(\{ u^{(j)} \} \) is called a Palais–Smale sequence.
Let U be a real Banach space. Define the symbol \(B_{\rho }(u)\) as the open ball in U about u of radius ρ, \(\partial B_{\rho }(u)\) as its boundary, and \(\bar{B}_{\rho }(u)\) as its closure.
Lemma 3.1
 \((J_{1})\) :

There are positive constants c and ρ such that \(J _{\partial B_{\rho }(0)\cap U_{2}}\geq c\).
 \((J_{2})\) :

There are \(\mu \in \partial B_{1}(0)\cap U_{2}\) and a positive constant \(\hat{c}\geq \rho \) such that \(J _{\partial \varOmega }\leq 0\), where \(\varOmega =(\bar{B}_{\hat{c}}(0)\cap U_{1})\oplus \{s\mu\mid 0< s< \hat{c}\}\).
Lemma 3.2
(Clark theorem [21])
Let U be a real Banach space, \(J\in C^{1}(U,\mathbb{R})\), with J being even, bounded from below and satisfying the Palais–Smale condition. Assume \(J(0)=0\), there is a set \(\varGamma \subset U\) such that Γ is homeomorphic to \(S^{k1}\) (\(k1\) dimension unit sphere) by an odd map, and \(\sup_{\varGamma }J<0\). Then J has at least k distinct pairs of nonzero critical points.
Lemma 3.3
Assume that \((r)\) and \((F_{1})\)–\((F_{3})\) are satisfied. Then the functional J satisfies the Palais–Smale condition.
Proof
4 Proofs of theorems
Proof of Theorem 1.1
Obviously, \(F(n,0)=0\) and \(f(n,0)=0\) for any \(n\in \mathbb{Z}[1,k]\) via \((F_{1})\) and \((F_{2})\). Hence, \(u=0\) is a trivial solution of BVP (1.1) with (1.2).
In the light of Lemmas 3.1 and 3.3, it is sufficient to verify condition \((J_{2})\).
Similar to the proof of Theorem 1.1 in [4], we can prove that BVP (1.1) with (1.2) admits at least three solutions, and so we omit it. □
Remark 4.1
Note that \((F_{3})\) implies \((F_{1})\). Similar to the above argument, we can also prove Theorem 1.2. For simplicity, we omit its proof.
Proof of Theorem 1.3
Obviously \(J\in C^{1}(U,\mathbb{R})\), J is even, and \(J(0)=0\). From Lemma 3.3, J satisfies the Palais–Smale condition. By the proof of Theorem 1.1, we have that J is bounded from below. On account of Lemma 3.2, it is sufficient to find a set Γ and an odd map such that Γ is homeomorphic to \(S^{q1}\) by an odd map.
5 Examples
Firstly, our example illustrates Theorem 1.1.
Example 5.1
As an example of Theorem 1.3, we have the following.
Example 5.2
Declarations
Acknowledgements
The authors are extremely grateful to the referees and the editors for their careful reading and making some valuable comments and suggestions on the manuscript. This work was carried out while visiting Central South University. The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.
Funding
This project is supported by the National Natural Science Foundation of China (No. 11501194).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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