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Existence and multiple solutions to a discrete fourth order boundary value problem
Advances in Difference Equations volume 2018, Article number: 427 (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.
Introduction and statement of the main results
In this article, we are interested in the existence and multiple solutions to the discrete fourth order nonlinear equation
with boundary value conditions
where \(\Delta^{j} u_{n}=\Delta (\Delta^{j1} u_{n})\) (\(j=2,3,4\)), \(\Delta^{0} u_{n}=u_{n}\), \(\Delta u_{n}=u_{n+1}u_{n}\), \(f(s,u)\in C( \mathbb{R}^{2},\mathbb{R})\), \(r_{n}>0\) is realvalued for each \(n\in \mathbb{Z}[0,k]\), \(r_{0}=r_{k}\), \(k\geq 1\) is an integer. Here, \(\mathbb{Z}\) denotes the sets of integers, \(\mathbb{R}\) denotes the sets of real numbers, \(\mathbb{N}\) denotes the sets of natural numbers. Given \(a\leq b\) in \(\mathbb{Z}\), let \(\mathbb{Z}[a,b]:=\mathbb{Z}\cap [a,b]\). Let u* denote the transpose of a vector u.
Boundary value problem (1.1) with (1.2) can be regarded as being a discrete analogue of the fourth order differential equation
with boundary value conditions
(1.3) includes the following equation:
which is used to describe the stationary states of the deflection of an elastic beam [29]. Differential equations similar to (1.3) and special cases of it have been studied using a number of different methods in the literature, we refer the reader to papers [1, 2, 11,12,13,14, 24, 25] and the references contained therein.
Difference equations [1,2,3,4,5,6,7,8,9,10, 15,16,17,18,19,20, 22, 26,27,28, 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.
If \(f(n,u_{n})=q_{n}u_{n}\), Peterson and Ridenhour [22] considered the fourth order difference equation
and gave some conditions on \(q_{n}\) that ensure (1.6) is (2,2)disconjugate on \([a, b+4]\) utilizing an appropriately defined quadratic form.
Making use of the symmetric mountain pass lemma, Chen and Tang [5] established some existence criteria to guarantee the fourth order difference system
has infinitely many homoclinic orbits.
In [16], the existence, multiplicity, and nonexistence results of nontrivial solutions for discrete nonlinear fourth order boundary value problems
with
are obtained. The methods used here are based on the critical point theory and monotone operator theory.
Positive solutions of the following fourth order nonlinear difference equations with a deviating argument
are investigated. Došlá, Krejčová, and Marini [8] introduced for (1.8) the notions of a minimal solution and a maximal solution, and gave necessary and sufficient conditions for their existence. Some relationships with nonoscillatory solutions, which have a different growth at infinity, were presented as well.
Graef, Kong, and Wang [10] studied the discrete fourth order periodic boundary value problem with a parameter
By using variational methods and the mountain pass lemma, sufficient conditions are found under which the above problem has at least two nontrivial solutions.
In 2015, Liu, Zhang, and Shi [19] considered the following fourth order nonlinear difference equation:
with boundary value conditions
Using the critical point theory, the authors established various sets of sufficient conditions for the existence and nonexistence of solutions for the Dirichlet boundary value problem and gave some new results.
By using the invariant set of descending flow and variational method, Long and Chen [20] in 2018 established the existence of multiple solutions to a class of second order discrete Neumann boundary value problem
The solutions included signchanging solutions, positive solutions, and negative solutions. Moreover, an example was given to illustrate our results.
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.
Throughout this article, assume that there is a function \(F(s,u)\) such that
for any \((s,u)\in \mathbb{R}^{2}\).
Our main results are the following theorems.
Theorem 1.1
Assume that the function \(F(s,u)\geq 0\) satisfies the following assumptions:
 \((F_{1})\) :

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

There exist two constants \(a_{2}\in ( \frac{\lambda_{ \max }}{2},+\infty ) \) and \(a_{3}>0\) such that
$$ F(s,u)\geq a_{2}u^{2}a_{3}, \quad \forall (s,u) \in \mathbb{R}^{2}, $$where \(\lambda_{\min }\) and \(\lambda_{\max }\) are constants which can be referred to (2.4) and (2.5).
Remark 1.1
In [10], the authors considered the discrete fourth order periodic boundary value problem with a parameter
The following hypotheses are satisfied in [10]:
 \((H1)\) :

\(p_{n}>0\) 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
Assume that the function \(F(s,u)\geq 0\) satisfies the following assumptions:
 \((F_{3})\) :

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

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
Assume that the function \(F(s,u)\geq 0\), \((F_{1})\) and \((F_{2})\) and the following assumptions are satisfied:
Then BVP (1.1) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
Remark 1.2
In [9], the authors considered the fourth order nonlinear difference equation
Define \(f_{0}=\liminf_{u\rightarrow 0}\min_{n\in \mathbb{Z}(1,N)}\frac{f(n,u)}{u}\) and \(f^{\infty }= \limsup_{u\rightarrow 0}\max_{n\in \mathbb{Z}(1,N)} \frac{f(n,u)}{u}\).
The following hypotheses are satisfied in [9]:
 \((H_{1})\) :

\(p_{n}\geq 0\) 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 \(f_{0}> \lambda_{m}\).
If \(f(n,u_{n})=\tau_{n}\psi (u_{n})\), (1.1) reduces to the following fourth order nonlinear equation:
where \(\psi \in C(\mathbb{R},\mathbb{R})\), \(\tau_{n}>0\) is realvalued for each \(n\in \mathbb{Z}[1,k]\). Therefore, we can easily obtain the following results.
Theorem 1.4
Assume that the following assumptions are satisfied:
 \((\varPsi_{1})\) :

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
$$ \varPsi (u)\leq a_{6}u^{2}, \quad \forall s\in \mathbb{R}^{2}, \vert u \vert \leq \delta _{2}. $$  \((\varPsi_{3})\) :

There exist two constants \(a_{7}\in ( \frac{ \lambda_{\max }}{2},+\infty ) \) and \(a_{8}>0\) such that
$$ \varPsi (u)\geq a_{7}u^{2}a_{8},\quad \forall (s,u)\in \mathbb{R}^{2}, $$where \(\lambda_{\min }\) and \(\lambda_{\max }\) are constants which can be referred to (2.4) and (2.5).
Corollary 1.1
Assume that \((\varPsi_{1})\), \((\varPsi_{2})\), \((\varPsi_{3})\) and the following assumption are satisfied.
Then BVP (1.10) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
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.
In order to apply the critical point theory, we define a kdimensional Hilbert space U by
and equip it with the inner product
and the induced norm
Remark 2.1
It is obvious that
As a matter of fact, U is isomorphic to \(\mathbb{R}^{k}\). Throughout this article, when we say \(u= ( u_{1},u_{2},\ldots,u_{k} ) \in \mathbb{R}^{k}\), we always imply that u can be extended to a vector in U so that (2.1) holds.
Define a functional J on U by
After a careful computation, we have
Therefore, \(J'(u)=0\) if and only if
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 \(S=R=(0)\).
For \(k=2\), let
and
For \(k=3\), let
For \(k=4\), let
For \(k\geq 5\), let
For \(k\geq 3\), let
Let \(M:=S+R\). We rewrite \(J(u)\) as
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.
Set
and
Let \(P=\{(c,c,\ldots,c)^{*}\in U\mid c\in \mathbb{R}\}\), then P is an invariant subspace of U. Denote Q by
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
Let U be a real Banach space, \(U=U_{1}\oplus U_{2}\), where \(U_{1}\) is finite dimensional. Suppose that \(J\in C^{1}(U,\mathbb{R})\) satisfies the Palais–Smale condition and the following:
 \((J_{1})\) :

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 \(\varOmega =(\bar{B}_{\hat{c}}(0)\cap U_{1})\oplus \{s\mu\mid 0< s< \hat{c}\}\).
and \(\varUpsilon =\{d\in C(\bar{\varOmega },U)\mid d _{\partial \varOmega }=id \}\), where id denotes the identity operator.
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
Let \(\{ u^{(j)} \} _{j\in \mathbb{N}}\subset U\) be such that \(\{ J ( u^{(j)} ) \} _{j\in \mathbb{N}}\) is bounded and \(J' ( u^{(j)} ) \rightarrow 0\) as \(j\rightarrow \infty \). Then there is a constant \(A>0\) such that
From \((F_{2})\) and (2.3), for any \(\{ u^{(j)} \} _{j\in \mathbb{N}}\subset U\), we have
Then
It comes from \(a_{2}\in ( \frac{\lambda_{\max }}{2},+\infty ) \) that we can find a constant \(B>0\) such that, for any \(j\in \mathbb{N}\), \(\Vert u^{(j)} \Vert \leq B\). Thus, we know that the sequence \(\{ u^{(j)} \} _{j\in \mathbb{N}}\) is bounded in the k dimensional space U. Therefore, the Palais–Smale condition holds. □
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).
It comes from Lemma 3.3 that \(J(u)\) is bounded from above in U. Let
Therefore, there exists a sequence \(\{ u^{(j)} \} \) on U such that
What is more, from the proof of Lemma 3.3, we have
This implies that \(\lim_{ \Vert u \Vert \rightarrow +\infty }J(u)= \infty \). Thus, \(\{ u^{(j)} \} \) is bounded. Then \(\{ u ^{(j)} \} \) has a convergent subsequence defined by \(\{ u^{(j _{n})} \} \). Set
Due to the continuity of \(J(u)\) in u, there must be a point \(\bar{u}\in U\), \(J(\bar{u})=\bar{J}\). Clearly, \(\bar{u}\in U\) is a critical point of \(J(u)\).
From \((F_{1})\), for any \(u\in Q\), \(\Vert u \Vert \leq \delta_{1}\), we have
Denote
We have
Thus, there are constants \(c>0\) and \(\delta_{1}>0\) such that \(J _{\partial B_{\delta_{1}}(0)\cap Q}\geq c\). Assumption \((J_{1})\) of the linking theorem is satisfied.
In view of \(Mu=0\), for all \(u\in P\), we have
Hence, \(\bar{u}\notin P\) and the critical point ū of \(J(u)\) corresponding to the critical value J̄ is a nontrivial solution of BVP (1.1) with (1.2).
In the light of Lemmas 3.1 and 3.3, it is sufficient to verify condition \((J_{2})\).
Choose \(\alpha \in \partial B_{1}(0)\cap Q\), for any \(\beta \in P\) and \(s\in \mathbb{R}\), let \(u=s\alpha +\beta \). By \((F_{2})\), we have
Consequently, there is some positive constant \(\chi >\delta_{1}\) such that
where \(\varOmega =(\bar{B}_{\chi }(0)\cap Q)\oplus \{s\alpha\mid 0< s<\chi \}\). Applying the linking theorem, \(J(u)\) has a critical value \(c_{0}\geq c>0\), where
and \(\varUpsilon =\{d\in C(\bar{\varOmega },U)\mid d _{\partial \varOmega }=id \}\).
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.
Choose
Clearly, Γ is homeomorphic to \(S^{q1}\) by an odd map. It is comes from (4.2) that \(\sup_{\varGamma }(J)<0\). On the basis of Lemma 3.2, J has at least q distinct pairs of nonzero critical points. For this reason, BVP (1.1) with (1.2) admits at least q distinct pairs of nontrivial solutions. The proof of Theorem 1.3 is complete. □
Remark 4.2
By virtue of Theorem 1.1, the conclusion of Theorem 1.4 is clearly right. As a result of Corollary Theorem 1.3, the results of Corollary 1.1 are evidently correct.
Examples
Firstly, our example illustrates Theorem 1.1.
Example 5.1
Consider the equation
with boundary value conditions
We have
and
Also,
and the eigenvalues of M are \(\lambda_{1}=0\), \(\lambda_{2}=12\), and \(\lambda_{3}=12\). It is easy to verify that all the conditions of Theorem 1.1 are satisfied and then BVP (5.1) with (5.2) admits at least three solutions.
As an example of Theorem 1.3, we have the following.
Example 5.2
Consider the equation
with boundary value conditions
We have
and
Also,
and the eigenvalues of M are \(\lambda_{1}=0\), \(\lambda_{2}=8\), \(\lambda_{3}=8\), and \(\lambda_{4}=24\). It is easy to verify that all the conditions of Theorem 1.3 are satisfied, and then BVP (5.3) with (5.4) admits at least three distinct pairs of nontrivial solutions.
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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).
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Correspondence to Xia Liu.
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MSC
 39A10
 34B05
 58E05
 65L10
Keywords
 Boundary value problems
 Existence and multiple solutions
 Fourth order
 Critical point theory
 Discrete