# Existence of Solutions to Boundary Value Problems for a Fourth-Order Difference Equation

Authors: Xia Liu; Tao Zhou; Haiping Shi
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

In this article, some new existence criteria of at least one solution to boundary value problems for a fourth-order difference equation are obtained by using the critical point theory. In a special case, a necessary and sufficient condition for the existence and uniqueness of solution is also established. An example of the main result is given.


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## Body

## 1. Introduction

Throughout this article, the sets of all natural numbers, integers, and real numbers are defined asN, Z, and R, respectively. The transpose of a vector x is defined as x∗. For a,b∈Z with a<b, the discrete interval {a,a+1,…,b} is denoted as [a,b]Z.Consider the existence of solutions to boundary value problem (BVP) for a fourth-order difference equation:(1)Δ2pt-2Δ2xt-2-Δqt-1Δxt-1-rtxt+ft=0,t∈1,TZ,satisfying the boundary value conditions(2)Δkx-1=ΔkxT-1,k=0,1,2,3,where T∈N and T≥1, Δ is the forward difference operator denoted as Δx(t)=x(t+1)-x(t), Δkx(t)=Δ(Δk-1x(t))(k=2,3,4), Δ0x(t)=x(t), p(t)∈C[-1,T]Z,R with p(-1)=p(T-1), p(0)=p(T), q(t)∈C[0,T]Z,R with q(0)=q(T), r(t)∈C[1,T]Z,R, f(t)∈C([1,T]Z,R).And (1), (2) can be regarded as a discrete analogue of(3)psx′′s′′-qsx′s′-rsxs+fs=0,s∈0,1,with boundary value conditions (4)xi0=xi1,i=0,1,2,3.Equations similar in structure to (3) arise in the study of the existence of solutions to differential equations [1–11].Difference equations are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Many authors were interested in difference equations and obtained some significant results [12–29].Consider the fourth-order nonlinear difference equation(5)Δ2ptΔ2xt+ft,xt=0,t∈Z.Thandapani and Arockiasamy [24] in 2001 established some new criteria for the oscillation and nonoscillation of solutions.Xia [26] in 2017 considered the following second-order nonlinear difference equation with Jacobi operators (6)Lxt-ωxt=ft,xt+Γ,…,xt,…,xt-Γ,t∈Z,containing both many advances and retardations. By using variational methods and critical point theory, some new criteria are obtained for the existence of a nontrivial homoclinic solution.By Krasnoselskii’s fixed point theorems in cones, Cabada and Dimitrov [13] obtained some existence, multiplicity, and nonexistence results for the nonlinear singular and nonsingular fourth-order equation depending on a real parameter (7)xt+4+Mxt=λgtfxt+ct,t∈0,1,…,T-1.In [18], a higher order nonlinear difference equation (8)∑i=0nriXk-i+Xk+i+fk,Xk+Γ,…,Xk,…,Xk-Γ=0,n∈N,k∈Z,is studied. By using critical point theory, sufficient conditions for the existence of periodic solutions are established.In [23], Raafat determined the forbidden set, introduced an explicit formula for the solutions, and discussed the global behavior of solutions of the difference equation (9)xt+1=axt-3b-cxt-1xt-3,t=0,1,2,…,where a,b,c are positive real numbers and the initial conditions x(3),x(2),x(1),x(0) are real numbers.By using the symmetric mountain pass theorem, Chen and Tang [15] established some existence criteria to guarantee the fourth-order difference equation of the form (10)Δ4xt-2+qtxt=ft,xt+1,xt,xt-1,t∈Z,having infinitely many homoclinic orbits.Using the direct method of the calculus of variations and the mountain pass technique, Leszczyński [20] obtained the existence of at least one and at least two solutions of the difference equation: (11)Δ2γt-1ϕptΔ2xt-2=ft,xt+1,xt,xt-1,t∈1,TZ,with boundary value conditions (12)Δx-1=Δx0=0,xT+1=xT+2=0.Our purpose in this article is to use the critical point theory to explore some existence criteria of solutions to the boundary value problem (1), (2) for a fourth-order nonlinear difference equation. In a special case, a necessary and sufficient condition for the existence and uniqueness of solution is also established. Results obtained here are motivated by the recent papers [18, 25].Define(13)rmin=minr1,r2,…,rT,rmax=maxr1,r2,…,rT.The rest of this article is organized as follows. In Section2, we shall introduce some basic notations, set up the variational framework of BVP (1), (2), and give a lemma which will be useful in the proofs of our theorems. Section 3 contains our results of at least one solution. In Section 4, we shall finish proving the main results. In Section 5, we shall provide an example to illustrate our main theorem.


## 2. Preliminaries

LetX be a vector space: (14)X≔x:-1,T+2Z→R∣Δkx-1=ΔkxT-1,k=0,1,2,3.For any x,y∈X, denote (15)x,y≔∑t=1Txtyt,x≔∑t=1Tx2t1/2.Remark 1.
It is obvious that(16)x-1=xT-1,x0=xT,x1=xT+1,x2=xT+2,∀x∈X. In reality, X is isomorphic to RT. In the following and in the sequel, when we write x=x(1),x(2),…,x(T)∈RT, we always mean that x can be extended to a vector in X so that (16) is satisfied.For anyx∈X, define the functional I as (17)Ix≔12∑t=1Tpt-2Δ2xt-22+12∑t=1TqtΔxt2-12∑t=1Trtxt2+∑t=1Tftxt.It is easy to see that I∈C1(X,R) and (18)∂I∂xt=Δ2pt-2Δ2xt-2-Δqt-1Δxt-1-rtxt+ft,t∈1,TZ.For this reason,I′(x)=0 when and only when (19)Δ2pt-2Δ2xt-2-Δqt-1Δxt-1-rtxt+ft=0,t∈1,TZ.Thereby a functionx∈X is a critical point of the functional I on X when and only when x is a solution of the BVP (1), (2).LetP and Q be the T×T matrices as follows.IfT=1, let P=Q=(0).IfT=2, let (20)P=p-1+4p0+3p1-2p0-4p1-2p2-2p0-4p1-2p2p0+4p1+3p2,Q=q0+q1-q0-q1-q0-q1q0+q1.IfT=3, let (21)P=p-1+4p0+p1p2-2p0+p1p1-2p2+p3p2-2p0+p1p0+4p1+p2p3-2p1+p2p1-2p2+p3p3-2p1+p2p1+4p2+p3.IfT=4, let (22)P=p-1+4p0+p1-2p0+p1p1+p3-2p3+p4-2p0+p1p0+4p1+p2-2p1+p2p2+p4p1+p3-2p1+p2p1+4p2+p3-2p2+p3-2p3+p4p2+p4-2p2+p3p2+4p3+p4.IfT≥5, let (23)P=n1m1p10⋯0pT-1mTm1n2m2p2⋯00pTp1m2n3m3⋯0000p2m3n4⋯000⋯⋯⋯⋯⋯⋯⋯⋯0000⋯mT-3pT-300000⋯nT-2mT-2pT-2pT-1000⋯mT-2nT-1mT-1mTpT00⋯pT-2mT-1nT,where n(k)=p(k)+4p(k-1)+p(k-2), m(k)=-2(p(k-1)+p(k)),k=1,2,…,T.IfT≥3, let(24)Q=q0+q1-q10⋯-q0-q1q1+q2-q2⋯00-q2q2+q3⋯0⋯⋯⋯⋯⋯000⋯-qT-1-q000⋯qT-1+q0.Let M≔P+Q. Therefore, we rewrite I(x) by(25)Ix=12x∗Mx-12∑t=1Trtxt2+∑t=1Tftxt.Assume thatX is a real Banach space and I∈C1(X,R) is a continuously Fréchet differentiable functional defined on X. As usual, I is said to satisfy the Palais-Smale condition if any sequence xkk=1∞⊂X for which Ixkk=1∞ is bounded and I′xk→0 as k→∞ possesses a convergent subsequence. Here, the sequence xkk=1∞ is called a Palais-Smale sequence.LetBρ be the open ball in X about 0 of radius ρ, ∂Bρ be its boundary, and B¯ρ be its closure.Lemma 2 (saddle point theorem [30]).
Suppose thatX is a real Banach space, X=X1⊕X2, where X1≠{0} and is finite dimensional. Suppose that I∈C1(X,R) satisfies the Palais-Smale condition and
( I 1 ) there are constants ε,η>0 such that I∂Bη∩X1≤ε
( I 2 ) there is ξ∈Bη∩X1 and a constant χ>ε such that Iξ+X2≥χ.
ThenI admits a critical value c≥χ, where (26)c=infg∈Ωmaxx∈Bη∩X1Igx,Ω=g∈CB¯η∩X1,X∣g∂Bη∩X1=id and id is defined as the identity operator.


## 3. Main Theorems

Our main theorems are as follows.Theorem 3.
Assume that the following assumptions are satisfied.
( p ) For any t∈[-1,T]Z, p(t)>0.
( q ) For any t∈[0,T]Z, q(t)>0.
( r 1 ) For any t∈[1,T]Z, r(t)<0.
( M )   n ( 1 ) + m ( 1 ) + p ( 1 ) + p ( T - 1 ) + m ( T ) = 0, n(2)+m(2)+p(2)+p(T)+m(1)=0, n(k)+m(k)+p(k)+p(k-2)+m(k-1)=0,k=3,4,…,T.
( f 1 ) For any t∈[1,T]Z, (27)-∑t=1Tf2t∑t=1Trt<λmin-rmax∑t=1Tft, where λmin can be referred to (32).
Then the BVP (1), (2) has at least one solution.Ifr(t)>0, we obtain the theorem as follows.Theorem 4.
Assume that(p), (q), (M) and the following assumptions are satisfied.
( r 2 ) For any t∈[1,T]Z, r(t)>0.
( r 3 )   r m i n > λ m a x, where λmax can be referred to (33).
( f 2 ) For any t∈[1,T]Z, (28)∑t=1Tf2t∑t=1Trt<rmin-λmax∑t=1Tft2. Then the BVP (1), (2) has at least one solution.Ifr(t)=0, consider the following equation:(29)Δ2pt-2Δ2xt-2-Δqt-1Δxt-1=ft,t∈1,TZ,with boundary value condition (2).By a similar argument to that in Section2, we define the functional I as (30)Ix=12x∗Mx-f~,x,where f~=(f(1),f(2),…,f(T))∗. Hence, a function x∈X is a critical point of the functional I on X when and only when x is a solution of the BVP (29), (2).It is obvious to find that the critical point ofI is just the solution of the following linear equation: (31)Mx=f~.LetΩ=(M,f~). Making use of the linear algebraic theory, we can obtain the following necessary and sufficient conditions.Theorem 5.
The BVP (29), (2) has at least one solution when and only when rank(M)=rank(Ω), where rank(M) defines the rank of the matrix M.Theorem 6.
The BVP (29), (2) has a unique solution when and only when rank(M)=T.


## 4. Proofs of the Theorems

In this section, we shall finish proofs of the theorems by using critical point theory.By(M), it is evident that M is positive semidefinite. λ1=0 is an eigenvalue of M and (1,1,…,1)∗ is an eigenvector corresponding to 0. Let λ2,λ3,…,λT be the other eigenvalues of M.Define(32)λmin=minλ2,λ3,…,λT,(33)λmax=maxλ2,λ3,…,λT.Let the spaceX2={(x1,x2,…,xT)∗∈X:xj≡c,c∈R,j=1,2,…,T} and X2⊥=X1 such that (34)X=X1⊕X2.Proof of Theorem3.
Letxkk∈N⊂X be such that Ixkk∈N is bounded and I′xk→0 as k→∞. Thus, for any k∈N, there exists a positive constant K1 such that (35)Ixk≤K1.
It comes from Hölder inequality that(36)∑t=1Tftxkt≤∑t=1Tf2t1/2∑t=1Txkt21/2.
By (25) and (36), we have (37)K1≥Ixk=12xk∗Mxk-12∑t=1Trtxkt2+∑t=1Tftxkt≥-12∑t=1Trtxkt2-∑t=1Tftxkt≥-rmax2∑t=1Txkt2-∑t=1Tf2t1/2∑t=1Txkt21/2=-rmax2xk2-∑t=1Tf2t1/2xk. That is,(38)-rmax2xk2-∑t=1Tf2t1/2xk≤K1.
From the assumption(r1), (38) implies that xkk∈N is bounded in X. Since the dimension of X is finite, xkk∈N has a convergent subsequence. As a result, I(x) satisfies the Palais-Smale condition. Hence, it is sufficient to prove that I(x) satisfies the conditions (I1) and (I2) of Lemma 2.
For anyu=(c,c,…,c)∗∈X2, we have (39)Iu=-12∑t=1Trtc2+∑t=1Tftc. Choose (40)c=∑t=1Tft∑t=1Trt,(41)η=u=T∑t=1Tft∑t=1Trt. Therefore, (42)Iu=∑t=1Tft22∑t=1Trt. Let (43)ε=∑t=1Tft22∑t=1Trt. Thus, (44)Iu=ε,∀u∈∂Bη∩X1. The condition (I1) of Lemma 2 is proved.
Then, we prove the condition(I2) of Lemma 2. For any v∈X1, by Hölder inequality, we have (45)Iv=12v∗Mv-12∑t=1Trtvt2+∑t=1Tftvt≥λmin2v2-rmax2∑t=1Tvt2-∑t=1Tftvt≥λmin2v2-rmax2v2-∑t=1Tf2t1/2v=λmin-rmax2v2-∑t=1Tf2t1/2v≥-∑t=1Tf2t2λmin-rmax, as one finds by minimization with respect to v. In other words, (46)Iv≥-∑t=1Tf2t2λmin-rmax.
Let(47)χ=-∑t=1Tf2t2λmin-rmax. It follows from (f1) that (48)Iv≥χ>ε.
All the assumptions of Lemma2 are proved. According to saddle point theorem, the proof of Theorem 3 is complete.Proof of Theorem4.
Letxkk∈N⊂X be such that Ixkk∈N is bounded and I′xk→0 as k→∞. Thus, for any k∈N, there exists a positive constant K2 such that (49)Ixk≤K2.
By (25), (33) and Hölder inequality, we have (50)-K2≤Ixk=12xk∗Mxk-12∑t=1Trtxkt2+∑t=1Tftxkt≤λmax2xk2-rmin2xk2+∑t=1Tftxkt≤λmax-rmin2xk2+∑t=1Tf2t1/2∑t=1Txkt21/2=λmax-rmin2xk2+∑t=1Tf2t1/2xk. That is,(51)rmin-λmax2xk2-∑t=1Tf2t1/2xk≤K2.
By condition(r3), (51) implies that xkk∈N is a bounded in X. Since the dimension of X is finite, xkk∈N has a convergent subsequence. Consequently, I(x) satisfies the Palais-Smale condition. Hence, it is sufficient to prove that I(x) satisfies conditions (I1) and (I2) of Lemma 2.
For anyu=(c,c,…,c)∗∈X2, let (52)ε=∑t=1Tft22∑t=1Trt, and similar to the proof of Theorem1.1 we have (53)Iu=ε,∀u∈∂Bη∩X1. Condition (I1) of Lemma 2 is proved.
Then, we prove the condition(I2) of Lemma 2. For any v∈X1, by Hölder inequality, we have (54)Iv=12v∗Mv-12∑t=1Trtvt2+∑t=1Tftvt≤λmax2v2-rmin2∑t=1Tvt2+∑t=1Tftvt≤λmax2v2-rmin2v2+∑t=1Tf2t1/2v=λmax-rmin2v2+∑t=1Tf2t1/2v≤-∑t=1Tf2t2λmax-rmin, as one finds by maximization with respect to v. That is to say, (55)Iv≤-∑t=1Tf2t2λmax-rmin.
Let(56)χ=-∑t=1Tf2t2λmax-rmin. By (f2), we have (57)Iv≤χ<ε. Therefore, -I satisfies the condition of saddle point theorem. By Lemma 2, the proof of Theorem 4 is finished.


## 5. Examples

In this section, we shall provide two examples to illustrate our main theorem.Example 1.
Fort∈[1,3]Z, assume that(58)Δ2t-2Δ2xt-2-Δt-12Δxt-1+t3xt+0.1t=0, satisfying the boundary value conditions(59)x-1=x2,Δx-1=Δx2,Δ2x-1=Δ2x2,Δ3x-1=Δ3x2. We have (60)pt=t>0,qt=t2>0,rt=-t3<0,ft=0.1t,t∈1,3Z, with (61)p-1=2,p0=3,q0=9,M=25-7-18-714-7-18-725, and the eigenvalues of M are λ1=0,λ2=21, and λ3=43. Thus, λmin=21,rmax=-1 and (62)-∑t=13f2t∑t=13rt=-0.01+0.04+0.09×-36=5.04<λmin-rmax∑t=13ft=21--1×0.1+0.2+0.3=13.2. From the above argument, we see that all the suppositions of Theorem 3 are satisfied; then the BVP (58), (59) has at least one solution.Example 2.
Fort∈[1,4]Z, assume that(63)Δ2t-2Δ2xt-2-Δt-13Δxt-1+2txt+0.2t=0, satisfying the boundary value conditions(64)x-1=x3,Δx-1=Δx3,Δ2x-1=Δ2x3,Δ3x-1=Δ3x3. We have (65)pt=t>0,qt=t3>0,rt=-2t<0,ft=0.2t,t∈1,4Z, with (66)p-1=3,p0=4,q0=256,M=277-114-270-1119-1464-1484-74-2706-74338,and the eigenvalues of M are λ1=0,λ2≈23.9633,λ3≈107.7880 and λ4≈586.2487. Thus, λmin≈23.9633,rmax=-2 and (67)-∑t=14f2t∑t=14rt=-0.04+0.16+0.36+0.64×-20=24<λmin-rmax∑t=14ft≈23.9633--2×0.2+0.4+0.6+0.8=51.9266.From the above argument, we see that all the suppositions of Theorem 3 are satisfied; then the BVP (63), (64) has at least one solution.


## 6. Conclusions

Difference equations, the discrete analogue of differential equations, occur widely in numerous settings and forms, both in mathematics itself and in its applications in theory and practice. The boundary value problem discussed in this paper has important analogue in the continuous case of the fourth-order differential equation. Such problem is of special significance for the study of beam equations which are used to describe the bending of an elastic beam. The problem discussed in this paper can be extended to2nth-order difference boundary value problem.



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