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International Journal of Nonlinear Sciences and Numerical Simulation

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Volume 19, Issue 5

Issues

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Difference Systems

Tao Zhou / Xia Liu
  • Corresponding author
  • Oriental Science and Technology College, Hunan Agricultural University, Changsha 410128, China
  • Science College, Hunan Agricultural University, Changsha 410128, Changsha, China
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/ Haiping Shi
  • Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou 510440, China
  • School of Mathematics and Statistics, Central South University, Changsha 410083, China
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Published Online: 2018-06-26 | DOI: https://doi.org/10.1515/ijnsns-2017-0138

Abstract

This paper is devoted to investigate a question of the existence of solutions to boundary value problems for a class of nonlinear difference systems. The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

Keywords: boundary value problems; nonlinear difference systems; mountain pass lemma; critical point theory

MSC 2010: 39A10; 47J30; 58E05

1 Introduction

In the following and in the sequel, we denote by N, Z and R the sets of all natural numbers, integers and real numbers, respectively. For all a, b Z, we define Z(a)={a,a+1,},Z(a,b)={a,a+1,,b} when ab. Also, the symbol * will denote the transpose of a vector.

Consider the nonlinear second-order difference system Δpn(Δxn1)δqnxnδ+fnxn=0,nZ(1,k),(1)

with boundary value conditions αx0βΔx0=0=γxk+1+σΔxk,(2)

where Δ is the forward difference operator, Δxn= xn+1xn, δ>0 is the ratio of odd positive integers, pn and qn are real valued for each nZ, k is a given positive integer, α,β,γ and σ are constants, f C(R2, R). The boundary value problem eq. (1) with eq. (2) contains the following special Dirichlet boundary value conditions, mixed boundary value conditions and Neumann boundary value conditions: x0=0,xk+1=0;x0=0,Δxk=0;Δx0=0,xk+1=0;

and Δx0=0,Δxk=0.

We may regard eq. (1) as being a discrete analog of the following second-order differential equation p(t)φ(x)q(t)x(t)+f(t,x(t))=0,t[1,k].(3)

Equation (3) includes the following equation p(t)φ(x)+f(t,x(t))=0,tR,(4)

which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems and adiabatic reactor [1]. Equations similar in structure to eq. (4) arise in the study of periodic solutions and homoclinic orbits [10, 11, 12] of differential equations [21, 27, 28, 31].

Difference equations [1, 2, 3, 4, 5, 8, 9, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 29, 30, 32, 33, 34] have attracted the interest of many researchers in the past 20 years since they provide a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as finance insurance, biological populations, disease control, genetic study, physical field and computer application technology. Therefore, it is worthwhile to explore this topic.

When fnxn=0, Thandapani and Ravi [26] obtained results on the asymptotic behavior of solutions of Δpn(Δxn1)δ+qnxnδ=0(5)

including sufficient conditions for all solutions to be bounded or unbounded. Some results on the existence and behavior of nonincreasing solutions of eq. (5) are also obtained.

Cai and Yu [5] in 2006 concerned with the existence of solutions of boundary value problems for nonlinear second-order difference equations of the type Δpn(Δxn1)δ+qnxnδ+fnxn=0,nZ(1,k).

They applied the linking theorem and the mountain pass lemma in the critical point theory and give some new results for the existence of solutions.

Recently, Shi, Liu and Zhang [23] considered the following forward and backward difference equation Δpn(Δxn1)δ+qnxnδ+fnxn+1,xn,xn1=0,nZ,(6)

and obtained the existence of a nontrivial homoclinic orbit for eq. (6).

Motivated by the recent papers [6, 7, 18], the intention of this paper is to investigate a question of the existence of solutions to boundary value problems for a class of difference systems eq. (1) with eq. (2). The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

Let F(t,z)=0zf(t,s)ds0,

and pmin=minpn:nZ(1,k+1),pmax=maxpn:nZ(1,k+1),qmin=minqn:=nZ(1,k),qmax=maxqn:nZ(1,k).

For basic knowledge of variational methods, the reader is referred to [19, 20].

Let E be a real Banach space, JC1(E,R), i.e. J is a continuously Fréchet-differentiable functional defined on E. J is said to satisfy the Palais–Smale condition (P.S. condition for short) if any sequence x(l)lNE for which Jx(l)lN is bounded and Jx(l)0(l) possesses a convergent subsequence in E.

Let Bρ denote the open ball in E about 0 of radius ρ and let Bρ denote its boundary.

The rest of this paper is organized as follows. First of all, Section 2 presents variational structure. Next, in Section 3, we shall recall some related fundamental results and present some lemmas. Then, Section 4 is dedicated to the proof of main results. Finally, in Section 5, we shall give two examples to illustrate the applicability of the main results.

2 Variational structure

Our main tool is the critical point theory. We shall establish the corresponding variational framework for the boundary value problem (1) with eq. (2). We start by some basic notations for the reader’s convenience.

On one hand, let Rk be the real Euclidean space with dimension k. Rk can be equipped with the inner product x,y and norm x as follows, x,y:=j=1kxjyj,x,yRk,(7)

and x:=j=1kxj212,xRk.(8)

On the other hand, we define the norm s on Rk as follows: xs=j=1k|xj|s1s,

for all xRk and s>1.

Since xs and x2 are equivalent, there exist constants k1,k2 such that k2k1>0, and k1x2xsk2x2,xRk.(9)

For all xRk, define the functional J on Rk as follows: J(x)=1δ+1n=1k+1pn(Δxn1)δ+1+1δ+1n=1kqnxnδ+1n=1kFn(xn)+γσδpk+1xk+1δ+1δ+1+αβδp1x0δ+1δ+1

where x={xn}n=1k=(x1,x2,,xk),αx0βΔx0=0=γxk+1+σΔxk.

It is easy to see that JC1(Rk,R) and for any x={xn}n=1k=(x1,x2,,xk), by using αx0βΔx0=0=γxk+1+σΔxk and the summation by parts n=1kynΔxn1=ykxky1x0n=1kΔynxn,

we can compute the partial derivative as Jxn=Δpn(Δxn1)δ+qnxnδfnxn,nZ(1,k).

Thus, x is a critical point of J on Rk if and only if Δpn(Δxn1)δqnxnδ+fnxn=0,nZ(1,k).

We reduce the existence of the boundary value problem (1) with eq. (2) to the existence of critical points of J on Rk. That is, the functional J is just the variational framework of the boundary value problem (1) with eq. (2).

Let P=1100012100012000002100012

be a k×k matrix.

For convenience, we identify x Rk with x= (x1, x2,,xk).

3 Main results

In this section, we shall state and prove our main results by using the variational methods.

Theorem 3.1:

Suppose that the following assumptions are satisfied:

(B1) α0,β>0,γ0 and σ>0;

(p) for any nZ(1,k+1), pn>0;

(q1) for any nZ(1,k), qn>0;

(F1) there exist constants c1>0,c2>0 and ω>δ+1 such that F(t,y)c1|y|ωc2,(t,y)R2.

Then, the boundary value problem (1) with eq. (2) possesses at least one solution.

Proof

For any x=(x1,x2,,xk)Rk, combining with (F1), it is easy to see that J(x)=1δ+1n=1k+1pn(Δxn1)δ+1+1δ+1n=1kqnxnδ+1n=1kFn(xn)+γσδpk+1xk+1δ+1δ+1+αβδp1x0δ+1δ+1 pmax2δ+1δ+1n=1k+1xnδ+1+xn1δ+1+qmaxxδ+1δ+1δ+1c1n=1kxnω+c2k +γσδpmaxxδ+1δ+1δ+1+αβδpmaxxδ+1δ+1δ+1 3pmax2δ+1δ+1xδ+1δ+1+qmaxxδ+1δ+1δ+1+γσδpmaxxδ+1δ+1δ+1+αβδpmaxxδ+1δ+1δ+1c1n=1kxnω+c2k k2δ+1δ+13pmax2δ+1+qmax+γσδpmax+αβδpmaxxδ+1c1k1ωxω+c2k

as x+. By the continuity of J(x), we have from the above inequality that there exist upper bounds of values of functional J. Classical calculus shows that J attains its maximal value at some point which is just the critical point of J and the result follows. This completes the proof of Theorem 3.1.   □

Remark 3.1:

The results of Theorem 1.1 ensure that the boundary value problem (1) with eq. (2) possesses at least one solution. However, in some cases, we are interested in the existence of nontrivial solutions.

Theorem 3.2:

Assume that (p), (q1), (F1) and the following assumptions: (B2) α=0,β>0,γ=0 and σ>0;

(F2) for all (t,y)R2, limy0F(t,y)yδ+1=0.

Then, the boundary value problem (1) with eq. (2) possesses at least two nontrivial solutions.

Lemma 3.1:

(Mountain pass lemma [19, 20]). Let E be a real Banach space and JC1(E,R) satisfy the P.S. condition. If J(0)=0 and

(J1) there exist constants ρ,a>0 such that

J|Bρa, and

(J2) there exists eEBρ such that J(e)0.

Then, J possesses a critical value ca given by c=infgΥmaxθ[0,1]J(g(θ)),(10)

where Υ={gC([0,1],E)|g(0)=0,g(1)=e}.(11)

Lemma 3.2:

Assume that (p), (q1), (B2), (F1) and (F2) are satisfied. Then J satisfies the P.S. condition.

Proof

By (B2), we have that J(x)=1δ+1n=1k+1pn(Δxn1)δ+1+1δ+1n=1kqnxnδ+1n=1kFn(xn).

It is obvious that P is positive definite. Denote λmax=maxλj|j=1,2,,k,(12) λmin=minλj|j=1,2,,k.(13)

Let x(l)lNRk be such that Jx(l)lN is bounded and Jx(l)0 as l. Then, there exists a positive constant A such that Jx(l)A. It comes from (F2), eq. (12) that AJx(l)=1δ+1n=1k+1pnΔxn1(l)δ+1+1δ+1n=1kqnxn(l)δ+1n=1kFnxn(l) pmax2δ+1δ+1n=1k+1xn(l)δ+1+xn1(l)δ+1+qmaxδ+1x(l)δ+1δ+1c1n=1kxn(l)ω+c2k 3pmax2δ+1k2δ+1δ+1x(l)δ+1+qmaxk2δ+1δ+1x(l)δ+1c1k1ωx(l)ω+c2k.

That is, c1k1ωx(l)ω3pmax2δ+1+qmaxk2δ+1δ+1x(l)δ+1A+c2k.

Since ω>δ+1, there exists a positive constant B such that x(l)B,nN.

Therefore, x(l)nN is bounded on Rk. As a consequence, x(l)nN possesses a convergent subsequence in Rk. Therefore, the P.S. condition is satisfied.   □

Proof of Theorem 3.2

It follows from (F2) that for any ε=pminλminδ+12+qmin2(δ+1)k1δ+1, there exists δ>0, such that F(t,y)pminλminδ+12+qmin2(δ+1)k1δ+1yδ+1,

for |y|ρ.

For any x=(x1,x2,,xk)Rk and xρ, we have |xn|ρ,nZ(1,k). Then, J(x)=1δ+1n=1k+1pn(Δxn1)δ+1+1δ+1n=1kqnxnδ+1n=1kFn(xn)pminδ+1n=1k+1(Δxn1)δ+1+qminδ+1xδ+1δ+1pminλminδ+12+qmin2(δ+1)n=1kxnδ+1pmink1δ+1δ+1xPxδ+12+qmink1δ+1δ+1xδ+1pminλminδ+12+qmin2(δ+1)k1δ+1xδ+1pmink1δ+1λminδ+12δ+1xδ+1+qmink1δ+1δ+1xδ+1pminλminδ+12+qmin2(δ+1)k1δ+1xδ+1=pminλminδ+12+qmin2(δ+1)k1δ+1xδ+1.

Take apminλminδ+12+qmin2(δ+1)k1δ+1ρδ+1>0. Therefore, J(x)a>0,xBρ. At the same time, we have also proved that there exist constants a>0 and ρ>0 such that J|Bρa. That is to say, J satisfies the condition (J1) of the mountain pass lemma.

The rest of the proof is similar to that of [16, Theorem 1.2], but for the sake of completeness, we give the details.

For our setting, clearly J(0)=0. In order to exploit the mountain pass lemma in critical point theory, we need to verify other conditions of the mountain pass lemma. By Lemma 3.2, J satisfies the P.S. condition. So it suffices to verify the condition (J2).

From the proof of the P.S. condition, we know J(x)3pmax2δ+1+qmaxk2δ+1δ+1xδ+1c1k1ωxω+c2k. Since ω>δ+1, we can choose xˉ large enough to ensure that J(xˉ)<0. By the mountain pass lemma, J possesses a critical value ca>0, where c=infgΥsups[0,1]J(g(θ)) and Υ={gC([0,1],Rk)g(0)=0,g(1)=xˉ}.

Let x˜Rk be a critical point associated to the critical value c of J, i.e., J(x˜)=c. Similar to the proof of the P.S. condition, we know that there exists xˆRk such that J(xˆ)=cmax=maxθ[0,1]J(g(θ)).

Clearly, xˆ0. If x˜xˆ, then the conclusion of Theorem 1.2 holds. Otherwise, x˜=xˆ. Then, c=J(x˜)=cmax=maxθ[0,1]J(g(θ)). That is, supxRkJ(x)=infgΥsupθ[0,1]J(g(θ)). Therefore, cmax=maxθ[0,1]J(g(θ)),gΥ.

By the continuity of J(g(θ)) with respect to θ, J(0)=0 and J(Xˉ)<0 imply that there exists θ0(0,1) such that Jgθ0=cmax. Choose g1,g2Υ such that g1(θ)θ(0,1)g1(θ)θ(0,1) is empty, then there exists θ1,θ2(0,1) such that Jg1θ1=Jg2θ2=cmax. Thus, we get two different critical points of J on Rk denoted by x1=g1θ1,x2=g2θ2. The above argument implies that the boundary value problem (1) with eq. (2) possesses at least two nontrivial solutions.   □

Theorem 3.3:

Assume that (p), (q1), (B2) and the following assumptions are satisfied:

(q2) for any nZ(1,k), qn0;

(F3) yf(t,y)>0, for y0,tR.

Then, the boundary value problem (1) with eq. (2) has no nontrivial solution.

Proof

For the sake of contradiction, suppose that the boundary value problem (1) with eq. (2) has a nontrivial solution. Then, J has a nonzero critical point x. On one hand, since Jxn=ΔpnΔxn1δ+qnxnδfnxn,

we have n=1kfnxnxn=n=1kΔpnΔxn1δxn+n=1kqnxnδ+1 =n=1kpnΔxn1δ+1+n=1kqnxnδ+10.(14)

On the other hand, it follows from (F3) that n=1kfnxnxn>0.(15)

This contradicts eq. (14) and hence Theorem 3.3 is proved.   □

4 Examples

As an application of Theorems 3.2 and 3.3, we give two examples to illustrate our main results.

Example 4.1:

For nZ(1,k), consider the boundary value problem: Δ(Δxn1)δn2xnδ+(α+1)xnα=0(16)

with boundary value conditions (2) and α0,β>0,γ0, σ>0, where δ>0 is the ratio of odd positive integers, α>δ+1. We have pn1,qn=n2

and Fn(xn)=xnα+1.

It is easy to verify that all the conditions of Theorem 3.2 are satisfied and then the boundary value problem (16) with eq. (2) possesses at least two nontrivial solutions.

Example 4.2:

For nZ(1,k), consider the boundary value problem: Δen(Δxn1)δ+n6xnδ+8xn7=0(17)

with boundary value conditions (2) and α=0,β>0,γ=0, σ>0, where δ>0 is the ratio of odd positive integers.

We have pn=en,qn=n6

and fn(xn)=8xn7.

It is easy to verify that all the conditions of Theorem 3.3 are satisfied and then the boundary value problem (17) with eq. (2) has no nontrivial solution.

5 Conclusions

Difference equations occur widely in numerous settings and forms both in mathematics itself and in its applications. The boundary value problem discussed in this paper has important analog in the continuous case of the second-order differential equation. Such problem is of special significance for the study of a result that describes dynamically changing phenomena, evolution and variation. The problem discussed in this paper can be extended to boundary value problem for 2n-order ϕc-Laplacian [30] difference equations.

Acknowledgements

The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.

References

  • [1]

    R. P. Agarwal, D. O’Regan, P. J. Y. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic, Dordrecht, 1999. Google Scholar

  • [2]

    C. D. Ahlbrandt, A. C. Peterson. Discrete hamiltonian systems: difference equations, continued fraction and Riccati equations, Kluwer Academic Publishers, Dordrecht, 1996. Google Scholar

  • [3]

    Z. AlSharawi, J. M. Cushing, S. Elaydi, Theory and applications of difference equations and discrete dynamical systems, Springer, New York, 2014. Google Scholar

  • [4]

    G. M. Bisci, D. Repovš, Existence of solutions for p-Laplacian discrete equations, Appl. Math. Comput. 242 (2014), 454–461. Google Scholar

  • [5]

    X. C. Cai, J. S. Yu, Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320 (2) (2006), 649–661. CrossrefGoogle Scholar

  • [6]

    P. Chen, X. F. He, X. H. Tang, Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory, Math. Methods Appl. Sci. 39 (5) (2016), 1005–1019. Web of ScienceCrossrefGoogle Scholar

  • [7]

    P. Chen, X. H. Tang, Existence of solutions for a class of second-order p-Laplacian systems with impulsive effects, Appl. Math. 59 (5) (2014), 543–570. Web of ScienceCrossrefGoogle Scholar

  • [8]

    P. Chen, Z. M. Wang, Infinitely many homoclinic solutions for a class of nonlinear difference equations, Electron. J. Qual. Theory Differ. Equ. 2012 (47) (2012), 1–18. Google Scholar

  • [9]

    X. Q. Deng, Nonexistence and existence results for a class of fourth-order difference mixed boundary value problems, J. Appl. Math. Comput. 45 (1) (2014), 1–14. CrossrefGoogle Scholar

  • [10]

    C. J. Guo, R. P. Agarwal, C. J. Wang, D. O’Regan, The existence of homoclinic orbits for a class of first order superquadratic Hamiltonian systems, Mem. Differential Equations Math. Phys. 61 (2014), 83–102. Google Scholar

  • [11]

    C. J. Guo, D. O’Regan, Y. T. Xu, R. P. Agarwal. Existence of periodic solutions for a class of second-order superquadratic delay differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 21 (5) (2014) 405–419. Google Scholar

  • [12]

    C. J. Guo, D. O’Regan, Y. T. Xu, R. P. Agarwal, Existence of homoclinic orbits of a class of second-order differential difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 20 (2013), 675–690. Google Scholar

  • [13]

    M. Jia, Standing waves for the discrete nonlinear Schrödinger equations, Electron. J. Differ. Equ. 2016 (183) (2016), 1–9. Google Scholar

  • [14]

    J. H. Leng, Existence of periodic solutions for a higher order nonlinear difference equation, Electron. J. Differ. Equ. 2016 (134) (2016), 1–10. Google Scholar

  • [15]

    J. H. Leng, Periodic and subharmonic solutions for 2nth-order ϕc-Laplacian difference equations containing both advance and retardation, Indag. Math. (N.S.) 27 (4) (2016), 902–913. CrossrefWeb of Science

  • [16]

    X. Liu, Y. B. Zhang, H. P. Shi, Nonexistence and existence results for a class of fourth-order difference Neumann boundary value problems, Indag. Math. (N.S.) 26 (1) (2015), 293–305. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    X. Liu, Y. B. Zhang, H. P. Shi, Existence of periodic solutions for a class of nonlinear difference equations, Qual. Theory Dyn. Syst. 14 (1) (2015), 51–69. Web of ScienceCrossrefGoogle Scholar

  • [18]

    X. Liu, Y. B. Zhang, H. P. Shi, Nonexistence and existence results for a class of fourth-order difference Dirichlet boundary value problems, Math. Methods Appl. Sci. 38 (4) (2015), 691–700. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    J. Mawhin, M. Willem, Critical point theory and hamiltonian systems, Springer, New York, 1989. Google Scholar

  • [20]

    P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Amer. Math. Soc., Providence, RI, New York, 1986. Google Scholar

  • [21]

    S. Salahshour, A. Ahmadian, M. Senu, D. Baleanu, P. Agarwal, On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy 17 (2) (2015), 885–902. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    H. P. Shi, X. Liu, Y. B. Zhang, Periodic solutions for a class of nonlinear difference equations, Hokkaido Math. J. 45 (1) (2016), 109–126. CrossrefGoogle Scholar

  • [23]

    H. P. Shi, X. Liu, Y. B. Zhang, Homoclinic orbits for a class of nonlinear difference equations, Azerb. J. Math. 6 (1) (2016), 2218–6816. Google Scholar

  • [24]

    H. P. Shi, Y. B. Zhang, Standing wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials, Appl. Math. Lett. 58 (2016), 95–102. CrossrefGoogle Scholar

  • [25]

    A. N. Sharkovsky, Y. L. Maistrenko, E. Y. Romanenko. Difference equations and their applications, Kluwer Academic Publishers, Dordrecht, 1993. Google Scholar

  • [26]

    T. Thandapani, K. Ravi, Bounded and monotone properties of solutions of second-order quasilinear forced difference equations, Comput. Math. Appl. 38 (2) (1999), 113–121. CrossrefGoogle Scholar

  • [27]

    X. M. Zhang, P. Agarwal, Z. H. Liu, H. Peng, The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2), Open Math. 13 (1) (2015), 908–923. Web of ScienceGoogle Scholar

  • [28]

    X. M. Zhang, P. Agarwal, Z. H. Liu, H. Peng, F. You, Y. J. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1 with finite delays, Adv. Difference Equ. 2017 (123) (2017), 1–18. Web of ScienceCrossrefGoogle Scholar

  • [29]

    Z. Zhou, D. F. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math. 58 (4) (2015), 781–790. CrossrefGoogle Scholar

  • [30]

    Z. Zhou, M. T. Su, Boundary value problems for 2n-order ϕc-Laplacian difference equations containing both advance and retardation, Appl. Math. Lett. 41 (2015), 7–11. Web of ScienceCrossrefGoogle Scholar

  • [31]

    H. Zhou, L. Yang, P. Agarwal, Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval, J. Appl. Math. Comput. 53 (1–2) (2017), 51–76. CrossrefWeb of ScienceGoogle Scholar

  • [32]

    Z. Zhou, J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Sin. (Engl. Ser.) 29 (9) (2013), 1809–1822. CrossrefGoogle Scholar

  • [33]

    Z. Zhou, J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differ. Equ. 249 (5) (2010), 1199–1212. Web of ScienceCrossrefGoogle Scholar

  • [34]

    Z. Zhou, J. S. Yu, Y. M. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math. 54 (1) (2011), 83–93. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-06-29

Accepted: 2018-05-20

Published Online: 2018-06-26

Published in Print: 2018-07-26


This project is supported by the National Natural Science Foundation of China (No. 11501194). This work was carried out while visiting Central South University.


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 5, Pages 531–537, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0138.

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