Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system

Xia Liu; Tao Zhou; Haiping Shi

doi:10.1515/math-2019-0102

Article Open Access

Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system

Xia Liu , Tao Zhou and Haiping Shi

Published/Copyright: December 10, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we concern with a 2nth-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system.

Keywords: minimal period; 2nth-order; nonlinear; discrete system; critical point

MSC 2010: 39A23; 34C25; 58E05; 37J45

1 Introduction

Below ℕ, ℤ, ℝ denote the sets of all natural numbers, integers and real numbers, respectively. * denotes the transpose of a vector. [⋅] is denoted by the greatest-integer function. Let a, b ∈ ℤ, we define ℤ(a) = {a, a + 1, ⋯} and when a < b, ℤ(a, b) = {a, a + 1, ⋯, b}.

In this paper, we shall study the 2nth-order discrete system

Δ2nuk−n=(−1)nf(k,uk),n∈Z(3),k∈Z, (1.1)

where Δ is the forward difference operator Δ u_k = u_k+1 − u_k, Δⁱ u_k = Δ(Δⁱ⁻¹ u_k) for i ≥ 2, f ∈ C¹(ℝ², ℝ), f(k + T, u) = f(k, u) for a given integer T ≥ 3.

He and Chen [1] in 2008 concerned with the existence of a periodic solution for the following second order discrete convex systems involving the p-Laplacian:

Δϕpun−1+∇F(k,uk)=0,k∈Z.

Some existence theorems are obtained by using the dual least action principle.

In 2007, Cai and Yu [2] considered the 2nth-order difference equation

Δnrk−nΔnuk−n+f(k,uk)=0,n∈Z(3),k∈Z.

By the Linking Theorem, some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation.

By establishing a proper variational framework and using the critical point theory, Chen and Tang [3] established some new existence criteria to guarantee the 2nth-order nonlinear difference equation containing both many advances and retardations

Δnrk−nΔnuk−n+qnuk=f(k,uk+n,⋯,uk,⋯,uk−n),n∈Z(3),k∈Z,

has at least one or infinitely many homoclinic orbits. Their conditions on the potential are rather relaxed and some existing results in the literature are improved.

Leng in 2016 considered the 2nth-order difference equation with ϕ_c-Laplacian

Δnrk−nϕcΔnuk−n=(−1)nf(k,uk+1,uk,uk−1),k∈Z,

where n is a fixed positive integer, Δ is the forward difference operator Δ u_k = u_k+1 − u_k, Δⁿ u_k = Δ(Δⁿ⁻¹ u_k), r_k is real valued for each k ∈ ℤ, ϕ_c is a special ϕ-Laplacian operator defined by ϕc(s)=s1+s2, f ∈ C(ℤ × ℝ³, ℝ), r_k and f(k, v₁, v₂, v₃) are T-periodic in k for a given positive integer T. By using the critical point theory, some new criteria for the existence and multiplicity of periodic and subharmonic solutions are established.

In the aforementioned references, most of the results are periodic solutions or homoclinic orbits of difference equations. Yu, Long and Guo [4] established some existence criteria to periodic solutions with prescribed minimal period of second-order difference equation

Δ2uk−1+Asin⁡uk=f(k),k∈Z,

by making use of the variational methods.

Existence of solutions of higher-order nonlinear differential equations has been the subject of many investigations [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Difference equations, the discrete analogs of differential equations, occur widely in numerous settings and forms, not only in mathematics itself but also in several economical and population problems. However, to the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system [1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 22]. The difficulty lies in the fact that there are very scarce techniques to study the existence of periodic solutions with prescribed minimal period. The purpose of this paper is to establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period to a 2nth-order nonlinear discrete system. The main approach used in our paper is a variational technique. The motivation for the present work stems from the recent papers [18, 20].

Set

ω=2πT.

The rest of this paper is organized as follows. Firstly, in Section 2, we shall establish the variational framework associated with (1.1) and reduce the existence of periodic solutions of (1.1) to seeking the existence of critical points of the corresponding functional. Then, in Section 3, we shall give auxiliary results which will be of fundamental importance in proving our main results. Finally, in Section 4, we shall prove the existence results by using the variational methods.

To conclude the Introduction, the reader is referred to [23, 24, 25] for the general background on difference equations and [26] for the basic knowledge of variational methods.

2 Variational framework

The purpose of this section is to establish the variational framework associated with (1.1) and to state some basic notations for the coming discussion.

Let U be a pT-dimensional Euclidean space consisting of functions ℤ → ℝ and for any k ∈ ℤ,

U={u|uk+pT=uk}.

U is equipped with a norm

∥u∥=∑i=1pTui212,

and an inner product

u,v=∑i=1pTuivi.

Let us consider a functional J defined on U as follows

J(u)=12∑k=1pTΔnuk−12−∑k=1pTF(k,uk). (2.2)

It is obvious that J ∈ C¹(U, ℝ) and for any u ∈ U, we can calculate

∂J∂uk=(−1)nΔ2nuk−n−f(k,uk),∀k∈Z(1,pT).

As a consequence, u is a critical point of J(u) on U if and only if

Δ2nuk−n=(−1)nf(k,uk),∀k∈Z(1,pT).

Set

M=2−10⋯0−1−12−1⋯000−12⋯00⋯⋯⋯⋯⋯⋯000⋯2−1−100⋯−12pT×pT.

It is easy to know that the eigenvalues of M are

λk=4sin2⁡kπpT,k=0,1,2,⋯,pT−1,

0 is an eigenvalue of M and

min{λ1,λ2,⋯,λpT−1}=4sin2⁡πpT,max{λ1,λ2,⋯,λpT−1}≤4.

Let

K=ker⁡M=u∈U|Mu=0,

and the eigenvectors of M corresponding to λ_k by

μi=cos⁡2iπpT,cos⁡2iπ⋅2pT,⋯,cos⁡2iπ⋅pTpT∗,i=1,2,⋯,pT−12,

and

νi=sin⁡2iπpT,sin⁡2iπ⋅2pT,⋯,sin⁡2iπ⋅pTpT∗,i=1,2,⋯,pT−12.

We define

A=span {μi},i=1,2,⋯,pT−12,

and

B=span {νi},i=1,2,⋯,pT−12.

If pT is odd, then U = K ⊕ A ⊕ B. For any u ∈ U and k ∈ ℤ,

uk=a+(−1)kb+∑i=1pT−12aicos⁡ωipk+bisin⁡ωipk,

where a, b, a_i and b_i are constants.

If pT is even, then 4 is the eigenvalue of M. Let ξ be the eigenvector corresponding to 4, and D = span {ξ}. We have U = K ⊕ A ⊕ B ⊕ D. For any u ∈ U and k ∈ ℤ,

uk=a+∑i=1pT−12aicos⁡ωipk+bisin⁡ωipk,

where a, a_i and b_i are constants.

Now, we give the existence results of at least one periodic solution with minimal period pT as follows.

Theorem 2.1

Assume that the following conditions are satisfied:

there is a function F(s, u) ∈ C²(ℝ², ℝ) with F(s + T, u) = F(s, u), F(–s, –u) = F(s, u), F(s, u) ≥ 0 and it satisfies

∂F(s,u)∂u=f(s,u),∀s∈R;
lim|u|→+∞F(s,u)u2=0 uniformly for s ∈ ℝ;
there exist three constants η > 0 and ϵ > ε > 0 such that

∂2F(s,u)∂u2θ,θ≤ϵθ2,∀(s,u)∈R2,θ∈R

and

∂2F(s,u)∂u2θ,θ≥εθ2,∀|u|≤η,s∈R,θ∈R;
if u is a solution of (1.1) with a minimal period ϕT, ϕ is a rational number, and f(s, u) also has a minimal period ϕT, then ϕ must be an integer;
let p > 1 be a given positive integer and l_p denote the least prime factor of p,

4sin2⁡ωlp2pn>ϵ,4sin2⁡ω2pn<ε

and

∑k=1pTf2(k,0)<4πη24sin2⁡ωlp2pn−ϵε−4sin2⁡ω2pnω.

Then (1.1) has at least one periodic solution with minimal period pT.

Remark 2.1

The assumption (F₂) implies that

(F̃₂) there is a constant C > 0$ such that

F(s,u)≤C,∀(s,u)∈R2.

Corollary 2.1

Suppose that (F₁)–(F₄) are satisfied and

4sin2⁡ω2n>ϵ,4sin2⁡ω2pn<ε.

∑k=1pTf2(k,0)<4πη24sin2⁡ω2n−ϵεω,

then there is P > 0 such that for any prime integer p > P, (1.1) has at least one periodic solution with minimal period pT.

Theorem 2.2

Assume that the assumptions (F₁) – (F₄) hold. If

f(k,0)=0,∀k∈Z,

and

4sin2⁡ωlp2pn>ϵ,4sin2⁡ω2pn<ε,

then (1.1) has at least one periodic solution with minimal period pT.

Corollary 2.2

Suppose that (F₁) – (F₄) are satisfied. If

f(k,0)=0,∀k∈Z,

and

4sin2⁡ω2n>ϵ,4sin2⁡ω2pn<ε,

then there is P > 0 such that for any prime integer p > P, (1.1) has at least one periodic solution with minimal period pT.

3 Auxiliary results

In this section, we shall give auxiliary results which will be of fundamental importance in proving our main results.

Let B_r denote the open ball in U about 0 of radius r and ∂B_r denote its boundary.

Lemma 3.1

[26] Let U be a finite dimensional Hilbert space, J(u) ∈ C¹(U, ℝ) is coercive, i.e., J(u) → +∞, as ∥u∥ → +∞. Then J(u) attains its minimal at some ũ on U.

Set

U~=u∈U|u−k=uk,∀k∈Z.

We have Ũ = B, then

uk=∑i=1pT−12bisin⁡ωipk,∀k∈Z.

Lemma 3.2

Assume that the assumptions (F₁) – (F₅) hold. Then J(u) attains its minimal at some ũ on Ũ.

Proof

From (F̃₂), for any u ∈ Ū,

J(u)=12∑k=1pTΔnuk−1,Δnuk−1−∑k=1pTF(k,uk)=12∑k=1pTΔnuk,Δnuk−∑k=1pTF(k,uk)=12x∗Mx−∑k=1pTF(k,uk)≥12×4sin2⁡πpT∥x∥2−pTC=2sin2⁡πpT∥x∥2−pTC,

where x = (Δ^n–1 u₁, Δ^n–1 u₂, ⋯, Δ^n–1 u_pT)^*. Since

∥x∥2=∑k=1pTΔn−2uk+1−Δn−2uk2≥4sin2⁡πpT∑k=1pTΔn−2uk2≥⋯≥4sin2⁡πpTn−1∥u∥2,

we have

J(u)≥124sin2⁡πpTn∥u∥2−pTC→+∞,

as ∥u∥ → +∞. By Lemma 3.1, the conclusion of Lemma 3.2 is true.□

Lemma 3.3

Assume that u is a critical point of J(u) on Ũ. Then u is a critical point of J(u) on U.

The proof of Lemma 3.3 is similar to that of Lemma 2.2 in [4]. For the simplicity, we omit its proof.

Let

Ψϕ=−p24sin2⁡ωϕ2pn−ϵ∑k=1pTf2(k,0).

Lemma 3.4

Assume that the assumptions (F₁) – (F₅) hold and J(u) < Ψ_{l_p}. If u is a critical point of J(u) on Ũ, then u has a minimal period pT.

Proof

Assume, for the sake of contradiction, that u exists a minimal period pTϕ . It comes from (F₅) that ϕ ≥ l_p.

Similarly, for any u ∈ Ũ,

uk=∑j=1pT−ϕ2ϕbjsin⁡ωϕjpk,

and then

J(u)=12x∗Mx−∑k=1pTF(k,uk)≥2sin2⁡ωϕ2p∥x∥2−∑k=1pTF(k,uk),

where x = (Δx₁, Δx₂, ⋯, Δx_pT)^*. It is easy to see that

∥x∥2=∑k=1pTΔn−2uk+1−Δn−2uk2≥4sin2⁡ωϕ2p∑k=1pTΔn−2uk2≥⋯≥4sin2⁡ωϕ2pn−1∥u∥2.

We have

J(u)≥124sin2⁡ωϕ2pn∥u∥2−∑k=1pTf2(k,0)12∥u∥−ϵ2∥u∥2≥−124sin2⁡ωϕ2pn−ϵ∑k=1pTf2(k,0)≥−p24sin2⁡ωϕ2pn−ϵ∑k=1pTf2(k,0)=Ψϕ,

which is a contradiction to the assumption J(u) < Ψ_{l_p}. The result is obtained.□

4 Proofs of the existence results

In this section, we shall give the proofs of the existence results by making use of the variational method.

Proof of Theorem 2.1

We shall prove that (1.1) has at least one periodic solution with minimal period pT. Lemmas 3.1-3.3 imply that (1.1) has at least one pT-periodic solution. Hence, by Lemma 3.4, it suffices to prove that

J(u)<Ψlp,∀u∈U~.

According to the condition (F₃), we have

F(k,u)=f(k,0)u+12×∂2F(k,ζu)∂u2u2≥f(k,0)u+ε2u2,∀|u|≤η.

Then

J(u)=12∑k=1pTΔ2uk,Δ2uk−∑n=1pTF(k,uk)≤12∑k=1pTΔ2uk,Δ2uk−ε2∑k=1pTuk2−∑k=1pTf(k,0)uk.

Make a choice that

uk=ηsin⁡ωkp.

Combining with f(–k, 0) = f(k, 0) and f(k + T, 0) = f(k, 0), we have

f(k,0)=∑j=1[T−12]ajsin⁡2jπTk=∑j=1[T−12]ajsin⁡2jπpTpk,

where a_j is a constant. Therefore

∑k=1pTf(k,0)uk=∑j=1[T−12]ηaj∑k=1pTsin⁡2jπpTpk⋅sin⁡2πpTk=0.

Similarly, we get

J(u)≤24sin2⁡ω2pn−ε∥u∥2.

Obviously,

∥u∥=ηpπω12.

Thereby,

J(u)=24sin2⁡ω2pn−εη2pπω<Ψlp.

The desired result follows.□

Proof of Corollary 2.1

For the reason that p is a positive prime integer, it is easy to see that l_p = p. Therefore

∑k=1pTf2(k,0)<4πη24sin2⁡ω2n−ϵε−4sin2⁡ω2pnω.

Due to Theorem 2.1, the conclusion of Corollary 2.1 is obviously true. The proof of Corollary 2.1 is finished.

Remark 3.1

Similarly to the proofs of Theorem 2.1 and Corollary 2.1, we can also prove Theorem 2.2 and Corollary 2.2. For simplicity, we omit their proofs.

Acknowledgements

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).

References

[1] He T.S., Chen W.G., Periodic solutions of second order discrete convex systems involving the p-Laplacian, Appl. Math. Comput., 2008, 206(1), 124–132.10.1016/j.amc.2008.08.037Search in Google Scholar

[2] Cai X.C., Yu J.S., Existence of periodic solutions for a 2nth-order nonlinear difference equation, J. Math. Anal. Appl., 2007, 329(2), 870–878.10.1016/j.jmaa.2006.07.022Search in Google Scholar

[3] Chen P., Tang X.H., Existence of solutions for a class of second-order p-Laplacian systems with impulsive effects, 2014, Appl. Math., 59(5), 543–570.10.1007/s10492-014-0071-5Search in Google Scholar

[4] Yu J.S., Long Y.H., Guo Z.M., Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation, J. Dynam. Differential Equations, 2004, 16(2), 575–586.10.1007/s10884-004-4292-2Search in Google Scholar

[5] Chen P., Tang X.H., Existence and multiplicity of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations, J. Math. Anal. Appl., 2011, 381(2), 485–505.10.1016/j.jmaa.2011.02.016Search in Google Scholar

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

[7] Guo C.J., Guo C.X., Ahmed S., Liu X.F., Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 2016, 21(8), 2473–2489.10.3934/dcdsb.2016056Search in Google Scholar

[8] Guo C.J., O’Regan D., Xu Y.T., Agarwal R.P., Existence of subharmonic solutions and homoclinic orbits for a class of high-order differential equations, Appl. Anal., 2011, 90(7), 1169–1183.10.1080/00036811.2010.524154Search in Google Scholar

[9] He T.S., Huang Y.H., Liang K.H., Lei Y.F., Nodal solutions for noncoercive nonlinear Neumann problems with indefinite potential, Appl. Math. Lett., 2017, 71, 67–73.10.1016/j.aml.2017.03.015Search in Google Scholar

[10] He T.S., Wu D.Q., Sun H.Y., Liang K.H., Sign-changing solutions for resonant Neumann problems, J. Math. Anal. Appl., 2017, 454(2), 659–672.10.1016/j.jmaa.2017.05.017Search in Google Scholar

[11] Rabinowitz R.H., Heteroclinic orbits for a Hamiltonian system of double pendulum type, Topol. Methods Nonlinear Anal., 1997, 9(1), 41–76.10.12775/TMNA.1997.004Search in Google Scholar

[12] Tang X.H., Chen S.T., Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 2017, 37(9), 4973–5002.10.3934/dcds.2017214Search in Google Scholar

[13] Tang X.H., Chen S.T., Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 2017, 56(4) 110–134.10.1007/s00526-017-1214-9Search in Google Scholar

[14] Zhang X.M., Agarwal P., Liu Z.H., Peng H., You F., Zhu Y.J., Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1$ with finite delays, Adv. Difference Equ., 2017, 2017:123.10.1186/s13662-017-1169-3Search in Google Scholar

[15] Zhou H., Yang L., Agarwal P., Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval, J. Appl. Math. Comput., 2017, 53(1-2), 51–76.10.1007/s12190-015-0957-8Search in Google Scholar

[16] Candito P., Bisci G.M., Existence of two solutions for a second-order discrete boundary value problem, Adv. Nonlinear Stud., 2011, 11(2), 443–453.10.1515/ans-2011-0212Search in Google Scholar

[17] Chen P., Tang X.H., Existence of homoclinic solutions for some second-order discrete Hamiltonian systems, J. Difference Equ. Appl., 2013, 19(4), 633–648.10.1080/10236198.2012.666239Search in Google Scholar

[18] Leng J.H., Periodic and subharmonic solutions for 2nth-order ϕ_c-Laplacian difference equations containing both advance and retardation, Indag. Math. (N.S.), 2016, 27(4), 902–913.10.1016/j.indag.2016.05.002Search in Google Scholar

[19] Tang X.H., Lin X.Y., Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, J. Difference Equ. Appl., 2013, 19(5), 796–813.10.1080/10236198.2012.691168Search in Google Scholar

[20] Xia F., Existence of periodic solutions for higher order difference equations containing both many advances and retardations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 2018, 112(1), 239–249.10.1007/s13398-017-0376-9Search in Google Scholar

[21] Zhou Z., Ma D.F., Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 2015, 58(4), 781–790.10.1007/s11425-014-4883-2Search in Google Scholar

[22] Zhou Z., Yu J.S., Chen Y.M., Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 2011, 54(1), 83–93.10.1007/s11425-010-4101-9Search in Google Scholar

[23] Cull P., Flahive M., Robson R., Difference Equations: From Rabbits to Chaos, Springer, New York, 2005.Search in Google Scholar

[24] Elaydi S., An Introduction to Difference Equations, Springer, New York, 2005.Search in Google Scholar

[25] Kocic V.L., Ladas G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.10.1007/978-94-017-1703-8Search in Google Scholar

[26] Rabinowitz P.H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc., Providence, RI, New York, 1986.10.1090/cbms/065Search in Google Scholar

Received: 2018-06-07

Accepted: 2019-05-30

Published Online: 2019-12-10

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0102

Keywords for this article

minimal period; 2nth-order; nonlinear; discrete system; critical point

Creative Commons

BY 4.0