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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations

Peng Mei / Zhan Zhou
Published Online: 2018-12-29 | DOI: https://doi.org/10.1515/math-2018-0123

Abstract

We consider a 2nth-order nonlinear difference equation containing both many advances and retardations with p-Laplacian. Using the critical point theory, we obtain some new explicit criteria for the existence and multiplicity of periodic and subharmonic solutions. Our results generalize and improve some known related ones.

Keywords: Periodic and subharmonic solution; 2nth-order; Nonlinear difference equation; p-Laplacian; Critical point theory

MSC 2010: 39A23

1 Introduction

Let N, Z and R denote the sets of all natural numbers, integers and real numbers, respectively. For abZ, define Z(a) = {a, a + 1, ⋯}, Z(a, b) = {a, a + 1, ⋯, b}.

Consider the following 2nth-order nonlinear difference equation

ΔnrknφpΔnukn=(1)nf(k,uk+τ,,uk+1,uk,uk1,,ukτ),kZ,(1.1)

where τ, nZ(1), Δ is the forward difference operator defined by Δuk = uk+1-uk, Δ2 uk = Δ(Δ uk), rk > 0 is real valued for each kZ, rk and f(k, vτ, ⋯, vτ) are T-periodic in k for a given positive integer T, and fC(Z × R2τ+1, R). φp(s) is the p-Laplacian operator given by φp(s) = |s|p–2s (1 < p < ∞). For any integer m ⩾ 2, a solution to Eq. (1.1) is called a mth-order subharmonic solution if it is a mT-periodic solution.

Earlier, the main methods were all kinds of fixed point theorems in cones for the study of periodic solutions and boundary value problems of difference equations. It was not until 2003 that the critical point theory was used to establish sufficient conditions for the existence of periodic solutions. Guo and Yu [1, 2] first established sufficient conditions on the existence of periodic solutions of second-order nonlinear difference equations by using the critical point theory.

In 2007, Cai and Yu [3] obtained some criteria for the existence of periodic solutions of a 2nth-order difference equation

ΔnrknΔnukn+f(k,uk)=0,nZ(3),kZ,(1.2)

where f grows superlinearly at both 0 and ∞. Using Linking Theorem and Saddle Point Theorem, Zhou [4] improved the results of [3] and extended f in Eq. (1.2) into sublinear or asymptotically linear.

In fact, there are some papers which studied the periodic solutions of difference equations involving both advances and retardations which have important background and meaning in the field of cybernetics and biological mathematics. Chen and Fang [5] in 2007 considered the following second-order nonlinear difference equation containing both advance and retardation with p-Laplacian

ΔφpΔxn1+f(n,xn+1,xn,xn1)=0,nZ.(1.3)

In 2014, Lin and Zhou [6] obtained some new sufficient conditions on the existence and multiplicity of periodic solutions of the following ϕ-Laplacian difference equation

ΔnrknϕΔnuk1=(1)nf(k,uk+1,uk,uk1),kZ.(1.4)

In the past literature, when it comes to a special Eq. (1.1) that τ = 1, many excellent works have been done (e.g. see [5, 6, 7, 8, 9]). Using the critical point theory, they obtained some sufficient conditions on the existence and multiplicity of periodic solutions in the special case of Eq. (1.1).

Nevertheless, to the best of our knowledge, the results on periodic solutions of nonlinear difference equations containing both many advances and retardations with p-Laplacian are very scare. To fill this gap, this paper gives some sufficient conditions for the existence and multiplicity of periodic and subharmonic solutions to Eq. (1.1).

We mention that, in recent years, the critical point theory is also used on the study of homoclinic solutions [10, 11, 12, 13, 14, 15, 16, 17] and boundary value problems [18, 19, 20, 21, 22, 23] for difference equations.

Let

r_=minkZ(1,T){rk},r¯=maxkZ(1,T){rk}.

Our main result is as follows.

Theorem 1.1

Assume that the following hypotheses hold:

  • (T1)

    there exists a functional F(k, v0, v1, ⋯, vτ) ∈ C1(Z × Rτ+1, R) with F(k, v0, v1, ⋯, vτ)⩾0 and it satisfies

    F(k+T,v0,v1,,vτ)=F(k,v0,v1,,vτ),kZ,

    j=kk+τF(j,vj,vj1,,vjτ)vk=f(k,vk+τ,,vk+1,vk,vk1,,vkτ);

  • (T2)

    there exists a constant α0,r_(τ+1)p/2p(mT)|2p|2(2sinπmT)np such that

    lim supδ10F(k,v0,v1,,vτ)δ1pα,forkZandδ1=j=0τvj2;

  • (T3)

    there exists a constant βr¯(τ+1)p/2p(mT)|2p|2(2cos1(1)mT2mTπ))np,+ such that

    lim infδ1F(k,v0,v1,,vτ)δ1pβ,forkZandδ1=j=0τvj2.

Then for any given positive integer m, Eq. (1.1) possesses at least two mT-periodic nontrivial solutions.

It is worth pointing out that our sufficient conditions are based on the limit superior and limit inferior which are more applicable. Moreover, we also extend the conclusions to a more general form. As far as we know, in most of the previous results (e.g. see [7]), the values of c1 and c2 cannot be determined, that is not operable, but we present their specific values. In fact, if τ = 1, the assumptions in Theorem 1.1 are more explicit and easier to verify than those in Theorem 1.1 in [7]. For the sake of clarity, we put the remaining results at the end of the article. Our results complement the existing ones. See Remarks 4.6 and 4.7 for details.

The outline of this paper is as follows. In Section 2 we establish the variational framework associated with Eq. (1.1) and transfer the problem of the existence of periodic solutions of Eq. (1.1) into that of the existence of critical points of the corresponding functional. In Section 3, some related fundamental results are recalled for convenience, and some lemmas are proven. Then, we complete the proof of our main result by using Linking Theorem in Section 4. Finally, in Section 5, we illustrate our results with an example.

2 Variational structure

This section is to establish the corresponding variational framework for Eq. (1.1) and cite some basic conclusions for the forthcoming discussion.

Let S be the set of all two-side sequences, that is

S={{uk}|ukR,kZ}.

For any u, vS, a, bR, au + bv is defined by

au+bv={auk+bvk}k=+.

Then S is a vector space. For any given positive integers m and T, we define the subspace EmT of S as

EmT={uS|uk+mT=uk,kZ}.

It is trivial to show that, EmT is isomorphic to RmT and can be endowed with the inner product

u,v=j=1mTujvj,u,vEmT,

and corresponding norm

||u||=j=1mTuj212,||uEmT.

On the other hand, we define the norm ‖⋅‖p on EmT as follows:

||u||p=j=1mT|uj|p1p,

for all uEmT. Similarly to the derivation of [6], by Hölder inequality and Jensen inequality, we have

||u||||u||p(mT)2p2p||u||,1<p<2,(mT)2p2p||u||||u||p||u||,2p.

Let

c1(p)=1,1<p<2,(mT)2p2p,2p,c2(p)=(mT)2p2p,1<p<2,1,2p,

then

c1(p)||u||||u||pc2(p)||u||,uEmT,(2.1)

and

c1(p)c2(p)=(mT)|2p|2p.

Define the functional J on EmT as

J(u)=1pk=1mTrk1Δnuk1pk=1mTF(k,uk,uk1,,ukτ),uEmT.

Clearly, JC1(EmT, R) and by the fact that u0 = umT, u1 = umT+1, after a careful computation, we can find

Juk=(1)nΔnrknφpΔnuknf(k,uk+τ,,uk+1,uk,uk1,,ukτ),kZ(1,mT).

Thus, u is a critical point of J on EmT if and only if Eq. (1.1) holds.

So, we reduce the existence of periodic solutions of Eq. (1.1) to that of the critical points of the functional J on EmT. Indeed, uEmT can be identified with u = (u1, u2, ⋯, umT), where denotes the transpose of the vector.

Let P be the corresponding mT × mT matrix to the quadratic form Juk=(1)nΔnrknφpΔnuknf(k,uk+τ,,uk+1,uk,uk1,,ukτ),kZ(1,mT). with uk+mT = uk for kZ, which is defined by

P=2100112100012000002110012.

By the matrix theory, we obtain that the eigenvalues of P are

λj=4sin2jπmT,j=0,1,2,,mT1.

This implies λ0 = 0, λ1 > 0, λ2 > 0, ⋯, λmT–1 > 0. Therefore,

λ_=min{λ1,λ2,,λmT1}=4sin2πmT,λ¯=max{λ1,λ2,,λmT1}=4cos21(1)mT2mTπ.

Let

W=kerP={uEmT|Pu=0,uRmT}.

Then

W={uEmT|u={c},cR}.

Let V be the direct orthogonal complement of EmT to W, i.e., EmT = VW.

3 Some results and lemmas

For the reader’s convenience, we give some basic notations and some known results about the critical point theory.

Definition 3.1

Let E be a real Banach space, JC1(E, R), i.e., J is a continuously Fréchet-differentiable functional defined on E. If any sequence {u(i)E for which {J(u(i)) is bounded and J (u(i)) → 0 (i → ∞) possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition (P.S. condition for short).

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

Lemma 3.2

(Linking Theorem [24]). Let E be a real Banach space, E = E1E2, where E1 is finite dimensional. Suppose that JC1(E, R}) satisfies the P.S. condition and

  • (J1)

    There exist constants a > 0 and ρ > 0 such that J| BρE2a;

  • (J2)

    (J2) There exists an e B1E2 and a constant R0ρ such that J| Q ⩽ 0, where Q = (R0E1)⊕{se|0 < s < R0}.

Then J possesses a critical value ca, where

c=infhΓsupuQJ(h(u)),

and Γ = {hC(}, E)∣ h| Q = id}, where id denotes the identity operator.

Then we prove some lemmas which are useful in the proof of Theorem 1.1. First, similarly to the derivation of [3], we can find the following lemma.

Lemma 3.3

Let x = (Δn–1 u1, Δn–1 u2, ⋯, Δn–1 umT). For any uEmT, one has

λ_(n1)p2||u||p||x||pλ¯(n1)p2||u||p.

Let G(u)=1pk=1mTrk1Δnuk1p, it follows from Lemma 3.2 that

G(u)r¯p(k=1mTΔnuk1p)1ppr¯pc2(p)(k=1mTΔnuk12)12pr¯pc2p(p)λ¯p2||x||pr¯pc2p(p)λ¯np2||u||p

and

G(u)r_p(k=1mTΔnuk1p)1ppr_pc1(p)(k=1mTΔnuk12)12pr_pc1p(p)λ_p2||x||pr_pc1p(p)λ_np2||u||p.

Lemma 3.4

Assume that (T1) and (T3) hold. Then the functional J is bounded from above on EmT.

Proof

By (T3), there exist constants ζ > 0 and βr¯(τ+1)p/2p(mT)|2p|2(2cos1(1)mT2mTπ))np,β such that

F(k,v0,v1,,vτ)βj=0τvj2pζ,(k,v0,v1,,vτ)Z×Rτ+1.

For any uEmT, by Lemma 3.3 we have

J(u)=1pk=1mTrk1Δnuk1pk=1mTF(k,uk,uk1,,ukτ)r¯pc2p(p)λ¯np2||u||pk=1mTβi=τ0uk+i2pζr¯pc2p(p)λ¯np2||u||pβc1p(p)k=1mTi=τ0uk+i2p2+mTζ=r¯pc2p(p)λ¯np2||u||p(τ+1)p2βc1p(p)||u||p+mTζ=r¯pc2p(p)λ¯np2(τ+1)p2βc1p(p)||u||p+mTζmTζ.

The proof of Lemma 3.4 is complete. □

Remark 3.5

The case mT = 1 is trivial. For the case mT = 2, P has a different form, namely,

P=2222.

However, in this special case, the argument need not to be changed and we omit it.

Lemma 3.6

Assume that (T1) and (T3) hold. Then the functional J satisfies the P.S. condition.

Proof

Let {J(u(i)}) be a bounded sequence from the lower bound, i.e., there exists a positive constant M such that

MJu(i),iN.

By the proof of Lemma 3.4, it is easy to see that

MJu(i)r¯pc2p(p)λ¯np2(τ+1)p2βc1p(p)|u(i)|p+mTζ,iN.

Therefore,

(τ+1)p2βc1p(p)r¯pc2p(p)λ¯np2|u(i)|pM+mTζ.

Since β>r¯(τ+1)p/2pc2(p)c1(p)pλ¯np2, it is not difficult to know that {u(i)} is a bounded sequence on EmT. As a consequence, {u(i)} possesses a convergence subsequence and J satisfies the P.S. condition. □

4 Proof of the main result

In this section, we shall prove our main results by using Linking Theorem.

Assumptions (T1) and (T2) imply that F(k, 0, ⋯, 0) = 0 and f(k, 0, ⋯, 0) = 0 for kZ. Then u = 0 is a trivial mT-periodic solution of Eq. (1.1). It suffices to prove that J has at least two nontrivial critical points on EmT.

Firstly, we show the existence of one nontrivial critical point. By Lemma 3.4, J is bounded from above on EmT. The proof of it implies lim|u|+J(u)=. This means that –J(u) is coercive. Let we define c0=supuEmTJ(u).

There exists ūEmT such that J(ū) = c0 by the continuity of J(u). Clearly, ū is a critical point of J.

We claim that c0 > 0, which implies that ū is a nontrivial crtical point of J. Indeed, by (T2), there exist two positive constants δ and α′ ∈ α,r_(τ+1)p/2p(mT)|2p|2(2sinπmT)np such that F(k,u0,u1,,uτ)αj=0τuj2p,forkZandj=0τuj2δ2.

For any uV with ‖u‖ ⩽ δ, we have

J(u)r_pc1p(p)λ_np2(τ+1)p2αc2p(p)up.(4.1)

Taking σ=r_pc1p(p)λ_np2(τ+1)p2αc2p(p)δp, then by Eq. (4.1),

J(u)σ,uVBδ.

This implies that c0=supuEmTJ(u)σ>0. Hence the claim is proved. At the same time, we have proved that there exist constants σ > 0 and δ > 0 such that J|BδVσ. In the other word, J satisfies the condition (J1) of Linking Theorem.

In the remaining of the proof, we shall use Lemma 3.2 to obtain another nontrivial critical point. We have known that J satisfies the P.S. condition on EmT. In the following, we shall verify the condition (J2).

Take e B1V. For any zW and sR, we denote u = se + z. Then we have

J(u)=1pk=1mTrk1Δnuk1pk=1mTF(k,uk,uk1,,ukτ)r¯pspk=1mTΔnekpk=1mTF(k,sek+zk,sek1+zk1,,sekτ+zkτ)r¯pspc2p(p)k=1mTΔnek2p2βc1p(p)k=1mTi=τ0(sek+i+zk+i)2p2+mTζr¯pspc2p(p)λ¯np2βc1p(p)(τ+1)k=1mTsek+zk2p2+mTζr¯pc2p(p)λ¯np2sp(τ+1)p2βc1p(p)s2+z2p2+mTζ.

Noting that for all uW,k=1mTrk1Δnuk1p=0, we have

J(u)=1pk=1mTrk1Δnuk1pk=1mTF(k,uk,uk1,,ukτ)=k=1mTF(k,uk,uk1,,ukτ)0.

Thus, there exists a positive constant R1 > δ such that J(u) ⩽ 0 for any u Q, where Q = (R1W)⊕{se|0 < s < R1}. This verifies the condition (J2) of Lemma 3.2. Therefore, J possesses a critical value cσ > 0 with

c=infhΓsupuQJ(h(u))andΓ={hC(Q¯,EmT)h|Q=id}.

Let ũEmT be a critical point associated to the critical value c of J, i.e., J(ũ) = c. If ũū, then we are done. Otherwise, if ũ = ū, it follows that

c0=supuEmTJ(u)=infhΓsupuQJ(h(u)).

In particular, choosing h = id, we have supuQJ(u)=c0. Since the choice of e B1V is arbitrary, we can take –e B1V. Similarly, there exists a positive number R2 > δ, so that J(u) ⩽ 0 for any u Q1, where Q1 = (R2W) ⊕ {–se|0 < s < R2}.

Using Linking Theorem again, J possesses a critical value c′⩾ σ > 0, where

c=infhΓ1supuQ1J(h(u))andΓ1={hC(Q¯1,EmT)h|Q1=id}.

We claim that c′ ≠ c0. Otherwise, suppose that c′ = c0, then supuQ1J(u)=c0. Due to the facts J|Q ⩽ 0 and J|Q1 ⩽ 0, J attains its maximum at some points in the interior of sets Q and Q1. However, QQ1W and J(u) ⩽0 for any uW. This implies that c0 ⩽0, which contradicts c0 > 0. This proves the claim and hence the proof is complete.

According to Theorem 1.1, it is easy to obtain the following theorems and corollaries.

Theorem 4.1

Assume that (T1) and the following conditions hold:

  • (T4)

    there exist constants ρ1>0,α0,r_(τ+1)p/2p(mT)|2p|2(2sinπmT)npsuch that

    F(k,v0,v1,,vτ)αj=0τvj2p,forkZandj=0τvj2ρ12;

  • T5)

    there exist constants ρ2>0,ζ1>0,βr¯(τ+1)p/2p(mT)|2p|2(2cos1(1)mT2mTπ))np,+such that

    F(k,v0,v1,,vτ)βj=0τvj2pζ1,forkZandj=0τvj2ρ22.

    Then for any given positive integer m, Eq. (1.1) has at least two mT-periodic nontrivial solutions.

Theorem 4.2

Assume that (T1) iand the following conditions hold:

  • (T6)

    limδ10F(k,v0,v1,,vτ)δ1p=0,δ1=j=0τvj2,(k,v0,v1,,vτ)Z×Rτ+1;

  • (T7)

    there exist constants ρ3 > 0 and p′ > p such that for kZ and j=0τvj2ρ32,

    0<pF(k,v0,v1,,vτ)j=0τF(k,v0,v1,,vτ)vjvj.

Then for any given positive integer m, Eq. (1.1) has at least two mT-periodic nontrivial solutions.

If τ = 0 and f(k, uk) = qkg (uk), Eq. (1.1) reduces to the following 2nth-order nonlinear equation with p-Laplacian,

ΔnrknφpΔnukn=(1)nqkguk,kZ,(4.2)

where gC( R, R), qk+T} = qk > 0, for all kZ. Then, we have the following results.

Corollary 4.3

Assume that the following hypotheses hold:

  • (G1)

    there exists a functional G(v) ∈ C1(R, R) with G(v)⩾0 and it satisfies

    G(v)=g(v);

  • (G2)

    there exists a constant α0,r_(τ+1)p/2p(mT)|2p|2(2sinπmT)np such that

    lim sup|v|0G(v)|v|pα;

  • (G3)

    there exists a constant βr¯(τ+1)p/2p(mT)|2p|2(2cos1(1)mT2mTπ))np,+ such that

    lim inf|v|G(v)|v|pβ.

Then for any given positive integer m, Eq. (4.2) has at least two mT-periodic nontrivial solutions.

Corollary 4.4

Assume that (G1) and the following conditions hold:

  • (G4)

    there exist constants ρ1>0,α0,r_p(mT)|2p|2(2sinπmT)np such that

    G(v)α|v|p,for|v|ρ1;

  • (G5)

    there exist constants ρ2>0,ζ1>0,βr¯p(mT)|2p|2(2cos1(1)mT2mTπ))np,+ such that

    G(v)β|v|pζ1,for|v|ρ2.

Then for any given positive integer m, Eq. (4.2) has at least two mT-periodic nontrivial solutions.

Corollary 4.5

Assume that (G1) and the following conditions hold:

  • (G6)

    lim|v|0G(v)|v|p=0,vR;

  • (G7)

    there exist constants ρ3 > 0 and ′ > p such that for kZ and |v|⩾ ρ3,

    0<p~G(v)vg(v).

Then for any given positive integer m, Eq. (4.2) has at least two mT-periodic nontrivial solutions.

Remark 4.6

If τ = 1, rk ≡ 1 and n = 1, Theorem 3.1 reduces to Theorem 3.1 in [5].

Remark 4.7

If τ = 0 and p = 2, Theorem 4.2 reduces to Theorem 1.1, Corollary 1.1 reduces to Corollary 1.1 in [3].

5 Example

Finally, as an application of Theorem 1.1, we give an example to illustrate our main results.

Example 5.1

For given nZ(1), consider the following difference equation

ΔnrknφpΔnukn=(1)nμukj=kk+τ2+sinjπT(i=τ0ui+j2)μ21,kZ(5.1)

where {rk}kZ is a real sequence and rk+T = rk > 0, 1 < p < ∞, μ > p, T is a given positive integer. Here

f(k,vk+τ,,vk+1,vk,vk1,,vkτ)=μvkj=kk+τ2+sinjπTi=τ0vi+j2μ21

and

F(k,v0,v1,,vτ)=2+sinkπTi=τ0vi2μ2.

Then

j=kk+τF(j,vj,vj1,,vjτ)vk=μvkj=kk+τ2+sinjπTi=τ0vi+j2μ21.

It is easy to verify all the assumptions of Theorem 1.1 are satisfied. Consequently, for any given positive integer m, Eq. (5.1) has at least two mT-periodic nontrivial solutions.

Acknowledgement

We are grateful to the anonymous referee for his/her valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11571084) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_16R16).

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About the article

Received: 2018-06-24

Accepted: 2018-11-08

Published Online: 2018-12-29


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1435–1444, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0123.

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© 2018 Mei and Zhou, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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