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

Peng Mei; Zhan Zhou

doi:10.1515/math-2018-0123

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

Peng Mei and Zhan Zhou

Published/Copyright: December 29, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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 a ⩽ b ∈ Z, define Z(a) = {a, a + 1, ⋯}, Z(a, b) = {a, a + 1, ⋯, b}.

Consider the following 2nth-order nonlinear difference equation

Δnrk−nφpΔnuk−n=(−1)nf(k,uk+τ,⋯,uk+1,uk,uk−1,⋯,uk−τ),k∈Z,(1.1)

where τ, n ∈ Z(1), Δ is the forward difference operator defined by Δu_k = u_k+1-u_k, Δ²u_k = Δ(Δu_k), r_k > 0 is real valued for each k ∈ Z, r_k and f(k, v_τ, ⋯, v_–τ) are T-periodic in k for a given positive integer T, and f ∈ C(Z × R²τ+1, R). φ_p(s) is the p-Laplacian operator given by φ_p(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

Δnrk−nΔnuk−n+f(k,uk)=0,n∈Z(3),k∈Z,(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

ΔφpΔxn−1+f(n,xn+1,xn,xn−1)=0,n∈Z.(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

Δnrk−nϕΔnuk−1=(−1)nf(k,uk+1,uk,uk−1),k∈Z.(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_=mink∈Z(1,T){rk},r¯=maxk∈Z(1,T){rk}.

Our main result is as follows.

Theorem 1.1

Assume that the following hypotheses hold:

there exists a functional F(k, v₀, v₁, ⋯, v_τ) ∈ C¹(Z × R^τ+1, R) with F(k, v₀, v₁, ⋯, v_τ)⩾0 and it satisfies
F(k+T,v0,v1,⋯,vτ)=F(k,v0,v1,⋯,vτ),k∈Z,
∑j=kk+τ∂F(j,vj,vj−1,⋯,vj−τ)∂vk=f(k,vk+τ,⋯,vk+1,vk,vk−1,⋯,vk−τ);
there exists a constantα∈0,r_(τ+1)p/2p(mT)−|2−p|2(2sin⁡πmT)npsuch that
lim supδ1→0F(k,v0,v1,⋯,vτ)δ1p⩽α,fork∈Zandδ1=∑j=0τvj2;
there exists a constantβ∈r¯(τ+1)p/2p(mT)|2−p|2(2cos⁡1−(−1)mT2mTπ))np,+∞such that
lim infδ1→∞F(k,v0,v1,⋯,vτ)δ1p⩾β,fork∈Zandδ1=∑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}|uk∈R,k∈Z}.

For any u, v ∈ S, a, b ∈ R, 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 E_mT of S as

EmT={u∈S|uk+mT=uk,k∈Z}.

It is trivial to show that, E_mT is isomorphic to R^mT and can be endowed with the inner product

u,v=∑j=1mTujvj,∀u,v∈EmT,

and corresponding norm

||u||=∑j=1mTuj212,|∀|u∈EmT.

On the other hand, we define the norm ‖⋅‖_p on E_mT as follows:

||u||p=∑j=1mT|uj|p1p,

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

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

Let

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

then

c1(p)||u||⩽||u||p⩽c2(p)||u||,∀u∈EmT,(2.1)

and

c1(p)c2(p)=(mT)−|2−p|2p.

Define the functional J on E_mT as

J(u)=1p∑k=1mTrk−1Δnuk−1p−∑k=1mTF(k,uk,uk−1,⋯,uk−τ),u∈EmT.

Clearly, J ∈ C¹(E_mT, R) and by the fact that u₀ = u_mT, u₁ = u_mT+1, after a careful computation, we can find

∂J∂uk=(−1)nΔnrk−nφpΔnuk−n−f(k,uk+τ,⋯,uk+1,uk,uk−1,⋯,uk−τ),∀k∈Z(1,mT).

Thus, u is a critical point of J on E_mT 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 E_mT. Indeed, u ∈ E_mT can be identified with u = (u₁, u₂, ⋯, u_mT)^∗, where ^∗ denotes the transpose of the vector.

Let P be the corresponding mT × mT matrix to the quadratic form ∂J∂uk=(−1)nΔnrk−nφpΔnuk−n−f(k,uk+τ,⋯,uk+1,uk,uk−1,⋯,uk−τ),∀k∈Z(1,mT). with u_k+mT = u_k for k ∈ Z, which is defined by

P=2−10⋯0−1−12−1⋯000−12⋯00⋯⋯⋯⋯⋯⋯000⋯2−1−100⋯−12.

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

λj=4sin2⁡jπmT,j=0,1,2,⋯,mT−1.

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

λ_=min{λ1,λ2,⋯,λmT−1}=4sin2⁡πmT,λ¯=max{λ1,λ2,⋯,λmT−1}=4cos2⁡1−(−1)mT2mTπ.

Let

W=ker⁡P={u∈EmT|Pu=0,u∈RmT}.

Then

W={u∈EmT|u={c},c∈R}.

Let V be the direct orthogonal complement of E_mT to W, i.e., E_mT = V ⊕ W.

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, J ∈ C¹(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 B_ρ denote the open ball in Eabout 0 of radius ρ and let ∂ B_ρ denote its boundary.

Lemma 3.2

(Linking Theorem [24]). Let E be a real Banach space, E = E₁ ⊕ E₂, where E₁is finite dimensional. Suppose that J ∈ C¹(E, R}) satisfies the P.S. condition and

There exist constants a > 0 and ρ > 0 such that J|_∂ B_ρ ∩ E₂ ⩾ a;
(J₂) There exists an e ∈ ∂B₁ ∩ E₂and a constant R₀ ⩾ ρ such that J|_∂Q ⩽ 0, where Q = (B̄_R0 ∩ E₁)⊕{se|0 < s < R₀}.

Then J possesses a critical value c⩾ a, where

c=infh∈Γsupu∈QJ(h(u)),

and Γ = {h ∈ C(Q̄}, 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 = (Δ^n–1u₁, Δ^n–1u₂, ⋯, Δ^n–1u_mT)^∗. For any u ∈ E_mT, one has

λ_(n−1)p2||u||p⩽||x||p⩽λ¯(n−1)p2||u||p.

Let G(u)=1p∑k=1mTrk−1Δnuk−1p, it follows from Lemma 3.2 that

G(u)⩽r¯p(∑k=1mTΔnuk−1p)1pp⩽r¯pc2(p)(∑k=1mTΔnuk−12)12p⩽r¯pc2p(p)λ¯p2||x||p⩽r¯pc2p(p)λ¯np2||u||p

and

G(u)⩾r_p(∑k=1mTΔnuk−1p)1pp⩾r_pc1(p)(∑k=1mTΔnuk−12)12p⩾r_pc1p(p)λ_p2||x||p⩾r_pc1p(p)λ_np2||u||p.

Lemma 3.4

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

Proof

By (T₃), there exist constants ζ > 0 and β′∈r¯(τ+1)p/2p(mT)|2−p|2(2cos⁡1−(−1)mT2mTπ))np,β such that

F(k,v0,v1,⋯,vτ)⩾β′∑j=0τvj2p−ζ,∀(k,v0,v1,⋯,vτ)∈Z×Rτ+1.

For any u ∈ E_mT, by Lemma 3.3 we have

J(u)=1p∑k=1mTrk−1Δnuk−1p−∑k=1mTF(k,uk,uk−1,⋯,uk−τ) ⩽r¯pc2p(p)λ¯np2||u||p−∑k=1mTβ′∑i=−τ0uk+i2p−ζ ⩽r¯pc2p(p)λ¯np2||u||p−β′c1p(p)∑k=1mT∑i=−τ0uk+i2p2+mTζ =r¯pc2p(p)λ¯np2||u||p−(τ+1)p2β′c1p(p)||u||p+mTζ =r¯pc2p(p)λ¯np2−(τ+1)p2β′c1p(p)||u||p+mTζ ⩽mTζ.

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=2−2−22.

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

−M⩽Ju(i),∀i∈N.

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

−M⩽Ju(i)⩽r¯pc2p(p)λ¯np2−(τ+1)p2β′c1p(p)|u(i)|p+mTζ,∀i∈N.

Therefore,

(τ+1)p2β′c1p(p)−r¯pc2p(p)λ¯np2|u(i)|p⩽M+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 E_mT. 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 k ∈ Z. 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 E_mT.

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

There exists ū ∈ E_mT 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)−|2−p|2(2sin⁡πmT)np such that F(k,u0,u1,⋯,uτ)⩽α′∑j=0τuj2p,fork∈Zand∑j=0τuj2⩽δ2.

For any u ∈ V with ‖u‖ ⩽ δ, we have

J(u)⩾r_pc1p(p)λ_np2−(τ+1)p2α′c2p(p)∥u∥p.(4.1)

Taking σ=r_pc1p(p)λ_np2−(τ+1)p2α′c2p(p)δp, then by Eq. (4.1),

J(u)⩾σ,∀u∈V∩∂Bδ.

This implies that c0=supu∈EmTJ(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|_{∂B_δ} ∩ V ⩾ σ. 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 E_mT. In the following, we shall verify the condition (J₂).

Take e ∈∂B₁ ∩ V. For any z ∈ W and s ∈R, we denote u = se + z. Then we have

J(u)=1p∑k=1mTrk−1Δnuk−1p−∑k=1mTF(k,uk,uk−1,⋯,uk−τ) ⩽r¯psp∑k=1mTΔnekp−∑k=1mTF(k,sek+zk,sek−1+zk−1,⋯,sek−τ+zk−τ) ⩽r¯pspc2p(p)∑k=1mTΔnek2p2−β′c1p(p)∑k=1mT∑i=−τ0(sek+i+zk+i)2p2+mTζ ⩽r¯pspc2p(p)λ¯np2−β′c1p(p)(τ+1)∑k=1mTsek+zk2p2+mTζ ⩽r¯pc2p(p)λ¯np2sp−(τ+1)p2β′c1p(p)s2+∥z∥2p2+mTζ.

Noting that for all u∈W,∑k=1mTrk−1Δnuk−1p=0, we have

J(u)=1p∑k=1mTrk−1Δnuk−1p−∑k=1mTF(k,uk,uk−1,⋯,uk−τ)=−∑k=1mTF(k,uk,uk−1,⋯,uk−τ)⩽0.

Thus, there exists a positive constant R₁ > δ such that J(u) ⩽ 0 for any u ∈∂Q, where Q = (B̄_R₁ ∩ W)⊕{se|0 < s < R₁}. This verifies the condition (J₂) of Lemma 3.2. Therefore, J possesses a critical value c ⩾ σ > 0 with

c=infh∈Γsupu∈QJ(h(u))andΓ={h∈C(Q¯,EmT)∣h|∂Q=id}.

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

c0=supu∈EmTJ(u)=infh∈Γsupu∈QJ(h(u)).

In particular, choosing h = id, we have supu∈QJ(u)=c0. Since the choice of e ∈ ∂B₁ ∩ V is arbitrary, we can take –e ∈∂B₁ ∩ V. Similarly, there exists a positive number R₂ > δ, so that J(u) ⩽ 0 for any u ∈∂Q₁, where Q₁ = (B̄_R₂ ∩ W) ⊕ {–se|0 < s < R₂}.

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

c′=infh∈Γ1supu∈Q1J(h(u))andΓ1={h∈C(Q¯1,EmT)∣h|∂Q1=id}.

We claim that c′ ≠ c₀. Otherwise, suppose that c′ = c₀, then supu∈Q1J(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, Q ∩ Q₁⊂ W and J(u) ⩽0 for any u ∈ W. 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:

there exist constantsρ1>0,α∈0,r_(τ+1)p/2p(mT)−|2−p|2(2sin⁡πmT)npsuch that
F(k,v0,v1,⋯,vτ)⩽α∑j=0τvj2p,fork∈Zand∑j=0τvj2⩽ρ12;
there exist constantsρ2>0,ζ1>0,β∈r¯(τ+1)p/2p(mT)|2−p|2(2cos⁡1−(−1)mT2mTπ))np,+∞such that
F(k,v0,v1,⋯,vτ)⩾β∑j=0τvj2p−ζ1,fork∈Zand∑j=0τvj2⩾ρ22.
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:

limδ1→0F(k,v0,v1,⋯,vτ)δ1p=0,δ1=∑j=0τvj2,∀(k,v0,v1,⋯,vτ)∈Z×Rτ+1;
there exist constants ρ₃ > 0 and p′ > p such that for k ∈ Z and ∑j=0τvj2⩾ρ32,
0<p′F(k,v0,v1,⋯,vτ)⩽∑j=0τ∂F(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, u_k) = q_kg (u_k), Eq. (1.1) reduces to the following 2nth-order nonlinear equation with p-Laplacian,

Δnrk−nφpΔnuk−n=(−1)nqkguk,k∈Z,(4.2)

where g ∈ C( R, R), q_k+T} = q_k > 0, for all k ∈ Z. Then, we have the following results.

Corollary 4.3

Assume that the following hypotheses hold:

there exists a functional G(v) ∈ C¹(R, R) with G(v)⩾0 and it satisfies
G′(v)=g(v);
there exists a constantα∈0,r_(τ+1)p/2p(mT)−|2−p|2(2sin⁡πmT)npsuch that
lim sup|v|→0G(v)|v|p⩽α;
there exists a constantβ∈r¯(τ+1)p/2p(mT)|2−p|2(2cos⁡1−(−1)mT2mTπ))np,+∞such that
lim inf|v|→∞G(v)|v|p⩾β.

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:

there exist constantsρ1′>0,α∈0,r_p(mT)−|2−p|2(2sin⁡πmT)npsuch that
G(v)⩽α|v|p,for|v|⩽ρ1′;
there exist constantsρ2′>0,ζ1′>0,β∈r¯p(mT)|2−p|2(2cos⁡1−(−1)mT2mTπ))np,+∞such that
G(v)⩾β|v|p−ζ1′,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:

lim|v|→0G(v)|v|p=0,∀v∈R;
there exist constantsρ3′ > 0 and p͠′ > p such that for k ∈ Z 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, r_k ≡ 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 n ∈ Z(1), consider the following difference equation

Δnrk−nφpΔnuk−n=(−1)nμuk∑j=kk+τ2+sinjπT(∑i=−τ0ui+j2)μ2−1,k∈Z(5.1)

where {r_k}_k∈Zis a real sequence andr_k+T = r_k > 0, 1 < p < ∞, μ > p, T is a given positive integer. Here

f(k,vk+τ,⋯,vk+1,vk,vk−1,⋯,vk−τ)=μvk∑j=kk+τ2+sinjπT∑i=−τ0vi+j2μ2−1

and

F(k,v0,v−1,⋯,v−τ)=2+sinkπT∑i=−τ0vi2μ2.

Then

∑j=kk+τ∂F(j,vj,vj−1,⋯,vj−τ)∂vk=μvk∑j=kk+τ2+sinjπT∑i=−τ0vi+j2μ2−1.

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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Received: 2018-06-24

Accepted: 2018-11-08

Published Online: 2018-12-29

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0123

Keywords for this article

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

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