Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

Yuhua Long; Yuanbiao Zhang; Haiping Shi

doi:10.1515/math-2016-0046

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Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

Yuhua Long , Yuanbiao Zhang and Haiping Shi

Published/Copyright: July 22, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

By using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.

Keywords: Homoclinic solutions; 2nth-order; Nonlinear difference equations; Discrete variational theory

MSC 2010: 34C37; 37J45; 39A12

1 Introduction

The problem of homoclinic orbits for differential equations has been the subject of many investigations. As is known to us, homoclinic orbits play an important role in analyzing the chaos of dynamical systems. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation, its perturbed system probably produce chaotic phenomenon. So homoclinic orbits have been extensively investigated since the time of Poincaré, see for instance [1–8] and the references therein.

On the other hand, difference equations [9] are closely related to differential equations in the sense that a differential equation model is often derived from a difference equation, and numerical solutions of a differential equation have to be obtained by discretizing the differential equation. Therefore, the study of homoclinic orbits [10–21] of difference equation is meaningful.

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} when a < b. l² denotes the space of all real functions whose second powers are summable on Z. Also, * denotes the transpose of a vector.

The present paper considers the 2nth-order difference equation

Δnγk−nΔnuk−n+(−1)nχkuk=(−1)nf(k,uk+1,uk,uk−1),k∈Z,(1)

where n is a fixed positive integer, Δ is the forward difference operator Δuk=uk+1−uk,Δnuk=Δ(Δn−1uk), γk and χk are positive real valued for each k ∈ Z, f ∈ C(Z × R³, R), γk, χk and f(k, v₁, v₂, v₃) are T-periodic in k for a given positive integer T.

Difference equations containing both advance and retardation have many applications in theory and practice [9–11, 22]. We may think of (1) as a discrete analogue of the following 2nth-order functional differential equation

dndtnγ(t)dnu(t)dtn+(−1)nX(t)u(t)=(−1)nf(t,u(t+1),u(t),u(t−1)),t∈R.(2)

Equations similar in structure to (2) arise in the study of homoclinic orbits [2, 4–6] of functional differential equations for which the evolution of the function depends on its current state, its history, and its future as well. Such a problem is of special significance for the study of master equations in stochastic process [2–6].

Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [23] established sufficient conditions on the existence of periodic solutions of second-order nonlinear difference equations. Compared to first-order or second-order difference equations, the study of higher-order equations has received considerably less attention (see, for example, [10–25] and the references contained therein). Peil and Peterson [26] in 1994 studied the asymptotic behavior of solutions of 2nth-order difference equation

∑i=0nΔirik−iΔiuk−i=0(3)

with r_i (k) = 0 for 1 ≤ i ≤ n – 1. In 1998, Anderson [27] considered (3) for k ∈ Z(a), and obtained a formulation of generalized zeros and (n, n)-disconjugacy for (3). Migda [28] in 2004 studied an mth-order linear difference equation. Cai, Yu [24] in 2007 and Zhou, Yu, Chen [25] in 2010 obtained some criteria for the existence of periodic solutions of the following difference equation

Δnrk−nΔnuk−n+fk,uk=0.(4)

In 2011, Chen and Tang [11] established some new existence criteria to guarantee that the 2nth-order nonlinear difference equation

Δnrk−nΔnuk−n+qkuk=fk,uk+n,⋅⋅⋅,uk.⋯,uk−n(5)

has at least one or infinitely many homoclinic solutions.

However, to the best of our knowledge, the results on homoclinic solutions of higher-order nonlinear difference equations are scarce in the literature [10, 11, 15, 20, 24, 27]. Furthermore, since (1) contains both advance and retardation, there are very few manuscripts dealing with this subject. The purpose of this paper is two-folded. On one hand, we shall further demonstrate the powerfulness of critical point theory in the study of homoclinic orbits for difference equations. On the other hand, we shall extend and improve some existing results. In fact, one can see the following Remarks 1.3 and 1.4 for details. The proof is based on the notable Mountain Pass Lemma in combination with variational technique. The motivation for the present work stems from the recent papers [1, 11, 20].

Let

γ_=mink∈Z(1,T){γk},γ¯=maxk∈Z(1,T)⁡{γk},χ_=mink∈Z(1,T){χk},χ¯=maxk∈Z(1,T)χk.

Our main results are as follows.

Theorem 1.1

Assume that the following hypotheses are satisfied:

(F₁) there exists a function F(k, v₁, V₂) ∈ C¹(Z × R², R) with F(k + T, v₁, V₂) = F(k, v₁, V₂) and it satisfies
∂Fk−1,v2,v3∂v2+∂Fk,v1,v2∂v2=fk,v1,v2,v3;
(F₂) there exist positive constants ϱ anda<χ_4such thatF(k,v1,v2)≤av12+v22for allk ∈ Zandv12+v22≤ϱ;
(F₃) there exist constantsρ,c>4n−1γ¯+χ¯4and b such thatF(k,v1,v2)≥cv12+v22+bfor all k ∈ Zand(v12+v22)≥ρ;
(F₄) ∂F(k,v1,v2)∂v1v1+∂F(k,v1,v2)∂v2v2−2F(k,v1,v2)>0, for all (k, v₁, v₂) ∈ Z × R² \ {(0, 0)};
(F₅) ∂F(k,v1,v2)∂v1v1+∂F(k,v1,v2)∂v2v2−2F(k,v1,v2)→+∞asv12+v22→+∞.

Then (1) has a nontrivial homoclinic solution.

Remark 1.2

By (F₃), it is easy to see that there exists a constant ζ > 0 such that

F3′Fk,v1,v2≥cv12+v22+b−ζ,∀k,v1,v2∈Z×R2.

As a matter of fact, let ζ=maxF(k,v1,v2)−c(v12+v22)−b:k∈Z,v12+v22≤ρ, we can easily get the desired result.

Remark 1.3

Theorem 1.1 extends Theorem 1.1 in [19] which is the special case of our Theorem 1.1 by letting n = 1.

Remark 1.4

In many studies (see e.g. [16, 18, 19, 22, 23]) of second order difference equations, the following classical Ambrosetti-Rabinowitz condition is assumed.

(AR) there exists a constant β > 2 such that 0 < βF(k, u) ≤ uf(k, u) for all k ∈ Zand u ∈ R \ {0}.

Note that (F₃) – (F₅) are much weaker than (AR). Thus our result improves the existing ones.

Theorem 1.5

Assume that (F₁) – (F₅) and the following hypothesis are satisfied:

(F₆) γ–k = γk, χ–k = χk, F(–k, v₁, v₂) = F(k, v₁, v₂).

Then (1) has a nontrivial even homoclinic solution.

For the basic knowledge of variational methods, the reader is referred to [29, 30].

2 Preliminaries

In order to apply the critical point theory, we shall establish the corresponding variational framework for (1) and give some lemmas which will be of fundamental importance in proving our results. We start by some basic notations.

Let S be the set of sequences u=(⋯,u−k,⋯,u−1,u0,u1,⋯,uk,⋯)={uk}k=−∞+∞, 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, is defined as a subspace of S by

Em={u∈S|uk+2mT=uk,∀k∈Z}.

Clearly, E_m is isomorphic to R^2mT. E_m can be equipped with the inner product

u,v=∑j=−mTmT−1ujvj,∀u,v∈Em,(6)

by which the norm || · || can be induced by

u=∑j=−mTmT−1uj212,∀u∈Em.(7)

It is obvious that E_m with the inner product (6) is a finite dimensional Hilbert space and linearly homeomorphic to R^2mT

For all u ∈ E_m, define the functional J on E_m as follows:

Ju=12∑k=−mTmT−1γk−1(Δnuk−1)2+12∑k=−mTmT−1χkuk2−∑k=−mTmT−1F(k,uk+1,uk).(8)

Clearly, J ∈ C¹(E_m, R) and for any u = {u_k}_kez ∈ E_m, by the periodicity of {u_k}_n∈Z, we can compute the partial derivative as

∂J∂uk=(−1)nΔnγk−nΔnuk−n+χkuk−fk,uk+1,uk,uk−1.(9)

Thus, u is a critical point of J on E_m if and only if

Δn(γk−nΔnuk−n)+(−1)nχkuk=(−1)nf(k,uk+1,uk,uk+1),∀k∈Z(−mT,mT−1).

Due to the periodicity of u = {uk}_k∈Z ∈ E_m and f(k, v₁, v₂, v₃) in the first variable k, we reduce the existence of periodic solutions of (1) to the existence of critical points of J on E_m. That is, the functional J is just the variational framework of (1).

In what follows, we define a norm ‖·‖_∞ in E_m by

u∞=maxj∈Z(−mT,mT−1)uj,∀u∈Em.

Let E be a real Banach space, J ∈ C¹ (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 {u⁽ⁱ⁾} ⊂ E for which {J (u⁽ⁱ⁾)} is bounded and J′ (u⁽ⁱ⁾) → 0 (i → ∞) possesses a convergent subsequence in E.

Let P be the 2mT × 2mT matrix defined by

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

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

λj=21−cosjmTπ,j=0,1,2,⋯,2mT−1.(10)

Thus, λ₀ = 0, λ₁ > 0, λ₂ >0, …, λ_2mT–1 > 0. Therefore, λmax=max{λ1,λ2,⋯,λ2mT−1}=4.

For convenience, we identify u ∈ E_m with u = (u_–mT, u_–mT +1, …, u_mT–1)*- Let B_ρ denote the open ball in E about 0 of radius ρ and let ∂B_ρ denote its boundary.

Lemma 2.1

(Mountain Pass Lemma [29, 30]). LetE be a real Banach space and J ∈ C¹(E, R) satisfy the P.S. condition. If J(0) = 0 and

(J₁) there exist constantsρ, α > 0 such that ∂B_ρ ≥ α, and
(J₂) there exists e ∈ E \ B_ρsuch thatJ(e) ≤ 0.

Then J possesses a critical value c ≥ α given by

c=infg∈Γmaxs∈[0,1]J(g(s)),(11)

where

Γ={g∈C([0;1],E)|g(0)=0,g(1)=e}.(12)

Lemma 2.2

The following inequality is true:

12∑k=−mTmT−1γk−1(Δnuk−1)2≤4nγ¯2u2.(13)

Proof

From the definition of P,

12∑k=−mTmT−1γk−1(Δnuk−1)212∑k=−mTmT−1γk(Δnuk,Δnuk)≤γ¯2x∗Px≤γ¯2⋅4x2,

where x=(Δn−1u−mT,Δn−1u−mT+1,⋯,Δn−1umT−1)∗. Since

x2=∑k=−mTmT−1Δn−2uk+1−Δn−2uk2≤4∑k=−mTmT−1Δn−2uk2≤4n−1u2,

we have

12∑k=−mTmT−1γk−1(Δnuk−1)2≤4nγ¯2u2.

□

3 Proofs of theorems

In this section, we shall prove our main results by using the critical point method.

Lemma 3.1

Assume that (F₁) – (F₅) are satisfied. Then J satisfies the PS. condition.

Proof

Assume that {u⁽ⁱ⁾}_i∈N in E_m is a sequence suchthat {J (u⁽ⁱ⁾)}_i∈N is bounded. Then there exists a constant K > 0 such that – K ≤ J (u⁽ⁱ⁾). By (13) and (F′₃), we have

−K≤Ju(i)≤4nγ¯2u(i)2+χ¯2u(i)2−∑k=−mTmT−1cuk+1(i)2+uk(i)2+b−ζ≤4nγ¯2+χ¯2−2cu(i)2+2mT(ζ−b).

2c−4nγ¯2−χ¯2u(i)2≤2mT(ζ−b)+K.(14)

Therefore,

Since c>4n−1γ¯+χ¯4, (14) implies that {u⁽ⁱ)}_i∈N is bounded in E_m. Thus, {u⁽ⁱ)}_i∈N possesses a convergence subsequence in E_m. The desired result follows. □

Lemma 3.2

Assume that (F₁) – (F₅) are satisfied. Then for any given positive integer m, (1) possesses a 2mT-periodic solution u^(m) ∈ E_m.

Proof

In our case, it is clear that J(0) = 0. By Lemma 3.1, J satisfies the P.S. condition. By (F₂), we have

Ju≥=χ_2u2−a∑k=−mTmT−1uk+12+uk2≥χ_2u2−2au2=χ_2−2au2.

Taking α=χ_2−2aϱ2>0, we obtain

J(u)∂Bϱ≥α>0,

which implies that J satisfies the condition (J₁) of the Mountain Pass Lemma.

Next, we shall verify the condition (J₂).

There exists a sufficiently large number ε > max{ϱ, ρ} such that

2c−4nγ¯2−χ¯2ε2≥b.(15)

Let e ∈ E_m and

ek=ε,ifk=0,0,ifk∈{j∈Z:−mT≤j≤mT−1andj≠0},ek+1=ε,ifk=0,0,ifk∈{j∈Z:−mT≤j≤mT−1andj≠0}.

Then

F(k,ek+1,ek)=F(0,ε,ε),ifk=0,0,ifk∈{j∈Z:−mT≤j≤mT−1andj≠0}.

With (15) and (F₃), we have

J(e)=12∑k=−mTmT−1γk−1(Δnek−1)2+12∑k=−mTmT−1χkek2−∑k=−mTmT−1F(k,ek+1,ek)≤4nγ¯2e2+χ¯2e2−2ce2−b=−2c−4nγ¯2−χ¯2ε2−b≤0.(16)

All the assumptions of the Mountain Pass Lemma have been verified. Consequently, J possesses a critical value c_m given by (11) and (12) with E = E_m and Γ = Γ_m, where Γ_m = {g_m ∈ C([0,1], E_m)|g_m(0), g_m(1) = e, e ∈ E_m\B_ε}.

Let u^(m) denote the corresponding critical point of J on E_m. Note that ‖u^(m)‖ ≠ 0 since c_m > 0. □

Lemma 3.3

Assume that (F₁) – (F₅) are satisfied. Then there exist positive constants ϱ and η independent of m such that

ϱ≤u(m)∞≤η.(17)

Proof

The continuity of F(0,v₁, v₂) with respect to the second and third variables implies that there exists a constant τ > 0 such that |F(0, v₁, v₂)| ≤ τ for v12+v22≤ϱ. It is clear that

Ju(m)≤max0≤s≤112∑k=−mTmT−1γk−1(Δn(se)k−1)2+χk(se)k2−∑k=−mTmT−1F(k,(se)k+1,(se)k)≤4nγ¯2+χ¯2e2+τ=4nγ¯2+χ¯2ε2+τ.

Let ξ=4nγ¯2+χ¯2ε2+τ, we have that J (u^(m)) ≤ ξ, which is independent of m. From (8) and (9), we have

Ju(m)=12∑k=−mTmT−1∂F(k−1,uk(m),uk−1(m))∂v2uk(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m)−∑k=−mTmT−1F(k,uk+1(m),uk(m))=12∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v1uk+1(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m)−∑k=−mTmT−1F(k,uk+1(m),uk(m))≤ξ.

By (F₄) and (F₅), there exists a constant η > 0 such that 12∂F(k,v1,v2)∂v1v1+∂F(k,v1,v2)∂v2v2−F(k,v1,v2)>ξ, for all k ∈ Z and v12+v22≥η, which implies that uk(m)≤η for all k ∈ Z, that is u(m)∞≤η.

From the definition of J, we have

0=J′(u(m)),u(m)≥χ_∑k=−mTmT−1un(m)2−∑k=−mTmT−1∂F(k−1,uk(m),uk−1(m))∂v2uk(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m)≥χ_u(m)2−∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v1uk+1(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m).

Therefore, combined with (F₂), we get

χ_u(m)2≤∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v1uk+1(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m)≤∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v1212u(m)+∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v2212u(m).

That is,

χ_u(m)2≤∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v1212+∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v2212.

Thus,

χ_u(m)2≤∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v1212+∑k=−mTmT−1∂F(k,uk+1(m),uk(m))∂v22122.(18)

Combined with (F₂), we get

χ_u(m)2≤∑k=−mTmT−12auk+1(m)212+∑k=−mTmT−12auk(m)2122≤16a2u(m)2.

Thus, we have u^(m) = 0. But this contradicts ‖u^(m)‖ ≠ 0, which shows that

u(m)∞≥ϱ,

and the proof of Lemma 3.3 is finished. □

Proof of Theorem 1.1

In the following, we shall give the existence of a nontrivial homoclinic solution.

Consider the sequence uk(m)k∈Z of 2mT-periodic solutions found in Lemma 3.2. First, by (17), for any m ∈ N, there exists a constant k_m ∈ Z independent of m such that

ukm(m)≥ϱ.(19)

Since γ_k, χ_k and f(k,v₁,v₂,v₃) are all T-periodic in k, uk+jT(m)(∀j∈N) is also 2mT-periodic solution of (1). Hence, making such shifts, we can assume that k_m ∈ Z(0, T − 1) in (19). Moreover, passing to a subsequence of ms, we can even assume that k_m = k_o is independent of m.

Next, we extract a subsequence, still denote by u^(m), such that

uk(m)→uk,m→∞,∀k∈Z.

Inequality (19) implies that uk0≥ξ and, hence, u = {u_k} is a nonzero sequence. Moreover,

Δn(γk−nΔnuk−n)+(−1)nχkuk−(−1)nf(k,uk+1,uk,uk−1)

=limn→∞Δnγk−nΔnuk−n(m)+(−1)nχkuk(m)−(−1)nfk,uk+1(m),uk(m),uk−1(m)=0.

So u = {u_k} is a solution of (1).

Finally, we show that u ∈ l². For u_m ∈ E_m, let

Pm=k∈Z:uk(m)<22ϱ,−mT≤k≤mT−1,Qm=k∈Z:uk(m)≥22ϱ,−mT≤k≤mT−1.

Since F(k,v₁, v₂) ∈ C¹(Z × R², R), there exist constants ξ¯>0,ξ_>0 such that

max∑k=−mTmT−1∂F(k,v1,v2)∂v1212+∑k=−mTmT−1∂F(k,v1,v2)∂v22122:ϱ≤v12+v22≤η,k∈Z≤ξ¯,min12∂F(k,v1,v2)∂v1v1+∂F(k,v1,v2)∂v2v2−F(k,v1,v2):ϱ≤v12+v22≤η,k∈Z≥ξ_.

For k ∈ Q_m,

∂F(k,uk+1(m),uk(m))∂v1212+∂F(k,uk+1(m),uk(m))∂v2212≤ξ¯ξ_12∂F(k,uk+1(m),uk(m))∂v1uk+1(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m))−F(k,uk+1(m),uk(m)).

By (18), we have

χ_2u(m)2≤∑k∈Pm∂F(k,uk+1(m),uk(m))∂v1212+∑k∈Pm∂F(k,uk+1(m),uk(m))∂v22122+∑k∈Qm∂F(k,uk+1(m),uk(m))∂v1212+∑k∈Qm∂F(k,uk+1(m),uk(m))∂v22122≤∑k∈Pm2auk+1(m)212+∑k∈Pm2auk(m)2122+ξ¯ξ_12∑k∈Qm∂F(k,uk+1(m),uk(m))∂v1uk+1(m)+∂F(k,uk+1(m),uk(m))∂v2uk(m)−F(k,uk+1(m),uk(m))≤16a2u(m)2+ξ¯ξξ_.

Thus,

u(m)2≤ξ¯ξξ_(χ_2−16a2).

For any fixed D ∈ Z and m large enough, we have that

∑k=−DDuk(m)2≤u(m)2≤ξ¯ξξ_(χ_2−16a2).

Since ξ¯,ξ_,ξ,χ_ and a are constants independent of m, passing to the limit, we have that

∑k=−DDuk2≤ξ¯ξξ_(χ_2−16a2).

Due to the arbitrariness of D, u ∈ l². Therefore, u satisfies u_k → 0 as |k| → ∞. The existence of a nontrivial homoclinic solution is obtained. □

Proof of Theorem 1.5

Consider the following boundary problem:

Δn(γk−nΔnuk−n)+(−1)nχkuk=(−1)nf(k,uk+1,uk,uk−1),k∈Z(−mT,mT),γ−mT=γmT=0,χ−mT=χmT=0,γ−k=γk,χ−k=χk,k∈Z(−mT,mT).

Let S be the set of sequences u=(⋯,u−k,⋯,u−1,u0,u1,⋯,uk,⋯)={uk}k=−∞+∞ 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, E~m is defined as a subspace of S by

E~m={u∈S|u−k=uk,∀k∈Z}.

Clearly, E~m is isomorphic to R2mT+1.E~m can be equipped with the inner product

u,v=∑j=−mTmTujvj,∀u,v∈E~m,

by which the norm ‖·‖ can be induced by

u=∑j=−mTmTuj212,∀u∈E~m.

It is obvious that E~m is Hilbert space with 2mT + 1-periodicity and linearly homeomorphic to R^{2mT +1}.

Similarly to the proof of Theorem 1.1, we can also prove Theorem 1.5. For simplicity, we omit its proof. □

4 Example

As an application of Theorem 1.1, we give an example to illustrate our main result.

Example 4.1

For all k ∈ Z, assume that

Δn(γk−nΔnuk−n)+(−1)nχkuk=(−1)nλukuk−12+uk2uk−12+uk2+1+uk2+uk+12uk2+uk+12+1,(20)

where λ>4n−1γ¯+χ¯4. We have

f(k,v1,v2,v3)=λv2v12+v22v12+v22+1+v22+v32v22+v32+1

and

F(k,v1,v2)=λ2v12+v22−lnv12+v22+1.

It is easy to verify that all the assumptions of Theorem 1.1 are satisfied. Consequently, (20) has a nontrivial homoclinic solution.

Remark 4.2

It is easy to check that f doesn’t satisfy the classical Ambrosetti-Rabinowitz condition. Thus, our result improves the existing ones.

Acknowledgement

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

This work was supported by PCSIR (no. IRT1226), the Foundation of Guangzhou Education Bureau (no. 2012A019 ) and National Natural Science Foundation of China (no. 11401121).

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Received: 2015-5-7

Accepted: 2016-3-9

Published Online: 2016-7-22

Published in Print: 2016-1-1

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