# Existence of Periodic Solutions for a Class of Asymptoticallyp-Linear Discrete Systems Involving p-Laplacian

Authors: Kai Chen; Qiongfen Zhang
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

By applying Mountain Pass Theorem in critical point theory,
two existence results are obtained for the following asymptoticallyp-linear p-Laplacian discrete system Δ(|Δu(t−1)|p−2Δu(t−1))+∇[−K(t,u(t))+W(t,u(t))]=0. The results obtained generalize some known works.


---

## Body

## 1. Introduction

Consider the periodic solutions of the following ordinaryp-Laplacian discrete system
(1.1)Δ(|Δu(t-1)|p-2Δu(t-1))+∇F(t,u(t))=0,
where Δ is the forward difference operator defined by Δu(t)=u(t+1)-u(t), Δ2u(t)=Δ(Δu(t)), p∈(1,+∞),t∈ℤ,u∈ℝN, F:ℤ×ℝN→ℝ,F(t,x) is continuously differentiable in x for every t∈ℤ and T-periodic in t for all x∈ℝN, and F satisfies (F) F(t,x)=-K(t,x)+W(t,x), and K and W are T-periodic in t for all x∈ℝN with T>1.Difference equations provide a natural description of many discrete models in real world. Discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control; so it is very important to study the solutions of difference equations. For more details about difference equations, we refer the readers to the books [1–3].In some recent papers [4–17], the authors investigated the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers imply that the critical point theory is a useful method to the study of periodic solutions for difference equations. Motivated by the above papers and the paper [18], we will generalize the results of [18] to p-Laplacian systems (1.1). Here are our main results.Theorem 1.1.
Assume thatF satisfies (F) and K and W satisfy the following conditions:
(A1) there exist constants a1>0 and γ∈(p-1,p] such that
(1.2)K(t,0)=0,K(t,x)≥a1|x|γ,∀(t,x)∈Z[1,T]×RN;
(A2)  (∇K(t,x),x)≤pK(t,x) for all (t,x)∈ℤ[1,T]×ℝN, where ℤ[a,b]:=ℤ∩[a,b] for every a,b∈ℤ with a≤b;
(A3)  W(t,0)=0, limsup|x|→0(W(t,x)/|x|p)<a1 uniformly for t∈ℤ[1,T];
(A4)  there exists a function g∈l1(ℤ[1,T],ℝ) such that
(1.3)(∇W(t,x),x)-pW(t,x)≥g(t)for(t,x)∈Z[1,T]×RN,(1.4)lim⁡|x|→∞[(∇W(t,x),x)-pW(t,x)]=+∞∀t∈Z[1,T];
(A5)  there exist constants a2>0 and d>0 such that
(1.5)W(t,x)≤a2|x|p+d,∀(t,x)∈Z[1,T]×RN;
(A6)  there exists x0∈ℝN such that
(1.6)∑t=1T[K(t,x0)-W(t,x0)-g(t)p]<0.
Then problem (1.1) possesses one nontrivial periodic solution.Corollary 1.2.
Assuming thatF satisfies (F) and that K and W satisfy (A1)–(A4) and
(A7) there exists a function V∞∈l∞(ℤ[1,T],ℝ) such that
(1.7)lim⁡|x|→∞|∇W(t,x)-V∞(t)|x|p-2x||x|p-1=0uniformlyfort∈Z[1,T],(1.8)∑t=1T[max⁡|x|=1K(t,x)-V∞(t)p]<0.
Then problem (1.1) possesses one nontrivial periodic solution.Remark 1.3.
As far as we know, similar results of discrete system (1.1) which satisfies (F) and is asymptotically p-linear at infinity cannot be found in the literature. From this point, our results are new.


## 2. Preliminaries

LetET be the Sobolev space defined by(2.1)ET={u:Z⟶RN∣u(t+T)=u(t),t∈Z},
with the norm(2.2)‖u‖=(∑t=1T(|Δu(t)|p+|u(t)|p))1/p,u∈ET,
where |·| denote the usual norm in ℝN. It is easy to see that (ET,∥·∥) is a finite dimensional Banach space and linear homeomorphic to ℝNT. As usual, let
(2.3)‖u‖∞=sup⁡{|u(t)|:t∈Z[1,T]},∀u∈l∞(Z[1,T],RN).
Since ET is finite dimensional Banach space, there exists a positive constant C0 such that(2.4)‖u‖∞≤C0‖u‖.For anyu∈ET, let
(2.5)φ(u)=1p∑t=1T|Δu(t)|p+∑t=1T[K(t,u(t))-W(t,u(t))].
We can compute the Fréchet derivative of (2.5) as
(2.6)∂φ(u)∂u=Δ(|Δu(t-1)|p-2Δu(t-1))+∇[K(t,u(t))-W(t,u(t))],t∈Z[1,T].
Hence, u is a critical point of φ on ET if and only if(2.7)Δ(|Δu(t-1)|p-2Δu(t-1))+∇[K(t,u(t))-W(t,u(t))]=0,t∈Z[1,T],u∈RN.
So, the critical points of φ are classical solutions of (1.1). We will use the following lemma to prove our main results.Lemma 2.1 (see [19]).
LetE be a real Banach space and  φ∈C1(E,ℝ) satisfying the (PS) condition. Suppose  φ(0)=0 and (a)
there exist constantsρ,α>0 such that  φ|∂Bρ(0)≥α;(b)
there exists ane∈E∖B̅ρ(0) such that  φ(e)≤0.Then  φ possesses a critical value  c≥α which can be characterized as  c=infh∈Γmaxs∈[0,1]φ(h(s)), where  Γ={h∈C([0,1],E)∣h(0)=0,h(1)=e} and  Bρ(0) is an open ball in  E of radius  ρ centered at  0.
It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma2.1 holds true under condition (C).


## 3. Proofs of Main Results

Proof of Theorem1.1.
The proof is divided into three steps.
Step 1.
The functionalφ satisfies condition (C). Let {un}⊂ET satisfying (1+∥un∥)∥φ′(un)∥→0 as n→∞ and φ(un) is bounded. Hence, there exists a positive constant C1 such that
(3.1)|φ(un)|≤C1,(1+‖un‖)‖φ′(un)‖≤C1.
We prove {un} is bounded by contradiction. If {un} is unbounded, without loss of generality, we can assume that ∥un∥→∞ as n→∞. Let zn=un/∥un∥, then we have ∥zn∥=1. Going to a subsequence if necessary, we may assume that zn⇀z weakly in ET and so zn→z strongly in l1(ℤ[1,T],ℝ). It follows from (3.1) and (A2) that
(3.2)(p+1)C1≥pφ(un)-〈φ′(un),un〉=∑t=1T[(∇W(t,un),un)-pW(t,un)]+∑t=1T[pK(t,un)-(∇K(t,un),un)]≥∑t=1T[(∇W(t,un),un)-pW(t,un)].
From (A1) and (A5), we obtain
(3.3)φ(un)=1p∑t=1T|Δun(t)|p+∑t=1T[K(t,un(t))-W(t,un(t))]≥1p∑t=1T|Δun(t)|p-a2∑t=1T|un(t)|p-dT=1p‖un‖-(1p+a2)∑t=1T|un(t)|p-dT.
Hence, we have
(3.4)φ(un)‖un‖p≥1p-(1p+a2)∑t=1T|zn(t)|p-dT‖un‖p.
Passing to the limit in the above inequality, by using the fact that φ(un) is bounded and {zn(t)} converges uniformly to z(t) on ℤ[1,T], we obtain
(3.5)(1p+a2)∑t=1T|z(t)|p≥1p,
which implies that z≠0. Let Ω⊂ℤ[1,T] be the set on which z≠0, then the measure of Ω is positive. Moreover, |un(t)|→∞ as n→∞ for t∈Ω. Thus, from (A4), we get
(3.6)∑t=1T[(∇W(t,un),un)-pW(t,un)]=∑Ω[(∇W(t,un),un)-pW(t,un)]+∑Z[1,T]∖Ω[(∇W(t,un),un)-pW(t,un)]≥∑Ω[(∇W(t,un),un)-pW(t,un)]+∑Z[1,T]∖Ωg(t).
It follows from Fatou’s lemma and (A4) that
(3.7)lim⁡n→∞∑t=1T[(∇W(t,un),un)-pW(t,un)]=+∞,
which contradicts with (3.2). Therefore, {un} is bounded in ET. Hence, there exists a subsequence, still denoted by {un}, such that
(3.8)un⇀u0weaklyinET.
Since ET is finite dimensional space, we have un→u0 in ET. Therefore, the functional φ satisfies condition (C).Step 2.
From (A3) and (A5), there exist constants0<ε<1/p,q>p and C2>1/TC0p such that
(3.9)W(t,u)≤(a1-ε)|u|p+C2|u|qforu∈RN,t∈Z[1,T].
Let
(3.10)δ=(pεqTC2C0p)1/(q-p).
Then 0<δ<1. If u∈ET and ∥u∥=δ/C0:=ρ, then it follows from (2.4) that |u(t)|≤δ for t∈ℤ[1,T]. Set
(3.11)α=(q-p)εqρp.
Then from (A1), (2.4), (3.9), (3.10), and (3.11), we have(3.12)φ(u)=1p∑t=1T|Δu(t)|p+∑t=1TK(t,u(t))-∑t=1TW(t,u(t))≥1p∑t=1T|Δu(t)|p+a1∑t=1T|u(t)|γ-(a1-ε)∑t=1T|u(t)|p-C2∑t=1T|u(t)|q≥1p∑t=1T|Δu(t)|p+ε∑t=1T|u(t)|p-C2∑t=1T|u(t)|q≥ε(∑t=1T|Δu(t)|p+∑t=1T|u(t)|p)-C2T‖u‖∞q≥ε‖u‖p-C2TC0q‖u‖q=(q-p)εqρp=α∀u∈ETwith‖u‖=ρ.Step 3.
Setf(s)=s-pW(t,sx0) for s>0. Then it follows from (A4) that
(3.13)f′(s)=s-p-1[-pW(t,sx0)+(∇W(t,sx0),sx0)]≥s-p-1g(t)∀t∈Z[1,T],s>0.
Integrating the above inequality from 1 to ξ>1, we have
(3.14)W(t,ξx0)≥ξpW(t,x0)+g(t)p(ξp-1)∀t∈Z[1,T],ξ>1.
From (A2), it is easy to see that
(3.15)K(t,ξx0)≤ξpK(t,x0)∀t∈Z[1,T],ξ>1.
From (3.14), (3.15), and (A6), we have
(3.16)φ(ξx0)=∑t=1T[K(t,ξx0)-W(t,ξx0)]≤ξp∑t=1T[K(t,x0)-W(t,x0)-g(t)p]+1p∑t=1Tg(t)≤0forlargeenoughξ>1.
Choose ξ1>1 such that T1/p|ξ1x0|>ρ and φ(ξ1x0)≤0. Let e=ξ1x0, then ∥e∥=T1/p|ξ1x0|>ρ and φ(e)≤0. It is easy to see that φ(0)=0. Hence, by Lemma 2.1, there exists u∈ET such that
(3.17)φ(u)=c,φ′(u)=0.
Then the function u is a desired nontrivial T-periodic solution of (1.1). The proof is complete.Proof of Corollary1.2.
LetC3=-(1/3T)∑t=1T[max|x|=1K(t,x)-(V∞(t)/p)]. Then it follows from (A7) that C3>0 and there exists a positive constant C4>0 such that
(3.18)|∇W(t,x)-V∞(t)|x|p-2x|≤C3|x|p-1fort∈Z[1,T],|x|≥C4.
For any x∈ℝN∖{0}, let x*=C4x/|x|. Then it follows from (3.18) that for all t∈ℤ[1,T] and x∈ℝN with |x|>C4(3.19)W(t,x)-V∞(t)p|x|p=W(t,x*)-1pV∞(t)|x*|p+∫01(∇W(t,x*+s(x-x*))-V∞(t)(x*+s(x-x*)),x-x*)ds≤W(t,x*)-C4ppV∞(t)+C3∫01|x*+s(x-x*)||x-x*|ds≤W(t,x*)-C4ppV∞(t)+2C3|x|p,
which implies that
(3.20)W(t,x)≤(V∞(t)p+2C3)|x|p+W(t,x*)-C4ppV∞(t)fort∈Z[1,T],|x|>C4,
which together with (A3) shows that (A5) holds. Similarly, we have
(3.21)W(t,x)≥(V∞(t)p-2C3)|x|p+W(t,x*)-C4ppV∞(t)fort∈Z[1,T],|x|>C4.
From (A2), it is easy to show that
(3.22)K(t,x)≤K(t,x|x|)|x|pfort∈Z[1,T],|x|>1.
Choose x0∈ℝN such that |x0|>C4+1 and
(3.23)C3T|x0|p+∑t=1T[min⁡|x|=C4W(t,x)+g(t)p-C4ppV∞(t)]>0.
It follows from (3.21), (3.22), and (3.23) that
(3.24)∑t=1T[K(t,x0)-W(t,x0)-g(t)p]≤|x0|p∑t=1T[K(t,x0|x0|)-V∞(t)p+2C3]-∑t=1T[W(t,x*)+g(t)p-C4ppV∞(t)]≤|x0|p∑t=1T[max⁡|x|=1K(t,x)-V∞(t)p+2C3]-∑t=1T[min⁡|x|=C4W(t,x)+g(t)p-C4ppV∞(t)]≤-C3T|x0|p-∑t=1T[min⁡|x|=C4W(t,x)+g(t)p-C4ppV∞(t)]<0.
This implies that (A6) holds. By Theorem 1.1, the conclusion of Corollary 1.2 holds true. The proof is complete.


## 4. An Example

In this section, we give an example to illustrate our results.Example 4.1.
In problem (1.1), let p=4/3 and
(4.1)W(t,x)=a(t)|x|4/3(1-1ln⁡(e+|x|)),K(t,x)=b|x|θ+c(t)|x|σ,
where b>0,a,c∈l1(ℤ,[0,+∞)),1<θ<σ≤4/3,a(t+T)=a(t),c(t+T)=c(t). It is easy to check that (F), (A1)–(A3), and (A5) hold. On the one hand, we have
(4.2)(∇W(t,x),x)-43W(t,x)=(4/3)a(t)|x|7/3(e+|x|)(ln⁡(e+|x|))2.
Then, it is easy to check that condition (A4) holds. On the other hand, we have
(4.3)∑t=1T[K(t,x)-W(t,x)-g(t)p]=∑t=1T[b|x|θ+c(t)|x|σ-a(t)|x|4/3(1-1ln⁡(e+|x|))-g(t)p]=bT|x|θ+|x|σ∑t=1Tc(t)-‖g‖l1p-|x|4/3(1-1ln⁡(e+|x|))∑t=1Ta(t),
which implies that there exists x0∈ℝN such that (A6) holds if
(4.4)∑t=1Ta(t)>∑t=1Tc(t).
Hence, from Theorem 1.1, problem (1.1) with W and K as in (4.1) has one nontrivial T-periodic solution if (4.4) holds.



---

*Source: 948345-2012-05-07.xml*