R E S E A R C H
Open Access
A note on existence of infinitely many
periodic solutions for second-order nonlinear
difference systems
Qin Jiang and Sheng Ma
**Correspondence: [email protected] Department of Mathematics, Huanggang Normal University, Hubei, 438000, China
Abstract
The paper deals with the existence of infinitely many periodic solutions for second-order nonlinear difference systems. A variational approach is applied using the saddle point theorem.
Keywords: second-order difference systems; infinitely many periodic solutions; variational methods; saddle point theorem
1 Introduction and statement of main results The interest in the second-order difference systems
⎧ ⎨ ⎩
u(t– ) +∇F(t,u(t)) = , t∈Z[,T],
u() =u(T), ()
has been aroused, whereu(t) =u(t+ ) –u(t),u(t) =(u(t)),∇F(t,x) = ∂F∂(tx,x),T is a positive integer, andZandRdenote the set of integers and the set of real numbers, respectively, andZ[a,b] ={a,a+ , . . . ,b}, fora,b∈Zanda≤b. Assume thatF(t,x) is T-periodic intfor allx∈RN andF(t,x)∈C(Z×RN,R).
In , Yu and Guo [–] established a variational structure and variational methods to study discrete Hamiltonian systems and obtain the solvability condition of a periodic solution for discrete systems, based on operator theory. Since then more and more authors have contributed to study second-order discrete Hamiltonian systems, with an effective tool named the critical point theory, and one obtained many interesting results [–]. In [], with operator theory, Xue and Tang constructed a variational setting unlike the one in [] to study the second-order superquadratic discrete Hamiltonian systems () and obtained the existence of periodic solutions. This result generalized the one in []. In [], Xue and Tang obtained the existence of one periodic solution of systems () under the hypothesis there existM> ,M> and ≤α< such that
∇F(t,x)≤M|x|α+M,
for all (t,x)∈Z[,T]×RN, and the condition
|x|–α
T
t=
F(t,x)→+∞ as|x| → ∞.
Subsequently, in [], Yan, Wu and Tang extended these results in [] to the case that F(t,x) isTi-periodic inxiand obtained the existence of multiple periodic solutions, where
xi is theith component ofx= (x,x, . . . ,xN),i∈[,N]. Especially in [], Che and Xue
obtained the existence of infinitely many periodic solutions for systems in the case that: (F) there existf,g:Z[,T]→R+andα∈[, ) such that
∇F(t,x)≤f(t)|x|α
+g(t) for all (t,x)∈Z[,T]×RN,
and a suitable oscillating behaviour at infinity, (F)lim infr→∞supx∈RN,|x|=r|x|–αTt=F(t,x) = –∞, (H)lim supr→∞infx∈RN,|x|=rTt=F(t,x) = +∞.
Consequently, it is natural to ask if infinitely many solutions still exist whenα= . With the fact thatα= , (F) and (F), respectively, change to the linearly bounded gradient condition:
(F) there existf,g:Z[,T]→R+such that
∇F(t,x)≤f(t)|x|+g(t) for all (t,x)∈Z[,T]×RN,
and the condition
(F)lim infr→∞supx∈RN,|x|=r|x|– T
t=F(t,x) = –∞.
However, similarly to what was pointed in [], (F) does not hold if Tt=f(t) is bounded. Therefore, it is necessary to improve condition (F). Inspired by [, , ], in this paper, we will use minmax methods to further study systems () under the following assumptions:
(H) there existf,g:Z[,T]→R+withT t=f(t) <
λ
such that
∇F(t,x)≤f(t)|x|+g(t) for all(t,x)∈Z[,T]×RN,
(H) lim infr→+∞supx∈RN,|x|=r|x|– T
t=F(t,x) < –
Tt=f(t) λ , whereλk= – coskTπ satisfy the eigenvalue problem
–u(t– ) =λku(t), k∈Z
,
T
,
and we note
=λ<λ<· · ·<λ[T]≤.
Theorem . Under the hypotheses of (H), (H),and(H):
(a) discrete systems()have a sequence{un}of solutions such that{un}is a critical point
of the functionalϕandϕ(un)→+∞asn→ ∞;
(b) discrete systems()have a sequence{u∗n}of solutions such that{u∗n}is a local
minimizer ofϕandϕ(u∗n)→–∞asn→ ∞,
where the variational functionalϕis
ϕ(u) =
T
t=
u(t)–
T
t=
Ft,u(t),
on Hilbert space HTdefined by
HT=
u:Z→RN|u(t) =u(t+T),t∈Z,
with the inner product and the norm,respectively,written as
u,v=
T
t=
u(t),v(t), ∀u,v∈HT
and
u= T
t=
u(t)
.
Remark . As pointed out in[],the nonlinearity potential F does not need the symme-try condition in the paper.Moreover,Theorem.is a complement to and development of Theorem.in[]corresponding toα= .
2 Proof of main result
For allu∈HT,u∞=supt∈Z[,T]|u(t)|is defined. It is obvious that
u∞≤ u= T
t=
u(t)
. ()
By the definition ofHT,HTis finite dimensional. By (H), one getsϕ∈C(HT,R) and
ϕ(u),v=
T
t=
u(t),v(t)–
T
t=
∇Ft,u(t),v(t),
for allu,v∈HT.
Subsequently, an important lemma in [] is shown for the reader’s convenience. The lemma is constructed in a variational setting, with the operator theory, unlike the one in []. Details can be found in [].
Lemma ([]) Let Nkbe a subspace of HTwritten as
Nk:=
u∈HT|–u(t– ) =λku(t)
,
(i) Nk⊥Nj,k=j,k,j∈Z[, [T]]. (ii) HT=
[T/] k= Nk.
Let V =Nand W=N⊥=
[T/]
k= Nk.Via Lemmain[],we have
HT=N⊕N⊥=V⊕W
and
T
t=
u(t)≥λu, ∀u∈W=N⊥. ()
From the definition of N,one gets u(t) =u() =C∈RN,for all u∈Nand t∈Z[,T].By
Lemma,one rewrites u as u=u¯+u˜∈V⊕W=N⊕N⊥,where
¯
u= T
T
t=
u(t).
From the fact that
u= T
t=
u(t)
=
T
t=
u¯+u(t)˜
= T
t=
¯u+u,˜ u¯+u˜
= T
t=
|¯u|+u(t)˜
=T|¯u|+˜u,
one has
u ≤√T+ |¯u|+˜u and u ≥|¯u|+˜u.
Thus one obtains thatu → ∞if and only if(|¯u|+˜u) → ∞.
Proof of Theorem . The proof of Theorem . relies on a minimax theorem (Corol-lary .) in []. We complete the proof with a series of statements below.
Step , we claim thatϕis coercive in the subspaceW.
Due to (H), there exists a constantCsatisfying the following inequality:
F(t,x)≤
∇F(t,sx),xds+F(t, )
≤
∇F(t,sx)|x|ds+C≤
f(t) |x|
+g(t)|x|+C
for allt∈Z[,T] andx∈RN. Hence, using the Hölder inequality, (), (), and (), for all
The detailed proof of (c) can be founded in []. On the other hand, by (H), one can take a constant
a>
λ
.
Thus one gets
≤ |¯u|
clusion (d) is achieved.
Now we have a family of mapsnexpressed as
n=γ∈C(Ban,HT)|γ|∂Ban=Id|∂Ban
and minimax valuescnformulated as
Step , we claim that, for sufficiently largen, there exist sequences{γk} ⊂nand{νk}in
HT, respectively, satisfying
max
u∈Ban
ϕγk(u)
→cn,
ϕ(νk)→cn, ϕ(νk)→, dist
νk,γk(Ban)
→ ask→ ∞. ()
By Step , we knowϕ(u)→+∞asu →+∞,u∈W. Therefore there exists a constant Csatisfying
max
u∈Banϕ
γ(u)≥ inf
u∈Wϕ(u)≥C.
Furthermore, one has
cn≥ inf
u∈Wϕ(u)≥C,
for sufficiently largen. By the factγ(Ban)∩W=∅and the conclusion of Step , one obtains cn> max
u∈∂Banϕ(u)
for sufficiently largen. Therefore, for a fixedn, this claim is proved from Theorem . and Corollary . in [].
Step , we draw the conclusion that the sequence{νk}is bounded inHT.
For sufficiently largek, by (), one has
cn≤max u∈Banϕ
γk(u)
≤cn+ .
We choosewk∈γk(Ban) satisfying
νk–wk ≤. ()
From the conclusion (d) of Step , for a fixedn, a sufficiently largemexists, rendering the formula
bm>an and inf
u∈Hbmϕ(u) >cn+ .
These inequalities imply thatγk(Ban)∩Hbm=∅for eachk. We now writewk=w¯k+w˜k, wherew¯k∈V andw˜k∈W. Then one has
¯wk<bm ()
for eachk. Moreover, by (), (), (), and (), one gets
+cn≥ϕ(wk) =
T
t=
wk(t)
–
T
t=
Ft,wk(t)
≥
λ ˜wk
– T
t=
f(t)wk(t)
+g(t)wk(t)+C
≥ bounded inHTvia (). The conclusion is proved.
Step , we claim thatcnis a critical value ofϕ.
Since{νk} is bounded andHT is finite dimensional space,{νk}contains a convergent
subsequence that is still denoted as{νk}for convenience, meeting
lim
k→∞νk=un.
Then, by (), one has
ϕ(un) =cn and ϕ(un) = .
Thusϕhas a critical pointun.
We prove part (a) of Theorem .. One chooses sufficiently largensatisfyingan>bm,
then one hasγ(Ban)∩Hbm=∅for anyγ ∈n. It follows that
max
u∈Ban
ϕγ(u)≥ inf
u∈Hbmϕ(u).
With this and the conclusion (d) of Step ,
lim
n→∞cn= +∞
is implied. Part (a) of Theorem . is proved.
For allu∈Pm, similar to (), one obtains
ϕ(u) =
T
t=
u(t)–
T
t=
Ft,u(t)
≥
λ˜u
– T
t=
f(t)u(t)
+g(t)u(t)+C
≥
λ
–
T
t=
f(t)
˜u–˜u
T
t=
g(t) –bm
T
t=
f(t) –bm T
t=
g(t) –CT. ()
Due to (),ϕis bounded below onPm. Take
μm= inf u∈Pm
ϕ(u)
and a sequence{uk} ⊂Pm, satisfying ϕ(uk)→μm ask→ ∞.
Similar to the proof of the boundedness of{wk}in Step ,{uk}is bounded inHTvia ().
Then{uk}contains a convergent subsequence that is still denoted{uk}for convenience,
satisfying
uku∗m weakly inHT, ask→ ∞.
Noting thatPm is convex and closed inHT, one hasu∗m∈Pm. Moreover, in view of the
weakly lower semi-continuity ofϕ, one has
μm= lim
k→∞ϕ(uk)≥ϕ
u∗m
and
μm=ϕ
u∗m.
Next, we draw the conclusion thatu∗mis an interior point ofPm. Thusu∗mis a critical
point ofϕ. Taking
u∗m=u¯∗m+u˜∗m,
whereu¯∗m∈V,u˜∗m∈W. Ifan<bm, one has∂Ban⊂Pm, which implies that ϕu∗m= inf
u∈Pm
ϕ(u)≤ sup
u∈∂Ban
ϕ(u).
From the inequality above and the result (d) of Step , one gets
By the conclusion of Step , one hasu¯∗m=bmfor largem. From this one deduces thatu∗m
is an interior point ofPmandu∗mis a critical point ofϕ. Then, the proof of Theorem . is
completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thanks the editors and the referee for their many valuable comments, which helped improving the article. This research was partly supported by the Science Foundation of Huanggang Normal University, China (No. 201617503).
Received: 10 July 2016 Accepted: 10 November 2016 References
1. Guo, ZM, Yu, JS: The existence of periodic and subharmonic solutions of subquadratic second order difference equations. J. London Math. Soc. (2)68(2), 419-430 (2003)
2. Guo, ZM, Yu, JS: Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. Nonlinear Anal.55(7-8), 969-983 (2003)
3. Guo, ZM, Yu, JS: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China Ser. A46(4), 506-515 (2003)
4. Bin, HH, Yu, JS, Guo, ZM: Nontrivial periodic solutions for asymptotically linear resonant difference problem. J. Math. Anal. Appl.322(1), 477-488 (2006)
5. Guo, ZM, Yu, JS: Applications of critical theory to difference equations. In: Differences and Differential Equations. Fields Inst. Commun., vol. 42, pp. 187-200. Amer. Math. Soc., Providence, RI (2004)
6. Ma, MJ, Yu, JS: Existence of multiple positive periodic solutions for nonlinear functional difference equations. J. Math. Anal. Appl.305(2), 483-490 (2005)
7. Xue, YF, Tang, CL: Existence of a periodic solution for subquadratic second-order discrete Hamiltonian systems. Nonlinear Anal.67(7), 2072-2080 (2007)
8. Xue, YF, Tang, CL: Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems. Appl. Math. Comput.196(2), 494-500 (2008)
9. Yan, SH, Wu, XP, Tang, CL: Multiple periodic solutions of second order discrete Hamiltonian systems. Appl. Math. Comput.234, 142-149 (2014)
10. Yu, JS, Guo, ZM, Zou, XF: Periodic solutions of second order self-adjoint difference equations. J. London Math. Soc. (2)
71(1), 146-160 (2005)
11. Zhou, Z, Yu, JS, Guo, ZM: Periodic solutions of higher-dimensional discrete systems. Proc. R. Soc. Edinb. A134(5), 1013-1022 (2004)
12. Che, CF, Xue, XP: Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential. Adv. Differ. Equ.2012, 50 (2012)
13. Tang, XH, Meng, Q: Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal.11, 3722-3733 (2010)
