R E S E A R C H
Open Access
Asymptotically almost periodic solution to a
class of Volterra difference equations
Wei Long
*and Wen-Hai Pan
*Correspondence:
[email protected] College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China
Abstract
This paper is concerned with an asymptotically almost periodic solution to a class of Volterra-type difference equations. We establish a compactness criterion for the sets of asymptotically almost periodic sequences. Then, by using the compactness criterion and Schauder’s fixed point theorem, we present an existence theorem for an asymptotically almost periodic solution to the addressed Volterra-type difference equation. Our existence theorem extends and complements a recent result due to (Dinget al.in Electron. J. Qual. Theory Differ. Equ. 6:1-13, 2012).
MSC: 39A24; 34K14
Keywords: asymptotically almost periodic; Volterra difference equation
1 Introduction and preliminaries
In this paper, we consider the following nonlinear Volterra-type difference equation:
x(n) = λ
i=
fi
n,x(n)· m∈Z
ai(n,m)gi
m,x(m), n∈Z, (.)
whereλis a fixed positive integer andfi:Z×R→R,ai:Z×Z→R,gi:Z×R→R (i= , , . . . ,λ) satisfy some conditions recalled in Section .
For the background of discrete Volterra equations, we refer the reader to the well-known monograph [] by Agarwal. The first motivation for this paper is some recent work on asymptotical periodicity for Volterra-type difference equations in [–] by Diblíket al. In fact, asymptotical behavior for Volterra-type difference equations, including periodicity, asymptotical periodicity,etc., has been of great interest for many mathematicians. How-ever, to the best of our knowledge, there is seldom literature available about asymptotically almost periodicity for Equation (.). Thus, in this paper, we will investigate this problem. In addition, it is needed to note that compared with asymptotically periodic sequences, in general, it is more difficult to obtain the compactness for a set of asymptotically almost periodic sequences.
On the other hand, in a recent work [], by using the classical Schauder fixed point the-orem, Dinget al.established an interesting existence theorem for the following functional integral equation:
y(t) =et,yα(t)+gt,yβ(t)h(t) +
Rk(t,s)f
s,yγ(s)ds
, t∈R. (.)
In fact, the existence of almost periodic type solutions has been an interesting and impor-tant topic in the study of qualitative theory of difference equations. We refer the reader to [–] and references therein for some recent developments on this topic. Equation (.) can be seen as a discrete analogue (but more general) of Equation (.). That is another main motivation for this work.
Throughout the rest of this paper, we denote byZ(Z+) the set of (nonnegative) integers, byNthe set of positive integers, byR(R+) the set of (nonnegative) real numbers, bya subset ofR, and byXa Banach space.
First, let us recall some notations and basic results of almost periodic type sequences (for more details, see [, , ]).
Definition .[] A functionf :Z→Xis called almost periodic if∀ε,∃N(ε)∈Nsuch that among anyN(ε) consecutive integers there exists an integerpwith the property that
f(k+p) –f(k)<ε, ∀k∈Z.
Denote byAP(Z,X) the set of all such functions. Moreover, we denoteAP(Z,R) byAP(Z) for convenience.
Lemma . [, Theorem .] A necessary and sufficient condition for the sequence f :Z→Rto be almost periodic is that for any integer sequence{nk},one can extract a subsequence{nk}such that{f(n+nk)}converges uniformly with respect to n∈Z.
Remark . Letf,g∈AP(Z). By Lemma ., it is not difficult to show that∀ε,∃N(ε)∈N such that among anyN(ε) consecutive integers there exists acommonintegerpwith the property that
f(k+p) –f(k) <ε and g(k+p) –g(k) <ε
for allk∈Z.
Next, we denote by C(Z,X) the space of all the functions f : Z→ X such that
lim|n|→∞f(n)= .
Definition . A functionf :Z→Xis called asymptotically almost periodic if it admits a decompositionf=g+h, whereg∈AP(Z,X) andh∈C(Z,X). Denote byAAP(Z,X) the set of all such functions. Moreover, we denoteAAP(Z,R) byAAP(Z) for convenience.
Definition . Let⊂Randf be a function fromZ×toRsuch thatf(n,·) is contin-uous for eachn∈Z. Then f is called almost periodic in n∈Zuniformly forω∈if for everyε> and every compact ⊂, there corresponds an integerNε( ) > such that amongNε( ) consecutive integers there exists an integerpwith the property that
f(k+p,ω) –f(k,ω) <ε
Similarly, for each subset⊂R, we denote byC(Z×) the space of all the functionsf :
Z×→Rsuch thatf(n,·) is continuous for eachn∈Z, andlim|n|→∞f(n,x) = uniformly forxin any compact subset of.
Definition . A functionf :Z×→Ris called asymptotically almost periodic inn uniformly for x∈if it admits a decompositionf =g+h, whereg ∈AP(Z×) and h∈C(Z×). Denote byAAP(Z×) the set of all such functions.
Lemma . Let E∈ {AP(Z,X),AAP(Z,X)}.Then the following hold true: (a) f ∈Eimplies thatf is bounded.
(b) f,g∈Eimplies thatf +g∈E.Moreover,f ·g∈EifX=R. (c) Eis a Banach space equipped with the supremum norm.
Proof The proof is similar to that of the continuous case (cf.[, ]). So, we omit the
details.
2 A compactness criterion
The following theorem is a well-known result for the continuous case (see,e.g., [, p., Theorem .]). Here, we give a discrete version.
Theorem . Let f be a function from Z to R. Then f ∈ AAP(Z) if and only if ∀ε,
∃M(ε),N(ε)∈Nsuch that among any N(ε)consecutive integers there exists an integer p with the property that
f(k+p) –f(k) <ε
for all k∈Zwith|k| ≥M(ε)and|k+p| ≥M(ε).
Proof We first show the ‘only if ’ part. Letf ∈AAP(Z). Then there existg∈AP(Z) and h∈C(Z,R) such thatf =g+h. Byg∈AP(Z), for eachε> ,∃N(ε)∈Nsuch that among anyN(ε) consecutive integers there exists an integerpwith the property that
g(k+p) –g(k) <ε
, ∀k∈Z.
In addition, sinceh∈C(Z,R), for the aboveε> , there existsM(ε)∈Nsuch that|h(k)|< ε
for allk∈Zwith|k| ≥M(ε). Thus, we have
f(k+p) –f(k) ≤ g(k+p) –g(k) + h(k+p) + h(k) <ε
for allk∈Zwith|k| ≥M(ε) and|k+p| ≥M(ε).
Next, let us prove the ‘if ’ part. First, let us show thatf is bounded. Lettingε= , there existsM(),N()∈Nsuch that among anyN() consecutive integers there exists an integer pwith the property that
for allk∈Zwith|k| ≥M() and|k+p| ≥M(). Then, for eachk∈Zwith|k| ≥M(), there existspk∈[M() –k,M() +N() –k]∩Zsuch that
f(k+pk) –f(k) < .
Noting thatk+pk∈[M(),M() +N()], we get
f(k) ≤ f(k+pk) + ≤ max
k∈[M(),M()+N()]
f(k) +
for allk∈Zwith|k| ≥M(). Thus,
sup
k∈Z
f(k) ≤ max
k∈[–M(),M()+N()]
f(k) + < +∞.
Now, let us show thatf∈AAP(Z). We divide the remaining proof into three steps. Step . Sincef is bounded, we can choose a sequence{sn} ⊂Nsuch thatlimn→+∞sn= +∞andlimn→+∞f(k+sn) exists for eachk∈Z. Let
g(k) = lim
n→+∞f(k+sn), k∈Z.
For each ε> , among anyN(ε) consecutive integers there exists an integerpwith the property that
f(k+p) –f(k) <ε
for allk∈Zwith|k| ≥M(ε) and|k+p| ≥M(ε). Then, for each fixedk∈Z, we have
f(k+sn+p) –f(k+sn) <ε
for sufficiently largen, which yields that
g(k+p) –g(k) ≤ε.
Thus,g∈AP(Z).
Step . Now fixε> . Then, for eachn∈N, there existstn∈[sn–N(ε),sn]∩Zsuch that
f(k+tn) –f(k) <ε (.)
for allk∈Zwith|k| ≥M(ε) and|k+tn| ≥M(ε). Letrn=sn–tn. Thenrn∈ {, , , . . . ,N(ε)}, which means that there exist a subsequence{rn} ⊂ {rn}andr(ε)∈ {, , , . . . ,N(ε)}such that
rn≡r(ε).
Thus, for allk∈Zwith|k| ≥M(ε), we have
Combining this with (.),limn→+∞tn = +∞, and g(k) = lim
n→+∞f(k+sn), ∀k∈Z, we conclude
f(k) –gk–r(ε) ≤ε
for allk∈Zwith|k| ≥M(ε).
Step . By Step , we know that for eachε> , there existsr(ε)∈ {, , , . . . ,N(ε)}such that
f(k) –gk–r(ε) ≤ε
for allk∈Zwith|k| ≥M(ε). Takingε= , /, . . . , we get a sequence{r(/m)}. On the other hand, it follows from Step thatg∈AP(Z). Thus, going to a subsequence, if necessary, we may assume thatg(·–r(/m)) is uniformly convergent onZ. Let
g(k) = lim
m→+∞g
k–r(/m), k∈Z. Theng∈AP(Z). In addition, noting that
f(k) –g(k) ≤ f(k) –gk–r(/m) + gk–r(/m)–g(k)
≤
m+ g
k–r(/m)–g(k)
for allk∈Zwith|k| ≥M(/m), we know thatf–g∈C(Z). This completes the proof.
Definition . F⊆AAP(Z) is said to be equi-asymptotically almost periodic if for each ε> , there existM(ε),N(ε)∈Nsuch that among anyN(ε) consecutive integers there exists an integerpwith the property that
sup
f∈F
f(k+p) –f(k) <ε
for allk∈Zwith|k| ≥M(ε) and|k+p| ≥M(ε).
Theorem . Let F⊆AAP(Z).Then F is precompact in AAP(Z)if and only if the following two conditions hold:
(i) for eachk∈Z,{f(k) :f ∈F}is bounded; (ii) Fis equi-asymptotically almost periodic.
Proof ‘only if ’ part
LetF⊂AAP(Z) be precompact. ThenFis bounded inAAP(Z). So, (i) obviously holds. In addition,∀ε> , there existsN∈Nandf,f, . . . ,fN∈Fsuch that
F⊂ N
i=
By Remark ., we can get that{f,f, . . . ,fN}is equi-asymptotically almost periodic. Comb-ing this with (.), we can show thatFis equi-asymptotically almost periodic,i.e., (ii) holds.
‘if part’
Let{fn} ⊂F. Since{fn(k)}is bounded for eachk∈Z, we can assume that (if necessary going to a subsequence){fn(k)}is convergent for eachk∈Z. On the other hand, sinceFis equi-asymptotically almost periodic, for eachε> , there existM(ε),N(ε)∈Nsuch that among anyN(ε) consecutive integers there exists an integerpwith the property that
sup
n∈Z
fn(k+p) –fn(k) <ε/ (.)
for allk∈Zwith|k| ≥M(ε) and|k+p| ≥M(ε). For the aboveε> , there exists a positive integerKsuch that for alln,m>K, the following hold:
fn(k) –fm(k) <ε/, k∈
–M(ε),M(ε) +N(ε)∩Z. (.) For allk∈Zwith|k| ≥M(ε), takingp∈[–k+M(ε), –k+M(ε) +N(ε)]∩Z, by (.) and (.), we get
fn(k) –fm(k) ≤ fn(k) –fn(k+p) + fn(k+p) –fm(k+p) + fm(k+p) –fm(k)
<ε/ +ε/ +ε/ =ε, n,m>K;
also, for allk∈Zwith|k|<M(ε), by (.), we have
fn(k) –fm(k) <ε/ <ε, n,m>K.
Thus, we get
sup
k∈Z
fn(k) –fm(k) ≤ε, n,m>K,
which means that{fn(k)}is uniformly convergent onZ,i.e.,{fn}is convergent inAAP(Z).
So,Fis precompact inAAP(Z).
3 Application to Volterra difference equations
In this section, we discuss the existence of an asymptotically almost periodic solution to Volterra difference equation (.). Throughout the rest of this paper,p,q≥ are two fixed real numbers and
p+
q= .
In addition, we denote bylp(Z) (resp.lq(Z)) the space of all the functionsf :Z→R satis-fying
fp:=
k∈Z f(k) p
/p < +∞
resp.fq:=
k∈Z f(k) q
/q < +∞
.
(H) For eachi∈ {, , . . . ,λ},fi(·,x)∈AAP(Z)for any fixedx∈R, and there exists a constantLi≥such that
fi(k,x) –fi(k,y) ≤Li|x–y|, ∀k∈Z,∀x,y∈R.
(H) For eachi∈ {, , . . . ,λ},gi(k,·)is continuous for eachk∈Z, and for eachr> , there exists a sequence{μri} ⊂lp(Z)such that
gi(k,x) ≤μri(k), |x| ≤r,k∈Z.
(H) For eachi∈ {, , . . . ,λ},ai∈AAP(Z,lq(Z)), where[ai(k)](l) =ai(k,l),∀k,l∈Z. (H) There exists a constantM> such that
λ
i=
αiLiμMi p< ,
whereαi=supn∈Zai(n)q; and λ
i=
sup
n∈Z,|x|≤K
fi(n,x) ·αi·μMi p
<K, ∀K>M,
Theorem . Assume that (H)-(H)hold. Then Equation(.) has an asymptotically almost periodic solution.
Proof We denote
(Aix)(n) =fin,x(n), n∈Z,x∈AAP(Z),i= , , . . . ,λ; (Bix)(n) =
m∈Z
ai(n,m)gim,x(m), n∈Z,x∈AAP(Z),i= , , . . . ,λ;
and
(Mx)(n) = λ
i=
(Aix)(n)·(Bix)(n), n∈Z,x∈AAP(Z).
It suffices to prove thatMhas a fixed point inAAP(Z). We give the proof in three steps. Step .AiandBiboth mapAAP(Z) intoAAP(Z),i= , , . . . ,λ.
Sincefiis Lipschitz, by Remark ., we can first show that for each compact subsetK⊂R and eachi∈ {, , . . . ,λ},{fi(·,x) :x∈K}is equi-asymptotically almost periodic. Then it is easy to show thatAix∈AAP(Z) for eachx∈AAP(Z).
Sinceai∈AAP(Z,lq(Z)), there existbi∈AP(Z,lq(Z)) andci∈C
(Z,lq(Z)) such thatai= bi+ci. For eachx∈AAP(Z), noting that forn,p∈Z,
(Bix)(n) =
m∈Z
ai(n,m)gi
m,x(m)
= m∈Z
bi(n)(m)gim,x(m)+ m∈Z
where prove eachBi(E) is precompact inAAP(Z). By a direct calculation, we can get
(Biy)(n) ≤αi·μMi p, i= , , . . . ,λ,
Finally, we give a simple example to illustrate our result.
+n. In addition, (H) can be easily verified. By a direct calculation, we can get
Thus, (H) holds withM= . Then, by using Theorem ., Equation (.) has an asymp-totically almost periodic solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WL completed the main study, carried out the results of this article and drafted the manuscript. WP checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Acknowledgements
The work was supported by the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province, the Foundation of Jiangxi Provincial Education Department (GJJ12205), and the Research Project of Jiangxi Normal University (2012-114).
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doi:10.1186/1687-1847-2012-199
Cite this article as:Long and Pan:Asymptotically almost periodic solution to a class of Volterra difference equations.
