R E S E A R C H
Open Access
Multiple periodic solutions in shifts
δ
±
for an
impulsive functional dynamic equation on
time scales
Meng Hu
*and Lili Wang
*Correspondence:
[email protected] School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, People’s Republic of China
Abstract
In this paper, based on the theory of calculus on time scales, by using a multiple fixed point theorem in cones, some criteria are established for the existence and
multiplicity of positive periodic solutions in shifts
δ
±for an impulsive functional dynamic equation on time scales of the following form:x(t) = –a(t)x(t) +b(t)f(t,x(g(t))),t=tj,t∈T,x(t+
j ) =x(tj–) +Ij(x(tj)), whereT⊂Rbe a periodic time scale in shifts
δ
±with periodP∈[t0,∞)Tandt0∈Tis nonnegative andfixed. Finally, some numerical examples are presented to illustrate the feasibility and effectiveness of the results.
MSC: 34K13; 34K45; 34N05
Keywords: periodic solution; functional dynamic equation; impulse; shift operator; time scale
1 Introduction
The time scales approach unifies differential, difference,h-difference, andq-differences equations and more under dynamic equations on time scales. The theory of dynamic equa-tions on time scales was introduced by Hilger in his PhD thesis in []. The existence problem of periodic solutions is an important topic in qualitative analysis of functional dynamic equations. Up to now, there are only a few results concerning periodic solutions of dynamic equations on time scales; see, for example, [, ]. In these papers, authors con-sidered the existence of periodic solutions for dynamic equations on time scales satisfying the condition ‘there exists aω> such thatt±ω∈T,∀t∈T’. Under this condition all pe-riodic time scales are unbounded above and below. However, there are many time scales such asqZ={qn:n∈Z} ∪ {}and√N={√n:n∈N}which do not satisfy the condition. Adıvar and Raffoul introduced a new periodicity concept on time scales which does not oblige the time scale to be closed under the operationt±ωfor a fixedω> . They defined a new periodicity concept with the aid of shift operatorsδ±which are first defined in [] and then generalized in [].
Recently, based on a fixed point theorem in cones, Çetinet al.studied the existence of positive periodic solutions in shiftsδ±for some nonlinear first-order functional dynamic equation on time scales; see [, ].
However, to the best of our knowledge, there are few papers published on the existence of positive periodic solutions in shiftsδ±for a functional dynamic equation with impulses.
As we know, impulsive functional dynamic equation on time scales plays an important role in applications; see, for example, [, ].
Motivated by the above, in the present paper, we consider the following system:
x(t) = –a(t)x(t) +b(t)f(t,x(g(t))), t=t j,t∈T, x(t+
j) =x(tj–) +Ij(x(tj)),
(.)
whereT⊂Rbe a periodic time scale in shiftsδ±with periodP∈[t,∞)Tandt∈Tis
nonnegative and fixed;a,b∈C(T, (,∞)) are-periodic in shiftsδ±with periodωand –a∈R+;f ∈C(T×(,∞), (,∞)) is periodic in shiftsδ
±with periodωwith respect to the first variable;g∈C(T,T) is periodic in shiftsδ±with periodω;x(t+j) andx(t–
j) represent the right and the left limit ofx(tj) in the sense of time scales, in addition, iftjis right-scattered, thenx(tj+) =x(tj), whereas, iftj is left-scattered, thenx(t–j) =x(tj);Ij∈C((,∞), [,∞)), j∈Z. Assume that there exists a positive constantqsuch thattj+q=δω+(tj),Ij+q=Ij,j∈Z. For each intervalIofR, we denoteIT=I∩T, without loss of generality, set [t,δ+ω(t))T∩
{tj,j∈Z}={t,t, . . . ,tq}.
The main purpose of this paper is to establish some sufficient conditions for the exis-tence of at least three positive periodic solutions in shiftsδ±of system (.) using a multiple fixed point theorem (Avery-Peterson fixed point theorem) in cones.
The organization of this paper is as follows. In Section , we introduce some notations and definitions and state some preliminary results needed in later sections; then we give the Green’s function of system (.), which plays an important role in this paper. In Sec-tion , we establish our main results for positive periodic soluSec-tions in shiftsδ±by applying Avery-Peterson fixed point theorem. In Section , some numerical examples are presented to illustrate that our results are feasible and more general.
2 Preliminaries
LetTbe a nonempty closed subset (time scale) ofR. The forward and backward jump operatorsσ,ρ:T→Tand the graininessμ:T→R+are defined, respectively, by
σ(t) =inf{s∈T:s>t}, ρ(t) =sup{s∈T:s<t} and μ(t) =σ(t) –t.
A pointt∈Tis called left-dense ift>infTandρ(t) =t, left-scattered ifρ(t) <t, right-dense ift<supTandσ(t) =t, and right-scattered ifσ(t) >t. IfThas a left-scattered max-imumm, thenTk=T\{m}; otherwiseTk=T. IfThas a right-scattered minimumm, then
Tk=T\{m}; otherwiseTk=T.
A functionf :T→Ris right-dense continuous provided it is continuous at right-dense point inTand its left-side limits exist at left-dense points inT. Iff is continuous at each right-dense point and each left-dense point, thenf is said to be a continuous function onT. The set of continuous functionsf:T→Rwill be denoted byC(T) =C(T,R).
For the basic theories of calculus on time scales, see [].
A functionp:T→Ris called regressive provided +μ(t)p(t)= for allt∈Tk. The set of all regressive and rd-continuous functionsp:T→Rwill be denoted byR=R(T,R). Define the setR+=R+(T,R) ={p∈R: +μ(t)p(t) > ,∀t∈T}.
Ifris a regressive function, then the generalized exponential functioneris defined by
er(t,s) =exp
t
s
ξμ(τ)
for alls,t∈T, with the cylinder transformation
ξh(z) =
Log
(+hz)
h ifh= ,
z ifh= .
Letp,q:T→Rbe two regressive functions, define
p⊕q=p+q+μpq, p= – p
+μp, pq=p⊕(q).
Lemma .[] Assume that p,q:T→Rbe two regressive functions,then (i) e(t,s)≡andep(t,t)≡;
(ii) ep(σ(t),s) = ( +μ(t)p(t))ep(t,s); (iii) ep(t,s) =ep(s,t)=ep(s,t); (iv) ep(t,s)ep(s,r) =ep(t,r);
(v) (ep(t,s))= (p)(t)ep(t,s).
The following definitions and lemmas about the shift operators and the new periodicity concept for time scales can be found in [, ].
LetT∗be a nonempty subset of the time scaleTandt∈T∗be a fixed number, define
operatorsδ±: [t,∞)×T∗→T∗. The operatorsδ+andδ–associated witht∈T∗(called
the initial point) are said to be forward and backward shift operators on the setT∗, re-spectively. The variables∈[t,∞)Tinδ±(s,t) is called the shift size. The valuesδ+(s,t) and
δ–(s,t) inT∗indicatesunits translation of the termt∈T∗to the right and left, respectively.
The sets
D±:=(s,t)∈[t,∞)T×T∗:δ∓(s,t)∈T∗
are the domains of the shift operatorδ±, respectively. Hereafter,T∗is the largest subset of the time scaleTsuch that the shift operatorsδ±: [t,∞)×T∗→T∗exist.
Definition .(Periodicity in shiftsδ±[]) LetTbe a time scale with the shift operators
δ±associated with the initial pointt∈T∗. The time scaleTis said to be periodic in shifts
δ±if there existsp∈(t,∞)T∗such that (p,t)∈D±for allt∈T∗. Furthermore, if P:=infp∈(t,∞)T∗: (p,t)∈δ±,∀t∈T∗ =t,
thenPis called the period of the time scaleT.
Definition .(Periodic function in shiftsδ±[]) LetTbe a time scale that is periodic in shiftsδ±with the periodP. We say that a real-valued functionf defined onT∗is periodic in shiftsδ±if there existsω∈[P,∞)T∗such that (ω,t)∈D±andf(δ±ω(t)) =f(t) for allt∈T∗, whereδω±:=δ±(ω,t). The smallest numberω∈[P,∞)T∗is called the period off.
Definition .(-periodic function in shiftsδ±[]) LetTbe a time scale that is peri-odic in shiftsδ±with the periodP. We say that a real-valued functionf defined onT∗is
-periodic in shiftsδ±if there existsω∈[P,∞)T∗such that (ω,t)∈D±for allt∈T∗, the shiftsδω
±are-differentiable with rd-continuous derivatives andf(δω±(t))δ±ω(t) =f(t) for allt∈T∗, whereδω
Lemma .[] δω
+(σ(t)) =σ(δ+ω(t))andδ–ω(σ(t)) =σ(δ–ω(t))for all t∈T∗.
Lemma .[] LetTbe a time scale that is periodic in shiftsδ±with the period P.Suppose that the shiftsδ±ωare-differentiable on t∈T∗whereω∈[P,∞)T∗and p∈Ris-periodic in shiftsδ±with the periodω.Then
(i) ep(δω±(t),δ±ω(t)) =ep(t,t)fort,t∈T∗;
(ii) ep(δω±(t),σ(δω±(s))) =ep(t,σ(s)) =+eμp((tt),sp)(t) fort,s∈T∗.
Lemma .[] LetTbe a time scale that is periodic in shiftsδ±with the period P,and let f be a-periodic function in shiftsδ±with the periodω∈[P,∞)T∗.Suppose that f ∈Crd(T),
then
t
t
f(s)s=
δ±ω(t)
δω±(t)
f(s)s.
Lemma .[] Suppose that r is regressive and f :T→Ris rd-continuous.Let t∈T,
y∈R,then the unique solution of the initial value problem
y=r(t)y+f(t), y(t) =y
is given by
y(t) =er(t,t)y+
t
t
er
t,σ(τ)f(τ)τ.
Define
PC(T) =x:T→R|x|(tj,tj+)∈C(tj,tj+),∃x
tj–=x(tj),x
tj+,j∈Z .
Set
X=x:x∈PC(T),xδ+ω(t)=x(t)
with the normx=supt∈[t,δω
+(t)]T|x(t)|, thenXis a Banach space.
Lemma . x(t)∈X is anω-periodic solution in shiftsδ±of system(.)if and only if x(t) is anω-periodic solution in shiftsδ±of
x(t) =
δω+(t)
t
G(t,s)b(s)fs,xg(s)s+ j:tj∈[t,δω+(t))T
G(t,tj)e–a
σ(tj),tj
Ij
x(tj)
, (.)
where
G(t,s) = e–a(t,σ(s))
e–a(t,δ+ω(t)) –
.
Proof Ifx(t) is anω-periodic solution in shiftsδ±of system (.). For anyt∈T, there exists
–G(ti,ti)e–a
σ(ti),ti
Ii
x(ti)
=Ii
x(ti)
.
So,xis anω-periodic solution in shiftsδ±of system (.). This completes the proof.
It is easy to verify that the Green’s functionG(t,s) satisfies the property
<
ξ– ≤G(t,s)≤
ξ
ξ– , ∀s∈
t,δω+(t)T, (.)
whereξ=e–a(t,δ+ω(t)). By Lemma ., we have
Gδ+ω(t),δ+ω(s)=G(t,s), ∀t∈T∗,s∈t,δ+ω(t)T. (.)
In order to obtain the existence of periodic solutions in shiftsδ±of system (.), we need the following concepts and Avery-Peterson fixed point theorem.
LetXbe a Banach space andKbe a cone inX, defineKr={x∈K| x ≤r}. A map
α is said to be a nonnegative continuous concave functional onK ifα:K→[, +∞) is continuous and
αλx+ ( –λ)y≥λα(x) + ( –λ)α(y) for allx,y∈K, <λ< .
Letγ andθ be nonnegative continuous convex functionals onK,α be a nonnegative continuous concave functional onK, andψbe a nonnegative continuous functional onK. Then for positive real numbersa,b,c, andd, we define the following convex sets:
K(γ,d) =x∈K|γ(x) <d ,
K(γ,α,b,d) =x∈K|b≤α(x),γ(x)≤d ,
K(γ,θ,α,b,c,d) =x∈K|b≤α(x),θ(x)≤c,γ(x)≤d ,
and a closed setR(γ,ψ,a,d) ={x∈K|a≤ψ(x),γ(x)≤d}.
Lemma .(Avery-Peterson fixed point theorem []) Letγ andθ be nonnegative con-tinuous convex functionals on K,αbe a nonnegative continuous concave functional on K,
andψbe a nonnegative continuous functional on K satisfyingψ(ρx)≤ρψ(x)for≤ρ≤,
such that for some positive numbers E and d,
α(x)≤ψ(x) and x ≤Eγ(x) (.)
for all x∈K(γ,d).Suppose H:K(γ,d)→K(γ,d)is completely continuous and there exist positive numbers a,b,and c with a<b such that:
() {x∈K(γ,θ,α,b,c,d)|α(x) >b} =∅andα(Hx) >bforx∈K(γ,θ,α,b,c,d), () α(Hx) >b,forx∈K(γ,α,b,d)withθ(Hx) >c,
Then H has at least three fixed points x,x,x∈K(γ,d)such that
Furthermore, for anyx∈K,∀t∈[t,δ+ω(t)]T, we have
Next, we show thatHmaps any bounded sets inKinto relatively compact sets. We first prove thatf maps bounded sets into bounded sets. Indeed, letε= , for anyν> , there existsδ> such that{x,y∈K,x ≤ν,y ≤ν,x–y<δ,s∈[t,δω+(t)]T}imply Choose a positive integer N such that ν
Furthermore, fort∈T, we have
(Hx)(t) = –a(t)(Hx)(t) +b(t)ft,xg(t)
and
(Hx)(t)≤ sup t∈[t,δω+(t)]T
–a(t)(Hx)(t) +b(t)ft,xg(t)≤AuD+BuW.
To sum up,{Hx:x∈K,x ≤ν}is a family of uniformly bounded and equicontinuous functionals on [t,δω+(t)]T. By a theorem of Arzela-Ascoli, the functionalHis completely
continuous. This completes the proof.
3 Main result
Now, we fixη,l∈[t,δ+ω(t)]T, η≤l, and let the nonnegative continuous concave
func-tionalα, the nonnegative continuous convex functionalsθ,γ, and the nonnegative con-tinuous functionalψbe defined on the coneKby
α(x) = inf
t∈[η,l]Tx(t), ψ(x) =θ(x) =t∈[tsup
,δ+ω(t)]T
x(t),
γ(x) = sup t∈[t,δω+(t)]T
(x)(t),
(.)
respectively, where (x)(t) =δ+ω(t)
t h(t,s)x(s)s,h(t,s)∈C(T
, (,∞)).
The functionals defined above satisfy the following relations:
α(x)≤ψ(x) =θ(x), ∀x∈K. (.)
Lemma . For x∈K,there exists a constant E> such that
sup t∈[t,δ+ω(t)]T
x(t)≤E sup t∈[t,δ+ω(t)]T
(x)(t).
Proof Forx∈K, we have
sup t∈[t,δ+ω(t)]T
(x)(t) = sup t∈[t,δω+(t)]T
δ+ω(t)
t
h(t,s)x(s)s
≥
ξxt∈[tsup,δω
+(t)]T δ+ω(t)
t
h(t,s)s
= L
ξ t∈[tsup,δ+ω(t)]T
x(t),
whereL:=supt∈[t ,δω+(t)]T
δ+ω(t)
t h(t,s)s. SettingE:=
ξ
L. This completes the proof.
Moreover, for eachx∈K,
x= sup t∈[t,δω+(t)]T
x(t)≤ξ
Lt∈[tsup,δω+(t)]T
(x)(t) =Eγ(x), (.)
To check condition () in Lemma ., we take x˜ =bξ. It is easy to verify that x˜ ∈
To sum up, all conditions in Lemma . are satisfied. Hence,Hhas at least three fixed points, that is, system (.) has at least three positiveω-periodic solutions in shiftsδ±. This
completes the proof.
4 Numerical examples
Consider the following system with impulses:
x(t) = –a(t)x(t) +b(t)f(t,x(g(t))), t=t j,t∈T, x(t+
j) =x(tj–) +Ij(x(tj)).
(.)
Example Let
a(t) = ., b(t) = – .sinπt, f(t,x) =
|sinπt|
+
x
, x≤,
++|xsinπt|, x> ,
Ij
x(tj)
= .sinx(tj), j= , , . . . , ,
in system (.), then
ω= ; IM ,IM,Im∈[, .].
CaseI: LetT=R,t= , thenδ+ω(t) =t+ . It is easy to verifya(t),b(t),f(t,x) satisfy
aδ+ω(t)δω+ (t) =a(t), bδω+(t)δ+ω(t) =b(t), fδ+ω(t),x=f(t,x), ∀t∈T∗, and –a∈R+.
By a direct calculation, we can get
ξ=e.= ., B=
( – .sinπt)t=
t+cosπt π
= .
Choosea= ,b= ,L= ,d= , then
f(t,x)≤ + . = . < . –IM forx∈[, .],
f(t,x) > +
= . > . –I
m forx∈[, .],
f(t,x) < +
= . < . –I M
forx∈[, ].
According to Theorem ., whenT=R, for system (.) there exist at least three positive periodic solutionsxˆ,xˆ,xˆin shiftsδ±with periodω= , and
sup t∈[t,δ+ω(t)]T
ˆ
x(t) < < sup
t∈[t,δω+(t)]T ˆ
x(t), inf
t∈[η,l]Txˆ(t) < <
inf t∈[t,δω+(t)]Tˆ
x(t).
CaseII: LetT=Z,t= , thenδ+ω(t) =t+ . It is easy to verifya(t),b(t),f(t,x) satisfy
By a direct calculation, we can get
ξ= (.)–= , B=
( – .sinπt)t= (t– .tsinπt)|= .
Choosea= ,b= ,L= ,d= , then
f(t,x)≤ + = < –IM forx∈[, ],
f(t,x) > +
= . > –I
m forx∈[, ],
f(t,x) < +
= . < . –I M
forx∈[, ].
According to Theorem ., whenT=Z, for system (.) there exist at least three positive periodic solutionsx˜,x˜,x˜in shiftsδ±with periodω= , and
sup t∈[t,δ+ω(t)]T
˜
x(t) < < sup
t∈[t,δω+(t)]T ˜
x(t), inf
t∈[η,l]Tx˜(t) < <
inf t∈[t,δω+(t)]T˜
x(t).
Example Let
a(t) =
t, b(t) =
bt
, f(t,x) =
|sinπ
t|
+
x
, x≤,
++|xsinπ
t|, x> ,
Ij
x(tj)
= .sinx(tj), j= , , . . . , ,
in system (.), whereb=
tt, then
ω= ; IM,IM,Im∈[, .].
LetT= N,t
= , thenδ+ω(t) = t. It is easy to verifya(t),b(t),f(t,x) satisfy
aδ+ω(t)δω+ (t) =a(t), bδω+(t)δ+ω(t) =b(t), fδ+ω(t),x=f(t,x), ∀t∈T∗,
and –a∈R+.
By a direct calculation, we can get
ξ= t∈[,)
+ t –t
= ., B=
bt
t= .
Choosea= ,b= ,L= ,d= , then
f(t,x)≤ + . = . < –IM forx∈[, .],
f(t,x) > +
= . > . –I
m forx∈[, .],
f(t,x) < +
= . < . –I M
According to Theorem ., whenT= N, for system (.) there exist at least three pos-itive periodic solutionsxˆ,xˆ,xˆin shiftsδ±with periodω= , and
sup t∈[t,δ+ω(t)]T
ˆ
x(t) < < sup
t∈[t,δω+(t)]T ˆ
x(t), inf
t∈[η,l]Txˆ(t) < <t∈[tinf,δω+(t)]Tˆ
x(t).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
Acknowledgements
This work is supported by the Basic and Frontier Technology Research Project of Henan Province (Grant No. 142300410113).
Received: 22 March 2014 Accepted: 12 May 2014 Published:22 May 2014
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