Delay dynamic equations with stability

DOUGLAS R. ANDERSON, ROBERT J. KRUEGER, AND ALLAN C. PETERSON Received 13 August 2005; Accepted 23 October 2005

We first give conditions which guarantee that every solution of a first order linear delay dynamic equation for isolated time scales vanishes at infinity. Several interesting examples are given. In the last half of the paper, we give conditions under which the trivial solution of a nonlinear delay dynamic equation is asymptotically stable, for arbitrary time scales. Copyright © 2006 Douglas R. Anderson et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited. 1. Preliminaries

The unification and extension of continuous calculus, discrete calculus,q-calculus, and indeed arbitrary real-number calculus to time-scale calculus, where a time scale is sim-ply any nonempty closed set of real numbers, were first accomplished by Hilger in [4]. Since then, time-scale calculus has made steady inroads in explaining the interconnec-tions that exist among the various calculuses, and in extending our understanding to a new, more general and overarching theory. The purpose of this work is to illustrate this new understanding by extending some continuous and discrete delay equations to cer-tain time scales. Examples will include specific cases in differential equations, difference equations,q-difference equations, and harmonic-number equations. The definitions that follow here will serve as a short primer on the time-scale calculus; they can be found in [1,2] and the references therein.

Definition 1.1. Define the forward (backward) jump operatorσ(t) attfort <supT(resp., ρ(t) attfort >infT) by

σ(t)=inf{τ > t:τ∈T}, ρ(t)=sup{τ < t:τ∈T}, ∀t∈T. (1.1) Also defineσ(supT)=supT, if supT<∞, andρ(infT)=infT, if infT>−∞. Define the graininess functionμ:T→Rbyμ(t)=σ(t)−t.

Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 94051, Pages1–19

Throughout this work the assumption is made thatTis unbounded above and has the topology that it inherits from the standard topology on the real numbersR. Also assume throughout thata < bare points inTand define the time scale interval [a,b]T= {t∈T:

a≤t≤b}. Other time scale intervals are defined similarly. The jump operatorsσ andρ allow the classification of points in a time scale in the following way: ifσ(t)> tthen call the pointtright-scattered; while ifρ(t)< tthen we saytis left-scattered. Ifσ(t)=tthen call the pointtright-dense; while ift >infTandρ(t)=tthen we saytis left-dense. We next define the so-called delta derivative. The novice could skip this definition and look at the results stated inTheorem 1.4. In particular in part (2) ofTheorem 1.4we see what the delta derivative is at right-scattered points and in part (3) ofTheorem 1.4we see that at right-dense points the derivative is similar to the definition given in calculus.

Definition 1.2. Fixt∈Tand lety:T→R. DefineyΔ(t) to be the number (if it exists) with the property that given>0 there is a neighbourhoodUoftsuch that, for alls∈U,

_y

σ(t)−y(s)−yΔ_{(t)σ(t)−s≤σ(t)−s. (1.2)} CallyΔ(t) the (delta) derivative ofy(t) att.

Definition 1.3. IfFΔ(t)=f(t) then define the (Cauchy) delta integral by

t

a f(s)Δs=F(t)−F(a). (1.3) The following theorem is due to Hilger [4].

Theorem1.4. Assume that f :T→Rand lett∈T. (1)If f is differentiable att, then f is continuous att.

(2)If f is continuous attandtis right-scattered, then f is differentiable attwith fΔ_(t)= f

σ(t)−f(t)

σ(t)−t . (1.4)

(3)If f is differentiable andtis right-dense, then fΔ_(t)=lim

s→t

f(t)−f(s)

t−s . (1.5)

(4)If f is differentiable att, then f(σ(t))= f(t) +μ(t)fΔ(t).

Next we define the important concept of right-dense continuity. An important fact concerning right-dense continuity is that every right-dense continuous function has a delta antiderivative [1, Theorem 1.74]. This implies that the delta definite integral of any right-dense continuous function exists.

Definition 1.5. We say thatf :T→Ris right-dense continuous (and writef ∈Crd(T;R))

provided f is continuous at every right-dense pointt∈T, and lims→t− f(s) exists and is

We saypis regressive provided 1 +μ(t)p(t)=0,∀t∈T. Let

᏾:=p∈Crd(T;R) : 1 +μ(t)p(t)=0, t∈T . (1.6)

Also, p∈᏾+_{if and only if p∈᏾and 1 +μ(t)p(t)>0,∀t∈T. Then if p∈᏾,t} 0∈T,

one can define the generalized exponential functionep(t,t0) to be the unique solution of

the initial value problem

xΔ_{=p(t)x, xt}

0

=1. (1.7)

We will use many of the properties of this generalized exponential functionep(t,t0) listed

inTheorem 1.6.

Theorem1.6 ([1, Theorem 2.36]). Ifp,q∈᏾ands,t∈T, then (1)e0(t,s)≡1andep(t,t)≡1;

(2)ep(σ(t),s)=(1 +μ(t)p(t))ep(t,s);

(3) 1/ep(t,s)=e p(t,s), where p:= −p/(1 +μp); (4)ep(t,s)=1/ep(s,t)=e p(s,t);

(5)ep(t,s)ep(s,r)=ep(t,r);

(6)ep(t,s)eq(t,s)=ep⊕q(t,s), wherep⊕q:=p+q+μpq; (7)ep(t,s)/eq(t,s)=ep q(t,s).

2. Introduction to a delay dynamic equation

Since we are interested in the asymptotic properties of solutions we assume as mentioned earlier that our time scaleTis unbounded above. Consider the delay dynamic equation

xΔ_{(t)= −a(t)xδ(t)δ}Δ_{(t), t∈t}

0,∞_T, (2.1)

where the delay functionδ: [t0,∞)T→[δ(t0),∞)Tis strictly increasing and delta diff

er-entiable withδ(t)< tfort∈[t0,∞)Tand limt→∞δ(t)= ∞. For example, ifT=[−m,∞),

andδ(t) :=t−m,t∈[0,∞), wherem >0, then (2.1) becomes the well-studied delay dif-ferential equation

x_{(t)= −a(t)x(t−m). (2.2)} IfT= {−m,−m+ 1,..., 0, 1, 2,...}, andδ(t) :=t−m,t∈N0, wheremis a positive integer,

then (2.1) becomes

Δx(t)= −a(t)x(t−m), (2.3) whereΔis the forward difference operator defined byΔx(t)=x(t+ 1)−x(t). IfT=qN0∪ {q−1_,q−2_,...,q−m_}whereqN0:= {1,q_,q2_,...},q >_{1, and}δ₍t) :=₍₁/qm₎t_,t∈qN0_{, where} m∈N, then (2.1) becomes the delay quantum equation

Dqx(t)= −_q1_ma(t)x

₁

qmt

where

Dqx(t) :=x(qt)−x(t)

(q−1)t (2.5)

is the so-called quantum derivative studied in Kac and Cheung [5]. More examples will be given later. We will use the following three lemmas to proveTheorem 3.1.

Lemma2.1 (chain rule). AssumeTis an isolated time scale, andg(σ(t))=σ(g(t))fort∈T. Ifg:T→Tandh:T→R, then

g(t)

t0

h(s)Δs

Δ

=hg(t)gΔ(t). (2.6) Proof. Sincetis right-scattered,

g(t)

t0

h(s)Δs

Δ =_μ1

(t)

g(σ(t))

t0

h(s)Δs−

g(t)

t0

h(s)Δs

=_μ1 (t)

g(σ(t))

g(t) h(s)Δs

=_μ1 (t)

σ(g(t))

g(t) h(s)Δs

=_μ1 (t)h

g(t)σg(t)−g(t)

=hg(t)g

σ(t)−g(t) μ(t) =hg(t)gΔ(t).

(2.7)

Lemma2.2. AssumeTis an isolated time scale and the delayδ satisfiesδ◦σ=σ◦δ, or

T=R. Then the delay equation (2.1) is equivalent to the delay equation

xΔ_{(t)= −aδ}−1_{(t)x(t) +}

t

δ(t)a

_δ−1

(s)x(s)Δs

Δ

. (2.8)

Proof. Assumexis a solution of (2.8). Then using the chain rule (Lemma 2.1) for isolated time scales or the regular chain rule forT=R,

xΔ_{(t)= −aδ}−1_{(t)x(t) +}

t

δ(t)a

δ−1_(s)x(s)Δs

Δ

= −aδ−1_{(t)x(t) +aδ}−1_{(t)x(t)−a(t)xδ(t)δ}Δ_(t) = −a(t)xδ(t)δΔ(t).

(2.9)

Lemma2.3. Ifxis a solution of (2.1) with initial functionψ, then

Proof. We use the variation of constants formula [1, page 77] for (2.8), to obtain x(t)=e−a(δ−1₎t,t₀xt₀+ Using integration by parts [1, page 28],

x(t)=e−a(δ−1₎t,t₀xt₀+e_−a(δ−1₎(t,τ)

3. Asymptotic properties of the delay equation

The results in this section generalize some of the results by Raffoul in [9]. Letψ: [δ(t0),

t0]T→Rbe rd-continuous and letx(t) :=x(t,t0,ψ) be the solution of (2.1) on [t0,∞)T

withx(t)=ψ(t) on [δ(t0),t0]T. Letφ =sup|φ(t)|fort∈[δ(t0),∞)T, and define the

Banach spaceB= {φ∈C([δ(t0),∞)T:φ(t)→0 ast→ ∞}, with

S:=φ∈B:φ(t)=ψ(t)∀t∈δt0

,t0

T

. (3.1)

In the following we assume

e−a(δ−1₎t,t₀−→0 ast−→ ∞, (3.2) and takeD: [t0,∞)T→Rto be the function

D(t) :=

t

t0

_1−μa_(τδ₎−_a1(τ)

δ−1₍τ₎

e−a(δ−1₎(t,τ)

τ

δ(τ)

_a

δ−1_(s)ΔsΔτ

+

t

δ(t)

_a

δ−1_(s)Δs.

(3.3)

To enable the use of the contraction mapping theorem, we in fact assume there exists α∈(0, 1) such that

D(t)≤α, t∈t0,∞

T. (3.4)

Theorem3.1. AssumeT=RorTis an isolated time scale. If (3.2) and (3.4) hold and δ◦σ=σ◦δ, then every solution of (2.1) goes to zero at infinity.

Proof. AssumeTis an isolated time scale. Fixψ: [δ(t0),t0]→Rand defineP:S→Bby

(Pφ)(t) :=ψ(t) fort≤t0and fort≥t0,

(Pφ)(t)=ψt0e−a(δ−1₎t,t₀+

t

δ(t)a

_δ−1

(s)φ(s)Δs −e−a(δ−1₎t,t₀

t0

δ(t0)

aδ−1_(s)ψ(s)Δs

−

t

t0

aδ−1_(τ)

1−μ(τ)aδ−1₍τ₎e−a(δ−1)(t,τ)

τ

δ(τ)a

δ−1_(s)φ(s)Δs

Δτ.

(3.5)

Then byLemma 2.3, it suffices to show thatPhas a fixed point. We will use the contrac-tion mapping theorem to showP has a fixed point. To show that (Pφ)(t)→0 ast→ ∞, note that the first and third terms on the right-hand side of (Pφ)(t) go to zero by (3.2). From (3.3) and (3.4) and the fact thatφ(t)→0 ast→ ∞, we have that

_φ(t)t δ(t)

_a

Hence (Pφ)(t)→0 ast→ ∞and therefore,PmapsSintoS. It remains to show thatPis a contraction under the sup norm. Letx,y∈S. Then

_{(Px)(t)−(Py)(t)}

≤

t

t0

a

δ−1_(τ)

1−μ(τ)aδ−1₍τ₎

e−a(δ−1₎(t,τ)

τ

δ(τ)

_a

δ−1_{(s)x(s)−y(s)Δs}

Δτ

+

t

δ(t)

_a

δ−1_{(s)x(s)−y(s)Δs}

≤ x−y

t

δ(t)

_a

δ−1_(s)Δs

+

t

t0

a

δ−1_(τ)

1−μ(τ)aδ−1₍τ₎

e−a(δ−1₎(t,τ)

τ

δ(τ)

_a

δ−1_(s)Δs

Δτ

≤αx−y.

(3.10) Therefore, by the contraction mapping principle [6, page 300],Phas a unique fixed point inS. This completes the proof in the isolated time scale case. See Raffoul [9] for the proof of theT=Zcase and a reference for a proof of the continuous case. Example 3.2. For any real numberq >1 and positive integerm, define

T=q−m_,q−m+1_,...,q−1_{, 1,q,q}2_{,... . (3.11)}

We show if 0< c < qm/2m(q−1), then for any initial functionψ(t),t∈[q−m, 1]T, the

solution of the delay initial value problem Dqx(t)= −_q1_mc_tx

₁

qmt

, t∈[1,∞]T, (3.12)

x(t)=ψ(t), t∈q−m_{, 1}

T (3.13)

goes to zero ast→ ∞.

To obtain (3.12) from (2.1), takea(t)=c/tandδ(t)=q−m_{twhich impliesa(δ}−1_(t))=

c/qm_{tand δ}Δ_(t)=q−m_{. To useTheorem 3.1, we verify that conditions (3.2) and (3.4)} hold. Note that

e−a(δ−1₎(t, 1)=

s∈[1,t)T

1−s(q−1)aqms=1−q−m_c(q−1)n _(3.14)

fort=qn_{. Ifc∈(0,q}m_{/2m(q−1)), thenc∈(0, 2q}m_{/(q−1)) so that 1−q}−m_c(q−1)∈ (−1, 1) and

lim

t→∞e−a(δ−1)(t, 1)=n→∞lim

1−q−m_c(q−1)n

Thus, (3.2) is satisfied. Now considerD(t) as defined in (3.3). We seekα∈(0, 1) such that D(t)≤α,∀t∈[1,∞)T. Here we havet0=1,μ(t)=(q−1)t, and

e−a(δ−1₎(t,τ)=1−q−mc(q−1)n−k (3.16) fort=qn_,τ=qk_{withk < n. For the second integral inD(t), note that}

qu

u f(ζ)Δζ=(qu−u)f(u), (3.17) whence

t

δ(t)a

δ−1_(s)Δs=

q−m+1_t

q−m_t +

q−m+2_t

q−m+1_t+···+

t

q−1_t

_c

qm_s

Δs

=

mc

qm_q−m_tq−mt

(q−1) =mc_qm(q−1),

(3.18)

which is independent oft. It follows that D(t)=mc_qm(q−1) +mc_qm(q−1)

t

1

c qm_τ·

1

1−(q−1)τc/qmτe−a(δ−1)(t,τ)

Δτ

=mc_qm(q−1) +mc qm(q−1)

n−1

k=0

c

qm_qk_−qk_(q−1)c

1−q−m_c(q−1)n−k

(q−1)qk

=mc_qm(q−1) +mc qm

n−1

k=0

c(q−1)2

qm_−(q−1)c

1−q−m_c(q−1)n−k

=mc_qm(q−1) +mc

2_q−m_(q−1)2

qm_−c(q−1)

1−q−m_c(q−1) −q−m_(q−1)c

1−q−m_c(q−1)n −1

=mc(_qq_m−1)+ mcq

−m_(q−1) 1−cq−m_(q−1)

1−q−m_{c(q−1)1−1−q}−m_c(q−1)n_. (3.19) Consequently,

D(t)=mc(_qq_m−1)2−1−q−m_c(q−1)n_<2mc(q−1)

qm , ∀t=qn∈[1,∞)T. (3.20)

Since 0< c < qm/2m(q−1), by takingα:=2mc(q−1)/qmcondition (3.4) is satisfied by D(t)< α <1, ∀t∈[1,∞)T. (3.21)

Example 3.3. Consider the time scale of harmonic numbers

for allt=Hn∈[0,∞)T. By choosingc∈(0,Hm/2m),D(t)< α:=2cm/Hm<1,∀t=Hn∈ [0,∞)T, satisfying (3.4). Thus byTheorem 3.1, for any given initial functionψ, the

solu-tion of the IVP (2.1), (3.25) goes to zero ast=Hngoes to infinity.

Example 3.4. LetT= {−mh,...,−h, 0,h, 2h,...}wherem∈N, then (2.1) becomes Δhx(t)= −cx(t−hm), t∈hN, (3.30) whereΔhx(t) :=(x(t+h)−x(t))/h. Our results give that if

0< c < 1

2mh, (3.31)

then all solutions go to zero ast→ ∞.

In this case,a(t)=c,δ(t)=t−mh, andδ−1_{(t)=t+mh. It can be shown that}

e−a(δ−1₎(t, 0)=(1−ch)t/h. (3.32) Note for 0< ch <2, condition (3.2) is satisfied because

e−a(δ−1₎(t, 0)−→0 (3.33)

ast→ ∞. It also can be shown that

D(t)=mhc+mhc1− |1−ch|t/h≤2mhc (3.34) if 0< ch <1. Hence (3.4) holds if 0< c <1/2mh. Therefore, we useTheorem 3.1to con-clude that all solutions of (3.30) go to zero ast→ ∞. It can be shown that ifc≤0, then there isλ≥1 such thatx(t)=λht_{is a solution of (3.30). However,x(t) does not approach} zero ast→ ∞. Hence our lower estimate forcis sharp. Ifm=1, then it can be shown that all solutions of (3.30) go to zero if 0< c <1/h. Ifm=2, our example shows that if 0< c <1/4hall solutions go to zero ast→ ∞. It can be shown that ifc=1/2h, then there is a solution that does not go to zero ast→ ∞. Next we give an elementary example whereTis the real interval [−m,∞),m >0. Delay differential equations have been studied extensively; for example, see [8].

Example 3.5. LetT=[−m,∞). For the delay differential equation

x_{(t)= −cx(t−m), t∈[0,∞), (3.35)} if 0< c <1/2m, then all solutionsx(t) approach zero ast→ ∞.

Note that (2.1) reduces to (3.35) ifa(t)=c,δ(t)=t−m, andT=[−m,∞). It can be shown that

e−a(δ−1₎(t,τ)=e−c(t−τ), D(t)=cm

2−e−ct_{. (3.36)}

4. Asymptotic stability of a nonlinear delay dynamic equation

In this section we consider, on arbitrary time scales, the nonlinear delay dynamic equa-tion The initial condition associated with (4.1) takes the form

x(t)=ψ(t), t∈δt0 techniques in this section are motivated by those in [7]. See also a related discussion in [3]. In other words, the trivial solution of (4.1) is asymptotically stable.

By property (iii),

By Gronwall’s inequality [1, page 257],

Therefore by (H1) and (4.1), there exists ¯t∈[δ(t∗),t∗)T such thatx(¯t)≤0. Integrate

which also contradicts (i) and provides the desired result. Now we show the limit ofx(t) goes to zero ast→ ∞.

which goes to∞ast→ ∞by (H1) and properties (ii) and (iv). This contradicts (4.20)

Without loss of generality, we will assume lim sup

t→∞ x(t)=x¯=0. (4.25)

Since the solution is oscillatory, there exists a sequence{tj}∞j=1inTsuch that limj→∞tj=

ForT=Zand f1as above, (4.1) becomes the (delay) difference equation

x(t+ 1)=x(t)− t−1

s=t−mkx

(s), t∈Z. (4.28)

Fixξ∈(0, 1),m∈N, and 0< k≤ξ/m(m+ 1). To check (i), note that t

τ=t−m 1 ckc

τ−(τ−m)=km(m+ 1)≤ξ <1, t∈Z. (4.29)

In (ii),∞τ=t0k|c|m= ∞. For (iii),|f1(τ,c)| =k|c|, so we takea1(τ)≡k >0, which is rd-continuous. Finally, (iv) is met as|f1(τ,x)| =k|x| ≤ky= f1(τ,y) for|x| ≤y≤Mwith

τ∈[t0,∞)T. Therefore the trivial solution of (4.1) is asymptotically stable byTheorem

4.1.

Lemma4.3. Assumexis a global solution for (4.1), (4.2). Then eitherxis bounded orxis oscillatory.

Proof. Ifxis nonoscillatory, then there existsT >0 such that fort > T,xdoes not change sign. Without loss of generality, we supposex(t)>0 fort > T. By (H1),i=n1fi(t,x(t))>0

fort > T, which together with (4.1) yields thatxΔ(t)<0 for larget. Therefore,xis strictly decreasing on each interval, so by continuity,xis bounded.

Theorem4.4. Assume there existsM >0andt1≥t0such that for|c| ≥M,

(i)_δσ((tt))

n

i=1(1/c)fi(τ,c)(τ−δ(τ))Δτ≤ξ <1, fort∈[t1,∞)T;

(ii)|ni=1fi(τ,c)| ≤

n

i=1ai(τ)|c|where theaiare rd-continuous and nonnegative for

τ≥t0;

(iii)|ni=1fi(t,u)| ≤

n

i=1fi(t,y)for|u| ≤y,t∈[δ(t0),∞)T.

Then every solutionxof (4.1), (4.2) for bounded initial functionψis bounded and satisfies

lim sup t→∞

_{x(t)≤M. (4.30)}

Proof. Appealing to assumption (ii),xΔ(t) exists and is finite for eacht∈[t0,∞)T, so

that solutions of (4.1), (4.2) are global. Suppose xis an unbounded solution of (4.1), (4.2) with bounded initial functionψ. ByLemma 4.3,xis also oscillatory. As inCase 2

of the proof ofTheorem 4.1, without loss of generality there exists a sequence{tj}∞ j=1in

Tsuch that limj→∞tj= ∞,M≤x(σ(tj)) with|x(s)| ≤x(σ(tj)) for alls∈[δ2(tj),tj], and x(σ(tj))→ ∞as j→ ∞. Moreover there exist correspondingt∗j ∈[δ(tj),tj]T satisfying

x(t∗j)≤0. Then

xσtj≤xσtj−xt∗j= −

σ(tj)

t∗

j

τ

δ(τ)

n

i=1

fiτ,x(s)Δs

so that by (iii) we have

a contradiction of (i). Thereforexmust be bounded. To prove the last assertion of the theorem, suppose

lim sup t→∞

_{x(t)=M > M.}¯ _(4.34)

As in the proof ofTheorem 4.1 there are two cases to consider, nonoscillatory and os-cillatory. Assuming the former leads to a contradiction; in the latter case, there exists a sequence {¯_tj}∞

j=1 as before, which would likewise lead to a contradiction, whereby the

conclusion of the theorem holds.

Finally we consider the general nonlinear delay dynamic equation xΔ_{(t)= −}t

where for simplicity we define

b forward to generalizeTheorem 4.1to get the following result.

Theorem4.5. Assume there existsM >0such that for|c| ≤M,

Then the trivial solution of (4.35) is asymptotically stable.

IfTis an isolated time scale, then the delay dynamic equation (4.35) with f1(t,x)=

a(t)δΔ(t)xand fi(t,x)=0, 2≤i≤nbecomes xΔ_{(t)= −}t

τ(t)a(t)δ

Δ_{(t)x(s)Δsg(t,s), (4.37)}

whereτ(t)≤δ(t)< tare delay functions andgis (the Heaviside function) defined by

g(t,s)=

⎧ ⎨ ⎩

0, τ(t)≤s≤δ(t),

1, s > δ(t). (4.38) Then (the Dirac delta function)

gΔs₍t_,s₎₌

⎧ ⎪ ⎨ ⎪ ⎩

1

μ(s), s=δ(t), 0, otherwise.

(4.39)

Using the above we obtain

xΔ_{(t)= −}t τ(t)a(t)δ

Δ_{(t)x(s)Δsg(t,s)}

= −a(t)δΔ(t)

t

τ(t)x(s)g

Δs₍t_,s₎Δs

= −a(t)δΔ(t)

σ(δ(t))

δ(t) x(s)g

Δs₍t_,s₎Δs

= −a(t)δΔ(t)xδ(t)_μδ1 (t)μ

δ(t) = −a(t)δΔ(t)xδ(t).

(4.40)

Acknowledgment

This research was supported by NSF Grant 0354281. References

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Douglas R. Anderson: Department of Mathematics, Concordia College, Moorhead, MN 56562, USA E-mail address:[email protected]

Robert J. Krueger: Department of Mathematics, Concordia University, St. Paul, MN 55104, USA E-mail address:[email protected]

Allan C. Peterson: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA