Distributions of zeros of solutions to first order delay dynamic equations

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R E S E A R C H

Open Access

Distributions of zeros of solutions to ﬁrst

order delay dynamic equations

D O’Regan

1

_{, SH Saker}

2

_{, AM Elshenhab}

2

_{and RP Agarwal}

3*

*_{Correspondence:} [email protected]

3_{Department of Mathematics, Texas} A&M University, Kingsvilie, TX 78363, USA

Full list of author information is available at the end of the article

Abstract

This paper is concerned with the distributions of zeros of solutions to ﬁrst order delay dynamic equations on time scales. The results are obtained using iterative sequences.

MSC: 26A15; 26D10; 26D15; 39A13; 34A40; 34N05

Keywords: distribution of zeros; delay equations; time scales

1 Introduction

The oscillation and distributions of zeros of solutions of first order delay differential and difference equations are studied widely in the literature; see [–] and the references therein. However, there are only a few papers considering the distribution of zeros of so-lutions of first order delay and advanced dynamic equations on time scales (see [, ]). In [], Zhou considered the first order delay differential equation

x(t) +p(t)x(t–τ) = , fort∈[t,∞), (.)

where

t t–τ

p(s)ds≥ρ>

e, and ρ< , (.)

and established lower and upper bounds for the quotientx(t–τ)/x(t). In particular the author proved thatx(t–τ)/x(t)≥fn(ρ) andx(t–τ)/x(t) <gm(ρ), where the sequences

fn(ρ) andgn(ρ) are deﬁned by

⎧ ⎨ ⎩

f(ρ) =eρ, fn+(ρ) =eρfn(ρ), n= , , . . . ,

g(ρ) =(–_ρρ), gm+(ρ) =(–

ρ)gm(ρ)

gm(ρ)ρ+, m= , , . . . ,

and using these sequences the author studied the distribution of zeros of solutions of (.). In [], Zhang and Zhou considered the ﬁrst order delay diﬀerential equation

x(t) +p(t)xτ(t)= , fort∈[t,∞), (.)

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and studied the distribution of zeros of solutions using the two sequencesfn(ρ) andgm(ρ)

where ⎧ ⎨ ⎩

f(ρ) = , fn+(ρ) =eρfn(ρ), n= , , , . . . ,

g(ρ) =(–_ρρ), gm+(ρ) =(–ρ)g



m(ρ)

gm(ρ)ρ+, m= , , . . . , and

t

τ(t)

p(s)ds≥ρ> , and  <ρ< . (.)

Zhang and Lian in [] initiated the study of the distribution of zeros of dynamic equations on time scales and in particular, they considered the ﬁrst order delay dynamic equation

x(t) +p(t)xτ(t)= , fort∈[t,∞)T, (.)

on a time scaleT, wherep∈Crd(T,R+) is a non-negative rd-continuous function,τ ∈

Crd(T,T) is strictly increasing,τ(t) <tfort∈Tandlimt→∞τ(t) =∞. In [] the authors

established lower and the upper bounds for the quotientx(τ(t))/x(t) using the sequences

fnandgmwhere

f(ρ) = , fn(ρ) =e(–ρ)fn–(ρ), n= , , . . . , (.)

and ⎧ ⎪ ⎨ ⎪ ⎩

g(ρ) =_(–_ρ₎_–ρ_M_(–_ρ₎, gm(ρ) =_(–_ρ₎_–_M_(–ρ_ρ₎₊ 

g_m_–(ρ)

, m= , , . . . , (.)

and whereM< ( –ρ)/ and ≤ρ<  satisﬁes the condition

sup λ∈E

λexp

t

τ(t)

ζμ(s)

–λp(s)s

≤ρ,

whereE={λ:λ> ,  –λp(t)μ(t) > };ζμ(s)andμ(s) will be deﬁned later.

Motivated by these papers, we study the distribution of zeros of oscillatory solutions of the delay dynamic equation (.) on a time scaleTby considering new sequencesfnandgm.

In the next section, we present some basic ideas on time scales. In Section , we establish lower and upper bounds forx(τ(t))/x(t) and in Section , we study the distribution of zeros of solutions of (.).

2 Some preliminaries and lemmas

In this section, we present some preliminaries; see [, ]. A time scaleTis an arbitrary nonempty closed subset of the real numbersR. The forward and backward jump operators are deﬁned by

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withinf∅=supTandsup∅=infT. The graininess functionμon a time scaleTis deﬁned byμ(t) :=σ(t) –t. For a functionf :T→Rthe (delta) derivative is deﬁned by

f(t) =f(σ(t)) –f(t)

σ(t) –t ,

iff is continuous attandtis right-scattered. Iftis not right-scattered then the derivative is deﬁned by

f(t) =lim

s→t

f(t) –f(s)

t–s ,

provided this limit exists. A functionf is said to be-diﬀerentiable if its-derivative exists. A useful formula isfσ₌_f₍_σ₍_t_{)) =}_f₍_t_{) +}_μ₍_t₎_f₍_t_{). We will make use of the following}

product and quotient rules for the derivative of the productfgand the quotientf/g(where

ggσ_{= , and here}_gσ₌_g_◦_σ_{) of two}_{-diﬀerentiable functions}_f _and_g_:

(fg)=fg+fσg=fg+fgσ,

f g

=f

_g_–_fg ggσ .

Letf :R→Rbe continuously differentiable and supposeg:T→Ris delta differentiable. Thenf ◦g:T→Ris delta differentiable and the chain rule,

(f ◦g)₍_t_{) =}





fg(t) +hμ(t)g₍_t₎_dh

g₍_t_), _(.)

holds. A special case of (.) is

xλ(t)=λ





hxσ + ( –h)xλ–x(t)dh.

Fors,t∈T, a functionF:T→Ris called an antiderivative off :T→RprovidedF₌ f(t) holds for allt∈T. In this case we define the integral off by_stf(τ)τ=F(t) –F(s). Fora,b∈T, and a-differentiable functionf, the Cauchy integral off _{is defined by}

b a f

₍_τ₎_τ₌_f₍_b_{) –}_f₍_a_{). The integration by parts formula reads}

b a

f(t)g(t)t=f(t)g(t)b_a– b

a

f(t)gσ(t)t,

and inﬁnite integrals are deﬁned as

∞

a

f(t)t= lim

b→∞

b a

f(t)t.

A functionp:T→Ris called regressive if  +μ(t)p(t)=  fort∈T. A functionp:T→Ris called positively regressive (we writep∈R+) if it is rd-continuous function and satisﬁes  +

μ(t)p(t) >  for allt∈T. Hilger in [] showed that forp(t) rd-continuous and regressive, the solution of the initial value problem

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is given by the generalized exponential functionep(t,t), which is deﬁned by

ep(t,t) =exp

t t

ζμ(s)

p(s)s

,

wheret,t∈T, and the cylinder transformationζh(z) is deﬁned by

ζh(z) =

⎧ ⎨ ⎩

log(+hz)

h , ifh= ,

z, ifh= ,

wherez∈Randh∈R+_.

The next lemma can be found in [].

Lemma . Assume t,t∈T.

(i)For a non-negativeϕwith–ϕ∈R+_,_{we have the following inequality}_:

 – t

t

ϕ(s)s≤e–ϕ(t,t)≤exp – t

t ϕ(s)s

. (.)

(ii)Ifϕis rd-continuous and non-negative,then

 + t

t

ϕ(s)s≤eϕ(t,t)≤exp

t t

ϕ(s)s

. (.)

Lemma . Assume thatTis a time scale with t∈T.If f(t) > on[t,∞)T,then

lnf(t)≤f

₍_t₎

f(t) , for t∈[t,∞)T.

Proof Fixt. We consider two cases: (i)f₍_t₎_≤_{ and (ii)}_f₍_t₎_≥_.

In the ﬁrst case, we see that

hμ(t)f(t) +f(t)≤f(t).

Now recallf(σ(t)) =f(t) +μ(t)f₍_t_{) so}_hμ₍_t₎_f₍_t_{) +}_f₍_t_{) =}_hf₍_σ₍_t_{)) + ( –}_h₎_f₍_t_{) >  and as}

a result



hμ(t)f₍_t_{) +}_f₍_t₎≥



f(t).

Apply the chain rule (.), and we get (notef₍_t₎_≤_)

lnf(t)= 





hμ(t)f₍_t_{) +}_f₍_t₎dh

f₍_t₎_≤f ₍_t₎ f(t) .

In the second case, we see that

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Applying the chain rule (.), we get

lnf(t)= 





hμ(t)f₍_t_{) +}_f₍_t₎dh

f(t)≤f

₍_t₎ f(t) .

Thus, we deduce in both cases that

lnf(t)≤f

₍_t₎ f(t) .

The proof is complete.

Lemma . Assume that Tis a time scale with t∈T.If f(t) > and f(t)≥for t∈ [t,∞)T,then forα> 

e–αf(t)≥–αe–αf(t)f(t).

Proof Sincef(t) >  andf₍_t₎_≥_{ for}_t_∈_[_t

,∞)T, we have forh∈(, )

f(t)≤hμ(t)f(t) +f(t). (.)

Applying the chain rule (.) and using (.), we see that

e–αf(t)= –α





e–α(f(t)+hμ(t)f(t))dh

f(t)≥–αe–αf(t)f(t).

3 Lower and upper bounds forx(

τ

(t))/x(t)

In this section, we establish lower and upper bounds forx(τ(t))/x(t) wherex(t) is a solution of equation (.). We use the notationτ(t) =tand inductively deﬁne the iterates ofτ–i(t) by

τ–i(t) =τ–◦τ–(i–)(t), fori= , , . . . ,

whereτ–(t) is the inverse function ofτ(t). From the deﬁnition it is clear that

τ(t) <t<τ–(t) <· · ·<τ–(n–)(t) <τ–n(t) <· · ·.

To ﬁnd the lower bound forx(τ(t))/x(t) we deﬁne for  <ρ<  a sequencefn(ρ) by

⎧ ⎨ ⎩

f(ρ) = , f(ρ) = /ρ,

fn+(ρ) =_f fn(ρ)

n(ρ)+–e(–ρ)fn(ρ), n= , , , , . . . .

(.)

We note some properties offn(ρ) for the reader’s interest (see [] or use an elementary

argument using x

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so there exists a functionf(ρ) such

lim

n→∞fn(ρ) =f(ρ), ≤f(ρ)≤e,

wheref(ρ) satisﬁes

f(ρ) =e(–ρ)f(ρ)_. _(.)

If ( –ρ) > /e, then eitherfn(ρ) is nondecreasing andlimn→∞fn(ρ) = +∞orfn(ρ) is

nega-tive orfn(ρ) is∞after a ﬁnite numbers of terms.

Theorem . Assume thatTis a time scale and t, t,t∈T, t≥t,t≥τ–(t),x(t)

is a solution of(.)on[t,∞)T,x(t)is positive on[t,t]Tand there existsρ∈(, )with

∞>fn(ρ) > for n∈ {, , . . .}and

sup λ∈E

λexp

t

τ(t)

ζμ(s)

–λp(s)s

≤ρ for t∈τ–(t),t

T; (.)

here E={λ:λ> ,  –λp(t)μ(t) > for t∈[τ–₍_t

),t]T}.Then for n≥whenτ–(+n)(t)≤

twe have

x(τ(t))

x(t) ≥fn(ρ), for t∈

τ–(+n)(t),t

T,

where fn(ρ)is deﬁned in(.).

Proof From (.), we see that

x(t) = –p(t)xτ(t)≤, fort∈τ–(t),t

T, (.)

so sincex(t) is nonincreasing on [τ–₍_t

),t]Twe have

x(τ(t))

x(t) ≥, fort∈

τ–(t),t

T. (.)

Note (.) and the fact thatxis positive on [t,t]T, so fort∈[τ–(t),t]Twe have (note

x(σ(t)) >  sinceσ(t)≥t≥τ–₍_t ) >t)

 = –μ(t)x₍_t_{) +}_p₍_t₎_x_τ₍_t₎

=x(t) –xσ(t)–μ(t)p(t)xτ(t)

<x(t) –μ(t)p(t)x(t)

= –μ(t)p(t)x(t).

Hence  –μ(t)p(t) >  fort∈[τ–(t),t]Tso –p∈R+on the interval [τ–(t),t]T. Using Lemma . (with the time scale [τ–₍_t

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τ(t)∈[τ–₍_t

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=x(t) +xτ(t)

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thusgk(ρ) >_(–_ρ₎_–ρ_M_(–_ρ₎wherek= , , . . . . Then there exists a functiong(ρ) with

lim

m→∞gm(ρ) =g(ρ) =



ρ–(M– ) – (M– )ρ–ρ.

for  <  –ρ≤/e(note (M– ) – (M– )ρ–ρ>  if  <  –ρ≤/e).

Theorem . Assume thatTis a time scale and t,t∈T,t≥t,x(t)is a solution of(.)

on[t,∞)T,there exists a positive integer N≥such that x(t)is positive on[t,τ–N(t)]T

and there existsρ∈(, )with gm(ρ) > for m∈ {, , . . . ,N– }and

sup λ∈E

λexp

t

τ(t)

ζμ(s)

–λp(s)s

≤ρ for t∈τ–(t),τ–N(t)

T, (.)

where E={λ:λ> ,  –λp(t)μ(t) > for t∈[τ–₍_t

),τ–N(t)]T}and

M= sup

s∈[τ–₍_t

),τ–N(t)]T

p(s)μ(s) < –ρ  .

Then for m∈ {, . . . ,N– }we have

x(τ(t))

x(t) <gm(ρ), for t∈

τ–(t),τ–(N–m)(t)

T,

where gm(ρ)is deﬁned in(.).

Proof From (.), we see that

x(t)≤, fort∈τ–(t),τ–N(t)

T, (.)

and as in Theorem . notice  –μ(t)p(t) >  fort∈[τ–(t),τ–N(t)]Tso –p∈R+on the

interval [τ–₍_t

),τ–N(t)]T. From Lemma . (with the time scale [τ–(t),τ–N(t)]T) and

(.), we have fort∈[τ–₍_t

),τ–(N–)(t)]T(noteτ–(t)≤τ–N(t)) t

τ(t)

p(s)s≥ –ρ and

τ–(t)

t

p(s)s≥ –ρ. (.)

Lett∈[τ–₍_t

),τ–(N–)(t)]Tand consider G(r) :=

r t

p(s)s–  +ρ, forr∈t,τ–(t)_T.

NoteG: [t,τ–(t)]→Ris nondecreasing,G(t) = – +ρ< , and

Gτ–(t)= τ–(t)

t

p(s)s–  +ρ≥ –ρ–  +ρ= .

IfG(τ–₍_t_{)) = , then}

τ–(t)

t

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whereas ifG(τ–₍_t_{)) >  then}_G₍_t_{) <  <}_G₍_τ–₍_t_)).

and this together withxbeing nonincreasing on [τ–₍_t

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Repeating the above procedure, we obtain fort∈[τ–₍_t

),τ–(N–m)(t)]T x(τ(t))

x(t) <

(ρ–_g  m–(ρ))

( –ρ)_{– }_M_{( –}_ρ₎:=gm(ρ).

4 Distributions of zeros of solutions

In this section, we study the distribution of zeros of solutions of (.) using the lower and upper bounds forx(τ(t))/x(t) in Section .

Theorem . Assume thatTis a time scale and t,t∈T,t≥t,x(t)is a solution of(.)

on[t,∞)T,and there existρ∈(, )and n,m∈ {, , . . .}with fn(ρ)≥gm(ρ),and with

N=  + min

n≥,m≥

n+m:fn(ρ)≥gm(ρ)

=  +n+m

assume∞>fk(ρ) > ,gk(ρ) > for n∈ {, , . . . ,N– }and

sup λ∈E

λexp

t

τ(t)

ζμ(s)

–λp(s)s

≤ρ for t∈τ–(t),τ–N(t)

T,

where E={λ:λ> ,  –λp(t)μ(t) > for t∈[τ–₍_t

),τ–N(t)]T}and

M= sup

s∈[τ–(t),τ–N(t)]T

p(s)μ(s) < –ρ  .

Then every solution of(.)cannot be totally positive or totally negative on[t,τ–N(t)]T. Proof Note

fn(ρ)≥g_m(ρ). (.)

Without loss of generality assumexis positive on [t,τ–N(t)]T. From Theorem . we

have

x(τ(t))

x(t) ≥fn(ρ), fort∈

τ–(+n)(t),τ–N(t)

T

and from Theorem . we have (notem₌_N_{– ( +}_n₎_≤_N_{– )}

x(τ(t))

x(t) <gm(ρ), fort∈

τ–(t),τ–(N–m

₎

(t)

T.

Note sinceN=  +n₊_m_{we have (take}_t₌_τ–(N–m₎

(t))

fn(ρ)≤x(

τ–(+n)₍_t ))

x(τ–(+n₎

(t))

<gm(ρ),

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Theorem . Assume thatTis a time scale and t,t∈T,t≥t,x(t)is a solution of(.)

on[t,∞)T,and there existρ∈(, )and a positive integer N≥and m∈ {, , . . . ,N– }

with

tm τ(tm_)

p(s)s>  – 

gm(ρ)

where tm=τ

–(N–m)₍_t

),

and with

m= min

m∈{,...,N–} m: tm

τ(tm)

p(s)s>  – 

gm(ρ)

where tm=τ–(N–m)(t)

assume∞>fk(ρ) > ,gk(ρ) > for n∈ {, , . . . ,N– }and

sup λ∈E

λexp

t

τ(t)

ζμ(s)

–λp(s)s

≤ρ for t∈τ–(t),τ–N(t)

T,

where E={λ:λ> ,  –λp(t)μ(t) > for t∈[τ–(t),τ–N(t)]T}and

M= sup

s∈[τ–₍_t

),τ–N(t)]T

p(s)μ(s) < –ρ  .

Then every solution of(.)cannot be totally positive or totally negative on[t,τ–N(t)]T.

Proof Note

tm

τ(tm)

p(s)s>  – 

gm(ρ) wheretm

=τ–(N–m ₎

(t). (.)

Without loss of generality assumexis positive on [t,τ–N(t)]T. From Theorem ., we

have

x(τ(t))

x(t) <gm(ρ), fort∈

τ–(t),τ–(N–m

₎

(t)

T,

so in particular

x(τ(tm)) x(tm) <gm

(ρ). (.)

Integrating (.) fromτ(tm) tot_m, we obtain

xτ(tm)–x(t_m) =

tm

τ(tm)

p(s)xτ(s)s≥xτ(tm)

tm

τ(tm)

p(s)s,

and this together with (.) gives

tm τ(tm)

p(s)s≤ – x(tm)

x(τ(tm))≤ –



gm(ρ),

(14)

Theorem . Assume thatTis a time scale and t,t∈T,t≥t,x(t)is a solution of(.)

and from Theorem ., we have

(15)

From (.) andfn∗(ρ) >fn∗–(ρ) we have

(16)

Remark . WhenT=Requation (.) is the delay diﬀerential equation

x(t) +p(t)xτ(t)= , t∈R.

Theorem . and Theorem . are related to the results in [], Lemma . and Lemma ., and Theorem . is motivated from results in [], Theorem .

Acknowledgements

The authors are grateful to the anonymous referees and the editor for their careful reading, valuable comments and correcting some errors, which have greatly improved the quality of the paper.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have contributed equally to this manuscript. They read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland.}2_Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt. 3_{Department of Mathematics, Texas A&M} University, Kingsvilie, TX 78363, USA.

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Received: 26 April 2017 Accepted: 29 June 2017 References

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