Oscillation criteria for second order nonlinear neutral dynamic equations with distributed deviating arguments on time scales

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R E S E A R C H

Open Access

Oscillation criteria for second-order nonlinear

neutral dynamic equations with distributed

deviating arguments on time scales

Tuncay Candan

*

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Art and Science, Ni ˘gde University, Ni ˘gde, 51200, Turkey

Abstract

In this article, we establish some new oscillation criteria and give suﬃcient conditions to ensure that all solutions of nonlinear neutral dynamic equation of the form

(

r(t)

((

y(t) +p(t)y

(τ

(t)

))

)

γ

)

+ b

a

f

(

t,y

(δ

(t,

ξ

)

))

ξ

= 0

are oscillatory on a time scaleT, where

γ

≥1 is a quotient of odd positive integers.

Keywords: oscillation; dynamic equations; time scales; distributed deviating arguments

1 Introduction

The aim of this article is to develop some oscillation theorems for a second-order nonlin-ear neutral dynamic equation

r(t)y(t) +p(t)yτ(t)γ+

b

a

ft,yδ(t,ξ)ξ=  ()

on a time scaleT. Throughout this paper, it is assumed thatγ ≥ is a quotient of odd positive integers,  <a<b,τ(t) :T→T, is rd-continuous function such thatτ(t)≤tand

τ(t)→ ∞ast→ ∞,δ(t,ξ) :T×[a,b]→Tis rd-continuous function such that decreasing with respect toξ,δ(t,ξ)≤tforξ∈[a,b],δ(t,ξ)→ ∞ast→ ∞,r(t) >  and ≤p(t) <  are real valued rd-continuous functions deﬁned onT,p(t) is increasing and

(H) ∞

t ( 

r(t)) 

γ_t₌∞_,

(H) f :T×R→Ris a continuous function such thatuf(t,u) > for allu= and there exists a positive functionq(t)deﬁned onTsuch that|f(t,u)| ≥q(t)|uγ_|_.

A nontrivial functiony(t) is said to be a solution of () ify(t) +p(t)y(τ(t))∈C_rd [ty,∞] and r(t)((y(t) +p(t)y(τ(t))))γ _∈_C

rd[ty,∞] forty≥tandy(t) satisﬁes equation () forty≥t.

A solution of (), which is nontrivial for all larget, is called oscillatory if it has no last zero. Otherwise, a solution is called nonoscillatory.

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We note that ifT=R, we haveσ(t) =t,μ(t) = ,y(t) =y(t) and, therefore, () becomes a second-order neutral diﬀerential equation with distributed deviating arguments

r(t)y(t) +p(t)yτ(t) γ +

b

a

ft,yδ(t,ξ)dξ= .

If T=N, we have σ(t) =t+ , μ(t) = , y(t) =y(t) = y(t+ ) –y(t) and therefore () becomes a second-order neutral diﬀerence equation with distributed deviating argu-ments

r(t)y(t) +p(t)yτ(t)γ+

b–

ξ=a

ft,yδ(t,ξ)= 

and ifT=hN,h> , we haveσ(t) =t+h,μ(t) =h,y(t) =hy(t) =y(t+h_h)–y(t) and, there-fore, () becomes a second-order neutral diﬀerence equation with distributed deviating arguments

hr(t)hy(t) +p(t)yτ(t)γ+

b h–

k=a_h

ft,yδ(t,kh)h= .

In recent years, there has been important research activity about the oscillatory be-havior of second-order neutral diﬀerential, diﬀerence and dynamic equations. For ex-ample, Grace and Lalli [] considered the following second-order neutral delay equa-tion

a(t)x(t) +p(t)x(t–τ) +q(t)fx(t–τ)= , t≥t

and Graefet al.[] considered the nonlinear second-order neutral delay equation

y(t) +p(t)yτ(t) +q(t)fy(t–δ)= , t≥t.

Recently, Agarwal et al.[] considered second-order nonlinear neutral delay dynamic equation

r(t)y(t) +p(t)yτ(t)γ+ft,y(t–δ)= . ()

Later, Saker [] considered () but he used diﬀerent technique to prove his results. In [] and [], the authors considered the second order neutral functional dynamic equation of the form

r(t)y(t) +p(t)yτ(t)γ+ft,yδ(t)= ,

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the distributed deviating arguments on time scales. The books [, ] gives time scale calculus and some applications.

2 Main results

Throughout the paper, we use the following notations for simplicity:

x(t) =y(t) +p(t)yτ(t), x[]=rxγ, x[]=x[] ()

andθ(t) =δ(t,a) andθ(t) =δ(t,b).

Theorem . Assume that(H)and(H)hold.In addition,assume that r(t)≥.Then

every solution of()oscillates if the inequality

x[](t) +A(t)x[]θ(t)

≤, ()

where

A(t) = (b–a)q(t)( –p(θ(t)))

γ

r(θ(t))

θ(t)



γ

has no eventually positive solution.

Proof Lety(t) be a nonoscillatory solution of (), without loss of generality, we assume thaty(t) >  fort≥t, theny(τ(t)) >  andy(δ(t,ξ)) >  fort≥t>tandb≥ξ≥a. In the

case wheny(t) is negative, the proof is similar. In view of (), (H) and ()

x[](t) +

b

a

q(t)yγδ(t,ξ)ξ≤ ()

for allt≥t, and we see thatx[](t) is an eventually decreasing function. We claim that

x[]₍_t_{) >  eventually. Assume not then there exists a}_t

≥tsuch thatx[](t) =c< , then

we havex[]₍_t₎_≤_c_for_t_≥_t

and it follows that

x(t)≤

c r(t)

/γ

. ()

Now integrating () fromttotand using (H), we obtain

x(t)≤x(t) +c/γ t

t



r(s)

/γ

s→–∞ ast→ ∞

which contradicts the fact thatx(t) >  for allt≥t. Hence,x[](t) is positive. Therefore,

one sees that there is at≥tsuch that

x(t) > , x(t) > , x[](t) > , x[](t) < , t≥t. ()

Fort≥t≥t, this implies that

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then we conclude that

yγ_δ₍_t_,_ξ₎_≥_{ –}_p_δ₍_t_,_ξ₎γ_xγ_δ₍_t_,_ξ₎_, _t_≥_t

≥t,ξ∈[a,b]. ()

Multiplying () byq(t) and integrating both sides fromatob, we have

b

a

q(t)yγδ(t,ξ)ξ≥

b

a

q(t) –pδ(t,ξ)γxγδ(t,ξ)ξ. ()

Substituting () into (), we obtain

x[](t) +

b

a

q(t) –pδ(t,ξ)γxγδ(t,ξ)ξ≤. ()

On the other hand, we can verify thatx(t)≤ fort≥tand, therefore, we obtain

x(t) =x(t) + t

t

x(s)s≥(t–t)x(t)≥

t

x

₍_t_), _t_≥_t ≥t.

From the last inequality, it can be easily seen that

xδ(t,ξ)≥

δ(t,ξ) 

xδ(t,ξ)≥

θ(t)



xδ(t,ξ), t≥t≥t,ξ∈[a,b].

Substituting the last inequality into (), we have

x[](t) +

b

a

q(t) –pδ(t,ξ)γ

θ(t)



γ

xδ(t,ξ)γξ≤

and it can be found

x[](t) + (b–a)q(t) –pθ(t)

γθ(t)



γ

xθ(t) γ

≤,

or

x[](t) +(b–a)q(t)( –p(θ(t)))

γ

r(θ(t))

θ(t)



γ

x[]θ(t)

≤,

which is the inequality (). As a consequence of this, we have a contradiction and therefore

every solution of () oscillates.

Theorem . Assume that(H)and(H)hold.In addition,assume that r(t)≥,δ(t,ξ)

is increasing with respect to t and that the inequality

lim sup t→∞

t

θ(t)

A(s)s>  ()

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Proof Lety(t) be a nonoscillatory solution of (). We can proceed as in the proof of

The-By making use of (), we reach to a contradiction therefore the proof is complete.

Theorem . Assume that(H)and(H)hold.In addition,assume that r(t)≥,δ(t,ξ)

is increasing with respect to t and there exists a positive rd-continuous-diﬀerentiable functionα(t)such that

Proof Suppose to the contrary thaty(t) is nonoscillatory solution of (). We may assume without loss of generality thaty(t) >  fort≥t, then y(τ(t)) >  andy(δ(t,ξ)) >  for

t≥t>tandb≥ξ≥a. Proceeding as in the proof of Theorem ., we obtain () and the

inequality (). Using () and Pötzsche’s chain rule [, Theorem ], we obtain

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Now using () in (), we obtain

On the other hand, as in the proof of Theorem ., it can be shown that for suﬃciently larget≥t

Substituting () into (), we obtain

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Using the factu–mu_≤ 

Integrating the last inequality fromttot, we obtain

–z(t) <z(t) –z(t)≤–

which contradicts (). Therefore, the proof is complete.

Theorem . Assume that(H)and(H)hold andσ(t)= t for each t∈T.Letα(t),δ(t,ξ),

Proof Following the same lines as in the proof of Theorem ., we get () and (). Using the inequality,

The remaining part of the proof is similar to that of Theorem ., hence it is omitted.

Example . Consider the following second-order neutral nonlinear dynamic equation

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whereγ =_,r(t) = ,p(t) = (t+_t₊a_a–),q(t) =t–/_{. One can verify that the conditions of}

The-orem . are satisﬁed. Note that takingα(s) =s, we see that

lim sup t→∞

t

t

α(s)Q(s) –((α

₍_s₎₎

+)r(θ(s))

γ(θ(s)  )γ–α(s)

s

=lim sup t→∞

t

t

(b–a)s–_–  

(

s–b

 )/s

s=∞.

Therefore, () is oscillatory.

Competing interests

The author declares that they have no competing interests.

Received: 26 April 2012 Accepted: 3 April 2013 Published: 18 April 2013 References

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doi:10.1186/1687-1847-2013-112