3. Asymptotic behavior and oscillation of solutions of third order neutral dynamic equations with distributed deviating arguments

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ISSN: 1821-1291, URL: http://bmathaa.org Volume 10 Issue 2(2018), Pages 31-52.

ASYMPTOTIC BEHAVIOR AND OSCILLATION OF SOLUTIONS OF THIRD ORDER NEUTRAL DYNAMIC EQUATIONS WITH

DISTRIBUTED DEVIATING ARGUMENTS

ORHAN ¨OZDEMIR∗AND ERCAN TUNC¸

Abstract. The authors obtain some new results on the asymptotic proper-ties of solutions of a third order nonlinear neutral dynamic equation with dis-tributed deviating arguments on an arbitrary time scaleT. Several examples are provided to illustrate the results.

1. Introduction

This article deals with the oscillation and asymptotic behavior of solutions of the third order nonlinear neutral dynamic equation with distributed deviating ar-guments

r2(t)

r1(t) z∆(t)

α1∆

α2∆

+

Z b

a

q(t, ξ)f(x(φ(t, ξ))) ∆ξ= 0, t∈[t0,∞)T, (1.1) where _T is a time scale unbounded above with t0 ∈T, z(t) = x(t) +p(t)x(g(t)),

αi are quotients of positive odd integers fori= 1,2, and a, b∈Rwith 0< a < b. Standard notation and terminology for equations on time scales such as that found in [9, 10] will be used here, and we assume that the following conditions hold without further mention:

(C1) ri∈Crd([t0,∞)T,R

+_{) and}

Z ∞

t0 1 (ri(t))1/αi

∆t=∞, i= 1,2;

(C2) p∈Crd([t0,∞)T,R) withp(t)≥1, andp(t)6≡1 eventually;

(C3) g∈Crd([t0,∞)T,T) is strictly increasing,g(t)< t, and limt→∞g(t) =∞;

(C4) q(t, ξ) ∈ Crd([t0,∞)T×[a, b]T,[0,∞)), φ(t, ξ) ∈ Crd([t0,∞)T×[a, b]T,T) is non-increasing with respect toξ, and

lim t→∞ξmin∈[a,b]

φ(t, ξ) =∞;

2010Mathematics Subject Classification. 34K11, 34K40, 34N05, 39A21.

Key words and phrases. Oscillation, asymptotic behavior, third-order, neutral dynamic equa-tions, distributed deviating arguments.

c

2018 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted April 27, 2018. Published May 28, 2018. Communicate by Tongxing Li.

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(C5) f ∈C(R,R) satisfies uf(u)>0 foru6= 0 and there exist constantsκ >0 andβ=α1α2 such thatf(u)/uβ≥κforu6= 0.

The cases

g(σ(t))≥φ(t, ξ), ξ∈[a, b], (1.2) and

g(σ(t))≤φ(t, ξ), ξ∈[a, b], (1.3) are both considered, respectively.

For notational purposes, we let

z[1](t) :=r1(t)

z∆(t)α1

and z[2](t) :=r2(t)

z[1](t)

∆α2

.

By a solution of (1.1) we mean a nontrivial real valued functionx∈C_rd1 ([tx,∞)_T,R),

tx∈[t0,∞)T, which has the propertiesz∈Crd1 ([tx,∞)_T,R),z[1]∈Crd1 ([tx,∞)_T,R),

z[2]_∈_C1

rd([tx,∞)_T,R), and satisfies (1.1) on [tx,∞)_T. Our attention is restricted to those solutions of (1.1) which exist on some half line [tx,∞)_Tand satisfy sup{|x(t)|:

t ∈ [T1,∞)T} > 0 for any T1 ∈[tx,∞)T. Moreover, we tacitly assume that (1.1) possesses such solutions. Such a solution is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory.

The oscillation and asymptotic behavior of solutions for different classes of neu-tral differential equations and neuneu-tral dynamic equations on time scales is an active and important area of research, and we refer the reader to the papers ([1], [7], [8], [11]-[14], [16]-[19], [22]-[26], [29]-[39]) as examples of recent results on this topic. However, oscillation and asymptotic behavior results for third order neutral dy-namic equations with distributed deviating arguments are not very prevalent in the literature, and most of the literature for dynamic equations of type (1.1) is devoted to the cases where 0≤p(t)≤p0<1 and/or 0≤p(t)≡

Rb

a p(t, η) ∆η ≤p0<1; see, e.g., ([12], [17], [22]) and the references cited therein.

To the best of our knowledge, there are few such results for third order neutral dynamic equations with distributed deviating arguments of type (1.1) in the case where p(t)≥1, see, e.g, ([25], [33]), where the results obtained are for the special case T = R. Motivated by the papers mentioned above and (see also, [2]-[6], [15], [21], [27]), we shall establish some new sufficient conditions which guarantee that any solutionx(t) of (1.1) either oscillates or converges to zero ast→ ∞on an arbitrary time scaleTin the case whenp(t)≥1, and moreover, the results obtained here can easily be extended to more general third order neutral dynamic equations with distributed deviating arguments. It is therefore hoped that the present paper will contribute significantly to the growing body of research on third order neutral dynamic equations with distributed deviating arguments.

2. Some Preliminary Lemmas

We begin with the some preliminary lemmas that are essential in the proofs of our theorems. It will be convenient to employ the following notations:

φ1(t) :=φ(t, a), φ2(t) :=φ(t, b), d+(t) := max (0, d(t)),

λ= α2+ 1

α2

, R1(t, t1) :=

Z t

t1 ∆s r1/α2

2 (s)

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R2(t, t2) :=

Z t

t2

_R

1(s, t1)

r1(s)

1/α1

∆s f or t≥t2≥t1.

Throughout this paper, we assume that

ϕ1(t) :=

1

p(g−1₍_t₎₎

1− 1

p(g−1₍_g−1₍_t₎₎₎

>0 (2.1)

and

ϕ2(t) :=

1

p(g−1₍_t₎₎ 1−

1

p(g−1₍_g−1₍_t₎₎₎

R2 g−1 g−1(t)

, t2

R2(g−1(t), t2)

!

>0 (2.2)

for all sufficiently larget, where g−1_{denotes the inverse function of}_g_{, and we let}

q1(t) :=

Z b

a

q(t, ξ) (ϕ1(φ(t, ξ)))

β

∆ξ, q2(t) :=

Z b

a

q(t, ξ) (ϕ2(φ(t, ξ)))

β ∆ξ,

ψ(t) =

δ(t), if 0< α2≤1

δα2₍_t₎_, _if _α

2>1

, andδ(t) = R1(t, t1)

R1(σ(t), t1)

.

Lemma 2.1. Let x(t) be an eventually positive solution of (1.1). Then z(t) only satisfies the following two cases, fort sufficiently large,

(I) z(t)>0,z∆₍_t₎_>₀_, _z[1]₍_t₎∆

>0 and z[2]₍_t₎∆ <0,

(II) z(t)>0,z∆₍_t₎_<₀_, _z[1]₍_t₎∆

>0 and z[2]₍_t₎∆ <0.

The proof of the above lemma is standard; we omit its proof.

Lemma 2.2. Assume (2.1) and letx(t)be an eventually positive solution of (1.1) withz(t)satisfying case (II) of Lemma 2.1. If

Z ∞

t0 1

r1(v)

Z ∞

v

₁ r2(u)

Z ∞

u

q1(s) ∆s

1/α2 ∆u

!1/α1

∆v=∞, (2.3)

then the solution x(t)converges to zero ast→ ∞.

Proof. Let x(t) be an eventually positive solution of (1.1). Then, there exists a

t1 ∈[t0,∞)_T such thatx(t)>0, x(g(t))> 0 and x(φ(t, ξ))> 0 for t ≥t1 and

ξ∈[a, b]. From the definition ofz(t), we have, (see also (8.6) in [1]),

x(t) = 1

p(g−1₍_t₎₎ z g

−1₍_t₎

−x g−1(t)

= z g

−1₍_t₎ p(g−1₍_t₎₎ −

z g−1 _g−1₍_t₎

−x g−1 _g−1₍_t₎ p(g−1₍_t₎₎_p₍_g−1₍_g−1₍_t₎₎₎

≥ z g −1₍_t₎ p(g−1₍_t₎₎ −

z g−1 g−1(t)

p(g−1₍_t₎₎_p₍_g−1₍_g−1₍_t₎₎₎. (2.4)

Sinceg(t)< tandz(t) is decreasing, we have

z g−1(t)

≥z g−1 g−1(t) .

Substituting this into (2.4) gives

x(t)≥ϕ1(t)z g−1(t) for t≥t1. (2.5)

From (C4), we can chooset2≥t1 such thatφ(t, ξ)≥t1 for allt≥t2andξ∈[a, b].

Hence, from (2.5) we obtain

x(φ(t, ξ))≥ϕ1(φ(t, ξ))z g−1(φ(t, ξ))

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Now, from (2.6), conditions (C4)-(C5) and the fact thatz(t) is decreasing, equation (1.1) can be written as

z[2](t)

∆

+κq1(t)zβ g−1(φ1(t))

≤0 for t≥t2. (2.7)

Sincez(t)>0 andz∆(t)<0, there exists a constantLsuch that lim

t→∞z(t) =L <∞,

whereL≥0. If L >0 then there existst3≥t2 such thatg−1(φ1(t))> t2 and

z(t)≥L for t≥t3.

Integrating (2.7) two times fromt to∞gives,

−z∆(t)≥γ 1 r1(t)

Z ∞

t

₁ r2(u)

Z ∞

u

q1(s) ∆s

1/α2 ∆u

!1/α1

whereγ >0 is a constant. An integration of the last inequality fromt3to∞yields

z(t3)≥γ

Z ∞

t3 1

r1(v)

Z ∞

v

₁

r2(u)

Z ∞

u

q1(s) ∆s

1/α2 ∆u

!1/α1 ∆v,

which contradicts (2.3) and so we have L = 0. Thus, limt→∞z(t) = 0. From

the fact that 0< x(t)≤z(t) on [t1,∞)_T, we conclude that limt→∞x(t) = 0 and

completes the proof.

Lemma 2.3. Assume (2.2) and letx(t)be an eventually positive solution of (1.1) with z(t) satisfying case (I) of Lemma 2.1. Then z(t) satisfies the following in-equality

z[2](t)

∆

+κq2(t)zβ g−1(φ2(t))

≤0, (2.8)

for sufficiently larget.

Proof. Let x(t) be an eventually positive solution of (1.1) such that x(t) > 0,

x(g(t))>0,x(φ(t, ξ))>0 and z(t) satisfies case (I) of Lemma 2.1 fort ≥t1 ∈

[t0,∞)_T and ξ∈[a, b]. Proceeding as in the proof of Lemma 2.2, we again arrive

at (2.4). Since

z[1](t) = z[1](t1) +

Z t

t1

z[2](s)1/α2

r1/α2

2 (s)

∆s (2.9)

andz[2](t) is decreasing, we see that

z[1](t) ≥ z[2](t)

1/α2Z t

t1 1

r1/α2

2 (s)

∆s (2.10)

or

z[1](t)≥z[2](t)

1/α2

R1(t, t1) (2.11)

fort≥t1. Thus,

_z[1]₍_t₎

R1(t, t1)

∆

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Hence there exists at2∈[t1,∞)_T such that

z(t) = z(t2) +

Z t

t2

z[1](s)

R1(s, t1)

1/α1

R1(s, t1)

r1(s)

1/α1 ∆s

≥ R2(t, t2) R1/α1

1 (t, t1)

z[1](t)

1/α1

, (2.13)

which implies that

_z₍_t₎ R2(t, t2)

∆

≤0 for t≥t2. (2.14)

From (2.14) and the fact thatg−1₍_t₎_{< g}−1 _g−1₍_t₎

, we obtain

R2 g−1 g−1(t), t2

R2(g−1(t), t2)

z g−1(t)

≥z g−1 g−1(t)

. (2.15)

Using (2.15) in (2.4) gives

x(t)≥ϕ2(t)z g−1(t) for t≥t2. (2.16)

Since lim t→∞ξmin∈[a,b]

φ(t, ξ) =∞, we can choose at3≥t2 such that φ(t, ξ)≥t2 for all

t≥t3, and hence, from (2.16) we have

x(φ(t, ξ))≥ϕ2(φ(t, ξ))z g−1(φ(t, ξ))

for t≥t3. (2.17)

Substituting (2.17) into (1.1) gives (2.8) and completes the proof.

Lemma 2.4. [20] IfD andE are nonnegative andλ >1, then

λDEλ−1−Dλ≤(λ−1)Eλ,

where equality holds if and only if D=E.

Lemma 2.5. [9, p. 259, Theorem 6.13] Let a, b ∈ T and a < b. Then for

rd-continuous functionsf, g: [a, b]

T→R, we have

Z b

a

|f(t)g(t)|∆t≤ Z b

a

|f(t)|p∆t !1/p

Z b

a

|g(t)|q∆t !1/q

,

wherep >1and1/p+ 1/q= 1.

3. _{Main results}

Theorem 3.1. Assume (1.2) and (2.1)-(2.3). If there exists a positive function

η∈C1

rd([t0,∞)T,R)such that

lim sup t→∞

Z t

T

χ1(s)−

η∆ +(s)

Rα2

1 (s, t1)

∆s=∞ (3.1)

for all sufficiently large t1∈[t0,∞)_T, where

χ1(t) =κη(σ(t))q2(t)

Rβ₂ g−1₍_φ

2(t)), t2

Rα2

1 (σ(t), t1)

,

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Proof. Let x(t) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that there exists t1 ∈ [t0,∞)_T such that x(t) >0, x(g(t)) >0,

x(φ(t, ξ)) > 0, (2.1)-(2.2) hold and z(t) satisfies either case (I) or case (II) of Lemma 2.1 fort ≥t1 and ξ∈ [a, b]. If case (II) holds, from Lemma 2.2, we have

limt→∞x(t) = 0.

Next, assume that case (I) holds. Proceeding as in the proof of Lemma 2.3, we again arrive at (2.8)-(2.14). Define Riccati-type substitution by

ω(t) =η(t) z

[2]₍_t₎

z[1]₍_t₎α2 for t≥t1. (3.2) Clearly,ω(t)>0, and from (2.8) and (3.2) we obtain

ω∆(t) ≤ η+∆(t)

z[2](t)

z[1]₍_t₎α2 −κη(σ(t))q2(t)

zβ g−1(φ2(t))

zβ₍_σ₍_t₎₎

zβ(σ(t))

z[1]₍_σ₍_t₎₎α2

−η(σ(t))

z[1]₍_t₎α2∆

z[2]₍_t₎

z[1]₍_t₎α2

z[1]₍_σ₍_t₎₎α2. (3.3)

From the conditions (C3),(C4) and (1.2), we have

g−1(φ2(t))≤σ(t),

which together with (2.14) gives

z g−1₍_φ 2(t))

z(σ(t)) ≥

R2 g−1(φ2(t)), t2

R2(σ(t), t2)

. (3.4)

By the virtue of (2.13) and the fact thatt≤σ(t), we have

(z(σ(t)))β

z[1]₍_σ₍_t₎₎α2 ≥

Rβ₂(σ(t), t2)

Rα2

1 (σ(t), t1)

. (3.5)

Using (3.4) and (3.5) in (3.3), we obtain

ω∆(t) ≤ η∆₊(t) z

[2]₍_t₎

z[1]₍_t₎α2 −χ1(t)−η(σ(t))

z[1](t)α2

∆

z[2](t)

z[1]₍_t₎α2

z[1]₍_σ₍_t₎₎α2(3.6). From ([9], Theorem 1.90), we have

z[1](t)

α2∆ ≥

(

α2 z[1](σ(t))

α2−1

z[1]₍_t₎∆

, if 0< α2≤1,

α2 z[1](t)

α2−1

z[1](t)∆, if α2>1.

(3.7)

If 0< α2≤1, from (3.6) and (3.7) we arrive at

ω∆(t) ≤ η₊∆(t) z

[2]₍_t₎

z[1]₍_t₎α2 −χ1(t)−

α2η(σ(t))

r1/α2

2 (t)

z[2]₍_t₎λ z[1]₍_t₎α2+1

z[1]₍_t₎

z[1]₍_σ₍_t₎₎(3.8).

Ifα2>1, from (3.6) and (3.7) we arrive at

ω∆(t) ≤ η₊∆(t) z

[2]₍_t₎

z[1]₍_t₎α2 −χ1(t)−

α2η(σ(t))

r1/α2

2 (t)

z[2]₍_t₎λ z[1]₍_t₎α2+1

z[1]₍_t₎α2

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By the fact thatt≤σ(t), it follows from (2.12) that

z[1](t)

z[1]₍_σ₍_t₎₎ ≥

R1(t, t1)

R1(σ(t), t1)

. (3.10)

In view of (3.10), combining (3.8) and (3.9) yields, forα2>0 andt≥t3,

ω∆(t) ≤ η∆₊(t) z

[2]₍_t₎

z[1]₍_t₎α2 −χ1(t)−

α2η(σ(t))ψ(t)

r1/α2

2 (t)

z[2]₍_t₎λ

z[1]₍_t₎α2+1.(3.11) From (2.11), we have

z[2](t)

z[1]₍_t₎α2 ≤ 1

Rα2

1 (t, t1)

. (3.12)

Hence, from (3.12),z[1]₍_t₎_>_{0 and}_z[2]₍_t₎_>_{0, inequality (3.11) takes the form}

ω∆(t)≤ −χ1(t) +

η∆ +(t)

Rα2

1 (t, t1)

. (3.13)

An integration of (3.13) fromt3 tot yields

Z t

t3

χ1(s)−

η∆ +(s)

Rα2

1 (s, t1)

∆s≤ω(t3) (3.14)

which contradicts (3.1) and completes the proof.

η∈C_rd1 ([t0,∞)T,R)such that

lim sup t→∞

Z t

T

χ1(s)−

1

(α2+ 1)

α2+1

r2(s) η∆+(s)

α2+1

(η(σ(s))ψ(s))α2

!

∆s=∞ (3.15)

for all sufficiently large t1 ∈[t0,∞)_T, where χ1(t)is as in Theorem 3.1 and T >

t2≥t1, then any solution of equation (1.1) either oscillates or converges to zero as

t→ ∞.

x(φ(t, ξ)) > 0, (2.1)-(2.2) hold and z(t) satisfies either case (I) or case (II) of Lemma 2.1 fort≥t1 andξ∈[a, b]. If case (II) holds, we have limt→∞x(t) = 0 by

Lemma 2.2.

Assume that case (I) holds. Proceeding exactly as in the proof of Theorem 3.1, we again arrive at (3.11). In view of (3.2), inequality (3.11) takes the form

ω∆(t)≤ η

∆ +(t)

η(t) ω(t)−χ1(t)−

α2η(σ(t))ψ(t)

ηλ₍_t₎_r1/α2

2 (t)

ωλ(t). (3.16)

If we apply Lemma 2.4 with

D=[α2η(σ(t))ψ(t)]

1/λ

h r1/α2

2 (t)ηλ(t)

i1/λω(t) and E= 

 

α2

α2+ 1

h r1/α2

2 (t)ηλ(t)

i1/λ

[α2η(σ(t))ψ(t)] 1/λ

η∆ +(t)

η(t)



 

α2

,

we see that

η+∆(t)

η(t) ω(t)−

α2η(σ(t))ψ(t)

ηλ₍_t₎_r1/α2

2 (t)

ωλ(t)≤ 1

(α2+ 1)α2+1

r2(t) η+∆(t)

α2+1

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Using (3.17) in (3.16) gives

ω∆(t)≤ 1

(α2+ 1)

α2+1

r2(t) η+∆(t)

α2+1

[η(σ(t))ψ(t)]α2 −κη(σ(t))q2(t)

R₂β g−1₍_φ

2(t)), t2

Rα2

1 (σ(t), t1)

.

Integrating the last inequality fromt3 tot yields

Z t

t3

χ1(s)−

1

(α2+ 1)

α2+1

r2(s) η∆+(s)

α2+1

[η(σ(s))ψ(s)]α2

!

∆s≤ω(t3),

Theorem 3.3. Let α2 ≥ 1 and assume (1.2) and (2.1)-(2.3). If there exists a positive function η∈C1

lim sup t→∞

Z t

T

χ1(s)−

r1/α2

2 (s) η ∆ +(s)

2

4α2η(σ(s))ψ(s) [R1(s, t1)]

α2−1

!

∆s=∞ (3.18)

t→ ∞.

x(φ(t, ξ)) > 0, (2.1)-(2.2) hold and z(t) satisfies either case (I) or case (II) of Lemma 2.1 fort≥t1andξ∈[a, b]. If case (II) holds, we again have limt→∞x(t) =

0.

Next, suppose that case (I) holds. Proceeding exactly as in the proof of Theorem 3.2, we again arrive at (3.16) which can be written as

ω∆(t)≤η

∆ +(t)

η(t) ω(t)−χ1(t)−

α2η(σ(t))ψ(t) (ω(t))

1 α2−1

ηλ₍_t₎_r1/α2

2 (t)

ω2(t), (3.19)

fort≥t3. From (3.2) and (2.11),

(ω(t))α12−1 = (η(t)) 1 α2−1

z[2](t) 1 α2−1

z[1]₍_t₎1−α2

= (η(t))α1₂−1 z

[1]₍_t₎

z[2]₍_t₎1/α2

!α2−1

≥ (η(t))α12−1(R₁(t, t₁))α2−1. (3.20)

Using (3.20) in (3.19), we conclude that

ω∆(t) ≤ η

∆ +(t)

η(t) ω(t)−χ1(t)−

α2η(σ(t))ψ(t) (R1(t, t1))

α2−1

η2₍_t₎_r1/α2

2 (t)

ω2(t)(3.21)

fort≥t3. Completing square with respect toω, it follows from (3.21) that

ω∆(t)≤ −χ1(t) +

r1/α2

2 (t) η+∆(t)

2

4α2η(σ(t))ψ(t) [R1(t, t1)]

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Integrating this inequality fromt3 tot yields

Z t

t3

χ1(s)−

r1/α2

2 (s) η∆+(s)

2

4α2η(σ(s))ψ(s) [R1(s, t1)]

α2−1

!

∆s≤ω(t3),

which contradicts (3.18). The proof is complete.

Next, we present three results for the case when (1.3) holds.

η∈C1

lim sup t→∞

Z t

T

χ2(s)−

η∆ +(s)

Rα2

1 (s, t1)

∆s=∞ (3.23)

for all sufficiently large t1∈[t0,∞)_T, where

χ2(t) =κη(σ(t))q2(t)

Rβ₂(σ(t), t2)

Rα2

1 (σ(t), t1)

,

and T > t2 ≥ t1, then any solution of (1.1) either oscillates or tends to zero as

t→ ∞.

Lemma 2.2.

Next, assume that case (I) holds. Proceeding as in the proof of Theorem 3.1, we again arrive at (3.3) and (3.5). Since

σ(t)≤g−1(φ2(t)), (3.24)

fromz∆₍_t₎_>_{0 we see that}

z g−1(φ2(t))

z(σ(t)) ≥1. (3.25)

Using (3.25) and (3.5) in (3.3), we obtain

ω∆(t)≤η₊∆(t) z

[2]₍_t₎

z[1]₍_t₎α2 −χ2(t)−η(σ(t))

z[1](t)α2

∆

z[2](t)

z[1]₍_t₎α2

z[1]₍_σ₍_t₎₎α2. (3.26) The remainder of the proof is similar to that of Theorem 3.1, and so the details are

omitted.

η∈C_rd1 ([t0,∞)T,R)such that

lim sup t→∞

Z t

T

χ2(s)−

1

(α2+ 1)α2+1

r2(s) η+∆(s)

α2+1

(η(σ(s))ψ(s))α2

!

∆s=∞ (3.27)

t→ ∞.

(10)

Theorem 3.6. Let α2 ≥ 1 and assume (1.3) and (2.1)-(2.3). If there exists a positive function η∈C_rd1 ([t0,∞)T,R)such that

lim sup t→∞

Z t

T

χ2(s)−

r1/α2

2 (s) η ∆ +(s)

2

4α2η(σ(s))ψ(s) [R1(s, t1)]

α2−1

!

∆s=∞ (3.28)

t→ ∞.

The above theorem follows from (3.25) and Theorem 3.3; so its proof is omitted. The following sequel gives Philos-type oscillation criteria for equation (1.1). First, we need to introduce the class of functions P which will be used in the sequel.

Let D0 ≡ {(t, s) ∈ T2 : t > s ≥ t0}, D ≡ {(t, s) ∈ T2 : t ≥ s ≥ t0} and

H, h∈Crd(D,R). The functionH∈Crd(D,R) is said to belongs to the classP if

(i) H(t, t) = 0 fort≥t0andH(t, s)>0 onD0,

(ii) H has a nonpositive rd-continuous ∆-partial derivative H∆s₍_{t, s}_{) on} _D

0

with respect to second variable and satisfies

H∆s₍_{t, s}_{) +}_H₍_{t, s}₎ η

∆₍_s₎

η(σ(s)) =

h(t, s)

η(σ(s))H

1/λ (t, s),

where the functionη is as in Theorem 3.1.

Theorem 3.7. Assume (1.2) and (2.1)-(2.3). Suppose also that there exist func-tions η ∈C1

rd([t0,∞)T,R

+₎ _and_{H, h} _∈_C

rd(D,R) with H belongs to the classP

such that

lim sup t→∞

1

H(t, t∗) Z t

t∗

"

H(t, s)χ3(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)α2+1(η(s)ψ(s))α2

#

∆s=∞,

(3.29)

where

χ3(t) =κη(t)q2(t)

Rβ₂ g−1₍_φ

2(t)), t2

Rα2

1 (σ(t), t1)

,

andt∗> t2≥t1 for sufficiently large t1∈[t0,∞)T , then any solution of equation

(1.1) is either oscillatory or converges to zero ast→ ∞.

Lemma 2.2.

(11)

arrive at

ω∆(t) = η(t)

z[1]₍_t₎α2

z[2](t)

∆

+ η(t)

z[1]₍_t₎α2

!∆

z[2](σ(t))

≤ −κη(t)q2(t)

Rβ₂ g−1₍_φ

2(t)), t2

Rα2

1 (σ(t), t1)

+η

∆₍_t₎_ω₍_σ₍_t₎₎

η(σ(t))

−η(t)

z[2](σ(t)) z[1](t)α2

∆

z[1]₍_t₎α2

z[1]₍_σ₍_t₎₎α2 fort≥t3. (3.30)

If 0< α2≤1, from (3.7) and (3.30) we see that

ω∆(t)≤ −χ3(t) +

η∆₍_t₎_ω₍_σ₍_t₎₎

η(σ(t)) −α2η(t)

z[2]₍_σ₍_t₎₎ _z[1]₍_t₎∆ z[1]₍_t₎α2

z[1]₍_σ₍_t₎₎. (3.31)

Ifα2>1, from (3.7) and (3.30) we see that

ω∆(t)≤ −χ3(t) +

η∆₍_t₎_ω₍_σ₍_t₎₎

η(σ(t)) −α2η(t)

z[2]₍_σ₍_t₎₎ _z[1]₍_t₎∆

z[1]₍_t₎ _z[1]₍_σ₍_t₎₎α2. (3.32)

Using the fact that z[1](t) is increasing and z[2](t) is decreasing, we get z[1](t)≤ z[1]₍_σ₍_t_{)) and} _z[1]₍_t₎∆

≥ z[2]₍_σ₍_t₎₎1/α2

/(r2(t))

1/α2_{, respectively.}

Thus, (3.31) and (3.32) can be written as

ω∆(t)≤ −χ3(t) +

η∆₍_t₎_ω₍_σ₍_t₎₎

η(σ(t)) −

α2η(t) z[2](σ(t))

λ

r1/α2

2 (t) z[1](σ(t))

α2+1

z[1]₍_t₎

z[1]₍_σ₍_t₎₎ (3.33)

and

ω∆(t)≤ −χ3(t) +

η∆(t)ω(σ(t))

η(σ(t)) −

α2η(t) z[2](σ(t))

λ

r1/α2

2 (t) z[1](σ(t))

α2+1

z[1](t)α2

z[1]₍_σ₍_t₎₎α2,(3.34)

respectively.

Combining (3.33) and (3.34) and using (3.10), we obtain, forα2>0 andt≥t3,

ω∆(t)≤ −χ3(t) +

η∆₍_t₎_ω₍_σ₍_t₎₎

η(σ(t)) −

α2η(t)ψ(t)

r1/α2

2 (t)

ωλ₍_σ₍_t₎₎

ηλ₍_σ₍_t₎₎, (3.35)

and hence, in view of (i) and (ii), fort≥T ≥t3, we have

Z t

T

H(t, s)χ3(s) ∆s ≤ −

Z t

T

H(t, s)ω∆(s) ∆s+

Z t

T

H(t, s) η

∆₍_s₎

η(σ(s))ω(σ(s)) ∆s

− Z t

T

H(t, s)α2η(s)ψ(s)

r1/α2

2 (s)

ωλ₍_σ₍_s₎₎

(12)

Using integrating by parts formula on time scales, (3.36) yields

Z t

T

H(t, s)χ3(s) ∆s≤H(t, T)ω(T) +

Z t

T

H∆s₍_{t, s}₎_ω₍_σ₍_s_{)) ∆}_s

+

Z t

T

H(t, s)η

∆₍_s₎_ω₍_σ₍_s₎₎

η(σ(s)) ∆s−

Z t

T

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s

≤H(t, T)ω(T) +

Z t

T

h+(t, s)

η(σ(s))H

1/λ₍_{t, s}₎_ω₍_σ₍_s_{)) ∆}_s

− Z t

T

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s. (3.37)

Applying Lemma 2.4 with

D= [α2η(s)ψ(s)H(t, s)]

1/λ

ω(σ(s))

h r1/α2

2 (s)ηλ(σ(s))

i1/λ andE= 

 

α2

α2+ 1

h r1/α2

2 (s)ηλ(σ(s))

i1/λ

[α2η(s)ψ(s)] 1/λ

h+(t, s)

η(σ(s))



 

α2

,

we obtain,

h+(t, s)

η(σ(s))H

1/λ₍_{t, s}₎_ω₍_σ₍_s₎₎₋_H₍_{t, s}₎_α

2η(s)ψ(s)

1

r1/α2

2 (s)

ωλ(σ(s))

ηλ₍_σ₍_s₎₎

≤ 1

(α2+ 1)

α2+1

r2(s) (h+(t, s))α2+1

(η(s)ψ(s))α2 . (3.38)

Substituting (3.38) into (3.37) gives

Z t

T

"

H(t, s)χ3(s)−

r2(s) (h+(t, s))α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

∆s≤H(t, T)ω(T)(3.39).

So, for everyt≥t3, we have

Z t

t3

"

H(t, s)χ3(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

∆s≤H(t, t3)ω(t3),

which contradicts (3.29). The proof is complete.

Theorem 3.8. Assume (1.2) and (2.1)-(2.3). Let H andhbe as in Theorem 3.7 and suppose that

0< inf s≥t0

lim inf t→∞

H(t, s)

H(t, t0)

≤ ∞. (3.40)

Suppose also that there exist a positive function η ∈C_rd1 ([t0,∞)T,R) and Ψ (t)∈

Crd([t0,∞)_T,R)such that

lim sup t→∞

1

H(t, t∗) Z t

t∗

r2(s) (h+(t, s))

α2+1

(η(s)ψ(s))α2 ∆s <∞, (3.41)

lim sup t→∞

1

H(t, T)

Z t

T

"

H(t, s)χ3(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

(13)

forT ≥t∗, and

Z ∞

t∗

η(s)ψ(s)

r1/α2

2 (s)ηλ(σ(s))

Ψλ₊(σ(s)) ∆s=∞, (3.43)

whereχ3(t)is as in Theorem 3.7 andt∗> t2≥t1for sufficiently larget1∈[t0,∞)T.

Then any solution of equation (1.1) is either oscillatory or converges to zero as

t→ ∞.

Lemma 2.2.

Assume that case (I) holds and proceeding as in the proof of Theorem 3.7, we again arrive at (3.37) and (3.39). In view of (3.39) and (3.42), we have, for

t > T ≥t3,

Ψ (T)≤ω(T) (3.44)

and

lim sup t→∞

1

H(t, T)

Z t

T

H(t, s)χ3(s) ∆s≥Ψ (T) forT ≥t3. (3.45)

On the other hand, setting

A(t) = 1

H(t, t3)

Z t

t3

h+(t, s)

η(σ(s))H

1/λ₍_{t, s}₎_ω₍_σ₍_s_{)) ∆}_s _for_{t > t}

3

and

B(t) = 1

H(t, t3)

Z t

t3

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s fort > t3, it follows from (3.37) that

lim inf

t→∞ [B(t)−A(t)] ≤ ω(t3)−lim supt→∞

1

H(t, t3)

Z t

t3

H(t, s)χ3(s) ∆s

≤ ω(t3)−Ψ (t3)<∞. (3.46)

Now, we claim that

Z ∞

t3

η(s)ψ(s)

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s <∞. (3.47) To prove it, suppose to the contrary that

Z ∞

t3

η(s)ψ(s)

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s=∞. (3.48) By (3.40), there exists a constantε1>0 such that

inf s≥t0

lim inf t→∞

H(t, s)

H(t, t0)

> ε1>0. (3.49)

Let K1 > 0 be arbitrary number. Then, it follows from (3.48) that there exists

t4> t3 such that

Z t

t3

η(s)ψ(s)

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s≥

K1

α2ε1

(14)

Now, we have, for everyt≥t4> t3,

B(t) = 1

H(t, t3)

Z t

t3

α2H(t, s)

η(s)ψ(s)

r1/α2

2 (s)

ηλ₍_σ₍_s₎₎∆s

= 1

H(t, t3)

Z t

t3

α2H(t, s)

Z s

t3

η(u)ψ(u)

r1/α2

2 (u)

ωλ₍_σ₍_u₎₎

ηλ₍_σ₍_u₎₎∆u

!∆s ∆s

= 1

H(t, t3)

Z t

t3

"

−α2H∆s(t, s)

Z σ(s)

t3

η(u)ψ(u)

r1/α2

2 (u)

ωλ(σ(u))

#

∆s

≥ 1

H(t, t3)

Z t

t4

"

−α2H∆s(t, s)

Z s

t3

η(u)ψ(u)

r1/α2

2 (u)

ωλ(σ(u))

#

∆s

≥ 1

H(t, t3)

Z t

t4

−α2H∆s(t, s)

K1

α2ε1

∆s= K1

ε1

H(t, t4)

H(t, t3)

.

From (3.49), we have lim inf t→∞

H(t,t4)

H(t,t0) > ε1 and so we can choose at5≥t4 such that H(t,t4)

H(t,t3) ≥ε1for everyt≥t5. Hence,B(t)≥K1for allt≥t5. SinceK1is arbitrary, we have

lim

t→∞B(t) =∞. (3.50)

Next, we consider a sequence {Tn}∞n=1 in (t3,∞)_T with limn→∞Tn =∞and such that

lim

n→∞[B(Tn)−A(Tn)] = lim inft→∞ [B(t)−A(t)].

Then, from (3.46), there exists a constantK2such that

B(Tn)−A(Tn)≤K2, (3.51)

for all sufficiently large integern. Since (3.50) ensures that lim

n→∞B(Tn) =∞, (3.52)

(3.51) implies that

lim

n→∞A(Tn) =∞. (3.53)

From (3.51) and (3.52), we have

A(Tn)

B(Tn)

−1≥ −K2 B(Tn)

> −K2

2K2

=−1 2 , i.e.

A(Tn)

B(Tn)

> 1

2

for large enough positive integern, which together with (3.53) implies that

lim n→∞

[A(Tn)]α2+1 [B(Tn)]

α2 = lim_n_→∞

_A₍_T_n) B(Tn)

α2

A(Tn) =∞. (3.54)

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A(Tn) =

1

H(Tn, t3)

Z Tn

t3

h+(Tn, s)

η(σ(s)) H

1/λ₍_T

n, s)ω(σ(s)) ∆s

=

Z Tn

t3      _α

2H(Tn, s)η(s)ψ(s)

H(Tn, t3)

1/λ

× ω(σ(s)) h

r1/α2

2 (s)ηλ(σ(s))

i1/λ   

 

×h+(Tn, s)H

1/λ₍_T

n, s) (r2(s)) 1/(α2+1)

H(Tn, t3)

×

α2H(Tn, s)η(s)ψ(s)

H(Tn, t3)

−1/λ ∆s

≤ "

Z Tn

t3

α2H(Tn, s)η(s)ψ(s)

H(Tn, t3)r 1/α2

2 (s)ηλ(σ(s))

ωλ(σ(s)) ∆s #1/λ

× 

 Z Tn

t3

h+(Tn, s)H1/λ(Tn, s) (r2(s)) 1/(α2+1)

H(Tn, t3)

!α2+1

× _α

2H(Tn, s)η(s)ψ(s)

H(Tn, t3)

−α2 ∆s

#

1 α2 +1

= B1/λ(Tn)

" α−α2

2

H(Tn, t3)

Z Tn

t3

r2(s) (h+(Tn, s)) α2+1

(η(s)ψ(s))α2 ∆s

#

1 α2 +1

, (3.55)

and accordingly

[A(Tn)] α2+1

[B(Tn)] α2 ≤

α−α2

2

H(Tn, t3)

Z Tn

t3

r2(s) (h+(Tn, s)) α2+1

(η(s)ψ(s))α2 ∆s. (3.56)

Now, in view of (3.54), it follows from (3.56) that

lim n→∞

1

H(Tn, t3)

Z Tn

t3

r2(s) (h+(Tn, s)) α2+1

(η(s)ψ(s))α2 ∆s=∞, (3.57) from which, we arrive that

lim sup t→∞

1

H(t, t3)

Z t

t3

r2(s) (h+(t, s))

α2+1

(η(s)ψ(s))α2 ∆s=∞,

which contradicts (3.41) and so (3.47) holds. Thus, from (3.44) and (3.47) we get

Z ∞

t3

η(s)ψ(s)

r1/α2

2 (s)ηλ(σ(s))

Ψλ₊(σ(s)) ∆s≤ Z ∞

t3

η(s)ψ(s)ωλ₍_σ₍_s₎₎

r1/α2

2 (s)ηλ(σ(s))

∆s <∞ (3.58)

Theorem 3.9. Assume (1.2) and (2.1)-(2.3). Let H andhbe as in Theorem 3.7 and (3.40) holds. Suppose also that there exist a positive functionη∈C1

rd([t0,∞)T,R)

andΨ (t)∈Crd([t0,∞)_T,R)such that (3.43) and the following conditions hold:

lim inf t→∞

1

H(t, t∗)

Z t

t∗

(16)

and

lim inf t→∞

1

H(t, T)

Z t

T

"

H(t, s)χ3(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

∆s≥Ψ (T)(3.60)

forT ≥t∗, whereχ3(t)is as in Theorem 3.7 and t∗> t2≥t1. Then any solution of equation (1.1) is either oscillatory or converges to zero ast→ ∞.

Lemma 2.2.

Next, assume that case (I) holds and proceeding as in the proof of Theorem 3.7, we again arrive at (3.37) and (3.39). In view of (3.39) and (3.60), we have

Ψ (T)≤ω(T) (3.61)

and

lim inf t→∞

1

H(t, T)

Z t

T

H(t, s)χ3(s) ∆s≥Ψ (T) for T ≥t3. (3.62)

Define again the functions A(t) and B(t) as in Theorem 3.8, we see from (3.37) that

lim sup t→∞

[B(t)−A(t)] ≤ ω(t3)−lim inf

t→∞

1

H(t, t3)

Z t

t3

H(t, s)χ3(s) ∆s

≤ ω(t3)−Ψ (t3)<∞. (3.63)

Next, by using condition (3.60) and (3.62), we obtain

Ψ (t3) ≤ lim inf

t→∞

1

H(t, t3)

Z t

t3

"

H(t, s)χ3(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)α2+1(η(s)ψ(s))α2

#

∆s

≤ lim inf t→∞

1

H(t, t3)

Z t

t3

H(t, s)χ3(s) ∆s

−lim inf t→∞

1

H(t, t3)

Z t

t3

r2(s) (h+(t, s))

α2+1

(α2+ 1)α2+1(η(s)ψ(s))α2

∆s. (3.64)

Hence from (3.59) and (3.64), we get

lim inf t→∞

1

H(t, t3)

Z t

t3

r2(s) (h+(t, s))

α2+1

(η(s)ψ(s))α2 ∆s <∞. (3.65)

Therefore, there exists a sequence{Tn}∞n=1 in (t3,∞)_T with limn→∞Tn =∞ and such that

lim n→∞

1

H(Tn, t3)

Z Tn

t3

r2(s) (h+(Tn, s)) α2+1

(η(s)ψ(s))α2 ∆s <∞. (3.66)

Following the procedure of the proof of Theorem 3.8, we see that (3.57) holds, which contradicts (3.66). This contradiction proves that (3.48) fails. The rest of the proof

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Theorem 3.10. Assume (1.3) and (2.1)-(2.3) hold and let η, H and h be as in Theorem 3.7. If

lim sup t→∞

1

H(t, t∗) Z t

t∗

"

H(t, s)χ4(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

∆s=∞,(3.67)

where

χ4(t) =κη(t)q2(t)

Rβ₂(σ(t), t2)

Rα2

1 (σ(t), t1)

.

and t∗ > t2 ≥t1 for sufficiently large t1 ∈[t0,∞)_T, then any solution of equation (1.1) is either oscillatory or converges to zero ast→ ∞.

Lemma 2.2.

Next, assume that case (I) holds. Then again (2.8), (3.5), (3.7), (3.10), (3.24) and (3.25) hold. Define the functionwas in (3.2) and using (2.8), (3.5), (3.24) and (3.25), we arrive at

ω∆(t) = η(t)

z[1]₍_t₎α2

z[2](t)

∆

+ η(t)

z[1]₍_t₎α2

!∆

z[2](σ(t))

≤ −κη(t)q2(t)

Rβ₂(σ(t), t2)

Rα2

1 (σ(t), t1)

+η

∆₍_t₎_ω₍_σ₍_t₎₎

η(σ(t))

−η(t)

z[2](σ(t)) z[1](t)α2

∆

z[1]₍_t₎α2

z[1]₍_σ₍_t₎₎α2. (3.68)

The remainder of the proof is similar to that of Theorem 3.7, and so the details are

omitted.

The proof of the the following two theorems follows from Theorems 3.7-3.10; we omit the details.

Theorem 3.11. Assume (1.3) and (2.1)-(2.3). Let η, H and h be as in The-orem 3.8 such that (3.40) and (3.41) holds. If there exists a function Ψ (t) ∈ Crd([t0,∞)_T,R)such that (3.43) holds and

lim sup t→∞

1

H(t, T)

Z t

T

"

H(t, s)χ4(s)−

r2(s) (h+(t, s))α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

∆s≥Ψ (T)(3.69)

forT ≥t∗, whereχ4(t)is as in Theorem 3.10 and t∗ > t2≥t1, then any solution of equation (1.1) either oscillates or converges to zero ast→ ∞.

Theorem 3.12. Assume (1.3) and (2.1)-(2.3). Let H and h be as in Theo-rem 3.9 and (3.40) holds. Suppose also that there exist a positive function η ∈ C1

rd([t0,∞)T,R) and Ψ (t) ∈ Crd([t0,∞)_T,R) such that (3.43) and the following

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lim inf t→∞

1

H(t, t∗)

Z t

t∗

H(t, s)χ4(s) ∆s <∞ (3.70)

and

lim inf t→∞

1

H(t, T)

Z t

T

"

H(t, s)χ4(s)−

r2(s) (h+(t, s))

α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2

#

∆s≥Ψ (T)(3.71)

forT ≥t∗, where χ4(t)is as in Theorem 3.10 andt∗> t2≥t1. Then any solution of equation (1.1) is either oscillatory or converges to zero ast→ ∞.

Example 3.13. LetT:=qZ₌_{_qk _:_k_∈_Z_{, q >}₁_{} ∪ {}₀_}_{and consider the third order}

neutral dynamic equation

(x(t) + 8x(t/2))∆∆

3∆

+

Z b

a

t2+ξ

x3(t/2−ξ)∆ξ= 0, (3.72)

for t ∈2Z _with_t _≥ _t₀ _{:= 2}_{. Here we have} _α₁ _{= 1}_, _α₂ _{= 3}_, _g₍_t_{) =} _t/₂_, _q₍_{t, ξ}_{) =}

t2+ξ,r1(t) =r2(t) = 1,φ(t, ξ) =t/2−ξ,f(u) =uβ andp(t) = 8. It is clear that conditions (C1)-(C5) and (1.2) hold withκ= 1,β = 3and

ϕ1(t) = 7/64>0. (3.73)

Since

1− 1

p(g−1₍_g−1₍_t₎₎₎

R2(g−1(g−1(t)), t2)

R2(g−1(t), t2)

=2t−7 4t−8,

we see that

ϕ2(t)≥

1

64 fort≥t2= 4. (3.74)

In view of (3.73) and (3.74), we see that

q1(t) =

b

Z

a

(t2+ξ)(7 64)

3

∆ξ= (b−a)(7 64)

3

(t2+b+a

3 ), (3.75)

q2(t)≥

b

Z

a

(t2+ξ)( 1 64)

3_∆_ξ_{= (}_b₋_a₎₍1

64)

3₍_t2₊b+a

3 ) fort≥t2= 4. (3.76)

With (3.75), condition (2.3) becomes

Z ∞

t0 1

r1(v)

Z ∞

v

₁

r2(u)

Z ∞

u

q1(s) ∆s

1/α2 ∆u

!1/α1 ∆v

=

∞ Z

2

∞ Z

v



 ∞ Z

u

(7/64)3(b−a)

s2+b+a 3

∆s 



1/3

∆u∆v=∞

due to

∞ R

u

s2₊b+a

3

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Withη(t) =t and the fact that (3.76), we see that

lim sup t→∞

t

Z

T

χ1(s)∆s≥lim sup

t→∞

t

Z

4

(b−a)2s

(64)3

s2+b+a 3

s2₋₍₄_b_{+ 6)}_s_{+ 4}_b2_{+ 12}_b_{+ 8}

6s−6

3

∆s

≥lim sup t→∞

2(b−a) (384)3

t

Z

4

s2−(4b+ 6)s+ 4b2+ 12b+ 83

∆s=∞

and

lim sup t→∞

t

Z

T

η∆ +(s)

Rα2

1 (s, t1)

∆s= lim sup t→∞

t

Z

4

1

(s−2)3∆s <∞,

so condition (3.1) holds. Thus, all conditions of Theorem 3.1 are satisfied. There-fore, by Theorem 3.1, any solution of (3.72) is either oscillatory or converges to zero.

Example 3.14. Consider the neutral differential equation





1

t5

" t+ 2

2

x(t) +10t+ 11

t+ 1 x(t−2)

0!0#

5

 0

+

Z 2

1

(t+ξ)x5(t−2−ξ)dξ = 0,

(3.77)

fort≥2. Here we haveT=R,α1= 1,α2= 5,g(t) =t−2,q(t, ξ) =t+ξ,r1(t) =

(t+ 2)/2,r2(t) = 1/t5,φ(t, ξ) =t−2−ξ,f(u) =uβ andp(t) = (10t+ 11)/(t+ 1). It is clear that conditions (C1)-(C5) and (1.2) hold withκ= 1andβ = 5. In view of the fact that

10≤p(t)<11,

we see that

ϕ1(t)≥

9

110 >0. (3.78)

Since

1

p(g−1₍_g−1₍_t₎₎₎

R2(g−1(g−1(t)), t2)

R2(g−1(t), t2)

≤ 1

10

t+ 3

t−1 ≤ 3 10,

fort≥t2= 3, we obtain

ϕ2(t)≥

7

110. (3.79)

In view of (3.78) and (3.79), we see that

q1(t)≥ 2

Z

1

(t+ξ)( 9 110)

5_dξ _{= (} 9

110)

5₍_t_{+ 3}_/₂₎_, _(3.80)

q2(t)≥ 2

Z

1

(t+ξ)( 7 110)

5_dξ _{= (} 7

110)

5₍_t_{+ 3}_/₂₎ _for_t_≥_t

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By (3.80), condition (2.3) becomes

Z ∞

t0 1

r1(v)

Z ∞

v

₁

r2(u)

Z ∞

u

q1(s) ∆s

1/α2 ∆u

!1/α1 ∆v

≥ ∞ Z

2



 

2

v+ 2

∞ Z

v





1 1/u5

∞ Z

u

(9/110)5(s+ 3/2)ds 



1/5

du 



dv=∞,

due to

∞ R

u

(s+ 3/2)ds=∞foru≥2, and so condition (2.3) holds.

Withη(t) =t,H(t, s) = (t−s)2 _{and the fact that (3.81), it is clear that}

lim sup t→∞

1

H(t, t∗) Z t

t∗

H(t, s)κη(s)q2(s)

Rβ₂ g−1₍_φ

2(s)), t2

Rα2

1 (σ(s), t1)

∆s

≥lim sup t→∞

₇

110

5 ₁

(t−4)2

t

Z

4

s(t−s)2(s+ 3/2)

_s2₋₈_s_{+ 15}

s2₋₄

5

ds=∞

and

lim sup t→∞

1

H(t, t∗) Z t

t∗

r2(s) (h+(t, s))

α2+1

(α2+ 1)

α2+1₍_η₍_s₎_ψ₍_s₎₎α2∆s

= lim sup t→∞

1 (t−4)2

t

Z

4

(t−3s)6

66_s10₍_t₋_s₎4ds≤lim sup

t→∞

1 (t−4)2

t

Z

4

(t−s)2

66_s10 ds <∞,

so condition (3.29) holds. Hence, by Theorem 3.7, any solution of (3.77) either oscillates or converges to zero.

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∗

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa Univer-sity, 60240, Tokat, Turkey

E-mail address:[email protected]

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa Univer-sity, 60240, Tokat, Turkey