Research Article Asymptotic Properties of Solutions to Third … · 2019. 7. 31. · Research Article Asymptotic Properties of Solutions to Third-Order Nonlinear Neutral Differential

Research ArticleAsymptotic Properties of Solutions to Third-Order NonlinearNeutral Differential Equations

Qi Li1 Rui Wang1 Fanwei Meng2 and Jianxin Han3

1 Qingdao Technological University Feixian Shandong 273400 China2Department of Mathematics Qufu Normal University Qufu Shandong 273165 China3 College of Mathematics and Information Science Xinyang Normal University Xinyang 464000 China

Correspondence should be addressed to Qi Li liqi 2006163com and Rui Wang rwang 2005163com

Received 26 March 2014 Accepted 12 May 2014 Published 25 May 2014

Academic Editor Tongxing Li

Copyright copy 2014 Qi Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of this work is to discuss asymptotic properties of a class of third-order nonlinear neutral functional differential equationsThe results obtained extend and improve some related known results Two examples are given to illustrate the main results

1 Introduction

In this work we study the asymptotic behavior of solutions ofthe third-order neutral differential equation

(119886(119905)(119887(119905)(119909 (119905) + 119901 (119905) 119909 (120590 (119905)))1015840)1015840)1015840

+ 119902 (119905) 119891 (119909 (120591 (119905))) 119892 (1199091015840 (119905)) = 0(1)

We always assume that the following conditions hold

(H1) 119886(119905) 119887(119905) 119901(119905) 119902(119905) isin 119862([119905

0infin) [0infin)) 0 le 119901(119905) le

1199010lt 1

(H2) 120590(119905) 120591(119905) isin 119862([119905

0infin) [0infin)) 120590(119905) le 119905 120591(119905) le 119905

lim119905rarrinfin

120590(119905) = lim119905rarrinfin

120591(119905) = infin(H3) 119891 isin 119862(119877 119877) 119891(119909)119909 ge 119870 gt 0 for all 119909 = 0

(H4) 119892 isin 119862(119877 [119871infin)) 119871 gt 0

(H5) intinfin1199050

(1119886(119905))d119905 = intinfin1199050

(1119887(119905))d119905 = infin

Set 119911(119905) = 119909(119905) + 119901(119905)119909(120590(119905)) By a solution of (1) wemean a nontrivial function 119909(119905) isin 119862([119879

119909infin) 119877) 119879

119909ge 1199050

which has the properties 119911(119905) isin 1198621([119879119909infin) 119877) 119887(119905)1199111015840(119905) isin

1198621([119879119909infin)) and 119886(119905)(119887(119905)1199111015840(119905))1015840 isin 1198621([119879

119909infin)) and satisfies

(1) on [119879119909infin) We consider only those solutions 119909(119905) of (1)

which satisfies sup|119909(119905)| 119905 ge 119879 gt 0 for all 119879 ge 119879119909 We

assume that (1) possesses such a solution A solution of (1) iscalled oscillatory if it has arbitrarily large zeros on [119879

119909infin)

otherwise it is called nonoscillatoryRecently great attention has been devoted to the oscil-

lation of various classes of differential equations See forexample [1ndash19] Hartman andWintner [1] and Erbe et al [3]studied the third-order differential equation

119909101584010158401015840 (119905) + 119902 (119905) 119909 (119905) = 0 (2)

Paper [5] studied the oscillation of third-order trinomialdelay differential equation

119909101584010158401015840 (119905) + 119901 (119905) 1199091015840 (119905) + 119892 (119905) 119909 (120591 (119905)) = 0 (3)

Li et al [7] discussed (1) with 119891(119909(120591(119905))) = 119909(120591(119905)) and119892(1199091015840(119905)) = 1 Han [8] examined the oscillation of (1) with119887(119905) = 1

In this work we establish some oscillation criteria for (1)which extend and improve the results in [7 8]

2 Main Results

In the following all functional inequalities considered areassumed to hold eventually for all 119905 large enough Withoutloss of generality we deal only with the positive solutions of(1)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 618183 4 pageshttpdxdoiorg1011552014618183

2 Abstract and Applied Analysis

Theorem 1 Suppose that

intinfin

1199050

1119887 (V) int

infin

V

1119886 (119906) int

infin

119906

119902 (119904) d119904 d119906 dV = infin (4)

If for some function 120588 isin 1198621([1199050infin) (0infin)) for all sufficiently

large 1199052gt 1199051gt 1199050 one has

lim sup119905rarrinfin

int119905

1199052

[[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 = infin (5)

where

119876 (119905) =119870119871 (1 minus 119901

0) int120591(119905)1199052

intV1199051

(1 (119886 (119906) 119887 (V))) d119906 dVint1199051199051

(1119887 (119906)) d119906 (6)

then all solutions of (1) are oscillatory or convergent to zeroasymptotically

Proof Assume that 119909 is a positive solution of (1) Based oncondition (119867

5) there are two possible cases

(1) 119911(119905) gt 0 1199111015840(119905) gt 0 (119887(119905)1199111015840(119905))1015840 gt 0[119886(119905)(119887(119905)1199111015840(119905))1015840]1015840 lt 0

(2) 119911(119905) gt 0 1199111015840(119905) lt 0 (119887(119905)1199111015840(119905))1015840 gt 0[119886(119905)(119887(119905)1199111015840(119905))1015840]1015840 lt 0

First consider that 119911(119905) satisfies (1) We have

119909 (119905) = 119911 (119905) minus 119901 (119905) 119909 (120590 (119905)) ge (1 minus 1199010) 119911 (119905) (7)

From (1) (1198673) and (119867

4) we get

[119886(119905)(119887(119905)1199111015840(119905))1015840]1015840

le minus119870119871 (1 minus 1199010) 119902 (119905) 119911 (120591 (119905)) le 0 (8)

Define a function 120596 by

120596 (119905) = 120588 (119905) 119886 (119905) (119887(119905)1199111015840(119905))1015840119887 (119905) 1199111015840 (119905) 119905 ge 119905

1 (9)

we obtain 120596(119905) gt 0 Then

1205961015840 (119905)

= 1205881015840 (119905) 119886 (119905) (119887(119905)1199111015840(119905))1015840119887 (119905) 1199111015840 (119905) + 120588 (119905) (119886(119905)(119887(119905)1199111015840(119905))1015840)1015840

119887 (119905) 1199111015840 (119905)

minus 120588 (119905)119886 (119905) [(119887 (119905) 1199111015840 (119905))1015840]

2

(119887 (119905) 1199111015840 (119905))2

= 120588 (119905)(119886 (119905) (119887 (119905) 1199111015840 (119905))1015840)

1015840

119887 (119905) 1199111015840 (119905)

+ 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1205962 (119905)

120588 (119905) 119886 (119905)

le minus119870119871 (1 minus 1199010) 120588 (119905) 119902 (119905) 119911 (120591 (119905))

119887 (119905) 1199111015840 (119905)

+ 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1205962 (119905)

120588 (119905) 119886 (119905)

le minus119870119871 (1 minus 1199010) 120588 (119905) 119902 (119905) 119911 (120591 (119905))

119887 (119905) 1199111015840 (119905)

minus [ 120596(119905)radic120588(119905)119886(119905) minus 1

2radic 119886(119905)120588(119905)1205881015840(119905)]2

+(1205881015840 (119905))2119886 (119905)

4120588 (119905) (10)

By the proof of [7 Theorem 21] we have

1205961015840 (119905) le minus[[120588 (119905) 119876 (119905) minus

(1205881015840 (119905))2119886 (119905)4120588 (119905)

]]

(11)

where 119876(119905) is defined as in (6) We obtain

int119905

1199051

[[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 le minusint

119905

1199051

1205961015840 (119904) d119904 (12)

That is

int119905

1199051

[[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le 120596 (1199051) minus 120596 (119905) lt 120596 (119905

1) lt infin

(13)

which contradicts (5) Assume that case (2) holds Using thesimilar proof of [8 Lemma 4] we can get lim

119905rarrinfin119909(119905) = 0

This completes the proof

Based on Theorem 1 we present a Kamenev-type crite-rion for (1)

Abstract and Applied Analysis 3

Theorem 2 Assume that (4) holds If for some function 120588 isin1198621([1199050infin) (0infin)) for all sufficiently large 119905

1gt 1199050 one has


1119905119899 int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 = infin

(14)


Proof Assume that 119909(119905) is a positive solution of (1) Then bythe proof of Theorem 1 we have cases (1) and (2) Let case (1)hold Proceeding as in the proof of Theorem 1 we have (11)Then we have

int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le minusint119905

1199051

(119905 minus 119904)1198991205961015840 (119904) d119904

(15)

That is

1119905119899 int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le minus 119899119905119899 int119905

1199051

(119905 minus 119904)119899minus1120596 (119904) d119904 + (1 minus 1199051

119905 )119899

120596 (1199051)

lt (1 minus 1199051

119905 )119899

120596 (1199051) lt infin

(16)

which contradicts (14) Assume that case (2) holds We canget lim

119905rarrinfin119909(119905) = 0 The proof is completed

Next we present a Philos-type criterion for (1) Let

119863 = (119905 119904) 119905 ge 119904 ge 1199050 119863

0= (119905 119904) 119905 gt 119904 ge 119905

0 (17)

We say that a function 119867 isin 119862(119863 119877) belongs to a functionclass 119875 if it satisfies

(i) 119867(119905 119905) = 0 119905 ge 1199050 119867(119905 119904) gt 0 (119905 119904) isin 119863

0

(ii) 119867 has a continuous and nonpositive partial derivativeon 1198630with respect to the second variable and such

that

minus120597119867 (119905 119904)120597119904 = ℎ (119905 119904) radic119867 (119905 119904) (119905 119904) isin 119863

0 (18)


1gt 1199050 one has


1119867 (119905 119905

1) int119905

1199051

[119867 (119905 119904) 120588 (119904) 119876 (119904) minus ℎ21(119905 119904)

4119861 (119904) ] d119904 = infin(19)

where 119876(119905) is defined as in (6) 119861(119905) = 1120588(119905)119886(119905) and

ℎ1(119905 119904) = ℎ (119905 119904) minus 1205881015840 (119904)

120588 (119904) radic119867 (119905 119904) (20)


Proof Assume that 119909(119905) is a positive solution of (1) and 119911(119905)has the case of (1) 120596(119905) is defined as in (9) Then

1205961015840 (119905) le minus120588 (119905) 119876 (119905) + 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1

119886 (119905) 120588 (119905)1205962 (119905) (21)

Let 119861(119905) = 1120588(119905)119886(119905) we have

120588 (119905) 119876 (119905) le minus1205961015840 (119905) + 1205881015840 (119905)120588 (119905) 120596 (119905) minus 119861 (119905) 1205962 (119905) (22)

We obtain

int119905

1199051

119867(119905 119904) 120588 (119904) 119876 (119904) d119904

le int119905

1199051

119867(119905 119904) [minus1205961015840 (119904) + 1205881015840 (119904)120588 (119904) 120596 (119904) minus 119861 (119904) 1205962 (119904)] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[(ℎ (119905 119904) radic119867 (119905 119904) minus 119867 (119905 119904) 1205881015840 (119904)120588 (119904) )120596 (119904)

+119867 (119905 119904) 119861 (119904) 1205962 (119904) ] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[radic119867 (119905 119904)ℎ1(119905 119904) 120596 (119904) + 119867 (119905 119904) 119861 (119904) 1205962 (119904)] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[radic119867 (119905 119904) 119861 (119904)120596 (119904) + ℎ1(119905 119904)

2radic119861 (119904)]2

d119904

+ int119905

1199051

ℎ21(119905 119904)

4119861 (119904) d119904 lt 119867 (119905 1199051) 120596 (1199051) + int119905

1199051

ℎ21(119905 119904)

4119861 (119904) d119904(23)

That is

1119867 (119905 119905

1) int119905

1199051

[119867 (119905 119904) 120588 (119904) 119876 (119904) minus ℎ21(119905 119904)

4119861 (119904) ] d119904 le 120596 (1199051) (24)

which contradicts (19) Assume that (2) holds We can getlim119905rarrinfin

119909(119905) = 0 The proof is completed

3 Examples

In this section we will present two examples to illustrate themain results


Example 4 Consider the third-order nonlinear neutral dif-ferential equation

(119905(119909(119905) + 1199011119909( 119905

2))10158401015840

)1015840

+ 1205821199052 119909 (119905) (1 + (1199091015840 (119905))2) = 0

120582 gt 0 119905 ge 1(25)

where 1199011

isin [0 1) and 119870 = 119871 = 1 Let 120588(119905) = 119905 It followsfromTheorem 1 that every solution 119909(119905) of (25) is oscillatoryor convergent to zero asymptotically


(1119905 (11990512(119909(119905) + 1

2119909(119905 minus 12)1015840

)1015840

)1015840

+ 119905120582 (1205822 minus cos 119905119905 + 2 + sin 119905)

times 119909 (119905 minus 1) (1 + 1199092 (119905 minus 1)) (1 + (1199091015840 (119905))2) = 0

(26)

where 120582 gt 0 119905 ge 1 We have

int119905

1199050

119902 (119904) d119904 = int119905

1199051

119904120582 (1205822 minus cos 119904119904 + 2 + sin 119904) d119904

ge int119905

1199051

119904120582 (1205822 minus cos 119904119904 + sin 119904) d119904

= int119905

1199051

119889 [119904120582 (2 minus cos 119904)]

= 119905120582 (2 minus cos 119905) minus 1199051205820(2 minus cos 119905

0)

ge 119905120582 minus 1198700997888rarr infin(119905 997888rarr infin)

(27)

we see that (4) and (H1)ndash(H5) hold Let 119867(119905 119904) = (119905 minus 119904)2

120588(119905) = 1 Then ℎ1(119905 119904) = 2 It follows from Theorem 3

that the solutions of (26) are oscillatory or convergent to zeroasymptotically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was partially supported by the NNSF of China(11171178)

References

[1] P Hartman and A Wintner ldquoLinear differential and differenceequations withmonotone solutionsrdquoAmerican Journal ofMath-ematics vol 75 pp 731ndash743 1953

[2] L Erbe ldquoExistence of oscillatory solutions and asymptoticbehavior for a class of third order linear differential equationsrdquoPacific Journal of Mathematics vol 64 no 2 pp 369ndash385 1976

[3] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 of Monographs andTextbooks in Pure and Applied Mathematics Marcel DekkerNew York NY USA 1995

[4] R P Agarwal S R Grace and D OrsquoRegan ldquoOscillation criteriafor certain 119899th order differential equations with deviating argu-mentsrdquo Journal of Mathematical Analysis and Applications vol262 no 2 pp 601ndash622 2001

[5] J Dzurina ldquoAsymptotic properties of the third order delaydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 26 no 1 pp 33ndash39 1996

[6] T Candan and R S Dahiya ldquoFunctional differential equationsof third orderrdquo in Proceedings of the Conference on DifferentialEquations and Applications in Mathematical Biology vol 12 pp47ndash56 2005

[7] T Li C Zhang and G Xing ldquoOscillation of third-order neutraldelay differential equationsrdquo Abstract and Applied Analysis vol2012 Article ID 569201 11 pages 2012

[8] Z-Y Han ldquoOscillations of a class of third order nonlinearneutral functional differential equationsrdquo Journal of East ChinaNormal University no 1 pp 113ndash120 2012

[9] B Baculıkova and J Dzurina ldquoOscillation of third-order func-tional differential equationsrdquo Electronic Journal of QualitativeTheory of Differential Equations vol 43 pp 1ndash10 2010

[10] M F Aktas A Tiryaki and A Zafer ldquoOscillation criteriafor third-order nonlinear functional differential equationsrdquoApplied Mathematics Letters vol 23 no 7 pp 756ndash762 2010

[11] S R Grace R P Agarwal R Pavani and E ThandapanildquoOn the oscillation of certain third order nonlinear functionaldifferential equationsrdquo Applied Mathematics and Computationvol 202 no 1 pp 102ndash112 2008

[12] R P Agarwal S R Grace and D OrsquoRegan ldquoThe oscillation ofcertain higher-order functional differential equationsrdquo Mathe-matical and Computer Modelling vol 37 no 7-8 pp 705ndash7282003

[13] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[14] S H Saker and J Dzurina ldquoOn the oscillation of certain class ofthird-order nonlinear delay differential equationsrdquo Mathemat-ica Bohemica vol 135 no 3 pp 225ndash237 2010

[15] R P Agarwal M F Aktas and A Tiryaki ldquoOn oscillationcriteria for third order nonlinear delay differential equationsrdquoArchivum Mathematicum vol 45 no 1 pp 1ndash18 2009

[16] T S Hassan ldquoOscillation of third order nonlinear delaydynamic equations on time scalesrdquoMathematical andComputerModelling vol 49 no 7-8 pp 1573ndash1586 2009

[17] T Li Z Han S Sun and Y Zhao ldquoOscillation results forthird order nonlinear delay dynamic equations on time scalesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 34no 3 pp 639ndash648 2011

[18] E Thandapani and T Li ldquoOn the oscillation of third-order quasi-linear neutral functional differential equationsrdquoArchivum Mathematicum vol 47 no 3 pp 181ndash199 2011

[19] R P Agarwal B Baculikova J Dzurina and T Li ldquoOscillationof third-order nonlinear functional differential equations withmixed argumentsrdquo Acta Mathematica Hungarica vol 134 no1-2 pp 54ndash67 2012

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Stochastic AnalysisInternational Journal of


Theorem 1 Suppose that

intinfin

1199050

1119887 (V) int

infin

V

1119886 (119906) int

infin

119906

119902 (119904) d119904 d119906 dV = infin (4)

If for some function 120588 isin 1198621([1199050infin) (0infin)) for all sufficiently

large 1199052gt 1199051gt 1199050 one has


int119905

1199052

[[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 = infin (5)

where

119876 (119905) =119870119871 (1 minus 119901

0) int120591(119905)1199052

intV1199051

(1 (119886 (119906) 119887 (V))) d119906 dVint1199051199051

(1119887 (119906)) d119906 (6)


Proof Assume that 119909 is a positive solution of (1) Based oncondition (119867

5) there are two possible cases

(1) 119911(119905) gt 0 1199111015840(119905) gt 0 (119887(119905)1199111015840(119905))1015840 gt 0[119886(119905)(119887(119905)1199111015840(119905))1015840]1015840 lt 0

(2) 119911(119905) gt 0 1199111015840(119905) lt 0 (119887(119905)1199111015840(119905))1015840 gt 0[119886(119905)(119887(119905)1199111015840(119905))1015840]1015840 lt 0

First consider that 119911(119905) satisfies (1) We have

119909 (119905) = 119911 (119905) minus 119901 (119905) 119909 (120590 (119905)) ge (1 minus 1199010) 119911 (119905) (7)

From (1) (1198673) and (119867

4) we get

[119886(119905)(119887(119905)1199111015840(119905))1015840]1015840

le minus119870119871 (1 minus 1199010) 119902 (119905) 119911 (120591 (119905)) le 0 (8)

Define a function 120596 by

120596 (119905) = 120588 (119905) 119886 (119905) (119887(119905)1199111015840(119905))1015840119887 (119905) 1199111015840 (119905) 119905 ge 119905

1 (9)

we obtain 120596(119905) gt 0 Then

1205961015840 (119905)

= 1205881015840 (119905) 119886 (119905) (119887(119905)1199111015840(119905))1015840119887 (119905) 1199111015840 (119905) + 120588 (119905) (119886(119905)(119887(119905)1199111015840(119905))1015840)1015840

119887 (119905) 1199111015840 (119905)

minus 120588 (119905)119886 (119905) [(119887 (119905) 1199111015840 (119905))1015840]

2

(119887 (119905) 1199111015840 (119905))2

= 120588 (119905)(119886 (119905) (119887 (119905) 1199111015840 (119905))1015840)

1015840

119887 (119905) 1199111015840 (119905)

+ 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1205962 (119905)

120588 (119905) 119886 (119905)

le minus119870119871 (1 minus 1199010) 120588 (119905) 119902 (119905) 119911 (120591 (119905))

119887 (119905) 1199111015840 (119905)

+ 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1205962 (119905)

120588 (119905) 119886 (119905)

le minus119870119871 (1 minus 1199010) 120588 (119905) 119902 (119905) 119911 (120591 (119905))

119887 (119905) 1199111015840 (119905)

minus [ 120596(119905)radic120588(119905)119886(119905) minus 1

2radic 119886(119905)120588(119905)1205881015840(119905)]2

+(1205881015840 (119905))2119886 (119905)

4120588 (119905) (10)

By the proof of [7 Theorem 21] we have

1205961015840 (119905) le minus[[120588 (119905) 119876 (119905) minus

(1205881015840 (119905))2119886 (119905)4120588 (119905)

]]

(11)

where 119876(119905) is defined as in (6) We obtain

int119905

1199051

[[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 le minusint

119905

1199051

1205961015840 (119904) d119904 (12)

That is

int119905

1199051

[[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le 120596 (1199051) minus 120596 (119905) lt 120596 (119905

1) lt infin

(13)

which contradicts (5) Assume that case (2) holds Using thesimilar proof of [8 Lemma 4] we can get lim

119905rarrinfin119909(119905) = 0

This completes the proof

Based on Theorem 1 we present a Kamenev-type crite-rion for (1)



1gt 1199050 one has


1119905119899 int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 = infin

(14)



int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le minusint119905

1199051

(119905 minus 119904)1198991205961015840 (119904) d119904

(15)

That is

1119905119899 int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le minus 119899119905119899 int119905

1199051


119905 )119899

120596 (1199051)

lt (1 minus 1199051

119905 )119899

120596 (1199051) lt infin

(16)




119863 = (119905 119904) 119905 ge 119904 ge 1199050 119863

0= (119905 119904) 119905 gt 119904 ge 119905

0 (17)


(i) 119867(119905 119905) = 0 119905 ge 1199050 119867(119905 119904) gt 0 (119905 119904) isin 119863

0


that


0 (18)


1gt 1199050 one has


1119867 (119905 119905

1) int119905

1199051

[119867 (119905 119904) 120588 (119904) 119876 (119904) minus ℎ21(119905 119904)

4119861 (119904) ] d119904 = infin(19)


ℎ1(119905 119904) = ℎ (119905 119904) minus 1205881015840 (119904)

120588 (119904) radic119867 (119905 119904) (20)



1205961015840 (119905) le minus120588 (119905) 119876 (119905) + 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1

119886 (119905) 120588 (119905)1205962 (119905) (21)

Let 119861(119905) = 1120588(119905)119886(119905) we have

120588 (119905) 119876 (119905) le minus1205961015840 (119905) + 1205881015840 (119905)120588 (119905) 120596 (119905) minus 119861 (119905) 1205962 (119905) (22)

We obtain

int119905

1199051

119867(119905 119904) 120588 (119904) 119876 (119904) d119904

le int119905

1199051

119867(119905 119904) [minus1205961015840 (119904) + 1205881015840 (119904)120588 (119904) 120596 (119904) minus 119861 (119904) 1205962 (119904)] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[(ℎ (119905 119904) radic119867 (119905 119904) minus 119867 (119905 119904) 1205881015840 (119904)120588 (119904) )120596 (119904)

+119867 (119905 119904) 119861 (119904) 1205962 (119904) ] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[radic119867 (119905 119904)ℎ1(119905 119904) 120596 (119904) + 119867 (119905 119904) 119861 (119904) 1205962 (119904)] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[radic119867 (119905 119904) 119861 (119904)120596 (119904) + ℎ1(119905 119904)

2radic119861 (119904)]2

d119904

+ int119905

1199051

ℎ21(119905 119904)

4119861 (119904) d119904 lt 119867 (119905 1199051) 120596 (1199051) + int119905

1199051

ℎ21(119905 119904)

4119861 (119904) d119904(23)

That is

1119867 (119905 119905

1) int119905

1199051

[119867 (119905 119904) 120588 (119904) 119876 (119904) minus ℎ21(119905 119904)

4119861 (119904) ] d119904 le 120596 (1199051) (24)



3 Examples




(119905(119909(119905) + 1199011119909( 119905

2))10158401015840

)1015840

+ 1205821199052 119909 (119905) (1 + (1199091015840 (119905))2) = 0

120582 gt 0 119905 ge 1(25)

where 1199011



(1119905 (11990512(119909(119905) + 1

2119909(119905 minus 12)1015840

)1015840

)1015840

+ 119905120582 (1205822 minus cos 119905119905 + 2 + sin 119905)


(26)


int119905

1199050

119902 (119904) d119904 = int119905

1199051

119904120582 (1205822 minus cos 119904119904 + 2 + sin 119904) d119904

ge int119905

1199051

119904120582 (1205822 minus cos 119904119904 + sin 119904) d119904

= int119905

1199051

119889 [119904120582 (2 minus cos 119904)]


0)


(27)






Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1gt 1199050 one has


1119905119899 int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904 = infin

(14)



int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le minusint119905

1199051

(119905 minus 119904)1198991205961015840 (119904) d119904

(15)

That is

1119905119899 int119905

1199051

(119905 minus 119904)119899 [[120588 (119904) 119876 (119904) minus

(1205881015840 (119904))2119886 (119904)4120588 (119904)

]]d119904

le minus 119899119905119899 int119905

1199051


119905 )119899

120596 (1199051)

lt (1 minus 1199051

119905 )119899

120596 (1199051) lt infin

(16)




119863 = (119905 119904) 119905 ge 119904 ge 1199050 119863

0= (119905 119904) 119905 gt 119904 ge 119905

0 (17)


(i) 119867(119905 119905) = 0 119905 ge 1199050 119867(119905 119904) gt 0 (119905 119904) isin 119863

0


that


0 (18)


1gt 1199050 one has


1119867 (119905 119905

1) int119905

1199051

[119867 (119905 119904) 120588 (119904) 119876 (119904) minus ℎ21(119905 119904)

4119861 (119904) ] d119904 = infin(19)


ℎ1(119905 119904) = ℎ (119905 119904) minus 1205881015840 (119904)

120588 (119904) radic119867 (119905 119904) (20)



1205961015840 (119905) le minus120588 (119905) 119876 (119905) + 1205881015840 (119905)120588 (119905) 120596 (119905) minus 1

119886 (119905) 120588 (119905)1205962 (119905) (21)

Let 119861(119905) = 1120588(119905)119886(119905) we have

120588 (119905) 119876 (119905) le minus1205961015840 (119905) + 1205881015840 (119905)120588 (119905) 120596 (119905) minus 119861 (119905) 1205962 (119905) (22)

We obtain

int119905

1199051

119867(119905 119904) 120588 (119904) 119876 (119904) d119904

le int119905

1199051

119867(119905 119904) [minus1205961015840 (119904) + 1205881015840 (119904)120588 (119904) 120596 (119904) minus 119861 (119904) 1205962 (119904)] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[(ℎ (119905 119904) radic119867 (119905 119904) minus 119867 (119905 119904) 1205881015840 (119904)120588 (119904) )120596 (119904)

+119867 (119905 119904) 119861 (119904) 1205962 (119904) ] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[radic119867 (119905 119904)ℎ1(119905 119904) 120596 (119904) + 119867 (119905 119904) 119861 (119904) 1205962 (119904)] d119904

= 119867 (119905 1199051) 120596 (1199051)

minus int119905

1199051

[radic119867 (119905 119904) 119861 (119904)120596 (119904) + ℎ1(119905 119904)

2radic119861 (119904)]2

d119904

+ int119905

1199051

ℎ21(119905 119904)

4119861 (119904) d119904 lt 119867 (119905 1199051) 120596 (1199051) + int119905

1199051

ℎ21(119905 119904)

4119861 (119904) d119904(23)

That is

1119867 (119905 119905

1) int119905

1199051

[119867 (119905 119904) 120588 (119904) 119876 (119904) minus ℎ21(119905 119904)

4119861 (119904) ] d119904 le 120596 (1199051) (24)



3 Examples




(119905(119909(119905) + 1199011119909( 119905

2))10158401015840

)1015840

+ 1205821199052 119909 (119905) (1 + (1199091015840 (119905))2) = 0

120582 gt 0 119905 ge 1(25)

where 1199011



(1119905 (11990512(119909(119905) + 1

2119909(119905 minus 12)1015840

)1015840

)1015840

+ 119905120582 (1205822 minus cos 119905119905 + 2 + sin 119905)


(26)


int119905

1199050

119902 (119904) d119904 = int119905

1199051

119904120582 (1205822 minus cos 119904119904 + 2 + sin 119904) d119904

ge int119905

1199051

119904120582 (1205822 minus cos 119904119904 + sin 119904) d119904

= int119905

1199051

119889 [119904120582 (2 minus cos 119904)]


0)


(27)






Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905(119909(119905) + 1199011119909( 119905

2))10158401015840

)1015840

+ 1205821199052 119909 (119905) (1 + (1199091015840 (119905))2) = 0

120582 gt 0 119905 ge 1(25)

where 1199011



(1119905 (11990512(119909(119905) + 1

2119909(119905 minus 12)1015840

)1015840

)1015840

+ 119905120582 (1205822 minus cos 119905119905 + 2 + sin 119905)


(26)


int119905

1199050

119902 (119904) d119904 = int119905

1199051

119904120582 (1205822 minus cos 119904119904 + 2 + sin 119904) d119904

ge int119905

1199051

119904120582 (1205822 minus cos 119904119904 + sin 119904) d119904

= int119905

1199051

119889 [119904120582 (2 minus cos 119904)]


0)


(27)






Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Asymptotic Properties of Solutions to Third … · 2019. 7. 31. · Research Article Asymptotic Properties of Solutions to Third-Order Nonlinear Neutral Differential