OSCILLATION CRITERIA FOR THIRD ORDER NEUTRAL NONLINEAR DYNAMIC EQUATIONS WITH DISTRIBUTED DEVIATING ARGUMENTS ON TIME SCALES. 1.

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1 t m Mthemticl Pblictions DOI: /tmmp Ttr Mt. Mth. Pbl ), OSCILLATION CRITERIA FOR THIRD ORDER NEUTRAL NONLINEAR DYNAMIC EQUATIONS WITH DISTRIBUTED DEVIATING ARGUMENTS ON TIME SCALES Ther S. Hssn Sid R. Grce ABSTRACT. Some new oscilltion criteri for third order netrl nonliner dynmic eqtions with distribted deviting rgments on time scles re estblished. The obtined reslts extend, improve nd correlte mny known oscilltion reslts for third order dynmic eqtions. 1. Introdction This pper is concerned with the oscilltory behvior of third order netrl nonliner dynmic eqtions with distribted deviting rgments on time scles d ΔΔ αδ t) xt)+ pt, θ)x ht, θ) ) b Δθ + qt, τ)x λ gt, τ) ) Δτ =0 1.1) c nd t) x β h 1 t) ) rt)x γ h 2 t) )) ΔΔ ) α) Δ + b qt, τ)x λ gt, τ) ) Δτ =0, 1.2) on n rbitrry time scle T R with sp T =, 0 <<bnd 0 <c<d. We ssme tht: i) α, β, γ nd λ re the rtios of positive odd integers; c 2014 Mthemticl Institte, Slovk Acdemy of Sciences M t h e m t i c s Sbject Clssifiction: 34C10, 34C15, 34N05, 34K11, 39A10. K e y w o r d s: oscilltion, third order, netrl, dynmic eqtions, time scles. 141

2 TAHER S. HASSAN SAID R. GRACE ii), r : T [0, ), t) > 0 re rel vled, rd-continos fnctions nd t 0 s)δs =, t 0 T; 1.3) iii) q : T [, b] [0, ) ndp : T [c, d] [0, ) re rel vled, rd-continos fnctions; iv) g : T [, b] T nd h : T [c, d] T re nonincresing with respect to second vrible, gt, τ) t nd lim gt, τ) =, τ [, b], t nd ht, θ) t nd lim ht, θ) =, θ [c, d]; t v) h i : T T for i = 1, 2, re rel vled, rd-continos nondecresing fnctions sch tht h i t) t for t t 0 T nd lim t h i t) = for i =1, 2. We recll tht soltion x of the eqtion 1.1) respectively the eqtion 1.2)) is sid to be nonoscilltory if there exists t 0 T sch tht xt)x σt) ) > 0 for ll t [t 0, ) T ; otherwise, it is sid to be oscilltory. The eqtion 1.1) respectively the eqtion 1.2)) is sid to be oscilltory if ll its extendible soltions re oscilltory. Netrl differentil eqtions pper in modelling of the networks contining lossless trnsmission lines, in the stdy of vibrting msses ttched to n elstic br, s the Eler eqtion in some vritionl problems. In the theory of tomtic control nd in nero-mechmil systems in which inerti plys n importnt role; see [11]. In recent yers, there hs been mch reserch ctivity concerning the oscilltion theory nd pplictions of dynmic eqtions, see [1] [10], [13] [18] nd the references contined therein. Prticlrly, the stdy content of oscilltory criteri of first nd second dynmic eqtions on time scles is rich. In contrst, the stdy of oscilltion criteri of third order dynmic eqtions is reltively less. Some interesting reslts hve been obtined concerning the oscilltory nd symptotic behvior of some specil cses of the eqtions 1.1) nd 1.2); see [9], [12]. To the best of or knowledge, the oscilltory behvior of 1.1) nd 1.2) hve not been stdied p to now. The prpose of this pper is to estblish some new criteri for the eqtions 1.1) nd 1.2) by sing the pproch to redce the problem is sch wy tht specific oscilltion reslts for first nd second order dynmic eqtions cn be 142

3 THIRD ORDER NONLINEAR OSCILLATION dpted for the third order cse. In Section 2, we investigte the oscilltory behvior of the eqtion 1.1) while Section 3 is devoted to stdy of oscilltory properties of the eqtion 1.2). The obtined reslts extend, improve nd correlte mny of the known oscilltion reslts ppered in the litertre tht del with specil cses of the eqtions 1.1) nd 1.2). 2. Oscilltion of the eqtion 1.1) In this section we begin with the following lemms tht re essentil in the proofs of or reslts. For simplicity in wht follows, whenever we write t we men t [, ) T. It will be convenient to set yt) :=xt)+ d c pt, θ)x ht, θ) ) Δθ. 2.1) Eqtion 1.1) cn be written s t) y ΔΔ t) ) ) b α Δ + qt, τ)x λ gt, τ) ) Δτ =0. 2.2) Lemm 2.1. Let the condition 1.3) hold nd ssme tht x is n eventlly positive soltion of the eqtion 1.1). Then there re only two possible cses for the behvior of y for lrge t t 0 : I) yt) > 0, y Δ t) > 0, y ΔΔ t) > 0nd t) y ΔΔ t) ) α) Δ 0; II) yt) > 0, y Δ t) < 0, y ΔΔ t) > 0nd t) y ΔΔ t) ) α) Δ 0. Lemm 2.2. Let condition 1.3) hold nd ssme tht x is n eventlly positive soltion of eqtion 1.1) nd the corresponding y stisfies Cse I) of Lemm 2.1. If d 0 P t) := pt, θ)δθ p < 1, 2.3) then c xt) 1 p )yt), eventlly. 2.4) Proof. Since x is n eventlly positive soltion of the eqtion 1.1) nd the corresponding y is stisfying Cse I) of Lemm 2.1. Then there exists t 0 T sch tht xt) > 0, x ht, θ) ) > 0 nd y Δ t) > 0 for t nd θ [c, d]. 143

4 TAHER S. HASSAN SAID R. GRACE Now, d xt) = yt) pt, θ)x ht, θ) ) Δθ c d yt) pt, θ)y ht, θ) ) Δθ c yt) = 1 d c d c pt, θ)δθ y t) pt, θ)δθ y t) for t. This completes the proof. Lemm 2.3. Let the condition 1.3) hold nd ssme tht x is n eventlly positive soltion of the eqtion 1.1) nd the corresponding y stisfies Cse I) of Lemm 2.1. Then for t T yt) t) y ΔΔ t) ) ) α t s)δsδ. 2.5) Proof. Since x is n eventlly positive soltion of the eqtion 1.1) nd the corresponding y is stisfying Cse I) of Lemm 2.1. Then there exists t 0 T sch tht xt) > 0 nd x ht, θ) ) > 0 for t nd θ [c, d]. By sing the fct tht t) y ΔΔ t) ) α is decresing for t t1,wehve y Δ t) y Δ t) y Δ ) = s) y ΔΔ s) ) ) α s)δs t) y ΔΔ t) ) ) α s)δs. Integrting this ineqlity from to t, we obtin the desired reslt. 144

5 Let Qt) := b THIRD ORDER NONLINEAR OSCILLATION qt, τ)δτ, g 1 t) :=gt, ) nd g 2 t) :=gt, b). 2.6) In the following reslt, we employ the following xiliry eqtion g 2 t) λ z Δ t)+1 p ) λ Qt) z λ/α g 2 t) ) =0 2.7) for t, [t 0, ) T. s)δsδ Lemm 2.4. Let the conditions 1.3) nd 2.3) hold nd the eqtion 2.7) be oscilltory, then the eqtion 1.1) hs no eventlly positive soltion x sch tht y defined by 2.1) stisfies Cse I) of Lemm 2.1. Proof. Let t 0 T be sfficiently lrge sch tht xt) > 0, x ht, θ) ) > 0nd x gt, τ) ) > 0 for t t 0, θ [c, d] ndτ [, b] nd ssme tht y stisfies Cse I) of Lemm 2.1. Using 2.4) nd 2.5) in the eqtion 2.2), we get g 2 t) λ z Δ t)+1 p ) λ Qt) z λ/α g 2 t) ) 0, 2.8) s)δsδ for [t 0, ) T,wherezt) :=t) y ΔΔ t) ) α > 0. Integrting 2.8) from t to t nd letting,wehve zt) G t, zt) ), where G t, zt) ) g 2 v) λ := 1 p ) λ Qv) z λ/α g 2 v) ) Δv. t s)δsδ Now, we define seqence of sccessive pproximtions { w j t) } s follows: w 0 t) :=zt), w j+1 t) :=G t, w j t) ), j =0, 1, 2,... It is esy to show tht 0 <w j t) zt) nd w j+1 t) w j t), j =0, 1, 2,... Then, the seqence { w j t) } is nonincresing nd bonded for ech t. This mens tht we my define wt) := lim j w j t) 0. Since 0 wt) w j t) zt) for ll j

6 TAHER S. HASSAN SAID R. GRACE By the Lebesge s dominted convergence theorem on time scle, one cn esily find wt) =G t, wt) ). Therefore, g 2 t) λ w Δ t) = 1 p ) λ Qt) w λ/α g 2 t) ). s)δsδ Hence the eqtion 2.7) hs positive soltion wt). This completes the proof. In the cse when λ = α in the ineqlity 2.8), we stte the following lemm. Lemm 2.5 [1]). If λ = α, lim sp t { sp ξe ξη t, g2 t) )} < 1, ξ E where E := { ξ : ξ>0, 1 ξηt)μt) > 0 }, nd g 2 t) α ηt) :=1 p ) α Qt) s)δsδ, then the ineqlity 2.8) hs no eventlly positive soltion. Lemm 2.6. Let the condition 1.3) hold nd ssme tht xt) is n eventlly positive soltion the eqtion 1.1) nd the corresponding y stisfies Cse I) of Lemm 2.1. Then there exists [t 0, ) T sch tht yt) y Δ t)at, ) for t, ) T, 2.9) where t At, ):= 1 s)δs Δ. s)δs Proof. Let t 0 T be sfficiently lrge sch tht xt) > 0, x ht, θ) ) > 0nd x gt, τ) ) > 0 for t t 0, θ [c, d] ndτ [, b] nd ssme tht y stisfies Cse I) of Lemm 2.1 for t t 0. From Lemm 2.3, we hve y Δ t) Note tht [ y Δ t) s)δs 146 t) y ΔΔ t) ) ) α t s)δs for t t 0. ] Δ = t) [ t) y ΔΔ t) ) α) s)δs y Δ t) ] s)δs σt) s)δs,

7 THIRD ORDER NONLINEAR OSCILLATION we hve [ ] Δ y Δ t) < 0 s)δs for t, ) T. Then yt) yt) y ) y Δ ) = s)δs Δ s)δs y Δ t) s)δs = y Δ t)at, ). This completes the proof. s)δsδ For g 2 t) >t 0,weset Qt) :=1 p ) λ A λ ) g 2 t),t 0 Qt). The hypotheses of next two lemms inclde knowledge of the behvior of the soltion of the second order dynmic eqtion t) z Δ t) ) ) α Δ + λ Qt)z g 2 t) ) =0, 2.10) Lemm 2.7. If the conditions 1.3) nd 2.3) hold nd the eqtion 2.10) is oscilltory, then the eqtion 1.1) hs no eventlly positive soltion x sch tht y is defined by 2.1) which stisfies Cse I) of Lemm 2.1. Proof. Let xt) be n eventlly positive soltion of the eqtion 1.1), sy xt) > 0ndx ht, θ) ) > 0ndx gt, τ) ) > 0 for t for some [t 0, ) T, θ [c, d] ndτ [, b] nd ssme tht y stisfies Cse I) of Lemm 2.1. From 2.9), there exist constnt k 1,0<k 1 < 1ndt 2 > sch tht yt) At, )y Δ t) for t t ) From 2.4), we see tht xt) 1 p ) yt) for t t 3, 2.12) for some t 3 t 2. Hence, there exists t 4 t 3 sch tht y g 2 t) ) A ) g 2 t), y Δ g 2 t) ) for t t ) Using 2.12) nd 2.13) in the eqtion 2.2) we hve t) z Δ t) ) ) α Δ + λ Qt)z g 2 t) ) 0, 2.14) 147

8 TAHER S. HASSAN SAID R. GRACE for t t 4,wherezt) :=y Δ t) > 0. Integrting 2.14) from t to t t 4 nd letting,weobtin z Δ t) 1 Qs)z λ g 2 s) ) Δs. 2.15) t) Integrting 2.15) from t 4 to t t 4,weobtin zt) zt 4 )+ 1 Qs)z λ g 2 s) ) Δs ) t 4 Next, we define seqence {w m t)} m N0 by w 0 t) =zt), w m+1 t) =zt 4 )+ 1 Qs)z λ g 2 s) ) Δs ) t 4 t Δ. Δ, m N 0. It is esy to check by indction tht {w m t)} is well-defined decresing seqence stisfying zt 4 ) w m t) zt) for t t 4 nd m N 0. Ths, there exists fnction w on [t 4, ) T sch tht lim w mt) =wt) nd zt 4 ) wt) zt). m By the Lebesge s dominted convergence theorem on time scle, it follows tht wt) =zt 4 )+ 1 Qs)w λ g 2 s) ) Δs Δ. 2.16) ) t 4 Differentiting 2.16) twice, we conclde tht w is nonoscilltory soltion of the eqtion 2.10) with the desired property. This completes the proof of the lemm. The following lemm mkes se of the xiliry eqtion 1 v Δ t) ) ) Δ α +1 p ) λ Qt)v λ g Ā α 2 t) ) = 0 for t t 2, 2.17) t) where Āt) := s)δs for sfficiently lrge t 2, ) T for some [t 0, ) T. Lemm 2.8. If the conditions 1.3) nd 2.3) hold nd the eqtion 2.17) is oscilltory, then the conclsion of Lemm 2.7 holds. 148

9 THIRD ORDER NONLINEAR OSCILLATION Proof. Let xt) be n eventlly positive soltion of the eqtion 1.1), sy xt) > 0ndx ht, θ) ) > 0ndx gt, τ) ) > 0 for t for some [t 0, ) T, θ [c, d] ndτ [, b] ndlety stisfy Cse I) of Lemm 2.1. Hence y Δ t) =y Δ )+ s)y ΔΔ s) s) Δs t)y ΔΔ t) Δs s) )Δ. 2.18) = t)y ΔΔ t)āt) for t. An integrtion yields yt) y )+ Ā) )y ΔΔ Using 2.4) in the eqtion 2.2), integrting from to v nd letting v,wehve y ΔΔ 1 ) 1 p ) λ Qs)y λ g 2 s) ) Δs. 2.19) ) Sbstitting 2.19) into 2.18) gives yt) yt 2 )+ Ā) 1 p ) λ Qs)y λ g 2 s) ) Δs Δ Next we define the seqences {v m t)} m N0 by v 0 t) =yt), v m+1 t) =y )+ Ā) 1 p ) λ Qs)y λ g 2 s) ) Δs Δ, m N 0. The reminder of the proof is similr to the proof of Lemm 2.7 nd is omitted. Next, we present the following reslt. Lemm 2.9. Let the conditions 1.3) nd 2.3) hold nd ssme tht x is n eventlly positive soltion of the eqtion 1.1) nd the corresponding y stisfies Cse II) of Lemm 2.1. Then either 1 p ) xt) yt), 2.20) eventlly, where >1 is ny constnt with p <1 nd p is s in 2.3), or lim t xt) =0. 149

10 TAHER S. HASSAN SAID R. GRACE Proof. Pick t 0 T sch tht xt) > 0 nd x ht, θ) ) > 0 for t t 0 nd θ [c, d]. Also, since yt) stisfies Cse II) of Lemm 2.1, then there exists constnt k sch tht lim yt) =k<. t i) Assme tht k>0, then we hve k<yt) <k, for ll >1 nd t t ) Now, d xt) =yt) pt, θ)x ht, θ) ) Δθ, nd so c 1 p xt) k kp ) 1 p ) = k yt) for t t 0. ii) Assme tht k = 0, then lim t yt) = 0. Since 0 <xt) yt) on[t 0, ) T, then lim t xt) = 0. This completes the proof of the lemm. Lemm Let the conditions 1.3) nd 2.3) hold nd ssme tht xt) is n eventlly positive soltion of the eqtion 1.1) nd the corresponding y stisfies Cse II) of Lemm 2.1. If 1 Qs)Δs ΔΔv =, 2.22) ) then lim t xt) =0. t 0 v P r o o f. Proceeding s in the proof of Lemm 2.9, we obtin either 2.20) holds or lim t xt) = 0. We sppose tht 1 p ) xt) yt), for t t ) Using 2.23) in the eqtion 2.2), we hve t) y ΔΔ t) ) ) b α Δ = qt, τ) x λ gt, τ) ) Δτ p ) λ b qt, τ) y λ gt, τ) ) Δτ 1 p ) λ Qt)y λ g 1 t) ) for t t )

11 THIRD ORDER NONLINEAR OSCILLATION Integrting this ineqlity from t to t nd letting,wehve 1 p y ΔΔ ) λ/α t) 1 Qs)y λ g 1 s) ) Δs. 2.25) t) Using 2.21) in 2.25), we get y ΔΔ t) c 1 t) t t Qs)Δs, 2.26) where c := k [ ]) 1 p λ/α. Integrting 2.26) twice, we obtin >y ) c 1 Qs)Δs ΔΔv s t, ) v which is contrdiction. This completes the proof of the lemm. Lemm Let g 1 be nondecresing on [t 0, ) T, conditions 1.3) nd 2.3) hold nd ssme tht xt) is n eventlly positive soltion of the eqtion 1.1) nd the corresponding y stisfies Cse II) of Lemm 2.1. If lim sp t g 1 t) 1 ) Qs) [ g 1 t) g 1 s) ] λ Δs where Q nd g 1 re s in 2.6), c := lim t xt) =0. Δ > { c if λ = α, 0 if λ < α, 2.27) 1 p, p nd re s in Lemm 2.9, then P r o o f. Proceeding s in the proof of Lemm 2.10, we obtin 2.24). We lso hve y g 1 s) ) y g 1 t) ) y g 1 s) ) g 1 t) = y Δ τ)δτ g 1 s) y Δ g 1 t) ) g1t) Δτ = y Δ g 1 t) )[ g 1 t) g 1 s) ], 2.28) g 1 s) for t s t 0. Integrting 2.24) from to t t 0,weobtin 1 p y ΔΔ ) λ/α ) 1 Qs)y λ g 1 s) ) Δs. 2.29) ) 151

12 TAHER S. HASSAN SAID R. GRACE Sbstitting 2.28) into 2.29), we obtin 1 p y ΔΔ ) λ/α ) 1 Qs) [ g 1 t) g 1 s) ] λ Δs y Δ g 1 t) )) λ/α. ) Integrting from g 1 t) t 0 to t gives y Δ g 1 t) ) y Δ t) y Δ g 1 t) ) 1 p ) λ/α y Δ g 1 t) )) λ/α t 1 ) g 1 t) Qs) [ g 1 t) g 1 s) ] λ Δs Δ. So y Δ g 1 t) )) 1 λ/α 1 p ) t λ/α g 1 t) 1 ) Qs) [ g 1 t) g 1 s) ] λ Δs Δ. 2.30) Tking lim sp s t of both sides of the bove ineqlity. If λ = α, the contrdiction is obvios. If λ<α, then the left hnd side of 2.30) is positive nd mst decrese to zero to prevent contrdiction to the positivity of yt)). This contrdicts 2.27) nd completes the proof of the lemm. Lemm Let the hypotheses of Lemm 2.11 hold with the condition 2.27) be replced by lim sp 1 /α { Qs)Δs c if λ = α, ΔΔv > 2.31) t ) 0 if λ < α. g 1 t) v Then the conclsion of Lemm 2.11 holds. P r o o f. As in the proof of Lemm 2.11, we obtin 2.29) nd integrting we hve y Δ v) y Δ t) y Δ v) 1 p ) λ/α y λ/α g 1 t) ) 1 ) Integrting from g 1 t) t 0 to t yields 152 y 1 λ/α g 1 t) ) 1 p ) λ/α t g 1 t) v v 1 ) Qs)Δs Δ. Qs)Δs ΔΔv.

13 THIRD ORDER NONLINEAR OSCILLATION Tking lim sp s t gives contrdiction to the condition 2.31). This completes the proof of the lemm. We re now redy to present the min reslts in this section. Theorem 2.1. Let 1.3), 2.3), 2.22) nd either one of the eqtions 2.7), 2.10) or 2.17) be oscilltory, then every soltion xt) of the eqtion 1.1) oscilltes or stisfies lim t xt) =0. Proof. Let xt) be n eventlly positive soltion of the eqtion 1.1), sy xt) > 0 nd x ht, θ) ) > 0 nd x gt, τ) ) > 0 for t for some [t 0, ) T, θ [c, d] ndτ [, b]. Then yt) stisfies one of the two cses of Lemm 2.1. By either Lemms 2.4, 2.7 or 2.8, Cse I) cnnot hold. If Cse II) holds, Lemm 2.10 implies lim t xt) =0.Thisprovesthe theorem. Next, we estblish nother new oscilltion criteri for the eqtion 1.1). Theorem 2.2. Let λ α, g 1 be nondecresing on [t 0, ) T, 1.3) nd 2.3) hold. If either one of the eqtions 2.7), 2.10) or 2.17) is oscilltory, nd condition 2.27) or 2.31) holds, then every soltion xt) of the eqtion 1.1) oscilltes or stisfies lim t xt) =0. Proof. Let xt) be n eventlly positive soltion of the eqtion 1.1), sy xt)>0 nd x ht, θ) ) >0 nd x gt, τ) ) >0 for t for some [t 0, ) T, θ [c, d] ndτ [, b]. Then yt) stisfies one of the two cses of Lemm 2.1. By either Lemms 2.4, 2.7 or 2.8, Cse I) cnnot hold. By Lemm 2.11 or Lemm 2.12), Cse II) does not hold. This completes the proof of the theorem. Remrk 1. Or reslts of this section remin vlid of gt, τ) is nondecresing in the second vrible. In this cse, we replce nd g 1 t) by ḡ 1 t) =gt, b) g 2 t) by ḡ 2 t) =gt, ). Remrk 2. Wemypply Lemm2.5toeqtion 2.7) with λ = α. This detils re left to the reders. 153

14 TAHER S. HASSAN SAID R. GRACE 3. Oscilltion of the eqtion 1.2) We begin with the following lemms tht re essentil in the proof of or theorems. It will be convenient to set yt) =x β h 1 t) ) rt)x γ h 2 t) ). 3.1) The eqtion 1.2) cn then be written s t) y ΔΔ t) ) ) b α Δ + q t, τ) x λ gt, τ) ) Δτ =0. 3.2) Lemm 3.1. Let the condition 1.3) hold nd ssme tht x is n eventlly positive soltion of the eqtion 1.2). Then there re only three possible cses for the behvior of y for lrge t t 0 : I) yt) > 0, y Δ t) > 0, y ΔΔ t) > 0, nd t) y ΔΔ t) ) α) Δ 0; II) yt) > 0, y Δ t) < 0, y ΔΔ t) > 0, nd t) y ΔΔ t) ) α) Δ 0; III) yt) < 0, y Δ t) < 0, y ΔΔ t) > 0, nd t) y ΔΔ t) ) α) Δ 0. If cse I) or II) holds, then we find xt) y 1/β h 1 1 t)). 3.3) Using 3.3) in the eqtion 3.2), we hve t) y ΔΔ t) ) ) b α Δ + q t, τ) y λ/β h 1 1 gt, τ) ) Δτ 0, 3.4) nd when Cse III) holds, we see tht 0 <zt) = yt) =rt)x γ h 2 t) ) x β h 1 t) ) rt)x γ h 2 t) ), nd so z h 1 2 xt) t)) ) 1/γ r h 1 2 t)), 3.5) nd the eqtion 3.2) becomes t) z ΔΔ t) ) ) b α Δ q t, τ) r λ/γ h 1 2 gt, τ) ) z λ/γ h 1 2 gt, τ) ) Δτ. 3.6) Next, we ssme tht vi) ζ 1 t, τ) =h 1 1 gt, τ) t for τ [, b], ζ 1 is nonincresing with respect to the second vrible nd lim t ζ 1 t, τ) = ; 154

15 THIRD ORDER NONLINEAR OSCILLATION vii) ζ 2 t, τ) =h 1 2 gt, τ) t for τ [, b], ζ 2 is nonincresing with respect to the second vrible nd lim t ζ 2 t, τ) =. Also, we set Qt) := b qt, τ)δτ, ˆζ1 t) :=ζ 1 t, ), ζ1 t) :=ζ 1 t, b), 3.7) nd ˆζ2 t) :=ζ 2 t, ), ζ 2 t) :=ζ 2 t, b). 3.8) Now, if y stisfies Cse I) of Lemm 3.1, then 3.4) becomes t) y ΔΔ t) ) ) α Δ + Q t) y λ/β ζ1 t) ) 0, 3.9) where Q is s in 2.6). When y stisfies Cse II) of Lemm 3.1, then 3.4) becomes t) y ΔΔ t) ) α ) Δ + Q t) y λ/β ˆζ1 t) ) ) If y stisfies Cse III) of Lemm 3.1, then 3.6) tkes the form t) z ΔΔ t) ) α ) Δ ˆQ t) z λ/γ ζ2 t) ), 3.11) where ˆQ t) := b q t, τ) r λ/γ ζ 2 t, τ) ) Δτ. 3.12) As direct conseqence of Lemms 2.4, 2.6, 2.7, 2.8, 2.10, 2.11 nd 2.12, we get the following reslts. Lemm 3.2. Let the condition 1.3) hold nd eqtion ζ 1 t) λ/β z Δ t)+qt) z λ/αβ) g 2 t) ) =0 for t, 3.13) s)δsδ for [t 0, ) T, is oscilltory, then the eqtion 1.2) hs no eventlly positive soltion x sch tht y defined by 3.1) stisfies Cse I) of Lemm 3.1. Lemm 3.3. Let the condition 1.3) hold nd ssme tht xt) is n eventlly positive soltion of the eqtion 1.2) nd the corresponding y stisfies Cse I) of Lemm 3.1. Then there exists [t 0, ) T sch tht 2.9) holds. Lemm 3.4. If the condition 1.3) holds nd the eqtion t) z Δ t) ) α ) Δ + Qt)z λ g 2 t) ) =0, 3.14) where for ζ 1 t) >t 0,weset Qt) :=A λ ζ1 t),t 0 ) Qt), 155

16 TAHER S. HASSAN SAID R. GRACE is oscilltory, then the eqtion 1.2) hs no eventlly positive soltion x sch tht y is defined by 3.1) stisfies Cse I) of Lemm 3.1. Lemm 3.5. If the condition 1.3) hold nd the eqtion 1 v Δ t) ) ) Δ α + Qt)v λ g Ā α 2 t) ) =0 for t t 2, 3.15) t) where Āt) := s)δs for sfficiently lrge t 2, ) T for some [t 0, ) T, is oscilltory, then the conclsion of Lemm 3.4 holds. Lemm 3.6. Let the condition 1.3) hold nd ssme tht xt) is n eventlly positive soltion of the eqtion 1.2) nd the corresponding y stisfies Cse II) of Lemm 3.1. If 2.22) holds, then lim t xt) =0. Lemm 3.7. Let ˆζ 1 be nondecresing on [t 0, ) T, the condition 1.3) hold nd ssme tht xt) is n eventlly positive soltion of the eqtion 1.2) nd the corresponding y stisfies Cse II) of Lemm 3.1. If lim sp 1 Qs) [ˆζ1 t) t ) ˆζ 1 s) ] { λ/β Δs 1 if λ = αβ, Δ > 3.16) 0 if λ < αβ, ˆζ 1 t) then lim t xt) =0. Lemm 3.8. Let the hypotheses of Lemm 3.7 hold with the condition 3.16) be replced by lim sp 1 /α { Qs)Δs 1 if λ = αβ, ΔΔv > 3.17) t ) 0 if λ < αβ. ˆζ 1 t) v Then the conclsion of Lemm 3.7 holds. In the following two lemms, we consider the second order dely dynmic eqtion t) w Δ t) ) α ) Δ = d ζ2 t) ) λ/γ ˆQ t) w λ/γ ζ2 t) ), 3.18) where ζ 2 nd ˆQ re s in 3.8) nd 3.12) respectively,, α, γ, λ re s in the eqtion 1.1) nd d >0isconstnt. Lemm 3.9. Let ζ 2 be nondecresing on [t 0, ) T nd the condition 1.3) hold. If λγ ζ lim sp ζ2 s) ) 2 t) λ/γ ˆQ s) τ)δτ Δs >{ 1 d if λ = αγ, 3.19) t 0 if λ < αγ, ζ 2 t) ζ 2 s) then ll bonded soltions of the eqtion 3.18) re oscilltory. 156

17 THIRD ORDER NONLINEAR OSCILLATION Proof. Letwt) be bonded nonoscilltory soltion of the eqtion 3.18), sy wt) > 0 for t for some t 0 T. Then there exists t 2 [, ) T sch tht wt) > 0, w Δ t) < 0 nd t) w Δ t) ) ) α Δ > 0 for t t ) Now for v t 2,wehve w) w) wv) = v v) w Δ v) ) ) α w Δ τ)δτ = v v τ) τ) w Δ τ) ) α ) Δτ τ)δτ. 3.21) For t s t 2, setting = ζ 2 s) ndv = ζ 2 t) in the ineqlity 3.21) gives w ζ2 s) ) ζ 2 t)) w Δ ζ 2 t)) ) ) ζ 2 t) α τ)δτ. 3.22) ζ 2 s) Integrting the eqtion 3.18) from ζ 2 t) t 2 to t, weobtin ζ2 t) ) w Δ ζ2 t) )) α t) w Δ t) ) α ζ2 t) ) w Δ ζ2 t) )) α = ζ 2 t) d ζ2 s) ) λ/γ ˆQs)w λ/γ ζ2 s) ) Δs. 3.23) Using 3.22) in 3.23), one cn esily see tht ζ2 t) ) w Δ ζ2 t) )) α ζ2 t) ) w Δ ζ2 t) )) α ) λ αγ d ζ2 s) ) λ/γ ˆQs) λ/γ ζ 2 t) τ)δτ Δs, or [ ζ2 t) ) w Δ ζ2 t) )) ] α 1 λ ζ 2 t) αγ t ζ 2 t) ζ 2 s) λ/γ ζ d ζ2 s) ) 2 t) λ/γ ˆQs) τ)δτ Δs. 3.24) Now tke the lim sp s t of both sides of the bove ineqlity. If λ = αγ the contrdiction is obvios. If λ<αγthe left hnd side of 3.24) is positive nd mst decrese to zero to present contrdiction to the positivity of wt)). This contrdicts 3.19) nd completes the proof of the lemm. ζ 2 s) 157

18 TAHER S. HASSAN SAID R. GRACE Lemm Let ζ 2 be nondecresing on [t 0, ) T nd the condition 3.19) in Lemm 3.9 be replced by lim sp 1 ζ2 τ) ) { λ/γ ˆQ τ)δτ Δs d if λ = αγ, > 3.25) t s) 0 if λ < αγ. ζ 2 t) s Then the conclsion of Lemm 3.9 holds. Proof. Letwt) be bonded nonoscilltory soltion of the eqtion 3.18), sy wt) > 0 for t t 0 T. As in the proof of Lemm 3.9, we obtin 3.20) for t t 2 for some t 2 [, ) T. Integrting 3.18) from t 2 to t, we hve t) w Δ t) ) α ) w Δ ) ) t α = d ζ2 s) ) λ/γ λ/γ ˆQs)w ζ2 s) ) Δs, or w Δ ) w λ αγ ζ2 t) ) 1 ) d ζ2 s) ) λ/γ ˆQs)Δs. Integrting this ineqlity from ζ 2 t) tot, weobtin w ζ2 t) ) w λ αγ ζ2 t) ) 1 d ζ2 s) ) λ/γ ˆQs)Δs Δ, ) or w 1 λ αγ ) ζ2 t) ) ζ 2 t) ζ 2 t) 1 ) d ζ2 s) ) λ/γ ˆQs)Δs Δ. Tking the lim sp s t of both sides of the bove ineqlity, we gin obtin contrdiction s in the previos lemm. We re now redy to estblish the min reslts of this section. Theorem 3.1. Let λ αγ, λ αβ, ˆζ 1 nd ζ 2, i = 1, 2 be nondecresing on [t 0, ) T nd the condition 1.3) hold. If either one of the dynmic eqtions 3.13), 3.14) or 3.15) is oscilltory, the condition 3.16) or the condition 3.17)) holds nd the condition 3.19) or the condition 3.25)) is stisfied with 0 < d <1, then every soltion xt) of the eqtion 1.2) oscilltes or stisfies lim t xt) =0. Proof. Let xt) be n eventlly positive soltion of the eqtion 1.2), sy xt) > 0 nd x h i t) ) > 0, i = 1, 2 nd x gt, τ) ) > 0 for t 158

19 THIRD ORDER NONLINEAR OSCILLATION for some [t 0, ) T nd τ [, b]. Then yt) defined by 3.1) stisfies one of the three cses of Lemm 3.1 nd stisfies the eqtion 3.2). If Cse I) holds, then 3.3) holds. Now sing 3.3) in the eqtion 3.2), we obtin the ineqlity 3.9). By either Lemm 3.2, Lemm 3.4 or Lemm 3.5, Cse I) cnnot hold. Similrly, when Cse II) one cn esily obtin the ineqlity 3.10), nd by Lemm 3.7 or Lemm 3.8, Cse II) cnnot hold. Finlly, if Cse III) holds, we let 0 <zt) = yt) nd proceed s bove to obtin the ineqlity 3.11). It is esy to check tht z stisfies zt) > 0, z Δ t) > 0, z ΔΔ t) < 0 nd t) z ΔΔ t) ) ) α Δ 0, for t t 2. Now, there exists constnt d 0, 1) sch tht Using 3.26) in 3.11), we get z ζ2 t) ) d ζ 2 t)z Δ ζ2 t) ) for t t 3 t ) or t) z ΔΔ t) ) α ) Δ ˆQt) d ζ2 t)z Δ ζ2 t) )) λ/γ for t t 3, t) w Δ t) ) α ) Δ d) λ/γ ζ2 t) ) λ/γ ˆQt)w λ/γ ζ2 t) ) for t t 3, where wt) :=z Δ t). Proceeding s in the proofs of Lemm 3.9 nd Lemm 3.10, we rrive t the desired conclsion completing the proof of the theorem. Remrk 3. We my note tht Theorem 2.1 is lso pplicble to the eqtion 1.2). The detils re omitted. When p t, θ) 0 in the eqtion 1.1) or β =1,rt) =0ndh 1 t) =t in the eqtion 1.2), both eqtions re redced to t) x ΔΔ t) ) ) b α Δ + qt, τ)x λ gt, τ) ) Δτ = ) In this cse we hve the following new reslt. Theorem 3.2. Let λ α nd 1.3) hold. If either one of the eqtions 2.7) with p 0, 2.10) or 2.17) with p 0 is oscilltory nd the condition 2.27) or the condition 2.31)) holds, then every soltion xt) of the eqtion 3.27) oscilltes or stisfies lim t xt) =0. P r o o f. The conclsion follows from Theorem 2.2 or 3.1 nd is omitted. 159

20 TAHER S. HASSAN SAID R. GRACE 4. Generl remrks 1) The reslts of this pper re presented in form tht is essentilly new nd of high degree of generlity. 2) We note tht there re mny criteri in the litertre of first nd second order dynmic eqtions nd so by pplying these reslts to the eqtions 2.7), 2.10) nd 2.17), we cn obtin mny oscilltion reslts, more tht those known in the litertre. Here we omit the detils. 3) The reslts here re vlid for vrios type of time scles, e.g., T = R, T = Z, T = hz with h>0, T = q N 0 with q>1, T = N 2 0, etc. see [2]). 4) We note tht or reslts on the symptotic behvior of soltions re pplicble to the eqtions 1.1) nd 1.2) if gt, τ) t, τ [, b] while or oscilltion reslts re pplicble to the eqtions 1.1) nd 1.2) if gt, τ) <t, τ [, b]. Ths s it hs been known, there is the dely in the eqtions 1.1) nd 1.2) which cn generte oscilltions. 5) Or reslts of Section 2 re new nd or reslts of Section 3 inclde, extend nd improve the reslts in [9] nd [12]. 6) Finlly, it wold be of interest to consider the eqtions 1.1) nd 1.2) try to obtin some oscilltion criteri if other pproprite conditions on the fnctions pt, θ), θ [c, d], rt), etc. REFERENCES [1] BOHNER, M.: Some oscilltion criteri for first order dely dynmic eqtions, FrEst J. Appl. Mth ), [2] BOHNER, M. PETERSON, A.: Dynmic Eqtions on Time Scles: An Introdction with Applictions. Birkhäser, Boston, [3] DURINA, J. BACULIKOVA, B.: Property A) of third-order dvnced differentil eqtions, Mth. Slovc ), [4] GRACE, S. R. AGARWAL, R. P. AKTAS, M. F.: On the oscilltion of third order fnctionl differentil eqtions, Indin J. Pre Appl. Mth ), [5] GRACE, S. R. AGARWAL, R. P. PAVANI, R. THANDAPANI, E.: On the oscilltion of certin third order nonliner fnctionl differentil eqtions, Appl. Mth. Compt ), [6] GRACE, S. R. AGARWAL, R. P. BOHNER, M. O REGAN, D.: Oscilltion of second-order strongly sperliner nd strongly sbliner dynmic eqtions, Commn. Nonliner Sci. Nmer. Siml ), [7]GRACE,S.R. BOHNER,M. AGARWAL,R.P.:On the oscilltion of second-order hlf-liner dynmic eqtions, J. Difference Eq. Appl ), [8] GRACE, S. R. R. P. AGARWAL, R. P. KAYMAKALAN, B. SAE-JIE, W.: On the oscilltion of certin second order nonliner dynmic eqtions, Mth. Compt. Modelling ),

21 THIRD ORDER NONLINEAR OSCILLATION [9] GRACE, S. R. GRAEF, J. R. EL-BELTAGY, M. A.: On the oscilltion of third order netrl dely dynmic eqtions on time scles, Compt. Mth. Appl ), [10] GRAEF, J. R. PANIGRAHI, S. REDDY, P. R.: On oscilltory nd symptotic behvior of forth order nonliner netrl dely dynmic eqtions with positive nd negtive coefficients, Mth. Slovc ), [11] HALE, J. K.: Theory of Fnctionl Differentil Eqtions. Springer-Verlg, New York, [12] HAN, Z. LI, T. SUN, T. ZHANG, C.: Oscilltion behvior of third-order netrl Emden-Fowler dely dynmic eqtions on time scles, Adv. Difference Eq. 2010, Art. ID , 23 pp. [13] ELABBASY, E. M. HASSAN, T. S.: Oscilltion of soltions for third order fnctionl dynmic eqtions, Electron. J. Differentil Eqtions 2010, No. 131, 14 pp. [14] ERBE, L. PETERSON, A. SAKER, S. H.: Hille nd Nehri type criteri for third order dynmic eqtions, J. Mth. Anl. Appl ), [15] ERBE, L. HASSAN, T. S. PETERSON, A.: Oscilltion of third order fnctionl dynmic eqtions with mixed rgments on time scles, J. Appl. Mth. Compt ), [16] HASSAN, T. S.: Oscilltion criteri for hlf-liner dynmic eqtions on time scles, J. Mth. Anl. Appl ), [17] HASSAN, T. S.: Oscilltion of third order nonliner dely dynmic eqtions on time scles, Mth. Compt. Modelling ), [18] SAKER, S. H. GRACE, S. R.: Oscilltion criteri for qsi-liner fnctionl dynmic eqtions on time scles, Mth. Slovc ), Received September 6, 2014 Ther S. Hssn Deprtment of Mthemtics Fclty of Science University of Hil Hil, 2440 SAUDI ARABIA Deprtment of Mthemtics Fclty of Science Mnsor University Mnsor, EGYPT E-mil: Sid R. Grce Deprtment of Engineering Mthemtics Fclty of Engineering Ciro University Ormn, Giz EGYPT E-mil: 161