Oscillation Criteria for Nonlinear Damped Dynamic Equations on Time Scales

Transcript

1 Oscillaion Crieria for Nonlinear Damped Dynamic Equaions on ime Scales Lynn Erbe, aher S Hassan, and Allan Peerson Absrac We presen new oscillaion crieria for he second order nonlinear damped delay dynamic equaion ((x + p(x σ + qf (x(τ = 0 Our resuls generalize and improve some known resuls for oscillaion of second order nonlinear delay dynamic equaion Our resuls are illusraed wih examples Inroducion In his paper, we are concerned wih oscillaion behavior of he second order nonlinear damped delay dynamic equaion ( ((x + p(x σ + qf (x(τ = 0, on an arbirary ime scale, where is a quoien of odd posiive inegers, r, p and q are posiive rd coninuous funcions on, and he so-called delay funcion τ : saisfies τ for and lim τ = he funcion f C (R, R is assume o saisfy uf(u > 0 and f(u u K, for u 0 and for some K > 0 Since we are ineresed in he oscillaory and asympoic behavior of soluions near infiniy, we assume ha sup =, and define he ime scale inerval [, by [, := [, By a soluion of ( we mean a nonrivial real valued funcion x Crd [ x,, x which has he propery ha ( x C rd [ x, and saisfies equaion ( on [ x,, where C rd is he space of rd coninuous funcions he soluions vanishing in some neighborhood of infiniy will be excluded from our consideraion A soluion x of ( is said o be oscillaory if i is neiher evenually posiive nor evenually negaive, oherwise i is nonoscillaory Recenly here has been a large number of papers devoed o second order nonlinear dynamic equaions on ime scales For example Agarwal e al [] considered he second order delay dynamic equaion (2 x + qx(τ = 0, 2000 Mahemaics Subjec Classificaion 34K, 39A0, 39A99 Key words and phrases Oscillaion, delay nonlinear dynamic equaions, ime scales

2 2 L ERBE, S HASSAN, AND A PEERSON and esablished some sufficien condiions for oscillaion of (2 Zhang e al [9] sudy he oscillaion of he second order nonlinear delay dynamic equaion (3 x + qf(x( τ = 0, and he second order nonlinear dynamic equaion (4 x + qf(x(σ = 0, and esablished he equivalence of he oscillaion of (3 and (4, from which hey obained some oscillaion crieria and comparison heorems for (3 Sahiner [5] considered he second order nonlinear delay dynamic equaion (5 x + qf(x(τ = 0, and obained some sufficien condiions for oscillaion of (5 by means of he Riccai ransformaion echnique Erbe e al [0] exended Sahiner s resul o he second order nonlinear delay dynamic equaion (x + qf(x(τ = 0, Erbe e al [8] considered he pair of second order dynamic equaions (6 ((x + qx = 0, ((x + qx (σ = 0, and esablished some necessary and sufficien condiions for nonoscillaion of Hille- Kneser ype Saker [6] examined oscillaion for half-linear dynamic equaions (6, where > is an odd posiive ineger and Agarwal e al [2] sudied oscillaion for he same equaion (6, where > is he quoien of odd posiive inegers hese resuls can no be applied when 0 < Very recenly Erbe e al [6] and [7] considered he half-linear delay dynamic equaion ((x + qx (τ = 0, for he cases and 0 < respecively In addiion, i was assumed ha (7 r 0, and (8 Boh of he wo cases (9 and (0 τ q = r =, r <, were considered herefore i is of grea ineres o sudy he equaion ( wihou necessarily assuming he condiions (7-(0 We will sill assume > 0 is he quoien of odd posiive inegers and, hence our resuls will improve and exend many known resuls on half-linear oscillaion

3 NONLINEAR DAMPED DYNAMIC EQUAION 3 Noe ha in he special case when = R, hen σ =, µ = 0, g = g, b a g = b a gd, and ( becomes he second order nonlinear damped delay differenial equaion ( ((x + p(x + qf (x(τ = 0 he oscillaion of equaion ( when r =, p = 0 and f(x(τ = x (τ, was sudied by Agarwal e al [3] and proved ha if ( τ lim sup q ds >, s hen every soluion of ( oscillaes When = Z, hen σ = +, µ =, g = g, b a b g = g, and ( becomes he second order nonlinear damped delay difference equaion (2 (( x + p( x + qf (x(τ = 0 he oscillaion of equaion (2 when r =, p = 0 and f(x(τ = x, was sudied by handapani e al [7] where > 0, p is a posiive sequence, and i was shown ha every soluion of (2 is oscillaory, if n=n 0 q(n = We will see ha our resuls no only unify some of he known oscillaion resuls for differenial and difference equaions bu can be applied o oher cases o deermine he oscillaory behavior Noe ha, if =hz, h > 0, hen σ = + h, µ = h, y = h y = b a g = b a h h k=0 g(a + khh, =a y( + h y, h and ( becomes he second order nonlinear damped delay difference equaion (3 h (( h x + p( h x + qf (x(τ = 0 If hen =q N0 = { : = q k, k N 0, q > }, σ = q, µ = (q, x = q x = (x(q x/(q, g = g(q k µ(q k, k=n 0 where = q n0, and ( becomes he second order q nonlinear damped delay difference equaion (4 q (( q x + p( q x + qf (x(τ = 0

4 4 L ERBE, S HASSAN, AND A PEERSON If hen = N 2 0 := {n 2 : n N 0 }, σ = ( + 2, µ = + 2, N y = y(( + 2 y + 2, and ( becomes he second order nonlinear damped delay difference equaion (5 N (( N x + p( N x + qf (x(τ = 0 If = {H n : n N 0 } where H n denoes he se of numbers defined by n H 0 = 0, H n = k, n N, hen σ(h n = H n+, µ(h n = n +, y = Hn y(h n = (n + y(h n, and ( becomes he second order nonlinear damped delay difference equaion (6 Hn (r(h n ( Hn x(h n +p(h n ( Hn x(h n +q(h n f (x(τ(h n = 0 If hen k= = 2 = { n : n N 0 }, σ = 2 +, µ = 2 +, x = 2 x = x( 2 + x, 2 + and ( becomes he second order nonlinear delay damped difference equaion (7 2 (( 2 x + p( 2 x + qf (x(τ = 0 We recall ha for a discree ime scale, we have b g = gµ a [a,b We will uilize in he following a Riccai ransformaion echnique o esablish oscillaion crieria for (, where is he quoien of odd posiive inegers hese resuls improve and generalize he resuls ha have been esablished in [2], [6], [7], and [6] and our resuls are essenially new for equaions ( (7 Also, ineresing examples ha illusrae he imporance of our resuls are included in secion 3 below hroughou his paper, we le 2 Main Resuls d + := max{0, d} and d := max {0, d}, θ(a, b; u := a u r b u r P := r σ (δ r σ δp,, α(, u := θ(τ, σ; u, a := e p (,, r σ

5 NONLINEAR DAMPED DYNAMIC EQUAION 5 and R := a, Q := aq, β := τ R he firs wo Lemmas give some sufficien condiions in order ha a posiive soluion of ( is evenually increasing (2 Lemma 2 Assume ha R =, holds and ( has a posiive soluion x on [, [,, sufficienly large, so ha hen here exiss a x > 0, and ( ( x < 0, on [, Proof Pick [, such ha x(τ > 0 on [, From (, we have (22 ( ( x + p(x σ = qf (x(τ < 0 For [, we can wrie he lef hand side of (22 in he form which implies ( ( x a + p r σ ( ( x σ a < 0, ( R ( x < 0 Now, we claim ha x > 0 on [, If no, hen here exiss 2 such ha x ( 2 < 0 Using he fac ha R ( x is decreasing, we obain, for [ 2, R(x < c := R( 2 (x ( 2 < 0 Inegraing from 2 o, we find ha x < x( 2 + c 2 R, for 2 Condiion (2 implies ha x is evenually negaive, which is a conradicion herefore, x > 0 on [, and hence, from (, we ge ( ( x < 0 on [, he proof is complee Lemma 22 Assume ha [ (23 Qβ R holds and ( has a posiive soluion x on [, [,, sufficienly large, so ha ] =, x > 0, and ( ( x < 0, on [, hen here exiss a

6 6 L ERBE, S HASSAN, AND A PEERSON Proof As in he proof of Lemma 2, assume here is a 2 such ha x < 0 on [ 2, Pick 3 2 so ha τ 2, for 3 Using he fac ha R ( x is decreasing, we obain x(τ < x ( x(τ = where L := (R( 2 ( x ( 2 (R(τ ( x (τ τ (R( 2 ( x ( 2 τ (R ( x R τ R = Lβ, R < 0 From (, we ge, for 3 (R ( x = Qf (x(τ KQx (τ Hence, for 3, we have KL Qβ, R(x R( 3 (x ( 3 + KL KL I follows from his las inequaliy ha x x( 3 K L 3 3 Q(uβ (u u [ R s 3 3 Q(uβ (u u ] Q(uβ (u u Hence by (23, we have lim x =, which conradics he fac ha x is a posiive soluion of ( hus x > 0 on [, and hence, from (, we ge ( ( x < 0 on [, Lemma 23 Assume ha [ (24 Q R ] =, holds and ( has a posiive soluion x on [, hen eiher here exiss a [,, sufficienly large, so ha or lim x = 0 x > 0, and ( ( x < 0, on [,, Proof As in he proof of Lemma 2, assume here is a 2 such ha x < 0 on [ 2, hus, we ge lim x = l 0 If we assume l > 0, hen x(τ l, for 3 From (, we ge (R ( x = Qx (τ l Q,

7 NONLINEAR DAMPED DYNAMIC EQUAION 7 Hence, for 3, we have R(x R( 3 (x ( 3 l l I follows from his las inequaliy ha x x( 3 l 3 3 Q(u u [ R s 3 3 Q(u u ] Q(u u Hence by (24, we have lim x =, which conradics he fac ha x is a posiive soluion of ( hus, l = 0 his complees he proof hen Lemma 24 Assume ha here exiss, sufficienly large, such ha x > 0, x > 0, and ( ( x < 0, on [, x(τ > α (, x σ, for Proof Since ( x is sricly decreasing on [, We can choose so ha τ, for hen, we have ha and so (25 Also, we see ha and hence (26 x σ x(τ = σ τ (r ( x r (r(τ ( x (τ x σ x(τ + (r(τ ( x (τ x(τ x(τ > x(τ x ( = τ (r(τ ( x (τ (r(τ ( x (τ x(τ herefore, (25 and (26 imply < σ τ σ τ r, r (r ( x τ ( τ r r, r x σ x(τ < σ and hence we ge he desired inequaliy his complees he proof r ( τ r, x(τ > α (, x σ, for

8 8 L ERBE, S HASSAN, AND A PEERSON From Lemmas 2-22, we ge he following oscillaion crieria for equaion ( heorem 2 Assume one of he condiions (2 or (23 holds Furhermore, suppose ha here exiss a posiive differeniable funcion δ such ha, for all sufficienly large, (27 lim sup [Kα (s, δq r(p + + ( + + δ hen every soluion of equaion ( is oscillaory on [, ] = Proof Assume ( has a nonoscillaory soluion on [, hen, wihou loss of generaliy, here is a [,, sufficienly large, so ha x saisfies he conclusions of Lemmas 2-22 on [, wih x(τ > 0 on [, In paricular, we have x(τ > 0, x > 0, ( ( x < 0, for Consider he generalized Riccai subsiuion ( x (28 w = δ x By he produc rule and hen he quoien rule w = ( (x δ σ ( (x x + δ x = ( (x δ σ x + δ ((x x σ δ (x (x x x σ From ( and he definiion of w and P, we have w P ( x(τ δ σ wσ Kδq x σ δ (x (x x x σ Using he fac ha (x is sricly decreasing and he definiion of w, we obain w P ( x(τ δ σ wσ Kδq x σ δwσ (x δ σ x From Lemma 24, we ge (29 w P δ σ wσ Kα (, δq δwσ δ σ By he Pözsche chain rule ([4, heorem 90], we obain (x = 0 [ x + hµx ] dh x (x x = [( h x + hx σ ] dh x 0 (x x, > (x σ x, 0 <

9 NONLINEAR DAMPED DYNAMIC EQUAION 9 If 0 <, we have ha (20 w P δ σ wσ Kα (, δq δwσ δ σ whereas if >, we have ha x x σ ( x σ, x (2 w P δ σ wσ Kα (, δq δwσ x x σ δ σ x σ x Using he fac ha x is sricly increasing and (x is sricly decreasing, we ge ha ( r (22 x σ x, x σ (x σ From (20, (2 and (22, we obain (23 w P + δ σ wσ Kα (, δq δ(wσ λ (δ σ λ r where λ := + Define A > 0 and B > 0 by hen, using he inequaliy A λ := δ(wσ λ, B λ := (r λ P + (δ σ λ r λ λ (δ λ (24 λab λ A λ (λ B λ, we ge ha P + δ σ wσ δ(wσ λ = λab λ A λ (δ σ λ r From his las inequaliy and (23 we ge (λ B λ = (P + + ( + + δ w (P + + ( + + δ Kδα (, q Inegraing boh sides from o we ge [Kα (s, δq r(p + + which leads o a conradicion o (27 ( + + δ ] w( w w(, By choosing δ = and δ =, in heorem 2 we have he following oscillaion resuls Corollary 2 Assume one of he condiions (2 or (23 holds and, for all sufficienly large, (25 α (, q = hen every soluion of equaion ( is oscillaory on [,

10 0 L ERBE, S HASSAN, AND A PEERSON Corollary 22 Assume one of he condiions (2 or (23 holds and, for all sufficienly large, (26 lim sup [Ksα (s, q r(p + + ] ( + + s =, where P = p r σ hen every soluion of equaion ( is oscillaory on [, We are now ready o sae and prove Philos-ype oscillaion crieria for he equaion ( heorem 22 Assume one of he condiions (2 or (23 holds Furhermore, suppose ha here exis funcions H, h C rd (D, R, where D {(, s : s } such ha (27 H (, = 0,, H (, s > 0, > s, and H has a nonposiive coninuous -parial derivaive H (, s wih respec o he second variable and saisfies (28 H (, s + H (, s P (, s δ σ = h δ σ + (H (, s, and, for all sufficienly large, (29 [ ] lim sup Kα (s, δ q H (, s (h (, s + r H (, ( + + δ =, where δ is a posiive differeniable funcion hen every soluion of equaion ( is oscillaory on [, Proof Assume ( has a nonoscillaory soluion on [, hen, wihou loss of generaliy, here is a [,, sufficienly large, so ha x saisfies he conclusions of Lemmas 2-22 on [, wih x(τ > 0 on [, In paricular, we have x(τ > 0, x > 0, ( ( x < 0, for We define w also, as in heorem 2 From (23 wih P + replaced by P, we have (220 Kα (, δq w + P δ δ σ wσ r (δσ λ (wσ λ Muliplying boh sides of (220, wih replaced by s, by H (, s, inegraing wih respec o s from o,, + H (, s Kα (s, δq H (, s P δ σ wσ H (, s H (, s w δ r (δσ λ (wσ λ

11 NONLINEAR DAMPED DYNAMIC EQUAION Inegraing by pars and using (27 and (28, we obain + + (22 + H (, s Kα (s, δq H (, w ( + H (, s P δ σ wσ H (, s w σ H (, s δ r (δσ λ (wσ λ H (, w ( [ ] h (, s δ σ (H (, s λ w σ δ H (, s r (δσ λ (wσ λ H (, w ( [ ] h (, s δ σ (H (, s λ w σ δ H (, s r (δσ λ (wσ λ Again, define A > 0 and B > 0 by + A λ := H (, s δ(wσ λ, B λ := h (, s r r (δσ λ λ (δ λ and using he inequaliy (24, we obain h (, s δ σ (H (, s λ w σ H (, s δ(wσ λ = λab λ A λ r (δσ λ (λ B λ = h+ (, s r ( + + δ From his las inequaliy and (22, we have [ Kα (s, δ q H (, s (h (, s + r ( + + δ ] H (, w (, and his implies ha [ ] Kα (s, δ q H (, s (h (, s + r H (, ( + + δ w (, which conradics assumpion (29 his complees he proof Also, by Lemma 23, we obain anoher oscillaion crierion for he equaion ( as in heorems 2 and 22 and Corollaries 2 and 22 as follows Corollary 23 Assume ha (24 and (27 hold hen every soluion of equaion ( is oscillaory on [, or ends o zero Corollary 24 Assume ha (24 and (25 hold hen every soluion of equaion ( is oscillaory on [, or ends o zero Corollary 25 Assume ha (24 and (26 hold hen every soluion of equaion ( is oscillaory on [, or ends o zero,

12 2 L ERBE, S HASSAN, AND A PEERSON Corollary 26 Assume ha (24, (27, (28 and (29 hold hen every soluion of equaion ( is oscillaory on [, or ends o zero Remark 2 Noe ha condiions (7 and (8 are no assumed o hold and condiion (9 is also no necessary in order ha ( be oscillaory, in conras o he resuls of [6] and [7] We inroduce he following noaion, for all sufficienly large, p := lim inf σ r := lim inf Q, q := lim inf w σ, R := lim sup r s + r Q, w σ, where Q = Kα (, q, and assume ha l := lim inf σ Noe ha 0 l In order for he definiion of p o make sense we assume ha (222 Q < heorem 23 Assume (2 holds and is a (dela differeniable funcion wih r 0 and (222 holds Furhermore, assume l > 0 and (223 p > l 2 ( +, + or (224 p + q > l (+ hen every soluion of equaion ( is oscillaory on [, Proof Assume ( has a nonoscillaory soluion on [, hen, wihou loss of generaliy, here is a [,, sufficienly large, so ha x saisfies he conclusions of Lemmas 2-22 on [, wih x(τ > 0 on [, In paricular, we have x(τ > 0, x > 0, ( ( x < 0, for We define w also, as in heorem 2 by puing δ = Noe in his case P + = 0 From (23, we have (225 w Q (w σ + r 0, for [, Firs, we assume (223 holds I follows from (28 and ( x is sricly decreasing ha ( x ( w = <, for [, x r Since (2 implies =, we have ha lim r w = 0 Inegraing (225 from σ o and using lim w = 0, we have (226 w σ σ Q + σ (w σ w σ r

13 NONLINEAR DAMPED DYNAMIC EQUAION 3 I follows from (226 ha (227 w σ σ Q + σ (w σ w σ r Le ɛ > 0, hen by he definiion of p and r we can pick [,, sufficienly large, so ha w σ (228 Q p ɛ, and r ɛ, σ for [, From (227 and (228 and using he fac r 0, we ge ha (229 w σ (p ɛ + σ (p ɛ + (r ɛ + (p ɛ + (r ɛ + s (w σ s w σ s + r r s+ σ σ Using he Pözsche chain rule ([4, heorem 90], we ge ( s = 0 [s + hµ] ( 0 s + dh (230 = s + hen from (229 and (230, we have w σ (p ɛ + (r ɛ + aking he lim inf of boh sides as we ge ha Since ɛ > 0 is arbirary, we ge r p ɛ + (r ɛ + l (23 p r r + l Using he inequaliy Bu Au + wih B = and A = l we ge ha p s+ + dh B + ( + + A l 2 ( + +, ( σ which conradics (223 Nex, we assume (224 holds Muliplying boh sides of (225 by +, and inegraing from o ( we ge s + s + ( s r w r Q w σ + r

14 4 L ERBE, S HASSAN, AND A PEERSON Using inegraion by pars, we obain + w + w( ( s + + w σ r( r ( s w σ + r By he quoien rule and applying he Pözsche chain rule, ( s + = (s+ r r σ s+ r rr σ ( + σ r σ (232 ( + σ r Hence + w + w( r( ( s w σ r s + r Q + + s + r Q ( σ w σ ( + r Le < ɛ l be given, hen using he definiion of l, we can assume, wihou loss of generaliy, ha is sufficienly large so ha s l ɛ, s σ I follows ha We hen ge ha Le hen + w σ Ls, s where L := l ɛ > 0 + w( r( + where λ = + I follows ha + w s + r Q {( + L s w σ r u := s w σ, r ( s u λ w σ λ = r + w( r( + ( s w σ r s + r Q {( + L u u λ } + }

15 NONLINEAR DAMPED DYNAMIC EQUAION 5 Again, using he inequaliy Bu Au λ where A, B are consans, we ge + w I follows from his ha w Since w 0, we ge ( + + B + A, + w( s + r( r Q [( + L ] + + ( w( r( + w( r( s + r Q + L (+ ( s + r Q + L (+ ( w σ + w( r( aking he lim sup of boh sides as we obain R q + L (+ = q + Since ɛ > 0 is arbirary, we ge ha R q + Using his and he inequaliy (23 we ge herefore which conradics (224 s + r Q + L (+ ( l (+ (l ɛ (+ p r l r + r R q + l (+ p + q l (+ Remark 22 Noe ha condiion (8 is no assumed o hold, in conras o he resuls of [6] and [7] 3 Examples In his secion, we give some examples o illusrae our main resuls Example 3 Consider he nonlinear delay dynamic equaion (3 ( a ( x + (σ a σ ( x σ + α (, 2 x (τ = 0,

16 6 L ERBE, S HASSAN, AND A PEERSON where is he quoien of odd posiive inegers and a = e (, Here p = (σ a σ, q = α (, and = 2 a, hen, i is clear ha P = 0 and he condiion (2 holds since = =, R by Example 560 in [5] Also lim sup [Ksα (s, q r(p + + ] ( + + s = K lim sup s =, since α(s, = implies lim r α(, 0 soluion of (3 is oscillaory = hen by Corollary 22, every Example 32 Consider he nonlinear delay dynamic equaion ( (σ ( (32 x + (σ ( x σ + a α (, x (τ = 0, where 0 < is he quoien of odd posiive inegers, a = e p (, and we r σ assume (33 =, for 0 <, σ for hose ime scales [,, > 0 his holds for many ime scales, for example when =q N0 = { : = q k, k N 0, q > } I is clear saisfies ( R σ = < o see ha (23 holds noe ha [ ] Qβ since β = τ = R τ sσ = τ [ (σ [ 0 (σ ] ( sβ ] α (s,, ( = s τ α(, We can find 0 < k < such ha > k, for k > herefore, we ge [ ] Qβ > k (33 = K σ o apply Corollary 2, i remains o prove ha condiion (25 holds, hen α (, q = =,

17 where we use, as in Example 3, NONLINEAR DAMPED DYNAMIC EQUAION 7 r conclude ha if [,, > 0 is a ime scale where Corollary 2, every soluion of (32 is oscillaory = implies lim α(s, α(, σ = We =, hen, by Example 33 Consider he nonlinear dynamic equaion ( ( (34 x ( (σ ( + x σ a a σ η + α (, 2 x (τ = 0, where is he quoien of odd posiive inegers, η is a posiive consan and a = e p r σ Noe ha (, Here = ( (σ a, p = a σ and q = r = and, as in Example 3, if η > η > p = lim inf = R =, ( ( aa σ a ( 2 a ( ( 2 a σ ( 2 = 0, Kα (s, q σ α (s, = ηk lim inf a σ l 2 K(+ l 2 K(+ + ηk lim inf σ ηk lim inf σ s 2 α (s, s 2 sσ = ηlk > l 2 ( + +, η α (, 2 + hen, by heorem 23, we ge ha (34 is oscillaory if Addiional examples may be readily given We leave his o ineresed reader 4 Applicaions In his secion, we apply he oscillaion crieria o differen ypes of ime scales, for example if = R hen σ =, µ = 0, f = f, b a f = b a fd, and ( becomes he nonlinear damped delay differenial equaion (4 ((x + p(x + qf (x(τ = 0 hen we have from heorems 2-23 and Corollaries 2-26 he following oscillaion crieria for equaion (4 (42 heorem 4 Assume one of he condiions d R =,

18 8 L ERBE, S HASSAN, AND A PEERSON or [ ] (43 Qβ ds d =, R holds Furhermore, suppose ha here exiss a posiive differeniable funcion δ such ha, for all sufficienly large, (44 lim sup [Kα (s, δq r(p + + ( + + δ hen every soluion of equaion (4 is oscillaory on [, ] ds = Corollary 4 Assume one of he condiions (42 or (43 holds and, for all sufficienly large, (45 α (, qd = hen every soluion of equaion (4 is oscillaory on [, Corollary 42 Assume one of he condiions (42 or (43 holds and, for all sufficienly large, (46 lim sup [Ksα (s, q r(p + + ] ( + + s ds = hen every soluion of equaion (4 is oscillaory on [, heorem 42 Assume one of he condiions (42 or (43 holds Furhermore, suppose ha here exis funcions H, h C (D, R, where D {(, s : s } such ha (47 H (, = 0,, H (, s > 0, > s, (48 H (, s s + H (, s P (, s δ σ = h δ σ + (H (, s, and, for all sufficienly large, (49 [ ] lim sup Kα (s, δ q H (, s (h (, s + r H (, ( + + δ ds =, hen every soluion of equaion (4 is oscillaory on [, Corollary 43 Assume ha (44 holds and [ ] (40 Qds d =, R hen every soluion of equaion (4 is oscillaory on [, or ends o zero Corollary 44 Assume ha (45 and (40 hold hen every soluion of equaion (4 is oscillaory on [, or ends o zero Corollary 45 Assume ha (46 and (40 hold hen every soluion of equaion (4 is oscillaory on [, or ends o zero Corollary 46 Assume ha (47, (48, (49 and (40 hold hen every soluion of equaion (4 is oscillaory on [, or ends o zero

19 NONLINEAR DAMPED DYNAMIC EQUAION 9 heorem 43 Assume (42 holds and is differeniable funcion wih r 0 and hold Furhermore, assume ha or where, for all sufficienly large, p := lim inf K α q d < p > α q ds, ( + +, p + q >, q := lim inf hen every soluion of equaion (4 is oscillaory on [, K s + r α q ds If = Z, hen σ = +, µ =, f = f, b a f = b =a f, and ( becomes he nonlinear damped delay difference equaion (4 (( x + p( x( + + qf (x(τ = 0 hen we have from heorems 2-23 and Corollaries 2-26 he following oscillaion crieria for equaion (4 heorem 44 Assume one of he condiions (42 =, = R or [ (43 Qβ R s= = ] =, holds Furhermore, suppose ha here exiss a sequence δ such ha, for all sufficienly large N, (44 lim sup [Kα (s, N δq r(p + + ] ( + + δ = s=n hen every soluion of equaion (4 is oscillaory on [, Corollary 47 Assume one of he condiions (42 or (43 holds and, for all sufficienly large N, (45 α (, N q = =N hen every soluion of equaion (4 is oscillaory on [, Corollary 48 Assume one of he condiions (42 or (43 holds and, for all sufficienly large N, (46 lim sup [Ksα (s, N q r(p + + ] ( + + s = s=n hen every soluion of equaion (4 is oscillaory on [,

20 20 L ERBE, S HASSAN, AND A PEERSON heorem 45 Assume one of he condiions (42 or (43 holds Furhermore, suppose ha here exis wo sequences H, h on D, where D {(, s : s } such ha (47 H (, = 0,, H (, s > 0, > s, (48 s H (, s + H (, s P (, s δ σ = h δ σ + (H (, s, and, for all sufficienly large N, (49 [ ] lim sup Kα (s, N δ q H (, s (h (, s + r H (, N ( + + δ =, s=n hen every soluion of equaion (4 is oscillaory on [, Corollary 49 Assume ha (44 holds and [ ] (420 Q =, R s= = hen every soluion of equaion (4 is oscillaory on [, or ends o zero Corollary 40 Assume ha (45 and (420 hold hen every soluion of equaion (4 is oscillaory on [, or ends o zero Corollary 4 Assume ha (46 and (420 hold hen every soluion of equaion (4 is oscillaory on [, or ends o zero Corollary 42 Assume ha (47, (48, (49 and (420 hold hen every soluion of equaion (4 is oscillaory on [, or ends o zero heorem 46 Assume (42 holds, 0 and = α q < hold Furhermore, assume ha or p > where, for all sufficienly large N, K p := lim inf α q, =+ ( + +, p + q >, q := lim inf K =N hen every soluion of equaion (4 is oscillaory on [, s + r α q Similarly, we can sae oscillaion crieria for many oher ime scales, eg, =hz, h > 0, = { : = q k, k N 0, q > }, = N 2 0 := {n 2 : n N 0 }, or = {H n : n N} where H n is he so-called n-h harmonic number defined by H 0 = 0, H n = n k= k, n N 0

21 NONLINEAR DAMPED DYNAMIC EQUAION 2 References [] R P Agarwal, M Bohner and S H Saker, Oscillaion of second order delay dynamic equaion, Canadian Appl Mah Quar, 3 ( [2] R P Agarwal, D O Regan and S H Saker, Philos- ype oscillaion crieria for second order half linear dynamic equaions, Rocky Mounain J Mah 37 ( [3] R P Agarwal, S L Shien and C C Yeh, Oscillaion crieria for second-order rearded differenial equaions, Mah Comp Modelling 26 (997, [4] M Bohner and A Peerson, Dynamic Equaions on ime Scales: An Inroducion wih Applicaions, Birkhäuser, Boson, 200 [5] M Bohner, A Peerson, Advances in Dynamic Equaions on ime Scales, Birkhäuser, Boson, 2003 [6] L Erbe, S Hassan, A Peerson and S H Saker, Oscillaion crieria for half-linear delay dynamic equaions on ime scales, Nonlinear Dynam Sys h(submied [7] L Erbe, S Hassan, A Peerson and S H Saker, Oscillaion crieria for sublinear half-linear delay dynamic equaions on ime scales, J Comp Appl Mah (Submied [8] L Erbe, A Peerson and S H Saker, Hille-Kneser-ype crieria for second-order dynamic equaions on ime scales, Adv Diff Eq 2006 ( [9] L Erbe, A Peerson and S H Saker, Oscillaion crieria for second-order nonlinear dynamic equaions on ime scales, J London Mah Soc 76 ( [0] L Erbe, A Peerson and S H Saker, Oscillaion crieria for second-order nonlinear delay dynamic equaions on ime scales, J Mah Anal Appl 333 ( [] WB Fie, Concerning he zeros of soluions of cerain differenial equaions, rans Amer Mah Soc 9 ( [2] S Hilger, Analysis on measure chains a unified approach o coninuous and discree calculus, Resuls Mah 8 ( [3] V Kac and P Cheung, Quanum Calculus, Universiex, 2002 [4] J V Manojlovic, Oscillaion crieria for second-order half-linear differenial equaions, Mah Comp Mod 30 ( [5] Y Sahiner, Oscillaion of second-order delay dynamic equaions on ime scales, Nonlinear Analysis: h Meh Appl 63 ( [6] S H Saker, Oscillaion crieria of second-order half-linear dynamic equaions on ime scales, J Comp Appl Mah 77 ( [7] E handapani, K Ravi, and J Graef, Oscillaion and comparison heorems for half-linear second order difference equaions, Comp Mah Appl 42 (200, [8] A Winner, On he nonexisence of conjugae poins, Amer J Mah 73 ( [9] B G Zhang and S Zhu, Oscillaion of second-order nonlinear delay dynamic equaions on ime scales, Comp Mah Appl 49 ( Deparmen of Mahemaics, Universiy of Nebraska Lincoln, Lincoln, NE , USA address: lerbe2@mahunledu URL: hp://wwwmahunledu/~lerbe2 Deparmen of Mahemaics, Faculy of Science, Mansoura Universiy, Mansoura, 3556, Egyp Curren address: Deparmen of Mahemaics, Universiy of Nebraska Lincoln, Lincoln, NE , USA address: hassan2@mahunledu; shassan@mansedueg URL: hp://wwwmansedueg/pcvs/3040/ Deparmen of Mahemaics, Universiy of Nebraska Lincoln, Lincoln, NE , USA address: apeerson@mahunledu URL: hp://wwwmahunledu/~apeerson