Applied Mathematics Letters. Oscillation results for fourth-order nonlinear dynamic equations

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1 Applied Mahemaics Leers 5 (0) Conens liss available a SciVerse ScienceDirec Applied Mahemaics Leers jornal homepage: Oscillaion resls for forh-order nonlinear dynamic eqaions Chenghi Zhang a,, Tongxing Li a, Ravi P. Agarwal b, Marin Bohner c a School of Conrol Science and Engineering, Shandong Universiy, Jinan, Shandong 5006, PR China b Deparmen of Mahemaics, Texas A&M Universiy Kingsville, 700 Universiy Blvd., Kingsville, TX , USA c Deparmen of Mahemaics and Saisics, Missori S&T, Rolla, MO , USA a r i c l e i n f o a b s r a c Aricle hisory: Received 4 April 0 Acceped 5 April 0 This work is concerned wih he oscillaion of a cerain class of forh-order nonlinear dynamic eqaions on ime scales. A new oscillaion resl and an example are inclded. 0 Elsevier Ld. All righs reserved. Keywords: Oscillaion Forh-order dynamic eqaion Time scale. Inrodcion This work is concerned wih oscillaion of a forh-order nonlinear dynamic eqaion px 3 () + q()f x(σ ()) = 0 (.) on an arbirary ime scale T wih sp T =. Since we are ineresed in oscillaory behavior of solions, we assme ha he ime scale inerval akes he form [ 0, ) T := [ 0, ) T. Throgho his work, we assme ha p, q C rd (T, (0, )) and here exiss a posiive consan L sch ha f (y) L for all y 0. y Frher, we consider he case where p() <. 0 By a solion of (.) we mean a fncion x C 3 rd [T x, ) T, T x [ 0, ) T, which has he propery px 3 C rd [T x, ) T and saisfies (.) on [T x, ) T. We consider only hose solions x of (.) which saisfy sp{ x() : [T, ) T } > 0 for all T [T x, ) T. We assme ha (.) possesses sch solions. A solion of (.) is called oscillaory if i is neiher evenally posiive nor evenally negaive; oherwise i is called non-oscillaory. Eq. (.) is said o be oscillaory if all is solions are oscillaory. Following he developmen of he heory of dynamic eqaions on ime scales, e.g., in [ 3], here has been mch aciviy concerning oscillaory behavior of varios dynamic eqaions on ime scales. We refer he reader o he aricles [4 ]. Grace e al. [8] considered oscillaion of a forh-order nonlinear dynamic eqaion x 4 () + q()x γ () = 0. (.) Corresponding ahor. addresses: zchi@sd.ed.cn (C. Zhang), liongx007@63.com (T. Li), agarwal@amk.ed (R.P. Agarwal), bohner@ms.ed (M. Bohner) /$ see fron maer 0 Elsevier Ld. All righs reserved. doi:0.06/j.aml

2 C. Zhang e al. / Applied Mahemaics Leers 5 (0) Li e al. [9] invesigaed oscillaion of a forh-order delay dynamic eqaion px 3 () + q()x(τ()) = 0. (.3) By sing some comparison mehods, he ahors esablished a sfficien condiion which ensres ha every nbonded solion of (.3) is oscillaory when condiion (.) holds. In his work, we will se some Riccai sbsiions o obain some sfficien condiions which garanee ha all solions of (.) are oscillaory. In wha follows, all fncional ineqaliies considered in his noe are assmed o hold evenally, ha is, hey are saisfied for all large enogh.. The main resls In his secion, we will derive a new heorem for he oscillaion of (.). We begin wih he following lemma. Lemma.. Assme ha (.) holds and x is an evenally posiive solion of (.). Then here are he following for cases for [, ) T [ 0, ) T sfficienly large: (a) x() > 0, x () < 0, x () > 0, x 3 () < 0, (px 3 ) () < 0, (b) x() > 0, x () > 0, x () > 0, x 3 () < 0, (px 3 ) () < 0, (c) x() > 0, x () > 0, x () > 0, x 3 () > 0, (px 3 ) () < 0, (d) x() > 0, x () > 0, x () < 0, x 3 () > 0, (px 3 ) () < 0. Proof. The proof is obvios, and herefore is omied. In [, Secion.6], he Taylor monomials {h n (, s)} n=0 are defined recrsively by h 0 (, s) =, h n+ (, s) = s h n (τ, s) τ,, s T, n 0. I follows from [, Secion.6] ha h (, s) = s for any ime scale, b simple formlas in general do no hold for n. Now we esablish he following resls. Le R() := p(s) s. Theorem.. Assme ha one of he following condiions: R(s) s =, 0 R(s) s =, and 0 lim sp 0 Lq(v) R(s) s σ (v) 4 σ (v) v R(s) s R(s) s (.) (.) v = (.3) holds. If here exis wo posiive fncions δ, α C rd ([ 0, ) T, R) sch ha lim sp Lq(s)R(σ (s)) h (σ (s), 0 ) s =, 0 4R(σ (s))p(s) (.4) σ (s) z v lim sp Lq(s)δ(σ (s)) 3 p() v z p(s)(δ σ (s) + σ (s) p(z) z 4 p(z) z 4δ(σ (s)) s p(z) z s =, (.5) and lim sp 0 Lα(σ (s)) s q(v) v p() σ (s)(α + (s)) s = (.6) 4ksα(σ (s))

3 060 C. Zhang e al. / Applied Mahemaics Leers 5 (0) hold for all sfficienly large [ 0, ) T, for 4 > 3 > >, and for some consan k (0, ), where h + () := max{0, h()}, hen every solion of (.) is oscillaory. Proof. Sppose ha (.) has a non-oscillaory solion x. We may assme wiho loss of generaliy ha here exiss a [ 0, ) T sch ha x() > 0 for all [, ) T. From Lemma., we ge ha x saisfies for possible cases. Assme (a). Then px 3 is decreasing, and so p(s)x 3 (s) p()x 3 (), s [, ) T. Dividing he above ineqaliy by p(s) and inegraing he resling ineqaliy from o l, we obain x (l) x () + p()x 3 () Leing l, we ge l p(s) s. x () p()x 3 ()R(). (.7) Hence here exiss a consan k > 0 sch ha x () kr(). (.8) Inegraing (.8) from 0 o, we have x () x ( 0 ) k which implies ha x ( 0 ) k 0 0 R(s) s. R(s) s, This conradics (.). Nex, inegraing (.8) from o gives x () k R(s) s. Inegraing again from 0 o, we ge x() + x( 0 ) k R(s) s, 0 which implies ha x( 0 ) k R(s) s. 0 This conradics (.). Inegraing (.7) from o, we have x () p(s)x 3 (s)r(s) s p()x 3 () R(s) s. (.9) Inegraing (.9) from o, we ge x() p()x 3 () R(s) s p()x 3 () R(s) s. (.0) Now se ω() := p()x 3 () x() Then ω() < 0 for [, ) T and for [, ) T. (.) ω () = (px 3 ) () x(σ ()) p()x 3 ()x () x()x(σ ()) Lq() p()x 3 ()x (), x()x(σ ())

4 C. Zhang e al. / Applied Mahemaics Leers 5 (0) where i follows from (.9) ha ω () Lq() (px 3 ) () x()x(σ ()) Recalling (.) and (.), we ge R(s) s Lq() (px 3 ) () x () R(s) s. (.) ω () Lq() ω () In view of (.0), we have ω() From (.3), we obain R(s) s. R(s) s. (.3) (.4) ω () R(s) s Lq() σ () σ () Inegraing (.5) from o gives ω() R(s) s ω( ) ω(v) R(s) s v v v R(s) s v, 4 σ (v) R(s) s σ (v) R(s) s σ () R(s) s + R(s) s ω (v) Lq(v) σ (v) v R(s) s ω () R(s) s v R(s) s v which implies ha Lq(v) R(s) s v R(s) s v σ (v) 4 σ (v) R(s) s ω() R(s) s + ω( ) R(s) s + ω( ) R(s) s de o (.4). This conradics condiion (.3). Assme (b). Define R(s) s. (.5) ϕ() := p()x 3 () x () for [, ) T. (.6) Then ϕ() < 0 for [, ) T and ϕ () = (px 3 ) () x (σ ()) p()(x 3 ) () x(σ ()) Lq() x ()x (σ ()) x (σ ()) p()(x 3 ) () x ()x (). (.7) Recalling ha x > 0, x > 0, x > 0, and x 3 < 0, and sing [7, Lemma 4], we have x() d h (, 0 ) x () for [ d, ) T and for given d (/, ). On he oher hand, we obain (.8) x () = x ( ) + x (s) s ( )x () d x () (.9) for [, ) T, sfficienly large. I follows from (.8) and (.9) ha x() h (, 0 ) x (). (.0)

5 06 C. Zhang e al. / Applied Mahemaics Leers 5 (0) Sbsiing (.0) ino (.7) and sing (.6), we ge ϕ () Lq() h (σ (), 0 ) ϕ () p(). (.) Since px 3 is decreasing, we have (.7). Then ϕ()r(). (.) In view of (.), we ge R(σ ())ϕ () Lq()R(σ ()) h (σ (), 0 ) Inegraing (.3) from o, we have R()ϕ() R( )ϕ( ) R(σ ()) ϕ () p(). (.3) Lq(s)R(σ (s)) h (σ (s), 0 ) Lq(s)R(σ (s)) h (σ (s), 0 ) + s + ϕ(s) 4R(σ (s))p(s) which yields Lq(s)R(σ (s)) h (σ (s), 0 ) s + R( )ϕ( ) 4R(σ (s))p(s) de o (.). This conradics condiion (.4). Assme (c). Recalling x > 0, x 3 > 0, (px 3 ) < 0, we have p(s) R(σ (s) (s))ϕ s p(s) s, Ths x () x 3 ()p() x () 0. p(s) s p(s) s. Hence here exiss [, ) T sch ha (.4) x () = x ( ) + which implies ha x () s 0. p() s Ths, here exiss 3 [, ) T sch ha x() = x( 3 ) + 3 x (s) s p() s p() s x () p(s) s x (s) s I follows from (.5) and (.6) ha v p() v s s v p() v s p() x () s p() s s, (.5) s v 3 p() v s. (.6) x() We now se 3 s v p() v s p(s) s x (). (.7) ψ() := δ() p()x 3 () x () for [, ) T. (.8)

6 C. Zhang e al. / Applied Mahemaics Leers 5 (0) Then ψ() > 0 for [, ) T and ψ () = δ () p()x 3 () x () + δ(σ ()) px 3 () x = δ () 3 ) ()x () p()x 3 ()x 3 () ψ() + δ(σ ())(px δ() x ()x (σ ()) δ + () δ() = δ + () δ() 3 ) () ψ() + δ(σ ())(px x (σ ()) 3 ) () ψ() + δ(σ ())(px x (σ ()) p()x 3 ()x 3 () δ(σ ()) x ()x (σ ()) ψ () x () δ(σ ()) p()δ () x (σ ()) de o (.8). Then, from (.4) and (.7), we have σ () s v ψ () Lq()δ(σ ()) 3 p() v s δ+ σ () + () δ(σ ()) ψ() p(s) s δ() p()δ () Hence we ge ψ () Lq()δ(σ ()) σ () s v 3 p() v s σ () p(s) s + p()(δ + ()) σ () p(s) s 4δ(σ ()) p(s) s. Inegraing he las ineqaliy from 4 ( 4 [ 3, ) T ) o gives ha σ (s) z v Lq(s)δ(σ (s)) 3 p() v z p(s)(δ σ (s) + σ (s) (s)) 4 p(z) z 4δ(σ (s)) s p(z) z holds, which is a conradicion o (.5). Assme (d). In view of (.), we have p(z)x 3 (z) p()x 3 () + Lx(σ ()) Leing z in his ineqaliy, we ge Hence x 3 () + L x(σ ()) 0. p() x (z) + x () + Lx(σ ()) z z Leing z in his ineqaliy, we have Now define x () + Lx(σ ()) ζ () := α() x () x() p() Then ζ () > 0 for [, ) T and ζ () = α () x () x() = α () α() for [, ) T p() p(z) z p(s) s σ () p(s) s ψ (). s ψ( 4 ) 0. (.9) x ()x() (x ) () + α(σ ()) x()x(σ ()) x () ζ () + α(σ ()) x(σ ()) α(σ ()) α () x() x(σ ()) ζ () de o (.30). On he oher hand, from x > 0, x > 0, x < 0, we have ha x() ( )x (), (.30)

7 064 C. Zhang e al. / Applied Mahemaics Leers 5 (0) and so x 0. Hence x() x(σ ()) k σ () for each k (0, ) and for [ k, ) T sfficienly large. Ths, by (.9) and (.3), we obain Hence ζ () Lα(σ ()) ζ () Lα(σ ()) p() p() + α + () α() ζ () kα(σ ()) α () + σ ()(α + ()) 4kα(σ ()). Inegraing he las ineqaliy from k o yields ha q(v) v Lα(σ (s)) σ (s)(α + (s)) s ζ ( k ) p() 4ksα(σ (s)) k s σ () ζ (). (.3) holds for each k (0, ). This conradics condiion (.6). The proof is complee. 3. An example The following example illsraes applicaions of heoreical resls in he previos secion. Example 3.. Consider a forh-order dynamic eqaion λ x 3 () + x() = 0, T := Z := { k : k Z} {0}. (3.) Here λ > 0 is a consan. Le p() =, q() = λ/, f () =, and L =. Then R() = /. Using [, Theorem 5.68], we see ha R() 0 = (/) 0 =. Using [, Example.04], we have and so h (, 0 ) = ( 0)( 0 ), 3 h (σ (), 0 ) = h (, 0 ) = ( 0)( 0 ) 3. Ths, condiion (.4) holds if λ >. Le δ() =. Then condiion (.5) holds clearly. Noe ha s q(v) v p() = 4λ 3s. Hence condiion (.6) holds when we ake λ >, α() =, and k = /. From he above, we conclde ha every solion of (3.) is oscillaory when λ >. Acknowledgmen This research was sppored by NNSF of PR China (Gran Nos , , ). References [] Marin Bohner and, Allan Peerson, Dynamic Eqaions on Time Scales, an Inrodcion wih Applicaions, Birkhäser, Boson, 00. [] Marin Bohner, Allan Peerson, Advances in Dynamic Eqaions on Time Scales, Birkhäser, Boson, 003. [3] Sefan Hilger, Analysis on measre chains a nified approach o coninos and discree calcls, Resls Mah. 8 (990) [4] Ravi P. Agarwal, Marin Bohner, Samir H. Saker, Oscillaion of second order delay dynamic eqaions, Can. Appl. Mah. Q. 3 (005) 7. [5] Ravi P. Agarwal, Donal O Regan, Samir H. Saker, Oscillaion crieria for second-order nonlinear neral delay dynamic eqaions, J. Mah. Anal. Appl. 300 (004) 03 7.

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