HILLE AND NEHARI TYPE CRITERIA FOR THIRD-ORDER DYNAMIC EQUATIONS

HILLE AND NEHARI TYPE CRITERIA FOR THIRD-ORDER DYNAMIC EQUATIONS L. ERBE, A. PETERSON AND S. H. SAKER Absrac. In his paper, we exend he oscillaion crieria ha have been esablished by Hille [15] and Nehari [21] for second-order differenial equaions o hird order dynamic equaions on an arbirary ime scale T, which is unbounded above. Our resuls are essenially new even for hird order differenial and difference equaions, i.e. when T = R and T = N. We consider several examples o illusrae our resuls. Keywords and Phrases: Oscillaion, hird-order dynamic equaions, ime scales. 2000 AMS Subjec Classificaion: 34K11, 39A10, 39A99. 1. Inroducion The sudy of dynamic equaions on ime scales, which goes back o is founder Sefan Hilger [14], is an area of mahemaics ha has recenly received a lo of aenion. I has been creaed in order o unify he sudy of differenial and difference equaions. Many resuls concerning differenial equaions carry over quie easily o corresponding resuls for difference equaions, while oher resuls seem o be compleely differen from heir coninuous counerpars. The sudy of dynamic equaions on ime scales reveals such discrepancies, and helps avoid proving resuls wice - once for differenial equaions and once again for difference equaions. The general idea is o prove a resul for a dynamic equaion where he domain of he unknown funcion is a so-called ime scale T, which is a nonempy closed subse of he reals R. In his way resuls no only relaed o he se of real numbers or se of inegers bu hose peraining o more general ime scales are obained. The hree mos popular examples of calculus on ime scales are differenial calculus, difference calculus (see [17]), and quanum calculus (see Kac and Cheung [16]), i.e, when T = R, T = N and T = q N 0 = {q : N 0 }, where q > 1. Dynamic equaions on a ime scale have an enormous poenial for applicaions such as in populaion dynamics. For example, i can 1991 Mahemaics Subjec Classificaion. 34K11, 39A10, 39A99. Key words and phrases. Oscillaion, hird-order dynamic equaions, ime scales. 1

2 L. ERBE, A. PETERSON AND S. H. SAKER model insec populaions ha are coninuous while in season, die ou in say winer, while heir eggs are incubaing or dorman, and hen hach in a new season, giving rise o a nonoverlapping populaion (see [4]). There are applicaions of dynamic equaions on ime scales o quanum mechanics, elecrical engineering, neural neworks, hea ransfer, and combinaorics. A recen cover sory aricle in New Scienis [29] discusses several possible applicaions. The books on he subjec of ime scales by Bohner and Peerson [4] and [5] summarize and organize much of ime scale calculus and some applicaions. For compleeness, we recall he following conceps relaed o he noion of ime scales. A ime scale T is an arbirary nonempy closed subse of he real numbers R. We assume hroughou ha T has he opology ha i inheris from he sandard opology on he real numbers R. The forward jump operaor and he backward jump operaor are defined by: σ() := inf{s T : s > }, ρ() := sup{s T : s < }, where sup = inf T. A poin T, is said o be lef dense if ρ() = and > inf T, is righ dense if σ() =, is lef scaered if ρ() < and righ scaered if σ() >. A funcion g : T R is said o be righ dense coninuous (rd coninuous) provided g is coninuous a righ dense poins and a lef dense poins in T, lef hand limis exis and are finie. The se of all such rd coninuous funcions is denoed by C rd (T). The graininess funcion µ for a ime scale T is defined by µ() := σ(), and for any funcion f : T R he noaion f σ () denoes f(σ()). Definiion 1. Fix T and le x : T R. Define x () o be he number (if i exiss) wih he propery ha given any ɛ > 0 here is a neighbourhood U of wih [x(σ()) x(s)] x ()[σ() s] ɛ σ() s, for all s U. In his case, we say x () is he (dela) derivaive of x a and ha x is (dela) differeniable a. We will frequenly use he resuls in he following heorem which is due o Hilger [14]. Theorem 1. Assume ha g : T R and le T. (i) If g is differeniable a, hen g is coninuous a. (ii) If g is coninuous a and is righ-scaered, hen g is differeniable a wih g g(σ()) g() () =. µ()

HILLE AND NEHARI TYPE CRITERIA 3 (iii) If g is differeniable and is righ-dense, hen g g() g(s) () = lim. s s (iv) If g is differeniable a, hen g(σ()) = g() + µ()g (). In his paper we will refer o he (dela) inegral which we can define as follows: Definiion 2. If G () = g(), hen he Cauchy (dela) inegral of g is defined by a g(s) s := G() G(a). I can be shown (see [4]) ha if g C rd (T), hen he Cauchy inegral G() := 0 g(s) s exiss, 0 T, and saisfies G () = g(), T. For a more general definiion of he dela inegral see [4], [5]. In he las few years, here has been increasing ineres in obaining sufficien condiions for he oscillaion/nonoscillaion of soluions of differen classes of dynamic equaions on ime scales. We refer he reader o he papers [1-3], [6], [7], [9-12], [22-28] and he references cied herein. In his paper, we are concerned wih he oscillaory behavior of soluions of he hird-order linear dynamic equaion (1.1) x () + p()x() = 0, on an arbirary ime scale T, where p() is a posiive real-valued rd coninuous funcion defined on T. Since we are ineresed in he oscillaory and asympoic behavior of soluions near infiniy, we assume ha sup T =, and define he ime scale inerval [ 0, ) T by [ 0, ) T := [ 0, ) T. By a soluion of (1.1) we mean a nonrivial real valued funcions x() Cr 3 [T x, ), T x 0 where C r is he space of rd coninuous funcions. The soluions vanishing in some neighborhood of infiniy will be excluded from our consideraion. A soluion x of (1.1) is said o be oscillaory if i is neiher evenually posiive nor evenually negaive, oherwise i is nonoscillaory. Equaion (1.1) is said o be oscillaory in case here exiss a leas one oscillaory soluion. We noe ha, Equaion (1.1) in is general form covers several differen ypes of differenial and difference equaions depending on he choice of he ime scale T. For example, if T = R, hen σ() =, µ() = 0, x () = x (), b f() = b f()d and (1.1) becomes he hird order linear differenial a a equaion (1.2) x () + p()x() = 0.

4 L. ERBE, A. PETERSON AND S. H. SAKER If T = N, hen σ() = + 1, µ() = 1, x () = x() = x( + 1) x(), b f() = b 1 a =a f() and (1.1) becomes he hird-order difference equaion (1.3) 3 x() + p()x() = 0. If T =hz +, h > 0, hen σ() = +h, µ() = h, x () = h x() = x(+h) x() b a h, b a h h f() = k=0 f(a+kh)h and (1.1) becomes he hird-order difference equaion (1.4) 3 hx() + p()x() = 0. If T = q N = { : = q k, k N, q > 1, hen σ() = q, µ() = (q 1), x () = q x() = (his is he so-called quanum derivaive, see x(q ) x() (q 1) Kac and Cheung [16]), b f() = a (a,b) hird order q difference equaion (1.5) 3 qx() + p()x() = 0. f()µ() and (1.1) becomes he When T = N 2 0 = { = n 2 : n N 0 }, hen σ() = ( + 1) 2 and µ() = 1 + 2, N x() = x(( +1) 2 ) x() 1+2, b f() = a (a,b) f()µ() and (1.4) becomes he hird order equaion (1.6) 3 Nx(n) + p(n)x(n) = 0. If T = T n = { n : n N 0 } where { n } is he se of he harmonic numbers defined by n 1 0 = 0, n = k, n N 0, k=1 hen σ( n ) = n+1, µ( n ) = 1, n+1 x ( n ) = n x( n ) = (n+1)x( n ), b f() = a f()µ() and (1.1) becomes he hird order difference equaion (a,b) (1.7) 3 n x( n ) + p( n )x( n ) = 0. Leighon [19] sudied he oscillaory behavior of soluions of he second order linear differenial equaion (1.8) x () + p()x() = 0, and showed ha if (1.9) p()d =, 0 hen every soluion of equaion (1.8) oscillaes. Hille [15] improved he condiion (1.9) and proved ha every soluion of (1.8) oscillaes if (1.10) lim inf p(s)ds > 1 4.

HILLE AND NEHARI TYPE CRITERIA 5 Nehari [21] by a differen approach proved ha if (1.11) lim inf 1 0 s 2 p(s)ds > 1 4, hen every soluion of (1.8) oscillaes. The oscillaory behavior of he corresponding hird order equaion (1.2) has been sudied by a number of auhors including Hanan [13], Lazer [18] and Mehri[20]; and various well-known inegral and Kneser-ype ess exis. Mehri [20] exended he resul of Leighon [19] and proved ha (1.2) is oscillaory if and only if (1.9) holds. Bu one can easily see ha he condiion (1.9) can no be applied o he cases when p() = β and p() = β 2 3 for some β > 0. Hanan [13] improved he condiion (1.9) for equaion (1.2) and showed ha if (1.12) 0 2 p()d <, hen (1.2) is nonoscillaory. A corollary of a resul of Lazer [18, Theorem 3.1] implies ha (1.2) is oscillaory in case (1.13) 0 1+δ p()d =, for some 0 < δ < 1. which improves he condiion (1.9). By comparison wih he Euler Cauchy equaion i has been shown ha (cf Erbe [8]), if (1.14) lim sup 3 p() < 2 3 3, hen (1.2) is nonoscillaory and if (1.15) lim inf 3 p() > 2 3 3, hen (1.2) is oscillaory. The naural quesion now is: Do he oscillaion condiions (1.10) and (1.11) due o Hille and Nehari for second order differenial equaions exend o hird-order linear dynamic equaions on ime scales?. The purpose of his paper is o give an affirmaive answer o his quesion. We will esablish new oscillaion crieria for (1.1) which guaranee ha every soluion oscillaes or converges o zero. Our resuls improve he oscillaion condiion (1.9) and (1.13) ha has been esablished by Mehri [20] and Lazer [18]. The resuls are essenially new for equaions (1.3)-(1.7). Some examples which dwell upon he imporance of our main resuls are given. To he bes of he auhors knowledge his approach for he invesigaion of he oscillaory behavior of soluions of (1.1) has no been sudied before.

6 L. ERBE, A. PETERSON AND S. H. SAKER 2. Main Resuls Before saing our main resuls, we begin wih he following lemma which is exraced from [10] (also see [11]). Lemma 1. Suppose ha x() is an evenually posiive soluion of (1.1). Then here are only he following wo cases for 1 sufficienly large: (I) x() > 0, x () > 0, x () > 0, or (II) x() > 0, x () < 0, x () > 0. Lemma 2. Assume ha 0 p(s) s =, and le x() be a soluion of (1.1), hen x() is oscillaory or lim x() = lim x () = lim x () = 0. Proof. Assume he conrary and le x() be a nonoscillaory soluion which may be assumed o be posiive for [ 0, ) T. Then x () = p()x() < 0; hence x () is decreasing. If x () > 0 for [ 0, ) T, hen x() is increasing and x () = x ( 0 ) p(s)x(s) s x ( 0 ) x( 0 ) 0 p(s) s. 0 This implies ha lim x () = which is a conradicion by Lemma 1. Now assume here is a 1 [ 0, ) T, 1 1, such ha x () < 0 and by Lemma 1 we may also assume x () > 0 on [ 1, ) T. Since x () < 0 for [ 0, ) T, hen x() is decreasing and here are wo cases: Case 1. lim x() = α > 0. Then x() α for [ 1, ) T. Muliplying (1.1) by σ() and inegraing from 1 o we have x () 1 x ( 1 ) x () + x ( 1 ) + I follows ha 1 σ(s)p(s)x(s) s = 0. A = 1 x ( 1 ) x ( 1 ) = x () x () + σ(s)p(s)x(s) s 1 α σ(s)p(s) s α p(s) s, 1 1 which is a conradicion. Case 2. lim x() = 0. From he fac ha x () > 0 for [ 1, ) T i follows ha x () is increasing and lim x () = β where < β

HILLE AND NEHARI TYPE CRITERIA 7 0. This implies ha x () β for all [ 1, ) T, and hence x( 1 ) x() β( 1 ) which is impossible for β < 0. Therefore lim x () = 0. Now x () < 0 for [ 1, ) T implies ha x () is decreasing and lim x () = γ where 0 γ <. This implies ha x ( 1 ) x () γ( 1 ) which again is impossible for γ > 0, and hence γ = 0. This complees he proof. Remark 1. If we assume ha 0 p() <, hen i can easily be shown ha he exisence of a soluion of (1.1) saisfying case (II) of Lemma 1 is incompaible wih 0 p(s) s =. In Lemma 3 we consider wha happens if here is a soluion of (1.1) saisfying case (II) in Lemma 1 if 0 p() <, 0 p(s) s <, and (2.1) below holds. Lemma 3. Assume ha x() is a soluion of (1.1) which saisfies case (II) of Lemma 1. If (2.1) hen lim x() = 0. 0 z u p(s) s u z =, Proof. Le x() be a soluion of (1.1) such ha case (II) of Lemma 1 holds for [ 1, ) T. Since x() is posiive and decreasing, lim x() := l 0. Assume ha lim x() = l > 0. Inegraing boh sides of equaion (1.1) from o, we ge x () p(s)x(s) s. Inegraing again from o, we have x () p(s)x(s) s u. Inegraing again from u 0 o, we obain x( 0 ) Since x() l, we see ha 0 x( 0 ) l z u 0 z u p(s)x(s) s u z. p(s) s u z. This conradics (2.1). Thus l = 0 and he proof is complee. In [4, Secion 1.6] he Taylor monomials {h n (, s)} n=0 are defined recursively by h 0 (, s) = 1, h n+1 (, s) = s h n (τ, s) τ,, s T, n 1. I follows [4, Secion 1.6] ha h 1 (, s) = s for any ime scale, bu simple formulas in general do no hold for n 2. However, if T = R, hen h n (, s) = ( s) n ; if T = N n! 0, hen h n (, s) = ( s)n, where n = ( 1) ( n + 1) is n! he so-called falling (facorial) funcion (cf Kelley and Peerson [17]); and if

8 L. ERBE, A. PETERSON AND S. H. SAKER T = q N 0, hen h n (, s) = n 1 P q ν s ν=0 ν. We will use hese Taylor monomials µ=0 qµ in he res of his paper. Lemma 4. Assume x saisfies Then x() > 0, x () > 0, x () > 0, x 0, [T, ) T. (2.2) lim inf Proof. Le Then G(T ) = 0 and x() h 2 (, 0 )x () 1. G() := ( T )x() h 2 (, T )x (). G () = (σ() T )x () + x() h 2 (σ(), T )x () ( T )x () = µ()x () + x() h 2 (σ(), T )x () = x σ () h 2 (σ(), T )x () ( ) σ() = x σ () (τ T ) τ x (). By Taylor s Theorem ([4, Theorem 1.113]) x σ () = x(t ) + h 1 (σ(), T )x (T ) + T σ() x(t ) + h 1 (σ(), T )x (T ) + x () T h 1 (σ(), σ(τ))x (τ) τ σ() T h 1 (σ(), σ(τ)) τ, since x () is nonincreasing. I would follow ha G () > 0 on [T, ) T provided we can prove ha σ() T h 1 (σ(), σ(τ)) τ = σ() T ( τ) τ. To see his, we ge by using he inegraion by pars formula ([4, Theorem 1.77]) b a f σ (τ)g (τ) τ = f(τ)g(τ)] b a b a f (τ)g(τ) τ,

and hence σ() T h 1 (σ(), σ(τ)) τ = HILLE AND NEHARI TYPE CRITERIA 9 σ() T (σ() σ(τ)) τ = [(σ() τ)(τ T )] τ=σ() τ=t = σ() T (τ T ) τ, σ() T ( 1)(τ T ) τ which is he desired resul. Hence G () > 0 on [T, ) T. Since G(T ) = 0 we ge ha G() > 0 on (T, ) T. This implies ha Therefore, since and since we ge ha ( T )x() h 2 (, T )x () > 1, (T, ) T. x() h 2 (, 0 )x () = lim ( T )x() h 2 (, T )x () T = 1 = lim h 2 (, T ) h 2 (, 0 ), lim inf x() h 2 (, 0 )x () 1. T h2(, T ) h 2 (, 0 ), In he nex resul we will use he funcion Ψ() defined by Ψ() := h 2(, 0 ). σ() Lemma 5. Le x be a soluion of (1.1) saisfying Par (I) of Lemma 1 for [ 0, ) T and make he Riccai subsiuion Then (2.3) w () + w() = x () x (). x() x σ () p() + w 2 () 1 + µ()w() = 0, for [ 0, ) T. Furhermore given any 0 < k < 1, here is a T k [ 0, ) T such ha (2.4) w () + kψ()p() + k σ() w2 () 0,

10 L. ERBE, A. PETERSON AND S. H. SAKER (2.5) w () + kψ()p() + w()w σ () 0, and (2.6) hold for [T k, ) T. w () + kψ()p() + w 2 () 1 + µ()w() 0, Proof. Le x be as in he saemen of his lemma. Then by he quoien rule [4, Theorem 1.20] we have ( ) x w () = (2.7) Bu (2.8) x = x ()x () ( x ) 2 x ()x σ () = x ()p()x() ( x ) 2 x ()x σ () = x() x σ () p() x () x σ () w(). x () x σ () w() = x () x () x () x σ () w() = w 2 x () () x () + µ()x () w 2 () = 1 + µ()w(), so we ge ha (2.3) holds. Nex consider he coefficien of p() in (2.3). Noice ha Now since x() x σ () = x() x () x () x σ (). lim inf x() h 2 (, 0 )x () 1, given 0 < k < 1, here is an S k [ 0, ) T such ha x() x () k h 2(, 0 ), [S k, ) T.

HILLE AND NEHARI TYPE CRITERIA 11 Also, since x σ () = x () + µ()x () we have x σ () x () = 1 + µ()x () x (), and since x () = p()x() < 0, x () is decreasing and so x () = x ( 1 ) + 1 x (τ) τ x ( 1 ) + x ()( 1 ) > x ()( 1 ), for all > 1 S k. I follows ha here is a T k [S k, ) T such ha for [T k, ) T. Hence x () x () ( 1) k x σ () x () = 1 1 + µ() k k + σ() k σ() k. Hence, we have and so we have x () x σ () k σ(), x() x σ () = x() x () x () x σ () kψ(), for [T k, ) T. Hence (2.6) holds. Also, x () x σ () = x () x () x () x σ () k σ() w() k σ() w(), and so (2.4) follows from (2.7). Furhermore, since x () is decreasing, x () x σ () x σ () x σ () = wσ (), and so (2.5) follows from (2.7). This complees he proof of Lemma 5. Lemma 6. Le x be a soluion of (1.1) saisfying par (I) of Lemma 1 and le w() = x (). Then w() saisfies ( x () 1)w() < 1 for [ 1, ) T and lim w() = 0.

12 L. ERBE, A. PETERSON AND S. H. SAKER Proof. From (2.3), we see ha w w 2 () () 1 + µ()w() = x () w() (by (2.8)) x σ () x σ () x σ () w() = w()w σ (), (since x () is decreasing) for [ 1, ) T, and so ( 1 ) = w () w() w()w σ () 1 for [ 1, ) T. Therefore w (s) 1 w(s)w σ (s) s s, 1 and so 1 w() + 1 w( 1 ) ( 1), which implies ( 1 )w() < 1, since w( 1 ) > 0. Lemma 7. Le x(), w(), and Ψ() be as in Lemma 5. Define (2.9) p := lim inf 1 Ψ(s)p(s) s and q := lim inf σ 2 (s)ψ(s)p(s) s, 1 (2.10) r := lim inf w(), R := lim sup w(), and σ() (2.11) l := lim inf, l := lim sup Then 0 r R 1, 1 l l, and σ(). (2.12) p r r 2, q min{1 R, Rl r 2 l }. Proof. Muliplying (2.6) by (σ(s)) 2, and inegraing for 1 T k gives (2.13) 1 (σ(s)) 2 w (s) s + k + 1 (σ(s)) 2 Ψ(s)p(s) s 1 (σ(s)) 2 w 2 (s) 1 + µ(s)w(s) 0.

An inegraion by pars in (2.13) yields 2 w() 2 1w( 1 ) HILLE AND NEHARI TYPE CRITERIA 13 + k 1 (s 2 ) s w(s) s 1 (σ(s)) 2 Ψ(s)p(s) s + Since (s 2 ) s = s + σ(s), we obain afer rearranging (2.14) where 2 w() 2 1w( 1 ) k + 1 1 (σ(s)) 2 Ψ(s)p(s) s 1 H(s, w(s)) s, H(s, w(s)) = (s + σ(s))w(s) (σ(s))2 w 2 (s) 1 + µ(s)w(s). (σ(s)) 2 w 2 (s) s 0. 1 + µ(s)w(s) We claim ha H(s, w(s)) 1, for s [ 1, ) T. To see his observe ha if we le g(s, u) := (s + σ(s))u (σ(s))2 u 2 1 + µ(s)u, hen we have (since s + σ(s) = 2σ(s) µ(s)), afer some simplificaion g(s, u) = (2σ(s) µ(s))u(1 + µ(s)u) (σ(s))2 u 2 1 + µ(s)u = (2σ(s) µ(s))u s2 u 2. 1 + µ(s)u We noe ha if µ(s) = 0, hen he maximum of g(s, u) (wih respec o u) occurs a u 0 := 1. Moreover in he case µ(s) > 0, afer some calculaions, s one finds ha for fixed s > 0, he maximum of g(s, u) for u 0 occurs a u 0 = 1 also. Hence, we have s g(s, u) g(s, u 0 ) = (s + σ(s))u 0 (σ(s))2 u 2 0 1 + µ(s)u 0 = s + σ(s) s (σ(s))2 s(s + µ(s)) = 1, for u 0. Hence we conclude ha H(s, w(s)) 1 and so 1 H(s, w(s)) s 1.

14 L. ERBE, A. PETERSON AND S. H. SAKER Subsiuing his in (2.14) and dividing by we obain (2.15) w() 2 1w( 1 ) k 1 (σ(s)) 2 Ψ(s)p(s) s + (1 1 ). If we ake he lim sup of boh sides of (2.15) we ge R 1 kq. Thus we have R 1 kq for all 0 < k < 1 and so we have R 1 q. Inegraing (2.5) from o we ge (2.16) w() k Ψ(s)p(s) s + w(s)w σ (s) s for [T k, ) T, where 0 < k < 1, is arbirary. Hence, from (2.16) we have (2.17) w() k = k Ψ(s)p(s) s + Ψ(s)p(s) s + Now for any ɛ > 0 here exiss 2 1 such ha r ɛ < w(), for all [ 2, ) T. Therefore, from (2.17) we ge (2.18) w() k k k k Ψ(s)p(s) s + Ψ(s)p(s) s + (r ɛ) 2 Ψ(s)p(s) s + (r ɛ) 2 Ψ(s)p(s) s + (r ɛ) 2. w(s)w σ (s) s (sw(s))(σ(s)w σ (s)) s. sσ(s) (r ɛ) 2 sσ(s) s Therefore, aking he lim inf of boh sides of (2.18) gives r kp + (r ɛ) 2. s sσ(s) ( 1 s s s) Since ɛ > 0 is arbirary and 0 < k < 1 is arbirary, we obain Finally, we show ha r p + r 2. q Rl r 2 l.

HILLE AND NEHARI TYPE CRITERIA 15 To see his le ɛ > 0 be given, hen here exiss a 2 [ 1, ) T such ha r ɛ < w() < R + ɛ, l ɛ σ() l + ɛ, [ 2, ) T. Similar o how we used (2.5) o obain (2.14) we can use (2.4) o obain he inequaliy w() 2 2w( 2 ) k 2 (σ(s)) 2 Ψ(s)p(s) s + 1 (s + σ(s))w(s) s k sσ(s)w 2 (s) s 2 2 = 2 2w( 2 ) k (σ(s)) 2 Ψ(s)p(s) s 2 + 1 (1 + σ(s) 2 s )(sw(s)) s k 1 σ(s) 2 s (s2 w 2 (s)) s 2 2w( 2 ) k (σ(s)) 2 Ψ(s)p(s) s 2 + (R + ɛ) 1 (1 + σ(s) 2 s ) s k(r 1 σ(s) ɛ)2 2 s s 2 2w( 2 ) k (σ(s)) 2 Ψ(s)p(s) s 2 + (R + ɛ)(1 + l + ɛ) ( 2) k(r ɛ) 2 (l ɛ) ( 2). Taking he lim sup of boh sides we ge R kq + (R + ɛ)(1 + l + ɛ) k(r ɛ) 2 (l ɛ). Since ɛ > 0 is arbirary we ge R kq + R(1 + l ) kr 2 l and since 0 < k < 1 is also abirary we have finally ha which yields he desired resul. R q + R(1 + l ) r 2 l, As a consequence of he previous lemmas, we may now esablish some oscillaion crieria. Theorem 2. Assume ha (2.1) holds and le x() be a soluion of (1.1). If (2.19) p = lim inf Ψ(s)p(s) s > 1 4,

16 L. ERBE, A. PETERSON AND S. H. SAKER hen x() is oscillaory or saisfies lim x() = 0. Proof. Suppose ha x() is a nonoscillaory soluion of equaion (1.1) wih x() > 0 on [ 1, ) T. Then if par (I) of Lemma 1 holds, le w() be as defined in Lemma 5. From Lemma 7 we obain p r r 2 1 4 which conradics (2.19). Now if par (II) of Lemma (1) holds, hen by Lemma 3, lim x() = 0. This complees he proof. Theorem 3. Assume ha (2.1) holds and le x() be a soluion of (1.1). If (2.20) q = lim inf 1 0 σ(s)h 2 (s, 0 )p(s) s > hen x() is oscillaory or saisfies lim x() = 0. Proof. From Lemma 7 we have ha which implies ha q l 1 + l, q min{1 R, Rl r 2 l } min{1 R, Rl } l 1+l, which is a conradicion o (2.20). Theorem 4. Assume (2.1) holds, 0 p 1, and 4 q > l ( 1 p 2 1 2 1 4p ) l (2.21). 1 + l Then every soluion of (1.1) is oscillaory or saisfies lim x() = 0. Proof. Firs we use he fac ha a := p r r 2 o ge ha and so using (2.12), r r 0 := 1 1 4a 2, 2 for r 0 R 1. Noe ha when q min{1 R, l R r 2 l } min{1 R, l R r 2 0l }, 1 R = l R r 2 0l, R = R 0 := 1 + r2 0l 1 + l,

HILLE AND NEHARI TYPE CRITERIA 17 and so q 1 R 0 = 1 1 + r2 0l 1 + l = l ( 1 p 2 1 2 1 4p ) l, 1 + l afer some easy calculaions. This conradics (2.21) and he proof is complee. Remark 2. A close look a he proof of Lemma 7 shows ha he inequaliy holds, when we replace l and l, by λ := lim sup 1 1 q Rl r 2 l σ(s) s s and λ 1 := lim sup 1 σ(s) s, respecively. Then Theorem 3 and Theorem 4 hold wih l and l replaced by λ and λ respecively. Remark 3. We noe here ha our mehods of proof can be applied o he hird order linear equaion (2.22) x + p()x σ = 0 which also can be viewed as a generalizaion of he hird-order differenial equaion x + p()x = 0. In paricular, in Lemma 5 we can prove (2.3) wih he coefficien of p() replaced by xσ (). Also we ge (2.4) (2.6), wih Ψ() replaced by h 2(σ(), 0 ). x σ () σ() Finally we ge ha Theorems 2 4 hold wih p and q replaced by and respecively. ˆp = lim inf ˆq = lim inf h 2 (σ(s), 0 ) p(s) s, 0 σ(s) 1 σ(s)h 2 (σ(s), 0 )p(s) s. 1

18 L. ERBE, A. PETERSON AND S. H. SAKER 3. Examples Example 1. For examples where condiion (2.19) in Theorem 2 is saisfied we ge he following resuls. If T = [0, ), hen h 2 (, 0) = 2 and Ψ() = 2 so (2.19) holds if h 2 (, 0 ) = 2 lim inf sp(s)ds > 1 2. If T = N 0, hen h 2 (, 0) = 1 2 2, so (2.19) holds if lim inf n n kp(k) > 1 2. k=n If T = q N 0, hen h 2 (, 1) = ( 1)( q) 1+q and σ() = q so we (2.19) holds if lim inf sp(s) s > q(1 + q). 4 Example 2. For examples where condiion (2.20) in Theorem 3 is saisfied we ge he following resuls. If T = [0, ), hen (2.20) holds if lim inf If T = N 0, hen (2.20) holds if lim inf n If T = q N 0, hen (2.20) holds if lim inf 1 1 n 1 s 3 p(s)ds > 1. n 1 k=n 1 k 3 p(k) > 1. 1 s 3 p(s) s > 1. 1 Example 3. Consider he hird-order dynamic equaion (3.1) x () + β x() = 0, 3 for T := [1, ). Here p() = β. To apply Theorem 2 i is easy o show 3 ha (2.1) holds and p = β. Hence, by Theorem 2, if β > 1, hen every 2 2 soluion of (3.1) is oscillaory or converges o zero. As a specific example noe ha if β = 6, hen a basis of he soluion space of (3.1) is given by { 1, 2 cos( 2 log ), 2 sin( 2 log )}, which conains oscillaory soluions and saisfies he propery ha every nonoscillaory soluion converges o zero.

HILLE AND NEHARI TYPE CRITERIA 19 We wish o nex consider wo examples illusraing condiion (2.21). Example 4. Le T = q N 0 Then we have Ψ()p() = p() := and le α h 2 (, 1), 0 < α 1 4. α, so σ() p = lim inf = α lim inf Ψ(s)p(s) s and since (σ()) 2 Ψ()p() = αq, we have q = lim inf 1 1 = lim inf αq s 1 = αq > q = p. s sσ(s) = α, 1 (σ(s)) 2 Ψ(s)p(s) s q 1+q Since l = l = q, we see ha if q >, hen Theorem 3 applies. Tha is, if α > 1 1, all soluions are oscillaory or converge o zero. If 0 < α, 1+q 1+q hen condiion (2.21) of Theorem 4 is equivalen o q = αq > q ( 1 α 1 2 2 1 4α)q, 1 + q which in urn is equivalen o α > ( 1 + α + 1 2 2 1 4α). 1 + q Solving his inequaliy gives (3.2) q > 1 + 1 4α. 2α Therefore, for any 0 < α 1, Theorem 4 implies ha all soluions are 1+q oscillaory or converge o zero if (3.2) holds. For example, if α = 1 and 8 q > 4 + 4 2 6.82, hen Theorem 4 applies and Theorem 3 does no apply if 4 + 4 2 < q < 7. Example 5. We le T = q N 0 aq N 0 where 1 < a < q < a 2. Then T = {1, a, q, aq, q 2, aq 2, }.

20 L. ERBE, A. PETERSON AND S. H. SAKER Thus, 2n = q n, and 2n+1 = aq n, for n = 0, 1, 2,, so σ() = { q a = 2n+1 a, = 2n and so l = a and l = q. We have a l 1 + l = a 1 + a, and so if q < a, we can no apply Theorem 3. Likewise, if p 1+a < 1, 4 Theorem 2 does no apply. Therefore, if p = 1 and wih 8 l = 3, l 2 = 4, 3 and l ( 1 p 2 2 1 1 4p )l.588 <.6 = a 1 + l 1 + a. So if.588 < q <.6 and p = 1, hen Theorem 4 applies bu Theorem 2 and 8 3 do no. References [1] R. Agarwal, M. Bohner, and S. H. Saker, Oscillaion crieria for second order delay dynamic equaion, Canad. Appl. Mah. Quar., o appear. [2] R. P. Agarwal, D. O Regan and S. H. Saker, Oscillaion crieria for second-order nonlinear neural delay dynamic equaions, J. Mah. Anal. Appl., 300 (2004) 203 217. [3] E. Akin-Bohner, M. Bohner, and S. H. Saker, Oscillaion for a cerain of class of second order Emden-Fowler dynamic equaions, Elec. Trans. Numer. Anal., o appear. [4] M. Bohner and A. Peerson, Dynamic Equaions on Time Scales: An Inroducion wih Applicaions, Birkhäuser, Boson, 2001. [5] M. Bohner and A. Peerson, Advances in Dynamic Equaions on Time Scales, Birkhäuser, Boson, 2003. [6] M. Bohner and S. H. Saker, Oscillaion of second order nonlinear dynamic equaions on ime scales, Rocky Mounain J. Mah., 34(4) (2004) 1239 1254. [7] M. Bohner and S. H. Saker, Oscillaion crieria for perurbed nonlinear dynamic equaions, Mah. Comp. Modeling, 40 (2004) 249 260. [8] L. Erbe, Exisence of oscillaory soluions and asympoic behavior for a class of hird order linear differenial equaions, Pacific J. Mah., 64 (1976) 369 385. [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 (2003) 701 714. [10] L. Erbe, A. Peerson, and S. H. Saker, Asympoic behavior of soluions of a hirdorder nonlinear dynamic equaion on ime scales, J. Comp. Appl. Mah., 181 (2005) 92 102. [11] L. Erbe, A. Peerson, and S. H. Saker, Kamenev-ype oscillaion crieria for secondorder linear delay dynamic equaions, Dynamic Sys. & Appl., 15 (2006) 65 78. [12] L. Erbe, A. Peerson, and S. H. Saker, Oscillaion and asympoic behavior of a hird-order nonlinear dynamic equaion, Canad. Appl. Mah. Quar., o appear.

HILLE AND NEHARI TYPE CRITERIA 21 [13] M. Hanan, Oscillaion crieria for hird order differenial equaions, Pacific J. Mah., 11 (1961) 919 944. [14] S. Hilger, Analysis on measure chains a unified approach o coninuous and discree calculus, Resuls Mah., 18 (1990) 18 56. [15] E. Hille, Non-oscillaion heorems, Trans. Amer. Mah. Soc., 64 (1948) 234 252. [16] V. Kac and P. Cheung, Quanum Calculus, Universiex, Springer, New York, 2001. [17] W. Kelley and A. Peerson, Difference Equaions: An Inroducion Wih Applicaions, second ediion, Harcour/Academic Press, San Diego, 2001. [18] A. C. Lazar, The behavior of soluions of he differenial equaion y + p(x)y + q(x)y = 0, Pacific J. Mah., 17 (1966) 435 466. [19] W. Leighon, The deecion of he oscillaion of soluions of a second order linear differenial equaion, Duke J. Mah., 17 (1950) 57 62. [20] B. Mehri, On he condiions for he oscillaion of soluions of nonlinear hird order differenial equaions, Cas. Pes Mah., 101 (1976) 124 129. [21] Z. Nehari, Oscillaion crieria for second-order linear differenial equaions, Trans. Amer. Mah. Soc., 85 (1957) 428 445. [22] S. H. Saker, Oscillaion crieria of second-order half-linear dynamic equaions on ime scales, J. Comp. Appl. Mah., 177 (2005) 375 387. [23] S. H. Saker, Oscillaory behavior of linear neural delay dynamic equaions on ime scales, Kyungpook Mah. J., o appear. [24] S. H. Saker, Oscillaion of second-order nonlinear neural delay dynamic equaions on ime scales, J. Comp. Appl. Mah., 177 (2005) 375 387. [25] S. H. Saker, Oscillaion crieria for a cerain class of second-order neural delay dynamic equaions, Dynamics of Coninuous, Discree and Impulsive Sysems Series B: Applicaions & Algorihms, o appear. [26] S. H. Saker, New oscillaion crieria for second-order nonlinear dynamic equaions on ime scales, Nonlin. Func. Anal. Appl. (NFAA), o appear. [27] S. H. Saker, On oscillaion of second-order delay dynamic equaions on me scales, Ausral. J. Mah. Anal. Appl., o appear. [28] S. H. Saker, Oscillaion of second-order neural delay dynamic equaions of Emden Fowler ype, Dynam. Sys. Appl., o appear. [29] V. Spedding, Taming Naure s Numbers, New Scienis, July 19, 2003, 28 31. Deparmen of Mahemaics, Universiy of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A. lerbe@mah.unl.edu, apeerso@mah.unl.edu Deparmen of Mahemaics, Faculy of Science, Mansoura Universiy, Mansoura, 35516, Egyp. shsaker@mans.edu.eg