OSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES

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1 Dynamic Sysems and Applicaions 6 (2007) OSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES S. H. SAKER Deparmen of Mahemaics and Saisics, Universiy of Calgary, 2500 Calgary Drive, Calgary Albera T2N N4, Canada (saker@mah.ucalgary.ca) ABSTRACT. In his paper, we consider he second-order linear delay dynamic equaion on a ime scale T. x () + q()x(τ()) = 0, We will sudy he properies of he soluions and esablish some sufficien condiions for oscillaions. In he special case when T = R and τ() =, our resuls include some well-known resuls in he lieraure for differenial equaions. When, T = Z, T = hz, for h > 0 and T = T n = { n : n N 0 } where n } is he se of he harmonic numbers defined by = 0, n = n k= k for n N 0 our resuls are essenially new. The resuls will be applied on second-order neural delay dynamic equaions in ime scales o obain some sufficien condiions for oscillaions. An example is considered o illusrae he main resuls. Keywords. Oscillaion, delay dynamic equaions, neural delay dynamic equaions, ime scales. AMS (MOS) Subjec Classificaion. 34K, 39A0, 39A99 (34A99, 34C0, 39A).. INTRODUCTION In recen years, he sudy of dynamic equaions on ime scales has become and an area of mahemaics and has received a lo of aenion. I was creaed o unify he sudy of differenial and difference equaions, and i also exends hese classical cases o cases in beween, e.g., o he so-called q-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 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 an arbirary nonempy closed subse of he reals. The hree mos popular examples of calculus on ime scales are differenial calculus, difference calculus, and quanum calculus. The books by Bohner and Peerson 4, 5] summarize and organize much of ime scale calculus. Dynamic equaions on a ime scale have an enormous poenial for applicaions such Received Augus 22, $5.00 c Dynamic Publishers, Inc. On leave from Deparmen of Mahemaics, Faculy of Science, Mansoura Universiy, Mansoura 3556, Egyp. The auhor was parially suppored by AIF gran.

2 346 S. H. SAKER as in populaion dynamics. For example, i can 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. 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()). 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 neighborhood 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. Assume ha g : T R and le T. wih (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 g () := g(σ()) g(). µ() (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: 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() := g(s) s exiss, T, and saisfies G () = g(), T. For a more general definiion of he dela inegral see 4, 5].

3 OSCILLATION OF SECOND-ORDER DELAY 347 In recen years here has been much research aciviy concerning he qualiaive heory of dynamic equaions on ime scales. One of he main subjecs of he qualiaive analysis of he dynamic equaions is he oscillaory behavior. Recenly, some ineresing resuls have been esablished for oscillaion and nonoscillaion of dynamic equaions on ime scales, we refer o he papers 3, 6 2, 6, 7, 2] and he references cied herein. To he bes our knowledge he papers ha are concerned wih he oscillaion of delay dynamic equaions in ime scales are, 2,, 8, 9, 20, 22, 23]. In his paper, we are concerned wih oscillaion of he second-order linear delay dynamic equaion (.) x () + q()x(τ()) = 0, on a ime scale T, where he funcion q is an rd-coninuous funcion such ha q() > 0 for T, τ : T T, τ() and lim τ() =. Throughou his paper hese assumpions will be assumed. Le T 0 = min{τ() : 0} and τ () = sup{s 0 : τ(s) } for T 0. Clearly τ () for T 0, τ () is nondecreasing and coincides wih he inverse of τ when he laer exiss. By a soluion of (.) we mean a nonrivial real-valued funcions x() Cr 2 T x, ), T x τ ( ) 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 (.) is said o be oscillaory if i is neiher evenually posiive nor evenually negaive. Oherwise i is nonoscillaory. The equaion iself is called oscillaory if all is soluions are oscillaory. 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, ) T by, ) T :=, ) T. We noe ha, Equaion (.) in is general form covers several differen ypes of differenial and difference equaions depending on he choice of he ime scale T. When T = R, σ() =, µ() = 0, f = f, and b a f() = b a f()d, and (.) becomes he second-order delay differenial equaion (.2) x () + q()x(τ()) = 0, When T = Z, σ() = +, µ() =, f = f, and b a b f() = f(), and (.) becomes he general second-order delay difference equaion (.3) x( + 2) 2x( + ) + x() + q()x(τ()) = 0. =a

4 348 S. H. SAKER When T =hz, h > 0, σ() = + h, µ() = h, f = h f = f( + h) f(), and h b a f() = and (.) becomes he second-order delay difference equaion b a h h (.4) x( + 2h) 2x( + h) + x() + h 2 q()x(τ()) = 0. k=0 hf(a + kh) When T ={ n : n N 0 }, where { n } is he se of harmonic numbers defined by n = 0, n = k, n N, k= we have σ( n ) = n+, µ( n ) = <, n+ x ( n ) = (n + ) x( n ) and (.) becomes he difference equaion (.5) x( n+2 ) n + 3 n + 2 ] ( ) x( n+ ) + (n + ) n (n + 2)q( n) x(τ( n )) = 0. For oscillaion of second-order differenial equaions, Hille 4] considered he linear equaion (.6) x () + q()x() = 0, and proved ha: If (.7) q := lim inf or (.8) q := lim sup q(s)ds > 4, q(s)ds >, hen every soluion of equaion (.6) oscillaes. Nehari 5] considered also (.6) and proved ha: If (.9) lim inf s 2 q(s)ds > hen every soluion oscillaes. For oscillaion of dynamic equaions on ime scales, Erbe, Peerson and Saker 8] esablished some new oscillaion crieria for nonlinear dynamic equaions. As a linear version of heir resuls one can easily see ha if (.0) lim inf hen every soluion of he dynamic equaion q(s) s > 4, (.) x () + q()x σ = 0, 4,

5 OSCILLATION OF SECOND-ORDER DELAY 349 is oscillaory. We noe ha he condiion (.0) is a ime scale analogue of he Hille condiion (.7). Lomaidze 3], considered (.6) when (.7) and (.8) are no saisfied, and proved ha: If (.2) q, and lim 4 sup s 2 q(s)ds > hen every soluion of (.6) oscillaes. ( + ) 4q, 2 The naural quesion now is: Can he oscillaion condiion (.2) of Lomaidze be exended for (.) on ime scales and in he special case when T = R and τ() = include he condiion (.2), i.e., can we find new oscillaion crieria for (.) when (.3) where l := lim. σ() lim inf lim inf σ() τ(s) s q(s) s 4l s 2 ( τ(s) s and ) q(s) s 4, The purpose of his paper is o give an affirmaive answer o his quesion. The paper, is organized as follows: In Secion 2, by analyzing he Riccai dynamic inequaliy, we esablish some properies of he soluions of (.) and also esablish some new sufficien condiions for oscillaion of (.). In he special case when T = R and τ() = our resuls include he oscillaion condiion (.2) esablished by Lomaidze 3] for second-order differenial equaion. In he case, when T = Z, T =hz, h > 0, T = T n our resuls are essenially new. An example is considered o illusrae he main resuls. In Secion 3, we consider he second-order neural delay dynamic equaion (.4) y() + r()y(τ())] + q()y(δ()) = 0, on a ime scale T and exend he resuls in Secion 2 and esablish some new sufficien condiions for oscillaions. The echnique in his paper is differen from he echniques considered in, 2,, 8, 9, 20, 22, 23]. 2. MAIN RESULTS In his secion, we sudy he properies of he soluions of (.) and esablish some new sufficien condiions for oscillaions. In wha follows, we will assume ha he graininess funcion µ() saisfies max T µ() = h 0 0, and (2.) τ(s)q(s) s =.

6 350 S. H. SAKER In he nex resuls, for simpliciy, we will use he noaions P () := τ(s) q(s) s, Q() := ( ) σ() s s 2 τ(s) q(s) s, s P := lim inf P (), Q := lim inf Q(), P := lim sup P (), Q := lim sup Q(), We will need he following lemma in he proof of our main resuls. Lemma 2. (]). Le x be a posiive soluion of (.) on, ) and T = τ ( ). Then (i) x () 0, x() x ()for T, (ii) x is nondecreasing, while x()/ is nonincreasing on T, ). Before, we proceed o he formulaion of he oscillaion resuls, we esablish some properies of he soluion of (.). Lemma 2.2. Le x() be a nonoscillaory soluion of (.) such ha x(τ()) > 0 for τ ( ). Assume ha P /4l. Define w() := x () x(), hen (2.2) lim inf λ w() = 0, for λ <, (2.3) lim inf w σ () 2 ( 4P l). Proof. From he definiion of w() and in in view of Lemma 2., since x() be a nonoscillaory soluion of (.) such ha x(τ()) > 0 for τ ( ), we have w() > 0, and saisfies w () = ( x ) σ x() ( x )σ () = x σ () In view of (.), we ge w () + q() x(τ()) x() From Lemma 2., we have x(τ()) x() ] + x() x () = ( x ) σ () x () x()x σ () + x() x () x () x() + x() x (), for. + w()w σ () = 0, for. τ(). This implies ha (2.4) w () + p() + w()w σ () 0, for. where p(s) := τ(s) q(s) > 0. This implies ha s So ha w () w()w σ () <, for. ( ) <. w()

7 OSCILLATION OF SECOND-ORDER DELAY 35 Inegraing he las inequaliy from o, we have (2.5) ( )w() <, for, which implies ha (2.6) lim w() = 0, lim λ w() = 0, for λ < and lim w(s) s = 0, and his proves (2.2). Now, we prove (2.3). If P = 0, hen (2.3) is rivial. So, we may assume ha P > 0. Inegraing (2.4) from σ() o (σ() ) and using (2.6), we have (2.7) w σ (s) σ() p(s) s + σ() Se r := lim inf w σ (). By using (2.5), we see ha (2.8) 0 < r, and r r 2 > 0. w(s)w σ (s) s for. Then i follows ha for any ɛ (0, r) here exiss 2, such ha and from he definiion of P, we have From (2.7), we have w σ = σ() σ() σ() σ() This implies ha, Then p(s) s + σ() σ() r ɛ < w σ (), p(s) s P ɛ, for 2. w(s)w σ (s) s σ() p(s) s + (r ɛ) 2 σ() sσ(s) s ( ) p(s) s + (r ɛ) 2 s s σ() p(s) s + σ() (r ɛ)2. p(s) s + w σ () p(s) s + σ() σ() (r ɛ)2. (2.9) r P ɛ + l(r ɛ) 2 for 2. Since ɛ is an arbirary, we have (2.0) P r lr 2, which implies ha (2.) lr 2 r + P 0. σ() sw σ (s)sw σ (s) s s 2

8 352 S. H. SAKER Then, from (2.) since P /4l, we see ha (2.3) holds. The proof is complee. Lemma 2.3. Le x() be a nonoscillaory soluion of (.) such ha x(τ()) > 0 for τ ( ). Assume ha Q /4. Then (2.2) lim sup w σ () 2 ( + 4Q ). Proof. We proceed as in he proof of Lemma 2.2 o ge (2.4). From (2.4) we see ha w () 0. This implies ha w() w σ (), and hence (2.4) becomes (2.3) w () + p() + (w σ ()) 2 0, for. Muliplying (2.3) by 2, and inegraing from o ( ) and using he inegraion by pars, we obain s 2 p(s) s s 2 w (s) s s 2 (w σ (s)) 2 s I follows ha Then, we have (2.4) = 2 w() ] + = 2 w() + 2 w( ) + = 2 w() + 2 w( ) + + µ(s)w σ (s) s. w() 2 w( ) + (s 2 ) w σ (s) s w σ () 2 w( ) + From (2.6), since lim lim s 2 (w σ (s)) 2 s (s + σ(s))w σ (s) s 2sw σ (s) s w σ (s) s 2sw σ (s) s 2 (w σ (s)) 2] s. w σ (s) s s 2 (w σ (s)) 2 s s 2 (w σ (s)) 2 s s 2 p(s) s s 2 p(s) s 2sw σ (s) s 2 (w σ (s)) 2] s. w(s) s = 0, and w() w σ (), we have 2 w( ) + h ] 0 w σ (s) s = 0. Also, using he inequaliy a 2 + b 2 2ab, we have 2σ(s)w σ (s) σ 2 (s) (w σ (s)) 2],

9 OSCILLATION OF SECOND-ORDER DELAY 353 and his implies ha (2.5) Se 2sw σ (s) s 2 (w σ (s)) 2] s. (2.6) R := lim sup w σ (). Then from (2.4), we have R Q. The esimaion in (2.2) is valid for Q = 0. We may assume ha Q > 0. For an arbirary ɛ (R, Q ), here exiss 2 such ha (2.7) R ɛ < w σ () < R + ɛ. From he definiion of Q, we see ha (2.8) Then, from (2.4) (2.8), we obain s 2 p(s) s > Q ɛ, for 2. (2.9) lim sup w σ () Q + ɛ + (R + ɛ)(2 R ɛ), for 2. From (2.6) and (2.9), since ɛ is an arbirary, we ge (2.20) Q R R 2, and herefore (2.2) R 2 R + Q 0. Now since Q /4, we see ha R 2 ( + 4Q ) which is (2.2). The proof is complee. of (.). From Lemma 2.2 and Lemma 2.3, we have he following properies of he soluions Lemma 2.4. Le x() be a nonoscillaory soluion of (.) such ha x(τ()) > 0 for τ ( ). Assume ha P /4l and 0 Q /4. Then ( ) x σ lim sup () x 2 ( + 4Q ), and lim inf Theorem 2.2. Assume ha ( x x ) σ () 2 ( 4P l). (2.22) lim supp () + Q()] >, hen every soluion of (.) oscillaes.

10 354 S. H. SAKER Proof. Assume for he sake of conradicion ha (.) has a nonoscillaory soluion. Wihou loss of generaliy, we may assume ha here is a posiive soluion x() of (.) such ha x(τ()) > 0 for τ ( ). Le w() = x () and proceeding as x() in he proofs of Lemmas 2.2 and 2.3, o ge (2.23) and (2.24) σ() p(s) s w σ () σ() s 2 p(s) s w σ () + 2 w( ) + From (2.23) and (2.24), we obain P () + Q() 2 w( ) σ() w(s)w σ (s) s w(s)w σ (s) s, w σ (s) s 2sw σ (s) s 2 (w σ (s)) 2] s. w σ (s) s + Using he fac ha 2sw σ (s) s 2 (w σ (s)) 2], we have P () + Q() + 2 w( ) + 2 w( ) Now, since σ() (wσ (s)) 2 <, we have w σ (s) s w σ (s) s lim supp () + Q()] + lim sup2 w( ) Using (2.6) in he las inequaliy, we ge lim supp () + Q()], which conradics (2.22). The proof is complee. 2sw σ (s) s 2 (w σ (s)) 2] s σ() From Theorem 2.2, we have he following oscillaion resul. Corollary 2.2. Assume ha (2.25) Q >, or P >, hen every soluion of (.) oscillaes. σ() w(s)w σ (s) s (w σ (s)) 2 s. w σ (s) s]. Example 2.. Consider he second-order delay Euler dynamic equaion (2.26) x () + γ τ() x(τ()) = 0, for, ) T,

11 OSCILLATION OF SECOND-ORDER DELAY 355 where τ() and lim τ() =. I is clear ha τ(s)q(s) s = τ(s) γ sτ(s) s = γ s =, so he condiion (2.) holds. To apply Corollary 2.2, i remains s o saisfy condiion (2.25). In he case, we have Q = lim sup = γ lim inf s = γ. s 2 P (s) s = lim inf s 2 γ sτ(s) τ(s) s s So by Corollary 2.2, every soluion of (2.26) oscillaes if γ >. Also, we noe ha P = lim sup γ lim inf σ() σ() p(s) τ(s) s = γ lim s inf σ() s = γ lim sσ(s) inf σ() s s 2 ) s = γ. ( s So by Corollary 2.2, every soluion of (2.26) oscillaes if γ >. Remark 2.. Noe ha he oscillaion condiion P > on Corollary 2.2 is he ime scale analogue of he condiion (.8) of Hille 4]. Now, we concenrae our work o give an affirmaive answer o he quesion posed in he inroducion and consider he case when Q /4, and P /4l. Theorem 2.3. Assume ha P /4l. Then, every soluion of (.) oscillaes if (2.27) Q > 2 ( + 4P l). Proof. Assume for he sake of conradicion ha (.) has a nonoscillaory soluion. Wihou loss of generaliy, we may assume ha here is a posiive soluion x() of (.) such ha x(τ()) > 0 for τ ( ). Then from Lemma 2.2, we have w σ () > r ɛ, for 2 >, where ( ) x w σ σ () = () and r = x 2 ( 4P l). From Theorem 2.2, we have I follows ha s 2 p(s) s w σ () + 2 w( ) + w σ (s) s 2sw σ (s) s 2 (w σ (s)) 2] s r + ɛ w( ) Q 2 ( + 4P l), w σ (s) s.

12 356 S. H. SAKER which conradics he condiion (2.27). The proof is complee. Theorem 2.4. Assume ha Q /4. Then, every soluion of (.) oscillaes if (2.28) P > 2 ( + 4Q ). Proof. Assume for he sake of conradicion ha (.) has a nonoscillaory soluion. Wihou loss of generaliy, we may assume ha here is a posiive soluion x() of (.) such ha x(τ()) > 0 for τ ( ). Then from Lemma 2.3, we have where From Lemma 2.2, we have w σ () which implies ha σ() p(s) s + Since ɛ is an arbirary, so ha w σ < R + ɛ, for 2 >, R = 2 ( + 4Q ). σ() w(s)w σ (s) s p(s) s, for, σ() lim sup p(s) s R + ɛ, for 2 >. σ() P 2 ( + 4Q ), which conradics he assumpion (2.28). The proof is complee. Remark 2.2. In he special case when T = R and τ() =, we see ha he condiion (2.28) becomes he condiion (.2). So, our resuls in he special case involve he oscillaion resuls of differenial equaions esablished by Lomaidze 3], and are essenially new for equaions (.3) (.5) and can be applied o differen ypes of ime scales wih max T µ() = h 0 0. Also he resuls can be exended o he nonlinear delay dynamic equaions x () + q()f(x(τ())) = 0, when f(u) K u for K > APPLICATIONS ON NEUTRAL DYNAMIC EQUATIONS In his secion, we apply he oscillaion resuls esablished in Secion 2 on he neural delay dynamic equaion (.4). We assume ha he following assumpions are saisfied: (h ). τ(), δ() are defined on he ime scale T, and lim δ() = lim τ() =, (h 2 ). r() and q() are posiive real-valued rd-coninuous funcions defined on T and 0 r() r <.

13 (h 3 ). δ(s)q(s)( r(δ(s))) =. OSCILLATION OF SECOND-ORDER DELAY 357 To he bes of our knowledge no similar resuls are given for equaion (.4) even for differenial or difference equaions. Lemma 3.. Assume ha (h ) (h 3 ) hold. Le y() be a nonoscillaory soluion of (.4) such ha y(), y(τ()) and y(δ()) > 0 for sufficienly large. Le (3.) u() := y() + r()y(τ()), and b := lim inf B(s) s, B := lim inf s 2 B(s) s, σ() where B(s) := δ(s) q(s)( r(δ(s))). Assume ha b s /4l and 0 B /4, hen lim inf ( u u lim inf λ ( u ) () = 0, for λ <, u ) σ () 2 ( 4b l), and lim sup Proof. In view of (.4) and (3.), we have ( u (3.2) u () + q()y(δ()) 0, >, u ) σ () 2 (+ 4B ). and so u () is an evenually decreasing funcion. From Lemma 2., u () is evenually nonnegaive and hence y() = u() r()y(τ()) = u() r()u(τ()) r(τ())y(τ(τ()))] u() r()u(τ()) ( r())u(). Then, for 2 = δ ( ), we see ha (3.3) y(δ()) ( r(δ()))u(δ()). Then from (3.2) and (3.3), we have (3.4) u () + q()( r(δ()))u(δ()) 0, for 2. Define w() := u () and proceeding as in he proof of Lemma 2.2 by using (3.4) we u() have w() > 0 and saisfies he Riccai dynamic inequaliy (3.5) w () + B() + (w σ ()) 2 0, for 2. The remainder of he proof is similar o ha of he proofs of Lemmas 2.2 and 2.3 by using (3.5) and hence is omied. Now, we are ready o sae he main oscillaion resuls for (.4) based on Lemma 3. and he inequaliy (3.5). The proofs are similar o ha of he proofs of Theorems and hence are omied.

14 358 S. H. SAKER Theorem 3.. Assume ha (h ) (h 3 ) hold. If (3.6) b > 4, or B > 4, hen every soluion of (.4) oscillaes. Theorem 3.2. Assume ha (h ) (h 3 ) hold. If (3.7) b + B >, where b := lim sup B(s) s, and B := lim sup s 2 B(s) s. σ() hen every soluion of (.4) oscillaes. Theorem 3.3. Assume ha (h ) (h 3 ) hold and b /4l. If (3.8) B > 2 ( + 4b l), hen, every soluion of (.5) oscillaes. Theorem 3.4. Assume ha (h ) (h 3 ) hold and B /4. If (3.9) b > 2 ( + 4B ), hen every soluion of (.4) oscillaes. REFERENCES ] R. P. Agarwal, M. Bohner and S. H. Saker, Oscillaion of second-order delay dynamic equaions, Canadian 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), ] E. A. Bohner and J. Hoffacker, Oscillaion properies of an Emden-Fowler ype Equaions on Discree ime scales, J. Diff. Eqns. Appl. 9 (2003), ] M. Bohner and A. Peerson, Dynamic Equaions on Time Scales: An Inroducion wih Applicaions, Birkhäuser, Boson, ] M. Bohner and A. Peerson, Advances in Dynamic Equaions on Time Scales, Birkhäuser, Boson, ] M. Bohner and S. H. Saker, Oscillaion of second order nonlinear dynamic equaions on ime scales, Rocky Mounain J. Mah. 34 (2004), ] M. Bohner and S. H. Saker, Oscillaion crieria for perurbed nonlinear dynamic equaions, Mahl. Comp. Modelling 40 (2004), ] E. Akin-Bohner, M. Bohner and S. H. Saker, Oscillaion crieria for a cerain class of second order Emden-Fowler dynamic Equaions, Elec. Transc. Numerical. Anal. (o appear). 9] L. Erbe and A. Peerson, Posiive soluions for a nonlinear differenial equaion on a measure chain, Mahl. Comp. Modelling, Boundary Value Problems and Relaed Topics, 32 (2000),

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