Commun Nonlinear Sci Numer Simulat

Transcription

1 Commn Nonlinear Sci Nmer Simla 4 (2009) Conens liss available a ScienceDirec Commn Nonlinear Sci Nmer Simla jornal omepage: Oscillaion of second-order srongly sperlinear and srongly sblinear dynamic eqaions Said R. Grace a, Ravi P. Agarwal b, Marin Boner c, *,, Donal O Regan d a Deparmen of Engineering Maemaics, Facly of Engineering, Cairo Universiy, Orman, Giza 222, Egyp b Deparmen of Maemaical Sciences, Florida Insie of Tecnology, Melborne, FL 3290, USA c Deparmen of Economics and Finance, Missori Universiy of Science and Tecnology, Rolla, MO , USA d Deparmen of Maemaics, Naional Universiy of Ireland, Galway, Ireland aricle info absrac Aricle isory: Received 2 Ocober 2008 Acceped 9 Janary 2009 Available online 9 Janary 2009 We esablis some new crieria for e oscillaion of second-order nonlinear dynamic eqaions on a ime scale. We sdy e case of srongly sperlinear and e case of srongly sblinear eqaions sbjec o varios condiions. Ó 2009 Elsevier B.V. All rigs reserved. AMS Sbjec Classificaion: 34C0 39A0 PACS: Hq Ks Lj Bf Keywords: Dynamic eqaion Time scale Nonlinear Sperlinear Sblinear Oscillaion. Inrodcion Tis paper is concerned wi e oscillaory beavior of solions of second-order nonlinear dynamic eqaions of e form ðax D Þ D ðþþf ð; x r ðþþ ¼ 0; P ð:þ sbjec o e following ypoeses: * Corresponding aor. addresses: [email protected] (S.R. Grace), [email protected] (R.P. Agarwal), [email protected] (M. Boner), [email protected] (D. O Regan). Sppored by NSF Gran # /$ - see fron maer Ó 2009 Elsevier B.V. All rigs reserved. doi:0.06/j.cnsns

2 3464 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) (i) a is a posiive real-valed rd-coninos fncion saisfying eier Z ¼ ð:2þ or Z < : ð:3þ (ii) f : ½ ; Þ R! R is coninos saisfying sgnf ð; xþ ¼sgnx and f ð; xþ 6 f ð; yþ; x 6 y; P : ð:4þ By a solion of Eq. (.), we mean a nonrivial real-valed fncion x saisfying Eq. (.) for P x P. A solion x of Eq. (.) is called oscillaory if i is neier evenally posiive nor evenally negaive; oerwise i is called nonoscillaory. Eq. (.) is called oscillaory if all is solions are oscillaory. Te eory of ime scales, wic as recenly received a lo of aenion, was inrodced by Hilger [9]. Several aors ave exponded on varios aspecs of is new eory, see [5,,2]. In recen years, ere as been mc researc aciviy concerning e oscillaion and nonoscillaion of varios eqaions on ime scales, we refer e reader o [2,3,0,3,4,6,7]. Mos of e resls are obained for special cases of Eq. (.), e.g. wen a ¼ and f ð; xþ ¼qðÞx or a ¼ and f ð; xþ ¼qðÞf ðxþ, were f saisfies e condiion jf ðxþ=xj P k > 0 for x 0, or f 0 ðxþ P f ðxþ=x > 0 for x 0, see [5 7]. In[8], e aors considered e second-order Emden Fowler dynamic eqaion on ime scales x DD þ qðþx a ¼ 0, were a is e raio of odd inegers, and ey sed e Riccai ransformaion ecniqe o obain several oscillaion crieria for is eqaion. For e coninos case of (.), i.e., ðax 0 Þ 0 ðþþf ð; xðþþ ¼ 0; nmeros oscillaion and nonoscillaion crieria ave been esablised, see e.g. [,4,6 9]. Te main goal of is paper is o esablis new oscillaion crieria for (.) (wio employing e Riccai ransformaion ecniqe). Te paper is organized as follows: In Secion 2, we presen some basic preliminaries concerning calcls on ime scales and prove some axiliary resls a will be sed in e remainder of is paper. In Secion 3, we presen some oscillaion crieria wen condiion (.2) olds, and Secion 4 is devoed o e sdy of oscillaion of Eq. (.) wen condiion (.3) olds. In Secion 5, some applicaions are discssed. Te resls of is paper are presened in a form wic is essenially new and of a ig degree of generaliy. Te obained resls nify and improve many known oscillaion crieria wic appeared in e lierare. 2. Preliminaries A ime scale T is an arbirary nonempy closed sbse of e real nmbers R. Since we are ineresed in oscillaory beavior, we sppose a e ime scale nder consideraion is nbonded above. Te forward jmp operaor r : T! T is defined by rðþ ¼inffs 2 T : s > g. For a fncion f : T! R we wrie f r ¼ f r, and f D represens e dela derivaive of e fncion f as defined for example in []. Te reader nfamiliar wi ime scales may ink of f D as e sal derivaive f 0 if T ¼ R and as e sal forward difference Df if T ¼ Z. For e definiion of rd-coniniy and frer deails we refer o []. Besides e sal properies of e ime scales inegral, we reqire in is paper only e se of e cain rle [, Teorem.90] ðx a Þ D Z a ¼ xd ½x r þð ÞxŠ a d; ð2:þ 0 were a > 0 and x is sc a e rig-and side of (2.) is well defined. Inegraion by pars, e prodc rle, and e qoien rle are no needed in is paper. One conseqence of (2.), a will be sed in e proof of for resls rogo is paper, is as follows. Lemma 2.. Sppose jyj D is of one sign on ½ ; Þ and a > 0. Ten jyj D ðjyj r Þ a 6 ðjyj a Þ D a 6 jyjd jyj a on ½ ; Þ: Proof. Replacing x by jyj in (2.), we see a eier jyj D > 0 so a jyj is increasing and ence jyj 6 jyj r and s jyj 6 jyj r þð Þjyj 6 jyj r for all 2½0; Š, or oerwise jyj D < 0 so a jyj is decreasing and ence jyj P jyj r and s jyj P jyj r þð Þjyj P jyj r for all 2½0; Š. We sall obain resls for srongly sperlinear and srongly sblinear eqaions according o e following classificaion.

3 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) Definiion 2.2. Eq. (.) (or e fncion f) is said o be srongly sperlinear if ere exiss a consan b > sc a jf ð; xþj jf ð; yþj 6 for jxj 6 jyj; xy > 0; P jxj b jyj b 0 ; ð2:2þ and i is said o be srongly sblinear if ere exiss a consan c 2ð0; Þ sc a jf ð; xþj jf ð; yþj jxj c P jyj c for jxj 6 jyj; xy > 0; P : ð2:3þ If (2.2) olds wi b ¼, en (.) is called sperlinear and if (2.3) olds wi c ¼, en (.) is called sblinear. Lemma 2.3. Condiion (.4) implies a jf ð; xþj 6 jf ð; yþj for jxj 6 jyj; xy P 0; P : ð2:4þ Proof. Assme (.4). Sppose jxj 6 jyj and xy P 0. Ten eier 0 6 x 6 y and s 0 6 f ð; xþ 6 f ð; yþ so a jf ð; xþj 6 jf ð; yþj for all P,ory 6 x 6 0 and s f ð; yþ 6 f ð; xþ 6 0 so a again jf ð; xþj 6 jf ð; yþj for all P. Ts (2.4) olds. Te following lemma will be sed rogo. Lemma 2.4. Sppose x solves (.) and is of one sign on ½ ; Þ. Le ; v; P. Ten and jxj D ðvþ ¼x D ðvþsgnxðvþ jxjðþ ¼jxjðÞþaðvÞjxj D ðvþ Z v jf ðs; xðrðsþþþj: ð2:5þ ð2:6þ Proof. Le ;v; T wi ; v; P. Le s 2 T wi s P. Inegrae (.) from v o s and divide e resling eqaion by. Now inegrae e resling eqaion from o and mliply e resling eqaion wi sgnxðvþ. Ten observe sgnxðvþ¼sgnxðþ, jxðvþj ¼ xðvþsgnxðvþ, jxðþj ¼ xðþsgnxðþ, and so (.4) gives jxjðþ ¼jxjðÞþaðvÞx D ðvþsgnxðvþ v jf ðs; xðrðsþþþj: Now differeniae (2.7) wi respec o and en plg in ¼ v o obain (2.5). Finally se (2.5) in (2.7) o arrive a (2.6). ð2:7þ 3. Crieria nder condiion (.2) In is secion, we give some new oscillaion crieria for Eq. (.) wen condiion (.2) olds. We le AðÞ :¼ for P : Te following simple conseqence of Lemma 2.4 will be sed rogo is secion. Lemma 3.. Assme (.2). Sppose x solves (.) and is of one sign on ½ ; Þ. Ten on ½ ; Þ, jxj D P 0; ence jxj is increasing: ð3:þ Moreover, pick any > and le ~c ¼ xð Þ and c ¼ jxð 0Þj Að Þ þ aðþjxj D ð Þ sgnxð Þ: Ten jxj P j~cj on ½ ; Þ; were ~cx > 0 ð3:2þ and jxj 6 jc Aj on ½ ; Þ; were c Ax > 0: ð3:3þ Proof. Using (2.6) wi P v ¼ P, we find jxðþj 6 jxðþj þ aðþjxj D ðþ for all P ;

4 3466 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) wic is a conradicion o (.2) wen jxj D ðþ < 0. Tis complees e proof of (3.). Nex, (3.2) follows from (3.). Finally, sing (2.6) wi P and ¼ v ¼, we find jxðþj 6 jxð Þj þ að Þjxj D ð ÞAðÞ ¼ jxð 0Þj AðÞ þ aðþjxj D ð Þ AðÞ 6 jc AðÞj so a (3.3) follows. Te firs oscillaion resl abo (.) is immediae. Teorem 3.2. Assme (.2). If Z jf ðs; cþj ¼ for all c 0; ð3:4þ en Eq. (.) is oscillaory. Proof. Differeniaing (2.6) wi respec o and en leing ¼ and sing (3.2) and (2.4), we find, for all v P, jxj D ð Þ P jf ðs; xðrðsþþþj P ; að Þ að Þ wic conradics (3.4) and complees e proof. Te nex resl deals wi e oscillaion of all bonded solions of Eq. (.). Teorem 3.3. Assme (.2). If Z Z s jf ðs; cþj ¼ for all c 0; ð3:5þ en all bonded solions of Eq. (.) are oscillaory. Proof. Le x be a bonded nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ. Using (3.2) and (2.4), we obain jf ðs; xðrðsþþþj P for all s P : Ts, sing (2.6) wi v P P ¼, ogeer wi (3.), we find Z v Z v jxðþj P jf ðs; xðrðsþþþj P ; s s wic, since x is bonded, conradics (3.5) and complees e proof. Teorem 3.4. Assme (.2). Sppose (.) is sperlinear. If Z lim sp! AðÞ jf ðs; cþj > jcj for all c 0; ð3:6þ en Eq. (.) is oscillaory. Proof. Le x be a nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ. Using (3.2) and (2.2) (wi b ¼ ), we obain jf ðs; xðrðsþþþj P jxðrðsþþj j~cj for all s P : Ts, sing (2.6) wi v P P ¼, along wi (3.), we find Z v Z v jxðþj P jf ðs; xðrðsþþþj P jxðrðsþþj s s j~cj Z v Z Z v P jxðrðsþþj P jxðþj j~cj j~cj and ence j~cj P AðÞ ; wic conradics (3.6) and complees e proof. Nex, we presen e following resl for e srongly sperlinear Eq. (.).

5 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) Teorem 3.5. Assme (.2). Sppose (.) is srongly sperlinear. If Z Z jf ðs; cþj ¼ for all c 0; ð3:7þ s en Eq. (.) is oscillaory. Proof. Le x be a nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ. Using (3.2) and (2.2) (wi b > ), we obain jf ðs; xðrðsþþþj P for all s P jxðrðsþþj b j~cj b 0 : Ts, differeniaing (2.6) wi respec o and sing (3.) and Lemma 2. (e ineqaliy on e lef-and side), we find, for all v P, jxj D ðþ P jf ðs; xðrðsþþþj P jxðrðsþþj b aðþ aðþ j~cj b P jxðrðþþj b P ðb Þjxj D ðþ aðþ j~cj b aðþ j~cj b ðjxj b Þ D ðþ and ence ðjxj b Þ D ðþ P b j~cj b aðþ : Inegraing is ineqaliy from o P, we obain jxð Þj b P jxðþj b þ b Z v P b j~cj b s j~cj b wic conradics (3.7) and complees e proof. Finally, for e srongly sblinear Eq. (.), we ave e following resl. Teorem 3.6. Assme (.2). Sppose (.) is srongly sblinear. If Z ; s jf ðs; caðsþþj ¼ for all c 0; ð3:8þ en Eq. (.) is oscillaory. Proof. Le x be a nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ. Using (3.3) and (2.3) (wi 0 < c < ), we obain jf ðs; xðsþþj jxðsþj c P jf ðs; c AðsÞÞj jc AðsÞj c for all s P : Ts, sing (2.6) wi ¼ and v P P, we find Z v Z v jxðþj P jf ðs; xðrðsþþþj P jf ðs; xðrðsþþþj s ¼ AðÞ jf ðs; xðrðsþþþj P AðÞ jf ðs; xðsþþj jf ðs; c AðsÞÞj P AðÞ jc AðsÞj c jxðsþj c (were we ave sed (3.) and (2.4) in e second las ineqaliy) and ence Z xðþ v AðÞ P zðþ; were zðþ :¼ jc j c jf ðs; c AðsÞÞj xðsþ c AðsÞ : Ts, sing Lemma 2. (e ineqaliy on e rig-and side), jzj D ðsþ ¼ z D ðsþ ¼jc j c jf ðs; c AðsÞÞj xðsþ c AðsÞ P jc j c jf ðs; c AðsÞÞjjzðsÞj c P jc j c jf ðs; c AðsÞÞj ð cþð jzjd ðsþþ ðjzj c Þ D ðsþ

6 3468 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) and ence ðjzj c Þ D ðsþ P c jc j c jf ðs; c AðsÞÞj: Inegraing is ineqaliy from o P, we obain jzð Þj c P jzðþj c þ c jc j c jf ðs; c AðsÞÞj P c jc j c jf ðs; c AðsÞÞj; wic conradics (3.8) and complees e proof. 4. Crieria nder condiion (.3) Te prpose of is secion is o presen crieria for e oscillaion of Eq. (.) wen f is eier srongly sperlinear or srongly sblinear and wen (.3) olds. We le Z eaðþ :¼ for P : Te following conseqence of Lemma 2.4 will be sed rogo is secion. Lemma 4.. Assme (.3). Sppose x solves (.) and is of one sign on ½ ; Þ. Ten eier on ½ ; Þ jxj D P 0; ence jxj is increasing ð4:þ or ere exiss > sc a on ½ ; Þ jxj D 6 0; ence jxj is decreasing: ð4:2þ Moreover, le 8 xð n c ¼ jxð Þj þ að Þjx D ð ÞjAð e o 0 Þ < 0 Þ sgnxð Þ and ^c ¼ e Að0 Þ : að Þjx D ð Þjsgnxð Þ if ð4:þ olds; if ð4:2þ olds: Ten and jxj 6 jcj on ½ ; Þ; were cx > 0 ð4:3þ jxj P j^c e Aj on ½ ; Þ; were ^c e Ax > 0: ð4:4þ Proof. Assme a (4.) does no old. Ten ere exiss > sc a jxj D ð Þ < 0. Now, differeniaing (2.6) and sing v ¼, we find jxj D ðþ ¼ aðþjxj D ð Þ aðþ aðþ jf ðs; xðrðsþþþj 6 aðþ aðþ jxjd ð Þ 6 0 for all P, wic proves (4.2). Nex, sing (2.6) wi v ¼ ¼ 6 and aking ino accon (2.5), we find jxðþj 6 jxð Þj þ að Þjxj D ð Þ 6 jxðþj þ að Þjx D ð Þj 6 jcj for all P ; wic proves (4.3). Finally, o prove (4.4), we consider wo cases: If (4.) olds, en jxðþj P jxð Þj ¼ xðþ Að eað Þ e 0 Þ¼j^cj Að e 0 Þ P j^cj AðÞ: e If (4.2) olds, en, sing (2.6) wi P P ¼ v, we find jxðþj ¼ jxðþj að Þjxj D ð Þ Z þ jf ðs; xðrðsþþþj P j^cj ; and leing!sows jxðþj P j^cj e AðÞ for all P.

7 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) Teorem 4.2. Assme (.3). If Z jf ðs; c e AðrðsÞÞÞj ¼ for all c 0; ð4:5þ en Eq. (.) is oscillaory. Proof. Le x be a nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ. Using (4.4) and (2.4), we obain jf ðs; xðrðsþþþj P jf ðs; ^c e AðrðsÞÞÞj for all s P : Ts, sing (2.6) wi ¼ v ¼ 6, we find jxðþj ¼ jxð Þj þ að Þjxj D ð Þ 6 jxð Þj þ að Þjxj D ð Þ wic conradics (4.5) and complees e proof. jf ðs; xðrðsþþþj jf ðs; ^c e AðrðsÞÞÞj; For e srongly sperlinear Eq. (.), we presen e following resl. Teorem 4.3. Assme (.3). Sppose (.) is srongly sperlinear. If Z jf ðs; c e AðrðsÞÞÞj ¼ for all c 0; ð4:6þ en Eq. (.) is oscillaory. Proof. Le x be a nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ. Using (4.4) and (2.2) (wi b > ), we obain jf ðs; xðrðsþþþj P jf ðs; ^ce AðrðsÞÞÞj jxðrðsþþj b j^c AðrðsÞÞj e for all s P b 2 : Ts, sing (2.6) wi P P ¼ v, we find jxðþj ¼ jxðþj að Þjxj D ð Þ P að Þjxj D ð Þ þ so a, wi (2.4) and b ¼ að Þjx D ð Þj, and ence jxðþj P b e AðÞþ e AðÞ xðþ eaðþ P wðþ; Z þ Z jf ðs; xðrðsþþþj jf ðs; xðrðsþþþj jf ðs; xðrðsþþþj P b e AðÞþ e AðÞ Z Z were wðþ :¼ b þj^cj b jf ðs; ^c AðrðsÞÞÞj e xðrðsþþ b : eaðrðsþþ jf ðs; ^c AðrðsÞÞÞj e j^c AðrðsÞÞj e jxðrðsþþj b b Ts, sing Lemma 2. (e ineqaliy on e lef-and side), jwj D ðsþ ¼w D ðsþ ¼j^cj b jf ðs; ^c AðrðsÞÞÞj e xðrðsþþ b eaðrðsþþ and ence P j^cj b jf ðs; ^c e AðrðsÞÞÞjjwðrðsÞÞj b P j^cj b jf ðs; ^c e AðrðsÞÞÞj ðb ÞjwjD ðsþ ðjwj b Þ D ðsþ ðjwj b Þ D ðsþ P b j^cj jf ðs; ^ce AðrðsÞÞÞj: b

8 3470 S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) Inegraing is ineqaliy from o P, we obain jwð Þj b P jwðþj b þ b j^cj b wic conradics (4.6) and complees e proof. jf ðs; ^c e AðrðsÞÞÞj P b j^cj b Finally, for e srongly sblinear Eq. (.), we ave e following resl. Teorem 4.4. Assme (.3). Sppose (.) is srongly sblinear. If Z en Eq. (.) is oscillaory. jf ðs; ^c e AðrðsÞÞÞj; jf ðs; cþj ¼ for all c 0; ð4:7þ Proof. Le x be a nonoscillaory solion of (.) sc a x is of one sign on ½ ; Þ.ByLemma 4., eier (4.) or (4.2) olds. In e case of (4.), we ave jxðþj P jxð Þj for all P and s, by (2.6) wi ¼ v ¼ 6, ogeer wi (2.4), jxðþj ¼ jxð Þj þ að Þjxj D ð Þ 6 jxð Þj þ að Þjxj D ð Þ jf ðs; xðrðsþþþj jf ðs; xð ÞÞj; a conradicion o (4.7). In e case of (4.2), sing (4.3) and (2.3) (wi 0 < c < ), we obain jf ðs; xðrðsþþþj jf ðs; cþj jxðrðsþþj c P jcj c for all s P : Ts, differeniaing (2.6) wi respec o and leing v ¼ 6, we find jxj D ðþ ¼ aðþjxj D ð Þ aðþ aðþ 6 jcj c aðþ 6 jcj c aðþ Z 2 jf ðs; xðrðsþþþj jf ðs; cþjjxðrðsþþj c jf ðs; cþjjxðþj c ; were we ave sed again (4.2) in e las ineqaliy. Now, sing (2.) (e ineqaliy on e rig-and side), we obain jcj c aðþ jf ðs; cþj 6 jxj D ðþ jxðþj c 6 ðjxj c Þ D ðþ : c Inegraing is ineqaliy from o P, we obain jxð Þj c P jxðþj c þ c jcj c wic conradics (4.7) and complees e proof. jf ðs; cþj P c jcj c jf ðs; cþj; 5. Some applicaions We sall apply e obained resls for Eq. (.) o e second-order Emden Fowler dynamic eqaion on ime scales ðax D Þ D þ qðx r Þ a ¼ 0; were a and q are nonnegaive rd-coninos fncions and a is e raio of posiive odd inegers. By Teorems 3.2, 3.4,3.5 and 3.6, respecively, we obain e following. Teorem 5.. Le condiion (.2) old and define AðÞ ¼ R =. Eq. (5.) is oscillaory if one of e following condiions olds: R qðsþ ¼,ifa > 0; lim sp! faðþ R R qðsþg > c for any c > 0, ifa P ; R s qðsþ ¼,ifa > ; R ðaðsþþ a qðsþ ¼,if0 < a <. ð5:þ By Teorems 4.2, 4.3 and 4.4, respecively, we obain e following.

9 Teorem 5.2. Le condiion (.3) old and define AðÞ e ¼ R =. Eq. (5.) is oscillaory if one of e following condiions olds: R R s qðsþðaðrðsþþþ e a ¼,ifa > 0; R ðaðrðsþþþ e a qðsþ ¼,ifa > ; R R s qðsþ ¼,if0 < a <. S.R. Grace e al. / Commn Nonlinear Sci Nmer Simla 4 (2009) Remark 5.3. From e resls of is paper, we can obain some oscillaion crieria for Eq. (.) on differen ypes of ime scales. If T ¼ R, en rðþ ¼ and x D ¼ x 0. In is case, e resls of is paper are e same as ose in [20]. IfT ¼ Z, en rðþ ¼ þ and x D ðþ ¼DxðÞ ¼xð þ Þ xðþ: In is case, e resls of is paper are e discree analoges of ose in [20]. IfT ¼ Z wi > 0, en rðþ ¼ þ and x D ðþ ¼D xðþ ¼ðxð þ Þ xðþþ=. Te reformlaion of or resls are easy and lef o e reader. We may employ oer ypes of ime scales, e.g. T ¼ q N 0 wi q >, T ¼ N 2 0, and oers, see [,2]. Te deails are lef o e reader. Remark 5.4. Te resls of is paper can be exended o dynamic eqaions of ype (.) wi deviaing argmens, e.g. ðax D Þ D þ f ð; xðsðþþþ ¼ 0; were s : T! T wi lim! sðþ ¼. Te deails are lef o e reader. References [] Agarwal R, Boner M, Li W-T. Nonoscillaion and oscillaion eory for fncional differenial eqaions. Monograps and exbooks in pre and applied maemaics. New York: Marcel Dekker; [2] Agarwal RP, Boner M, Grace SR. On e oscillaion of second-order alf-linear dynamic eqaions. J Differ Eq Appl 2008 (o appear). [3] Agarwal RP, Boner M, Grace SR. Oscillaion crieria for firs-order forced nonlinear dynamic eqaions. Can Appl Ma Q 2008;5(3) (o appear). [4] Agarwal RP, Boner M, Grace SR, O Regan D. Discree oscillaion eory. New York: Hindawi Pblising; [5] Agarwal RP, Boner M, O Regan D, Peerson A. Dynamic eqaions on ime scales: a srvey. J Comp Appl Ma 2002;4( 2): 26. Agarwal RP, Boner M, O Regan D, ediors. Special isse on dynamic eqaions on ime scales. Preprin in Ulmer Seminare 5. [6] Agarwal RP, Grace SR, O Regan D. Oscillaion eory for difference and fncional differenial eqaions. Dordrec: Klwer Academic Pblisers; [7] Agarwal RP, Grace SR, O Regan D. Oscillaion eory for second order linear, alf-linear, sperlinear and sblinear dynamic eqaions. Dordrec: Klwer Academic Pblisers; [8] Agarwal RP, Grace SR, O Regan D. On e oscillaion of cerain second order difference eqaions. J Differ Eq Appl 2003;9():09 9. In onor of Professor Allan Peerson on e occasion of is 60 birday, Par II. [9] Agarwal RP, Grace SR, O Regan D. Oscillaion eory for second order dynamic eqaions. Series in maemaical analysis and applicaions, vol. 5. London: Taylor & Francis; [0] Akın-Boner E, Boner M, Saker SH. Oscillaion crieria for a cerain class of second order Emden Fowler dynamic eqaions. Elecron Trans Nmer Anal 2007;27: 2. [] Boner M, Peerson A. Dynamic eqaions on ime scales: an inrodcion wi applicaions. Boson: Birkäser; 200. [2] Boner M, Peerson A. Advances in dynamic eqaions on ime scales. Boson: Birkäser; [3] Boner M, Saker SH. Oscillaion crieria for perrbed nonlinear dynamic eqaions. Ma Comp Model 2004;40(3 4): [4] Boner M, Saker SH. Oscillaion of second order nonlinear dynamic eqaions on ime scales. Rocky Monain J Ma 2004;34(4): [5] Erbe L, Peerson A. Bondedness and oscillaion for nonlinear dynamic eqaions on a ime scale. Proc Amer Ma Soc 2004;32(3): [6] Erbe L, Peerson A, Řeák P. Comparison eorems for linear dynamic eqaions on ime scales. J Ma Anal Appl 2002;275(): [7] Erbe L, Peerson A, Saker SH. Oscillaion crieria for second-order nonlinear dynamic eqaions on ime scales. J London Ma Soc 2003;67(3):70 4. [8] Han Z, Sn S, Si B. Oscillaion crieria for a class of second-order Emden Fowler delay dynamic eqaions on ime scales. J Ma Anal Appl 2007;334(2): [9] Hilger S. Analysis on measre cains a nified approac o coninos and discree calcls. Resls Ma 990;8:8 56. [20] Ksano T, Ogaa A, Usami H. On e oscillaion of solions of second order qasilinear ordinary differenial eqaions. Hirosima Ma J 993;23(3):