ETNA Electronic Transactions on Numerical Analysis. Volume 27, pp. 1-12, 2007. Copyright 2007, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu OSCILLATION CRITERIA FOR A CERTAIN CLASS OF SECOND ORDER EMDEN–FOWLER DYNAMIC EQUATIONS ELVAN AKIN–BOHNER , MARTIN BOHNER , AND SAMIR H. SAKER Abstract. By means of Riccati transformation techniques we establish some oscillation criteria for the second order Emden–Fowler dynamic equation on a time scale. Such equations contain the classical Emden–Fowler equation as well as their discrete counterparts. The classical oscillation results of Atkinson (in the superlinear case) and Belohorec (in the sublinear case) are extended in this paper to Emden–Fowler dynamic equations on any time scale. Key words. oscillation, dynamic equation, time scale, Riccati transformation technique AMS subject classifications. 34K11, 39A10, 39A99, 34C10, 39A11 1. Introduction. In this paper we shall consider the second order Emden–Fowler dynamic equation (1.1) for "! #%$'&)( on a time scale, where and are positive, real-valued rd-continuous functions, and * is an odd positive integer. We shall also consider the two cases +, (1.2) -. 21 0 / and +3, (1.3) - . 16 054 / In the case of *87:9 , (1.1) is the prototype of a wide class of nonlinear dynamic equations called Emden–Fowler superlinear dynamic equations, and if 4 * 4 9 , then (1.1) is the prototype of dynamic equations called Emden–Fowler sublinear dynamic equations. It is in teresting to study (1.1) because the continuous version, i.e., when is a continuous variable, has several physical applications, see e.g., [20] and when is a discrete variable it is a difference equation of Emden–Fowler type and also is important in applications. By a;< solution >=?;8# of (1.1) we mean a nontrivial real-valued function satisfying equation (1.1) for =@;# for some . A solution of (1.1) is said to be oscillatory if it is neither eventually 7 positive nor eventually negative; otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Our attention is0restricted to=Mthose solutions ! AB$C1 L of (1.1) which exist on some half line and satisfy for any DEGFHJI IK 7 7 =N;3>A . Much recent attention has been given to dynamic equations on time scales (or measure chains), and we refer the reader to the landmark paper of Hilger [20] for a comprehensive treatment of the subject. Since then several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal, Bohner, O’Regan, and Peterson [1] and the references cited therein. Two books on the subject of time scales, by Bohner and Peterson [9, 10], summarize and organize much of the time scale calculus. O Received December 1, 2003. Accepted for publication March 8, 2004. Recommended by A. Ruffing. Department of Mathematics and Statistics, University of Missouri–Rolla, Rolla, MO 65409 (akine, bohner@umr.edu). Mansoura University, Department of Mathematics, Mansoura 35516, Egypt (shsaker@mans.edu.eg). 1 ETNA Kent State University etna@mcs.kent.edu 2 E. AKIN–BOHNER, M. BOHNER, AND S. SAKER In recent years there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales. We refer the reader to the papers [2, 3, 4, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23]. In [3], Akın-Bohner and Hoffacker considered the second order dynamic equation (1.4) PQ and gave necessary and sufficient conditions for oscillation of all solutions when *R7S9 and 4 4 * 9 . Their results cannot be applied in the case when * 9 and applied only to discrete time scales. In this paper we use the Riccati transformation technique to obtain some oscillation cri teria for (1.1) when (1.2) or (1.3) holds. Our results can be applied in the case * 9 and also for any time scale. So our results extend and improve the results established by Akın-Bohner and Hoffacker [3]. The paper is organized as follows. In the next section we present some basic definitions concerning the calculus on time scales. In Section 3 we develop a Riccati transformation technique and give some fundamental lemmas. These lemmas are used to superlinear case, i.e., give sufficient conditions for oscillation of all solutions of (1.1) in the ; 2PB$ when * (see Section 5). 9 (in Section 4) and in the sublinear case, i.e., when * 9 Our results when (1.2) holds are sufficient for oscillation of all solutions of (1.1), and when (1.3) holds our results ensure that all solutions are either oscillatory or converge to zero. For the superlinear case we present an extension of the classical Atkinson [5] result, and for the sublinear case we present an extension of the classical Belohorec [6] result. Since we are interested in oscillatory behavior, we suppose that the time scale under ! #%$)13 consideration is not bounded above, i.e., it is a time scale interval of the form . 2. Some Preliminaries on Time Scales. A time scale T is an arbitrary nonempty closed subset of the real numbers U . On any time scale T we define the forward and backward jump operators by WVYXZ K b )L H\[ and T<K][N7 ^ K )La6 DEBF_H`[ VcXBZ 4 T3K][ de d2 A point with is said to be left-dense if ^ and right-dense if , T 7 T 4 left-scattered if ^ and right-scattered if . The graininess function for a time 7 f g h scale T is defined by f . For a function i8K0TkjlU (the range U of i may K actually be replaced by any Banach space) the (delta) derivative is defined by Pmh a (2.1) i mhn i i if i is continuous at and is right-scattered. If is not right-scattered, then the derivative is defined by mh oepYVcq (2.2) i i mh - r>s i th kpYVcq [ i 0h - rs [ i [ [ ! #$)&'( provided this limit exists. A function i:K juU is said to be rd-continuous if it is continuous at each right-dense point and there exists a finite left limit at all left-dense points, and i is said to be differentiable if its derivative exists. A useful formula is o (2.3) i i f C6 i We will make use of the product and quotient rules for the derivative of the product 8y the quotient ixw\v (where vyvz|{ of two differentiable functions i and v (2.4) < iv i v i 3 z v n iv i v z and i h i v iv } va~ vyv z 6 iv and ETNA Kent State University etna@mcs.kent.edu 3 EMDEN–FOWLER DYNAMIC EQUATIONS For #$)& and a differentiable function i , the Cauchy integral of i T + o &th i is defined by #C6 i i / An integration by parts formula reads + i o! v ( i + h v / i C$ v / and improper integrals are defined as +3, Note that in the case T 8C$ o\ 9 $ a C6 i / + C$ i Q , / ^ and in the case T s we have U o + apcVYq i i + o i >C$ i / we have o8Mh $ ^ o 9 a i \ i h i 9 + C$ i x c i / a >C6 i / 3. A Riccati Transformation. Crucial for our calculations is the0following lemma. Q d L EMMA 3.1. If and are differentiable on a time scale T with for all { then we have (3.1) } d hG o d zn T , 6 ~ Proof. Using (2.3), (2.4), and RG } y z a a yz P , we obtain fx ~ P h fx ¢h ¡ y zn z £G hk G z ¢h f hn z z z 6 This shows that (3.1) holds. Using Lemma 3.1, we now derive the following result. 0 ¤ { T HEOREM 3.2. Suppose solves (1.1) such that for all differentiable function and define ¥ by the Riccati substitution (3.2) ¥ y 6 N T . Let be any ETNA Kent State University etna@mcs.kent.edu 4 E. AKIN–BOHNER, M. BOHNER, AND S. SAKER Then we have G h (3.3) d ¥ z G h z P 6 Proof. We use again (2.3) and (2.4) to find h <h ¥ hW¦ z y } y cG } ~ G y @§ ~ Q h¨G z } G Q y ~ h z } Now applying formula (3.1) from Lemma 3.1 (with 6 ~ replaced by ), we arrive at (3.3). We will use the above Theorem 3.2 several times in the remainder of this paper, in conjunction with the formula (3.4) + * ! ©G h¡©>B( Q z = J©x$ 9 which is a simple consequence of Keller’s chain rule [9, Theorem 1.90]. The following result is used frequently in the remainder of this paper. t L EMMA 3.3. Assume that (1.2) holds. If is a solution of (1.1) such that ;<= all , then %_ ;3 (3.5) ;< = for 7 for 6 Proof. In view of (1.1) we have ªG% >_kh>G (3.6) 4 ; = <G for all , and so «|K is an eventually decreasing function. We first show that « is ;e = 4 « eventuallynonnegative. Suppose there exists such that . Then from (3.6) ;< . Hence 4 « we have « for ¬ « 0 $ which implies by (1.2) that 0d¬<0 (3.7) x t « + -P­ [ t / [ h®1 as j 5;g = 1$ j @¯0 7 « contradicting the fact that for all . Hence = nonnegative. Therefore, we see that there exists some such that t B$° 7 Hence (3.5) holds. ;WB$ d;<B$ « « 4 for ; = 6 is eventually ETNA Kent State University etna@mcs.kent.edu 5 EMDEN–FOWLER DYNAMIC EQUATIONS 4. Oscillation Criteria in the Superlinear Case. In this section we give some oscilla²± is odd. tion criteria when * First we consider the case when (1.2) holds. T HEOREM 4.1. Assume that (1.2) holds. Furthermore, assume that there exists a differentiable function such that (4.1) + pcVYq - s DEBF , ´³ µ holds for all constants [ z hµ [ x 0 [ 21 ¶ [ [ / . Then every solution of (1.1) is oscillatory on 7 ! #%$C1 . loss of Proof. Suppose to the contrary that is a nonoscillatory solution of (1.1). Without 0 7 generality, we may assume that is an eventually positive solution of (1.1) such that ;3= # ¸h 7 for all . We shall consider only this case, since the substitution · transforms equation (1.1) into an equation of the same form. By Lemma 3.3 we obtain that (3.5) holds. ; Now note that * 9 and (3.4) imply + (3.4) ! ©G = +3 ; * xy© 9 h"©%t( ! ©G0 x y© 9 = h"©%t( z * t * (3.5) ; t=£ x * : ¹ =\3; ¥ - . [ [ / + - . [ h¸t [ t z + [ - .¿ [ P0 ¿ - . + ; [ h¸t [ [ z [ h ¹ 1 [ / [ t ® [ À [ [ [ as 18$ j [ / [  [ / j - - .bÁ (4.1) [ / À [ [ [ [ ® [ z z [ [ [ + ! =$C1 ¥ - + lh ¹ . Note N;¾7 = . Now define the function ¥ on for all . Therefore, using (3.3) from Theorem ¥ =£mh x ½; ¥ ; x>¼ t=£ º» where we put K * by (3.2). Then (3.5) implies 3.2, we obtain ¥ $ 9 ¹ ETNA Kent State University etna@mcs.kent.edu 6 E. AKIN–BOHNER, M. BOHNER, AND S. SAKER which is impossible. The proof is complete. C OROLLARY 4.2. Assume that (1.2) holds. Furthermore, assume that there exists a positive differentiable function à such that (4.2) + pcVYq - s DEBF , à z [ ÅÄÆ mh"µ t [ È [ à µ à bÇ [ CËÌ È [ à z 21 [ ÊÉ [ / holds for all constants eÍ Proof. Define . Then every solution of (1.1) is oscillatory on and note that [9] 7 Ã Í . 6 à ! #$C1 Í Ã Ã z Since (4.2) holds for à , (4.1) holds for . So the claim follows by Theorem 4.1. From Theorem 4.1 and Corollary 4.2 we can obtain different conditions for oscillation of all solutions of (1.1) bydifferent choices of à . For instance, we obtain the following two _Î _¯ corollaries if we choose à , respectively. The first choice confirms that 9 and à the Leighton–Wintner theorem is valid for Emden–Fowler dynamic equations. C OROLLARY 4.3 (Leighton–Wintner). Assume +, (4.3) +, 21 and 0 / a216 / ! #%$)13 Then every solution of (1.1) is oscillatory on . C OROLLARY 4.4. Assume that (1.2) holds. Furthermore, assume that (4.4) + pYVcq - s ˪РDEGF , µ [ ÄÏ h"µ x 0 [ Ì 9 [ Æ È Í [ [ Q1 [ / ! #$C1 holds for all constants 7 . Then every solution of (1.1) is oscillatory on . 0ÑÎ The next result is the same as [3, Theorem 5] when 9 . But we note that [3, Theorem 5] cannot be applied in the case when * and also not for the second order 9 Emden–Fowler differential equation, i.e., when T U . So the following result extends and improves in various ways the results established in [3]. Its classical version was given in 1955 by Atkinson [5]. T HEOREM 4.5. Assume that (1.2) holds. Define + Ò 6 [ t / [ If (4.5) + pcVYq Ò - s DEBF , > [ 218$ [ [ / ! #$C1 then every solution of (1.1) is oscillatory on . t _;¾>= Proof. Again we suppose is a solutionofÍ (1.1) such that for all . By 7 Ò Lemma 3.3 we obtain (3.5). Now we let and define the Riccati substitution ¥ by ETNA Kent State University etna@mcs.kent.edu 7 EMDEN–FOWLER DYNAMIC EQUATIONS (3.2). Using the product rule from (2.4), we calculate ¥ 9 G ¿ 8 Ò h h C G Ò z J$ z Ó (3.4) + h 9 ! ©G h +< * Q J© z 9 z J© ( h¡©> z * h¡©>0( 9 ! ©G = h 9 due to (3.4) and because Q = 9 Ó ¬W z * ¬Ô * where the last inequality is true because Ò z Ò h 9 À Ò z z C ¬ 6 Upon integration we arrive at + Ò P [ - . [ + ¬ [ / h [ ah ) * = ) / = =£ h ¥ 9 [ * h ¬ À ¥ 9 ) h h ¿ - . C ¥ 9 * =\6 h ¥ * 9 This contradicts (4.5) and finishes the proof. 0aÎ a Ò 9 , i.e., Putting in Theorem 4.5, we obtain the following corollary. taÎ 9 . If C OROLLARY 4.6. Assume (4.6) + pYVcq - s DEBF , > [ 218$ [ [ / ! #%$C1 then every solution of (1.1) is oscillatory on . E XAMPLE 4.7. Consider the dynamic equation (4.7) Here taÎ 9 o and z » - ¼ 9 > z ; for 6 9 . Using [8, Theorem 5.11], we find +, [ - . [ +3, e186 [ [ - . / [ / ! $)13 Hence, by Corollary 4.6, equation (4.7) is oscillatory on 9 . T HEOREM 4.8. Assume that (1.2) holds. If there exists a positive differentiable function ²± à and an odd integer Õ such that (4.8) + pYVYq - s DEBF , µ Ö 0h 9 >Ö [ Ã Æ ÄÏ [ > [ È ah à [ à [ È Ã t [ P [ ËªÐ Ì Q1 [ / ETNA Kent State University etna@mcs.kent.edu 8 E. AKIN–BOHNER, M. BOHNER, AND S. SAKER µ ! #%$)13 7 , then every solution of (1.1) oscillates on . holds for all Proof. The proof is similar to [23, Theorem 3.2] and hence is omitted. oÎ 9 , then (4.8) reduces to Note that when à + pYVcq DEBF , - s Ö mh 9 Ö > [ 218$ [ [ / which can be considered as an extension of Kamenev type oscillation criteria for second order differential equations (see [22]). Next we consider the case when (1.3) holds. Now we give some sufficient conditions when (1.3) holds, which guarantee that every solution of (1.1) oscillates or converges to zero ! #%$)13 on . T HEOREM 4.9. Assume that (1.3) holds. Furthermore, assume that there exists a positive function such that (4.1) holds, and + + , (4.9) 9 t [ ae186 [ / / ! #%$)13 Then every solution of equation (1.1) is oscillatory or converges to zero on . T HEOREM 4.10. Assume that (1.3) holds. Furthermore, assume that there exists a (4.9) hold. Then every solution of equation (1.1) is positive function à such that (4.8) and ! #%$)13 oscillatory or converges to zero on . 5. Oscillation Criteria in ¡ thePBSublinear Case. In this section we give some new oscil$ lation criteria for (1.1) when * is a quotient of odd positive integers. 9 First we consider the case when (1.2) holds. T HEOREM 5.1. Assume that (1.2) holds and suppose that is differentiable and nondecreasing. Furthermore, assume that there exists a differentiable function such that (5.1) + pYVYq - s DEBF , ³ [ z [ h"µ| C \0 [ [ ® e1 ¶ [ [ / µ ! #$C1 holds for all constants . Then every solution of (1.1) is oscillatory on . 7 Proof. Suppose to the contrary that is a nonoscillatory solution of (1.1). Without loss of 0 generality, we may assume that is an eventually positive solution of (1.1) such that 7 ;3= # for all . Hence, by Lemma 3.3, we obtain (3.5), which implies 7 ×;k aR Ñ > z so that, again by using (3.5), ! =M$)13 , and therefore we obtain ²¬Ø . Hence is nonincreasing on + 0o8t=£x ¨;Ù= for all -. ¬<Ú²¡ÛC$ [ [ / Ún0=£Bh¢= where we find that (5.2) =£ and Û8 td¬ =` Ü . By putting Ü for all ;< Ú I 6 ¨Û I and ;3q5Ý£Þ =Ê$ H L 9 , ETNA Kent State University etna@mcs.kent.edu 9 EMDEN–FOWLER DYNAMIC EQUATIONS "G$ Now note that * and (3.4) imply 9 + (3.4) ! ©B + * ! ©B (5.2) ; Ü x * ¹ ¹ x Ü _;Q º» 3; ¥ ¥ mh ! =$)13 [ / [ - -P­ [ h¸t [ t z ¿ + $ . Note b;Q7 = . Now define the function ¥ on by for all . Therefore, using (3.3) from Theorem 3.2, [ x ¥ - - ­ [ [ 0 > [ ¿ - ­ h¸t [ [ À [ [ / [ [ z [ [ [ / [ À [ [ [ / + ; z [ [ + ; + J© - - ­ z ¥ + lh ( h¡©% 9 ¹ x>¼ J© x z * where we put K * (3.2). Then (3.5) implies ¥ we obtain Q z = 9 = ; h¡©%0 ( Q z * [ - ­ Á z [ h ¹ P ) \0 [ ® [  [ [ / (5.1) 1 as j 18$ j which is impossible. The proof is complete. The next result follows as in the proof of Corollary 4.2. C OROLLARY 5.2. Assume that (1.2) holds and suppose that is differentiable and nondecreasing. Furthermore, assume that there exists a positive differentiable function à such that (5.3) + pcVYq - s DEBF , ÄÆ [ mh"µ à z t [ P [ ) È [ à µ à 0 Ç [ [ ËÌ È Ã z 21 [ [ É / ! #%$C1 holds for all constants . Then every solution of (1.1) is oscillatory on . 7 From Theorem 5.1 and Corollary 5.2 we can obtain different conditions for oscillation of all solutions of (1.1) bydifferent choices of à . For instance, we obtain the following two Î a8 corollaries if we choose à , respectively. 9 and à C OROLLARY 5.3 (Leighton–Wintner). Suppose is differentiable and nondecreasing and assume +3, (5.4) 21 0 / +, and / ae16 ETNA Kent State University etna@mcs.kent.edu 10 E. AKIN–BOHNER, M. BOHNER, AND S. SAKER ! #%$)13 . Then every solution of (1.1) is oscillatory on C OROLLARY 5.4. Assume that (1.2) holds and suppose that nondecreasing. Furthermore, assume that (5.5) - s is differentiable and + pYVcq ˪РDEBF , > ÄÏ ahµ [ P ) [ t [ Æ È Í µ Ì 9 [ [ [ e1 [ / ! #$C1 7 holds for all constants . Then every solution of (1.1) is oscillatory on . The next result gives a condition for oscillation in the sublinear case. Its classical version was given in 1961 by Belohorec [6]. T HEOREM 5.5. Assume that (1.2) holds. If (5.6) +3, } oe18$ 0 ~ / ! #$C1 then every solution of (1.1) is oscillatory on . t Proof. We suppose that is a solution of (1.1) satisfying G . Then by Lemma 3.3 we obtain that « and « « 7 observe that ;8>= for all for all 4 + 0o0=£ 7 ®;e= _ ;<_mh"=` [ - . ; _ [ / for all ;3 if £ = 7 « C « . Next note that Ó (3.4) + h 9 * ;Ô h * ! © Q 6 t ® « _ « ; « ~ 6 Ct } 9 h } 0 } ) * for all « z « ; J© « « R « ( h¡©% Using these two inequalities, we obtain after dividing (1.1) by ] J© « 9 * ( h¡©% 9 « = h 9 Q z « + 9 Ó ! © = ~ ~ Upon integration we arrive at + -ß [ } 0 [ + ¬ [ [ ~ ) * ) h ) [ / mh « « [ * h 9 9 ¬ C « * h 9 h -ß / 9 6 « * ;3 , and let . First ETNA Kent State University etna@mcs.kent.edu 11 EMDEN–FOWLER DYNAMIC EQUATIONS This contradicts (5.6) and finishes the proof. 0aÎ 9 in Theorem 5.5, we obtain the following corollary. Putting taÎ 9 . If C OROLLARY 5.6. Assume that +3, (5.7) y oQ1$ / ! #%$C1 then every solution of (1.1) is oscillatory on . E XAMPLE 5.7. Consider the dynamic equation (5.8) tdÎ º'á 9 à ' º á Q for z ; 6 9 à º'á d 9 and 9\w Here . As in Example 4.7 it follows from Corollary 5.6 that (5.8) ! $C1 is oscillatory on 9 . T HEOREM 5.8. Assume that is differentiable and nondecreasing and suppose that (1.2) ѱ such that holds. If there exists a positive differentiable function à and an odd integer Õ (5.9) + pYVcq - s DEGF , Ö P [ µ µ Ö mh 9 à ÄÏ x ah [ [ È Æ [ à C [ Ã È 0 [ P Ã ËªÐ Ì 21 [ [ [ / ! #%$)13 7 for all constants , then every solution of (1.1) oscillates on . Proof. The proof is similar to [23, Theorem 3.2] and hence is omitted. Next we consider the case when (1.3) holds. Now we give some sufficient conditions when (1.3) holds, which guarantee that every solution of (1.1) oscillates or converges to zero ! #%$)13 on . b;W T HEOREM 5.9. Assume that and that (1.3) holds. Furthermore, assume that there exists a positive function such that (5.1) holds, and + + , (5.10) 9 0 o216 [ / [ / ! #%$)13 Then every solution of equation (1.1) is oscillatory or converges to zero on . ; T HEOREM 5.10. Assume that and that (1.3) holds. Furthermore, assume that there exists a positive function à such that (5.9) ! and (5.10) hold. Then every solution of #%$)13 equation (1.1) is oscillatory or converges to zero on . R EMARK 5.11. Note that our results also can be extended to the more general equation 0%" where *n7 t I I XNP0PQ Dâ for ¡! #%$)&'($ to cover the case when * is even. REFERENCES [1] R. P. A GARWAL , M. B OHNER , D. O’R EGAN , AND A. P ETERSON , Dynamic equations on time scales: A survey, J. Comput. Appl. Math., 141 (2002), pp. 1–26. [2] E. A KIN , L. E RBE , B. K AYMAKÇALAN , AND A. P ETERSON , Oscillation results for a dynamic equation on a time scale, J. Differ. Equations Appl., 7 (2001), pp. 793–810. [3] E. A KIN -B OHNER AND J. H OFFACKER , Oscillation properties of an Emden–Fowler type equation on discrete time scales, J. Difference Equ. Appl., 9 (2003), pp. 603–612. [4] , Solution properties on discrete time scales, J. Difference Equ. Appl., 9 (2003), pp. 63–75. ETNA Kent State University etna@mcs.kent.edu 12 E. AKIN–BOHNER, M. BOHNER, AND S. SAKER [5] F. V. ATKINSON , On second-order non-linear oscillations, Pacific J. Math., 5 (1955), pp. 643–647. [6] Š. 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