ITERATIVE OSCILLATION RESULTS FOR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH ADVANCED ARGUMENT

Transcription

1 Elecronic Jornal of Differenial Eqaions, Vol (2017, No. 162, pp ISSN: URL: hp://ejde.mah.xsae.ed or hp://ejde.mah.n.ed ITERATIVE OSCILLATION RESULTS FOR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH ADVANCED ARGUMENT IRENA JADLOVSKÁ Absrac. This aricle concerns he oscillaion of solions o a linear secondorder differenial eqaion wih advanced argmen. Sfficien oscillaion condiions involving limi inferior are given which essenially improve known resls. We base or echniqe on he ieraive consrcion of solion esimaes and some of he recen ideas developed for firs-order advanced differenial eqaions. We demonsrae he advanage of or resls on Eler-ype advanced eqaion. Using MATLAB sofware, a comparison of he effeciveness of newly obained crieria as well as he necessary ieraion lengh in pariclar cases are discssed. 1. Inrodcion We consider he linear second-order advanced differenial eqaion y ( + q(y(σ( = 0, 0 > 0, (1.1 where q C([ 0, and σ C 1 ([ 0, are sch ha q( > 0, σ( and σ ( 0. By a solion of (1.1, we ndersand a nonrivial fncion y C 2 ([ 0,, which saisfies (1.1 on [ 0,. We resric or aenion o hose solions y of (1.1 which saisfy sp{ y( : T } > 0, for all T 0. We recall ha a solion of (1.1 is said o be oscillaory if i has arbirarily large zeros, and oherwise i is said o be nonoscillaory. Eqaion (1.1 is called oscillaory if all of is solions are oscillaory as well. Differenial eqaions wih deviaing argmen are deemed o be adeqae in modeling of he conless processes in all areas of science. As is well known, a disingishing feare of delay differenial eqaions nder consideraion is he dependence of he evolion rae of he processes described by sch eqaions on he pas hisory. This conseqenly resls in predicing he fre in a more reliable and efficien way, explaining a he same ime many qaliaive phenomena sch as periodiciy, oscillaion or insabiliy. The concep of he delay incorporaion ino sysems plays an essenial role in modeling o represen ime aken o complee some hidden processes, see [8, 11]. Conrariwise, advanced differenial eqaions can find se in many applied problems whose evolion rae depends no only on he presen, b also on he fre Mahemaics Sbjec Classificaion. 34C10, 34K11. Key words and phrases. Linear differenial eqaion; advanced argmen; second-order; oscillaion. c 2017 Texas Sae Universiy. Sbmied April 28, Pblished Jly 4,

2 2 IRENA JADLOVSKÁ EJDE-2017/162 Therefore, an advance cold be inrodced ino he eqaion o highligh he inflence of poenial fre acions, which are available a he presence and shold be beneficial in he process of decision making. For insance, poplaion dynamics, economical problems or mechanical conrol engineering are ypical fields where sch phenomena is believed o occr (see [8] for deails. The firs oscillaion resls for differenial eqaions wih deviaing argmen were obained in he classical paper by Fie [10] in Since hen, a grea deal of he effor has been made by many researchers in order o advance he knowledge frher (for he smmary of mos essenial conribions on he sbjec, see, e.g., monographs [1, 2, 11, 9, 18] and he references cied herein. Mos of he lierare, however, has been devoed o he invesigaion of differenial eqaions wih delay argmen, and very lile is known p o now abo hose wih advanced argmens. In pariclar, wo main approaches for he invesigaion of (1.1 have appeared (see [2, Chaper 2], [5, 15, 16]. Taking Ksano s and Naio s comparison heorem [16, Theorem 1] ino accon, he oscillaory behavior of (1.1 can be reaed as ha of he ordinary differenial eqaion y ( + q(y( = 0. (1.2 I seems obvios ha in sch a case, all impac of he advanced argmen is compleely negleced. On he oher hand, an anoher approach has been based on he comparison wih he firs-order advanced differenial eqaion ( y ( q(sds y(σ( = 0, (1.3 in he sense ha oscillaion of (1.1 is inheried from ha of (1.3 (see [2, Theorem ]. Here, he advance may generae oscillaions. In pariclar, by applying he famos Hille s resl [13] and he well-known oscillaion crierion de o Ladas [17] o (1.2 and (1.3, respecively, one can immediaely ge he following cople of oscillaion crieria for (1.1: The qesion narally arises: q(sds > 1 4, (1.4 σ( q(sds d > 1 e. (1.5 Is i possible o esablish an effecive oscillaion resl of Hille ype which simlaneosly akes ino accon he presence of he advance and he second order nare of he eqaion sdied as well? The prpose of his aricle is o give an affirmaive answer o his qasion, i.e., o propose an approach for invesigaion he (1.1 when boh above-menioned condiions (1.4 and (1.5 fail. The se is made of some of he recen resls developed for firs-order delay/advanced differenial eqaions which have been based on he ieraive applicaion of he Grönwall s ineqaliy (see [4, 7]. This echniqe enables one o obain sfficien condiions for oscillaion of (1.1 involving, which essenially se vale of he advanced argmen. Or mehod of he proof ha is qie differen from he very recen sdy [3] is essenially new. Finally, we demonsrae he advanage of or resls on Eler-ype advanced eqaions. Using MATLAB sofware, a comparison of he effeciveness of newly

3 EJDE-2017/162 ITERATIVE OSCILLATION RESULTS 3 obained crieria is provided as well as he necessary ieraion lengh in pariclar cases. 2. Main resls In his secion, we esablish a nmber of new oscillaion crieria for (1.1. In he seqel, all fncional ineqaliies are assmed o hold evenally, ha is, hey are saisfied for all large enogh. Remark 2.1. As y( is also a solion of (1.1, we may resric orselves only o he case where y( is evenally posiive. Remark 2.2. In view of he well-known Leighon s crierion [19] and he comparison heorem [16, Theorem 1], eqaion (1.1 is oscillaory if q(sds =. Therefore, we assme hrogho he paper ha q(sds <. We define ( q( = q( 1 + σ( q(sds d. Theorem 2.3. Assme ha he second-order differenial eqaion is oscillaory. Then (1.1 is oscillaory. y ( + q(y( = 0 (2.1 Proof. Sppose o he conrary ha y is a posiive solion of (1.1 on [ 0,. Obviosly, here exiss 1 0 sch ha y( > 0, y ( > 0, y ( 0, for 1. (2.2 An inegraion of (1.1 from o in view of (2.2 leads o y ( q(sy(σ(sds (2.3 ( y(σ( Inegraing (2.4 from o σ(, we have y(σ( y( + σ( y(σ( Using ha y(σ( y( in (2.5, one obains y(σ( y( + y(σ( ( y( 1 + q(sds. (2.4 σ( σ( Combining he las ineqaliy and (1.1 yields q(sds d. (2.5 q(sds d q(sds d. y ( + q(y( 0. (2.6 Define w( = y (/y( o see ha w( saisfies he firs-order Riccai ineqaliy w ( q( w 2 ( 0, which in rn implies (see [1, Lemma 2.2.1] ha he eqaion (2.1 has a posiive solion; a conradicion. The proof is complee.

4 4 IRENA JADLOVSKÁ EJDE-2017/162 Corollary 2.4. If hen (1.1 is oscillaory. q(sds > 1 4, (2.7 Remark 2.5. The crierion (2.7 of Hille ype akes he presence of he advanced argmen ino accon and hs can be applied even if he corresponding known one (1.4 fails. The lemma below is a sligh modificaion of [14, Lemma 1] originally given for he firs-order eqaion wih delayed argmen. For he sake of clariy, we also inclde is complee proof. Lemma 2.6. Le y( be an evenally posiive solion of (1.1. Then ρ := σ( y(σ( y( q(sdsd 1 e, (2.8 λ, (2.9 where λ is he smaller roo of he ranscendenal eqaion λ = e ρλ. Proof. Le y(σ( α =. y( Dividing (2.4 by y( and inegraing from o σ(, we have ( y(σ( σ( y(σ( ln q(sds d, y( y( or y(σ( ( σ( y(σ( exp q(sds d, y( y( which clearly implies α e ρα. (2.10 Noe ha (2.10 is impossible when ρ > 1/e, since λ < exp ρλ for all λ > 0 and so (1.1 has no posiive solions. If ρ 1/e, hen he eqaion λ = exp ρλ has roos λ λ, wih λ = λ = e if and only if ρ = 1/e and (2.10 holds if and only if λ α λ. As an immediae conseqence of Lemma 2.9, we have he following resl, which applies when (1.5 fails. Theorem 2.7. Le (2.8 hold and λ be as in Lemma 2.6. Assme ha he secondorder differenial eqaion y ( + kλq(y( = 0 (2.11 is oscillaory for some k (0, 1. Then (1.1 is oscillaory. Proof. Sppose o he conrary ha y is a posiive solion of (1.1 on [ 0,. Then i follows from Lemma 2.6 ha here exiss 1 [ 0, sch ha, for every k (0, 1, y(σ( kλ on [ 1,. (2.12 y(

5 EJDE-2017/162 ITERATIVE OSCILLATION RESULTS 5 Using (2.12 in (1.1, i is easy o see ha y is a posiive solion of he ineqaliy y ( + kλy( 0. The same as in he proof of Theorem 2.3, we can conclde ha he corresponding eqaion (2.11 also has a posiive solion, a conradicion. The proof is complee. Corollary 2.8. Le (2.8 hold and λ be as in Lemma 2.6. If hen (1.1 is oscillaory. q(sds > 1 4λ, (2.13 In he nex lemma, we derive some sefl esimaes which are based on he ieraive applicaion of he Grönwall ineqaliy and permi s o improve all he previos resls. Lemma 2.9. Le y( be an evenally posiive solion of (1.1. Define ( s a 1 (s, = exp q(xdx d, Then for large enogh. ( s a n+1 (s, = exp q(xa n (σ(x, dx d, n N. y(s y(a n (s,, s, (2.14 Proof. We will prove Lemma 2.9 by mahemaical indcion. Since y is an evenally posiive solion of (1.1, here exiss 1 0 sch ha y saisfies (2.2 on [ 1,. Ths y(σ( y( and by vire of (2.4, we have y ( y( q(sds. Applying he Grönwall ineqaliy, we obain ( s y(s y( exp q(xdx d, s 1, (2.15 ha is, he esimae (2.14 is valid for n = 1. Nex, we assme ha (2.14 holds for some n > 1. Then Sbsiing (2.16 ino (2.3 yields y ( y(σ(s y(a n (σ(s,, σ(s. (2.16 q(sy(σ(sds y( q(sa n (σ(s, ds. Again, applying he Grönwall ineqaliy, we have ( s y(s y( exp q(xa n (σ(x, dx d, (2.17 i.e., y(s y(a n+1 (s,. This esablished he indcion sep and complees he proof.

6 6 IRENA JADLOVSKÁ EJDE-2017/162 Theorem Le a n (, s be as in Lemma 2.9. Assme ha he firs-order advanced differenial eqaion ( y ( q(sa n (σ(s, σ(ds y(σ( = 0 (2.18 is oscillaory for some n N. Then (1.1 is oscillaory. Proof. Sppose o he conrary ha y is a posiive solion of (1.1 on [ 0,. Then here exiss 1 0 sch ha y saisfies (2.2 on [ 1,. I follows from Lemma 2.9 ha y(σ(s y(σ(a n (σ(s, σ(, s, (2.19 for some n N and large enogh. Inegraing (1.1 from o and sing (2.19, we are led o y ( q(sy(σ(sds y(σ( q(sa n (σ(s, σ(ds, (2.20 which means ha y is a posiive solion of he firs-order advanced differenial ineqaliy ( y ( q(sa n (σ(s, σ(ds y(σ( 0. In view of [20, Theorem 1], he eqaion (2.18 also has a posiive solion, a conradicion. The proof is complee. Corollary Le a n (, s be as in Lemma 2.9. If σ( for some n N, hen (1.1 is oscillaory. q(sa n (σ(s, σ(ds d > 1 e, (2.21 Remark The above heorem permis s o dedce oscillaion of (1.1 from ha of he firs-order advanced differenial eqaion (2.18. One can see ha, even for n = 1, he crierion (2.21 is sharper han (1.5 and hs provides a beer resl. Theorem Assme ha he second-order differenial eqaion y ( + q(a n (σ(, y( = 0 (2.22 is oscillaory for some n N. Then (1.1 is oscillaory. Proof. Sppose o he conrary ha y is a posiive solion of (1.1 on [ 0,. Then here exiss 1 0 sch ha y saisfies (2.2 on [ 1,. I follows from Lemma 2.9 ha y(σ( y(a n (σ(, (2.23 for some n N and large enogh. Using (2.23 in (1.1, we see ha y is a posiive solion of y ( + q(a n (σ(, y( 0. As in he proof of Theorem 2.3, we can see ha he corresponding eqaion (2.22 also has a posiive solion, a conradicion. The proof is complee.

7 EJDE-2017/162 ITERATIVE OSCILLATION RESULTS 7 Corollary If for some n N, hen (1.1 is oscillaory. We define ( q n ( = q( 1 + where a n (s, is as in Lemma 2.9. q(sa n (σ(s, sds > 1 4 σ( q(sa n (σ(s, ds d, n N, Theorem Assme ha he second-order differenial eqaion is oscillaory for some n N. Then (1.1 is oscillaory. (2.24 y ( + q n (y( = 0 (2.25 Proof. Sppose o he conrary ha y is a posiive solion of (1.1 on [ 0,. Then here exiss 1 0 sch ha y saisfies (2.2 on [ 1,. As in he proof of Theorem 2.10, we obain (2.20, ha is, y ( y(σ( Inegraing (2.26 from o σ( and sing (2.14, i.e., we obain y(σ( y( + ( y( 1 + q(sa n (σ(s, σ(ds. (2.26 y(σ( y(a n (σ(,, σ(, σ( σ( y(σ( a n (σ(, q(sa n (σ(s, σ(ds d q(sa n (σ(s, σ(ds d. The res of he proof is similar o ha of Theorem 2.3 and so we omi i. Corollary If q n (sds > 1 4 for some n N, hen (1.1 is oscillaory. Lemma Le y( be an evenally posiive solion of (1.1. Then ρ n := σ( (2.27 q(sa n (σ(s, σ(ds d 1 e, (2.28 and y(σ( λ n, y( where a n (, s is as in Lemma 2.9 and λ n is he smaller roo of he eqaion λ n = e ρnλn. Proof. We proceed as in he proof of Theorem 2.10 o obain ha y saisfies (2.20. The nex argmens are he same as in he proof of Lemma 2.6 so we can omi hem.

8 8 IRENA JADLOVSKÁ EJDE-2017/162 Theorem Le (2.28 hold and λ n be as in Lemma Assme ha he second-order differenial eqaion y ( + kλ n q(y( = 0 (2.29 is oscillaory for some n N and k (0, 1. Then (1.1 is oscillaory. Corollary Le (2.28 hold and λ n be as in Lemma If for some n N, hen (1.1 is oscillaory. q(sds > 1 4λ n, (2.30 Finally, we discss he efficiency of newly obained crieria on Eler-ype differenial eqaions. Example Consider he second-order advanced Eler differenial eqaion y ( + a y(c = 0, c 1, a > 0, 1. ( Known oscillaion crieria (1.4 and (1.5 give a > 1 4 (2.32 and a ln c > 1 e, (2.33 respecively. The recen resl [3, Corollary 1] gives ( c β 1 a + 1 β 1 a + cβ > 1, ( β where β = 1 1 4a 2 and a 1/4. From Corollary 2.4, we have ha (2.31 is oscillaory if a(1 + a ln c > 1 4. (2.35 To apply Corollary 2.8, we se ρ := a ln c 1/e. Then he smaller roo of he eqaion λ = e ρλ is λ = W ( ln eρ ln e ρ = W ( ρ, ρ where W ( denoes he principal branch of he Lamber fncion, see [6] for deails. Conseqenly, he oscillaion crierion (2.13 becomes a W ( ρ > 1 ρ 4, ha is, W ( a ln c > 1 ln c 4. (2.36 Now, we se n = 1. Afer simple calclaions, he following condiions for oscillaion of (2.31, i.e., a 1 a ln c > 1 e, (2.37 ac a > 1 4, (2.38

9 EJDE-2017/162 ITERATIVE OSCILLATION RESULTS 9 ( a 1 + ca (c a 1 > 1 1 a 4, (2.39 (a 1W ( a a 1 ln c > 1 ln c 4, where a ln c 1/e, (2.40 a 1 resl from Corollaries 2.11, 2.14, 2.16 and 2.19, respecively. A comparison of he effeciveness of he above-menioned crieria in erms of he reqired vale c for a given coefficien a = 0.23 is shown in he Table 1. Table 1. Comparison of he srengh of crieria (2.32 (2.40 for a given a = 0.23 crierion reqired c (2.32 inapplicable ( ( ( ( ( ( ( ( On he oher hand, if we se a = 0.19 and c = 2 in (2.31, hen i is easy o verify ha all crieria (2.33 (2.40 fail. In sch a case, i is ineresing o compare he lengh of he ieraion process in pariclar cases corresponding o Corollaries As can be seen from Table 2, 13 ieraion seps are necessary when applying Corollary 2.11, Corollary 2.14 reqires 7 seps, while Corollaries 2.16 and 2.19 ensre he oscillaion of (2.31 afer he same nmber of ieraions (6 seps. Acknowledgemens. This research was sppored by he inernal gran projec no. FEI References [1] R. P. Agarwal, S. R. Grace, D. O Regan; Oscillaion Theory for Second Order Linear, Half- Linear, Sperlinear and Sblinear Dynamic Eqaions, Klwer Academic, Dordrche, [2] R. P. Agarwal, S. R. Grace, D. O Regan; Oscillaion Theory for Second Order Dynamic Eqaions, Taylor & Francis, London and New York, [3] B. Baclíková; Oscillaory behavior of he second order fncional differenial eqaions, Applied Mahemaics Leers 72 (2017: [4] E. Braverman, G. E. Chazarakis, I. P. Savrolakis; Ieraive oscillaion ess for differenial eqaions wih several non-monoone argmens, Advances in Difference Eqaions (2016: 87. [5] J. Džrina; Oscillaion of second order differenial eqaions wih advanced argmen, Mahemaica Slovaca 45.3 (1995: [6] R. M. Corless, G. H. Gonne, D. E. Hare, D. J. Jeffrey, D. E. Knh; On he Lamber W fncion, Advances in Compaional mahemaics, 5(1, (1996: [7] G. E. Chazarakis, H. Péics; Differenial eqaions wih several non-monoone argmens: An oscillaion resl; Applied Mahemaics Leers, (2016. [8] L. E. Elsgols, S. B. Norkin; Inrodcion o he heory and applicaion of differenial eqaions wih deviaing argmens Elsevier, 1973.

10 10 IRENA JADLOVSKÁ EJDE-2017/162 Table 2. Comparison of ieraive processes for (2.31 resling from Corollaries 2.11, 2.14, 2.16, 2.19, respecively. n cri. val. 1/e (Cor n cri. val. 1/4 (Cor n cri. val. 1/4 (Cor n cri. val. 1/4 (Cor ndefined [9] L. H. Erbe, Q. Kong, B. G. Zhang; Oscillaion Theory for Fncional Differenial Eqaions, Marcel Dekker Inc., New York, [10] W. B. Fie; Properies of he solions of cerain fncional-differenial eqaions, Transacions of he American Mahemaical Sociey 22 (1921: [11] I. Gyori, G. Ladas; Oscillaion Theory of Delay Differenial Eqaions wih Applicaions, Clarendon Press, Oxford, [12] P. Harman; Ordinary Differenial Eqaions, Wiley, New York, [13] E. Hille; Nonoscillaion heorems, Trans. Amer. Mah. Soc., 64 (1948, [14] J. Jaroš, I. P. Savrolakis; Oscillaion ess for delay eqaions, Rocky Monain J. Mah., 29 (1999, [15] R. Koplaadze, G. Kvinikadze, I. P. Savrolakis; Properies A and B of n-h order linear differenial eqaions wih deviaing argmen, Georgian Mah. J. 6 (1999, [16] T. Ksano, M. Naio; Comparison heorems for fncional differenial eqaions wih deviaing argmens, J. Mah. Soc. Japan 3 (1981, [17] G. Ladas, V. Lakhshmikanham, J. S. Papadakis; Oscillaions of higher-order rearded differenial eqaions generaed by he rearded argmen, Delay and Fncional Differenial Eqaions and Their Applicaions, Academic Press, New York, 1972, [18] G. S. Ladde, V. Lakshmikanham, B. G. Zhang; Oscillaion Theory of Differenial Eqaions wih Deviaing Argmens, Marcel Dekker, New York, [19] W. Leighon; The deecion of he oscillaion of solions of a second order linear differenial eqaion, Dke J. Mah. 17 ( [20] Ch. G. Philos; On he exisence of nonoscillaory solions ending o zero a for differenial eqaions wih posiive delays, Arch. Mah. 36, (1981,

11 EJDE-2017/162 ITERATIVE OSCILLATION RESULTS 11 Irena Jadlovská Deparmen of Mahemaics and Theoreical Informaics, Facly of Elecrical Engineering and Informaics, Technical Universiy of Košice, B. Němcovej 32, Košice, Slovakia address: