Note on oscillation conditions for first-order delay differential equations

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1 Elecronic Journal of Qualiaive Theory of Differenial Equaions 2016, No. 2, 1 10; doi: /ejqde hp:// Noe on oscillaion condiions for firs-order delay differenial equaions Kirill Chudinov B Per Naional Research Polyechnic Universiy, 29 Kosool skii Ave, Per, , Russia Received 15 Noveber 2015, appeared 1 February 2016 Counicaed by Ivan Kiguradze Absrac. We consider explici condiions for all soluions o linear scalar differenial equaions wih several variable delays o be oscillaory. The considered condiions have he for of inequaliies bounding he upper lii of he su of inegrals of coefficiens over a subse of he real seiaxis, by he consan 1 fro below. The ain resul is a new oscillaion condiion, which sharpens several known condiions of he kind. Soe resuls are presened in he for of counerexaples. Keywords: differenial equaions, delay, oscillaion, sufficien condiions Maheaics Subjec Classificaion: 34K06, 34K11. 1 Inroducion I follows fro resuls by Ladas e al. [7] and Traov [12] ha all soluions of he equaion ẋ() + a()x( τ) = 0, 0, (1.1) where a() 0 and τ = cons > 0, are oscillaory in case li sup + a(s) ds > 1. τ For an equaion wih variable delay, Corollary 2.1 fro [7] presens he following oscillaion condiion. Suppose a C(R +, R + ), h C 1 (R +, R + ), h() and h () 0 for all R +, li h() =, and li sup h() a(s) ds > 1. Then all soluions of he equaion ẋ() + a()x(h()) = 0, 0, (1.2) are oscillaory. This resul is exended and sharpened in any publicaions. In alos all of he he condiion is iposed ha he delay funcion h is nondecreasing. The presen paper is devoed o condiions for all soluion of he equaion ẋ() + a k ()x(h k ()) = 0, 0, (1.3) where a k () 0, h k (), and h k () as, o be oscillaory. All new obained oscillaion condiions are generalizaions of he resuls forulaed above. We do no suppose B Eail: cyril@lis.ru

2 2 K. Chudinov ha he funcions h k are necessarily nondecreasing and accopany he obained resuls by a nuber of counerexaples in order o copare he new oscillaion condiions wih known ones. In Secion 2 we discuss published resuls concerning oscillaion condiions of he considered kind. In Secion 3 our ain resul is obained, and i is shown ha known resuls are is corollaries. In Secion 4 equaion (1.2) is discussed. In Secion 5 soe ideas fro he previous secion are exended o he case of equaion (1.3). Soe resuls in he las hree secions are represened in he for of counerexaples. 2 Known oscillaion condiions Theore fro he book [9] by Ladde e al. represens an oscillaion condiion for (1.2) ha sharpens slighly he cied resul fro [7], as i is supposed ha h C(R +, R + ), and he nonnegaiviy of h is replaced by he nondecrease of h. This resul is exended o he case of equaion (1.3) in Theore fro he book [5] by Győri and Ladas. The basic oscillaory condiion in he heore is he inequaliy li sup a k (s) ds > 1. ax k h k () I is no saed explicily ha he funcions h k are supposed o be nondecreasing, however, he auhors did no enion anyhing o replace his condiion. I is shown in Secion 4 of his paper ha he nondecrease is acually essenial. In [1, p. 36], here is an exaple showing ha he inequaliy li sup a k (s) ds 1, in k h k () in conras o ha conaining ax in place of in, is no necessary for a nonoscillaing soluion o exis. In Secion 3 of he presen work we sharpen his resul. Tang [11] obained an oscillaion condiion for he case of several consan delays ẋ() + a k ()x( τ k ) = 0, (2.1) which is no a consequence of he above condiions for (1.3). The basic inequaliy +τk li sup a k (s) ds > 1 is derived fro an oscillaion condiion obained for an equaion wih disribued delay. I is shown in Secion 3 ha he above inequaliy canno be replaced by li sup a k (s) ds > 1. τ k There are few published exensions of he considered oscillaion condiions for he case of nondecreasing delay. The following resul is by Traov [12]. If a() 0, h() h 0 > 0, li h() =, and +h0 li sup a(s) ds > 1,

3 Oscillaion condiions for delay differenial equaions 3 hen every soluion of (1.2) oscillaes. In [12] he auhor also presened an exaple showing he sharpness of he consan 1: if i is diinished by arbirary ε > 0, hen he condiion does no guaranee oscillaion. Koplaadze and Kvinikadze [6] obained anoher oscillaion condiion for he case of nononoone delay. Suppose a() 0, h C(R +, R + ), h(), and li h() =. Define δ() = ax{h(s) s [0, ]}. Then he inequaliy li sup a(s) ds > 1 δ() is sufficien for all soluions of (1.2) o be oscillaory. Noe ha he naure of he considered oscillaion condiions differs fro ha of he oscillaion condiions of 1/e-ype. This is expressed, in paricular, in he possibiliy o exend he above oscillaion condiion o equaions wih oscillaing coefficiens. Such exension was apparenly firs ade by Ladas a al. [8], heir resuls sharpened by Fukagai and Kusano [4]. Below we do no consider 1/e-ype heores and he proble of filling he gap beween 1/e and 1. A deailed discussion of his subjec is found in he onographs [1 3] and he review [10]. 3 Main resul Le paraeers of equaion (1.3) saisfy he following condiions for all k = 1,..., : he funcions a k : R + R are locally inegrable; he funcions h k : R + R are Lebesgue easurable; a k () 0 and h k () for all R +. We say ha a locally absoluely coninuous funcion x : R + R is a soluion o he equaion ẋ() + a k ()x(h k ()) = 0, 0, (1.3) if here exiss a Borel iniial funcion ϕ : (, 0] R such ha he equaliy (1.3) akes place for alos all 0, where x(ξ) = ϕ(ξ) for all ξ 0. Le us define a faily of ses E k () = {s h k (s) s}, 0, k = 1,...,. I follows fro he saed above ha all he ses of he faily are Lebesgue easurable. Theore 3.1. Suppose li h k () = for all k = 1,...,, and li sup Then every soluion of equaion (1.3) is oscillaory. E k () a k (s) ds > 1.

4 4 K. Chudinov Proof. Suppose he condiions of he heore are fulfilled and consider an arbirary soluion x of equaion (1.3). Assue ha x is no oscillaory. Wihou loss of generaliy, suppose ha here exiss 0 0 such ha x() > 0 for all 0. Then here exiss 1 0 such ha h k () 0 for all 1 and k = 1,...,. I is obvious ha x() is nonincreasing for all 1. Furher, here exiss 2 1 such ha x(h k ()) x() for all 2 and k = 1,...,, and E k ( 2 ) a k(s) ds > 1. There also exiss 3 > 2 such ha for all he ses S k = E k ( 2 ) [ 2, 3 ], k = 1,...,, we have a S k k (s) ds > 1. Therefore, 3 3 x( 3 ) = x( 2 ) + ẋ(s) ds = x( 2 ) 2 2 x( 2 ) S k which conradics he assupion. a k (s)x(h k (s)) ds x( 2 ) a k (s)x(h k (s)) ds ( 1 S k a k (s) ds Corollary 3.2. Suppose he funcions h k are coninuous and sricly increasing, li h k () = for k = 1,...,, and li sup Then every soluion of equaion (1.3) is oscillaory. h 1 k () ) < 0, a k (s) ds > 1. (3.1) Proof. For each k =1,..., here exiss he inverse funcion h 1 k, which is defined on [h k (0), ) and is sricly increasing. Hence E k () = [, h 1 k ()]. Corollary 3.3 ([11]). Suppose h k () = τ k, where τ k > 0, and Then every soluion of equaion (1.3) is oscillaory. Proof. We have h 1 k () = + τ k and E k () = [, + τ k ]. +τk li sup a k (s) ds > 1. (3.2) Corollary 3.4 ([5]). Suppose he funcions h k is nondecreasing, li h k () = for k = 1,...,, and li sup a k (s) ds > 1. (3.3) ax k h k () Then every soluion of equaion (1.3) is oscillaory. Proof. By virue of he nondecrease of h k we have ha [ax k h k (), ] E k (ax k h k ()). Since li h k () =, i follows fro (3.3) ha li sup E k () a k(s) ds > 1. The following exaple suppleens Corollaries 3.2, 3.3 and 3.4. Exaple 3.5. Consider he equaion ẋ() + a 1 ()x( 3) + a 2 ()x( 1) = 0, 0, (3.4)

5 Oscillaion condiions for delay differenial equaions 5 where for n = 0, 1, 2,... we pu 0, [6n, 6n + 3), a 1 () = 3/4, [6n + 3, 6n + 4), 0, [6n + 4, 6(n + 1)); a 2 () = { 0, [6n, 6n + 5), 3/4, [6n + 5, 6(n + 1)). We see ha ( li sup 3 ) 6(n+1) 6(n+1) a 1 (s) ds + a 2 (s) ds = a 1 (s) ds + a 2 (s) ds = 3/2 > n+3 6n+5 However, every soluion x of equaion (3.4) is nonincreasing on R +, and x(6(n + 1)) = x(6n)/16, n = 0, 1, 2,..., ha is x() > 0 for all 0. Exaple 3.5 shows ha inequaliy (3.1) canno be replaced by li sup h k () In paricular, his eans ha inequaliy (3.2) canno be replaced by li sup a k (s)ds > 1. τ k a k (s)ds > 1. (3.5) Inequaliy (3.3) also canno be replaced by (3.5). This srenghens he resul fro [1, p. 36] cied in Secion 2, since 4 Equaion wih single delay Consider he equaion wih single delay a k (s) ds a k (s) ds. h k () in k h k () ẋ() + a()x(h()) = 0, 0, (1.2), which is a special case of equaion (1.3). Define E() = {s h(s) s}. By Theore fro [9], if h is nondecreasing, li h() = and li sup a(s) ds > 1, h() hen all soluions of (1.2) are oscillaory. The onooniciy of h is here essenial. This fac can be shown by a very siple exaple in case he easure µ { E() a(s) ds > 1} = 0. The las is no assued in he following exaple. Exaple 4.1. Consider equaion (1.2), where a() α > 1. Pu ε (0, 1) and h() = {, [n, n + 1 ε), n, [n + 1 ε, n + 1),

6 6 K. Chudinov for n = 0, 1, 2,... Consider he soluion of (1.2) deerined by an iniial value x(0) = x 0 > 0. One ay choose ε so ha he soluion is posiive. Indeed, fix an arbirary posiive ineger n and consider x() for [n, n + 1). We have x() = { x(n)e α( n), [n, n + 1 ε); x(n)e α(1 ε) αx(n)( (n + 1 ε)), [n + 1 ε, n + 1). (4.1) Thus, x(n + 1) = x(n)(e α(1 ε) αε). To provide ha x(n) is posiive for all n i is sufficien o choose ε so ha ε < (e α(1 ε) )/α. Obviously, for soe ε 0 > 0 he inequaliy is valid for all ε (0, ε 0 ). Furher, i follows fro (4.1) ha x(n + 1) x() x(n) for (n, n + 1), hence for he chosen ε we have x() > 0 for all R +. On he oher hand, li sup h() a(s) ds = n+1 a(s) ds = α > 1. n I is obvious ha Exaple 4.1 ay be odified for he case ha h is coninuous. Consider Theore 3.1 for he case = 1. Corollary 4.2. Suppose li h() = and li sup E() a(s) ds > 1. Then every soluion of equaion (1.2) is oscillaory. The funcion h is no supposed o be nondecreasing in Corollary 4.2. The following corollaries represen an idea ha o prove ha all soluions o equaion (1.2) are oscillaory i ay be sufficien o consider an auxiliary equaion wih nondecreasing delay. In paricular, his allows o esablish oscillaion in case he funcion h is no defined precisely. Corollary 4.3. Le h 0 = 0, h n+1 > h n for n = 0, 1, 2,..., and li n h n =. Suppose h() h n for [h n, h n+1 ) and hn+1 li sup a(s) ds > 1. n h n Then every soluion of (1.2) is oscillaory. Proof. I is readily seen ha for n = 0, 1, 2... and [h n, h n+1 ) we have [, h n+1 ) E(). Therefore, hn+1 a(s) ds a(s) ds. h n E(h n ) Hence li sup E() a(s) ds li sup hn+1 n h n a(s) ds. I reains o apply Corollary 4.2. Corollary 4.4 ([6]). Pu g() = sup{h(s) s < }. Suppose li h() = and Then every soluion of (1.2) is oscillaory. li sup a(s) ds > 1. g() Proof. We have [g(), ) E(g()). Indeed, if r [g(), ), hen h(r) sup{h(s) s < } = g(), and hence, r {s g() h(s) g()} = E(g()).

7 Oscillaion condiions for delay differenial equaions 7 Obviously, g() as, herefore, li sup g() I reains o apply Corollary 4.2. a(s) ds li sup a(s) ds. E() Corollary 4.5. Pu G() = inf{s h(s) > }. Suppose li h() = and Then every soluion of (1.2) is oscillaory. G() li sup a(s) ds > 1. Proof. I is no hard o see ha [, G()) E(). Hence he resul follows fro Corollary 4.2. Noe ha boh he funcions g and G defined in Corollaries 4.4 and 4.5, respecively are nondecreasing. In Figure 4.1 he graphs of soe delay h and he corresponding g and G are represened. The secions of he graph of g(), where i differs fro ha of h(), are coloured red. The se E(T) is arked green in he axis O. s s G T s h s g T T G T Figure 4.1: The graphs of he funcions h, g and G, and he se E(T). Le us show ha he oscillaion condiions of Corollaries 4.4 and 4.5 are equipoen. Indeed, G(g()) = inf{s h(s) > sup{h(r) r < }},

8 8 K. Chudinov and since g() as, we have ha li sup g() On he oher hand, and G() as, hence, G() li sup G(g()) a(s) ds li sup g() G() a(s) ds li sup a(s) ds. g(g()) = sup{h(s) s < inf{r h(r) > }}, G() a(s) ds li sup g(g()) a(s) ds li sup a(s) ds. g() The applicaion of Corollaries 4.4 and 4.5 is illusraed by he following exaple. Exaple 4.6. Consider equaion (1.2), where a() α > 0. Suppose here exiss a sequence { n } n=1 such ha n as n and h() n for all [ n, n + 1/α]. We have G( n ) n + 1/α. Hence, G( n ) n a(s) ds n +1/α n a(s) ds > 1. By Corollary 4.5 every soluion is oscillaory. We also have g( n + 1/α) n. Hence, n +1/α g( n +1/α) a(s) ds > 1, and by Corollary 4.4 every soluion is oscillaory. The nex exaple shows ha Corollaries 4.4 and 4.5 are weaker han Corollary 4.2. Exaple 4.7. For n = 0, 1, 2,... pu in equaion (1.2) a() = { 1/4, [2n, 2n + 1), 2/3, [2n + 1, 2n + 2); h() = { 2n, [2n, 2n + 1), 2n 1, [2n + 1, 2n + 2). G() We have li sup a(s) ds = G(2n) a(s) ds = 1/4 + 2/3 < 1. Therefore, Corollary 4.5 (and Corollary 4.4 as well) does no allow o deerine if here exiss a nonoscillaing 2n soluion. In fac E(2n + 1) = [2n + 1, 2n + 2) [2n + 3, 2n + 4), li sup a(s) ds = a(s) ds = 4/3 > 1, E() E(2n+1) and by Corollary 4.2 every soluion is oscillaory. 5 Generalizaion Below we exend Corollaries 4.4 and 4.5 o he case of equaion (1.3). For all k = 1,..., pu g k () = sup{h k (s) s < } and G k () = inf{s h k (s) > }. Corollary 5.1. Suppose li h k () = for k = 1,...,, and Then every soluion of equaion (1.3) is oscillaory. Gk () li sup a k (s) ds > 1. (5.1)

9 Oscillaion condiions for delay differenial equaions 9 Proof. I is no hard o see ha [, G k ()) E k (). Corollary 5.2. Suppose li h k () = for k = 1,...,, and Then every soluion of equaion (1.3) is oscillaory. li sup a k (s) ds > 1. (5.2) ax k g k () Proof. Analogously o he case = 1 considered in secion 4, we have G k (g k ()). So, li sup Gk () a k (s) ds li sup Gk (g k ()) g k () Thus, Corollary 5.2 follows fro Corollary 5.1. a k (s) ds li sup ax k g k () a k (s) ds. The following exaple shows ha in case > 1 Corollary 5.1 is sharper han Corollary 5.2. Exaple 5.3. Consider he equaion ẋ() x( 1) + 1 x( 2) = 0, 0. (5.3) 3 We have g 1 () = 1, g 2 () = 2, G 1 () = + 1, G 2 () = + 2. Furher, li sup a k (s) ds = (a 1 (s) + a 2 (s)) ds = 1/2 + 1/3 < 1; ax k g k () 1 and Gk () li sup a k (s) ds = a 1 (s) ds + a 2 (s)) ds = 1/2 + 2/3 > 1. Thus, Corollary 5.1 does allow o esablish ha all soluions of (5.3) are oscillaory, while Corollary 5.2 does no. A las, noe ha Exaple 3.5 shows ha inequaliy (5.2) canno be replaced by li sup g k () a k (s) ds > 1. Acknowledgeens The auhor is graeful o Prof. Vera Malygina and he anonyous referee for several useful coens and suggesions. The research is perfored wihin he basic par of he public conrac wih he Minisry of Educaion and Science of he Russian Federaion (conrac 2014/152, projec 1890) and suppored by he Russian Foundaion for Basic Research (gran ).

10 10 K. Chudinov References [1] R. P. Agarwal, L. Berezansky, E. Braveran, A. Dooshnisky, Nonoscillaion heory of funcional differenial equaions wih applicaions, Springer, New York, MR ; url [2] R. P. Agarwal, M. Bohner, W.-T. Li, Nonoscillaion and oscillaion: heory for funcional differenial equaions, Monographs and Texbooks in Pure and Applied Maheaics, Vol. 267, Marcel Dekker, Inc., New York, MR ; url [3] L. H. Erbe, Q. Kong, B. G. Zhang, Oscillaion heory for funcional-differenial equaions, Monographs and Texbooks in Pure and Applied Maheaics, Vol. 190, Marcel Dekker, Inc., New York, MR [4] N. Fukagai, T. Kusano, Oscillaion heory of firs order funcional-differenial equaions wih deviaing arguens, Ann. Ma. Pura Appl. 136(1984), MR765918; url [5] I. Győri, G. Ladas, Oscillaion heory of delay differenial equaions, Oxford Maheaical Monographs, The Clarendon Press, Oxford Universiy Press, New York, MR [6] R. Koplaadze, G. Kvinikadze, On he oscillaion of soluions of firs-order delay differenial inequaliies and equaions, Georgian Mah. J. 1(1994), No. 6, MR ; url [7] G. Ladas, V. Lakshikanha, J. S. Papadakis, Oscillaions of higher-order rearded differenial equaions generaed by he rearded arguen, Delay and funcional differenial equaions and heir applicaions (Proc. Conf., Park Ciy, Uah, 1972), , Acadeic Press, New York, MR [8] G. Ladas, Y. G. Sficas, I. P. Savroulakis, Funcional differenial inequaliies and equaions wih oscillaing coefficiens, Trends in heory and pracice of nonlinear differenial equaions (Arlingon, Tex., 1982), , Lecure Noes in Pure and Appl. Mah., Vol. 90, Dekker, New York, MR [9] G. S. Ladde, V. Lakshikanha, B. G. Zhang, Oscillaion heory of differenial equaions wih deviaing arguens, Monographs and Texbooks in Pure and Applied Maheaics, Vol. 110, Marcel Dekker, Inc., New York, MR [10] Kh. Niri, I. P. Savroulakis, On he oscillaion of he soluions o delay and difference equaions, Tara M. Mah. Publ. 43(2009), MR ; url [11] X. H. Tang, Oscillaion of firs order delay differenial equaions wih disribued delay, J. Mah. Anal. Appl. 289(2004), No. 2, MR ; url [12] M. I. Traov, Condiions for he oscillaion of he soluions of firs order differenial equaions wih rearded arguen (in Russian), Izv. Vysš. Učebn. Zaved. Maeaika 1975, No. 3(154), MR