OSCILLATION OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Elecronic Journal of Qualiaive Theory of Differenial Equaions 2010, No. 43, 1-10; hp:// OSCILLATION OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS B. BACULÍKOVÁ AND J. DŽURINA Absrac. The aim of his paper is o sudy oscillaory and asympoic properies of he hird-order nonlinear delay differenial equaion (E) ˆa() ˆx () γ + q()f(x[τ()]) = 0. Applying suiable comparison heorems we presen new crieria for oscillaion or cerain asympoic behavior of nonoscillaory soluions of (E). Obained resuls essenially improve and complemen earlier ones. Various examples are considered o illusrae he main resuls. 1. Inroducion We are concerned wih oscillaory behavior of he hird-order funcional differenial equaions of he form [ [ (E) a() x () ] γ ] + q()f(x[τ()]) = 0 In he sequel we will assume ha he following condiions are always saisfied hroughou his paper: (H1) a(),q() C([ 0, )), a(),q() are posiive, τ() C([ 0, )), τ(), lim τ() =, (H2) a 1/γ (s)ds <, 0 (H3) γ is a quoien of odd posiive inegers, (H4) f(x) C(, ), xf(x) > 0, f (x) 0 for x 0 and f( xy) f(xy) f(x)f(y) for xy > 0. By a soluion of Eq.(E) we mean a funcion x() C 2 [T x, ), T x 0, which has he propery a()(x ()) γ C 1 [T x, ) and saisfies Eq. (E) on [T x, ). We consider only hose soluions x() of (E) which saisfy sup{ x() : T } > 0 for all T T x. We assume ha (E) possesses such a soluion. A soluion of (E) is called oscillaory if i has arbirarily large zeros on [T x, ) and oherwise i is called o be nonoscillaory. Equaion (E) is said o be oscillaory if all is soluions are oscillaory. Following Tanaka [23] we say ha a nonrivial soluion x() of (E) is srongly decreasing if i saisfies (1.1) x()x () < Mahemaics Subjec Classificaion. 34C10, 34K11. Key words and phrases. Third-order funcional differenial equaions, oscillaion, nonoscillaion, comparison heorem. EJQTDE, 2010 No. 43, p. 1

2 for all sufficienly large and i said o be srongly increasing if (1.2) x()x () > 0. Recenly differenial equaions of he form (E) and is special cases have been he subjec of inensive research (see enclosed references). Grace e al. in [9] have esablished a useful comparison principle for sudying properies of (E). They have compared Eq.(E) wih a couple of he firs order delay differenial equaions in he sense ha we deduce oscillaion of Eq.(E) from he oscillaion of his couple of equaions. Dzurina and Baculikova in [5] improve heir resuls for he case when 0 a 1/γ (s)ds =. Zhong e al. in [24] adaped Grace e al. s mehod and exended some of heir resuls o neural differenial equaion [ { (E 1 ) a() [x() + p()x(σ())] } γ] + q()f(x[τ()]) = 0. So ha again from oscillaion of a suiable firs order delay equaion we deduce oscillaion of (E 1 ) On he oher hand, Saker and Dzurina in [21] sudied a paricular case of Eq.(E), namely he differenial equaion [ [ (E 2 ) a() x () ] γ ] + q()x γ [τ()] = 0. They presened condiions under which every nonoscillaory soluion of (E 2 ) ends o zero as. Those resuls are applicable even if he crieria presened in [9] fail. I is useful o noice ha for a very special case of (E), ha is, for (E 3 ) x () + q()x() = 0, Harman and Winner in [11] have derived ha (E 3 ) always has a srongly decreasing soluion. Thus, he effor for obaining crieria for all nonoscillaory soluions o be srongly decreasing appeared. Therefore, from all above menioned resuls, we conclude ha if he gap beween and τ() is small, hen here exiss a nonoscillaory soluion of (E) and he Theorems from [9] are no applicable o deduce oscillaion of (E). In his case, our goal is o prove ha every nonoscillaory soluion of (E) ends o zero as. While if he difference τ() is large enough, hen we can sudy he oscillaory characer of (E). So our aim of his aricle is o provide a general classificaion of oscillaory and asympoic behavior of he sudied equaion. We presen crieria for (E) o be oscillaory or for every is nonoscillaory soluion o be eiher sricly decreasing or end o zero as. A firs we urn our aenion o Theorem 2.2 from [9], which is he main resul of he paper. Formulaion of Theorem 2.2 in [9] does no mach is proof and for all ha we provide a correced version of he heorem. EJQTDE, 2010 No. 43, p. 2

3 Theorem A. [Theorem 2.2 in [9]] Le (H1) hold and assume ha here exis wo funcions ξ() and η() C 1 ([ 0, ), R) such ha If (1.3) ξ () > 0, η () > 0 and τ() < ξ() < η() < for 0. 0 ( 1 a(u) ( ) 1/γ q(s)f(τ(s))f a 1/γ (v)dv ds) du =, 0 τ(s) u and boh he firs order delay equaions ( ) τ() (E I ) y () + cq()f sa 1/γ (s)ds f T ( ) y 1/γ [τ()] = 0, for any consan c, 0 < c < 1, and T 0, and ( ) η() (E II ) z () + q()f (ξ() τ()) f a 1/γ (s)ds f are oscillaory, hen every soluion of Eq.(E) is oscillaory. ξ() ( ) z 1/γ [η()] = 0 Remark 1. In he formulaion of Theorem 2.2 in [9], here is an excess erm f(g()) included in Eq. (E I ). Now, he reader can easily reconsruc he resuls from [9] peraining o Theorem Main resuls We sar our main resuls wih he classificaion of he possible nonoscillaory soluions of (E). Lemma 1. Le x() be a posiive soluion of (E). Then eiher (i) x () > 0, evenually and x() is eiher srongly increasing or srongly decreasing, or (ii) x () < 0, evenually and x() is srongly increasing. Proof. Le x() be a nonoscillaory soluion of Eq.(E). We may assume ha x() > 0, evenually (if i is an evenually negaive, he proof is similar). Then [a()[x ()] γ ] < 0, evenually. Thus, a()[x ()] γ is decreasing and of one sign and i follows from hypohesis (H1) and (H2) ha here exiss a 1 0 such ha x () is of fixed sign for 1. If we have x () > 0, hen x () is increasing and hen eiher (1.1) or (1.2) hold, evenually. On he oher hand, if x () < 0 hen x () is decreasing, hence x () is of fixed sign. If we have x () < 0, hen lim x() =. This conradics he posiiviy of x(). Whereupon x () > 0. The proof is complee. The following crierion eliminaes case (ii) of Lemma 1. Lemma 2. Le x() be a posiive soluion of (E). If [ 1 u ] 1/γ (2.1) q(s)f(τ(s))f a 1/γ (v)dv ds) du =, a(u) 0 τ(s) 0 hen x() does no saisfy case (ii) of Lemma 1. EJQTDE, 2010 No. 43, p. 3

4 Proof Le x() be a posiive soluion of Eq.(E). We assume ha x() saisfies case (ii) of Lemma 1. Tha is x () < 0 and x () > 0, evenually. Then here exis a 1 0 and a consan k, 0 < k < 1 such ha x() kx () for 1. Consequenly, (2.2) x[τ()] kτ()x [τ()] for 2 1. Now Eq.(E), in view of (H4) and (2.2), implies [a()[x ()] γ ] + f(k)q()f[τ()]f(x [τ()]) 0. An inegraion of his inequaliy yields (2.3) f(k) 2 q(s)f[τ(s)]f(x [τ(s)])ds a( 2 )[x ( 2 )] γ a()[x ()] γ. On he oher hand, since a 1/γ ()[x ()] is increasing, here exis a consan m > 0 such ha (2.4) a 1/γ ()x () m, for 2, which implies (2.5) x (τ()) τ() a 1/γ (s)x (s)a 1/γ (s)ds m a 1/γ (s)ds. τ() Combining (2.5) ogeher wih (2.3), and aking ino accoun (H3), we ge ( 1 ) 1/γ (2.6) c q(s)f[τ(s)]f a 1/γ (v)dv ds) x (), a() 2 τ(s) where c = [f(m)f(k)] 1/γ. Inegraing (2.6) from 3 o, we have 1 u ) 1/γ c q(s)f[τ(s)]f a 1/γ (v)dv ds) du x ( 3 ). a(u) 2 τ(s) 3 Leing we ge a conradicion o condiion (2.1). Therefore, we have eliminaed case (ii) of Lemma 1. Now we are prepared o provide oscillaion and asympoic crieria for soluions of Eq.(E). Theorem 1. Le (2.1) hold. If he firs order delay equaion [ ] τ() [ ] (E 4 ) y () + q()f (τ() u)a 1/γ (u)du f y 1/γ [τ()] = 0 0 is oscillaory, hen every soluion of Eq.(E) is eiher oscillaory or srongly decreasing. Proof. Le x() be a nonoscillaory soluion of Eq.(E). We may assume ha x() > 0 for 0. From Lemma 2 we see ha x () > 0 and x() is eiher srongly increasing or srongly decreasing. EJQTDE, 2010 No. 43, p. 4

5 Assume ha x() is srongly increasing, ha is x () > 0, evenually. Using he fac ha a()[x ()] γ is decreasing, we are lead o (2.7) x () 1 x (u)du = [ a()(x ()) γ] 1/γ Inegraing (2.7) from 1 o, we have x() 1 a 1/γ (u) [ a(u)(x (u)) γ] 1/γ du 1 [ a(s)(x (s)) γ] 1/γ 1 a 1/γ (u)du. s 1 a 1/γ (u)duds [ a()(x ()) γ] 1/γ ( u)a 1/γ (u)du. 1 There exiss a 2 1 such ha for all 2, one ges (2.8) x[τ()] y 1/γ [τ()] τ() 2 (τ() u)a 1/γ (u)du, where y() = a()(x ()) γ. Combining (2.8) ogeher wih (E), we see ha [ ] y () = q()f(x(τ())) q()f y 1/γ [τ()] τ() [ ] τ() q()f (τ() u)a 1/γ (u)du f 2 2 (τ() u)a 1/γ (u)du [ ] y 1/γ [τ()], where we have used (H3). Thus, y() is a posiive and decreasing soluion of he differenial inequaliy [ ] τ() [ ] y () + q()f (τ() u)a 1/γ (u)du f y 1/γ [τ()] 0. 2 Hence, by Theorem 1 in [19] we conclude ha he corresponding differenial equaion (E 4 ) also has a posiive soluion, which conradics o oscillaion of (E 4 ). Therefore x() is srongly decreasing. Adding an addiional condiion, we achieve sronger asympoic behavior of nonoscillaory soluions of Eq.(E). Lemma 3. Assume ha x() is a srongly decreasing soluion of Eq.(E). If [ 1 1/γ (2.9) q(s)ds] a 1/γ dudv =, (u) hen x() ends o zero as. 0 v u Proof. We may assume ha x() is posiive. I is clear ha here exiss a finie lim x() = l. We shall prove ha l = 0. Assume ha l > 0. Inegraing Eq.(E) from o and using x[τ()] > l and (H3), we obain a()(x ()) γ q(s)f(x[τ(s)])ds f(l) q(s)ds, EJQTDE, 2010 No. 43, p. 5

6 which implies x () l [ 1/γ 1 q(s)ds] a 1/γ, () where l 1 = f 1/γ (l) > 0. Inegraing he las inequaliy from o, we ge x () l 1 1 a 1/γ (u) [ 1/γ q(s)ds] du. Now inegraing from 1 o, we arrive a [ 1 1/γ x( 1 ) l 1 q(s)ds] a 1/γ dudv. (u) 1 v Leing we have a conradicion wih (2.9) and so we have verified ha lim x() = 0. Combining Theorem 1 and Lemma 3 we ge: Theorem 2. Assume ha (2.1) and (2.9) holds. If he equaion (E 4 ) is oscillaory hen every soluion of Eq.(E) is oscillaory or ends o zero as. For a special case of Eq.(E), we have: Corollary 1. Assume ha (2.9) holds and (2.10) 1 u β q(s)τ β (s) a (v)dv) 1/γ ds a(u) 0 0 τ(s) u u 1/γ du =. Assume ha β is a quoien of odd posiive inegers. If he delay equaion [ ] β τ() (E 5 ) y () + q() (τ() s)a 1/γ (s)ds y β/γ [τ()] = 0 0 is oscillaory hen every soluion of he equaion [ [ (E 6 ) a() x () ] γ ] + q()x β [τ()] = 0 is oscillaory or ends o zero as. In Theorems 1 and 2 and Corollary 1 we have esablished new comparison principles ha enable o deduce properies of he hird order nonlinear differenial equaion (E) from oscillaion of he firs order nonlinear delay equaion (E 4 ). Consequenly, aking ino accoun oscillaion crieria for (E 4 ), we immediaely obain resuls for (E). Corollary 2. Assume (2.9) and 1 u γ 1/γ (2.11) q(s)τ γ (s) a (v)dv) ds) 1/γ du = a(u) 0 τ(s) hold. If 0 (2.12) lim inf τ() [ ] γ τ(u) q(u) (τ(u) s)a 1/γ (s)ds du > 1 0 e, EJQTDE, 2010 No. 43, p. 6

7 hen every soluion of he equaion [ [ (E 7 ) a() x () ] γ] + q()x γ [τ()] = 0 is oscillaory or ends o zero as. Proof. Condiion (2.12) (see Theorem in [16]) guaranees oscillaion of (E 6 ) wih β = γ. Now we eliminae he srongly decreasing soluions of (E) o ge an oscillaion resul. We relax condiion (2.9) and employ anoher one. Our mehod is new and complemens he one presened in [9]. Theorem 3. Le (2.1) hold and τ () > 0. Assume ha here exis a funcion ξ() C 1 ([ 0, )) such ha (2.13) ξ () 0, ξ() >, and η() = τ(ξ(ξ())) <. If boh firs order delay equaions (E 4 ) and ξ() ) 1/γ (E 8 ) z () + 1 ξ(s2 ) a 1/γ q(s 1 )ds 1 ds 2 f 1/γ (z[η()]) = 0 (s 2 ) s 2 are oscillaory, hen Eq. (E) is oscillaory. Proof. Le x() be a nonoscillaory soluion of Eq.(E). We may assume ha x() > 0. From Theorem 1, we see ha x() is srongly decreasing (i.e., x () < 0). Inegraion of (E) from o ξ() yields Then a() ( x () ) ξ() γ q(s 1 )f(x(τ(s 1 )))ds 1 f(x[τ(ξ())]) x () f1/γ (x[τ(ξ())]) a 1/γ () Inegraing from o ξ() once more, we ge ξ() x f 1/γ (x[τ(ξ(s 2 ))]) ξ(s2 ) () a 1/γ (s 2 ) ξ() f 1/γ (x[η()]) ( ) 1/γ ξ() q(s 1 )ds 1. ξ() ) 1/γ ds 2 q(s 1 )ds 1 s 2 ( 1 ξ(s2 ) a 1/γ q(s 1 )ds 1 (s 2 ) s 2 q(s 1 )ds 1. ) 1/γ ds 2. Finally, inegraing from o, one ges ξ(s3 (2.14) x() f 1/γ ) ) 1/γ 1 ξ(s2 ) (x[η(s 3 )]) s 3 a 1/γ q(s 1 )ds 1 ds 2 ds 3. (s 2 ) s 2 Le us denoe he righ hand side of (2.14) by z(). Then z() > 0 and one can easily verify ha z() is a soluion of he differenial inequaliy ξ() ) 1/γ z () + 1 ξ(s2 ) a 1/γ q(s 1 )ds 1 ds 2 f 1/γ (z[η()]) 0 (s 2 ) s 2 EJQTDE, 2010 No. 43, p. 7

8 Then Theorem 1 in [19] shows ha he corresponding differenial equaion (E 8 ) has also a posiive soluion. This conradicion finishes he proof. For he special case of Eq. (E) wih f(u) = u β, we immediaely have: Corollary 3. Le (2.10) hold and τ () > 0. Le β be a quoien of odd posiive inegers. Assume ha here exis a funcion ξ() C 1 ([ 0, )) such ha (2.13) holds. If boh Eq.(E 5 ) and ξ() ) 1/γ (E 9 ) z () + 1 ξ(s2 ) a 1/γ q(s 1 )ds 1 ds 2 z β/γ [η()] = 0 (s 2 ) s 2 are oscillaory, hen equaion (E 6 ) is oscillaory. When choosing ξ() we are very paricular abou wo condiions ξ() > and τ(ξ(ξ())) < hold. Unforunaely here is no general rule how o choose funcion ξ() o obain he bes resul for oscillaion of (E 8 ). We sugges for funcion ξ() o be close o he inverse funcion of τ(). In he nex example he reader can see he deails. Example 1. Le us consider hird order differenial equaion (E 10 ) [ 2 x ()] + bx(λ) = 0, b > 0, λ (0,1), 1. I is easy o verify ha (2.10) holds for (E 10 ). Now (E 5 ) reduces o y () + b(λ lnλ 1)y(λ) = 0 and he oscillaion crierion (2.12) akes he form ] lim [ b 2λ λ3 + ln (λ 1) + ln λ(2λ 1) 2 > 1 e, which is evidenly fulfilled. Choosing ξ() = α wih 1 < α < 1 λ equaion (E 9 ) akes he form z () + b(α 1)ln α z[λα 2 ] = 0 and he oscillaion crierion (2.12) reduces o lim b(α 1)ln α(1 λα2 ) > 1 e, which is saisfied. All condiions of Corollary 3 are saisfied and hence Eq. (E 10 ) is oscillaory. Remark 2. In he proof of Lemma 2 we have recognized ha condiion(2.1) eliminaes case (ii) of Lemma 2.1. On he oher hand, Theorems 1 and 3 implies ha oscillaion of boh equaions (E 4 ) and (E 8 ) eliminaes case (ii) of Lemma 2.1. Therefore if (2.1) is no saisfied, we have he following resul: Theorem 4. Le τ () > 0. Assume ha here exis a funcion ξ() C 1 ([ 0, )) such ha (2.13) holds If boh firs order delay equaions (E 4 ) and (E 8 ) are oscillaory, hen every posiive soluion x() of Eq. (E) saisfies case (ii) of Lemma 2.1. EJQTDE, 2010 No. 43, p. 8

9 Example 2. Le us consider hird order delay differenial equaion [ (E 11 ) 5 (x ()) 3] b + x(λ) = 0, b > 0, λ (0,1), 1. 2 I is easy o verify ha (2.1) fails for Eq.(E 11 ). On he oher hand, he corresponding equaion (E 4 ), namely, y () + b ( λ 9 ) 3 2 (λ)1/3 + 3 z(λ) = 0 is oscillaory for every b. Moreover, seing ξ() = α, wih 1 < α < λ 1/2, we see ha he corresponding equaion (E 8 ), namely, ( ) z () + b1/3 α 1 4/3 y(α 2 λ) = 0 α is oscillaory if ( ) α 1 4/3 b 1/3 ln 1 α α 2 λ > 1 e, which for α = ( λ + 1)/(2 λ) reduces o ( (2.15) b 1/3 1 ) 4/3 λ ln λ λ + 2 λ + 1 > 1 e. Therefore Theorem 4 ensures ha every posiive soluion x() of Eq.(E 11 ) saisfies case (ii) of Lemma 2.1 provided ha (2.15) holds. We noe ha for b = 1/128λ 3/2 one such soluion of Eq.(E 11 ) is x() = 1/2. In his paper we provide a full classificaion of nonoscillaory soluions of (E). Our parial resuls guaranee described asympoic behavior of all nonoscillaory soluions (boundedness, convergence o zero and nonexisence). Our crieria improve and properly complemen known resuls even for simple cases of (E). Our conclusions are precedened by illusraive examples ha confirm upgrading of known oscillaion crieria. If we apply known/new crieria for boh nonlinear firs order equaions (E 4 ) and (E 8 ) o be oscillaory, we obain more general crieria for asympoic properies of nonlinear hird order equaion (E). References [1] R. P. Agarwal, S. L. Shien, C. C. Yeh, Oscillaion crieria for second order rearded differenial equaions, Mah. Compu. Modelling 26 (1997), No Zbl [2] M. Barušek, M. Cecchi, Z. Došlá and M. Marini, Oscillaion for hird-order nonlinear differenial equaions wih deviaing argumen Absr. Appl. Anal. 2010, Aricle ID , 19 p. (2010). Zbl pre [3] B. Baculíková, E. M. Elabbasy, S. H. Saker, J. Džurina, Oscillaion crieria for hird-order nonlinear differenial equaions, Mah. Slovaca 58 (2008), No.2, Zbl [4] B. Baculíková, J. Džurina, Oscillaion of hird-order neural differenial equaions, Mah. Compu. Modelling 52 (2010), [5] B. Baculíková, J. Džurina, Oscillaion of hird-order nonlinear differenial equaions, Appl. Mah. Leers (2010) (o appear) [6] J. Džurina, Asympoic properies of hird order delay differenial equaions,, Czech. Mah. J. 45(120), (1995), Zbl EJQTDE, 2010 No. 43, p. 9

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