Oscillation of solutions to delay differential equations with positive and negative coefficients

Transcription

1 Elecronic Journal of Differenial Equaions, Vol. 2000(2000), No. 13, pp ISSN: URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp ejde.mah.sw.edu fp ejde.mah.un.edu (login: fp) Oscillaion of soluions o delay differenial equaions wih posiive and negaive coefficiens El M. Elabbasy, A. S. Hegazi, & S. H. Saker Absrac In his aricle we presen infinie-inegral condiions for he oscillaion of all soluions of firs-order delay differenial equaions wih posiive and negaive coefficiens. 1 Inroducion Consider he firs-order delay differenial equaion ẋ()+p()x( σ) Q()x( τ)=0, (1.1) where P () andq() are posiive coninuous real funcions and σ, τ are posiive consans. Equaion (1.1) has he following general form ẋ()+ n m P i ()x( σ i ) Q j ()x( τ j )=0, (1.2) where P i (),Q j () C([ 0, ),R + )andσ i,τ j [0, ), for i =1,...,n and j = 1,...,m. By a soluion of (1.1) or (1.2), we mean a funcion x() C([ 0 ρ),r) ha for some 0 saisfies (1.1) (or (1.2)) for all 0,whereρ=max{σ, τ} (or ρ =max{max 1 i n σ i, max 1 j m τ j }). As usual a funcion x() is called oscillaory if i has arbirarily large zeros. Oherwise he soluion is called non-oscillaory. Qian and Ladas [1] obained for (1.1) he well-known oscillaion crierion lim inf ρ j=1 [P (s) Q(s + τ σ)] ds > 1 e. (1.3) Elabbasy and Saker [32] obained he oscillaion crierion for he generalized equaion, lim inf [P i (s) Q k (s + τ k σ i )] ds > 1 ρ e. (1.4) k J i 1991 Mahemaics Subjec Classificaions: 34K15, 34C10. Key words and phrases: Oscillaion, delay differenial equaions. c 2000 Souhwes exas Sae Universiy and Universiy of Norh exas. Submied Sepember 6, Published February 16,

2 2 Oscillaion of soluions o delay differenial equaions EJDE 2000/13 I is easy o see ha (1.3) is given by (1.4) when puing n = m =1. Many auhors have considered he delay differenial equaion, wih posiive coefficien, ẋ()+p()x(τ()) = 0. (1.5) he firs sysemaic sudy of oscillaion for all soluions of (1.5) was made by Myshkis [3]. He proved ha every soluion of (1.5) oscillaes if lim sup [ τ() <, lim inf [ τ()] lim inf P () > 1 e. (1.6) In 1972, Ladas, Laksmikaham and Papadakis [4] proved ha he same conclusion holds if lim sup P (s) ds > 1. (1.7) τ () In 1979, Ladas [5] and, in 1982, Kopladaze and Canurija [2] replaced (1.7) by lim inf P (s) ds > 1 τ () e. (1.8) Concerning he consan 1/e in (1.8), if he inequaliy P (s) ds 1 τ () e (1.9) holds evenually, hen according o a resul in [2], (1.5) has a non-oscillaory soluion. I is obvious ha here is a gap beween he condiions (1.7) and (1.8) when he limi lim P (s) ds (1.10) τ () does no exis. In 1995 Elber and Savrolakis [6] esablished infinie-inegral condiions for oscillaion (1.5) in he case where τ () P (s) ds 1 e and lim P (s) ds = 1 τ () e. (1.11) hey proved ha if [ i P (s) 1 ] ds =, (1.12) i 1 e hen every soluion of (1.5) oscillaes. In 1996, Li [7] showed ha if τ() P (s) ds > 1/e for some 0 > 0and 0 P() [ τ()p(s)ds 1 ] d =, (1.13) e

3 EJDE 2000/13 El M. Elabbasy, A. S. Hegazi, & S. H. Saker 3 hen every soluion of (1.5) oscillaes. Domshlak and Savrolakis [8] esablished sufficien condiions for he oscillaion, in he criical case where lim P () = 1 eτ, of he delay differenial equaion ẋ()+p()x( τ)=0. (1.14) Recenly Domshlak and Savrolakis [9] and Jaros and Savrolakis [10] considered he delay differenial equaion ẋ()+a 1 ()x( τ)+a 2 ()x( σ)=0 (1.15) and esablished sufficien condiions for he oscillaion of all soluions in he criical sae ha he corresponding limiing equaion admis a non-oscillaory soluion. he oscillaory properies of various funcional differenial equaions have been employed by many auhors. For some conribuion o he oscillaion heory of delay differenial equaions we refer o he aricles by Zhang and Goplsamy [11], Gyori and Ladas [12], Li [13], Arino, Ladas and Sficas [14], Ladas and Sficas [15], Ladas, Qian and Yan [16], Arino, Gyori and Jawhari [17], Hun and Yorke [18], Gyori [19], Cheng [20], Kwang [21], Kulenovic, Ladas and Meimardou [22], Kulenovic and Ladas [23, 24, 25], Goplsamy, Kulenovic and Ladas [26], Ladas and Qian [27, 28], Elabbasy, Saker and Al-Shemas [29], Elabbasy and Saker [30] and Elabbasy, Saker and Saif [31], Elabbasy and Saker [32]. o a large exen, he sudy of funcional differenial equaions is moivaed by having many applicaions in Physics [33], Biology [34], Ecology [35], and he sudy of spread of infecious diseases [36]. Our aim in his paper is o give an infinie-inegral condiions for oscillaion of all soluions of (1.1) and (1.2) by using he generalized characerisic equaion x() x( σ i). and he funcion of he form In secion 2, we presen an infinie-inegral condiion for oscillaion of (1.1) which indicaes ha condiion (1.3) is no longer necessary. In secion 3, we exended he resuls in secion 2 o esablish infinie sufficien condiions for oscillaion of (1.2) which indicaes ha condiion (1.4) is no longer necessary. As far as we known, here are no oher resuls for differenial equaions wih posiive and negaive coefficiens wih more han one delay. In he sequel, when we wrie a funcional inequaliy we will assume ha i holds for all sufficienly large values of. Lemma 1.1 ([12]) Le a (, 0), τ (0, ), 0 x() C[[ 0, ),R] saisfies he inequaliy R and suppose ha x() a + max x(s) for 0. τ s hen x() canno be a non-negaive funcion.

4 4 Oscillaion of soluions o delay differenial equaions EJDE 2000/13 Lemma 1.2 ([12]) Assume ha P i and τ i C[[ 0, ),R + ] for i =1,...,n. hen he differenial inequaliy n ẋ()+ P i ()x( τ i ()) 0, 0 (1.16) has an evenually posiive soluion if and only if he equaion n ẏ()+ P i ()y( τ i ()) = 0, 0 (1.17) has an evenually posiive soluion. Lemma 1.3 ([13]) Consider he delay differenial equaion n ẋ()+ R i ()x( τ i )=0, 0 (1.18) +τi and assume ha lim sup R i (s)ds > 0 for some i and x() is an evenually posiive soluion of (1.18), hen for he same i, lim inf x( τ i ) x() < (1.19) Lemma 1.4 ([13]) If (1.18) has an evenually posiive soluion, hen evenually. +τi R i (s) ds < 1,,...,n (1.20) 2 Oscillaion of soluions o (1.1) Now we obain an infinie-inegral condiions for oscillaion of all soluions of (1.1). We need he following Lemma. Lemma 2.1 Assume ha: (h1) P, Q C([ 0, ),R + ), σ, τ [0, ) and τ σ (h2) P () Q( + τ σ), for 0 +σ τ (h3) τ σ Q(s) ds 1 for 0 + σ Le x() be an evenually posiive soluion of (1.1) and se z() =x() τ σ Q(s+τ)x(s)ds, 0 + σ τ. (2.1) hen z() is a non-increasing posiive funcion and saisfies he inequaliy ż()+[p() Q(+τ σ)] z( σ) 0. (2.2)

5 EJDE 2000/13 El M. Elabbasy, A. S. Hegazi, & S. H. Saker 5 he proof of his lemma can be found as Lemma in [12]. heorem 2.2 Assume ha (h1), (h2) and (h3) from Lemma 2.1 are saisfied. Also assume ha for R() =P() Q(+τ σ), (h4) +σ R(s) ds > 0 for 0 for some 0 > 0. (h5) [ 0 R()ln e ] +σ R(s)ds d =. hen every soluion of (1.1) oscillaes. Proof. On he conrary assume ha 1.1) has an evenually posiive soluion x(). By Lemma 2.1 i follows ha he funcion z() is posiive and saisfies (2.2). So Lemma 1.2 yields ha he delay differenial equaion ẏ()+[p() Q(+τ σ)] y( σ) =0 (2.3) has an evenually posiive soluion. Le λ() = ẏ()/y(). hen λ() isnon- negaive and coninuous, hen here exiss 1 0 such ha y( 1 ) > 0and y()=y( 1 )exp( 1 λ(s)ds). Furhermore, if λ() saisfies he generalized characerisic equaion λ() =R()exp( λ(s)ds), σ we can show ha e rx x + ln(er) for r>0. (2.4) r Define A() = +σ R(s)ds. By using (2.4) we find ha or 1 λ() = R()exp(A() A() 1 R()[ A() σ σ λ(s)ds) λ(s) ds + ln(ea() ] A() +σ ( R(s)ds)λ() R() λ(s) ds R()(ln e σ hen for N>, λ()( +σ R(s) ds) d R()(ln e +σ +σ R(s) ds) (2.5) R() λ(s) ds d (2.6) σ R(s) ds) d.

6 6 Oscillaion of soluions o delay differenial equaions EJDE 2000/13 By inerchanging he order of inegraion, we find ha Hence hen Hence R()( λ(s) ds) d σ R()( λ(s) ds) d σ R()( λ(s) ds) d σ σ σ σ ( s+σ s λ(s)( λ()( R()λ(s)d) ds. s+σ s +σ R()d) ds. R(s) ds) d λ()( +σ λ()( +σ R(s) ds) d R(s) ds) d From (2.6) and (2.7), i follows ha N σ λ()( +σ R(s) ds)d σ On he oher hand, by Lemma 1.4, we have +σ evenually. hen by (2.8) and (2.9), we find or In view of (h5) ln N σ λ()d y(n σ) y(n) However, by Lemma 1.3, λ(s)( R() (R())(ln e s+σ s σ +σ R()d) ds (2.7) λ(s) ds d. R(s) ds)d. (2.8) R(s) ds < 1 (2.9) (R()) ln(e R()ln(e +σ +σ R(s) ds)d R(s)ds)d. (2.10) y( σ) lim =. (2.11) y() lim inf y( σ) y() < (2.12) which conradics (2.11), and his complees he presen proof. herefore, every soluion of (1.1) oscillaes.

7 EJDE 2000/13 El M. Elabbasy, A. S. Hegazi, & S. H. Saker 7 3 Oscillaion of soluions o (1.2) Our objecive in his secion is o esablish infinie-inegral condiions for oscillaion of all soluions of (1.2). We need he following heorem for he proof of he main resuls in his secion. heorem 3.1 Assume ha: (H1) P i,q j C([ 0, ),R + ), σ i,τ j [0, ) for i =1,...,n and j =1,...,m (H2) here exis a posiive number p n and a pariion of he se {1,...,m} ino p disjoin subses J 1,J 2,J 3,...,J p, such ha j J i implies ha τ j σ i (H3) P i () k J i Q k ( + τ k σ i ) for 0 + σ i τ k,and,...,p, (H4) p τk σ i Q k (s) ds 1 for 0 + σ i. Le x() be an evenually posiive soluion of (1.2) and se z() =x() τk hen z() is a non-increasing and posiive funcion. σ i Q k (s + τ k )x(s) ds, 0 + σ i τ k. (3.1) Proof Assume ha ρ is such ha x() is posiive for 1 ρ ρ =max 1 i n {σ i }. From (2.1) we have Hence ż() =ẋ() ż() =ẋ() From (1.2), we have ż() = As we know ha we have Q k ()x( τ k )+ m Q j ()x( τ j )+ j=1 Q k ( + τ k σ i )x( σ i ). Q k ( + τ k σ i )x( σ i ). P i ()x( σ i )+ Q k (+τ k σ i )x( σ i ) n i=p+1 P i ()x( σ i ) > 0, n i=p+1 P i ()x( σ i ). ż() [P i () Q k ( + τ k σ i )]x( σ i ) (3.2)

8 8 Oscillaion of soluions o delay differenial equaions EJDE 2000/13 By using (H3) we have ż() 0 for 1 +ρ. (3.3) his implies ha z() is a non-increasing funcion. Now we prove ha z() is posiive. For oherwise, here exiss a 2 1 such ha z( 2 ) 0. Since ż() 0for 1 +ρand ż() 0on[ 1 +ρ, ), here exiss a 3 2 such ha z() z( 3 )for 3. hus from (2.1) i follows ha for 3, Hence x() = z()+ z( 3 )+ z( 3 )+ x() z( 3 )+ Hypohesis (H4) yields τk σ i τk k J σ i j τk σ i τk σ i Q k (s + τ k )x(s) ds Q k (s + τ k )x(s) ds Q k (s + τ k ) ds( max ρ s x(s)). Q k (s + τ k ) ds( max ρ s x(s)). x() z( 3 )+ max x(s) for all 3, ρ s where z( 3 ) 0. Lemma 1.1 implies ha x() canno be non-negaive funcion on [ 3, ). hus conradicing x() > 0. herefore, z() is a non-increasing and posiive funcion. heorem 3.2 Assume ha (H1), (H2), (H3) and (H4) above are saisfied, σ p = max{σ 1,σ 2,σ 3,...,σ p }, p +σi R i (s) ds > 0 for 0 for some 0 > 0. Also assume ha +σp (H5) lim sup R p (s) ds > 0 (H6) ( [ p R i()) ln e p 0 +σi k J i Q k (+τ k σ i ). hen every soluion of (1.2) oscillaes. ] R i (s) ds d = where R i () =P i () Proof. On he conrary assume ha (1.2) has an evenually posiive soluion x(). By heorem 2.1 i follows ha he funcion z() defined by (3.1) is an evenually posiive funcion. Also by (3.2) we have ż()+ [P i () Q k ( + τ k σ i )]x( σ i ) 0. (3.4)

9 EJDE 2000/13 El M. Elabbasy, A. S. Hegazi, & S. H. Saker 9 From he fac ha evenually 0 <z() x(), we see ha z() isaposiive funcion and saisfies evenually ż()+ [P i () Q k ( + τ k σ i )]z( σ i ) 0. (3.5) hen by Lemma 1.2, we have ha he delay differenial equaion ẏ()+ [P i () Q k ( + τ k σ i )]y( σ i )=0 (3.6) has an evenually posiive soluion. Le λ() = ẏ()/y(). hen λ() isa non-negaive and coninuous, and here exiss 1 0 wih y( 1 ) > 0such ha y() =y( 1 )exp( 1 λ(s)ds). Furhermore, λ() saisfies he generalized characerisic equaion λ() = R i ()exp( λ(s)ds) σ i wih R i () =P i () p k J i Q k (+τ k σ i ) Le B() = p R i (s) ds. By using (2.4) we find ha or +σi hen for N>, λ() = +σi R i (s) dsλ() λ()( R i ()exp ( 1 B() λ(s)ds ) B() σ i R i () [ 1 λ(s) ds + ln(eb() ] B() σ i B() +σi R i () λ(s) ds σ i R i (s) ds)d +σi R i ()(ln e Inerchanging he order of inegraion, we find ha R i () λ(s) ds d σ i σi R i ()(ln e +σi R i () λ(s) ds d σ i R i (s) ds) R i (s) ds) d. (3.7) s+σi ( s R i ()λ(s)d) ds.

10 10 Oscillaion of soluions o delay differenial equaions EJDE 2000/13 Hence hen ( ( σi s+σi R i ()) λ(s) ds d λ(s)( σ i s R i ()) λ(s) ds d σ i From (3.7) and (3.8), i follows ha Hence λ()( +σi σi R i ()d) ds. +σi λ()( R i (s) ds) d. (3.8) σi +σi R i (s) ds)d λ() R i (s) ds d +σi R i ()(ln e R i (s) ds) d. (3.9) λ()( N σ i ( R i ())(ln e +σi +σi On he oher hand, by Lemma 1.4, we have +σi R i (s) ds)d (3.10) R i () ds) d. R i (s) ds < 1, i =1,...,p (3.11) evenually. hen by (3.10) and (3.11), we find +σi λ()d ( R i ())ln(e R i () ds) d N σ i or ln y(n σ i) +σi ( R i ())ln(e R i () ds) d. (3.12) y(n) In view of (H6) we have his implies ha lim p y( σ i ) y() =. (3.13) y( σ p ) lim =. (3.14) y()

11 EJDE 2000/13 El M. Elabbasy, A. S. Hegazi, & S. H. Saker 11 However by Lemma 1.3, we have lim inf y( σ p ) y() < his conradics (3.14) and complees he presen proof. herefore, every soluion of (1.2) oscillaes. References [1] C. Qian and G. Ladas, Oscillaion in differenial equaions wih posiive and negaive coefficiens, Canad. Mah. Bull. 33 (1990), [2] R. G. Kopllaadze and. A. Canurija, On oscillaory and monoonic soluion of firs order differenial equaions wih deviaing argumens, Differenialnye Uravenenija 18 (1982), , in Russian. [3] 3 A. R. Myshkis, Linear homogeneous differenial equaions of firs order wih deviaing argumens, Uspehi Ma. Nauk 5 (1950), , in Russian. [4] G. Ladas, V. Lakshmikanham and L. S. Papadakis,Oscillaion of higherorder rearded differenial equaions generaed by he rearded argumens, in delay and funcional differenial equaions and heir applicaions, Academic Press, New Your, [5] G. Ladas, Sharp condiions for oscillaion caused by delays, Appl. Anal. 9 (1979), [6] A. Elber and I. P. Savrolakis, Oscillaion and non-oscillaion crieria for delay differenial equaions, Proc. Amer. Mah. Sco., Vol. 124, no. 5 (1995) [7] B. Li. Oscillaions of delay differenial equaions wih variable coefficiens, J. Mah. Anal. Appl. 192 (1995), [8] Y. Domshlak and I. P. Savrolakis, Oscillaion of firs-order delay differenial equaions in a criical sae, Applicabl Analysis., Vol. 61 (1996) [9] Y. Domshlak and I. P. Savrolakis, Oscillaion of differenial equaions wih deviaing argumens in a criical sae, Dynamic Sysems of Appl. 7 (1998), [10] Y. Domshlak and I. P. Savrolakis. Oscillaion ess for delay equaions, Rocky Mounain Journal of Mahemaics, Vol. 29, no. 4 (1999), [11] B. G. Zhang and K. Gopalsamy, Oscillaion and non-oscillaion in a nonauonomous delay logisic model, Quar. Appl. Mah. 46 (1988), [12] I. Gyori and G Ladas, Oscillaion heory of Delay Differenial Equaions Wih applicaions, Clarendon Press, Oxford(1991).

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13 EJDE 2000/13 El M. Elabbasy, A. S. Hegazi, & S. H. Saker 13 [29] El. M. Elabbasy, S. H.Saker and E. H. Al-Shemas, Oscillaion of nonlinear delay differenial equaions wih applicaion o models exhibiing he Allee effec, Far Eas Journal of Mahemaical Sciences, Vol. 1 (1999), no. 4, [30] El. M. Elabbasy, S. H. Saker, Oscillaion of nonlinear delay differenial equaions wih several posiive and negaive coefficiens, Kyungpook Mahemaical Journal, Vol. 39 (1999), no. 2, [31] El. M. Elabbasy, S. H. Saker and K. Saif, Oscillaion in hos macroparasie model wih delay ime, Far Eas Journal of Applied Mahemaics, Vol. 3 (1999), no. 2, [32] El. M. Elabbasy, S. H. Saker, Oscillaion of soluions of delay differenial equaions, M. Sc. hesis, Mansoura Universiy, Egyp, (1997). [33] W. K. Ergen, Kineecs of he circulaing fuel neuclar reacion, Jornal of Applied Physics, 25 (1954), [34] N. Macdonald, Biological delay sysems: Linear sabiliy heory, Cambridge Universiy Press (1989). [35] P. I. Wangersky and J. W. Conningham, ime lag in prey-predaor populaion models, Ecology 38 (1957), [36] K. L. Cooke and J. A. Yorke, Some equaions modeling growh processes and gonorrhea epidemic, Mah. Biosc. 16 (1973), ElM.Elabbasy,A.S.Hegazi,&S.H.Saker Mahemaics Deparmen, Faculy of Science Mansoura Universiy Mansoura, EGYP mahsc@mum.mans.eun.eg