OSCILLATION BEHAVIOUR OF FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS (Gelagat Ayunan bagi Persamaan Pembezaan Tunda Neutral Peringkat Pertama)

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1 Journal of Qualiy Measuremen and Analysis Jurnal Pengukuran Kualii dan Analisis JQMA () 5, 6-67 OSCILLATION BEHAVIOUR OF FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS (Gelaga Ayunan bagi Persamaan Pembezaan Tunda Neural Peringka Perama) AHMAD, FATIMA N. AHMED, UMMUL KHAIR SALMA DIN & MOHD SALMI MD. NOORANI ABSTRACT In his paper, we obain some sufficien condiions for he oscillaion of all soluions of firs order neural delay differenial euaions. The resuls aained improve and generalise some of exising resuls in he lieraure. Some examples are presened o illusrae our resuls. Keywords: oscillaion; nonoscillaion; neural delay differenial euaions; firs order ABSTRAK Dalam makalah ini, diperoleh beberapa syara cukup unuk ayunan bagi penyelesaian persamaan pembezaan unda neural peringka perama. Hasil yang diperoleh dapa menambah baik dan mengilak beberapa kepuusan yang sedia ada dalam susasera. Beberapa conoh dikemukakan bagi menjelaskan hasil ersebu. Kaa kunci: ayunan; bukan ayunan; persamaan pembezaan unda neural; peringka perama. Inroducion A neural delay differenial euaion (NDDE) is a differenial euaion in which he highes order derivaive of he unknown funcion appears in he euaion boh wih and wihou delays. Recenly, increasing numbers of invesigaions have been carried ou in sudying he oscillaion of NDDEs Agarwal e al. (4), Candan and Dahiya (9), Gyori and Ladas (99), Karpuz and Ocalan (8), Zhou (999) sudied oscillaion crieria, while Parhi and Rah (), Gopalsamy and Zhang (99), Tanaka (999; ), Yu e al. (99) focused on he nonoscillaory soluions. The sudy of oscillaory behaviour of soluions of NDDEs has some imporan applicaions, such as neworks conaining lossless ransmission lines, modelling of he ransformaion of informaion, and he heory of auomaic conrol (Driver 984; Hale 977; Sficas & Savroulakis 987; Ocalan 9; Erbe e al. 995). Consider he firs order NDDE of he form where r ()( x () px ( τ) + ) + x () ( σ) =,, () [ ) ( ) [ r, ] C,,,, + p, τ, σ. () The oscillaory soluions of () have been invesigaed by a number of researchers and some sufficien condiions for he oscillaory and nonoscillaory soluions have been obained (Grammaikopoulos e al. 986; Kubiaczyk & Saker ; Saker & Elabbasy ; Graef e al. 986). Le m = max{ τ,σ}. By he soluion of (), a funcion x C [ m ),, for some such ha x () + px ( τ ) is coninuously differeniable and () is idenically saisfied

2 Ahmad, Faima N. Ahmed, Ummul Khair Salma Din & Mohd Salmi Md. Noorani for. Such a soluion of () is said o be oscillaory if i has arbirarily large zeros and nonoscillaory if i is evenually posiive or evenually negaive. The main objecive of his aricle is o give some new sufficien condiions for he oscillaory soluions of (). Wihou loss of generaliy, we will deal only wih he posiive soluions of (). We presen some of he well known lemmas, which will be needed in he proof of our main resuls. They may also have furher applicaions in he analysis. For he proofs see Chuanxi and Ladas (989), and Gyori and Ladas (99). Lemma.. Le be such ha [ ) f, g:,, ( ) = ( ) + + { } f g pg( c), max, c, where p, c, and p. Assume ha lim f( ) l exiss. Then he following saemens hold: () If liminf g ( ) a, () If limsup g ( ) b, hen l = ( + ) p a. hen l = ( + ) p b. Lemma.. Assume ha ρ is a posiive consan. Le [ ) Then liminf h( s) ds. > e ρ h C,, +, and suppose ha () he delay differenial ineualiy x ( ) + h( ) x( ρ ), has no evenually posiive soluion, () he advanced differenial ineualiy x ( ) h( ) x( ρ ), has no evenually posiive soluion. Lemma.3. Assume ha () s ds =, (3) holds and le x () be an evenually posiive soluion of NDDE () + ( ) + () ( ) = ;, (4) ( x px τ ) x σ where [ ) p p C +,,,, and τ (, ), + σ. 6

3 Oscillaion behaviour of firs order neural delay differenial euaions Le z () = x () + px ( τ ) and w () = z () + pz ( τ ). Then () z () is a decreasing funcion and eiher lim z ( ) =, (5) or lim z ( ) =. (6) () The following saemens are euivalen (i) (5) holds, (ii) p <, (iii) lim x ( ) =, (iv) w() >, w () > and w () >. (3) The following saemens are euivalen: (i) (6) holds, (ii) p >, (iii) lim x ( ) =, (iv) w() >, w () < and w () >.. Main Resuls In his secion, we give some new sufficien condiions for all soluions of () o be oscillaory. Theorem.. Consider he euaion () where r () r posiive consan and p =. Assume ha (3) holds. Then every soluion of () is oscillaory. Proof: Assume, for he sake of a conradicion, ha () has an evenually posiive soluion x ( ) >, >. Le z() = x () x ( τ ). Then x () ( σ ) z ( ) = <. r Hence for all, we have z () > or z ( ) <. Le z () >. This implies ha ()( s x s σ ) ds < r z( ) <. (7) 63

4 Ahmad, Faima N. Ahmed, Ummul Khair Salma Din & Mohd Salmi Md. Noorani On he oher hand, z () > gives x () > x ( τ ) and hence liminf x ( ) >. Thus, here exiss a posiive consan, k such ha x () > k>. Then which leads o () x( σ ) d > k () d, σ σ σ () x( σ ) d =. This is a conradicion wih (7). Therefore z () <, which implies x () < x ( τ ). Then x () is bounded and hence liminf x ( ) and liminf z ( ) exis. From Lemma., we ge lim z ( ) =. This conradics he fac ha z () is a negaive and monoonic decreasing funcion. The proof is compleed. Example.. Consider he NDDE Here we have x ( ) x ( ) + e x ( ) =, >. (8) ( π ) e r π π ( ) =, =, = and =. τ π σ Then all he hypohesis of Theorem. are saisfied, where d e d ( ) = =. Hence every soluion of (8) is oscillaory. Theorem.. Assume ha () holds wih p, r () ( σ τ) >, r( + p) e hen every soluion of () is oscillaory. rposiive consan, () > and Proof: Assume, for he sake of conradicion ha () has an evenually posiive soluion x ( ) >, >. Le z() = x () + px ( τ ) and w () = z () + pz ( τ ). By direc subsiuion where z() and w() are soluions of (), p and r are consans, we have rz ( ) + prz ( τ) + z( σ) =. (9) 64

5 Oscillaion behaviour of firs order neural delay differenial euaions From Lemma.3 we have w () >, w is evenually increasing and rw () + prw ( τ) + w( σ) = () Since z() is decreasing and eiher lim z ( ) = or lim z ( ) =, i is claimed ha w ( τ ) w ( ). Hence, and so r( + p) w ( τ) + w( σ) w + r( + p) ( τ) w ( σ). Leing = + τ, we obain w () + w ( ( σ τ) ) r( + p) if + p > () or w () w ( + ( τ σ) ) r( + p) if + p < () In view of Lemma. (i) and (ii), and also he condiion (9), i is impossible for () and () o have evenually posiive soluions. This conradics he fac ha w () >. Thus, he proof is compleed. Example.. Consider he NDDE e 3π x ( ) + x ( π ) + x ( ) =, >. 4 3πe 3πe (3) Here we have e p =,, r, τ π and σ 3π 3πe = 3πe = 4 = = Then all he hypohesis of Theorem. are saisfied, where 3 ( ) 3π e π π σ τ =. r( p) e π = > + e e + 4 3π e Hence every soluion of (3) is oscillaory. 65

6 Ahmad, Faima N. Ahmed, Ummul Khair Salma Din & Mohd Salmi Md. Noorani Remark.. Theorem. is an exen for Theorems 3 and 4 in Ladas and Sficas (986) and Theorem 6..3 in Gyori and Ladas (99). Theorem.3. Assume ha () and (3) hold, < p < and s () liminf ds, > rs ( σ ) e σ hen every soluion of () oscillaes. Proof: Assume, for he sake of conradicion ha () has an evenually posiive soluion x ( ) >, >. Le z() = x () + px ( τ ). Then by Lemma.3, we obain z( ) >. As x () > z (), i follows from () ha r()z() + ()z( σ ) ( ) Dividing he las ineualiy by r ( ) >, we obain r () () z ( ) + z( ) + z( σ ). (5) r () r () r ( s) ds r( s) Le z() = e y (). This implies ha y ( ) >. Subsiuing in (5) yields, for all ( ) y ( ) + y( σ ),. (6) r( σ ) In view of Lemma. (i) and (4), i is impossible for (6) o have an evenually posiive soluion. This conradics he fac ha y () > and he proof of Theorem.3 is compleed. (4) Example.. Consider he NDDE as follows, e π x ( ) x + x( π) =, > π. 5 π (7) Here we have e π r () =, () =, < p= <, τ = and σ = π. π 5 Then all he hypohesis of Theorem.3 are saisfied where liminf σ s () 5π ds = liminf ds = > rs ( σ ) e. π Hence every soluion of (7) is oscillaory. Remark.. Theorem.3 is an exen for Theorem 7 in Ladas and Sficas (986). 66

7 Oscillaion behaviour of firs order neural delay differenial euaions Acknowledgemens This research has been compleed wih he suppor of hese grans: FRGS//3/SG4/ UKM//3 and DLP-4-. References Agarwal R.P., Bohner M. & Li W.T. 4. Nonoscillaion and Oscillaion: Theory for Funcional Differenial Euaions. New York: Marcel Dekker. Candan T. & Dahiya R. S. 9. Posiive soluions of firs order neural differenial euaions. Appl. Mah. Le. : Chuanxi Q. & Ladas G Oscillaions of neural differenial euaions wih variable coefficiens. Appl. Anal. 3: 5-8. Driver R.D A mixed neural sysem. Nonlinear Anal. 8: Erbe L.H., Kong Q. & Zhang B.G Oscillaion Theory for Funcional Differenial Euaions. New York: Marcel Dekker. Gopalsamy K. & Zhang B.G. 99. Oscillaion and nonoscillaion in firs order neural differenial euaions. J. Mah. Anal. Appl. 5: Grammaikopoulos M.K., Grove E.A. & Ladas G Oscillaions of firs order neural delay differenial euaions. J. Mah. Anal. Appl. : 5-5. Graef J.R., Grammaikopoulos M.K. & Spikes P.W On he behavior of soluions of a firs order nonlinear neural delay differenial euaion. Appl. Anal. 4: -. Gyori I & Ladas G. 99. Oscillaion Theory of Delay Differenial Euaions. Oxford Mahemaical Monographs. New York: Clarendon Press. Hale J.K Theory of Funcional Differenial Euaions. New York: Springer. Karpuz B. & Ocalan O. 8. Oscillaion crieria for some classes of linear delay differenial euaions of firs order. Bull. Ins. Mah. Acad. Sin. 3(): Kubiaczyk I. & Saker S.H.. Oscillaion of soluions of neural delay differenial euaions. Mah. Slovaca 5: Ladas G. & Sficas Y.G Oscillaions of neural delay differenial euaions. Canad. Mah. Bull. 9: Ocalan O. 9. Exisence of posiive soluions for a neural differenial euaion wih posiive and negaive coefficiens. Appl. Mah. Le. : Parhi N. & Rah R.N.. On oscillaion and asympoic behaviour of soluions of forced firs order neural differenial euaions. Proc. Indian Acad. Sci.(Mah. Sci.) : Saker S.H. & Elabbasy E.M.. Oscillaion of firs order neural delay differenial euaions. Kyungpook Mah. J. 4: 3-3. Sficas Y.U. & Savroulakis I.P Necessary and sufficien condiions for oscillaions of neural differenial euaions. J. Mah. Anal. Appl. 3: Tanaka S.. Oscillaion of soluions of firs order neural differenial euaions. Hiroshima Mah. J. 3: Tanaka S Exisence of posiive soluions for a class of firs order neural funcional differenial euaions. J. Mah. Anal. App. 9: Yu J.S., Wang Z.C. & Chuanxi Q. 99. Oscillaion of neural delay differenial euaions. Bull. Ausral. Mah. Soc. 45: 95-. Zhang B.G Oscillaion of firs order neural funcional differenial euaions. J. Mah. Anal. Appl. 39: Zhou Y The disribuion of zeros of neural differenial euaions. Hiroshima Mah. J. 9: Pusa Pengajian Sains Maemaik Fakuli Sains dan Teknologi Universii Kebangsaan Malaysia 436 UKM Bangi Selangor DE, MALAYSIA rozy@ukm.edu.my*, zahra8zahra@yahoo.com, ummul@ukm.edu.my, msn@ukm.edu.my * Corresponding auhor 67