International Journal of Mathematical Archive-5(3), 2014, Available online through ISSN

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1 Iteatioal Joual of Mathematical Achive-5(3, 04, 7-75 Available olie though ISSN ON THE OSCILLATOY BEHAVIO FO A CETAIN CLASS OF SECOND ODE DELAY DIFFEENCE EQUATIONS P. Mohakuma ad A. ameh* Pofeo of Mathematic, Aaaupadaiveedu Ititute of Techoology, Viayaka Miio Uiveity, Kachipuam, Tamiladu, Idia. Seio Lectue i Mathematic, Ditict Ititute of Educatio ad Taiig, Uthamacholapuam, Salem , Tamiladu Idia. (eceived o: ; evied & Accepted o: ABSTACT I thi pape, we tudy the ocillatoy behavio fo a cetai cla of ecod ode delay diffeece equatio of the fom u+ qu σ ( = 0 a, Whee { a } { q } ae eal equece ad { } 0. a > Example ae ieted to illutate the eult. Keywod: Ocillatoy, Secod ode, Double Sequece, Delay Diffeece equatio. AMS Claificatio: 39A. (. INTODUCTION We coide the ecod ode delay diffeece equatio of the fom u+ qu σ ( = 0 a Whee i the fowad diffeece opeato i defied by + epected to the diffeece equatio (. thoughout we hall aume that the followig coditio hold. (C: { a },{ q } ae eal equece ad { a } > 0 (C: σ ( > 0 i a itege uch that lim σ ( (C3: a a = 0 = = By a olutio of equatio (. we mea a eal equece { } (. u = u u ad { a },{ q } ae eal equece. With. A olutio { u } u atifyig (. fo 0 i aid to be ocillatoy if it i eithe evetually poitive o evetually egative. Othewie it i called o-ocillatoy. Fo moe detail o ocillatoy behavio diffeece equatio we efe [-3]. Coepodig autho: A. ameh* Seio Lectue i Mathematic, Ditict Ititute of Educatio ad Taiig, Uthamacholapuam, Salem , Tamiladu Idia. Iteatioal Joual of Mathematical Achive- 5(3, Mach 04 7

2 P. Mohakuma ad A. ameh* / O The Ocillatoy Behavio Fo A Cetai Cla of Secod Ode Delay Diffeece Equatio/ IJMA- 5(3, Mach-04. MAIN ESULT I thi ectio, we peet ome ufficiet coditio fo the ocillatio of all the olutio of equatio (. Theoem: Aume the (C3 hold σ ( 0 ( ( σ lim up σ ( p + = (. = 0 + σ ( The evey olutio of equatio (. i ocillatoy. Poof: Let { u } be o-ocillatoy olutio of equatio (. without lo of geeality, we uppoe that u > 0, u σ ( > 0 ad fo fom the equatio i oiceaig thee exit a o-egative cotat k ad ka,, k > 0 Summig the iequality fo to u u k a = Lettig we have u, which i cotadictio to the fact that u < 0 fo. Sice a a u k a fo, k > 0 u i poitive. The u > 0 ad u > 0 a a Defie ω σ ( = (A au σ ( σ( + σ( = u+ uσ( a a + u σ( σ( σ( uσ( ( + σ σ( = u+ uσ( a a+ uσ( uσ( + σ( ( u ( + σ + σ σ( = u+ uσ( a a+ uσ( + a+ uσ( uσ( + σ( σ( + σ( = σ ( q + ω+ a u u I the view of (C ad (. σ( + + σ( σ( + ( u = + σ( σ( + σ ( q ω+ σ ( + a+ ( uσ ( + a = q + ω ( ω σ( + σ( + σ ( + σ( + ( σ( + That i 04, IJMA. All ight eeved 7

3 P. Mohakuma ad A. ameh* / O The Ocillatoy Behavio Fo A Cetai Cla of Secod Ode Delay Diffeece Equatio/ IJMA- 5(3, Mach-04. = ( + a σ( + σ ( σ( σ ( q ω+ + σ( σ( + a+ σ ( Thi implie that ( σ ( < σ ( q + σ ( (.3 Summig the iequality fo to we have ( ( σ ω ω σ ( q = + σ ( Lettig, we have, i view of (. that ω a, which cotadict ω > 0 complete. Theoem: Let all the aumptio of Theoem hold except the coditio (. which chaged to ( ( σ lim up ( q σ ( = = 0 + σ ( ad the poof i (.4 The evey olutio { u } of equatio (. i ocillatoy. Poof: Poceedig a i the poof of Theoem, we aume that equatio (. o-ocillatoy olutio, ay u > 0, u σ ( > 0 ad fo fom the equatio (.3 we have. ( σ ( ( σ ( q < ( ω (.5 = + σ ( = Sice ( = ( ω ω ( = = (.6 We get M ω = = Whee ( σ ( M = σ ( q ( ( ω (.7 + σ ( Lettig, we have ( M ω = The lim up = ( M ω Which cotadict the coditio (.4 Thi complete the poof. (.8 04, IJMA. All ight eeved 73

4 P. Mohakuma ad A. ameh* / O The Ocillatoy Behavio Fo A Cetai Cla of Secod Ode Delay Diffeece Equatio/ IJMA- 5(3, Mach-04. Next, we peet ome ew ocillatio eult fo equatio (. we itoduce a double equece { H( m, / m 0} Such that ( i H ( m, = 0 fo m o ( ii H ( m, > 0 fo m > o ad ( ii L H ( m, = h(m, H ( m, ; fo m o Theoem: 3 Aume that (C-(C3 hold ad let { u } be a poitive equece ad aume that thee exit a double equece H( m, / m 0 uch that { } ( ( σ lim up H (, q σ ( = (.9 H (, = 0 + σ ( h (, σ ( whee M (, = (.0 H (, σ ( + The evey olutio { u } of equatio (. i ocillatoy Poof: Let { u } be o-ocillatoy olutio of equatio (.. Let u fit aume the { } that u > 0, u σ ( > 0 ad fo I the view of (A ad (B Let u deote a γ = ad B = σ( + σ( σ( + σ( + a = q + ω ( ω σ( + σ( + σ ( + σ( + ( σ( + q a = + ω ( ω σ( + σ( + σ ( + σ( + ( σ( + H (, σ ( q H(, + γω + B( ω = = u i evetually poitive ad 04, IJMA. All ight eeved 74 { ( } H (, σ ( q [ H (, ω] = LH (, ω+ + H (, γω + B ω = = = H (, ω (,( (, ( (, ( H h Hγ ω+ + HB (, ω+ = It follow that M(, M ( H, (, = H (, ω (, H Bω+ + + = B 4 = B ( lim up (, H (, σ ( H q σ ( ω = 0 + σ (

5 P. Mohakuma ad A. ameh* / O The Ocillatoy Behavio Fo A Cetai Cla of Secod Ode Delay Diffeece Equatio/ IJMA- 5(3, Mach-04. Which clealy cotadict (.9. Thi cotadictio complete ou poof emak: By chooig vaiou pecific double equece{ H( m, } we ca deive eveal ocillatio citeia fo (. Let u coide the double equece { Hm (, } defied by H( m, = ( m, m o, Whee i a cotat. H ( m, = 0 fo m 0, H ( m, > 0 fo m > 0 ad L H ( m, 0 o m > 0 The Hece, we have the followig coollay Coollay: If ( ( σ lim up ( m q ( fo ome, σ = m = 0 + σ ( { u } of equatio (. i ocillatoy Example: Coide the delay diffeece equatio σ ( The evey olutio u + u = 0 ( E ( whee λ 0 = =, the equatio ( E i Ocillatoy. EFEENCES..P. Agawal, Diffeece Equatioad Iequalitie- Theoy, Methodad Applicatio- d editio...p. Agawal, Mati Bohe, Said. Gace, Doal O ega, Dicete Ocillatio Theoy-CMIA Book Seie,Volume, ISBN : P. Agawal,Mutafa F. Akta ad A. Tiyaki, O Ocillatio Citeia fo Thid ode Noliea Delay Diffeetial Equatio-Achivum Mathematicum(BANO- Tomu 45, -8 ( W. T. Li,. P. Agawal, Iteval Ocillatio Citical fo Secod Ode No liea Diffeetial quatio with Dampig-Comp. Math. Appl.40, 7-30 ( Joh. Geaf ad E. Thadapai, Ocillatoy ad Aymptotic Behavio of Solutio of Thid ode delay Diffeece Equatio-Fukcialaj Ekvacioj, 4, ( Savithi ad E.Thadapai, Ocillatoy Popetie of Thid ode Neutal Delay Diffeetial Equatio- Poceedig of the Fouth Iteatioal Cofeece o Dyamical Sytem ad Diffeetial Equatio- May 4-7,wilmig pp ( E.Thadapai ad B.S. Lalli, Ocillatio Citeia fo a Secod Ode Damped Diffeece Equatio-Appl. math. Lett. vol. 8, No., PP -6, ( E.Thadapai, I.Gyoi ad B.S. Lalli, A Applicatio of Dicete Iequality to Secod Ode No-liea Ocillatio.-J. Math. Aal. Appl. 86,00-08 ( E. Thadapai ad B. Selvaaj, Ocillatoy Behavio of Solutio of Thee dimeioal Delay Diffeece Sytem- adovi Mathematicki, vol. 3, 39-5 ( P. Mohakuma ad A. ameh, Ocillatoy Behaviou Of The Solutio Of The Thid Ode Noliea Neutal Delay Diffeece Equatio IJET ISSN: Vol. Iue 7, July 03 pp Souce of uppot: Nil, Coflict of iteet: Noe Declaed 04, IJMA. All ight eeved 75