ISSN (e): 50 3005 Volume, 05 Issue, 07 July 05 Iteratioal Joural of Computatioal Egieerig Research (IJCER) Some Oscillatio Properties of Third Order Liear Neutral Delay Differece Equatios AGeorge Maria Selvam, M Paul Logaatha, KRRajkumar 3,3 Sacred Heart College, Tirupattur 635 60, SIdia, Dravidia Uiversity, Kuppam ABSTRACT I this paper, we establish some sufficiet coditios for the oscillatio of solutios of third order liear eutral delay differece equatios of the form a ( x p x( )) q x( ) 0 I INTRODUCTION I this paper, we cosider the third order liear eutral delay differece equatios from a x p x( )) q x( ) 0 () where N( 0) 0, 0,, 0 H a p q are positive sequeces is a oegative iteger, subject to the followig coditios,, 0 p p,,,lim lim ad H as q() t t () 0 s0 0 0 R KM ( ( s)) a ( s ) H3 limsup q( s)( p( ( s))) s 4 sa( s) 0 We set z x p x( ) wwwijceroliecom Ope Access Joural Page s0 as as () The oscillatio theory of differece equatios ad their applicatios have received more attetio i the last few decades, see [[]-[4]], ad the refereces cited therei Especially the study of oscillatory behavior of secod order equatios of various types occupied a great deal of iterest However the study of third order differece equatios have received cosiderably less attetio eve though such equatios have wide applicatios I [[5]- [0]] the authors ivestigated the oscillatory properties of solutios of third order delay differece equatios ad i [[]-[5]] Motivated by the above observatios, i this paper, we ivestigate the oscillatory behavior of solutios of equatio () Let max lim, x0 satisfyig () for all 0 By a solutio of equatio () we mea a real sequece x() which is defied for all 0 A o-trivial solutio x() is said to be oscillatory if it is either evetually positive or evetually egative; otherwise, it is ooscillatory Equatio () is said to be oscillatory if all its solutios are oscillatory II MAIN RESULTS Lemma Let x() be a positive solutio of equatio () for all 0 such x 0, x 0, ad x 0 o [, ) for some 0 The for each k with 0 < k <, there exists such that x( ) k, () x Proof From the Lagrage s Mea value theorem, we have for, for some k
such that ad hece, usig equatio (3) x x( ) xk k x 0 ad x Some Oscillatio Properties of Third Order ; for some k (3) is o-icreasig, that x( k) x( ) x x( ) x( )( ) x x( ) ( ) (4) x( ) x( ) [, ] for Now Apply Lagrage s Mea value theorem oce agai for x() o x( ) x xc () for some c such that c ad x( c) x( ) x( ) x( )( ) Hece x( ) x( ) For K (0,), we ca fid x( ) K x( ), we have From equatio (4) ad for all Hece, x( ) x for (5) x ( ) x( ) K K K K K (6) Lemma Let x() be a positive solutio of equatio (), the the correspodig sequece z() satisfies the 3 followig coditio z() > 0, z 0, z 0, z exits such that Proof We defie a fuctio H() for By Taylor s Theorem, From (8) z M, z for each M, 0 < M <, as 0 for some 0 M ( ) H ( ) z z (7) M ( ) H z ( ) z z (8) ( ) z z z z ( ) ( ) ( ) The there ( ) M ( ) H z( ) ( ) z z ( ) z z H 0 ad H( ) H H( ) 0 for every from (7) wwwijceroliecom Ope Access Joural Page
Some Oscillatio Properties of Third Order z M z M ( ) ( ) z z 0 for Theorem 3 Assume that H to ( H ) hold, the equatio () is oscillatory Proof: Suppose, if possible that the equatio () has a ooscillatory solutio Without loss of geerality suppose that x() is a positive solutio of equatio () We shall discuss the followig cases for z() (i) z 0, z 0, (ii) z 0, z 0, z 0, z 0, 3 z 0, 3 z 0, Case z 0, z 0, z 0, 3 z 0, Sice z() > 0ad z 0, the there exists fiite limits lim z Assume that k > 0 The for ay 0 k x k p x( ) Whe k p( k) m ( k) wwwijceroliecom Ope Access Joural Page 3, we have k z k Let x k p( k) m( k ) x mz Now from the equatio () we have k We shall prove that k = 0 k( p) 0 p a x p x( q x( ) a z q mz( ( s)) Summig the above iequality from to we get, a( t) z( t) m q( s) z( ( s)) t Usig the fact that z( ) k we obtai, s ( ) a z m q s z s s a z mk q s s ( z) mk q( s) Summig from to we have, a s z( s) mk q( t) s s as () ts z mk q( t) s as () ts Summig the last iequality to z( ) mk q( t) () s as ts H Thus k = 0 Moreover, the iequality, 0 x z implies lim x 0 This cotradicts Case z 0, z 0, z 0, 3 z 0, We have x() = z() - p() x( ), we obtai further x( ) z( ) p( ) x( ) z( ) p( ) x( ), we have
Some Oscillatio Properties of Third Order From equatio () we have, ( p( )) z( ) a z q x ( ) a z q p z ( ( )) ( ) a z w, z a z w a z z( ) z w( ) ( a z) a z z z( ) z z( ) w ( ) ( a ( ) z ( )) ( ) z a z ( z( )) w( ) q( p( )) z( ) a w z ( ) a ( ) Also from Lemma () with x() = z x( ) K, ( ) x z( ) K () z K for z z( ) By Lemma () Usig () ad () i (0) z( ) K z( ) z z( ) K M z( ) KM ( ) z () w q p w ( ) a ( ) Usig the iequality Ad put KM w( ) a ( ( )) V Vx Ux, U 0 4 U a x w( ), V, U, ( ) a ( ) From equatio (3) we have w a a ( ) ( ) a ( ) 4 a w (9) (4) (0) (3) wwwijceroliecom Ope Access Joural Page 4
Some Oscillatio Properties of Third Order KM a w q( p( )) 4 a we obtai Summig the last iequality from to ( ) KM ( ( s)) a ( s ) q( s)( p( ( s))) w 4 sa( s) Takig lim sup i the above iequality, we obtai cotradictio with H 3 (5) REFERENCES [] R P Agarwal, Differece ad Equatios ad Iequalities, d Editio, Marcel Dekker, New York, 000 [] R P Agarwal, M Boher, S R Grace ad D O Rega, Discrete Oscillatio Theory, Hidawi, New York, 005 [3] R P Agarwal, S R Grace ad D O Rega, Oscillatio Theory for Differece ad Futioal Differetial Equatios, Kluwer, Dordrecht, 000 [4] S H Saker, Oscillatio theory of delay differetial ad differece equatios secod ad third orders, Lambert Academic Publishig (00) [5] R P Agarwal, M F Aktas, A Tiryaki, O oscillatio criteria for third order oliear delay differetial equatios, Arch Math (Bro), 45 (009), - 8 [6] S R Grace, R P Agarwal, R Pavai, E Thadapai, O the oscillatio of criteria third order oliear fuctioal differetial equatios, ApplMathComp,0(008),0- [7] S H Saker, Oscillatio criteria of third order oliear delay differetial equatios, Mathematics Solvaca, Vol56, o4 (006) pp 433-450 [8] J Dzuria, E Thadapai ad S Tamilvaa, Oscillatory solutios to third order half-liear etural differetial equatios, Elec Jdiff Eqs, Vol 0 (0), No9, pp - 9 [9] B Baculikova ad J Dzuria, Oscillatio of third order etural differetial equatios, Math Comp Modellig, 5(00), 5 6 [0] O the Oscillatio of Third Order Liear Neutral delay Differetial Equatios, K V V seshagiri rao, P V H S Sai Kumar I JMTT Vol4, Issue December 03 [] R P Agarwal, S R Grace ad D O Rega, O the oscillatio of certai third order differece equatios, Adv Differ Eq, 3(005), 345-367 [] S R Grace, R P Agarwal ad J R Graef, Oscillatio criteria for certai third order o-liear differece equatios, Appl Aal Discrete Math, 3(009), 7-38 [3] J R Graef ad E Thadapai, Oscillatory ad asymptotic behavior of solutios of third order delay differece equatios, Fukcial Ekvac4(999), 355-369 [4] S H Saker, Oscillatio of third order differece equatios, Port Math, 6(004), 49-57 [5] S H Saker, Oscillatio asymptotic behavior of third order oliear eutral delay differece equatios, DyaSys,5(006)549-568 wwwijceroliecom Ope Access Joural Page 5