International Journal of Trend in Scientific Research and Development (IJTSRD) International Open Access Journal ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 2 | Issue – 3 Oscillation off Even Order Nonlinear Neutral Differential Equations of E G. Pushpalatha Assistant Professor, Department of Mathematics, Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Namakkal, Tamilnadu, India S. A. Vijaya Lakshmi Research Scholar, Department of Mathematics, Vivekanandha College of Arts and Sciences Science for Women, Tiruchengode, Namakkal, Tamilnadu, India ABSTRACT This paper presently exhibits about the oscillation of even order nonlinear neutral differential equations of E of the form π(π‘)π§ ( ) (π‘) + π(π‘)π(β πΎ(π‘) + π£ π£(π‘)π πΏ(π‘) =0 Where π§(π‘) = π₯(π‘) + π(π‘)π₯ π(π‘) , π ≥ 2, is a even integer. The output we considered ∫ π (π‘)ππ‘ = ∞, (πΈ ) π ∈ πΆ 1([π‘ , ∞), πΈ), πΎ ∈ πΆ ([π‘ , ∞), πΈ), πΏ ∈ πΆ([π‘ , ∞), πΈ), π′ (π‘) ≥ π > 0, πΎ(π‘) ≤ π‘, πΏ(π‘) ≤ π‘ , π β πΎ = πΎ β π , π β πΏ = πΏ β π, lim →∞ πΎ(π‘) = ∞ , lim constant. (π‘) →∞ πΏ( = ∞ , where π is a and ∫ π (π‘)ππ‘ < ∞. This canon here extracted enhanced and developed a few known results in literature.. Some model are given to embellish our main results. π(π₯)⁄π₯ ≥ π , π > 0, for π₯ ≠ 0 , where π , π is constant. INTRODUCTION Then the two cases are We apprehensive with the oscillation theorems for the following half-linear even order neutral delay differential equation ∫ π(π‘)π§ ( 0, π‘ ≥ π‘ ) , ′ (π‘) + π(π‘)π(β πΎ(π‘) + π£((π‘)π πΏ(π‘) = (1) Whereπ§(π‘) = π₯(π‘) + π(π‘)π₯ π(π‘) , π ≥ 2, is a even integer .Every part of this paper, we assume that that: (πΈ ) π ∈ πΆ([π‘ , ∞), πΈ), π(π‘) > 0, π′(π‘) ≥ 0; (πΈ ) π, π ∈ πΆ([π‘ , ∞), πΈ), 0 ≤ π(π‘) ≤ π < ∞, π(π‘) > 0, where π is a constant; (πΈ ) π ∈ πΆ(πΈ, πΈ) and ∫ ∞ ( ) ∞ ( ) ππ‘ = ∞ (2) ππ‘ < ∞, (3) By a solution π§ of (1) a function be π∈πΆ ([π‘ , ∞), πΈ) for some π‘ ≥ π‘ , ( ) π(π‘) , has a property Where π§(π‘) = π₯(π‘) + π(π‘)π₯ 1 ππ§ ∈ πΆ ([π‘ , ∞), πΈ) and satisfies (1) on (π‘ , ∞). { π‘): π‘ ≥ π|} > 0 for all Then (1) satisfies π π’π{|π₯)π‘ π ≥ π‘ is called oscillatory. In certain case when π = 2 the equation (1) lessen to the following equations @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr Apr 2018 Page: 632 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 (π(π‘) π₯(π‘) + π(π‘)π₯ π(π‘) π£(π‘)π πΏ(π‘) = 0, π‘ ≥ π‘ ′ ′ + π(π‘)π(β πΎ(π‘) + , The main results which covenant that every solution of (1) is oscillatory π(π‘) ≤ π‘ , πΎ(π‘) ≤ π‘, πΏ(π‘) ≤ π‘, 0 ≤ π(π‘) ≤ π < ∞. (1) lim Then the oscillatory behavior of the solutions of the neutral differential equations of the second order (π(π‘) π₯(π‘) + π(π‘)π₯ π(π‘) π(π )ππ + lim πππ Main results (π‘)ππ‘ = ∞, ′ ′ →∞ πππ ∫ ( ) π½(π )ππ (2)lim π π’π + π(π‘)(β πΎ(π‘) + (5) ( ) B.Theorem. 2.1 Assume that ∫ (π )ππ = ∞, ∞ ( ) ∞ 0 ≤ π(π‘) ≤ π < ∞. πΎ(π‘) ≤ π‘, πΏ(π‘) ≤ π‘, 0 ≤ π(π‘) < 1, are not assumed. π Could be a advanced argument and πΎ, πΏ could be a delay argument , Some known expand results are seen in [1,5]. Then the Even-order nonlinear neutral functional differential equations (n) (π₯(π‘) + π(π‘)π₯ π(π‘) )′ π£(π‘)π πΏ(π‘) = 0, π‘ ≥ π‘ + π(π‘)π(β πΎ(π‘) + (6) , Where n is even 0 ≤ π(π‘) < 1 and π(π‘) ≤ π‘. (A) Lemma. 1 The oscillatory behavior of solutions of the following linear differential inequality π€ ′ (π‘) + π(π‘)β(πΎ)) + π£(π‘)π(πΏ(π‘)) ≤ 0 Where π, π£, πΎ, πΏ ∈ πΆ([π‘ , ∞)), πΎ(π‘)π‘ , πΏ(π‘) ≤ π‘ lim πΎ(π‘) = ∞ , lim πΏ(π‘) = ∞ →∞ →∞ Now integrating from πΎ(π‘) to π‘ and πΏ(π‘) π‘π π‘ ππ‘ = ∞ holds. If ∞ π (π‘) + The usual limitations on the coefficient of (5) be π(π‘) ≤ π‘, πΎ(π‘) ≤ π(π‘), πΏ(π‘) ≤ π(π‘), > π½(π )ππ > 1 →∞ π£(π‘) πΏ(π‘) = 0, π‘ ≥ π‘ Where ∫ π 1 π£(π )ππ > , →∞ →∞ π ( ) ( ) Then it has no finally positive solutions. lim πππ (4) Where ∫ π If π (π‘) ππ‘ = ∞ {π(π‘), π(π(π‘))} Where π (π‘)= min π (π‘)= min{π£(π‘), π£(π(π‘))}, then every solutions of (1) is oscillatory. Proof Suppose, on the contradictory,π₯ is a nonoscillatory solutions of (1). Without loss of generality, we may assume that there exists a constant π‘ ≥ π‘ , such that π₯(π‘) > 0, π₯(π(π‘)) > 0 and (πΎ(π‘)) > 0 , π₯(πΏ(π‘)) > 0 for all π‘ ≥ π‘ . Using the definitions of π§ and x is a eventually positive solution of (1). Then there exists π‘ ≥ π‘ , such that π§(π‘) > 0, π§ ′ (π‘) > 0, π§ ( all π‘ ≥ π‘ . .Hencelim →∞ π§(π‘) ) (π‘) > 0 and π§ (π‘) ≤ 0 for ≠ 0. Applying ( πΈ ) and (1) we get π(π‘)π§ ( ) π(π‘)π§ ( ) ′ (π‘) ≤ −π π(π‘)β πΎ(π‘) < 0, π‘≥π‘ ′ (π‘) ≤ −π π£(π‘)π πΏ(π‘) < 0, π‘ ≥ π‘ . Therefore (π(π‘)π§ ( ) (π‘)) is a nonincreasing function. Besides, from the above inequality and the definition of π§, we get @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 633 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Case (1) π(π‘)π§ ( ′( ) π(π‘)π§ ( ) ′ ′ (π‘) + π π(π‘)π§ πΏ(π‘) + (π π(π‘) π§ ( (π‘) + π π(π‘)β πΎ(π‘) + (π π(π‘) π§ ( π(π‘) + π(π )π§ ( ) (π(π )π§ ( ) (10) (π ))′ππ (7) +π + (π ))′ππ π π (π π(π ) π§ ( ) π ∫ π(π )π§ πΏ(π ) ππ ≤ − ∫ π(π )π§ ( π (π )π§ πΎ(π ) ππ π + π ≤0 π(π )π§ πΏ(π ) ππ ′ π(π ) ππ ≤ 0 Pointing that π′(π‘) ≥ π > 0 +π (π π(π ) π§ ( ) (π(π )))′ ∫ ) (π π(π ) π§ ( ) ) (π ))′ππ − (π(π )))′ ππ π(π(π )) ≤ (π(π‘ )π§ ( ) π‘ −(π(π‘)π§ ( ) (π‘)) + (π π(π‘ ) π§ ( ) (π(π‘ ) − π(π )π§ ( π (π )π§ πΎ(π ) ππ ≤ − 1 (π π(π ) π§ ( ′ π (π ) ′( ππ Pointing that π′(π‘) ≥ π > 0 ) ) (π ))′ππ (π(π )))′ ππ π(π(π ) π π(π‘) π§ ( ) (11) π(π‘) Since π§ ′ (π‘) > 0 for π‘ ≥ π‘ . We can find a invariable π > 0 such that π§ πΏ(π‘) ≥ π, π‘ ≥ π‘ . ≤ π(π‘ )π§ ( ) π‘ −(π(π‘)π§ ( (π π(π‘ ) π§ ( (π(π(π‘))π§ ( ) ) (π‘)) + ) (π(π‘ ) − (8) (π(π‘))) Since π§ ′ (π‘) > 0 for π‘ ≥ π‘ . We can find a invariable π > 0 such that Then from (8) and the fact that (π(π‘)π§ ( ) (π‘)) is ∞ nonincreasing, we obtain ∫ π (π‘) < ∞ (12) We get inconsistency with (9),(12). ∞ Then from (8) and the fact that (π(π‘)π§ ( increasing, we obtain ∞ ∫ π (π‘) < ∞ ) (π‘)) is non (9) ) (π‘))′ + π π£(π‘)π πΏ(π‘) π ′ + ′ (π π(π‘) π§ ( ) π(π‘) π (π‘) + π π π£((π(π‘))π(πΏ((π(π‘))) ≤ 0 π (π‘) = ∞ Theorem. 2.2 Assume that ∫ ∞ ( ) ππ‘ = ∞ holds and π(π‘) ≥ π‘. if either lim Case (2) ( ∞ π (π‘) + π§ πΎ(π‘) ≥ π, π‘ ≥ π‘ . (π(π‘)π§ ′ ) Integrating (7) from π‘ π‘π π‘, we have π − π π(π‘) Integrating (10) from π‘ π‘π π‘, we have (π(π‘)π§ (π‘))′ + π π (π‘)π§ πΎ(π‘) + (π π(π‘) π§ ( ) (π(π‘)))′ ≤ 0 π ) ′ ) π π π((π(π‘))β(πΎ((π(π‘))) ≤ 0 ( ) ∫( →∞ ) πππ(∫ ( ) ( ) ( ) ππ ) ( ( )) ( ) ( ) ππ ( ( )) ( )( > + )! , (13) Or when πΎ , πΏ is increasing, @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 634 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 lim ∫( →∞ π π’π(∫ ( ) ( ) ( ) ππ ) ( ( )) ) ( ) ( ) ππ ( ( )) ( )( > π₯(π‘) . π¦(π‘) ≤ 1 + + )! , (14) (16) By combining (15) and (16), we get Where (π‘) = min π π(π‘), π π π(π‘) , ( ) ( ) π¦ ′ (π‘) + then every solution of (1) is oscillatory . ( ( ) ( ) ( ) Proof )! π¦ πΏ(π‘) ( ) π¦ πΎ(π‘) + (17) ≤0 Suppose, on the contrary π₯ is a oscillatory solution of (1). Without loss of generality, we may assume that there exists a constant π‘ ≥ π‘ , such that t π₯(π‘) > 0, Therefore , π¦ is a non negative solutions of (17). π₯(π(π‘)) > 0 and π₯(πΎ(π‘)) > 0 , π₯(πΏ(π‘)) > 0 for all π‘≥π‘ . Case (1) Assume that π₯ ( ) (π‘) is not consonantly zero on any interval [π‘ , ∞), and there exists a π‘ > π‘ . Such ( ) ( ) (π‘)π’ (π‘) ≤ 0 thatπ₯ for all π‘≥π‘ . If lim →∞ π₯(π‘) ≠ 0,then for every π, 0 < π < 1, there existsπ ≥ π‘ , such that for all π‘ ≥ π, lim Then there will be two cases If ∫( π₯(π‘) ≥ ( )! π‘ π’ ) (t) forever ∫( 0 < π < 1 , we obtain (π(π‘)π§ ( ) πΎ ( )! ( πΏ )! (π‘))′ + (π π(π‘) π§ (π‘)π½(π‘)π§ ( ) (π‘)πΎ(π‘)π§ ( ( ) ′ π(π‘) + (πΎ((t)) + ) (πΎ((t)) ≤ 0, For every π‘ sufficiently large. Let π₯(π‘) = π(π‘)π§ ( enough, we have π π₯(π‘) + π₯ π(π‘) π ) (π‘) > 0. Then for all π‘ large ) πππ(∫ ( ) ( ) ( ) ππ ) ( ( )) ( ) ( ) ππ ( ( )) ( )( + )! > holds then a constant be 0 < π < 1, such that lim ( →∞ →∞ πππ(∫ ( ( ) ( ) ) ( ) )( ( )! ( ) ( ) ( ) ππ + ππ )) > , (18) By lemma (1), (18) holds that (17) has negative solutions, which is contradictory. Case (2) By the definition of π¦ and ( π(π‘)π§ ( ) (π‘))′ + π π (π‘)π§ πΎ(π‘) + (π π(π‘) π§ ( ) (π(π‘)))′ ≤ 0 we get , ′ π πΎ (π‘)π½(π‘) + π₯ πΎ(π‘) (π − 1)! π(πΎ(π‘)) π πΏ (π‘)πΎ(π‘) + π₯ πΏ(π‘) ≤ 0 (π − 1)! π(πΏ(π‘)) (15) Next , let us denote π¦(π‘) = π₯(π‘) + π₯ π(π‘) . Since π₯ is non increasing, it follows from π(π‘) ≥ π‘ that π¦ ′ (π‘) = π₯ ′ (π‘) + π₯ π(π‘) ′ ≤ −π½(π‘)π§ πΎ(π‘) − πΎ(π‘)π§(πΏ(π‘)) < 0 (19) pointing that πΎ(π‘) ≤ π‘, πΏ(π‘) ≤ π‘, there exists π‘ ≥ π‘ , such that π¦ πΎ(π‘) ≥ π¦(π‘), π¦ πΏ(π‘) ≥ π¦(π‘), π‘ ≥ π‘ . (20) Integrating (17) from πΎ(π‘)π‘π π‘ and πΏ(π‘) to π‘ and applying πΎ , πΏ is increasing, we have @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 635 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Or when π ππΎ , π ππΏ is increasing, π¦(π‘) − (π¦ πΎ(π‘) + π¦ πΏ(π‘) ) + ( ∫( ) ( ) ( ) ∫ )! ( ) ( ) ( ) ( ) ( π¦ πΎ(π ) + lim π¦ πΏ(π ) )ππ ≤ 0, π‘ ≥ π‘ . ( ) ∫ Thus )! ( ) ( ) ((π¦ πΎ(π‘) ∫ ( π¦ πΏ(π‘) ) ∫ ( ) ( ) ( ) ) (π¦ πΎ(π‘) + π¦(πΏ(π‘))) (π )π½(π ) π π πΎ + π + π (π − 1)! ( ) π πΎ(π ) (π )πΎ(π ) πΏ + ) ππ ≤ 0 π πΏ(π ) ( ) From (20), we have ( )! ∫( ) ( ) +∫( ( ) ( ) ) ( ) ) ππ ≤ ∫( →∞ π π’π(∫ ( ) ( ) ( ) ππ ) ( ( )) ) ( ) ( ) ππ ( ( )) ( )( ′ π₯(π‘) + ( )! ( )! π¦(π‘) ≤ (1 + ≤ )! , ∫( π π’π(∫ ( ( ) ( ) ) ( ) ( ) ( ) ππ ) > ( ) ( ππ + )( , ∫ ( ) πππ(∫ (25) )π₯ π(π‘) ( )! π¦ π ( ) πΏ(π‘) π¦ π πΎ(π‘) + (26) ≤0 If ∞ ( ) ππ‘ = ∞ holds andπΎ(π‘) ≤ π(π‘) ≤ If π‘, πΏ(π‘) ≤ π(π‘) ≤ π‘. →∞ again. Since π₯ is non Case (1) Theorem 2.3 lim π₯ π(π‘) Therefore, π¦ is a positive solution of (26). Now, we consider the following two cases , on (23) and (24) holds . )! Which is in contrary with (22). Assume that ∫ πΏ(π‘) ≤ 0. Let ( ) ( ) π¦ ′ (π‘) + ( ) ( ) ( ( )) ) πΎ(π‘) + By combining (15) and (24), we get (22) 0 < π < 1 such that →∞ ( ) ( ) π₯ ( ( )) ( ) ( ) π₯ ( ( )) + + If (14) holds, we choose a constant lim π₯ π(π‘) increasing, it follows fromπ(π‘)) ≤ 0 that Taking upper limits as π‘ → ∞ in (21) we get lim (πΎ(π‘)) > 0π₯(πΏ(π‘)) > 0 for all π‘ ≥ π‘ . Continuing as in the proof of the theorem (2.2), we have π¦(π‘) = π₯(π‘) + (21) 1,π‘ ≥ π‘ (24) π₯(π‘) > 0, π₯(π(π‘)) > 0 and −1 ( ) ( ) )! Suppose, on the contrary π₯ is a oscillatory solution of (1). Without loss of generality, we may assume that there exists a constant π‘ ≥ π‘ , such that ) ππ ≤ 0, π‘ ≥ π‘ From the above the inequality,we get π¦(π‘) ( ) + Proof + ( ) ( ) ( ) ( ) ππ ( ) ( ( )) ( ) ( ) ( )( ππ ) > ( ( )) π π’π(∫ Where π½, πΎ is defined as in theorem (2.2), then every solution of (1) is oscillatory. π¦(π‘) − (π¦ πΎ(π‘) + π¦(πΏ(π‘))) + ( →∞ ( ) ( ) ( ) ππ ) ( ( )) ( ) ( ) ππ ( ( )) ( )( > either + lim ∫( →∞ ) πππ(∫ ( ) ( ) ( ) ππ ) ( ( )) ( ) ( ) ππ ( ( )) ( )( > + )! holds then a constant be 0 < π < 1, such that )! , (23) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 636 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 lim →∞ πππ(∫ ( )( )! ( ) ( ) ( ( ) From (28) we get ππ + πΏπ‘π‘πΏπ−1π πΎπ ππΏπ ππ ))>1π (27) ( Therefore (27) holds (26) has no positive solutions which is a contradiction. π‘ ≥ π‘ , such that π¦ π (29) πΎ(π‘) )! πΎ(π‘) (28) ( ) ( ) π¦ π πΎ(π‘) + π−1πΏ(π‘)π‘πΏπ−1π πΎ(π )π(πΏπ )π¦π−1πΏπ‘ππ ≤ 0 , π‘≥π‘2. Thus π¦(π‘) − π¦ π πΎ(π‘) π¦ π )! + πΎ(π‘) ∫ ( ) ( ) ( ( )) ( ) + π¦π−1πΏπ‘π−1πΏ(π‘)π‘πΏπ−1π πΎ(π )π(πΏπ )ππ ≤0, π‘≥π‘2. From the inequality , we obtain π¦(π‘) π¦ π πΎ(π‘) π π π + π (π − 1)! πΏ + ( ) πΎ ( ) ) ( ) ( ) ππ ( ( )) + Reference + ( ) ( ) ∫ π π’π(∫ ( Then the proof is similar to that of the theorem (2.2) then it is contradiction to (24). ≥ π¦(π‘) , Integrating (26) from π πΎ(π‘) toπ‘, π πΏ(π‘) and applying π ππΎ is nondecreasing, then we get π¦(π‘) − π¦ π → (30) ≥ π¦(π‘) π‘ ≥ π‘ . πΏ(π‘) −1 + πΏ(π‘)π‘πΏπ−1π πΎ(π )π(πΏπ )ππ )≤(π0+π0)(π−1)!ππ0, πΎ(π‘) ≤ π(π‘), πΏ(π‘) ≤ π(π‘), there exists ( ( ) ( ) ( ( )) π−1πΏ(π‘)π‘πΏπ−1π πΎ(π )π(πΏπ )ππ ≤1, π‘≥π‘2. lim From (19) and the condition ( ( ) Taking the upper limit as π‘ → ∞ in (29), we get Case (2) π¦ π ∫ )! (π )π½(π ) π(πΎ(π )) 1) M.K.Grammatikopoulos, G. Ladas, A. Meimaridou, Oscillation of second order neutral delay differential equations, Rat. Mat. 1 (1985) 267-274. 2) J.Dzurina, I.P.Stavroulakis, Osillation criteria for second order neutral delay differential equations, Appl.Math.Comput. 140 (2003) 445-453. 3) Z. L. Han, T. X. Li, S. R. Sun, Y.B. Sun, Remarks on the paper [Appl. Math. Comput. 207(2009) 388-396], Appl. Math. Comput. 215 (2010) 39984007. 4) Q. X. Zhang, J.R. Yan, L. Gao, Oscillation behavior of even – order nonlinear neutral delay differential equations with variable coefficients, Comput. Math.Appl. 59 (2010) 426-430. 5) C. H.Zhang, T. X. Li, B. Sun, E. Thandapani, On the oscillation of hogher- order half-linear delay differential equations, Appl.Math. Lett. 24 (2011) 1618-1621. (π )πΎ(π ) ππ ≤ 0 . π(πΏ(π )) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 637