Oscillation of Even Order Nonlinear Neutral Differential Equations of E

1. International Research Research and Development (IJTSRD) International Open Access Journal f Even Order Nonlinear Neutral Differential Equations of E International Journal of Trend in Scientific Scientific (IJTSRD) International Open Access Journal ISSN No: 2456 ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume 6470 | www.ijtsrd.com | Volume - 2 | Issue – 3 Oscillation of Even Order Nonlinear Neutral Differential G. Pushpalatha Department of Mathematics, Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Namakkal, Tamilnadu, India S. A. Vijaya Lakshmi Vijaya Lakshmi Department of Mathematics, Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Namakkal, Tamilnadu, India Assistant Professor, Department of Mathematics, Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Namakkal, Tamilnadu, India Research Scholar, Department of Mathematics, Vivekanandha College of Arts and Science Women, Tiruchengode, Namakkal, Tamilnadu, India ABSTRACT This paper presently exhibits about the even order nonlinear neutral differential equations of E of the form (??) ? ∈ ?1([??,∞),?), ? ∈ ? exhibits about the oscillation of ([??,∞),?), even order nonlinear neutral differential equations of ? ∈ ?([??,∞),?), ?′(?) ≥ ?? > 0, ?(?) ≤ ?, ?(?) ≤ ? , ? ∘ ? = ? ? ∘ ? , ? ??(?)?(???)(?)? + ?(?)?(ℎ??(?)? + ? = 0 ?(?)???(?)? ? ? ∘ ? = ? ∘ ?, Where ?(?) = ?(?) + ?(?)???(?)?,? ≥ integer. The output we considered ∫ ? and ∫ ???(?)?? < ∞ ?? enhanced and developed a few known results in literature. Some model are given to embellish our main results. ≥ 2, is a even ???(?)?? = ∞, ? lim?→∞?(?) = ∞ ,lim?→∞?( constant. (?) = ∞ , where ?? is a ? ?? ? . This canon here extracted here extracted (??) ? ∈ ?(?,?) and enhanced and developed a few known results in . Some model are given to embellish our ⁄ ≥ ??,??> 0, for ? ? ≠ 0 , where ??,?? 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 We apprehensive with the oscillation theorems for the ? ?(?)?? = ∞ (2) ∫ neutral delay ∞ ?? ? ?(?)?? < ∞ , (3) ∫ ′ ??(?)?(???)(?)? 0,? ≥ ??, (1) + ?(?)?(ℎ??(?)? + ?( (?)???(?)? = ? By a solution ? of (1) a function be of (1) a function be ? ∈ ????([??,∞),?) for some for some ??≥ ??, Where?(?) = ?(?) + ?(?)???(?)?,? ≥ integer .Every part of this paper, we assume that 2, is a even ? we assume that: Where ?(?) = ?(?) + ?(?)?? ?????∈ ?1([??,∞),?) and satisfies Then (1) satisfies ???{|?)? ? ≥ ?? is called oscillatory. ( ) ??(?)?, has a property and satisfies (1) on (??,∞). { ?):? ≥ ?|} > 0 for all (??) ? ∈ ?([??,∞),?),?(?) > 0,?′(?) ≥ ≥ 0; (??) ?,? ∈ ?([??,∞),?), In certain case when ? = 2 the equation (1) lessen to the following equations the equation (1) lessen to 0 ≤ ?(?) ≤ ??< ∞,?(?) > 0, where ?? ? is a constant; @ IJTSRD | Available Online @ www.ijtsrd.com @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Apr 2018 Page: 632

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 ′ If ′? + ?(?)?(ℎ??(?)? + (?(?)??(?) + ?(?)???(?)? ?(?)???(?)? = 0,? ≥ ??, ? ? ?(?)?? >1 ?(?)?? + lim ?→∞??? ? Then it has no finally positive solutions. lim ?→∞??? ? ?, ?(?) ?(?) (4) Main results ? Where ∫ ???(?)?? = ∞ ?? , The main results which covenant that every solution of (1) is oscillatory ?(?) ≤ ? ,?(?) ≤ ?,?(?) ≤ ?, 0 ≤ ?(?) ≤ ??< ∞. ? ? ? (1) lim?→∞??? ∫ ?(?)?? > ?(?) Then the oscillatory behavior of the solutions of the neutral differential equations of the second order ? (2)lim ?→∞???? ?(?)?? > 1 ?(?) ′ ′? + ?(?)(ℎ??(?)? + (?(?)??(?) + ?(?)???(?)? ?(?)??(?)? = 0,? ≥ ?? (5) B.Theorem. 2.1 ∞ ?? ? Assume that ∫ ?(?)?? = ∞ holds. If ? Where ∫ ???(?)?? = ∞ ?? , ∞ ∞ ? ??(?) ?? = ∞ ?? ? ??(?) + ?? 0 ≤ ?(?) ≤ ??< ∞. The usual limitations on the coefficient of (5) be ?(?) ≤ ?,?(?) ≤ ?(?),?(?) ≤ ?(?), Where ??(?)= min{?(?),?(?(?))}, then every solutions of (1) is oscillatory. ??(?)= min {?(?),?(?(?))} ?(?) ≤ ?,?(?) ≤ ?,0 ≤ ?(?) < 1, are not assumed. Proof ? Could be a advanced argument and ?,? could be a delay argument , Suppose, on the contradictory,? is a nonoscillatory solutions of (1). Without loss of generality, we may assume that there exists a constant ??≥ ??, such that Some known expand results are seen in [1,5]. Then the ?(?) > 0,?(?(?)) > 0 and (?(?)) > 0 , Even-order nonlinear neutral functional differential equations ?(?(?)) > 0 for all ? ≥ ??. Using the definitions of ? and x is a eventually positive solution of (1). Then there exists ??≥ ??, such that (?(?) + ?(?)???(?)?)′(n) ?(?)???(?)? = 0,? ≥ ??, (6) + ?(?)?(ℎ??(?)? + ?(?) > 0,?′(?) > 0,?(???)(?) > 0 and ??(?) ≤ 0 for all ? ≥ ??. Where n is even 0 ≤ ?(?) < 1 and ?(?) ≤ ?. (A) Lemma. 1 .Hencelim?→∞?(?) ≠ 0. The oscillatory behavior of solutions of the following linear differential inequality Applying ( ??) and (1) we get ′ ?′(?) + ?(?)ℎ(?)) + ?(?)?(?(?)) ≤ 0 ??(?)?(???)(?)? ≤ −???(?)ℎ??(?)? < 0, ? ≥ ?? Where ?,?,?,? ∈ ?([??,∞ )), ′ ??(?)?(???)(?)? ≤ −???(?)???(?)? < 0, ? ≥ ??. ?(?)? ,?(?) ≤ ? (?(?)?(???)(?)) Therefore function. Besides, from the above inequality and the definition of ?, we get is a nonincreasing ?→∞?(?) = ∞ ,lim ?→∞?(?) = ∞ lim Now integrating from ?(?) to ? and ?(?) ?? ? @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 633

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 ′ Case (1) ??(?)?(???)(?)? ?? + ???(?)???(?)? + ′ ′(10) ??(???(?)??(???)??(?)? ??(?)?(???)(?)? + ???(?)ℎ??(?)? + ′+ ?? ?′(?)(???(?)??(???)??(?)? Integrating (10) from ?? ?? ?, we have ? ?????((?(?))ℎ(?((?(?))) ≤ 0 ? ?(?)?(???)(?))′?? ?? (?(?)?(???)(?))′ + ???(?)???(?)? + ?? ? ??(???(?)??(???)(?(?)))′≤ 0 (7) + ??? ?(?)???(?)??? ?? +?? ??? (???(?)??(???)??(?)? ?? ? Integrating (7) from ?? ?? ?, we have ′ ?? ≤ 0 ? ? (?(?)?(???)(?))′?? ?? Pointing that ?′(?) ≥ ??> 0 ? ? ? ∫ ?(?)?(???)(?))′?? ?? ?? ?(?(?)) + ??? ?(?)???(?)??? ?? +?? ??? (???(?)??(???)(?(?)))′ ?? ≤ 0 ??∫ ?(?)???(?)??? ≤ − ?? ?? ??∫ ?? − ? ? ?′(?)(???(?)??(???)(?(?)))′ ? ?? ≤ (?(??)?(???)??−(?(?)?(???)(?)) + ?? ?? ????(?)??(???)??(?)?? (11) ?(???(??)??(???)(?(??) − Pointing that ?′(?) ≥ ??> 0 ? ? ? ?(?)?(???)(?))′?? ?? ??? ?(?)???(?)??? ≤ − ?? −?? ??? ?? Since ?′(?) > 0 for ? ≥ ??. We can find a invariable ? > 0 such that ? 1 ?′(?)(???(?)??(???)(?(?)))′ ?? ?(?(?) ???(?)? ≥ ?,? ≥ ??. ≤ ?(??)?(???)??−(?(?)?(???)(?)) + ?? ?? (?(?(?))?(???)(?(?))) (8) Then from (8) and the fact that (?(?)?(???)(?)) is nonincreasing, we (12) ∞ ?(???(??)??(???)(?(??) − obtain ∫ ??(?) < ∞ ?? We get inconsistency with (9),(12). Since ?′(?) > 0 for ? ≥ ??. We can find a invariable ? > 0 such that ∞ ∞ ? ??(?) = ∞ ?? ? ??(?) + ?? ???(?)? ≥ ?,? ≥ ??. Theorem. 2.2 Then from (8) and the fact that (?(?)?(???)(?)) is non increasing, we obtain ∞ ?? ? Assume that ∫ either ?(?)?? = ∞ holds and ?(?) ≥ ?. if ∞ ∫ ??(?) < ∞ ?? (9) ????(?)?(?) ?(?(?)) ? lim?→∞ ???(∫ ????(?)?(?) ?(?(?)) ?(?) ?? + Case (2) ?(?) ? (?????)(???)! ?? , (13) ??) > ∫ (?(?)?(???)(?))′ + ???(?)???(?)? ?′(?)(???(?)??(???)??(?)? + ?????((?(?))?(?((?(?))) ≤ 0 ? ?? ′ + Or when ? ,? is increasing, @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 634

4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 ????(?)?(?) ?(?(?)) ?(?) ≤ ?1 +?? ? ????(?) . (16) lim?→∞ ???(∫ ????(?)?(?) ?(?(?)) ?(?) ?? + ?(?) ? (?????)(???)! ?? , (14) ??) > ∫ By combining (15) and (16), we get Where (?) = min????(?),?????(?)??, (???)!?????(?)?(?) ?? ? ?′(?) + ???(?)? + ???(?)? ????? then every solution of (1) is oscillatory . ????(?)?(?) ???(?)? ???(?)?? ≤ 0 (17) Proof Therefore , ? is a non negative solutions of (17). Suppose, on the contrary ? is a oscillatory solution of (1). Without loss of generality, we may assume that there exists a constant ??≥ ??, such that t ?(?) > 0, Then there will be two cases Case (1) ?(?(?)) > 0 and ?(?(?)) > 0 , ?(?(?)) > 0 for all ? ≥ ??. If ????(?)?(?) ?(?(?)) ? Assume that ?(?)(?) is not consonantly zero on any interval [??,∞), and there exists a ??> ??. Such that?(???)(?)?(?)(?) ≤ 0 lim?→∞?(?) ≠ 0,then for every ?, 0 < ? < 1, there exists? ≥ ??, such that for all ? ≥ ?, lim?→∞ ???(∫ ????(?)?(?) ?(?(?)) ?(?) constant be 0 < ??< 1, such that ?? + ?(?) ? (?????)(???)! ?? holds then a ??) > ∫ for all ? ≥ ??. If ? (???)!(????(?)?(?) ? ?, (18) ? ?? lim?→∞ ???(∫ ?? + ?(?) ???(?)? ? (???)!?????(???)(t) forever ?(?) ≥ ????(?)?(?) ???(?)? ? ??)) > ∫ ?(?) 0 < ? < 1 , we obtain By lemma (1), (18) holds that (17) has negative solutions, which is contradictory. ′+ (?(?)?(???)(?))′ + ?? ??(???(?)??(???)??(?)? ? Case (2) (???)!????(?)?(?)?(???)(?((t)) + By the definition of ? and ? (???)!????(?)?(?)?(???)(?((t)) ≤ 0, ( ?(?)?(???)(?))′ + ???(?)???(?)? + ?? ??(???(?)??(???)(?(?)))′≤ 0 For every ? sufficiently large. Let ?(?) = ??(?)?(???)(?)?> 0. Then for all ? large enough, we have we get , ′ ′ ?′(?) = ?′(?) +?? ?(?)?(?(?)) < 0 pointing that ?(?) ≤ ?,?(?) ≤ ?, there exists ?? ≥ ??, such that ??(?) +?? ??????(?)?? ≤ −?(?)???(?)? − ?????(?)?? (19) ????(?)?(?) ?(?(?)) ????(?)?(?) ?(?(?)) ? ???(?)? + (? − 1)! ? (? − 1)! ???(?)? ≥ ?(?),???(?)? ≥ ?(?),? ≥ ?? . (20) ???(?)? ≤ 0 + (15) Integrating (17) from ?(?)?? ? and ?(?) to ? and applying ? ,? is increasing, we have Next , let us denote ?(?) = ?(?) +?? it follows from ?(?) ≥ ? that ?????(?)?. Since ? is non increasing, @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 635

5. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Or when ????? ,????? is increasing, ?(?) − (???(?)? + ???(?)?) + ? (???)!∫ ???(?)? ?(?) ????(?)?(?) ???(?)? ???(?)? ????(?)?(?) ? ?? ( ???(?)? + ????(?)?(?) ?(?(?)) ? lim?→∞ ???(∫ ?? + ????? ? ?(?) ????(?) )?? ≤ 0,? ≥ ?? . ∫ ????(?)?(?) ?(?(?)) ? (?????)(???)! ?? (24) ??) > ∫ ????(?) Thus Where ?,? is defined as in theorem (2.2), then every solution of (1) is oscillatory. ?(?) − (???(?)? + ?(?(?))) + ?? ????? ???(?)?) ∫ ?(?) ????(?)?(?) ???(?)? ? ? Proof (???)!((???(?)?∫ ? + ?(?) ????(?)?(?) ???(?)? Suppose, on the contrary ? is a oscillatory solution of (1). Without loss of generality, we may assume that there exists a constant ??≥ ??, such that ?? ≤ 0,? ≥ ?? ) From the above the inequality,we get ?(?) > 0,?(?(?)) > 0 and ?(?) (???(?)? + ?(?(?)))− 1 (?(?)) > 0?(?(?)) > 0 for all ? ≥ ??. Continuing as in the proof of the theorem (2.2), we have ? ????(?)?(?) ???(?)? ?? ? + (? − 1)!? ????(?)?(?) ???(?)? ??+ ?? ? ?(?) ′ ??(?) +?? ?????(?)?? + ?? ≤ 0 + ? ) ?(?) ????(?)?(?) ?(?(?)) ????(?)?(?) ?(?(?)) ? ???(?)? + (???)! ? (???)! From (20), we have ???(?)? ≤ 0. Let ????(?)?(?) ???(?)? ????(?)?(?) ???(?)? ? ? ?? ? + ∫ ) ?? ≤ (???)!∫ ?(?) ?(?) ????? 1,? ≥ ?? (21) ?(?) = ?(?) +?? increasing, it follows from?(?)) ≤ 0 that ?????(?)? again. Since ? is non Taking upper limits as ? → ∞ in (21) we get ?(?) ≤ (1 +?? ??)???(?)? (25) ????(?)?(?) ?(?(?)) ? lim?→∞ ???(∫ ????(?)?(?) ?(?(?)) ?(?) ?? + ?(?) By combining (15) and (24), we get ? (?????)(???)! ??? , (22) ??) ≤ ∫ (???)!?????(?)?(?) ?? ? ?′(?) + ???????(?)?? + If (14) holds, we choose a constant ???(?)? ????? ????(?)?(?) ?(?(?)) 0 < ??< 1 such that ???????(?)??? ≤ 0 (26) ????(?)?(?) ???(?)? ? lim?→∞ ???(∫ ?? + ?(?) Therefore, ? is a positive solution of (26). Now, we consider the following two cases , on (23) and (24) holds . ????(?)?(?) ???(?)? ? (?????)(???)! ???? , ??) > ∫ ?(?) Which is in contrary with (22). Case (1) Theorem 2.3 If ????(?)?(?) ?(?(?)) ? ∞ ?? ? lim?→∞ ???(∫ ????(?)?(?) ?(?(?)) ?(?) constant be 0 < ??< 1, such that ?? + Assume that ∫ ?,?(?) ≤ ?(?) ≤ ?. ?(?)?? = ∞ holds and?(?) ≤ ?(?) ≤ If ????(?)?(?) ?(?(?)) ?? + (?????)(???)! ?? ?(?) ? (?????)(???)! ?? either holds then a ??) > ∫ ? ? lim?→∞ ???(∫ ????(?) ????(?)?(?) ?(?(?)) ? , (23) ??) > ∫ ????(?) ? @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 636

6. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 From (28) we get (???)!(????(?)?(?) ? ?? lim?→∞ ???(∫ ?????−1????????))>1? (27) ?? + ?(?) ???(?)? ????(?)?(?) ?(?(?)) ? ?? ? + (???)!∫ ????(?) ????? ?−1?(?)???−1??(?)?(??)??≤1, ?≥?2. (29) Therefore (27) holds (26) has no positive solutions which is a contradiction. Taking the upper limit as ? → ∞ in (29), we get Case (2) ????(?)?(?) ?(?(?)) ? lim?→? ???(∫ ?? + From (19) and the condition ?(?) ?(?)???−1??(?)?(??)??)≤(?0+?0)(?−1)!??0, (30) ?(?) ≤ ?(?),?(?) ≤ ?(?), there exists Then the proof is similar to that of the theorem (2.2) then it is contradiction to (24). ??≥ ??, such that ???????(?)?? ≥ ?(?) , ???????(?)?? ≥ ?(?) ? ≥ ??. (28) Reference Integrating (26) from ?????(?)?to?,?????(?)? and applying ????? is nondecreasing, then we get 1)M.K.Grammatikopoulos, Meimaridou, Oscillation of second order neutral delay differential equations, Rat. Mat. 1 (1985) 267-274. G. Ladas, A. ?(?) − ???????(?)?? + ????(?)?(?) ???(?)? ? ?? ? 2)J.Dzurina, I.P.Stavroulakis, Osillation criteria for second order neutral delay differential equations, Appl.Math.Comput. 140 (2003) 445-453. ???????(?)?? + (???)!∫ ????(?) ????? ?−1?(?)???−1??(?)?(??)??−1????≤ 0 , ?≥?2. 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) 3998- 4007. Thus ?(?) − ???????(?)?? + ????(?)?(?) ?(?(?)) ? ?? ? (???)!???????(?)??∫ + ????(?) ????? ??−1???−1?(?)???−1??(?)?(??)??≤0, ?≥?2. 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. From the inequality , we obtain 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. ?(?) ???????(?)?? ? ????(?)?(?) ?(?(?)) ?? ? − 1 (? − 1)!? ????(?)?(?) ?(?(?)) ??+ ?? ? ????(?) ?? ≤ 0 . + ? ????(?) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018 Page: 637