 Advances in Pure Mathematics, 2011, 1, 49-53 doi: 10.4236/apm.2011.13011 Published Online May 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM New Results on Oscillation of even Order Neutral Differential Equations with Deviating Arguments Lianzhong Li1, Fanwei Meng2 1School of Mathematical and System Sciences, Taishan College, Tai’an, China 2School of Mathematical Sciences, Qufu Normal University, Qufu, China E-mail: llz3497@163.com , fwmeng@qfnu.edu.cn Received January 19, 2011; revised March 15, 2011; accepted Ma rc h 20, 2011 Abstract In this paper, we point out some small mistakes in  and revise them, we obtain some new oscillation re-sults for certain even order neutral differential equations with deviating arguments. Our results extend and improve many known oscillation criteria because the article just generalizes Meng and Xu’s results. Keywords: Oscillation, Neutral Differential Equation, Deviating Argument 1. Introduction Oscillation of some even order differential equations have been studied by many authors. For instance, see [1-7] and the references therein. We deal with the oscil-latory behavior of the even order neutral differential eq-uations with deviating arguments of the form   1100,nmiiiljj jjxtp txtqtfx ttt (1) where 2n is even, throughout this paper, it is as-sumed that: (A1) 0,,,,,,0ijj jpqC tRfCRR ufu  for 0uand jfuis non-decreasing on R, 1, 2,,,1, 2,,;imjl (A2)0,, ,iiCtRttt and lim ,itt 1, 2,,im; (A3)1,, ,,limjjjtCtRt tt  and 0jt, 1, 2,,;jl (A4) There exists a constant M0 such that  sgn Mjfxx x for 0,1,2, ,xjl; (A5)  1,0,1miipt pp and there exists a func- tion 0,,qtC tRsuch that, min :jqtq t 1, 2,,jl. By a solution of Equation (1) we mean a function xtwhich has the pro perty that  1miiixtptxt ,,nxCt R for some 0xtt and satisfies Equa-tion (1) on ,xt. We restrict our attention to those solutions xt of Equation (1) which exist on some half-line ,xt with sup :0xttT for any xTt. A nontrivial solution of Equation (1) is called oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. Equation (1) is said to be os-cillatory if all of it’s nontrivial solutions are oscillatory. Recently, Meng and Xu  studied Equation (1) and obtained some sufficient conditions for oscillation of the Equation (1), we list the main results of  as follows. Following Philos , we say that a function ,HHts belongs to a function class W, denotes by HW, if ,HCDR, where0,:Dtstst, which satisfies: (H1) ,0Htt and ,0Hts for 0tst; (H2) H has a continuous non-positive partial derivative HS satisfying the condition:  ,,,Htsk shts HtsSks for some ,lochL DR10,,,0,kCt is a non-decreasing function. Theorem A ([6, Theo re m 2.1]). Assume that (A1) - (A5) hold, let the functions ,,Hhksatisfy (H1) and (H2), suppose  121limsup ,,4tCFtrGtrC (2) L. Z. LI ET AL. Copyright © 2011 SciRes. APM 50 holds fo r every 01 2,0,0,rtC C where      1221,,d,,,1,d,,ltjrjtnrjjFtrH tsk sqssHtrkshtsGtr sHtr Htss s and 1p , then every solution of Equation (1) is oscillatory. Theorem B ([6, The orem 2.2]). Assume that (A1)-(A5) hold, and ,,Hhk are the same as in Th eorem A, suppose that 00,inf 0,liminfst tHtsHtt (3) and 0limsup ,tGtt  (4) If there exists a function 0,,mCt R such that for all 0tTt.  121,,4liminftCFtTGtTm TC (5) and  022limsupd,1,2,,ntjjttmsjlkssss(6) where max ,0mtmt, then every solution of Equation (1) is oscillatory. In Theorem A and B, function ,Gtr should be ,jGtr, so each of the condition (2), (4), (5) and (6) has as many as l conditions. Meanwhile, the Riccati func-tion t is not well-defined and there exist some small errors in the proof of the theorems. The purpose of this paper is further to strengthen oscillation results ob-tained for Equation (1) by Meng and Xu . In our paper, we redefine the functions ,,,,tr tFrGt and provide some new oscillation criteria for oscillation of Equation (1). 2. Main Results In the sequel, we need the following lemmas: Lemma 2.1 (). Let xt be a n times differentiable function on 0,t of one sign, 0nxt on 0,t which satisfies  () 0nxtxt. Then: (I1) There exists a 10tt such that ,1,2,,1ixti n are of one sign on 1,t; (I2) There exists a number 1, 3, 5,,1hn when n is even, or 2,4,6, ,1hn when n is odd, such that 0ixtxt for 0,1, ,ih,1;tt  110ni ixtx tfor 11,2,,,ihhn tt. Lemma 2.2 (). If xt is as in Lemma 2.1 and 10nnxxtt for 0tt, then for every 01, there exists a constant 0N, such that 11nnNt txt X for all large t. Lemma 2.3(). Suppose that xt is an eventually positive solution of Equation (1), let  1miiizx pxttt t, then there exists a number 10tt such that 0,zt 10, 0nzztt and 10,nttzt. Theorem 2.1 Assume that (A1) - (A5) hold, let the functions ,,Hhk satisfy (H1) and (H2), suppose  limsup ,,4tMFtrGt rN (7) holds fo r every 0rt and for some 1, where        2211,,d,,1,d,,trtlrnjjjFtrHtsksqs sHtrksh tsGtr sHtr Htsss and 1p, then every solution of Equation (1) is oscillatory. Proof. Suppose to the contrary that xt is a nonoscillatory solution of Equation (1) and that xt is even- tually positive (when xt is eventually negative, the proof is similar). Let zt be defined as in Lemma 2.3, then following the proof of Theorem 2.1 in , without loss of generality, assume there exists a 10tt such that  (1)0, 0,0,0, 0njjxt ttttzzztzz  12nnjjtNtzt z  (by lemma 2.2) and  1lnjjjttzMqzt  for all 1tt. Let  11nljjzkztttt (not as ), then we have L. Z. LI ET AL. Copyright © 2011 SciRes. APM 51      2112,,lnjjjNkMk qtttttt tttkt kt t    (not as ). Multiplying the above equation, with t replaced by s, by ,Hts and integrating it from T to t, for all 1tTt, for some 1, we obtain                 2122212212121,, ,,,,,d, dd,d d41,4lnjjtttjTTTlnjjttjlTTnjjjlnjjjlnjjjMHkqsHtTTh sNHskNHkhHtT TsskNHssts s stsstssstss ssts sstss sts sNH khssststss skNH     2221d,,,,d4tTtlTnjjjsksshHt tsts sTT sNH s Hence, we have  ,,4MFtTGtT TN for all 1tTt, this gives  limsup ,,4tMFt rGtrN which contradicts (7). This completes the proof of the Theorem. The assumption (7) in Theorem 2.1 can fail, conse-quently, Theorem 2.1 does not apply. The following re-sults provide some essentially new oscillation criteria for Equation (1). Theorem 2.2 Assume that (A1)-(A5) hold, the functions ,,,HhkF and G be the same as in Theorem 2.1, suppose that 00,inf 0,liminfst tHtsHtt (8) If there exists a function 0,,mCt R such that for all 0tTt and for some 1,  l,m,isup4tMtT tTFmTGN (9) and  0221limsup dlnjjtjttmsssssk (10) where max ,0mt mt. Then every solution of Equation (1) is oscillato ry. Proof. Assume to the contrary that (1) is non-oscil- latory. Following the proof of Theo rem 2.1, without loss of generality, assume for all 0tTt and for some 1, we obtain        221212,,,,d,d4d1tTtlTnjjjlnjjtjTtsssststss stss sssMHk qsHtTTkh sNHNHsk So, we get  1,, ,4MFtTGtT TNBtTN where    2120,,,1d,lnjjtjrHBsHktss str str srt L. Z. LI ET AL. Copyright © 2011 SciRes. APM 52 then   ,,41liminms,lupfittMFtT GtTNTNBtT For all 0Tt and for any 1, by (9) we have  1liminf ,tTmTN BtT So  TmT (11) and especially  000liminf ,(1)tBttt mtN  (12) Now, we claim that   0212limsup dlnjjtjttsssssk (13) Suppose to the contrary that   0212limsup dlnjjtjttsssssk (14) By (8), there is a positive constant  satisfying 00,inf liminf0,st tHtsHtt (15) Let  be any arbitrary positive number, from (14) there exists a 10tt such that,   0212dlnjjtjtsssssk for all 1tt Then, for 1tt, we have     101021220001120,,,,,1d1,,ddlnjjtsjttlnjjtsjttttvvtsd vtt vvvts vtt vtBHvHkHvstHkHtHt By (15), there exists a 21tt such that, for all 2tt, 10,,HttHtt, which implies 0,Btt for all 2tt. Since  is arbitrary, we have 00liminf,lim ,ttBtt Btt  which contradicts (12), thus (13) holds. Then by (11) and (13) we get     00212212limsu plimsu pdd,lnjjtjtlnjjtttjtmskskssssssss which contradicts (10). This completes the proof. Remark 1 Let 1 in Theorem 2.1, Theorem 2.1 reduces to Theorem A ; we obtain the same result in Theorem 2.2 in which we omit the assumption (4) in Theorem B . Therefore, Theorem 2.1 and 2.2 are gen-eralizations and improvements of the results obtained in . Remark 2 With an appropriate choices of the func-tions ,Hh and k, one can derive a number of oscilla-tion criteria for Equation (1) from our theorems. Let ()1,0kt is a constant, ,tssHt ,  1,,ts sht 0tst , and we have 00,,limlim 1,,ttHts tsHtt tt  for any 0st. Consequently, let 2, using Theorem 2.2, we have: Corollary 2.1 Assume that (A1)-(A5) and (8) hold, suppose that there exists a function 0,,mCt R such that, fo r some 1,  220211limsup d,ltnjjjtTMtsqs stNssmTt Tt  (16) and (10) (with 1ks) hold. Then every solution of Equation (1) is oscillatory. Example 1 Let 4,t, consider the following second order neutral differential equation   0xt ptx tqtxt (17) L. Z. LI ET AL. Copyright © 2011 SciRes. APM 53where   1,max21sin,0,2ptqtttf xx ,   44d12sinttsss, in this case 1M, Let 1, 2N, by direct calculation, we get   22221limsup d1limsup1sin12 sindcossincos 1tTttTtMt s qssNsttsssss stmTTT TT  It is easy to verify that (10) holds, therefore, Equation (17) is oscillatory by Corollary 2.1. However, we can easily find that  0201limsup limsupd,12sinttttGstts st  so condition (4) in Theorem B is not satisfied, these show that Theorem B cannot be applied to Equatio n (17 ). Obviously our results are superior to the resu lts obtained before. 3. Acknowledegments The authors are very grateful to the referee for his/her valuable suggestions. 4. References  R. P. Agarwal, S. R. Grace and D. ORegan, “Oscillation Theory for Differential Equations,” Kluwer Academic, Dordrecht, 2000.  R. P. Agarwal and S. R. Grace, “The Oscillation of High-er Order Differential Equations with Deviating Argu-ments,” Computers & Mathematics with Applications, Vol. 38, No. 3-4, 1999, pp. 185-199. doi:10.1016/S0898-1221(99)00193-5  Y. Bolat and O. 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