163/0000775000076400007640000000000010724025731010430 5ustar juliojulio163/rath.dvi0000664000076400007640000031746410724025702012107 0ustar juliojulio; TeX output 2007.11.29:1417O! /DvipsToPDF { 72.27 mul Resolution div } def /PDFToDvips { 72.27 div Resolution mul } def /HyperBorder { 1 PDFToDvips } def /H.V {pdf@hoff pdf@voff null} def /H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def /H.S { currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF /pdf@hoff exch def HyperBorder sub /pdf@llx exch def } def /H.L { 2 sub dup /HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def /pdf@urx exch def } def /H.A { H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def } def /H.R { currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def } def systemdict /pdfmark known not {userdict /pdfmark systemdict /cleartomark get put} if ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endps:SDict begin [ /Title () /Subject () /Creator (LaTeX with hyperref package) /Author () /Producer (dvips + Distiller) /Keywords () /DOCINFO pdfmark end7 6ps:SDict begin H.S endps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (page.1) cvn H.B /DEST pdfmark endV6ठZps:SDict begin [ /Count -0 /Dest (section.1) cvn /Title (1. Introduction) /OUT pdfmark endZps:SDict begin [ /Count -0 /Dest (section.2) cvn /Title (2. Main Results) /OUT pdfmark endops:SDict begin [ /Count -0 /Dest (section.3) cvn /Title (3. Positive solution for p\(t\) = 1 ) /OUT pdfmark endVps:SDict begin [ /Count -0 /Dest (section*.1) cvn /Title (References) /OUT pdfmark endTps:SDict begin [ /Page 1 /View [ /Fit ] /PageMode /UseOutlines /DOCVIEW pdfmark endJps:SDict begin [ {Catalog} << /ViewerPreferences << >> >> /PUT pdfmark endps:SDict begin H.S endps:SDict begin 12 H.A endMps:SDict begin [ /View [/XYZ H.V] /Dest (Doc-Start) cvn H.B /DEST pdfmark endpapersize=614.295pt,794.96999pt j cmti9ElectronicN0gforxP6=0,nor$thatGisLipscÎhitzian.PHencetheresultsofthispap cmmi10n !", cmsy102,UUps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.1.1) cvn H.B /DEST pdfmark endS>ݟbu cmex10BrG(t)[y[ٲ(t)8p(t)y(!Dz(t))] O!cmsy709bٓRcmr7( 0ercmmi7n1)+q(t)G(y(h(t)))=f(t)Sq(1.1)32tohaveabGoundedpositivesolutionwhichdoGesnottendtozeroastZ5!1.zHereq[;h;߸2C([0;1);RDz)DVsuchthatq(t)0;h(t)DVand!Dz(t)areincreasingfunctionswhich aare4Elessthatnorequaltot,:andapproach1ast!1,r52C^(n1)([0;1);(0;1)),p2C^(n) h([0;1);& msbm10R),UUG2C(R;R). W*eUUneedsomeofthefollowingassumptionsinthesequel. (H1)" ThereUUexistsabGoundedfunctionFc(t)suchthatF^(n1)Q(t)=f(t). 荍 (H2)" īRj)q˷1 #'tZcmr503Xt^n2q[ٲ(t)dt<1.Z (H3)" īRj)q˷1 #'t0 7ʄdt4ȋ&fe 2r7(t)Eв=1.0 (H4)" īRj)q˷1 #'t0 7ʄdt4ȋ&fe 2r7(t)Eе<1.˳ (H5)" īRj)q˷1 #'t01갲( w133&fe 2r7(t)DīRj1 #t g(s8t)^n2q[ٲ(s)ds)dt<1.Ldps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.1.1) cvn H.B /DEST pdfmark end FRemarkT1.1.SinceUUrG(t)>0,itfollowsthat8(i)" eitherUU(H3)or(H4)holdsexclusively*.q(ii)" IfUU(H3)holdsthen(H5)implies(H2)butnotconversely*. ff< O[ 2000~ #fcmti8MathematicsSubjectClassi cation.@34C10,X34C15,34K40. Key~wordsandphrases.@OscillatoryXsolution;nonoscillatorysolution;asymptoticb0UUforx6=0,UUandGisnon-decreasing.6It'Visobviousthat(H6),(H7)and(H1)isweakerthanbGoth(H6)and(H7).In6(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end)NifweputrG(t)=1,"u!Dz(t)=t ,h(t)=tvthenitreducesto(ps:SDict begin H.S end1.4 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.4) cvn H.B /ANN pdfmark end).\W*e nd6almostunoresultwiththeNDDEu(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end)yintheliterature.]F*orexampleif!Dz(t)N=t=26and3h(t)=t=3thentheexistingresultsfailtoansweranything.0Sinceweformulate6ourresultswith(H1)anddonotassumeeither(H8)or(H9),thereforeourwork6extends,Jimproves#andgeneralizessomeoftheresultsof[ps:SDict begin H.S end19ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.d1) cvn H.B /ANN pdfmark end,ps:SDict begin H.S end49ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p1) cvn H.B /ANN pdfmark end $,ps:SDict begin H.S end59ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p2) cvn H.B /ANN pdfmark end $,ps:SDict begin H.S end69ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r1) cvn H.B /ANN pdfmark end $,ps:SDict begin H.S end79ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r2) cvn H.B /ANN pdfmark end $,ps:SDict begin H.S end89ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r3) cvn H.B /ANN pdfmark end $].1While6studying%theexistenceofapGositivesolutionofneutraldelaydi erentialequation6(ps:SDict begin H.S end1.4 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.4) cvn H.B /ANN pdfmark end)Nfor6n2, 0andT0 {=minfh(Ty·);!Dz(Ty)g.|SuppGose2C([T0|sTy];RDz).|Bya q6solution"of(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end)S_,wemeanarealvqaluedcontinuousfunctionyF2mC^(n) h([T0|s;1);RDz)6suchu0andt1C>t0>0UUsuchthatps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.2) cvn H.B /DEST pdfmark end,jFc(t)j< {forWtt1|s:Sqò(2.2)ThenUUusing(H4)we ndt2C>t1Ȳsuchthattt2ȲimpliesUUps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.3) cvn H.B /DEST pdfmark end\cZi]1@Tt<$=1wfeA2 (֍rG(s)ds<<$K18pKwfe@ߟ (֍10 n]:Sqò(2.3)F*romUU(ps:SDict begin H.S end2.2 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.2) cvn H.B /ANN pdfmark end)8andUU(ps:SDict begin H.S end2.3 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end)itUUfollowsthatfort>t3C>t2ȟps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.4) cvn H.B /DEST pdfmark endƍ{ecZi{f1@ t<$&jFc(s)j&wfe (֍JrG(s)yds<<$K18pKwfe@ߟ (֍ o10n]:Sqò(2.4)F*romUU(H5)we ndt4C>t3Ȳsuchthatt>t4ȲimpliesUUps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.5) cvn H.B /DEST pdfmark end<$RZYDyŸwfe!Ǵ (֍(n82)!iQcZisR1@nt<$1~vwfeA2 (֍rG(s)cZi 1@#Ys(u8s)n2q[ٲ(u)duds<<$K1pKwfe@ߟ (֍ o10n]:Sqò(2.5)LetKT+>[t4 andT0 Dβ=mins f!Dz(Tc);h(T)g. ThenKfortTc,H(ps:SDict begin H.S end2.4 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end)and(ps:SDict begin H.S end2.5 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.5) cvn H.B /ANN pdfmark end)hold. LetfXWL=jC([T0|s;1);RDz)bGethesetofallcontinuousfunctionswithnormkxkj=sup;qƴtT0#jx(t)j<1.qClearlyUUX7isaBanachspace.Letps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.6) cvn H.B /DEST pdfmark endԯHSZ=bou2BqC([T0|s;1);RDz):<$K3Kwfe (֍5 -(18p)u(t)1b W;Sqò(2.6)< >: Bx(Tc);5t2[T0|s;Tc];j&h ڱ(1)rO \cmmi5n 0ncmsy51 ڟʉfe5B(n2)!-fīRj41 #2Jt D1?џ&fe ޟr7(s)PĊīRjWo51 #U}ns_(u8s)^n2q[ٲ(u)G(x(h(u)))dudsLc īRjUS1 #ct&hF(s)ʉfe,r7(s)% bds;5tTV:Sqò(2.8)Cps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end44u-R.N.RAZTH,N.MISRA,P.P.MISHRA,L.N.PADHY7&EJDE-2007/163V6लFirstsweshowthatifx;y"2]ISCthenAxt+Bqy2]IS.!Insfact,foreveryx;y2]ISCand 6वtTc,UUweget3# %]CP(Ax)(t)8+(Bqy[ٲ)(t)p(t)x(!Dz(t))8+<$l4(1p)lwfe# (֍5)$cZi 81@t<$Fc(s)wfeH (֍ڵrG(s)- dsC+<$l(1)^n1lwfe$(n82)!*cZi41@0Wt<$Ew1?ޟwfeA2 (֍rG(s)ScZi]1@Y%sftѲ(u8s)n2q[ٲ(u)G(y(h(u)))duds卍p8+<$l4(1p)lwfe# (֍5)$+<$l1plwfe@ߟ (֍ o10+<$l1plwfe@ߟ (֍ o10=1:0# 6लOnUUtheotherhandfortTc,$CP(Ax)(t)8+(Bqy[ٲ)(t)<$K4(18p)Kwfe# (֍5*n\8cZi 81@t<$Fc(s)wfeH (֍ڵrG(s)- dsC+<$l(1)^n1lwfe$(n82)!*cZi41@0Wt<$Ew1?ޟwfeA2 (֍rG(s)ScZi]1@Y%sftѲ(u8s)n2q[ٲ(u)G(y(h(u)))duds卍<$K4(18p)Kwfe# (֍5*n\8 "cZi #1@B\t<$+|1 wfeA2 (֍rG(s))Hds.<$Llwfe!Ǵ (֍(n82)!(cZi21@-ܴt<$C1=hcwfeA2 (֍rG(s)QpcZi[q1@WsdV(u8s)n2q[ٲ(u)duds<$K4(18p)Kwfe# (֍5*n\<$l18plwfe@ߟ (֍ o10<$l18plwfe@ߟ (֍ o10==<$K3Kwfe (֍5 -(18p):`ލ6लHence򍍍<$3wfe (֍5+G(18p)(Ax)(t)8+(Bqy[ٲ)(t)1c6forUUtTc.qSothatAx8+Bqy"2Sforallx;y2S.BNextweshowthatAisacontractioninS.&Infact,forx;yw2+S%2andtTc,we6haveǸj(Ax)(t)8(Ay[ٲ)(t)jijp(t)fx(!Dz(t))8y[ٲ((t))gjijp(t)jjx(!Dz(t))8y[ٲ((t))jipkx8y"k:䍑6लSinceUU0"6वx(t)j!0UUask!1.BecauseSisclosed,x=x(t)2S.qF*ortTc,wehaveJʸj(Bqxk됲)(t)8(Bx)(t)j@M<$^%1Kwfe!Ǵ (֍(n82)!(ڟcZi2۷1@..t<$D41=wfeA2 (֍rG(s)RcZi\1@Wsd(u8s)n2q[ٲ(u)jG(x(h(u)))G(xk됲(h(u)))jduds:6लSince2forallt8VTV;xk됲(t);k=8V1;2:::,jDtend2uniformlytox(t)ast!1andGis6continuous,thereforerjG(x(h(u)))%G(xk됲(h(u)))j!0ask!1.CW*econcludethat6limD5k+B!1[L7j(Bqxk됲)(t)8(Bx)(t)j=0.qThisUUmeansthatBƲiscontinuous.BNext,?weyshowthatBqSisrelativelycompact.W~Itsucestoshowthatthefamily6of-functionsfBqx/:x2Sg-ײisuniformlybGoundedandequicontinuous-on[T0|s;1).6TheuniformbGoundednessisobvious.5MF*ortheequicontinuity*,0accordingtoLevitan's6result]weonlyneedtoshowthat,foranygiven5%>0;[T0|s;1)]canbGedecomposed6into nitesubintervqalsinsuchawaythatoneachsubintervqalallfunctionsofthe^ups:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark endEJDE-2007/163WTNON-OSCILLAZTORYBEHA#VIOURSXk5V6लfamilyhavechangeofamplitudelessthan.SbF*rom(H5)and(H4),Vitfollowsthat 6forUUany>0,weUUcan ndTc^ËTlargeenoughsothatFa<$l՟wfe!Ǵ (֍(n82)!dcZie1@.T<$1%wfeA2 (֍rG(s)2cZi31@פls(u8s)n2q[ٲ(u)duds<<$rKwfe (֍4 -;{E6लand!h,Ƶ "cZi #1@B\T<$Cds wfeA2 (֍rG(s),F`<<$rKwfe (֍4 -:a6लThenUUforx2SandUUt2C>t1Tc^s,-uIj(Bqx)(t2|s)8(Bx)(t1|s)j]xK=   <$ 3(1)^nOwfe!Ǵ (֍(n82)!+/cZi501@1it2<$Gl1AKwfeA2 (֍rG(s)UjcZi_j1@Z7sg(u8s)n2q[ٲ(u)G(x(h(u)))dudscZi 81@t2<$Fc(s)wfeH (֍ڵrG(s)- ds`UUO<$P(1)^nlwfe!Ǵ (֍(n82)!(cZi21@-ܴt1<$C1=hcwfeA2 (֍rG(s)QpcZi[q1@WsdV(u8s)n2q[ٲ(u)G(x(h(u)))duds+cZi 81@t1<$Fc(s)wfeH (֍ڵrG(s)- ds  CK<$Kwfe!Ǵ (֍(n82)!(ڟcZi2۷1@..t1<$D41=wfeA2 (֍rG(s)RcZi\1@Wsd(u8s)n2q[ٲ(u)duds+ "cZi #1@B\t1<$Cds wfeA2 (֍rG(s)UO+<$Llwfe!Ǵ (֍(n82)!(cZi21@-ܴt2<$C1=hcwfeA2 (֍rG(s)QpcZi[q1@WsdV(u8s)n2q[ٲ(u)duds+ "cZi #1@B\t2<$Cds wfeA2 (֍rG(s).K<4<$33wfe (֍4 -=:E6लF*orUUx2SandT*t1C0suchthat鐸j(Bqx)(t2|s)8(Bx)(t1|s)j< if*00suchthatforlar}'getps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.9) cvn H.B /DEST pdfmark end!p7rG(t)><$ٲ1Kwfeo (֍ ;Sqò(2.9)tRps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end64u-R.N.RAZTH,N.MISRA,P.P.MISHRA,L.N.PADHY7&EJDE-2007/163V6andps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.10) cvn H.B /DEST pdfmark end'%} } |cZi|1@ 70Fc(t)dt  m<1 8with&Fn1(t)=f(t):Rf(2.10)6Thenӊther}'eexistsaboundedsolutionof(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end)5Qwhichisboundedbelowbyapositive 6c}'onstant.d6Pr}'oof.V5Usingo(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end)JandLe(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end) 'weLecanget(ps:SDict begin H.S end2.4 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end)ڢ.nRestoftheproGofissimilartothat6ofUUtheTheoremps:SDict begin H.S end2.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark end. -6टps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.5) cvn H.B /DEST pdfmark endCorollary2.5.4L}'et`%(A1),j(H5),4(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end)!q,4(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end)"hold.OThenthereexistsaboundedsolutionof(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end) whichisb}'oundedbelowbyapositiveconstant.Pr}'oof.UIBy(Remarkps:SDict begin H.S end1.1 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.1.1) cvn H.B /ANN pdfmark end(i)wehaveeither(H3)holdsor(H4)holds.?If(H3)holdsthenlweproGceedasintheproofofTheoremps:SDict begin H.S end2.4 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.4) cvn H.B /ANN pdfmark end. Ontheotherhandif(H4)holdsthenfrom(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end)and(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end)weget(ps:SDict begin H.S end2.4 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end)andthenproGceedasintheproofofTheoremps:SDict begin H.S end2.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark endUUtogetthedesiredresult.ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.6) cvn H.B /DEST pdfmark end 9RemarkT2.6.IfUUin(H5)wetakerG(t)1UUthenitreducestops:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.11) cvn H.B /DEST pdfmark endWncZix1@t@t0cZi1@2δtz(u8t)n2q[ٲ(u)du<1:Nq²(2.11)ArThe abGoveconditionisrequiredforournextresultwhichfollowsfromCorollaryps:SDict begin H.S end2.5 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark endUUwhenrG(t)1.dps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.7) cvn H.B /DEST pdfmark end Corollaryw2.7.iIne}'quality>(ps:SDict begin H.S end2.11 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.11) cvn H.B /ANN pdfmark end)!(is\asucientconditionforthenthorderNDDEps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.12) cvn H.B /DEST pdfmark endRybW/еy[ٲ(t)8p(t)y(!Dz(t))bWߴn ?+q(t)G(y(h(t)))=f(t)Nq(2.12)toLhaveasolutionb}'oundedLbelowbyapositiveconstantundertheassumptions(A1),(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end)"$andp(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end)*.ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.8) cvn H.B /DEST pdfmark end RemarkT2.8.W*eUUclaimthattheconditionps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.2.13) cvn H.B /DEST pdfmark endZWscZis1@8t0un1q[ٲ(u)ds<1Nq²(2.13)蓍implies(ps:SDict begin H.S end2.11 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.11) cvn H.B /ANN pdfmark end).Itisclearthat(ps:SDict begin H.S end2.13 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.13) cvn H.B /ANN pdfmark end)!|isequivqalenttoīRj J1 #s*/(us)^n1q[ٲ(u)du<1.Let'%r2͵K(s)=cZi 1@URs?(u8s)n1q[ٲ(u)du:ThisUUimplieslim8s!1'K(s)=0.qDi erentiating,UUwegetW\K0U(s)=(n81)cZi 1@8s#(us)n2q[ٲ(u)du:ArF*romUUthisintegratingbGetweens=tands=T,weobtain? cZiI T@DCtQhK0U(s)ds=(n81)cZi T@8tcZi1@0As&(us)n2q[ٲ(u)duds:$Hence t0|s,de ne捑D1 t(t)=fsisar}'ealnumberPwK:stand!Dz(s)=tg:ps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.14) cvn H.B /DEST pdfmark end Remark.2.14.0~Thenfunction1 +8de nedabGovenistheinversenfunctionof!Dz(t).$Since!Dz(t)UUisincreasingitisone-one.Clearly1 t((t))=tUUfort>1 t(t0|s).ps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.15) cvn H.B /DEST pdfmark end Theorem2.15.L}'et(A3),(H1),(H4),(H5)hold.Thenther}'eexistsaboundedsolutionof(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end) whichisb}'oundedbelowbyapositiveconstant.}ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark end84u-R.N.RAZTH,N.MISRA,P.P.MISHRA,L.N.PADHY7&EJDE-2007/163V6Pr}'oof.V5IfkGnecessaryincrementdsuchthatdT>1++ %^2%^&fes4ɴc U.ChoGosekGpositivekGnumbers 嗍6व< Kc1K&fe Uq2Ӳ,UUh=(c81)andH=d1+ l2l&feQic .qThenH>h>0.qSetgP?S=maxcfjG(x)j:hxHg:6लF*romUU(H4)and(H5)onecan ndT*>0suchthattTimplies+,DžcZidž1@U1 Qϱ(t)<$ٵFc(s)ٟwfeH (֍ڵrG(s)> <<$rKwfe (֍2 -;J獑6लand[C<$# wfe!Ǵ (֍(n82)!cZi1@@U1 Qϱ(t)<$1owfeA2 (֍rG(s)|cZi}1@ܒs}b(u8s)n2q[ٲ(u)duds<<$rKwfe (֍2 -:1N6लDe ne 4SZ=fy[ٲ(t)2X:hy(t)HA;tT0|sg:M6लDe neYRAx(t)=\( SAx(Tc);ift2[T0|s;Tc];܍ x(1 Qϱ(t)) !fe!%4ɴp(1 Qϱ(t))0 ٴcd1l&fe p(1 Qϱ(t))'~+ #1l&fe p(1 Qϱ(t))&īRj-,1 #+e1 Qϱ(t)&hENF(s)ENʉfe,r7(s)Vεds;iftTV:Z6Bqx(t)=\( SBx(Tc);̲if#xt2[T0|s;Tc];k&h1q(1)rn ʉfe9(n2)!]p(1 Qϱ(t))HīRjOF÷1 #MT1 Qϱ(t) k1f&fe ޟr7(s)wϟīRj~vz1 #|s_(u8s)^n2q[ٲ(u)G(y(h(u)))duds;̲if#xtTV:!6लW*eUUshowthatifx;y"2S,thenAx8+Bqy2S.qF*orUUtTweobtainGr\4*M AAx8+Bqy"=<$f1֟wfe'N? (֍p(1 t(t))H`h 8x(1 t(t))cZi 81@1 Qϱ(t)<$!l4Fc(s)!l4wfeH (֍ڵrG(s)6Mds+(cd1)م+<$l(1)^n1lwfe$(n82)!*cZi41@0W1 Qϱ(t)<$O 1IqwfeA2 (֍rG(s)\-(cZi 1@:sx(u8s)n2q[ٲ(u)G(y(h(u)))du)ds`i's(<$K1Kwfe (֍Vc -b 8h+<$䅵lwfe (֍2 '+<$䅵lwfe (֍2+8(cd1)b(=<$K1Kwfe (֍Vc -bX+28+c(d1)bIJ=(d1)+<$l2lwfe  (֍]cm(HA:Dr[6लF*urther,G*5 vSOC(Ax8+Bqy"=<$M1ͽwfe'N? (֍p(1 t(t))O/`h@8x(1 t(t))cZi 81@1 Qϱ(t)<$!l4Fc(s)!l4wfeH (֍ڵrG(s)6Mds+(cd1)مj+<$l(1)^n1lwfe$(n82)!*cZi41@0W1 Qϱ(t)<$O 1IqwfeA2 (֍rG(s)]~cZig1@cfspQd(u8s)n2q[ٲ(u)G(y(h(u)))duds`i'sa<$1Kwfe4r (֍d ab|8H޸<$䅵lwfe (֍2 '+(cd1)<$䅵lwfe (֍2 Gb卍a=<$1Kwfe4r (֍d abd(c81)<$33(c+2)33wfeS؟ (֍ c>bЍa>c81=h:D)6लThusKAx%7+Bqy"2S.nNextweshowthatAisacontractioninS.Infactforx;y"2S 6लandUUtTwehave gy[kAx8Ay[ٸkj<$ZR133wfe'N? (֍p(1 t(t)))jjx(1 t(t))8y(1 t(t))j<$K1Kwfe (֍Vc -kxyk: ցps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endJps:SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark endEJDE-2007/163WTNON-OSCILLAZTORYBEHA#VIOURSXk9V6लHenceꎵAisacontractionbGecause,O0jt< 1&fes4ɴc7<1.NextweproveBjiscompletely 6continuousbasintheproGofofTheorenps:SDict begin H.S end2.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark end.DvThenbyLemmaps:SDict begin H.S end2.19ps:SDict begin H.R endrps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.1) cvn H.B /ANN pdfmark endthereisa xedpGoint6वx0ȲinUUSsuchthat 77ŝAx0S+8Bqx0C=x0|s:PЍ6लW*ritingеx0>W=y[ٲ(t)andmultiplyingbGothsidesoftheaboveequationbyp(1 t(t))6weUUobtain,&܍ ܍TI'p(1 t(t))y[ٲ(t)u=y[ٲ(1 t(t))8+cZi 81@1 Qϱ(t)<$!l4Fc(s)!l4wfeH (֍ڵrG(s)6Mds(cd1)5U>+<$P(1)^nlwfe!Ǵ (֍(n82)!(cZi21@-ܴ1 Qϱ(t)<$Me1GDwfeA2 (֍rG(s)[dcZied1@`=sm(u8s)n2q[ٲ(u)G(y(h(u)))duds:$ 6लThennwereplacetby!Dz(t),usethefactthat1 t((t))=tnand nallywithsome6rearrangementUUofterms,obtain%NȍBB+y[ٲ(t)8p(t)y(!Dz(t))=8cZi 81@t<$Fc(s)wfeH (֍ڵrG(s)/D+8(cd1)C+<$l(1)^n1lwfe$(n82)!*cZi41@0Wt<$Ew1?ޟwfeA2 (֍rG(s)ScZi]1@Y%sftѲ(u8s)n2q[ٲ(u)G(y(h(u)))duds:":6लFirstdi erentiatingtheabGoveequationonceandthenmultiplyingbGothsidesbyrG(t)6and_ nallydi erentiatingn<1_times,weseethat,x0fҲistherequiredsolutionof6(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end)Kn,UUwhichisbGoundedbelowbyapGositiveconstant.6टNps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.16) cvn H.B /DEST pdfmark endTheorem2.16.L}'et(A3),&(H3),(H5),?t(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end)ͱ,?t(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end)#hold.Thenther}'eexistsab}'oundedsolutionof(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end) whichisb}'oundedbelowbyapositiveconstant. TheUUproGofoftheaboveUUtheoremissimilartothatoftheaboveUUtheorem.ps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.17) cvn H.B /DEST pdfmark end-ƍCorollary2.17.D*L}'etl(A3),W(H5),(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end),(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end)!7hold.$Thenthereexistsaboundedsolutionof(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end) whichisb}'oundedbelowbyapositiveconstant.Pr}'oof.UIInviewofRemarkps:SDict begin H.S end1.1 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.1.1) cvn H.B /ANN pdfmark end(i)theproGoffollowslinessimilartothoseinTheoremps:SDict begin H.S end2.15 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.15) cvn H.B /ANN pdfmark endUUandps:SDict begin H.S end2.169ps:SDict begin H.R endsps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.16) cvn H.B /ANN pdfmark end. The/Dresultsfortherange(A4)aresimilartothoseundercondition(A3).eHenceweUUskipallproGofsexceptthefollowingone.@ps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.18) cvn H.B /DEST pdfmark end 9Theorem2.18.L}'et(A4),(H1),(H4),(H5)hold.Thenther}'eexistsaboundedsolutionof(ps:SDict begin H.S end1.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end) whichisb}'oundedbelowbyapositiveconstant.Pr}'oof.UIW*eR\proGceedasintheproofoftheTheoremps:SDict begin H.S end2.15 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.15) cvn H.B /ANN pdfmark endwiththefollowingchanges.ChoGoselڍjV=maxcfjG(x)j:<$Kc81Kwfe (֍,$dPx2g:sRSZ=fy"2X:<$Kc81Kwfe (֍,$dPy2g:ٍF*romUU(H1),(H4)and(H5)wecan ndT*>0suchthattTimplies.EgcZih1@g1 Qϱ(t)<$ jFc(s)j wfe (֍JrG(s)&<<$Kc81Kwfe (֍F]28;Nand^\<$M"?募wfe!Ǵ (֍(n82)!dcZin1@jT1 Qϱ(t)<$1nwfeA2 (֍rG(s){cZi|1@ksVa(u8s)n2q[ٲ(u)duds<<$Kc1Kwfe (֍F]28: ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endKps:SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end104u-R.N.RAZTH,N.MISRA,P.P.MISHRA,L.N.PADHY7&EJDE-2007/163V6लDe neԍRAx(t)=\( SAx(Tc);ift2[T0|s;Tc];܍ x(1 Qϱ(t)) !fe!%4ɴp(1 Qϱ(t))0 y2c2l&fe p(1 Qϱ(t))'~+ #1l&fe p(1 Qϱ(t))&īRj-,1 #+e1 Qϱ(t)&hENF(s)ENʉfe,r7(s)Vεds;iftTV:*g`˼Bqx(t)=򍓫8 >< >:o} Bx(Tc);jif t2[T0|s;Tc];k&hű(1)rn ڟʉfe9(n2)!]p(1 Qϱ(t))IqlīRjP1 #N*P1 Qϱ(t) l 1gj&fe ޟr7(s)x īRjUS1 #cs8(u8s)^n2q[ٲ(u)G(y(h(u)))duds;jif tTV:6लF*or6therestoftheproGofwemayrefertheproGofsoftheTheoremsps:SDict begin H.S end2.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark endandps:SDict begin H.S end2.15 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.15) cvn H.B /ANN pdfmark end. 6ट )ps:SDict begin H.S endps:SDict begin 12 H.A endMps:SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 9_3.kPositivesolutionforp(t)=1 InUUthissectionwe ndsucientconditionfortheNDDEps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.1) cvn H.B /DEST pdfmark endFˍG(rG(t)(y[ٲ(t)8+y(!Dz(t)))09)n1Ҳ+q(t)G(y(h(t)))=f(t);Sqò(3.1)orUUps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.2) cvn H.B /DEST pdfmark end G(rG(t)(y[ٲ(t)8y(!Dz(t)))09)n1Ҳ+q(t)G(y(h(t)))=f(t);Sqò(3.2)-2toUUhaveabGoundedpositivesolution. The8sresultswithNDDE89(ps:SDict begin H.S end3.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end)arerareintheliterature.!W*edont ndsucharesultUUin[ps:SDict begin H.S end19ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.d1) cvn H.B /ANN pdfmark end]or[ps:SDict begin H.S end49ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p1) cvn H.B /ANN pdfmark end,ps:SDict begin H.S end59ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p2) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end69ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r1) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end79ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r2) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end89ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r3) cvn H.B /ANN pdfmark endUV].qT*oachieveUUourresultweneedthefollowingLemma.ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.1) cvn H.B /DEST pdfmark end3Lemma3.1(Schauder'sm,FixedPointTheorem[ps:SDict begin H.S end29ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.e1) cvn H.B /ANN pdfmark end]).NL}'etղ beaclosedconvexandnonemptyTsubsetofaBanachsp}'aceTX.LetB:1 ! b}'eacontinuousmappingsuch(thatBq isar}'elativelycompactsubsetofX.XThenBjhasatleastone xedp}'ointin .Thatisthereexistsanx2 suchthatBqx=x. 3 F*or wtt0|s,;de ne^!DZ0l1 t(t)=t,;^!DZ1l1(t)=1(t);9^!DZ2l1(t)=1(1(t)).-F*or wanypGositiveUUintegeri>2,wede neFˍI!Ǵi፷1 t(t)=1(䍑!Ǵi11(t)):ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.2) cvn H.B /DEST pdfmark end DlTheorem3.2.Supp}'oseв(H1),0(H4),(H5)hold.QThenther}'eexistsasolutionof(ps:SDict begin H.S end3.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end) which9isb}'ounded9belowbyapositiveconstant,Kthatis,itneitheroscillatesnortendstozer}'oasttendsto1.Pr}'oof.UIW*eOproGceedasintheproofofTheoremps:SDict begin H.S end2.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark endwiththefollowingchanges.oLetFˍs(=maxcfjG(x)j:1x5g:F*romUU(H1),(H4)and(H5)thereexistsT*>0suchthatfortTimpliesps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.3) cvn H.B /DEST pdfmark end<$XئJwfe!Ǵ (֍(n82)!m m rcZi|1@x-t<$jM1IwfeA2 (֍rG(s)hcZih·1@s᧲(u8s)n2q[ٲ(u)duds  m<1;Sqò(3.3)andUUps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.4) cvn H.B /DEST pdfmark endWu{ { xcZiy1@yt<$B9Fc(s)B9wfeH (֍ڵrG(s)Rds  m<1:Sqò(3.4) ;F*orUUanycontinuousfunctiong[ٲ(t),itisclearthatps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.5) cvn H.B /DEST pdfmark endgӍoC21 lX l濴l `=1|#1cZi#2r2l1 Qϱ(t)@kk2l1ҍ1ޱ(t)ag[ٲ(s)ds[wfe!Ǵ (֍(n82)!e\1 bX cl `=1r[cZi|\r2l1 Qϱ(t)@xRk2l1ҍ1ޱ(t)<$W1wfeA2 (֍rG(s)˟cZi̷1@Bs,(u8s)n2q[ٲ(u)duds<1;Sqò(3.6)  ;ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endKps:SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark endEJDE-2007/163WTNON-OSCILLAZTORYBEHA#VIOURO[11V6लandUUps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.7) cvn H.B /DEST pdfmark endō!1 䢟X ġl `=1cZir2l1 Qϱ(t)@ȏMk2l1ҍ1ޱ(t)<$vjFc(s)jvwfe (֍JrG(s)ɵds<1:Rg(3.7)u6Set̵S=*4fy 2X:1y[ٲ(t)5;tT0|sg.$-Nextwede nethemappingB:S!X 6लas!>Bqy[ٲ(t)=8 >>< >>:_ By[ٲ(Tc);.T0CtTc;z 38Pލ 1% l `=1īR䍑"N9r2l1 Qϱ(t) &b \rk2l1ҍ1ޱ(t)&h@F(s)@ʉfe,r7(s)Qds+&hl(1)rn1lʉfe5B(n2)!$Pލ/ ڷ1%/ ڴl `=1=MīR䍑Dr2l1 Qϱ(t) &bB1k2l1ҍ1ޱ(t)a'b lvg1gz&fe ޟr7(s)P īRjUS1 #cs8(u8s)^n2q[ٲ(u)G(y(h(u))bWdubds;.tTV:$k6लThen݋using(ps:SDict begin H.S end3.6 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.6) cvn H.B /ANN pdfmark end)land(ps:SDict begin H.S end3.7 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.7) cvn H.B /ANN pdfmark end)ISwe ndthatBqy<5andBy>1. jHenceBy2Sqfor6वy"2S.NextMweshowBqSlڲisrelativelycompactasintheproGofofTheorem2.2.HoThen6byyLemma3.1thereisa xedpGointy0inS5suchthatBqy0q=Ey0|s.HencefortTc,6weUUobtainDJy0|s(t)=j38+<$l(1)^n1lwfe$(n2)!-1 *X +fl `=1:cZiDr2l1 Qϱ(t)@@0ȴk2l1ҍ1ޱ(t)<$f1`rwfeA2 (֍rG(s)tcZi~1@z 8s (us)n2q[ٲ(u)G(y0|s(h(u)))duds"'lAʸuR1 8X ߴl `=1UQcZiURr2l1 Qϱ(t)@㋴k2l1ҍ1ޱ(t)<$8%Fc(s)8%wfeH (֍ڵrG(s)M͵ds:u6लInatheabGoveawereplacetby!Dz(t)andnotethat^ml1 t((t))=䍑!Ǵm11ֲ(t)and^!DZ0l1 t(t)=t.6ThenUUItfollowsfortTthatA֭y0|s(!Dz(t))=nt38+<$l(1)^n1lwfe$(n2)!-1 *X +fl `=1:cZiDk2l1ҍ1ޱ(t)@@0ȴk2l2ҍ1ޱ(t)<$kQ1d丟wfeA2 (֍rG(s)yşcZiƷ1@~s|(us)n2q[ٲ(u)G(y0|s(h(u)))duds".puR1 8X ߴl `=1UQcZiURk2l1ҍ1ޱ(t)@㋴k2l2ҍ1ޱ(t)<$<{Fc(s)<{wfeH (֍ڵrG(s)S(n2)!*cZi41@0Wt<$Ew1?ޟwfeA2 (֍rG(s)ScZi]1@Y%sftѲ(us)n2q[ٲ(u)G(y0|s(h(u)))duds卍Ը8cZi 81@t<$Fc(s)wfeH (֍ڵrG(s)- ds:6लDi erentiatingitheabGoveequation rstandthenmultiplyingbyrG(t)tobothsides6andafterthatdi erentiatingagainforng1times,weseethaty0wistherequired6solutionof(ps:SDict begin H.S end3.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end)B\whichisbGoundedbelowbyapGositiveconstant.Hencethissolution6neitherUUoscillatesnortendstozeroast!1.qHenceUUthetheoremisproved.6ऩ ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.3) cvn H.B /DEST pdfmark end 9Corollary3.3.jIf7(H1),3(H2),(H4)hold,`thenther}'eexistsapositivesolutionof(ps:SDict begin H.S end3.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end)"$whichisb}'oundedbelowbyapositiveconstant.ҴPr}'oof.UITheUUproGoffollowsfromRemarkps:SDict begin H.S end1.1 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.1.1) cvn H.B /ANN pdfmark endandtheaboveUUTheorem.9?ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.4) cvn H.B /DEST pdfmark end 9Theorem3.4.L}'et}C(H3),A(H5),ʲ(ps:SDict begin H.S end2.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end)Jand̲(ps:SDict begin H.S end2.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end)!Mhold. Thenthereexistsapositivesolution)ofG(ps:SDict begin H.S end3.1 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end)whichisb}'ounded)belowbyapositiveconstantthatis,NGitneitheroscillatesnortendstozer}'oasttendsto1.Ҵ TheUUproGofoftheaboveUUtheoremissimilartothatofTheoremps:SDict begin H.S end3.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.3.2) cvn H.B /ANN pdfmark end.ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.5) cvn H.B /DEST pdfmark end .ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endKps:SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end124u-R.N.RAZTH,N.MISRA,P.P.MISHRA,L.N.PADHY7&EJDE-2007/163V6TheoremT3.5.Supp}'ose(H1)hold.Assumefortt0Zps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.8) cvn H.B /DEST pdfmark end&1 jX t)qi=1%cZi&1@_i{1 Qϱ(t)<$1ywfeA2 (֍rG(s)نcZiه1@gsRl(u8s)n2q[ٲ(u)duds<1;Rg(3.8)v6andps:SDict begin H.S endps:SDict begin 12 H.A endPps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.9) cvn H.B /DEST pdfmark endč:1  X tݴi=1cZi1@ͨ˴i{1 Qϱ(t)<$n~1MwfeA2 (֍rG(s)Jds<1:Rg(3.9)@6ThenXJ(ps:SDict begin H.S end3.2 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.2) cvn H.B /ANN pdfmark end)znhasasolutionb}'oundedbelowbyapositiveconstant.6Pr}'oof.V5W*eOproGceedasintheproofofTheoremps:SDict begin H.S end3.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.3.2) cvn H.B /ANN pdfmark endwiththefollowingchanges.oLete̵=maxcfjG(x)j:1x5g:6लThenUUfrom(H1),(ps:SDict begin H.S end3.8 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.8) cvn H.B /ANN pdfmark end)8and(ps:SDict begin H.S end3.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.9) cvn H.B /ANN pdfmark end),thereexistsT*>0suchthatfortT<$x1wfe!Ǵ (֍(n82)!1 X tҴi=1cZi1@i{1 Qϱ(t)<$Gs1&ڟwfeA2 (֍rG(s)EcZiE1@!sͲ(u8s)n2q[ٲ(u)duds<1;v6लandč61 FX ti=1cZi1@ԍi{1 Qϱ(t)<$J Fc(s)J wfeH (֍ڵrG(s)$ds<1:@6लLetUUSZ=fy"2X:1y5;UPtT0|sg.qThenUUde ne%uMBqy[ٲ(t)=8 >>< >>:܍ By[ٲ(Tc);؂forеt2[T0|s;Tc];k 38+&h(1)rnlʉfe(n2)!qPލ(1%(i=17īRj>SN1 #su(us)^n2h q[ٲ(u)G(y(h(u)))duds8+Pލ 1% i=1īRj"1 # i{1 Qϱ(t)&h:DF(s):Dʉfe,r7(s)K_ds;؂forеtTV:6लThen8asinTheoremps:SDict begin H.S end2.2 9ps:SDict begin H.R end rps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark endweprove(i)Bqy2(n82)!-1 *X t+Dڴi=1:cZiD1@@0ȴi{1 Qϱ(t)<$_{1YwfeA2 (֍rG(s)mcZiw1@s)smղ(u8s)n2q[ٲ(u)G(y(h(u)))duds!Pu޲+uR1 8X ti=1UQcZiUR1@㋴i{1 Qϱ(t)<$1Fc(s)1wfeH (֍ڵrG(s)Gds:m6लThen5replacingtby!Dz(t)intheabGoveandusing^!Ǵil1 t(!Dz(t))=䍘i11(t),we5mayobtain6वy[ٲ(!Dz(t)).qConsequentlyUUfortTc,UUwe nd Ry[ٲ(t)8y(!Dz(t))=<$7J(1)^n17Jwfe$(n82)!QTcZiQU1@ߎt<$Ȯ1ըwfeA2 (֍rG(s)"cZi#1@U\s@(u8s)n2q[ٲ(u)G(y(h(u)))duds卍<8cZi 81@t<$Fc(s)wfeH (֍ڵrG(s)- ds:n6लW*eMomaydi erentiatetheabGoveandthenmultiplybyrG(t)andthenagaindi eren-6tiaten}Ÿ1timestoarriveat(ps:SDict begin H.S end3.2 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.2) cvn H.B /ANN pdfmark end)J.ThissolutionisbGoundedbelowbyapGositive6constant.8q6ट ops:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.6) cvn H.B /DEST pdfmark end Mps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endKps:SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark endEJDE-2007/163WTNON-OSCILLAZTORYBEHA#VIOURO[13V6RemarkL3.6.ItisnotdiculttoverifythattheabGovetheoremstillholds,Aifwe 6replaceUU(ps:SDict begin H.S end3.9 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.9) cvn H.B /ANN pdfmark end)8andUU(H1)bythefollowingassumptionps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.10) cvn H.B /DEST pdfmark endۍ1 זX tSi=1cZi1@Ai{1 Qϱ(t)<$G1'[wfeA2 (֍rG(s)FhcZiFi1@ԢsܿN(u8s)n2f(u)duds<1:Rf(3.10)6Ofcourse,inthatcasewehavetomoGdifythede nitionofthemappingB,asfollows.-@2Bqy[ٲ(t)=썓8 >>>>< >>>>:Nƍ By[ٲ(Tc);forNt2[T0|s;Tc];j 38&hl(1)rn1lʉfe5B(n2)!$Pލ/ ڷ1%/ ڴi=1>,џīRjD|1 #B嵴i{1 Qϱ(t) a1\ϟ&fe ޟr7(s)m\īRjt31 #rls|(us)^n2F q[ٲ(u)G(y(h(u)))duds +&h33(1)rn133ʉfe5B(n2)!"FPލ,1%,i=1;īRjB1 #@մi{1 Qϱ(t) _M1ZQ&fe ޟr7(s)k#īRjqS1 #o܌szG8(u8s)^n2f(u)duds;forNtTV:.@6लIfUUweputrG(t)=1UUin(ps:SDict begin H.S end3.8 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.8) cvn H.B /ANN pdfmark end)8and(ps:SDict begin H.S end3.10 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.10) cvn H.B /ANN pdfmark end) 8thenweobtainps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.11) cvn H.B /DEST pdfmark endPF1 ԟX tґi=10EcZi0F1@i{1 Qϱ(t)0fcZi0g1@žsҩL(u8s)n2q[ٲ(u)duds<1;Rf(3.11)6andUUps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.12) cvn H.B /DEST pdfmark endK(1 X t?si=1'cZi(1@+ai{1 Qϱ(t)HcZiɝI1@+s.(u8s)n2f(u)duds<1:Rf(3.12)kBThenUUfromtheabGoveUUtheoremthefollowingresultfollowsdirectly*.6टps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.7) cvn H.B /DEST pdfmark end7iCorollary83.7.If{(ps:SDict begin H.S end3.11 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.11) cvn H.B /ANN pdfmark end)"Ԡandp(ps:SDict begin H.S end3.12 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.12) cvn H.B /ANN pdfmark end)!holdfort>t0|s,thentheNDDEps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.13) cvn H.B /DEST pdfmark endKSn(y[ٲ(t)8y(t!Dz))(n) `+q(t)G(y(t))=f(t)Nq(3.13)Jhasasolution,b}'oundedbelowbyapositiveconstant.)0 The9$abGovecorollaryimprovesandgeneralizes[ps:SDict begin H.S end59ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p2) cvn H.B /ANN pdfmark end,>Theorem3.1]and[ps:SDict begin H.S end79ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r2) cvn H.B /ANN pdfmark end,>Theorem 2.5],BbGecausef}inthesepapers,BtheauthorsassumethefollowingadditionalconditionsthatUUwedon'trequire.8(i)" nUUisoGdd.q(ii)" GUUisnon-decreasingandxG(x)>0UUforx6=0. BeforeweclosethisarticlewepresentaninterestingexamplewhichillustratesmostUUoftheresultsofthispapGer.ps:SDict begin H.S endps:SDict begin 12 H.A endOps:SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.8) cvn H.B /DEST pdfmark end7iExampleT3.8.ConsiderUUNDDEps:SDict begin H.S endps:SDict begin 12 H.A endQps:SDict begin [ /View [/XYZ H.V] /Dest (equation.3.14) cvn H.B /DEST pdfmark end"&h(rG(t)(y[ٲ(t)8py(t=2))09)n1Ҳ+<$ 1lwfe- (֍trn+29G(y(t=3))=0 forNtV&fes2 U,Zh(t)= >tV&fes3and 嗍q[ٲ(t)h= <1&fewvtn+2Fc.a-ItOisnotdiculttoverifythatq(t)satis es(H2),j(H5)and(ps:SDict begin H.S end3.8 9ps:SDict begin H.R end sps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.8) cvn H.B /ANN pdfmark end) . SuppGosethatG(u)=1u^3Tanditisdecreasing.lClearlytheNDDE(ps:SDict begin H.S end3.14 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.14) cvn H.B /ANN pdfmark end)!hasapGositivesolutiony[ٲ(t)S1.fHencethisexampleillustratesalltheresultsofthispapGer.\YHowever sinceGisdecreasingand!Dz(t)isnotoftheformtLkP,!the existingresultsUUof[ps:SDict begin H.S end19ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.d1) cvn H.B /ANN pdfmark end,ps:SDict begin H.S end49ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p1) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end59ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.p2) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end69ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r1) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end79ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r2) cvn H.B /ANN pdfmark endUV,ps:SDict begin H.S end89ps:SDict begin H.R endnps:SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.r3) cvn H.B /ANN pdfmark endUV]arenotapplicableto(ps:SDict begin H.S end3.14 9ps:SDict begin H.R end tps:SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.14) cvn H.B /ANN pdfmark end).hps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end7 6ps:SDict begin H.S endps:SDict begin H.R endKps:SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end144u-R.N.RAZTH,N.MISRA,P.P.MISHRA,L.N.PADHY7&EJDE-2007/163V͸References6टps:SDict begin H.S endps:SDict begin 12 H.A endNps:SDict begin [ /View [/XYZ H.V] /Dest (section*.1) cvn H.B /DEST pdfmark endps:SDict begin H.S endps:SDict begin 10 H.A endKps:SDict begin [ /View [/XYZ H.V] /Dest (cite.d1) cvn H.B /DEST pdfmark end @[1]xDas,qPJ.;Oscilp[lationsRandasymptoticbehaviourofsolutionsforsecondorderneutraldelayxdi erential~equations,XJ.Indian.Math.So cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5u cmex10163/rath.pdf0000664000076400007640000077754410724025702012107 0ustar juliojulio%PDF-1.4 5 0 obj << /S /GoTo /D (section.1) >> endobj 8 0 obj (1. Introduction) endobj 9 0 obj << /S /GoTo /D (section.2) >> endobj 12 0 obj (2. Main Results) endobj 13 0 obj << /S /GoTo /D (section.3) >> endobj 16 0 obj (3. 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Title: Non-oscillatory behaviour of higher order functional differential equations of neutral type Authors: Radhanath Rath (Khallikote Autonomous College, Orissa, India) Niyati Misra (Berhampur Univ., Orissa, India) Prayag Prasad Mishra (Silicin Inst. of Tech., Orissa, India} Laxmi Narayan Padhy (Berhampur Univ., Orissa, India) Abstract: In this paper, we obtain sufficient conditions so that the neutral functional differential equation $$\displaylines{ \big[r(t) [y(t)-p(t)y(\tau (t))]'\big]^{(n-1)} + q(t) G(y(h(t))) = f(t) }$$ has a bounded and positive solution. Here $n\geq 2$; $q,\tau, h$ are continuous functions with $q(t) \geq 0$; $h(t)$ and $\tau(t)$ are increasing functions which are less than $t$, and approach infinity as $t \to \infty$. In our work, $r(t) \equiv 1$ is admissible, and neither we assume that $G$ is non-decreasing, that $xG(x) > 0$ for $x \neq 0$, nor that $G$ is Lipschitzian. Hence the results of this paper generalize many results in [1] and [4]-[8]. Submitted September 24, 2007. Published November 30, 2007. Math Subject Classifications: 34C10, 34C15, 34K40. Key Words: Oscillatory solution; nonoscillatory solution; asymptotic behaviour. 163/abstr.html0000664000076400007640000000620010724025702012425 0ustar juliojulio Electronic Journal of Differential Equations Electron. J. Diff. Eqns., Vol. 2007(2007), No. 163, pp. 1-14.

Non-oscillatory behaviour of higher order functional differential equations of neutral type

Radhanath Rath, Niyati Misra,

Prayag Prasad Mishra, Laxmi Narayan Padhy Abstract:
In this paper, we obtain sufficient conditions so that the neutral functional differential equation
$$\displaylines{
   \big[r(t) [y(t)-p(t)y(\tau (t))]'\big]^{(n-1)} +
   q(t) G(y(h(t))) = f(t)
}$$
has a bounded and positive solution. Here $n\geq 2$; $q,\tau, h$ are continuous functions with $q(t) \geq 0$; $h(t)$ and $\tau(t)$ are increasing functions which are less than $t$, and approach infinity as $t \to \infty$. In our work, $r(t) \equiv 1$ is admissible, and neither we assume that $G$ is non-decreasing, that $xG(x) > 0$ for $x \neq 0$, nor that $G$ is Lipschitzian. Hence the results of this paper generalize many results in [1] and [4]-[8].

Submitted September 24, 2007. Published November 30, 2007.
Math Subject Classifications: 34C10, 34C15, 34K40.
Key Words: Oscillatory solution; nonoscillatory solution; asymptotic behaviour.

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Radhanath Rath
Department of Mathematics
Khallikote Autonomous College
Berhampur, 760001 Orissa, India
email: radhanathmath@yahoo.co.in
Niyati Misra
Department of Mathematics
Berhampur University
Berhampur, 760007 Orissa, India
email: niyatimath@yahoo.co.in
Prayag Prasad Mishra
Department of Mathematics
Silicin Institute of Technology
Bhubaneswar, Orissa, India
email: prayag@silicon.ac.in
Laxmi Narayan Padhy
Department of Computer Science and Engineering, K.I.S.T,
Bhubaneswar Orissa, India
email: ln_padhy_2006@yahoo.co.in

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URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/163\hfil Non-oscillatory behaviour] {Non-oscillatory behaviour of higher order functional differential equations \\ of neutral type} \author[R. N. Rath, N. Misra, P. P. Mishra, L. N. Padhy \hfil EJDE-2007/163\hfilneg] {Radhanath Rath, Niyati Misra, \\ Prayag Prasad Mishra, Laxmi Narayan Padhy} \address{Radhanath Rath \newline Department of Mathematics, Khallikote Autonomous College, Berhampur, 760001 Orissa, India} \email{radhanathmath@yahoo.co.in} \address{Niyati Misra \newline Department of Mathematics, Berhampur University, Berhampur, 760007 Orissa, India} \email{niyatimath@yahoo.co.in} \address{Prayag Prasad Mishra \newline Department of Mathematics, Silicin Institute of Technology, Bhubaneswar, Orissa, India} \email{prayag@silicon.ac.in} \address{Laxmi Narayan Padhy \newline Department Of Computer Science and Engineering, K.I.S.T, Bhubaneswar Orissa, India} \email{ln\_padhy\_2006@yahoo.co.in} \thanks{Submitted September 24, 2007. Published November 30, 2007.} \subjclass[2000]{34C10, 34C15, 34K40} \keywords{Oscillatory solution; nonoscillatory solution; asymptotic behaviour} \begin{abstract} In this paper, we obtain sufficient conditions so that the neutral functional differential equation \begin{equation*} \big[r(t) [y(t)-p(t)y(\tau (t))]'\big]^{(n-1)} + q(t) G(y(h(t))) = f(t) \end{equation*} has a bounded and positive solution. Here $n\geq 2$; $q,\tau, h$ are continuous functions with $q(t) \geq 0$; $h(t)$ and $\tau(t)$ are increasing functions which are less than $t$, and approach infinity as $t \to \infty$. In our work, $r(t) \equiv 1$ is admissible, and neither we assume that $G$ is non-decreasing, that $xG(x) > 0$ for $x \neq 0$, nor that $G$ is Lipschitzian. Hence the results of this paper generalize many results in \cite{d1} and \cite{p1}--\cite{r3}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \section{Introduction} In this paper we find sufficient conditions for the neutral delay differential equation (NDDE in short), of order $n\geq 2$, \begin{equation} \big[r(t) [y(t)-p(t)y(\tau (t))]'\big]^{(n-1)} + q(t) G(y(h(t))) = f(t) \label{eE} \end{equation} to have a bounded positive solution which does not tend to zero as $t \to \infty$. Here $q, h ,\tau\in C ([0, \infty),R)$ such that $q(t) \geq 0, h(t)$ and $\tau(t)$ are increasing functions which are less thatn or equal to $t$, and approach $\infty$ as $t \to \infty$, $r \in C^{(n-1)} ([0, \infty), (0, \infty))$, $p \in C^{(n)} ([0, \infty), \mathbb{R})$, $G \in C(\mathbb{R},\mathbb{R})$. We need some of the following assumptions in the sequel. \begin{itemize} \item[(H1)] There exists a bounded function $F(t)$ such that $F^{(n-1)} (t) = f(t)$. \item[(H2)] $\int^\infty_{t_0}t^{n-2} q(t) dt < \infty$. \item[(H3)] $\int^\infty_{t_0} {dt \over r(t)} = \infty$. \item[(H4)]$\int^\infty_{t_0} {dt \over r(t)} < \infty$. \item[(H5)] $\int^\infty_{t_0} ({1\over r(t)} \int^\infty_t (s-t)^{n-2}q(s) ds)dt < \infty$. \end{itemize} \begin{remark} \label{rmk1} \rm Since $r(t) > 0$, it follows that \begin{itemize} \item[(i)] either (H3) or (H4) holds exclusively. \item[(ii)] If (H3) holds then (H5) implies (H2) but not conversely. \item[(iii)] If (H4) holds then (H2) implies (H5) but not conversely. \end{itemize} \end{remark} The study of oscillation and non-oscillation properties of neutral delay differential equations has attracted the attention of many authors all over the world during the last two decades.In \cite{d1,p1,p2,r1,r2,r3} the authors have proved the existence of a bounded positive solution of neutral delay differential equations \begin{gather} (y(t) - p(t) y(t - \beta))' + q(t) G (y(t - \delta))= f(t) ,\label{e1}\\ (y(t) - p(t) y(t - \beta))'' + q(t) G (y(t - \delta))= f(t), \label{e2},\\ (y(t) - p(t) y(t - \beta))^{(n)} + q(t) G (y(t - \delta))= f(t), \label{e3} \end{gather} where $\beta$ and $\delta$ are constants. For that purpose the authors assume the following hypothesis. \begin{itemize} \item[(H6)] There exists a function $F(t)$ such that $F(t) \to 0$ as $t \to \infty$ and $F^{n} (t) = f(t)$. \item[(H7)] $\big| \int^\infty_{t_0}t^{n-1} f(t) dt\big|< \infty$. \item[(H8)] $G$ is Lipschitzian in every interval of the form $[a,b]$, with $0 < a < b$. \item[(H9)] $x G(x) > 0$ for $x \neq 0$, and $G$ is non-decreasing. \end{itemize} It is obvious that (H6) $\Leftrightarrow$ (H7) and (H1) is weaker than both (H6) and (H7).In \eqref{eE} if we put $r(t)=1$ , $\tau(t)=t-\beta$, $h(t)=t-\delta$ then it reduces to \eqref{e3}. We find almost no result with the NDDE \eqref{eE} in the literature. For example if $\tau(t)=t/2$ and $h(t)=t/3$ then the existing results fail to answer any thing. Since we formulate our results with (H1) and do not assume either (H8) or (H9), therefore our work extends, improves and generalizes some of the results of \cite{d1,p1,p2,r1,r2,r3}. While studying the existence of a positive solution of neutral delay differential equation \eqref{e3} for $n \geq 2$, the authors take $p(t)$ in different ranges. But some how we find no result when $p(t) \equiv -1$, in these papers. However, in this work we consider $p(t)$ in different ranges including $p(t) = \pm 1$. Our results hold good when $n$ is both odd or even but $\geq 2$. Let $T_y > 0$ and $T_0 = \min \{h(T_y), \tau(T_y)\}$. Suppose $\phi \in C ([T_0 T_y], R)$. By a solution of $\eqref{eE}$, we mean a real valued continuous function $y \in C^{(n)} ([T_0, \infty), R)$ such that $y(t) = \phi(t)$ for $T_0 \leq t \leq T_y$ and $y(t) - p(t) y(t - \tau)$ is differentiable, $r(t) (y(t) - p(t) y(t-\tau))'$ is $(n-1)$ times further differentiable and then for $t\geq T_y$ the neutral equation \eqref{eE} is satisfied.Such a solution is said to be oscillatory if it has arbitrarily large zeros, otherwise it is called non-oscillatory. So far as existence and uniqueness of solutions of \eqref{eE} are concerned one may refer \cite{g1}, but in this work we assume the existence of solutions of \eqref{eE} and study its non-oscillatory behaviour. \section{Main Results} In this section we assume that there exists positive real numbers $p,c,$ and $d$ such that $p(t)$ satisfies one of the following conditions. \begin{itemize} \item[(A1)] $0 \leq p(t) \leq p < 1$. \item[(A2)] $-1 < -p \leq p(t) \leq 0$. \item[(A3)] $-d < p(t) \leq -c < -1$. \item[(A4)] $1 < c \leq p(t) < d$. \end{itemize} For our work we need the following Lemma from [3]. \begin{lemma}[Krasnoselskiis Fixed point Theorem \cite{e1}] \label{lem1} Let $X$ be a Banach space. Let $\Omega$ be a bounded closed convex subset of $X$ and let $S_1, S_2$ be maps of $\Omega$ into $X$ such that $S_1 x+ S_2 y \in \Omega$ for every pair $x, y \in \Omega$. If $S_1$ is a contraction and $S_2$ is completely continuous, then the equation \begin{equation*} S_1 x + S_2x = x \end{equation*} has a solution in $\Omega$. \end{lemma} \begin{theorem}\label{thm2.2} Let {\rm (A1), (H1), (H4), (H5)} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant i.e there exists a solution of \eqref{eE} which neither oscillates nor tends to zero as $t\to\infty$. \end{theorem} \begin{proof} Since $G \in C(\mathbb{R},\mathbb{R})$, then let \begin{equation}\label{f2} \mu = \max \{G(x) : {3\over 5}(1 -p) \leq x \leq 1\}. \end{equation} From (H1) , we find $\alpha >0$ and $t_1 > t_0>0$ such that \begin{equation}\label{f1} |F(t)| < \alpha \quad \text{for } t \geq t_1. \end{equation} Then using (H4) we find $t_2>t_1$ such that $t \geq t_2$ implies \begin{equation}\label{f} \int^\infty_t {1 \over r(s)} ds < {1 - p \over 10\,\alpha}. \end{equation} From \eqref{f1} and \eqref{f} it follows that for $t>t_3>t_2$ \begin{equation}\label{f3} \int^\infty_t {|F(s)| \over r(s)} ds < {1 - p \over 10}. \end{equation} From (H5) we find $t_4>t_3$ such that $t > t_4$ implies \begin{equation}\label{f4} \frac{\mu}{(n-2)!} \int^\infty_t {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\, ds < {1-p \over 10}. \end{equation} Let $T >t_4$ and $T_0 = \min \{ \tau(T), h (T)\}$. Then for $t \geq T$, \eqref{f3} and \eqref{f4} hold. Let $X = C([T_0, \infty), R )$ be the set of all continuous functions with norm $\|x\| = \sup_{t \geq T_0} |x(t)| < \infty$. Clearly $X$ is a Banach space. Let \begin{equation}\label{f5} S = \big\{u \in BC ([T_0, \infty),R ): {3 \over 5}(1-p) \leq u(t) \leq 1\big\}, \end{equation} with the supremum norm $\|u \|= \sup \{ |u(t)|: t \geq T_0 \}$. Clearly $S$ is a closed, bounded and convex subset of $C([T_0, \infty), R)$. Define two maps $A$ and $B: S \to X$ as follows. For $x \in S$, \begin{equation} \label{f6} Ax(t)=\begin{cases}Ax(T), & t \in [T_0, T];\\ p(t) x(\tau(t)) + {4(1 -p)\over 5} ,& t \geq T, \end{cases} \end{equation} and \begin{equation}\label{f7} Bx (t)=\begin{cases}Bx(T), & t \in [T_0, T];\\ \frac{(-1)^{n-1}}{(n-2)!}\int^\infty_t {1\over r(s)}\int^\infty_s (u-s)^{n-2}q(u) G(x(h(u)))du\,ds \\ -\int^\infty_t {F(s)\over r(s)}ds, & t \geq T. \end{cases} \end{equation} First we show that if $x, y \in S$ then $Ax + By \in S$. In fact, for every $x, y \in S$ and $t \geq T$, we get \begin{equation*} \begin{split} (Ax)(t) + (By) (t) & \leq p(t) x(\tau(t))+ {4(1-p)\over 5} - \int^\infty_t {F(s) \over r(s)}ds\\ & \quad +\frac{(-1)^{n-1}}{(n-2)!}\int^\infty_t {1\over r(s)}\int^\infty_s (u-s)^{n-2}q(u) G(y(h(u)))du\,ds\\ & \leq p + {4(1 - p)\over 5} + {1 - p \over 10}+{1 - p \over 10} \leq 1. \end{split} \end{equation*} On the other hand for $t \geq T$, \begin{align*} (Ax)(t) + (By) (t) & \geq {4(1-p)\over 5} - \int^\infty_t {F(s)\over r(s)}ds\\ & \quad +\frac{(-1)^{n-1}}{(n-2)!} \int^\infty_t {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G(y(h(u)))du \,ds\\ & \geq {4(1-p)\over 5} - \alpha\int^\infty_t {1\over r(s)}ds\\ &\quad -\frac{\mu}{(n-2)!} \int^\infty_t {1\over r(s)}\int^\infty_s(u-s)^{n-2}q(u) du\,ds\\ & \geq {4(1-p)\over 5} - {1-p \over 10} - {1-p \over 10} = {3\over 5}(1-p). \end{align*} Hence \begin{equation*} {3\over 5}(1-p) \leq (Ax)(t) + (By)(t) \leq 1 \end{equation*} for $t \geq T$. So that $Ax + By\in S$ for all $x, y \in S$. Next we show that $A$ is a contraction in $S$. In fact, for $x, y \in S$ and $t \geq T$, we have \begin{align*} |(Ax)(t) - (Ay)(t)| &\leq |p(t) \{x(\tau(t)) - y(\tau(t))\}|\\ & \leq|p(t)| |x(\tau(t)) - y(\tau(t))|\\ & \leq p\parallel x -y \parallel. \end{align*} Since $0 < p < 1$, we conclude that $A$ is a contraction mapping on $S$. We now show that $B$ is completely continuous. First, we shall show that $B$ is continuous. Let $x_k = x_k(t) \in S$ for $k=1,2,\dots $. be such that $\sup_{t\geq T}|x_k(t)-x(t)|\to 0$ as $k \to \infty$.Because $S$ is closed, $x = x(t) \in S$. For $t \geq T$, we have \begin{align*} & |(B x_k)(t) - (Bx)(t)| \\ & \leq \frac{1}{(n-2)!}\int^\infty_t {1\over r(s)} \int^\infty_s (u-s)^{n-2}q(u) |G(x(h(u)))- G(x_k(h(u)))|du \, ds. \end{align*} Since for all $t\geq T , x_k(t),k=1,2...$, tend uniformly to $x(t)$ as $t\to\infty$ and $G$ is continuous, therefore $|G(x(h(u))) -G (x_k(h(u)))|\to 0$ as $k \to \infty$. We conclude that $\lim_{k \to \infty} |(B x_k)(t) - (Bx)(t)| = 0$. This means that $B$ is continuous. Next, we show that $BS$ is relatively compact. It suffices to show that the family of functions $\{Bx : x \in S\}$ is uniformly bounded and equicontinuous on $[T_0, \infty)$. The uniform boundedness is obvious. For the equicontinuity, according to Levitan's result we only need to show that, for any given $\epsilon > 0, [T_0, \infty)$ can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have change of amplitude less than $\epsilon$. From (H5) and (H4), it follows that for any $\epsilon > 0$,we can find $T^\ast \geq T$ large enough so that \begin{equation*} \frac{\mu}{(n-2)!} \int^\infty_{T^\ast} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\, ds < {\epsilon \over 4}, \end{equation*} and \begin{equation*} \alpha \int^\infty_{T^\ast} {ds \over r(s)} < {\epsilon \over 4}. \end{equation*} Then for $x \in S$ and $t_2> t_1 \geq T^*$, \begin{align*} &|(Bx)(t_2) - (Bx)(t_1)|\\ &= \Big| \frac{(-1)^n}{(n-2)!}\int^\infty_{t_2} {1 \over r(s)} \int^\infty_s(u-s)^{n-2} q(u) G(x(h(u)))du\,ds - \int^\infty_{t_2} {F(s)\over r(s)} ds\\ & \quad - \frac{(-1)^n}{(n-2)!}\int^\infty_{t_1} {1 \over r(s) } \int^\infty_s (u-s)^{n-2}q(u)G(x(h(u)))du\,ds + \int^\infty_{t_1} {F(s)\over r(s)}ds \Big| \\ & \leq \frac{\mu}{(n-2)!} \int^\infty_{t_1} {1 \over r(s)} \int^\infty_s(u-s)^{n-2} q(u)du\,ds + \alpha \int^\infty_{t_1} {ds\over r(s)}\\ & \quad + \frac{\mu}{(n-2)!} \int^\infty_{t_2} {1\over r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\,ds + \alpha \int^\infty_{t_2} {ds \over r(s)}\\ & < 4{\epsilon \over 4}=\epsilon. \end{align*} For $x \in S$ and $T \leq t_1 < t_2 \leq T^\ast$, \begin{align*} &|(Bx)(t_2) - (Bx)(t_1)|\\ & \leq \frac{\mu}{(n-2)!} \int^{t_2}_{t_1} {1\over r(s)} \int^\infty_s (u-s)^{n-2}q(u) \,du\,ds + \alpha \int^{t_2}_{t_1} {1\over r(s)} ds \\ &\leq \max_{T \leq s \leq T^\ast} \Big[{\mu \over (n-2)!r(s)} \int^\infty_s (u-s)^{n-2}q(u)\,du + {\alpha \over r(s)}\Big](t_2 - t_1). \end{align*} Thus there exists a $\delta > 0$ such that \begin{equation*} |(Bx)(t_2) - (Bx)(t_1)| < \epsilon \quad \text{if } 0 < |t_2 - t_1| < \delta. \end{equation*} For any $x \in S, T_0 \leq t_1 < t_2 \leq T$, it is easy to see that \begin{equation*} | (Bx) (t_2) - (Bx)(t_1)| = 0 < \epsilon. \end{equation*} Therefore, $\{Bx : x \in S\}$ is uniformly bounded and equicontinuous on $[T_0, \infty)$ and hence $BS$ is relatively compact. By Lemma \ref{lem1}, there is an $x_0 \in S$ such that $Ax_0 + Bx_0 = x_0$. It is easy to see that $x_0(t)$ is the required non oscillatory solution of the equation $\eqref{eE}$, which is bounded below by the positive constant ${3(1-p)\over 4}$. \end{proof} \begin{corollary}\label{cor2.3} Let {\rm (A1), (H1), (H2), (H4)} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{corollary} \begin{proof} By remark \ref{rmk1}(iii) (H2) and (H4) imply (H5). Hence the proof follows from the proof of the above theorem,. \end{proof} \begin{theorem}\label{thm2.4} Let {\rm (A1), (H3), (H5)} hold. Suppose there exists $\alpha > 0$ such that for large $t$ \begin{equation}\label{f8} r(t) > {1 \over \alpha}, \end{equation} and \begin{equation}\label{f9} \big|\int^\infty_0 F(t) dt\big| < \infty \quad \text{with}\quad F^{n-1} (t) = f(t). \end{equation} Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{theorem} \begin{proof} Using \eqref{f8} and \eqref{f9} we can get \eqref{f3}. Rest of the proof is similar to that of the Theorem \ref{thm2.2}. \end{proof} \begin{corollary}\label{cor2.5} Let {\rm (A1), (H5)}, \eqref{f8}, \eqref{f9} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{corollary} \begin{proof} By Remark \ref{rmk1}(i) we have either (H3) holds or (H4) holds. If (H3) holds then we proceed as in the proof of Theorem \ref{thm2.4}. On the other hand if (H4) holds then from \eqref{f8} and \eqref{f9} we get \eqref{f3} and then proceed as in the proof of Theorem \ref{thm2.2} to get the desired result. \end{proof} \begin{remark} \label{rmk3}\rm If in (H5) we take $r(t)\equiv 1$ then it reduces to \begin{equation}\label{f10} \int^\infty_{t_0} \int^\infty_t (u-t)^{n-2} q(u) du < \infty. \end{equation} The above condition is required for our next result which follows from Corollary \ref{cor2.5} when $r(t) \equiv 1$. \end{remark} \begin{corollary}\label{cor2.6} Inequality \eqref{f10} is a sufficient condition for the $n$th order NDDE \begin{equation}\label{f11} \big(y(t) - p(t) y(\tau(t))\big)^{n}+ q(t) G(y(h(t))) = f(t) \end{equation} to have a solution bounded below by a positive constant under the assumptions (A1), \eqref{f8} and \eqref{f9}. \end{corollary} \begin{remark} \label{rmk4} \rm We claim that the condition \begin{equation}\label{f12} \int^\infty_{t_0} u^{n-1} q(u) ds < \infty \end{equation} implies \eqref{f10}. It is clear that \eqref{f12} is equivalent to $\int^\infty_s (u - s)^{n-1} q(u) du < \infty$. Let $$ K(s) = \int^\infty_s (u -s)^{n-1} q(u) du. $$ This implies $\lim_{s \to \infty}K(s) = 0$. Differentiating, we get $$ K' (s) = - (n-1)\int^\infty_s (u-s)^{n-2}q(u) du. $$ From this integrating between s=t and s=T ,we obtain $$ \int^T_t K' (s) ds = -(n-1)\int^T_t \int^\infty_s (u-s)^{n-2}q(u) \,du\,ds. $$ Hence $$ -(n-1)\int^T_t \int^\infty_s (u-s)^{n-2}q(u) \,du\, ds = K(T) - K(t). $$ In the limit as $T \to \infty$, we obtain $$ \int^\infty_t \int^\infty_s (u-s)^{n-2} q(u) \,du\,ds ={ K(t)\over (n-1)} = {1 \over (n-1)}\int^\infty_t (u-t)^{n-1}q(u) du < \infty. $$ Hence the claim holds. \end{remark} \begin{remark} \label{rmk5} \rm Corollary \ref{cor2.6} improves \cite[Theorem 1]{d1} and \cite[Theorem 4.3]{p1} because the authors assumed $G$ to satisfy (H9) and to be Lipschizian. It may be noted in view of the Remark\ref{rmk4} that the condition \eqref{f10} used in Coprollary\ref{cor2.6} is weaker than the condition \eqref{f12} used in \cite{d1,p1}. \end{remark} \begin{theorem} \label{thm2.7} Let {\rm (A2), (H1), (H4), (H5)} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{theorem} \begin{proof} We proceed as in the proof of the Theorem \ref{thm2.2} with the following changes \begin{equation*} \mu = \max \{|G(x)|: {1-p \over 10} \leq x \leq 1 \}. \end{equation*} By {\rm (H1), (H4), (H5)}, we find $T$ such that for $t\geq T$ \begin{equation*} \frac{\mu}{(n-2)!} \int^\infty_t {1\over r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\,ds < {1 - p \over 10}, \end{equation*} and \begin{equation*} \int_t^\infty {|F(t)| \over r(t)} dt < \alpha \int_t^\infty {dt \over r(t)} < {1-p \over 10}. \end{equation*} Let $S = \{y \in X : {1-p \over 10} \leq y(t) \leq 1, t \geq T_0\}$. \begin{equation*} (Ay)(t) = \begin{cases} {7p+3 \over 10} + p(t) y(t - \tau)-\int^\infty_t {F(s)\over r(s)} ds, &\text{for } t \geq T;\\ Ay(T), & \text{for } T_0 \leq t \leq T. \end{cases} \end{equation*} \begin{equation*} (By)(t) = \begin{cases} By(T), \quad \text{for } T_0 \leq t \leq T;\\ \frac{(-1)^{n-1}}{(n-2)!}\int^\infty_t {1\over r(s)} \int^\infty_s (u-s)^{n-2}q(u)G(y(h(u)))du\,ds \quad \text{for } t \geq T. \end{cases} \end{equation*} Then as in Theorem\ref{thm2.2} we prove (i) $Ax + By \in S$ (ii) $A$ is a contraction, and finally (iii) $B$ is completely continuous. Then by Lemma \ref{lem1} there is a fixed point $x_0$ in $S$ such that $Ax_0 + Bx_0 = x_0$ which is the required solution bounded below by a positive constant. \end{proof} \begin{remark}\label{rmkn}\rm The above theorem substantially improves \cite[Theorem 3.1]{r3} where the authors obtained a positive bounded solution of \eqref{eE} with assumptions (A2), (H2), (H4), (H6), (H8), (H9). It may be noted that (H6) implies (H1) and (H2) with (H4)implies (H5). Further we did not require (H8) and (H9). \end{remark} \begin{theorem}\label{thm2.8} Let {\rm (A2), (H3), (H5)}, \eqref{f8} and \eqref{f9} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{theorem} The proof of the above Theorem is similar to that of Theorem \ref{thm2.7}. \begin{definition}\label{d1} For any $t>t_0$, define $$ \tau _{-1}(t) =\{s\text{ is a real number } : s \geq t \text{ and } \tau(s)=t \}. $$ \end{definition} \begin{remark}\label{rk2} \rm The function $\tau_{-1}$ defined above is the inverse function of $\tau (t)$. Since $\tau (t)$ is increasing it is one-one.Clearly $\tau _{-1}(\tau (t))=t$ for $t>\tau_{-1}(t_0)$. \end{remark} \begin{theorem}\label{thm2.9} Let {\rm (A3), (H1), (H4), (H5)} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{theorem} \begin{proof} If necessary increment $d$ such that $d>1+\frac{2}{c}$. Choose positive numbers $\epsilon < \frac{c-1}{2}$, $h=(c-1)-\epsilon$ and $H = d-1+\frac{2\epsilon}{c}$. Then $H > h > 0$. Set $$ \mu = \max \{|G(x)|: h \leq x \leq H\}. $$ From (H4) and (H5) one can find $T > 0$ such that $t \geq T$ implies \begin{equation*} \int^\infty_{\tau_{-1}(t)} {F(s) \over r(s)} < {\epsilon \over 2}, \end{equation*} and \begin{equation*} \frac{\mu}{(n-2)!} \int^\infty_{\tau_{-1}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u)du\,ds < {\epsilon \over 2}. \end{equation*} Define \begin{equation*} S = \{ y(t) \in X : h \leq y(t) \leq H, t \geq T_0\}. \end{equation*} Define \begin{equation*} Ax(t) = \begin{cases} Ax (T), & \text{if } t \in [T_0, T];\\ {x(\tau_{-1}(t))\over p(\tau_{-1}(t))}- {cd - 1\over p(\tau_{-1}(t))} + {1\over p(\tau_{-1}(t))} \int^\infty_{\tau_{-1}(t)} {F(s)\over r(s)} ds , & \text{if } t \geq T. \end{cases} \end{equation*} \begin{equation*} Bx(t) = \begin{cases} Bx(T) ,& \text{if } t \in [T_0, T];\\ \frac{(-1)^n}{(n-2)!\, p(\tau_{-1}(t))} \int^\infty_{\tau_{-1}(t)} {1\over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G (y(h(u)))du\,ds, &\text{if } t \geq T. \end{cases} \end{equation*} We show that if $x,y \in S$, then $Ax+By \in S$. For $t\geq T$ we obtain \begin{equation*} \begin{split} Ax+By= &\frac{-1}{p(\tau_{-1}(t))}\Big[-x(\tau_{-1}(t))-\int^\infty_{\tau_{-1}(t)} {F(s)\over r(s)} ds+(cd-1)\\ & +\frac{(-1)^{n-1}}{(n-2)!}\int^\infty_{\tau_{-1}(t)} {1\over r(s)} (\int^\infty_s (u-s)^{n-2} q(u)G (y(h(u)))du)ds\Big]\\ &\leq \frac{1}{c}\big[-h+\frac{\epsilon}{2}+\frac{\epsilon}{2}+(cd-1)\big] \\ & =\frac{1}{c}\big[2\epsilon+ c(d-1)\big]=(d-1)+\frac{2\epsilon}{c}\\ & \leq H. \end{split} \end{equation*} Further, \begin{equation*} \begin{split} Ax+By=&\frac{-1}{p(\tau_{-1}(t))}\Big[-x(\tau_{-1}(t))-\int^\infty_{\tau_{-1}(t)} {F(s)\over r(s)} ds+(cd-1)\\ & +\frac{(-1)^{n-1}}{(n-2)!}\int^\infty_{\tau_{-1}(t)} {1\over r(s)} \int^\infty_s (u-s)^{n-2} q(u)G (y(h(u)))du\,ds\Big]\\ & \geq \frac{1}{d} \big[-H -\frac{\epsilon}{2} + (cd-1)-\frac{\epsilon}{2}\big]\\ & =\frac{1}{d}\big[d(c-1)-\epsilon\frac{(c+2)}{c}\big]\\ & >c-1-\epsilon = h. \end{split} \end{equation*} Thus $Ax+By \in S$. Next we show that $A$ is a contraction in $S$.In fact for $x,y \in S$ and $t \geq T$ we have $$ \| Ax-Ay\|\leq |\frac{1}{p(\tau_{-1}(t))}| |x(\tau_{-1}(t))-y(\tau_{-1}(t))| \leq \frac{1}{c}\|x-y\|. $$ Hence $A$ is a contraction because, $0<\frac{1}{c}<1$.Next we prove $B$ is completely continuous as in the proof of Theoren\ref{thm2.2}. Then by Lemma \ref{lem1} there is a fixed point $x_0$ in $S$ such that $$ Ax_0 + Bx_0 = x_0. $$ Writing $x_0=y(t)$ and multiplying both sides of the above equation by $p(\tau _{-1}(t))$ we obtain, \begin{equation*} \begin{split} p(\tau_{-1}(t))y(t) &=y(\tau_{-1}(t))+\int^\infty_{\tau_{-1}(t)} {F(s)\over r(s)} ds -(cd-1)\\ &\quad +\frac{(-1)^n}{(n-2)!} \int^\infty_{\tau_{-1}(t)} {1\over r(s)} \int^\infty_s (u-s)^{n-2} q(u)G (y(h(u)))du\,ds. \end{split} \end{equation*} Then we replace $t$ by $\tau(t)$ ,use the fact that $\tau_{-1}(\tau(t))=t$ and finally with some rearrangement of terms, obtain \begin{equation*} \begin{split} y(t)-p(t)y(\tau(t))=&-\int_t^\infty\frac{F(s)}{r(s)}+(cd-1)\\ & +\frac{(-1)^{n-1}}{(n-2)!}\int_t^\infty \frac{1}{r(s)} \int_s^{\infty}(u-s)^{n-2}q(u)G(y(h(u)))du \,ds . \end{split} \end{equation*} First differentiating the above equation once and then multiplying bothsides by $r(t)$ and finally differentiating $n-1$ times ,we see that, $x_0$ is the required solution of \eqref{eE}, which is bounded below by a positive constant. \end{proof} \begin{theorem}\label{thm2.10} Let {\rm (A3), (H3), (H5)}, \eqref{f8}, \eqref{f9} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{theorem} The proof of the above theorem is similar to that of the above theorem. \begin{corollary}\label{cor2.11} Let {\rm (A3), (H5)}, \eqref{f8}, \eqref{f9} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{corollary} \begin{proof} In view of Remark \ref{rmk1} (i) the proof follows lines similar to those in Theorem \ref{thm2.9} and \ref{thm2.10}. The results for the range (A4) are similar to those under condition (A3). Hence we skip all proofs except the following one. \end{proof} \begin{theorem}\label{thm2.12} Let {\rm (A4), (H1), (H4), (H5)} hold. Then there exists a bounded solution of $\eqref{eE}$ which is bounded below by a positive constant. \end{theorem} \begin{proof} We proceed as in the proof of the Theorem\ref{thm2.9} with the following changes. Choose \begin{equation*} \mu = \max \{|G(x)|: {c-1 \over d} \leq x \leq 2 \}. \end{equation*} \begin{equation*} S = \{y \in X : {c-1 \over d} \leq y \leq 2\}. \end{equation*} From (H1), (H4) and (H5) we can find $T > 0$ such that $t \geq T$ implies \begin{equation*} \int^\infty_{\tau_{-1}(t)} {|F(s)| \over r(s)} < {c-1 \over 2}, \end{equation*} and \begin{equation*} \frac{\mu}{(n-2)!} \int^\infty_{\tau_{-1}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u)du\,ds < {c-1 \over 2}. \end{equation*} Define \begin{equation*} Ax(t) = \begin{cases} Ax (T), & \text{if } t \in [T_0, T];\\ {x(\tau_{-1}(t))\over p(\tau_{-1}(t))}- {2c - 2\over p(\tau_{-1}(t))} + {1\over p(\tau_{-1}(t))} \int^\infty_{\tau_{-1}(t)} {F(s)\over r(s)} ds , & \text{if } t \geq T. \end{cases} \end{equation*} \begin{equation*} Bx(t) = \begin{cases} Bx(T) ,& \text{if } t \in [T_0, T];\\ \frac{(-1)^n}{(n-2)! \, p(\tau_{-1}(t))} \int^\infty_{\tau_{-1}(t)} {1\over r(s)}\\ \times \int^\infty_s (u-s)^{n-2}q(u) G (y(h(u)))du\,ds, & \text{if } t \geq T. \end{cases} \end{equation*} For the rest of the proof we may refer the proofs of the Theorems \ref{thm2.2} and \ref{thm2.9}. \end{proof} \section{Positive solution for $p(t) = \pm 1$ } In this section we find sufficient condition for the NDDE \begin{equation}\label{g1} (r(t) (y(t) + y(\tau(t)))' )^{n-1} + q(t) G(y(h(t))) = f(t), \end{equation} or \begin{equation}\label{g2} (r(t) (y(t) - y(\tau(t)))' )^{n-1} + q(t) G(y(h(t))) = f(t),\end{equation} to have a bounded positive solution. The results with NDDE \eqref{g1} are rare in the literature. We dont find such a result in \cite{d1} or \cite{p1,p2,r1,r2,r3}. To achieve our result we need the following Lemma. \begin{lemma}[Schauder's Fixed Point Theorem \cite{e1}] \label{lem3.1} Let $\Omega$ be a closed convex and nonempty subset of a Banach space $X$. Let $B : \Omega \to \Omega$ be a continuous mapping such that $B \Omega$ is a relatively compact subset of $X$. Then $B$ has at least one fixed point in $\Omega$. That is there exists an $x \in \Omega$ such that $Bx = x$. \end{lemma} %\begin{definition}\label{d2}\rm For $t \geq t_0$, define $\tau_{-1}^{0}(t)=t$, $\tau_{-1}^{1}(t)=\tau_{-1}(t),$\, $\tau_{-1}^{2}(t)=\tau_{-1}(\tau_{-1}(t))$. For any positive integer $i >2$, we define $$ \tau_{-1}^{i}(t)=\tau_{-1}(\tau_{-1}^{i-1}(t)). $$ %\end{definition} \begin{theorem}\label{thm3.2} Suppose {\rm (H1), (H4), (H5)} hold. Then there exists a solution of \eqref{g1} which is bounded below by a positive constant, that is, it neither oscillates nor tends to zero as $t$ tends to $\infty$. \end{theorem} \begin{proof} We proceed as in the proof of Theorem \ref{thm2.2} with the following changes. Let \begin{equation*} \mu = \max \{|G(x)|: 1 \leq x \leq 5\}. \end{equation*} From (H1), (H4) and (H5) there exists $T > 0$ such that for $t \geq T$ implies \begin{equation}\label{new0} \frac{\mu}{(n-2)!}\big|\int^\infty_t {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\,ds\big|< 1, \end{equation} and \begin{equation}\label{new1} \big|\int^\infty_t {F(s) \over r(s)} ds\big| < 1. \end{equation} For any continuous function $g(t)$, it is clear that \begin{equation}\label{new2} \sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)} g(s)ds <\int^\infty_t g(s)ds. \end{equation} Hence using the above inequality in \eqref{new0} and \eqref{new1}, we conclude that for $t\geq T$ \begin{equation} \label{new3} \frac{\mu}{(n-2)!}\sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u)du \,ds <1, \end{equation} and \begin{equation}\label{new4} \sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)} {|F(s)| \over r(s)} ds <1. \end{equation} Set $S = \{y \in X : 1 \leq y(t) \leq 5,\; t \geq T_0 \}$. Next we define the mapping $B : S \to X$ as \begin{equation*} By(t) = \begin{cases} By(T), & T_0 \leq t \leq T;\\ 3-\sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)} {F(s) \over r(s)} ds +\frac{(-1)^{n-1}}{(n-2)!} \sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)}\big({1\over r(s)}\\ \times \int^\infty_s (u-s)^{n-2}q(u) G(y(h(u))\big)du \big)ds, & t \geq T. \end{cases} \end{equation*} Then using \eqref{new3} and \eqref{new4} we find that $By<5$ and $By>1$. Hence $By \in S$ for $y \in S$.Next we show $BS$ is relatively compact as in the proof of Theorem 2.2. Then by Lemma 3.1 there is a fixed point $y_0$ in $S$ such that $By_0 = y_0$.Hence for $t\geq T$, we obtain \begin{align*} y_0(t)=& 3+\frac{(-1)^{n-1}}{(n-2)!}\sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G(y_0(h(u)))du \,ds\\ &-\sum^\infty_{l = 1} \int^{\tau_{-1}^{2l}(t)}_{\tau_{-1}^{2l-1}(t)} {F(s) \over r(s)} ds. \end{align*} In the above we replace $t$ by $\tau(t)$ and note that $\tau_{-1}^{m}(\tau(t))=\tau_{-1}^{m-1}(t)$ and $\tau_{-1}^0(t)=t$. Then It follows for $t\geq T$ that \begin{align*} y_0(\tau(t))=& 3+\frac{(-1)^{n-1}}{(n-2)!}\sum^\infty_{l = 1} \int^{\tau_{-1}^{2l-1}(t)}_{\tau_{-1}^{2l-2}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G(y_0(h(u)))du \,ds\\ & -\sum^\infty_{l = 1} \int^{\tau_{-1}^{2l-1}(t)}_{\tau_{-1}^{2l-2}(t)} {F(s) \over r(s)}\, ds. \end{align*} Then for $t\geq T$ we have \begin{align*} y_0(t)+y_0(\tau(t))=& 6+\frac{(-1)^{n-1}}{(n-2)!}\int_t^\infty {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G(y_0(h(u)))du \,ds\\ &-\int_t^\infty {F(s) \over r(s)} ds. \end{align*} Differentiating the above equation first and then multiplying by $r(t)$ to both sides and after that differentiating again for $n-1$ times, we see that $y_0$ is the required solution of \eqref{g1} which is bounded below by a positive constant.Hence this solution neither oscillates nor tends to zero as $t\to\infty$. Hence the theorem is proved. \end{proof} \begin{corollary}\label{cor3.3} If {\rm (H1), (H2), (H4)} hold, then there exists a positive solution of \eqref{g1} which is bounded below by a positive constant. \end{corollary} \begin{proof} The proof follows from Remark \ref{rmk1} and the above Theorem. \end{proof} \begin{theorem}\label{thm3.4} Let {\rm (H3), (H5)}, \eqref{f8} and \eqref{f9} hold. Then there exists a positive solution of \eqref{g1} which is bounded below by a positive constant that is, it neither oscillates nor tends to zero as $t$ tends to $\infty$. \end{theorem} The proof of the above theorem is similar to that of Theorem \ref{thm3.2}. \begin{theorem}\label{thm3.5} Suppose {\rm (H1)} hold. Assume for $t \geq t_0$ \begin{equation}\label{g3} \sum^\infty_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\,ds < \infty, \end{equation} and \begin{equation}\label{g4} \sum^\infty_ {i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {1\over r(s)} ds < \infty. \end{equation} Then \eqref{g2} has a solution bounded below by a positive constant. \end{theorem} \begin{proof} We proceed as in the proof of Theorem \ref{thm3.2} with the following changes. Let \begin{equation*} \mu = \max \{|G(x)|: 1 \leq x \leq 5\}. \end{equation*} Then from (H1), \eqref{g3} and \eqref{g4}, there exists $T> 0$ such that for $t \geq T$ \begin{equation*} \frac{\mu}{(n-2)!} \sum^\infty_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)}\frac{1}{r(s)} \int^\infty_s (u-s)^{n-2}q(u) du\, ds < 1, \end{equation*} and \begin{equation*} \sum^\infty_{i=1} \int^\infty_{\tau_{-1}^{i}(t)} {F(s) \over r(s)} ds <1. \end{equation*} Let $ S = \{y \in X : 1 \leq y \leq 5 ,\, t \geq T_0\}$. Then define \begin{equation*} By(t) = \begin{cases} By (T) ,& \text{for } t \in [T_0, T];\\ 3 + \frac{(-1)^n}{(n-2)!}\sum^{\infty}_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}\\ \times q(u) G(y(h(u)))du\,ds + \sum^\infty_{i =1} \int^\infty_{\tau_{-1}^{i}(t)} {F(s)\over r(s)} ds ,& \text{for } t \geq T. \end{cases} \end{equation*} Then as in Theorem \ref{thm2.2} we prove (i) $By \in S$ for $y \in S$, and (ii) $BS$ is relatively compact. Then by lemma \ref{lem3.1} there exists a fixed point $y_0 \in S$ such that $By_0 = y_0$, Putting $y_0 = y(t)$, we get \begin{align*} y(t) = 3 &- \frac{(-1)^{n-1}}{(n-2)!} \sum^\infty_{i =1} \int^\infty_{\tau_{-1}^{i}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G(y(h(u)))du\,ds\\ &+ \sum^\infty_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {F(s)\over r(s)}ds. \end{align*} Then replacing $t$ by $\tau(t)$ in the above and using $\tau_{-1}^i(\tau(t))=\tau_{-1}^{i-1}(t)$, we may obtain $y(\tau(t))$. Consequently for $t \geq T$, we find \begin{align*} y(t) - y(\tau(t)) = &\frac{(-1)^{n-1}}{(n-2)!} \int^\infty_t {1 \over r(s)} \int^\infty_s (u-s)^{n-2}q(u) G(y(h(u)))du \,ds\\ &-\int^\infty_t {F(s)\over r(s)}ds. \end{align*} We may differentiate the above and then multiply by $r(t)$ and then again differentiate $n-1$ times to arrive at \eqref{g2}. This solution is bounded below by a positive constant. \end{proof} \begin{remark} \label{rmkn1} \rm It is not difficult to verify that the above theorem still holds, if we replace \eqref{g4} and (H1) by the following assumption \begin{equation}\label{g5} \sum^\infty_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}f(u) du\,ds < \infty. \end{equation} Of course, in that case we have to modify the definition of the mapping $B$ as follows. \begin{equation*} By(t) = \begin{cases} By (T), & \text{for } t \in [T_0, T];\\ 3 -\frac{(-1)^{n-1}}{(n-2)!}\sum^{\infty}_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}\\ \times q(u) G(y(h(u)))du\,ds\\ + \frac{(-1)^{n-1}}{(n-2)!} \sum^{\infty}_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} {1 \over r(s)} \int^\infty_s (u-s)^{n-2}f(u) du\,ds , &\text{for } t \geq T. \end{cases} \end{equation*} If we put $r(t)=1$ in \eqref{g3} and \eqref{g5} then we obtain \begin{equation}\label{g6} \sum^\infty_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} \int^\infty_s (u-s)^{n-2}q(u) du\,ds < \infty, \end{equation} and \begin{equation}\label{g7} \sum^\infty_{i = 1} \int^\infty_{\tau_{-1}^{i}(t)} \int^\infty_s (u-s)^{n-2}f(u) du\,ds < \infty. \end{equation} \end{remark} Then from the above theorem the following result follows directly. \begin{corollary}\label{cor3.6} If \eqref{g6} and \eqref{g7} hold for $t>t_0$, then the NDDE \begin{equation}\label{g8} (y(t) - y(t - \tau))^{(n)} + q(t) G (y(t - \sigma))= f(t) \end{equation} has a solution, bounded below by a positive constant. \end{corollary} The above corollary improves and generalizes \cite[Theorem 3.1]{p2} and \cite[Theorem 2.5]{r2}, because in these papers, the authors assume the following additional conditions that we don't require. \begin{itemize} \item[(i)] $n$ is odd. \item[(ii)] $G$ is non-decreasing and $xG(x)>0$ for $x \neq 0$. \end{itemize} Before we close this article we present an interesting example which illustrates most of the results of this paper. \begin{example} \label{ex1}\rm Consider NDDE \begin{equation}\label{g10} (r(t) (y(t) - py(t/2))')^{n-1} + \frac{1}{t^{n+2}} G(y(t/3)) = 0 \quad \text{for } t \geq t_0. \end{equation} In this example suppose that $p$ is any constant and $r(t)\equiv 1$ or $r(t)\equiv \frac{1}{t^2}$. If we compare this equation \eqref{g10} with NDDE \eqref{eE} then $\tau(t)=\frac{t}{2}$, $h(t)=\frac{t}{3}$ and $q(t)=\frac{1}{t^{n+2}}$. It is not difficult to verify that $q(t)$ satisfies {(H2), (H5)} and \eqref{g3}. Suppose that $G(u) = 1- u^3$ and it is decreasing. Clearly the NDDE \eqref{g10} has a positive solution $y(t) \equiv 1$. Hence this example illustrates all the results of this paper. However since $G$ is decreasing and $\tau(t)$ is not of the form $t-k$, the existing results of \cite{d1,p1,p2,r1,r2,r3} are not applicable to \eqref{g10}. \end{example} \begin{thebibliography}{99} \bibitem{d1} Das, P.; \emph{Oscillations and asymptotic behaviour of solutions for second order neutral delay differential equations}, J. Indian. Math. Soc., 60,(1994), 159-170. \bibitem{e1} Erbe, L. H. and Kong, Q. K. and Zhang, B. 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