TAKA KUSANO. laculty of Science Hrosh tlnlersty 1982) (n-l) + + Pn(t)x 0, (n-l) + + Pn(t)Y f(t,y), XR R are continuous functions.

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1 Iera. J. Mah. & Mah. Si. Vol. 6 No. 3 (1983) ASYMPTOTIC RELATIOHIPS BETWEEN TWO HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS TAKA KUSANO laculy of Sciece Hrosh llersy 1982) ABSTRACT. Some asympoic relaioships bevee he wo ordiary differeial equaios () () (-l) x + Pl ()x + + P()x 0 (2) y() + Pl()y(-l) + + P()Y f(y) are sudied. Codiios are give ha lead o a asympoic equivalece bewee ceral of he soluios of (i) ad cerai of he soluios of (2). The case where he perurbaio f(y) depeds o a fucioal argume is also discussed. KEY ORDS AND PHRASES. Ordiary differeial equaios aympoic relaios fucio differeial equaios. 980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 34A. 1. INTRODUCT ION We are cocered wih some asympoic relaioships bewee he followig wo differeial equaios x() + Pl()x(-1) + + P()x 0 (1.1) () Y + Pl ()y (-l) + + P()Y f(y) (1.2) where Pi: [0 ) R i _< i _< ad f: [0oo) XR R are coiuous fucios. We obai codiios ha lead o a asympoic equivalece bewee cerai of he soluios of he liear equaio (i.i) ad cerai of he soluios of he perurbed liear equaio (1.2). No resricio is placed o he behavior of soluios of (i.i) which may be oscillaory pr ooscillaory. The example give a he ed of his oe deals wih he case where (i.i) has boh oscillaory ad ooscillaory soluios. Our resuls geeralize hose of Hallam [i] for secod order equaios. 2. MAIN RESULTS. I wha follows we deoe by W(l "m)() he Wroskia of I () m ().

2 560 T. KUSANO Le a fudameal sysem of soluios of (I.i) Xl() x ()} be fixed ad suppose ha here exis posiive coiuous fucios xi() i() I _< i _< which saisfy he iequaliies Ixi() <_ xi() o [0) I < i < (2.1) W(x I Xi_lXi+ I x) () * < i() o [0oo) i <_ i <_. W(x I x) () Noe ha W(x I x) () is a cosa muliple of exp(- I Pl(S)ds)" 0 (2.2) Wih regard o (1.2) we assume ha f(y) saisfies he iequaliy where [0)X R+ R+ R+-- If(y) < ( IYl) o [0o)XR (2.3) [0oo) is coiuous ad odecreasig i he secod variable for each fixed. THEOREM i. Suppose codiios (2.1)o (2.2). ad (2.3) are saisfied. Also suppose ha for some k i < k < ad some cosa c > i (s)(sc_(s))ds < i < i < (2.4) 0 ad x i () I * * Xk() i(s)(s cxk(s))ds o(i) as / (2.5) for I < i < wih i k. The here exiss a soluio y() of equaio (1.2) such ha y() Xk() + o(xk()) a.s. (2.6) l addiio for ay soluio Yk() of (1.2) sais.fyig he iequaliy lyk() < c() o [0) here exis_s a soluio x() of (I.I) such ha x() Yk() + O(Xk()) a s. (2.7)

3 ASYMPTOTIC RELATIONSHIPS BETWEEN DIFFERENTIAL EQUATIONS 561 PROOF. I view of (2.4) ad (2.5) we ca choose I > O so large ha *(s)(s c(s))ds < C-1 k 1 (2.8) ad xi*() r * * * J i(s)#(s cxk(s))ds Xk() < C--i 2(-l) > I (2.9) for 1 < i < wih $ # k. Le C[lOO) be he locally covex space of coiuous fucios o [ oo) wih he compac ope opology ad cosider he closed covex subse F of C[l=o) defied by F {y e C[I) ly() < CXk() Defie he operaor : F- C[l by he formula _> l}. y() Xk( + 7. (-l)-i-i x i () f (x I Xi_l xi+i X) where i=l (s) f (sy(s))ds (2 I0) (Xl... xi_lxi+ 1 X) () W(x I Xi_lXi+1 x) () W(x I x) () (2.11) We seek for a fixed poi y y() of i F. Usig he ideiies 7. i--0 (-l)-ix %- J ()W 0 < - i (x I. xi+ I X) () 0 j < 2 (2.12) I (-l)-ix. (-l) I ()W(x i xi_lxi+ 1 x) () (x I X) () i--0 (2.13) we easily see ha a fixed poi of a soluio of equaio (1.2). is i) maps F io F. This follows immediaely from (2.1) (2.2) (2.3) (2.8) (2.9) ad (2.10). ii) i s coiuous. Le {yg} be a sequece i F covergig uiformly o y e F

4 562 T. KUSANO o every compac subierval of [lco). Cosider ay compac subierval of he form [l2]. Le g > 0 be give ad choose 3 > 2 so ha * * e i(s)(scxk(s))ds < 4M i < i < (2.14) 3 where M max max l<i< [l2] xi(). Take a ieger N such ha * i(s) If(syg(s)) f(sy(s)) < s e [l3] i < i < (2.15) 2ld(3- I) for all 9 >_ N; his is possible sice f(y) is coiuous ad {yg} coverges uiformly o y o [l3]. Usig (2.14) ad (2.15) we have y() 7y() _< f 3 Y. xi() J i (s) If(sY (s)) f(sy(s))ids i=l I f + 2 Z xi() i (s)(s cxk(s))ds < =1 3 for all g [ l 2] ad all 9 >_ N. This shows ha ] is coiuous o F. iii) F i s relaively compac. From (2.10) (2.11) ad (2.12) (wih J 0) we obai r * * l<y) (=)l _< li()[ + z lxi() j i(s)(scxk(s))ds. i=l O ay compac subierval of [0=) he righ-had side of he above iequaliy is bouded by a cosa idepede of y e F. Therefore F is equicoiuous o every compac subierval of [lm) ad is relaive compacess follows from he Ascoli- Arzela heorem. Applyig ow he Schauder-Tychooff fixed-poi heorem we see ha he operaor has a fixed poi y y() i F. As remarked earlier y() is a soluio of (1.2).

5 A rr1c RELATIONSHIPS BETWEEN DIFFERENTIAL EQUATIONS 563 Tha y() saisfies he order relaio (2.6) follows from he iequaliy [ i-i wih he help of hypoheses (2.4) ad (2.5). To prove he opposie relaioship bewee he soluios of (i.i) ad (1.2) le yk() be a soluio of (2.2) saisfyig he iequaliy lyk() < CXk() o [o). The i is esy o verify ha he fucio x() defied by x() Yk() +?. (-l)-ixi() (Xl...Xi_lXi+ 1 X)(S)f(sYk(S))ds i-i (2.16) is a soluio of (i.i) which saisfies he order relaios (2.7). Thus he proof of Theorem 1 is complee. THEOREM 2. Le codiios (2.1) (2.2) ad (2.3) be saisfied. Suppose ha for some ieger k 1 < k < ad some cosa c > i k(s)(scxk(s))ds < o (2.17) ad xi() Xk() I J gi(s)$(s cxk(s))ds o(1) as o (2.18) for I < i < wih i k. The here exiss a soluio y() of (I2) such ha (2.6) is saisfied. I add-- iio if Yk() is a soluio of (1.2) saisfyig he iequaliy [Yk()[ _< CXk() he here exiss a soluio x() of (i) such ha (2.7) is saisfied. PROOF. I suffices o proceed as i he proof of Theorem i by replacig (210) ad (2.16) respecively by -k-1 y() Xk() / (-ll Xk() (Xl Xk_lXk+ 1 x)(s)f(sy(s))ds +. f (-x)-xx()j (Sl...Xi_lXil...x)(s)f(sy(s))ds i=l 1 isk (2.19)

6 564 T. KUSANO ad x() Yk() + (-l)-kxk() I (x I Xk_lXk+l...X)(S)f(sYk(S))ds )-i-lxi + Z (-i () (x I Xi_lXi+ 1 X)(S)f(sYk(S))ds. (2.20) i=l i+k l The deails will be lef o he reader. I is o hard o see ha he above argumes ca be applied o esablish sireilar asympoic relaioship bewee (i.i) ad he fucioal differeial equaio () (-l) y () + Pl()y () + + P()y() f(y(g()) (2.21) where Pi() ad f(y) are as before ad g [0) R is a coiuous fucio such. ha lira g() For example we have he followig aalogue of Theorem i. THEOREM 3. Le codiios (2.1) (2.2) ad (2.3) be saisfied. Suppose ha for some k I < k < ad some cosa c > I 0 i(s)(scxk(g(s)))ds < 1 < i_< (2.22) x i () f J i(s)(scxk(g(s)))ds o(1) as Xk() (2.23) for I < i < wih i k. The (2.21) has a soluio y() which saisfies (2.6). l_ addiio i f Yk() i_s a soluio o_ (2.21) saisfyig lyk() _< CXk() he (1.1) has a soluio x() such ha (2.7) is saisfied. 3. EXAMPLE. Cosider he hird order differeial equaios x" + x 0 (3.1) y + y b()[y[sg y (3.2) where y > 0 is a cosa ad b [0 =o) R is a coiuous fucio. The fucios Xl() (2/3e - x2() e /2 cos2) x3() s/2 sio2) form a fudameal sysem of soluios of (3.1) such ha W(XlX2X3)() i. We ca ake

7 ASYMPTOTIC RELATIONSHIPS BETWEEN DIFFERENTIAL EQUATIONS 565 * - * * /2 * * * Xl() (2/)e x2() x3() e $1() (72)e 2() 3() -/2 (213)e Suppose ha 0 e(l-y)lb( Id < =. (3.3) Theorem 1 (wih k-- i) he esures ha (3.2) has a soluio Yl() such ha Yl() xl() + o(e-) as 0% (3.4) Suppose ex ha 0 (l+(y/2)) e Ib() Id < (3.5) The applyig Theorem i (wih k-- 2 ad k 3) we see ha (3.2) has soluios Y2() ad Y3() such ha Y2() x2() /2 + o(e as oo (3.6) ad Y3() x3() + o (e /2 as oo. (37) Obviously Yl() is ooscillaory whereas Y2() ad y3(_) are oscillaory. Sice (3.5) is sroger ha (3.3) (3.5) guaraees he exisece of all he hree soluios Yl() Y2() ad Y3() lised above. From Theorem 1 i also follows ha i case (3.5) holds if y() is a soluio of (3.2) saisfyig /2 ly() < c e (3.8) for some cosa c > i he here exiss a soluio x() of (3.1) such ha x() y() + o(e /2) as o% Noe ha o all soluios of (3.2) are subjec o his esimae. I fac equaio (3.2) wih y 3 ad b() 28e 6 has a soluio 3 y() e eve hough (3.5) is saisfied. Fially cosider he fucioal differeial equaio y () + y() b()ly( + si )l sg y( + si ) (3.9)

8 566 T. KUSANO where y ad b() are as above. Appealig o Theorem i we coclude ha if (3.5) holds (3.9) has hree soluios Yl() Y2( ad Y3() wih properies (3.4) (3.6) ad (3.7) respecively. ACKNOWLEDGEMENT. This work was doe while he auhor was visiig he Iowa Sae Uiversiy. The auhor wishes o express his sicere haks o Professor R. S. Dahiya for his iviaio ad hospialiy. I. HALLAM T.G. Asympoic relaioships bewee he soluios of wo secod order differeial equaios A. Polo. Mah. 24 (1971)

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