We investigate sufficient conditions (theorem 1) for the non-existence or periodic

solutions of equation (2. 1) with P=0 and sufficient conditions (theorem 2)

 for existence of periodic solutions of equation ( 1.1.1 )

Iyase,  S.A .

Tejumola, H.O.

Covenant University Repository

.. PERIODIC SOLUTIONS OF A CLASS OF EVEN ORDER DIFFERENTIAL EQUATIONS " H. 0. Tejumola (U niversi ty of lbadan, Ihadan, ~igeria) S. A. lyase (U niversity o f Ah11ja, Ahuja, Nip,eri<L) AhstnH:t W e investigaLC' sufTi cienL cond i tions (theorem .I) for t II<' non-<:Xi:iL<'nC"!' or JH'-riodic solutions of equation (2. 1) wi t h P:: 0 and sufl irir;nt conditions (Lhc•orc•nl 2) for existence of periodic solutions of eq nat ion ( 1.1.1 ). K ey w ords w-perio d1c so lution 1 Introduction 1.1 The aim of t.his pa per is to provide sufficient conditious for IIi<' exist\'IICC' of periodic solutions of the 2rth order di f[erC' nt ia l equa.t. ion .r(2•· ) + a 1x(2•·- t ) + a2x(2r - 2) + · · · + a2r-3 J: +.hr - 2(i)i·-!- hr-d.r).i· + J2,.(.r) = p(t 2 x 2x2 i: ... _7.('2r-tl) ( 1. 1. i 1 and the non-existeJt ce of non-t rivial per iodic solut iens in Lhl' nts<· p:::: U. ,,.h<'r< ' at , a 2, a3, · · · , a2r - 3 are constants and }2,--2, f2r-l, f 2r, p <u·e continllOIIS real-valued funct ions depending only on Lhc argumen ts show n, L! 1c fu nct ion fJ is w-periodic in t , that is p(t , .r:1, : · · , x'2·r) :::= p(t + w, .1: 1 , · · · . . 1'2,) for SOill<' w > 0 and for a rbitrary x 1 , · · ·, x2, . . Such periodic di fferential <'quat ions r~risc 'Manuscript received Nov. 10, 1996 :209 No.3 If. 0. Tejumo/a etc. P SOLUTfONS Of Ai\ FVEJ\ O IWL'I? !)IFF. 1-."i).c ~II From (1.2.4) we a.b~ derive that 6 2 (n:) > 0 i!'.,. i!, cvc11 c111d o 21 ._ 1 • 1 satisfy (Ll ~ 0, n:3 :::; 0, a.1 ~ 0. 0 'Lr· - J <0 or cP2 (n) < 0 if 1· is odd <w d o.2 ,_ 1. j = 1, '2.. · · · . r- I 'i<ll i!--. 1\ 0 J ::::: 0. 11:\ :::; 0. fL:; ::::: (). ' · ' 11'2,- I < () 1.:2 ... .I I • ·) -\ j • ._. ( i .l."' l Thus it. follow s from (1 .2.5) th u.t. if,. is CVl'Jl ( rcsp. !'rOJJl (l.l.(i) i!' I is cHid• t.hat. ( 1.2.1) does not have periodi c solutions except. .t = 0. The refore from the genf'ral theory, the pert mbcd equal ion x( 2'') + U. JX(2•· - J ) + 0.2:1'( 2•·-'.!) + · · · + 0.·2r-'2;j'- + U'l.r-l·j· + (!2 r·1: ;-_ fl(l) ( I.'!. .•J I m which p is a continuous w-peri ocl ic function of l h<ts a nuiqtH' ....._·-pc·J i.H1i( so lution. Our main object here is to obtain nou-iin('ar analogues or ( i .:~ .. -,) and ( 1.:2.6) from whicb we shal l dcrivf' om Jton-c '\ist C'J JIC' and ('Xi st ('nee· J'('~tdt '· Similar non-existence a.nd c~xist.enc<' results using non -llllca r ilJ Jrdogu(•c.; ol' ( 1.'.!. 7 ) and (l.2.o) a r<' liJJcl er preparation a.nd will <lppc-'<n elsewhere. 2 M ain results 2.1 \•Vc st.a.rt here with 1·.11(' difff're11Lia.l cqu;-1!.iou .1'( L,·) +a ).1.( 2'' - I)+ (L2L('l r - 2 ) + ... + U.2r-:\ :r + f-2 ··-·l( .i: ).i'· + I2r- I ( .r ).1- -j- I'2·· ( I', :.:: () U'2r(O) = Ol ( •) ' - w l . • j where a!; bf'fore a1 , a2, a3. · · ·, 02r- :1 a.r<' con~lant~ <lltd /2·-·2· f.~r -1· h an· continuous real valued !'unction~ d<'p<'ndiJl.f; <mh· 011 1 h<' argu nw111~ silo\\'JJ. 01;1 non-<'xis!.ence result. is as follows: Theorem 1 Lel t.hc constants a 2.7, J -=- I, :2 . · · · . r - :2 an(i tlw !'i1nct i<,,J Ih-'L and f.2, sati sfy 02 :::; 0. (J.~ ::::: 0. 01.; :::; Q. · · · . 112,- 1 ~ 0 } y .J;;' .h ·-·L(s)d.5:::; 0. !Jh ,(y) > 0 (,11 i 0) ~~ - ~-~' -· No.:l 11. 0. Trjumola etc. P. SOLUTIONS OF J\.'\' EVli:N ORDF:R /)IFf'. EQS'. :!l:l To prove theorem 1, i L suffices to show that. every w- periodic sol u Lion ( x d I). · · · . :r,h.(t)) of (2.L.l) c:;atisfies (x1(l), x 2 (t), · · ·, x 2,.( t)) = (0 . 0, 0, · · ·, 0) for all /. To this end we defiue a funct ion V(y 1, • • ·, Y2r) whirlt will play a cruci<:d role-in our proof as follows, ( Yl V = (-lr+1ytfJ'2 hr-2(s)ds+(-lr+t Jo shr-t(s)ds 2r-3 +( -l)r+ 1Yt(Y2r + 2::::: Clk'!J2,·-d k=l 2r+ 1-'lk + 2:::::( -1 )kYk(Y2r+1-k + 2::::: CljY2r+t-(k+i)) k=2 ,7=1 1'-1 l"' k 2 +? L..) -1) Cl2k-tY(r+1)-k - k=l (:2.2.:2) and show that 1·, = 0 and from th is we conclude that (y 1(L), y'2(t), · · ·. y2,(t)) = ( 0. 0, 0. · · · , 0) for all l. To prove theorem 2, we will consider the non-linear vector difi'crential equa-tion. x = A(t)x + G(t, x) (2.2.:l) whNcA(t)isannxn matrixcontinuouslyforallt E R. A(t) = !l(l+..v), (,'(t .. r) is continuous for all (L,:c) E R x Rn, G(T,x) = (; (L +w,x) and t.lw following Lheorcn1. TheorC'mi'IJ lf a. th<' homogeneous linear equation i = A(L) :r (L.1.1) ba~ no non-trivial ..,;-periodic solution a11d if b. t.lwre exists an a-priori estimate independent of f./· for the w-pcriodic sul11t ions or the equation x = A(L)x + J..LG(t, .c), 0 ~ f..L ~ 1, (2 •) :-) ·-· ·) t hf'n (2.2.3) admits a.L least om· w-periodic solution. In what follows for atl\' I ""-'1 ~.r( l.r ) l ·· ·'i'{i/ __ l'.c( <.rj7 -·'7f11- ~ -L + ' <.r Y n =:i _ 'i .r ( 1 -- ' (. - - . . -~: · (. • 1 :__ , \ 1 1 r I/ lll.IOj ·''!lil t (1 ·;-. T\ . >) 11.\1-ll \ ', ' " ' , ;\\ I' PIH' s~ .i .1 1 (() i= .1' ) ·o > '<r, .:; ------ ( .l) ·''I . .• , "·' · ' ··lljl'.\ , )'.,()'1,1\ )Ill! )SilO.! ~~ Sl ''I ' ( ,.:'I) q I~·'' ,.,Jj>i.ill~U.i \ i '(.): - i \' /'. . 1'- ( ' ., ) ,,, -( -)' . . . . ( . .,_ ,1-·1 (. • + (!-• l.)'/' 11) T ( .• ; )./ .I lJ ! t I ' ~ . ' lll.IO.J dli I I I ~ ( J'J ' J ) ,J) I.T.\\ ,> I ,)_\\ "<it _ - .. = l-/1 = (.{j = 1(1 = (I IUIJ) ( 1 -- -'(. · .. - ·~: "(. · 1 = !) 1+'/i = '(i ) jjl! .l<Jj 0 = (!)!\ • I ! ' . !. 11 1~ .I 0 .1 °;1 = ( I) / \ JM'lJ ,J,\1 'Ill .\.!I!.! l!lj I I' .\I !? !<) j .l >I iI 1... • ,. No.3 11 . 0. Tejumola etc. P. SOLUTIONS OF AN EVEN ORDI~R ViFf'. EQS. 417 Fro111 (3.2.9) a2r-2 = O.'lr- 1 = 0 (:LL I~ ) Similarly from (3.2.3) or (3 .2.4) we geL a2,· = b2r > 0 (3.2. 1:3 ) or a2r = b2r < 0 (:{.:2. 1·1) depending on whether 1· is even or odd. Combining (3.2. 1 0 ) - (:~.2._] ,1) we get that (3.2.8) has no non-trivial w-pcriodic solution. l-Ienee assumpLion (a.) of Lheorem of is saLisficd. Since dyL( e-wll - l ) :j:. 0, ] being the 2·r x 2r idcn1 il\· matrix a ll w-period ic ~lutions x(t) of (3.2.6) must saLisfy the integral equation !t+w x(L) = J.LT{x(t)} = J.L(e-w11 - 1)- 1 e-(s- t)AG(s , x (s ))d.~ ' t (3.2.1.5) To verify hypothesis (b) of theorem we must show that llxll = max(lx,(t)i + lx2(t)l + · · · + lx'lr- 1 (L )i) :::; Uo CL2.1 G) lt is sufficient in view of (3.2.7) and (3.2.15) to obtain t he following estimates . max lx1(L)i :::; D1 , max lx2('t) l :::; D2, max lx3(t)i :::; D3 t E[O,wJ tE[O,wJ tE[O,w] (3.:L17) For every w-periodic solution y(l) of (3.2.1 ) , we define a. function w = w(!}J · · · Y'lr ) by w = [Yl {Yl ( - 1r+'Y1 Jo J'2,._2(s)ds + (-1)"'+1J.L Jo s.f'2,·-J(8)'ds 2r - a +( - lr+1Yt (Y2·· + L aky2,·-k) k= l r 2r+ 1- 2k + L ( -l)kYdY2r+l - k + L a jY2r+ l-(k+; )] k=2 J=l r - 1 1 " ( )k 2 +2 ~ - 1 a2k- t !/(,·+l )- k k= l (:L2.1X) • .\"o.:l If. 0. Tejumo/a de. P. SOLUTION~ OV :'\ :\" EVI~S 0/WVU !)IfF /-.'()'. . :! I., and bv hypothesi!' (:2 . 1.6) we have iltxpl ~ ki.Ti Tl1u~ r-"1. 1c > L ((-l)ka2kx(r+l)-k) + ( - tr+"J.b2rx 2 - k lx! k=l 2 I L . 2 , 2 .2 ··: ··.'2 r+ l 1' > -a.2.x,. + a ".r.,._ 1 - 0. 0 X,._ 2 + · · · + a2,·-6 .I +<.l2T- 'i ·l. - (-I) --•1 h:lr Usiug conditioJJ (3.2.10) if r i:; even or (3.2.11) if r is odd , W<' get :2 ~:2 ' , 2 2 . ."2 ~·~ ... 2 T + :2 1\ w 2:: D, (x.,. + r,._ 1 + .l:r_2 + · · · + X +:t ) - (- I) -!J b,,. O.LU) where /J 1 = min( -a2, a1, -an,···, -a2,.-t;, a2,-4 ). lnt.cgrat.ing (:~.2.:22) from I = 0 to f =...,;and u!'ing the fart. t.hat w{t) = w(f + ...,;) we get .'2 • . 2 .:2 ::"1. ... :2 r+"l. I\' u) j·w . 1"1. () 2:: J)i 0 (.l ,. + :r ··-1 + J r-'2 + · · · + .1. +.1 )dl - (-I) -1- b'2, I i<'nc(' where Sine<' l w ... 2 (- 1 )'+"1. Pw :r; dt < = /) .2 . 0 - 'lb2r lw i 2dl ~ 0 :3 03 = max[D 02w2 2· 4;-2] l w :2 !'" ., '2 u) .... .l- dl ~ -1 2 .r dt u ' 7r • 0 Tlw j.Wriodicit.y condition .i:(O) = .i(w) on x impli<'s that .i: ('/\) = 0 a.t SOIIl<' '1 '1 E (O,w). Thus /·I i( l) = i('l'i) + J :i: ( s )ds · T1 (:l.Ltl l (:L2.:2 I) . (I I ('' \ .. ' o \ ' - · 1 I I . ( !. <.: ) I I :\ ' [ . )I'.\ w! \ , ! : [ . I.) ~ . I ' ) . pI J.)'H . I' ).) Ill ' I . z I' N . ! )\~ ) \ . I ll \ . . II() ! )I~ II h:. I [l!l ll [, l.1, ) jj l(j .l ·>jl .IO lj).H\0) ll ll').l,l.) JO <.; t iO I )fi[O<; .J!flO!.l .lcJ '() ' ]I 'l'j< •ll'lll,l, j [<J: ( -.:<)() j) (h ( I ) ldd \" P..lll d t:' J\ '1111 \' ' SilO! Jl!ll h . l Jl!i]ll.ll,lJII(I !.l[J.IO q1 ::1n 1 ":"·!.!.).) y1 ._,,o, ,n, os 1o ,( )I J!pou .>d P" '' "";>11p;>p1 111 nu ·o 11 · ": <llllll l'.l L [<_J ( I Ui 1) ' ' ·'PA,l'T '.!loqp.1oo.\i ·.l.>p.l() !.>' !'41! II .J n• ~'I I ()~ ) l!i ib:., [I? ! lli.J!.ljp ( I .W. l l l ~ [ l lOj\' ' H !')1 10:) ptH~ · ~) ',) fi ()O.: II I'S ' H ' 'ill ! t~<;(, ) H II ·] \·l· ' uos I'~·>! )P 111,l\['PT \ lll?l.l.)'<f l \ ;·"I' ,10 !t:tunor· ·o.: 11n ! )l'll iJ, l l~'lltt,l.l,lji!J ' t.!p.!O 'l t(i11 d ll n?).l.l.l J O •;r io ! p qo c; .>! POl l.>, I '\' 'S ·,,o.:t:.,· f [l:! i ;:;: • .. (' 1/ . ·J ·! !.-; .'I I rill(} \ '·1. \ :·1 \T ·I() .'-. \' fl { /, l /0.'·." d ·' .. 

Periodic Solutions of a Class of Even Order Differential Equations

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Periodic Solutions of a Class of Even Order Differential Equations

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