ARCHIVUM MAHEMAICUM (BRNO) omu 47 (20), 23 33 MINIMAL AND MAXIMAL SOLUIONS OF FOURH ORDER IERAED DIFFERENIAL EQUAIONS WIH SINGULAR NONLINEARIY Kritín Rotá Abtrct. In thi pper we re concerned with ufficient condition for the exitence of miniml nd mximl olution of differentil eqution of the form L 4 y + f(t, y) = 0, where L 4 y i the iterted liner differentil opertor of order 4 nd f : [, ) (0, ) (0, ) i continuou function.. Introduction he purpoe of thi pper i to tudy the exitence of poitive olution with pecific ymptotic behvior for differentil eqution of the form L 4 y + f(t, y) = 0, where L 4 y i the iterted liner differentil opertor of fourth order defined below nd f : [, ) (0, ) (0, ) i continuou function, nonincreing in the econd vrible. A prototype of uch eqution i the eqution with ingulr nonlinerity f(t, y) = Q(t)y λ, where λ > 0 nd Q: [, ) (0, ) i continuou. Such eqution of the econd order were tudied in [3], [4]. Differentil eqution with iterted liner differentil opertor were tudied, for intnce, in [5]. 2. Iterted differentil eqution of the fourth order If u nd v re linerly independent olution of (A 2 ) y + P (t)y = 0, where P C 2 [, ), then u, v C 4 [, ) nd the linerly independent function y (t) = u 3 (t), y 2 (t) = u 2 (t)v(t), y 3 (t) = u(t)v 2 (t), y 4 (t) = v 3 (t) 200 Mthemtic Subject Clifiction: primry 34C0. Key word nd phre: iterted differentil eqution, mximl nd miniml olution. Supported by the VEGA Grnt Agency of the Slovk Republic No. /048/08 Received April 5, 2008, revied Augut 200. Editor O. Došlý.
24 K. ROSÁS tify the fourth order liner differentil eqution (A 4 ) y IV + 0P (t)y + 0P (t)y + [3P (t) + 9P 2 (t)]y = 0, (ee []). Differentil eqution (A 4 ) i clled iterted liner differentil eqution of the fourth order. We my uppoe without lo of generlity tht W [u, v](t) for t, where W [u, v](t) denote Wronkin of function u nd v. An elementry clcultion how tht Wronkin of function tifie y (t) = u 3 (t), y 2 (t) = u 2 (t)v(t), y 3 (t) = u(t)v 2 (t), y 4 (t) = v 3 (t) W (u 3, u 2 v, uv 2, v 3 )(t) 2 for t. We uppoe tht the eqution (A 2 ) i nonocilltory nd the u(t) (rep. v(t)) denote principl (rep. nonprincipl) olution of (A 2 ) uch tht nd dt u 2 (t) = u(t) v(t) = 0 (rep. dt v 2 (t) < ). We my ume tht both u(t) nd v(t) re eventully poitive. Second, nonprincipl v(t) of (A 2 ) i given by d v(t) = u(t), t. In thi pper we re concerned with the behvior of olution of differentil eqution of the form L 4 y + f(t, y) = 0, where L 4 y i the iterted liner differentil opertor of order 4 nd f : [, ) (0, ) (0, ) i continuou function. From Póly fctoriztion theory it follow tht the opertor L 4 y cn be written in the form L 4 y = 4 (t) ( 3 (t) ( 2 (t) ( (t) ( 0 (t)y ) ) ) ), where 0 (t) = u 3 (t), (t) = u 2 (t), 2 (t) = u2 (t), 3 (t) = u2 (t), 4 (t) = 6 2 3 u 3 (t), ee [6].
MINIMAL AND MAXIMAL SOLUIONS 25 3. Clifiction of poitive olution Conider the fourth order differentil eqution (A) L 4 y(t) + f ( t, y(t) ) = 0 where L 4 y i the iterted liner differentil opertor of order 4 nd f : [, ) (0, ) (0, ) i continuou, nd nonincreing in the econd vrible. We ume tht the eqution (A 2 ) i nonocilltory nd put nd L i y(t) = u2 (t) i L 4 y = 6 ( u 2 (t) u 3 (t) 3 L 0 y(t) = y(t) u 3 (t), d ( Li y(t) ), i 3, dt ( u 2 (t) 2 ( ( ) ) ) ). u 2 (t) u 3 (t) y he domin D(L 4 ) of the opertor L 4 i defined to be the et of ll continuou function y : [ y, ) (0, ), y uch tht L i y(t) for 0 i 3 exit nd re continuouly differentible on [ y, ). hoe function which vnih in neighborhood of infinity will be excluded from our conidertion. Our purpoe here i to mke detiled nlyi of the tructure of the et of ll poible poitive olution of the eqution (A). We ue the following lemm which i the pecil ce of generlized Kigurdze lemm (ee [2]). Lemm. If y(t) i poitive olution of (A), then either () or (2) L 0 y(t) > 0, L y(t) > 0, L 2 y(t) < 0, L 3 y(t) > 0, L 4 y(t) < 0, L 0 y(t) > 0, L y(t) > 0, L 2 y(t) > 0, L 3 y(t) > 0, L 4 y(t) < 0, for ll ufficiently lrge t. Solution tifying () nd (2) re clled olution of Kigurdze degree nd 3, repectively. If we denote by P the et of ll poitive olution of (A) nd by P l the et of ll olution of degree l, then we hve: P = P P 3. Conider P l for l = {, 3}. For ny y P l the it L ly(t) = c l (finite), L l y(t) = c l (finite or infinite but not zero) both exit. Solution y P l i clled mximl in P l, if c l i nonzero nd miniml in P l, if c l i finite. he et of ll mximl olution in P l denote P l [mx] nd the et of ll miniml olution in P l denote P l [min].
26 K. ROSÁS If c l = 0 for olution y P l for l {, 3}, then y i clled intermedite in P l. he et of ll intermedite olution in P l denote P l [int]. hen P = P [min] P [int] P [mx] P 3 [min] P 3 [int] P 3 [mx]. Our objective i to give ufficient condition for the exitence of mximl nd miniml olution in P i for i =, 3. Crucil role will be plyed by integrl repreenttion for thoe fourth type of olution of (A) derived below. Firt we define: I 0 = nd I i (t, ; u) = u 2 (r) I i (r, ; u) dr, i 3. If the econd, linerly independent olution v(t) of (A 2 ) i given by v(t) = u(t) d for t, then the et of poitive function x 0 (t) = u 3 (t), x (t) = u 3 (t) x 2 (t) = u 3 (t) x 3 (t) = u 3 (t) d = u2 (t) v(t), 2 u 2 (r) dr d = u(t) v2 (t), 2 u 2 (r) r 3 u 2 (ξ) dξ dr d = v3 (t) defined on [, ) form fundmentl et of poitive olution for L 4 x = 0 (i.e. (A 4 ), which re ymptoticlly ordered in the ene tht x i (t) x j (t) = 0 for 0 i < j 3, ee [2]. It i ueful to note tht I i (t, ; u) = ( v(t) ) i for i =, 2, 3. i! u(t) he olution from the cle P 3 [mx], P 3 [min], P [mx] nd P [min] tify the propertie y(t) v 3 (t) = λ 3, y(t) u(t)v 2 (t) = λ 2, y(t) u 2 (t)v(t) = λ nd y(t) u 3 (t) = λ 0, repectively, where 0 < λ i <, i =, 2, 3.
MINIMAL AND MAXIMAL SOLUIONS 27 4. Integrl repreenttion for olution Now we cn derive integrl repreenttion for type P 3 [mx], P 3 [min], P [mx] nd P [min]. Let y be olution of (A) uch tht y(t) > 0 for t. Integrting (A) from t to give (3) L 3 y(t) = c 3 + t u 3 () 6 f (, y() ) d, t, where c 3 = L 3 y(t) 0. If y P 3 [mx], then we integrte (3) three time over [, t] to obtin y(t) = k 0 u 3 (t) + k u 3 (t) + 6c 3 u 3 (t) u 2 ( ) + u 3 (t) d + 2k 2u 3 (t) I 2 (t, ; u) u 2 ( 2 ) 2 u 2 ( ) u 2 ( 3 ) d 3 d 2 d u 3 (r) f ( r, y(r) ) dr d, u 2 ( 2 ) d 2 d for t, where k i = L i y( ) for i = 0,, 2 nd we ued Fubini theorem. If y i olution of type P 3 [min], then integrting (3) with c 3 = 0 from t to nd then integrting the reulting eqution twice from to t, we hve y(t) = k 0 u 3 (t) + k u 3 (t) u 3 (t) I (t, ; u) u 2 ( ) d + 2c 2 u 3 (t) u 2 ( ) I (r, ; u) u 3 (r) f ( r, y(r) ) dr d, u 2 ( 2 ) d 2 d for t, where c 2 = L 2 y(t). An integrl repreenttion for olution y of type P [mx] i derived by integrting (3) with c 3 = c 2 = 0 twice from t to nd once on [, t] y(t) = k 0 u 3 (t) + c u 3 (t) + u 3 (t) u 2 ( ) d I 2 (r, ; u)u 3 (r) f ( r, y(r) ) dr d, for t, where c = L y(t) nd we ued Fubini theorem. If y P [min], then integrtion of (3) with c 3 = c 2 = c = 0 three time on (t, ) yield y(t) = c 0 u 3 (t) u 3 (t) for t, where c 0 = L 0 y(t). t I 3 (, t; u)u 3 () f (, y() ) d,
28 K. ROSÁS 5. Exitence theorem We re now prepred to dicu the exitence of mximl nd miniml olution of eqution (A) of type P nd P 3. heorem. he eqution (A) h poitive olution of type P 3 [mx] if (4) u 3 (t) f ( t, cv 3 (t) ) dt <, for ome c > 0. Proof. We ume tht (4) hold. hen there i uch tht u 3 (t) f ( t, cv 3 (t) ) dt < c. Let C denote loclly convex pce of ll continuou function y : [, ) R with the topology of uniform convergence on compct ubintervl of [, ). Define the ubet Y 3 of C[, ) nd mpping Φ 3 : Y 3 C[, ) by nd Y 3 = {y C[, ) : cv 3 (t) y(t) 2cv 3 (t), t } Φ 3 y(t) = cv 3 (t) + u 3 (t) I 2 (t, ; u) u 3 (r) f ( r, y(r) ) dr d. We will how tht (i): Φ 3 mp Y 3 into Y 3, (ii): Φ 3 i continuou on Y 3, (iii): Φ 3 (Y 3 ) i reltively compct. then nd (i) Since 0 u 3 (t) u 3 (t) I 2 (t, ; u) I 2 (t, ; u) Φ 3 y(t) cv 3 (t) Φ 3 y(t) cv 3 (t) + u 3 (t) And o Φ 3 y Y 3. cv 3 (t) + cu 3 (t) I 2 (t, ; u) cv 3 (t) + cv 3 (t) = 2cv 3 (t). u 3 (r) f ( r, y(r) ) dr d u 3 (r) f ( r, cv 3 (r) ) dr d, I 2 (t, ; u) d u 3 (r) f ( r, cv 3 (r) ) dr d
MINIMAL AND MAXIMAL SOLUIONS 29 (ii) Suppoe tht {y n } Y 3 nd y Y 3, nd tht y n = y in the topology n of C[, ). We hve Φ3 y n (t) Φ 3 y(t) u 3 (t) 6v 3 (t) I 2 (t, ; u) u 3 (r) f ( r, y n (r) ) f ( r, y(r) ) dr d u 3 () f (, yn () ) f (, y() ) d Becue f (, y n () ) f (, y() ) 2f (, cv 3 () ) nd f(, y n ()) f(, y()) = 0 for, then pplying the Lebegue convergence theorem, we hve Φ 3 y n (t) Φ 3 y(t) 0 for n on every compct ubintervl of [, ], which implie tht Φ 3 y i continuou on Y 3. (iii) If y Y 3, then we hve for t (, ) d dt ( u 3 (t) Φ 3y(t)) 2c u 2 (t) I 2(t, ; u). hi how tht the function d dt( u 3 (t) Φ 3y(t) ) i uniformly bounded on ny compct ubintervl of [, ), nd o function u 3 (t) Φ 3y(t) i equicontinuou on (, ). Now for t, t 2 [, ) we ee tht Φ3 y(t 2 ) Φ 3 y(t ) u 3 (t 2 ) u 3 (t ) u 3 (t 2 ) Φ 3y(t 2 ) + u 3 (t ) u 3 (t 2 ) Φ 3y(t 2 ) u 3 (t ) Φ 3y(t ) 2c u 3 (t 2 ) v3 (t 2 ) u 3 (t 2 ) u 3 (t ) + u 3 (t ) u 3 (t 2 ) Φ 3y(t 2 ) u 3 (t ) Φ 3y(t ), nd hence Φ 3 (Y 3 ) i equicontinuou t every point of [, ). Since Φ 3 (Y 3 ) i clerly uniformly bounded on [, ), it follow from Acoli-Arzèl theorem tht Φ 3 (Y 3 ) i reltively compct. herefore, by the Schuder-ychonoff fixed point theorem, there exit fixed element y Y 3 of Φ 3, i.e. Φ 3 y = y, which tifie the integrl eqution y(t) = cv 3 (t) + u 3 (t) I 2 (t, ; u) u 3 (r) f ( r, y(r) ) dr d. A imple computtion how tht thi fixed point i olution of (A) of type P 3 [mx]. he proof of heorem i complete. heorem 2. he eqution (A) h poitive olution of type P 3 [min] if (5) u 2 (t)v(t) f ( t, cu(t)v 2 (t) ) dt <,
30 K. ROSÁS for ome c > 0. Proof. Suppoe tht (5) hold. Chooe o tht (6) u 2 (t)v(t) f ( t, cu(t)v 2 (t) ) dt < c. Conider the et Y 2 function y C[, ) nd mpping Ψ 3 : Y 3 C[, ) defined by Y 2 = {y C[, ) : cu(t)v 2 (t) y(t) 2cu(t)v 2 (t), t } nd Ψ 3 y(t) = 2cu(t)v 2 (t) u 3 (t)e I (t, ; u) I (r, ; u)u 3 (r) f ( r, y(r) ) dr d. ht Ψ 3 (Y 2 ) Y 2 i n immedite conequence of (6). Since the continuity of Ψ 3 nd the reltive compctne of Ψ 3 (Y 2 ) cn be proved in the proof of heorem, there exit n element y Y 2 uch tht Ψ 3 y = y, which tifie y(t) = 2cu(t)v 2 (t) u 3 (t) I (t, ; u) I (r, ; u)u 3 (r) f ( r, y(r) ) dr d for t. It i ey to verify tht thi fixed point i olution of degree 3 of (A) uch tht L 2 y(t) = c 2 exit nd i finite nd nonzero. hi complete the proof. heorem 3. he eqution (A) h poitive olution of type P [mx] if (7) for ome c > 0. u(t)v 2 (t) f ( t, cu 2 (t)v(t) ) dt <, Proof. Suppoe tht (7) hold. ke o lrge tht u(t)v 2 (t) f ( t, cu 2 (t)v(t) ) dt < c. Conider cloed convex ubet Y of C[, ) defined by Y = {y C[, ) : cu 2 (t)v(t) y(t) 2cu 2 (t)v(t), t }. Define the opertor Φ : Y C[, )by the following formul Φ y(t) = cu 2 (t) v(t) + u 3 (t) I 2 (r, ; u)u 3 (r) f ( r, y(r) ) dr d. Agin we cn how tht (i) Φ (Y ) Y, (ii) Φ i continuou opertor nd (iii) Φ (Y ) i reltively compct.
MINIMAL AND MAXIMAL SOLUIONS 3 herefore, Φ h fixed point y Y, which give rie to type P [mx] olution of (A) ince it tifie y(t) = cu 2 (t) v(t) + u 3 (t) I 2 (r, ; u)u 3 (r) f ( r, y(r) ) dr d for t. Note tht L y(t) = c. he proof i thu complete. heorem 4. he eqution (A) h poitive olution of type P [min] if (8) for ome c > 0. v 3 (t) f ( t, cu 3 (t) ) dt <, Proof. Suppoe now tht (8) hold. here exit contnt uch tht Define the mpping Ψ by Ψ y(t) = 2cu 3 (t) u 3 (t) v 3 (t) f ( t, cu 3 (t) ) dt < c. t I 3 (, t; u)u 3 () f (, y() ) d. hen, it cn be verified without difficulty tht Ψ h fixed element y in the et Y 0 = {y C[, ) : cu 3 (t) y(t) 2cu 3 (t), t }. hi fixed point give rie to required poitive olution of (A), ince it tifie y(t) = 2cu 3 (t) u 3 (t) t I 3 (, t; u)u 3 () f (, y() ) d. Note tht L 0 y(t) = 2c. hi complete the proof. 6. Specil ce nd exmple We conider eqution (A) with pecil function f(t, y) = Q(t)y λ (B) L 4 y(t) + Q(t)y λ = 0, where λ > 0 nd Q: [, ) (0, ) i continuou. he objective of thi ection i to ue bove theorem to etblih ufficient condition for eqution (B) to hve olution x i (t), i =, 2, 3, 4 defined in ome neighborhood of infinity with the me ymptotic behvior x i (t) = u 4 i (t)v i (t), i 4, repectively, t. We write they corollrie, where the ymbol i ued to denote the ymptotic equivlence f(t) g(t) t f(t) g(t) =.
32 K. ROSÁS Corollry. A ufficient condition for (B) to hve poitive olution y 4 (t) which tifie y 4 (t) mv 3 (t) for ome m > 0 i tht u 3 (t) v 3λ (t) Q(t) dt <. Corollry 2. A ufficient condition for (B) to hve poitive olution y 3 (t) which tifie y 3 (t) mu(t) v 2 (t) for ome m > 0 i tht u 2 λ (t) v 2λ (t) Q(t) dt <. Corollry 3. A ufficient condition for (B) to hve poitive olution y 2 (t) which tifie y 2 (t) mu 2 (t) v(t) for ome m > 0 i tht u 2λ (t) v 2 λ (t) Q(t) dt <. Corollry 4. A ufficient condition for (B) to hve poitive olution y (t) which tifie y (t) 2mu 3 (t) for ome m > 0 i tht u 3λ (t) v 3 (t) Q(t) dt <. We preent here n exmple which illutrte theorem proved bove nd the corollrie. Exmple. Conider the nonocilltory liner differentil eqution of the econd order x + 4t 2 x = 0, t. We know, tht thi eqution h principl olution nd nonprincipl olution nd tht the iterted eqution u(t) = t 2 v(t) = t 2 ln t x IV + 5 2t 2 x 5 t 3 x + 8 6t 4 x = 0, h independent olution in the form x (t) = t 3 2, x2 (t) = t 3 2 ln t, x3 (t) = t 3 2 ln 2 t, x 4 (t) = t 3 2 ln 3 t.
MINIMAL AND MAXIMAL SOLUIONS 33 hen eqution y IV + 5 2t 2 y 5 t 3 y + 8 6t 4 y + Q(t)y λ = 0, t, where λ > 0 nd Q: [, ) (0, ) i continuou, h poitive regulr olution ) y (t) tifying y (t) mt 3 2 if t 3 2 (λ ) (ln t) 3 Q(t) dt <, b) y 2 (t) tifying y 2 (t) mt 3 2 ln t if t 3 2 (λ ) (ln t) 2 λ Q(t) dt <, c) y 3 (t) tifying y 3 (t) mt 3 2 ln 2 t if t 3 2 (λ ) (ln t) 2λ Q(t) dt <, d) y 4 (t) tifying y 4 (t) mt 3 2 ln 3 t if t 3 2 (λ ) (ln t) 3λ Q(t) dt < for ome m > 0. Acknowledgement. he uthor i grteful to the referee for hi ueful uggetion. Reference [] Brret, J. H., Ocilltion theory of ordinry liner differentil eqution, Adv. Mth. 3 (969), 45 509. [2] Fink, A. M., Kuno,., Nonocilltion theorem for differentil eqution with generl deviting rgument, Lecture Note in Mth. 032 (983), 224 239. [3] Kuno,., Swnon, C. A., Aymptotic propertie of emiliner elliptic eqution, Funkcil. Ekvc. 26 (983), 5 29. [4] Kuno,., Swnon, C. A., Aymptotic theory of ingulr emiliner elliptic eqution, Cnd. Mth. Bull. 27 (984), 223 232. [5] Neumn, F., Ocilltory behvior of itertive liner ordinry differentil eqution depend on their order, Arch. Mth. (Brno) 22 (4) (986), 87 92. [6] Póly, G., On the men-vlue theorem correponding to given liner homogeneou differentil eqution, rn. Amer. Mth. Soc. 24 (922), 32 324. Deprtment of Mthemtic, Comeniu Univerity, Mlynká dolin, Brtilv, Slovki E-mil: kritin.rot@fmph.unib.k