A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly Lebesgue inegrble on he rel inervl /. A soluion y{) of he differenil equion iy{) = 0 is sid o be squreinegrble n end-poin b of / if here is neighbourhood N of b such h /()l < oo. / n N If every soluion of xy{) = 0 is squre-inegrble n end-poin b of/, hen we sy h he operor T is of limi-circle ype b; oherwise x is of limi-poin ype b. In his pper we furher reduce he lrge number ofindependen limi-poin crieri wih resul which simulneously generlizes he well-known crieri of Brinck [1], Hrmn [4] nd Sers'[6]. Noe h he firs of hese resuls ws originlly given s condiion for selfdjoinness. In priculr, Brinck proved h he closed liner operor T in L 2 (- oo, oo) defined by &i(t) = {/el 2 (-oo, oo): T/GL 2 (-OO, oo),/'is loclly bsoluely coninuous} T/=T/ for is self-djoin provided h here exiss sufficienly smooh (see ler) posiive funcion w() wih j w()q()d ^ C (2) for ll inervls c (-oo, oo) wih /() ^ 1, where l() is he lengh of he inervl. Now, since T is self-djoin if nd only if he forml operor T is of limi-poin ype boh oo nd oo, n equivlen semen of Brinck's resul sys h T will be of limi-poin ype oo provided h here exiss smooh posiive funcion w() sisfying (2) for ll inervls <= [, oo) wih 1() < 1, for some > oo. In he ligh of his informion, we shll henceforh resric our enion o he inervl/ = [, oo) where > co. 2. Preliminries LEMMA 2. // w is posiive nd loclly bsoluely coninuous on he hlf-line [, oo) nd sisfies \w'(s)\ ^ 1 lmos everywhere on [, oo), hen here exiss sequence Received 13 April, 1973. [. LONDON MATH. SOC. (2), 8 (1974), 719-727]
720 IAN KNOWLES { k } such h 0 =, k -* oo s k -* oo nd Proof. We firs prove h I {w(s)}~ 1 ds= oo. (4) Assume h he bove inegrl is finie. Then for ll s ^ log [w(s)/w()] = I w'(u)/w(u)du s oo < I {^(M)}" 1^ since \w'\ ^ 1 <. Thus {W()}" 1 is bounded wy from zero nd his conrdics our iniil ssumpion. Consider now he funcion A(0= j MsT'ds. I is cler from (4) nd he inermedie vlue heorem h here exiss poin h > h = Similrly, by considering he funcion / 2 (0= f Ms)} one cn find poin 2 > x wih i / 2 (' 2 )= nd so on. The sricly incresing sequence so produced is obviously unbounded, from he coninuiy of w.
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR 721 LEMMA 2. (Gnelius [3]). ///^ 0 nd g re funcions of bounded vriion on he compc inervl K <= R, hen f f(s)dg(s) ^ (inf/+vrf) sup f dg(s) \ K K / CK K where he supremum is ken over ll inervls <= K. The proof of his lemm will be omied. 3. The min resul THEOREM. // here exiss posiive, loclly bsoluely coninuous funcion w(s) defined on [, oo) nd such h () w(s)ds = oo, (b) w' is essenilly bounded on [, oo), (c) here exiss consn C > Ofor which f w(s)q(s)ds ^ C if f M*)}" 1 ds ^ 1, hen he forml operor T defined by equion (1) is of limi-poin ype oo. Proof I will be sufficien o show h no every rel soluion of xy() = 0 is in l}[, oo). We proceed by conrdicion. Assume wihou loss of generliy h \w'(s)\ < 1 lmos everywhere on [, oo) nd le f Gl}[, oo) be he rel soluion sisfying f x {) = 0 nd fi() = 1; le f 2 el3[, oo) be he rel soluion sisfying f 2 () = 1 nd f 2 '{) = 0. Using he sequence { k } given in Lemm 1, we define funcions 0 fc (s), k = 0, 1,..., on [, oo) by 'k+ 1 w' 1 (r)dr, k ^ s < / fc+1, 5 I is esily seen h ech funcion 0 & is loclly bsoluely coninuous, nd sisfies O<0 fc^l, (5) w(s)\^) k '(s)\ < 1.e. on [, oo), (6) (f) k (s) - 1 s k -+ co for ll se [o, oo). (7) 10
722 IAN KNOWLES Le (f) be ny one of hese funcions nd define \//(s) = <f) 2 (s) w 2 (s). Then, 0= /,(#(s)t/,(s)ds 'k+l 'k + 1 = j q(s)hs)a 2 (s)ds- j {f x '{s)} 2 il,(s)ds fer inegring by prs, f k + 1 "2 /i'(5)/( 5 ){^( S )f (5)>V 2 (5) +»V(S)»V'( 5 ) 'k + 1 'k + 1 'k + 1 < q(s)ilj(s)f 2 l (s)ds- j {f l '(s)} 2 Hs)d3 +4 / 1 '(s)/ 1 (s) 0(s)w(s)^s (8) using (5) nd (6) nd condiion (b), where f k+l f l+l f. 2 v f f 2 A l l i=o k / \ C < X I inf w fi 2 <f> 2 + vr w'/i 2^2) sup wq by Lemm 2, * /,. r +1 ^ C 2 I inf wfi <p w ' = 0 \[r (>»,+,] by equion (3) nd condiion (c), (< j 'k+
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR 723 Finlly, subsiuing his esime ino he inequliy (8), nd hen using he Cuchy- Schwrz inequliy gives Hs)Ui(s)} 2 ds ^ 0(1)11 + **(s) 1/,'Os)! U which proves h he expression on he lef-hnd side of he ls inequliy mus be bounded by number which is independen of he priculr \j = \/ k h we use. Hence, by Fou's lemm, leing k -> oo Similrly, oo Now i is cler h (w 2 (s){f i '(s)} 2 ds ^ liminf f w 2 {s)(l) k 2 (s){f l '(s)} 2 ds k-»oo <. oo»v 2 (S){/ 2 '(5)} 2 <OO. {wf x ')f 2 -{wf 2 ')h=w nd, since we hve shown h ech of he four fcors on he righ-hnd side is in L 2 [, oo), we clerly hve w(s)ds < oo. This conrdics condiion () nd complees he proof. COROLLARY 1 (Brinck [1]). // here exiss posiive funcion w(s) defined on [, oo) nd sisfying () w, w' nd w" re ll bounded nd coninuously differenible on [, oo) nd oo w(s)ds = oo. (P) here exiss consn C > 0 wih f w(s)q(s)ds ^ C for ll inervls <= [, oo) wih /() ^ 1, hen T 15 of limi-poin ype oo. j
724 IAN KNOWLES Proof. Assume wihou loss of generliy h w < 1. We hve only o prove h condiion (c) of he heorem is sisfied. This, however, follows immediely from condiion (/?) bove nd he inequliy K)< {w(s)y 1 ds. The crierion of Sers [6] is lso specil cse. I is in fc equivlen o he following COROLLARY 2. // here exiss posiive funcion w(s) which is loclly bsoluely coninuous on [, oo) nd sisfies () w' is essenilly bounded on [, oo) nd uu s = oo, (/?) w 2 q is essenilly bounded bove on [, oo), hen T is of limi-poin ype oo. Proof Assume wihou loss of generliy h he upper bound in condiion (/) is 1. Then for ny inervl, f w(s)q(s)ds^ I {^(s)}" 1^ / j nd condiion (c) of he heorem is once gin sisfied. COROLLARY 3 (Hrmn [4]). // here exiss posiive non-decresing funcion u() defined on [, oo) nd rel number 9, 0 < 6 < 1, such h () (w(r)} dr = oo, (f$) here exiss consn B such h for ll e [, oo) u(+0/u()) ^ (y) for ll, se [, oo) such h 0 ^ -s f/?e«t is of limi-poin ype oo. Bu{\ < 0/«(O «(r)dr< i/(0, s Proof. Our objecive here will be o consruc funcion w() which sisfies he requiremens of he previous heorem. Observe firs of ll h if I u(r)dr s% 0, ^ s < / < oo,
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR 725 hen u(s) ( s) ^ 0 nd herefore ^ s+6fu(s). Consequenly, by condiion (/?), i.e. w(0 < u(s+9/u(s)) ^ Bu(s), (/?') u()/u(s) < B if s < / nd f w(r)dr ^ 0. Now, since u is non-decresing, we cn immediely sser h u(r)dr = oo. Hence, by he mehod of Lemm 1, here mus exis sequence { k } wih * 0 =, k -+ oo s k -> oo nd 'k+ 1 u(r)fr = 0, fc = O,l,... (9) k Define /i k = / k+1 k for A; = 0, 1,... Then by equion (9) nd condiion (/?'), for A: = 0, 1,... The funcion w(s) is defined on [ k, k+1 ] by (10) The piecewise liner funcion so defined is obviously posiive, loclly bsoluely coninuous nd monoone non-incresing over [, oo). Also, from inequliy (10), nd ^ B l.e. on [, oo) 'k+ 1 f w(s)ds = Z f S A^k H '('k+ I) s^6 vv is decresing, = oo. k+i f {«(s)}- l ds by(js') Our finl sk is o show h w sisfies condiion (c) of he heorem. Observe h if s e [ k, k+1 ] hen w(s) M(S) ^ w( k ) u(s) = 6B~ 2 u(s)/u( k ) ^ 6B' 1. Furhermore, his is rue for every vlue of k; we my herefore conclude h he
726 IAN KNOWLES bove inequliy holds for ll s e [, oo). Now, choose ny inervl [s, ] wih I follows from he bove inequliy h nd hence h u(s) (/ s) ^6B~ l. Rerrnging nd using condiion (/?') we find h s^ 9/u(), from which we infer (vi condiion (y)) h Thus by Lemm 2, jq(r)dr^u(). s ' r * w(r)q(r)dr ^ infw+ \w'(r)\dr sup q(r)dr [s,] Kc[s,] K by (ll), (11) nd he proof is complee. As finl remrk, he uhor feels h similr resul should hold for he generl operor where he funcion p() is posiive nd is reciprocl is loclly Lebesgue inegrble on [, oo); his would hen generlize he resul of Levinson [5] nd possibly even h of Everi [2]. Unforunely, s he mehod used in he heorem does no immediely crry hrough o he generl operor, his will hve o remin conjecure for he presen. The uhor wishes o hnk Professor P. Hrmn for severl helpful suggesions. References 1. 1. Brinck, " Self-djoinness nd specr of Surm-Liouville operors ", Mh. Scnd., 7 (1959), 219-239. 2. W. N. Everi, " On he limi-poin clssificion of second-order differenil operors ", /. London Mh. Soc, 41 (1966), 531-534. 3. T. Gnelius, " Un heoreme uberien pour l rnsformion de Lplce ", C. R. Acd. Sci. Pris, 242 (1956), 719-721. 4. P. Hrmn, " On dichoomies for soluions of n-h order liner differenil equions ", Mh. Ann., 147 (1962), 378-421.
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR 727 5. N. Levinson, " Crieri for he limi-poin cse for second-order liner differenil operors ", Csopis pro pisovni memiky fysiky, 74 (1949), 17-20. 6. D. B. Sers, " Noe on he uniqueness of he Green's funcions ssocied wih cerin differenil equions ", Cnd.. Mh., 2 (1950), 314-325. School of Mhemicl Sciences, Flinders Universiy of Souh Ausrli, Bedford Prk, Souh Ausrli. Curren ddress: Deprmen of Mhemics, Universiy of he Wiwersrnd, ohnnesburg, Souh Afric.