LIAPUNOV-TYPE INTEGRAL INEQUALITIES FOR CERTAIN HIGHER-ORDER DIFFERENTIAL EQUATIONS

Transcription

1 Eletroni Journl of Differentil Equtions, Vol. 9(9, No. 8, pp ISSN: URL: or ftp ejde.mth.txstte.edu LIAPUNOV-TYPE INTEGRAL INEQUALITIES FOR CERTAIN HIGHER-ORDER DIFFERENTIAL EQUATIONS SAROJ PANIGRAHI Abstrt. In this pper, we obtin Lipunov-type integrl inequlities for ertin nonliner, nonhomogeneous differentil equtions of higher order with without ny restrition on the zeros of their higher-order derivtives of the solutions by using elementry nlysis. As n pplitions of our results, we show tht osilltory solutions of the eqution onverge to zero s t. Using these inequlities, it is lso shown tht (t m+k t m s m, where 1 k n 1 nd t m is n inresing sequene of zeros of n osilltory solution of D n y + yf(t, y y p =, t, provided tht W (., λ L σ ([,, R +, 1 σ nd for ll λ >. A riterion for disonjugy of nonliner homogeneous eqution is obtined in n intervl [, b]. 1. Introdution The Russin mthemtiin A. M. Lipunov [15] proved the following remrkble inequlity: If y(t is nontrivil solution of y + p(ty =, (1.1 with y( = = y(b ( < b nd y(t for t (, b, then 4 b < p(t dt, (1. where p L 1 lo. This inequlity provides lower bound for the distne between onseutive zeros of y(t. If p(t = p >, then (1. yields (b > / p. In [1], the inequlity (1. is strengthened to 4 b < p + (tdt, (1.3 where p + (t = mx{p(t, }. The inequlity (1.3 is the best possible in the sense tht if the onstnt 4 in (1.3 is repled by ny lrger onstnt, then there exists Mthemtis Subjet Clssifition. 34C1. Key words nd phrses. Lipunov-type inequlity; osilltory solution; disonjugy; higher order differentil equtions. 9 Texs Stte University - Sn Mros. Submitted Otober 19, 8. Published Februry 5, 9. Supported by Ntionl Bord of Higher Mthemtis, Deprtment of Atomi Energy, Indi. 1

2 S. PANIGRAHI EJDE-9/8 n exmple of (1.1 for whih (1.3 no longer holds (see [1, p. 345], [13]. However, stronger results were obtined in [, 13]. In [13] it is shown tht p + (tdt > 1 nd where (, b suh tht y ( =. Hene p + (tdt > 1 b, p + (tdt > b = (b ( (b 4 b. In [, Corollry 4.1], the uthors obtined 4 b < p(tdt from whih (1. n be obtined. The inequlity finds pplitions in the study of boundry vlue problems. It my be used to provide lower bound on the first positive proper vlue of the Sturm-Liouville problems nd y (t + λq(ty = y( = = y(d ( < d y (t + (λ + q(ty = y( = = y(d ( < d by letting p(t to denote λq(t nd λ + q(t respetively in (1.. The disonjugy of (1.1 lso depends on (1.. Indeed, eqution (1.1 is sid to be disonjugte if p(t dt 4/(b. Eqution (1.1 is sid to be disonjugte on [, b] if no non-trivil solution of (1.1 hs more thn one zero. Thus (1. my be regrded s neessry ondition for onjugy of (1.1. Inequlity (1. hs lots of pplitions in eigenvlue problems, stbility, et. A number of proofs re known nd generliztions nd improvements hve lso been given (see [1, 14,, 4, 5]. Inequlity (1.3 ws generlized to the ondition (t (b tp + (tdt > (b (1.4 by Hrtmn nd Wintner [11]. An lternte proof of the inequlity (1.4, due to Nihri [17], is given in [1, Theorem 5.1 Ch XI]. For the eqution y (t + q(ty + p(ty =, (1.5 where p, q C([,, R, Hrtmn nd Wintner [11] estblished the inequlity { } (t (b tp + (tdt + mx (t q(t, (b t q(t dt > (b (1.6 whih redues to (1.4 when q(t =. In prtiulr, (1.6 implies the de l vllee Poussin inequlity [3]. In [1], Glbrith hs shown tht if nd b re suessive zeros of (1.1 with p(t liner funtion, then (b p(tdt π.

3 EJDE-9/8 LIAPUNOV-TYPE INTEGRAL INEQUALITY 3 This inequlity provides n upper bound for two suessive zeros of n osilltory solution of (1.1. Indeed, if p(t = p >, then (b π/(p 1/. Fink [8], obtined both upper nd lower bounds of (b p(tdt, where p(t is liner. Indeed, he showed tht 9 b 8 λ (b p(tdt π nd tht these re the best possible bounds, where λ is the first positive zero of J 1/3 nd J n is the Bessel funtion. The onstnt 9 8 λ = nd π = , so tht it gives delite test for the sping of the zeros for liner p. Fink [9] investigted the behviour of the funtionl (b p(tdt, where p is in ertin lss of sub or supper funtions. Elison [4, 5] obtined upper nd lower bounds of the funtionl (b p(tdt, where p(t is onve or onvex. St Mrry nd Elison [16] onsidered the sme problem for (1.5. Biley nd Wltmn [1] pplied different tehniques to obtin both upper nd lower bounds for the distne between two suessive zeros of solution of (1.5. They lso onsidered nonliner equtions. In reent pper, Brown nd Hinton [] used Opil s inequlity to obtin lower bounds for the sping of the zeros of solution of (1.1 nd lower bounds of the sping β α, where y(t is solution of (1.1 stisfying y(α = = y (β nd y (α = = y(β(α < β. Inequlity (1. is generlized to seond order nonliner differentil eqution by Elison [5], to dely differentil equtions of seond order in [6, 7] nd by Dhiy nd Singh [3], nd to higher order differentil eqution by Phptte [18]. In reent work [], the uthors hve obtined Lipunov-type inequlity for third order differentil equtions of the form y + p(ty =, (1.7 where p L 1 lo. The inequlity is used to study mny interesting properties of the zeros of n osilltory solution of (1.7 (see [, Theorems 5, 6]. Indeed, Phptte derived Lipunov-type inequlities for the eqution of the form D n [r(td n 1 (p(tg(y (t] + y(tf(t, y(t = Q(t, D n [r(td n 1 (p(th(y(ty (t] + y(tf(t, y(t = Q(t, D n [r(td n 1 (p(th(y(tg(y (t] + y(tf(t, y(t = Q(t, (1.8 under pproprite onditions, where n is n integer nd D = d n /dt n. It is ler tht the results in [18] re not pplible to odd order equtions. Furthermore, he hs tken the restrition on the zeros of higher order derivtives [18, Theorem 1]. We my observe tht in [18, p.53, Exmple], y (3π/4 beuse y (t = e t (os t sin t. On the other hnd, y (π/4 = but π/4 / (π/, 3π/ nd y (5π/4 = but 5π/4 < π. Although this exmple does not illustrte [18, Theorem 1], it hs motivted us to remove the restrition on the zeros of higher order derivtives of the solution of (1.5. The objetive of this pper is to obtin Lipunov-type integrl inequlity for the nth-order differentil eqution ( 1 ( 1 ( 1 r n 1 (t... r (t r 1 (t y (t p y (t... + y(t p f(t, y(ty = Q(t, (1.9

4 4 S. PANIGRAHI EJDE-9/8 under pproprite ssumptions on r i (t, 1 i n 1, f nd Q. Here n, p > 1 re even nd odd integers. In this work we remove this restrition on the zeros of higher order derivtives. Further, we show tht every osilltory solution of (1.9 onverges to zero s t with the help of Lipunov-type inequlity. We lso generlize theorem of Ptul [, Theorem ] to higher order equtions. A riteri for dionjugy of nonliner homgeneous eqution is obtined in n intervl [, b] by the help of the inequlity. Eqution (1.9 my be written s where n is n integer,. Min results D n y + yf(t, y y(t p = Q(t, (.1 Dy = 1 r 1 (t y (t p y (t, D i y = 1 r i (t (Di 1 y, i n, nd r n (t 1. We ssume tht (C1 r i : I R is ontinuous nd r i (t >, 1 i n 1 nd Q : I R is ontinuous, where I is rel intervl. (C f : I R R is ontinuous suh tht f(t, y W (t, y, where W : I R + R + is ontinuous, W (t, u W (t, v for u v nd R + = [, ]. We define E(t, r (t, r 3 (s,..., r n 1 (s n ; z(s n 1 = r (t sn 3 α 1 r 3 (s α n 3 r n 1 (s n s α r 4 (s 3... sn α n z(s n 1 ds n 1 ds n... ds, where z(t is rel vlued ontinuous funtion defined on [, b] I( < b nd α 1, α,..., α n re suitble points in [, b], nd E(t, r (t, r 3 (s,..., r n 1 (s n ; z(s n 1 s sn 3 = r (t r 3 (s r 4 (s 3... r n 1 (s n α 1 α α n ds sn α n z(s n 1 ds n 1 ds n Theorem.1. Suppose tht (C1-(C hold. Let α 1, α,..., α n [, b], where α 1, α,..., α n re the zeros of D y(t, D 3 y(t,..., D n y(t, D n 1 y(t respetively, [, b] I( < b nd y(t is nontrivil solution of (.1 with y( = = y(b. If is point in (, b where y(t ttins mximum nd M = mx{ y(t : t [, b]} = y(, then 1 < ( 1 ( p p 1 ( (r 1 (s 1 1/(p 1 [ ds 1 E(s1, r (s 1, r 3 (s,..., r n 1 (s n ; W (s n 1, M + 1 M p 1 E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; Q(s n 1 ] ds 1, (.

5 EJDE-9/8 LIAPUNOV-TYPE INTEGRAL INEQUALITY 5 for n 3 nd 1 < ( 1 ( p p 1 [ (r 1 (t 1/(p 1 dt for n =. W (t, Mdt + 1 M p 1 Proof. Let n 3. Integrting (.1 from α n to t [, b], we obtin tht is, D n 1 y(t + = α n y(s n 1 f(s n 1, y(s n 1 y(s n 1 p ds n 1 α n Q(s n 1 ds n 1 ; ] Q(t dt, (.3 (D n y(t + r n 1 (t y(s n 1 f(s n 1, y(s n 1 y(s n 1 p ds n 1 α n = r n 1 (t Q(s n 1 ds n 1. α n Further integrtion from α n 3 to t [, b] yields D n y(t ( s n + r n 1 (s n y(s n 1 f(s n 1, y(s n 1 y(s n 1 p ds n 1 ds n α n 3 α n ( s n = r n 1 (s n α n 3 Q(s n 1 ds n 1 ds n. α n Proeeding s bove we obtin D y(t + sn 3 = α 1 r 3 (s α n 3 r n 1 (s n tht is, α 1 r 3 (s s s α r 4 (s 3... sn α n y(s n 1 f(s n 1, y(s n 1 y(s n 1 p ds n 1 ds n... ds, α r 4 (s 3... sn 3 α n 3 r n 1 (s n sn α n Q(s n 1 ds n 1 ds n... ds ; (Dy(t + E(t, r (t, r 3 (s,..., r n 1 (s n ; y(s n 1 f(s n 1, y(s n 1 y(s n 1 p = E(t, r (t, r 3 (s,..., r n 1 (s n ; Q(s n 1. Hene Sine (Dy(t M p 1 E(t, r (t, r 3 (s,..., r n 1 (s n ; W (s n 1, M + E(t, r (t, r 3 (s,..., r n 1 (s n ; Q(s n 1. M = y( = M = y( = y (s 1 ds 1 y (s 1 ds 1, y (s 1 ds 1 y (s 1 ds 1, (.4

6 6 S. PANIGRAHI EJDE-9/8 it follows tht M y (s 1 ds 1. First, using Hölders inequlity with indies p nd p/(p 1 nd then integrting by prts we obtin M p ( 1 ( p p y (s 1 ds 1 = ( 1 ( p p (r 1 (s 1 1/p (r 1 (s 1 1/p y (s 1 ds 1 ( 1 ( p p 1 ( (r 1 (s 1 1/(p 1 ds 1 (r 1 (s 1 1 y (s 1 p ds 1 = ( 1 ( p p 1 (r 1 (s 1 1/(p 1 ds 1 ([(r 1 (s 1 1 y (s 1 p y (s 1 y(s 1 ] b [(r 1 (s 1 1 y (s 1 p y (s 1 ] y(s 1 ds 1 = ( 1 ( p p 1 (r 1 (s 1 1/(p 1 ds 1 (Dy (s 1 y(s 1 ds 1 ( 1 ( p p 1 (r 1 (s 1 1/(p 1 ds 1 (Dy (s 1 y(s 1 ds 1. Using (.4, tht is, M p < ( 1 ( p p 1 (r 1 (s 1 1/(p 1 ds 1 [M p E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; W (s n 1, Mds 1 + M E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; Q(s n 1 ds 1 ]; 1 < ( 1 ( p p 1 (r 1 (s 1 1/(p 1 ds 1 [ E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; W (s n 1, Mds M p 1 When n =, (.1 hs the form Hene (.5 yields E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; Q(s n 1 ds 1 ]. (Dy (t + y(tf(t, y(t y(t p = Q(t. M p < ( 1 ( p p 1 [ (r 1 (s 1 1/(p 1 ds 1 y(t p f(t, y(t dt+ (.5 ] y(t Q(t dt ;

7 EJDE-9/8 LIAPUNOV-TYPE INTEGRAL INEQUALITY 7 tht is, 1 < ( 1 ( p p 1 [ (r 1 (t 1/(p 1 dt Thus the proof is omplete. W (t, Mdt + 1 M p 1 ] Q(t dt. Remrks. If r i (t = 1; i = 1,,..., n 1; p = ; f(t, y = p(t nd n =, 3; then inequlities (.3 nd (. redue respetively, to the inequlities (1. nd p(t dt > 4/(b. This inequlity provides lower bound of the distne between onseutive zeros of the solution y(t. For the vrious pplitions of this inequlity one n see []. Lipunov-type integrl inequlities for (1.8 n be obtined under suitble ssumptions on g nd h. If r i (t = 1; i = 1,,..., n 1; n = 3, p =, f(t, y = q(t y(t β 1 nd Q(t =, then (.1 redues to y (t + q(t y(t β 1 y =, t, (.6 where β is positive onstnt nd q : [, [, is ontinuous funtion is lled n Emden-Fowler equtions of third order. If y(t is solution of (.6 with y( = = y(b, ( < b nd y(t for t (, b, then the sping between zeros of solutions of (.6 my be omputed by using (.. Exmple.. Consider y (t + y (t = sin t os t, t. (.7 Clerly, y(t = sin t is solution of (.7 with y( = = y(π, y ( = = y (π. M = mx t [,π] sin t = 1. From Theorem.1 it follows tht where 1 < π 4 π [E(s 1, r (s 1, W (s, M + 1 M E(s 1, r (s 1, Q(s ]ds 1, E(s 1, r (s 1, W (s, M = E(s 1, r (s 1, Q(s = s1 s1 Mds = { s 1, s 1 >, s 1, s 1 <, { sin ds s os s = s 1, s 1 >, s 1, s 1 <. Hene π π { π /, s 1 >, E(s 1, r (s 1, W (s, Mds 1 = π /, s 1 <, { π, s 1 >, E(s 1, r (s 1, Q(s ds 1 = π, s 1 <.

8 8 S. PANIGRAHI EJDE-9/8 As E >, then s 1 > nd π π E(s 1, r (s 1, W (s, Mds 1 = π /, E(s 1, r (s 1, Q(s ds 1 = π. Thus by Theorem.1, 1 < 3π 3 /8 or 3π 3 > 8, whih is obviously true. Theorem.3. Suppose tht (C1-(C hold. Let α 1, α,..., α n 3, α n be the zeros of D y(t, D 3 y(t,..., D n y(t, D n 1 y(t respetively, in [, b] I( < b, where y(t is nontrivil solution of D n y + yf(t, y y(t p = with y( = = y(b. If is point in (, b, where y(t ttins mximum, then the point nnot be very lose to s well s b. Proof. Let M = mx{ y(t : t [, b]} = y(. Then y ( =. Sine y( = y (tdt, using Hölders inequlity with indies p nd p/(p 1 nd then integrting by prts we obtin M p ( 1 ( p p y (t dt = ( 1 ( p p r 1 (t 1/p r 1 (t 1/p y (t dt ( 1 ( p r 1 (t 1/(p 1 p 1( r 1 (t 1 y (t p dt = ( 1 ( p r 1 (t 1/(p 1 p 1([r 1 (t 1 y (t p y (ty(t ( 1 ( p r 1 (t 1/(p 1 p 1( (Dy (t y(t dt. Proeeding s Theorem.1 we obtin Hene ] (Dy (ty(tdt (Dy (t M p 1 E(t, r (t, r 3 (s,..., r n 1 (s n ; W (s n 1, M. 1 < ( 1 ( p r 1 (t 1/(p 1 p 1 ( E(t, r (t, r 3 (s 3,..., r n 1 (s n ; W (s n 1, Mdt ; tht is, [( r 1 (t 1/(p 1 p 1] 1 < ( 1 p ( E(t, r (t, r 3 (s,..., r n 1 (s n ; W (s n 1, Mdt <. (.8

9 EJDE-9/8 LIAPUNOV-TYPE INTEGRAL INEQUALITY 9 Thus nnot be very lose to beuse [( r 1 (t 1/(p 1 p 1] 1 =. lim + Next we hve to show tht nnot be very lose to b. Sine y( = y (tdt, then proeeding s bove to obtin p M p ( 1 ( p y (t dt = ( 1 ( p r 1 (t 1/(p 1 p 1([ (Dy (ty(tdt ] b r 1 (t p 1 y (t p y (ty(t ( 1 ( p r 1 (t 1/(p 1 p 1 b (Dy (t y(t dt < M p( 1 ( p r 1 (t 1/(p 1 p 1 ( E(t, r (t, r 3 (s,..., r n 1 (s n ; W (s n 1, Mdt. Hene [( r 1 (t 1/(p 1 p 1] 1 < ( 1 p ( E(t, r (t, r 3 (s,..., r n 1 (s n ; W (s n 1, Mdt <. Thus nnot be very lose to b beuse [( r 1 (t 1/(p 1 p 1] 1 =. lim b This ompletes the proof of the theorem. We remrk tht Theorem.3 need not hold if α i / [, b] for some i {1,,..., n }. 3. Applitions In this setion we present some of the pplitions of the Lipunov-type inequlity obtined in Theorem.1 to study the symptoti behviour of osilltory solution of (.1. Definition. A solution y(t of (.1 is sid to be osilltory if there exists sequene < t m > [, suh tht y(t m =, m 1 nd t m s m. Theorem 3.1. Suppose tht (C1-(C hold. Let W (t, λ L σ ([,, R + for ll λ >, where 1 σ <. Let r i (t K for t nd 1 i n 1, where K > is onstnt. If < t m > is n inresing sequene of zeros of n osilltory solution y(t of D n y + yf(t, y y(t p = t,

10 1 S. PANIGRAHI EJDE-9/8 suh tht α 1, α,..., α n (t m, t m+k, 1 k n 1, for every lrge m, then (t m+k t m, s m, where α 1,..., α n re the zeros of D y(t, D 3 y(t,..., D n y(t, D n 1 y(t, respetively. Proof. If possible, let there exist subsequene of t m suh tht (+k M for every i, where M > is onstnt. Let M mi = mx{ y(t : t [, +k]} = y(s mi, where s mi (, +k. Sine W (t, λ L σ ([,, R + for ll λ >, then Hene t W σ (t, λdt <, for ll λ >. W σ (t, λdt s t. Thus, for 1 < σ <, we my hve W σ (t, λdt < [K n 1 M n 1+ 1 µ ] 1 for lrge i, where 1 µ + 1 σ = 1. From (.8 we obtin [ s i ] 1 ((r 1 (t 1/(p 1 p 1 (1 pk < n ( tmi +k n +k W (t, M mi dt; tht is, 1 < ( 1 pk n 1 ( tmi +k n 1 +k W (t, M mi dt. The use of Hölder s inequlity yields 1 < ( 1 pk n 1 ( n 1 ( [ mi +k 1/µ +k tmi+k W σ (t, M mi dt ( 1 pk n 1 ( n 1+ 1 [ µ +k ] 1/σ W (t, M mi dt < ( 1 [ ] pk 1 n 1 M n 1+ 1 µ K n 1 M n µ = p,. ontrdition. For σ = 1, we n hoose i lrge enough suh tht W (t, M mi < [K n 1 M n 1 ] 1 nd 1 < ( 1 tmi pk n 1 (+k n 1 +k W (t, M mi dt < ( 1 pk n 1 M n 1 [K n 1 M n 1 ] 1 = 1 p, ontrdition. Hene the Theorem is proved. Theorem 3.. Suppose tht (C1-(C hold with I = [,. Let there exist ontinuous funtion H : I R + suh tht W (t, L H(t for every onstnt L >. Let r 1 (t 1/(p 1 ds 1 <. ] 1/σ

11 EJDE-9/8 LIAPUNOV-TYPE INTEGRAL INEQUALITY 11 If for n 3, nd E(t, r (t, r 3 (s,..., r n 1 (s n ; Q(s n 1 dt <, E(t, r (t, r 3 (s,..., r n 1 (s n ; H(s n 1 dt <, H(tdt <, Q(t dt < for n = ; then every osilltory solution of (.1 onverges to zero s t. Proof. Let y(t be n osilltory solution of (.1 on [T y,, T y. Hene lim inf t y(t =. To omplete the proof of the theorem it is suffiient to show tht limsup t y(t =. If possible, let limsup t y(t = λ >. Choose < d < λ/. From the given ssumptions it follows tht it is possible to hoose lrge T > suh tht, for t T, t for n 3, nd t t r 1 (s 1 1/(p 1 ds 1 < p/(p 1, E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; Q(s n 1 ds 1 < d p 1, E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; H(s n 1 ds 1 < 1 t H(sds < d p 1, t Q(s ds < d for n =. Sine y(t is osilltory, we n find t 1 > T suh tht y(t 1 =. Let T > t 1 be suh tht α 1, α,..., α n 3, α n [t 1, T ], where α 1, α,..., α n 3, α n re the zeros, respetively, of D y(t,..., D n y(t. Further, lim sup t y(t > d implies tht we n find T > t 1 suh tht sup{ y(t : t [t 1, T ]} > d. Let T 1 = mx{t, T }. Let t > T 1 suh tht y(t =. Let M = mx{ y(t : t [t 1, t ]}, then M > d. From Theorem.1 we obtin (. for n 3 nd (.3 for n =, with = t 1 nd b = t. Hene, For n 3, p 1 1 < ( 1 ( p ((r 1 (s 1 1/(p 1 ds 1 t 1 [ E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; H(s n 1 t ] M p 1 E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; Q(s n 1 ds 1 < ( 1 p ( p/(p 1 p 1[ ( d p 1 ] 1 + <, M ontrdition. Hene lim sup t y(t =. Thus the proof of the theorem is omplete. Exmple 3.3. Consider (e t (e t y y + y 3 = e 4t (8os 3 t + 13sin 3 t + 1 os t 6 sin t + e 6t sin 3 t, (3.1

12 1 S. PANIGRAHI EJDE-9/8 where t. Thus r 1 (t = e t, r (t = e t, f(t, y = 1, nd hene H(t = 1. Clerly, y(t = e t sin t is solution of (3.1 with y( = nd (e t y (ty (t = for t =, π. Hene α 1 =, π. Let α 1 =. Sine it follows tht E(s 1, r (s 1 ; H(s = s 1 e s1 for s 1 >, E(s 1, r (s 1 ; Q(s 38s 1 e s1 for s 1 >, Agin tking α 1 = π, we obtin Then E(s 1, r (s 1 ; H(s ds 1 = 1, E(s 1, r (s 1 ; Q(s ds E(s 1, r (s 1 ; H(s = (s 1 πe s1 for s 1 > π, E(s 1, r (s 1 ; Q(s 38(s 1 πe s1 for s 1. > π, π π E(s 1, r (s 1 ; H(s ds 1 = e π, E(s 1, r (s 1 ; Q(s ds 1 38e π. From Theorem 3. it follows tht every osilltory solution of (3.1 tends to zero s t tends to infinity. Theorem 3.4. If (1 ( p p 1 r 1 (s 1 1/(p 1 ds 1 E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; p(s n 1 ds 1 1, (3. then D n y + p(ty y p = (3.3 is disonjugte on [, b], where p(t is rel-vlued ontinuous funtion on [, b]. Definition. Eqution (3.3 is sid to be disonjugte in [, b] if no non-trivil solution of (3.3 hs more thn n 1 zeros (ounting multipliities. Proof of Theorem 3.4. Indeed, if (3.3 is not disonjugte on [, b], then it dmits nontrivil solution y(t hs n zeros in [, b]. Let these zeros be given by 1 < < < n 1 < n b. Then D y(t, D 3 y(t,..., D n 1 y(t hve zeros in

13 EJDE-9/8 LIAPUNOV-TYPE INTEGRAL INEQUALITY 13 [ 1, n ]. From Theorem.1, it follows tht 1 < ( 1 ( n p r 1 (s 1 1/(p 1 ds 1 1 n p 1 E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; p(s n 1 ds 1 1 ( 1 ( p p 1 r 1 (s 1 1/(p 1 ds 1 E(s 1, r (s 1, r 3 (s,..., r n 1 (s n ; p(s n 1 ds 1, ontrdition. Hene (3.3 is disonjugte on [, b]. Remrk. If r i (t = 1; i = 1,,..., n 1; p =, n = 3, then (3. redues to p(t dt 4/(b. Thus the bove inequlity my be regrded s suffiieny ondition for the disonjugy of the eqution (1.7. As finl remrk, we note tht the results obtined in this pper generlize the results by Phptte [19]. Aknowledgements. The uthor would like to thnk the nonymous referee for his/her vluble suggestions. Referenes [1] P. Bily nd P. Wltmn; On the distne between onseutive zeros for seond order differentil equtions, J. Mth. Anl. Appl. 14 (1996, 3-3. [] R. C. Brown nd D. B. Hinton; Opil s inequlity nd osilltion of nd order equtions, Pro. Amer. Mth. So. 15 (1997, [3] R. S. Dhiy nd B. Singh; A Lipunov inequlity nd nonosilltion theorem for seond order nonliner differentil-differene eqution, J. Mth. Phys. Si. 7 (1973, [4] S. B. Elison; The integrl T R T / T / p(tdt nd the boundry vlue problem x (t + p(tx =, x( T = x( T =, J. Diff. Equtions. 4 (1968, [5] S. B. Elison; A Lipunov inequlity for ertin seond order nonliner differentil eqution, J. London Mth. So. (197, [6] S. B. Elison; Lipunov-type inequlity for ertin seond order funtionl differentil equtions, SIAM J. Appl. Mth. 7 (1974, [7] S. B. Elison; Distne between zeros of ertin differentil equtions hving delyed rguments, Ann. Mt. Pur Appl. 16 (1975, [8] A. M. Fink; On the zeros of y (t + p(ty = with liner ovex nd onve p, J. Mth. Pures et Appl. 46 (1967, 1-1. [9] A. M. Fink; Comprison theorem for R b p with p n dmissible sub or super funtion, J. Diff. Eqs. 5 (1969, [1] A. Glbrith; On the zeros of solutions of ordinry differentil equtions of the seond order, Pro. Amer. Mth. So. 17 (1966, [11] P. Hrtmn nd A. Wintner; On n osilltion riterion of de l vlee Poussion, Qurt. Appl. Mth. 13 (1955, MR [1] P. Hrtmn; Ordinry Differentil Equtions, Wiley, New York, 1964 n Birkhuser, Boston, 198. MR 3, 17. [13] M. K. Kowng; On Lipunov s inequlity for disfolity, J. Mth. Anl. Appl. 83(1981,

14 14 S. PANIGRAHI EJDE-9/8 [14] A. Levin; A omprision priniple for seond order differentil equtions, Dokl. Akd. NuuSSSR 135 (196, (Russin( Trnsltion, Sov. Mth. Dokl. 1 (1961, [15] A. M. Lipunov; Probleme generl de l stbilitie du mouvement, Ann. of Mth. Stud. Vol. 17, Prineton Univ. Press, Prineton, NJ, [16] D. F. St. Mrry nd S. B. Elison; Upper bounds T R T / p(tdt nd the differentil eqution T / x (t + p(tx =, J. diff. Eqs 6 (1969, [17] Z. Nihri; On the zeros of solutions of seond order liner differentil equtions, Amer. J. Mth. 76 (1954, MR [18] B. G. Phptte; On Lipunov-type inequlities for ertin higher order differentil equtions, J. Mth. Anl. Appl. 195 (1995, [19] B. G. Phptte; Inequlities relted to the zeros of solutions of ertin seond order differentil equtions, Ft Universittis Ser. Mth. Inform. 16 (1, [] N. Prhi nd S. Pnigrhi; On Lipunov-type inequlity for third order differentil equtions, J. Mth. Anl. Appl. 33 (1999, [1] N. Prhi nd S. Pnigrhi; On distne between zeros of solutions of third order differentil equtions, Annl. Polon. Mth. 7 (1, [] W.T. Ptul; On the distne between zeros, Pro. Amer. Mth. So. 5 (1975, [3] W. T. Reid; Sturmin Theory for Ordinry Differentil Equtions, Springer - Verlg, Newyork, 198. [4] D. Willet; Generlized de l vllee Poussin disonjugy test for liner differentil equtions, Cnd. Mth. Bull. 14 (1971, [5] P. K. Wong; Leture Notes, Mihign Stte University. Sroj Pnigrhi Deprtment of Mthemtis nd Sttistis, University of Hyderbd, Hyderbd 5 46, Indi E-mil ddress: spsm@uohyd.ernet.in, pnigrhi8@gmil.om