OPIAL S INEQUALITY AND OSCILLATION OF 2ND ORDER EQUATIONS. 1. Introduction We consider the second-order linear differential equation.

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Numer, Aril 997, Pges 3 9 S ) OPIAL S INEQUALITY AND OSCILLATION OF ND ORDER EQUATIONS R C BROWN AND D B HINTON Communicted y Hl L Smith) Astrct For second-order differentil eqution, we otin from Oil s inequlity lower ounds for the scing etween two zeros of solution or etween zero of solution nd zero of its derivtive These ounds re exressed in terms of ntiderivtives of the otentil, nd in rticulr we derive some new Liunov tye inequlities from them Introduction We consider the second-order liner differentil eqution ) y + qx)y =0, x, where q is rel mesurle function on [, ] stisfying qx) dx < Two rolems of interest re: i) otin lower ounds for the scing of zeros of solution of ), nd ii) otin lower ounds for the scing β α where y is solution of ) stisfying yα) =y β)=0ory α)=yβ) = 0 Of rticulr interest in this er is when q is oscilltory nd this ehvior ffects the ounds Our motivtion for this work comes from recent er of Hrris nd Kong [3] Two of their results [3, Theorems nd ] stte tht if y is solution of ) with no zeros in α, β) ndsuchthty α)=yβ)=0,then ) β α) mx qt)dt > α x β If insted yα) =y β)=0,then β 3) β α) mx qt)dt α x β x > Finlly, if there re no extreme vlues of y in α, β), then in either ) or 3) the solute vlue my e droed Their method of roof is to use Riccti eqution techniques Here, y using Oil s inequlity, we rove severl results which relte to rolems i) nd ii) ove In rticulr we otin ) nd 3) s consequence of Oil s inequlity In section we stte the two forms of Oil s inequlity tht we emloy In section 3 we ly them to rolem ii) ove, nd in section we ly them to Received y the editors Octoer, Mthemtics Suject Clssifiction Primry 3C0; Secondry 3L05, 3L5 α 3 c 997 Americn Mthemticl Society License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use

2 R C BROWN AND D B HINTON rolem i) Section 5 contins some further imlictions of section L, ) nd L, ) denote the usul Leesgue sces Two Oil tye inequlities A secil cse of n inequlity otined y Beesck nd Ds [] is the following see lso [, 9]) Theorem A If f is solutely continuous on [, ] with f) =0,nds L, ), then ) where ) sx) fx) f x) dx κ κ = st) t ) dt f x) dx with equlity if nd only if f 0 or f is liner nd s is constnt) If we relce f) =0inTheoremAyf) = 0, then ) holds where κ in ) is given y 3) κ = st) t) dt We lso use nother Oil inequlity which is secil cse of more generl result due to Boyd [] Theorem B If f is solutely continuous on [, ] with f) =0or f) =0nd, then ) ) fx) f x) dx K) ) f x) dx where 5) with I = 0 only for f liner {+ ), =, K) = π, =, ) I, <<, t} {+ )t} dt For=, equlity holds in ) Theorem B hs immedite liction to the cse where f) = f)= 0Choose c=+)/ nd ly ) to [, c] nd[c, ] nd then dd to otin tht ) { c ) } fx) f x) dx K) f x) dx) + f x) dx c ) { 6) K) f x) dx} For =, 6) is strict unless f is liner in ech of the suintervls [, c] nd[c, ] License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use

3 OPIAL S INEQUALITY AND OSCILLATION OF ND ORDER EQUATIONS 5 3 Disfocl rolems Consider the differentil eqution 3) y + qx)y =0, x, where q is rel nd q L, ) Theorem 3 Suose y is nontrivil solution of 3) which stisfies y) = y )=0Then 3) < Qx) x )dx where Qx) = x qt)dt Ify )=y)=0,then 33) < where Qx) = qt)dt Proof We first estlish 3) gives 3) y x) dx = = Qx) x)dx Multilying 3) y y nd integrting y rts qx)yx) dx = Qx)yx)y x) dx Qx) yx)y x) dx Q x)yx) dx < Qx) x ) dx y x) dx y ) nd ) of Theorem A The inequlity is strict since y liner imlies y 0 s y) =y ) = 0 By cncelling y x) dx nd squring we otin 3) The roof of 33) is similr using integrtion y rts nd ) of Theorem A nd 3) insted of ) By using the mximum of Q on [, ] in 3) nd 33), integrting, nd then tking squre root we see tht 35) < ) mx qt)dt x x when y) =y ) = 0, nd 36) < ) mx qt)dt x when y ) =y) = 0, which re the inequlities otined y Hrris nd Kong Note lso tht if y hs no extreme vlues in, ), then yy > 0 in the second line of 3) It follows tht Qx) Q + x) mx{0,qx)} nd we cn relce Q y Q + in the derivtion of 35) This mens tht the solute vlue signs my e omitted in 35) With minor chnges in the rgument yy is now negtive on, )), the sme conclusion lies to 36) if y hs no extreme vlue on, ) License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use

4 6 R C BROWN AND D B HINTON Results similr to Theorem 3 my e otined y liction of Boyd s theorem: Theorem 3 Suose y is nontrivil solution of 3) which stisfies y) = y )=0,,nd is the conjugte index of, ie, + =Then 37) K ) Qx) dx where Qx) = x qt)dt; if y ) = y) = 0, then 37) is true with Qx) = qt)dt In either cse K) is given y 5) For =the inequlity is strict For =the norm of Q in 37) ecomes mx Qx), x Proof In the cse y) =y ) = 0 from the roof of Theorem 3 we hve tht 38) y x) dx Qx) yx) y x) dx By liction of Hölder s inequlity nd Theorem B to 38), we get tht ) y x) dx Qx) dx yx)y x) dx K ) Qx) dx y x) dx with strict inequlity for = Cncelling y x) dx yields 38) A similr rgument yields 37) with Qx) = qt)dt when y ) =y)=0 Note tht for =, 37) in the y) =y ) = 0 cse is the sme s 35) nd in the y ) =y) cse the sme s 36) Theorems 3 nd 3 yield sufficient conditions for disfoclity of 3), ie, sufficient conditions so tht there does not exist nontrivil solution y of 3) stisfying either y) =y )=0ory )=y)=0 Disconjugcy conditions Aliction of 6) llows the use of n ritrry nti-derivtive Q in the ove rguments From this we re le to rove new Lyunov tye inequlities Theorem Suose y is nontrivil solution of 3) which stisfies y) = y)=0,,ndq x)=qx)on [, ] Then K ) Qx) dx with K) given y 5) For =the inequlity is strict Proof As in the roof of Theorem 3, multilying 3) y y nd integrtion y rts yields tht ) y x) dx = qx)yx) dx = Qx)yx)y x) dx License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use

5 OPIAL S INEQUALITY AND OSCILLATION OF ND ORDER EQUATIONS 7 By liction of Hölder s inequlity nd 6) to ) we get tht 3) y x) dx Qx) dx yx)y x) dx K Qx) dx y x) dx from which ) follows For = the inequlity is strict since solution of 3) cnnot e liner on ech of the intervls [, + )/], [ + )/,] s this imlies discontinuity of y We exmine ) further in the cses =, x qt)dt + µ ThensinceK) =, For =,wetkeqx)= ) x ) < mx x qt) dt + µ We now minimize ) over µ This is seen to occur with µ = M + m)/, where so tht M = mx x inf µ mx x qt)dt, m = min x ) qt) dt + µ = M m We hve therefore the following corollry of Theorem qt)dt, Corollry If y is nontrivil solution of 3) which stisfies y) =y)=0, then there exist t nd t in [, ] such tht < t 5) qt) dt For =, we squre ) to otin tht 8 π ) qt) dt + µ) dx Agin we minimize with resect to µ nd find tht µ = ) qt) dt dx ) Using this vlue of µ yields nother corollry t Corollry If y is nontrivil solution of 3) which stisfies y) =y)=0, then π ) 8 ) qt) dt dx ) qt) dt dx) License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use

6 8 R C BROWN AND D B HINTON 5 Alictions First we consider the eqution 5) y + λcos kx)y =0, 0 x B, k>0 Let y e nontrivil solution of 5) with y) =y) = 0 By 5), < t λ cos kt dt = λsin kt sin kt ) k λ k so tht 5) t k < λ ) λ B If we view 5) s n eigenvlue rolem with indefinite weight function nd oundry conditions y0) = yπ) = 0, then 5) sys tht rel eigenvlue λ hs λ lrge if k is lrge More generlly it is n inequlity which gives lower ound for the scing of zeros in the form ) > k/ λ With λ =,5ndk=,5,0, 0 we hve comuted the first ositive zero of the solution of 5) with y0) = 0, y 0) = using MAPLE V suroutine The results rounded to deciml lces) re given elow: k λ = λ = Thus 5) gives lower ound for which for k lrge is consistently slightly less thn 50% of the true vlue However this is still much etter thn the estimtes for 5) otinle y the usul Lyunov inequlity which for λ =,=0is < cos kx) + dx, 0 where + denotes the ositive rt of function This inequlity gives lower ound for the scing of zeros only of order k rther thn k s in 5) Theorem lso llows for counting of the numer of zeros of solution of 3) If solution y of 3) hs consecutive zeros = 0 < < < n =, then ) yields tht for Q = q, 53) K i i i i Qx) dx If we sum 53) nd ly Hölder s inequlity we otin tht n + = # zeros of y in [, ] K Qx) dx + As finl liction we show how Theorem my e lied to yield lower ound for the first eigenvlue λ 0 of 5) y + qx)y = λy, y) =y)=0 License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use

7 OPIAL S INEQUALITY AND OSCILLATION OF ND ORDER EQUATIONS 9 Let y e the eigenfunction of 5) corresonding to λ 0 Now choose µ<λ 0 nd let Q x) =qx) µ Proceeding s in the roof of Theorem yields tht 55) λ 0 µ) = yx) dx = y x) dx + y x) dx Qx)yx)y x) dx y x) dx Qx) dx Q x)yx) dx yx)y x) dx Since the first eigenvlue of y = λy, y) =y)=0isπ / ),wehvethe Wirtinger tye inequlity 56) yx) dx ) π y x) dx Using now 6) nd 56) in 55) yields tht λ 0 µ) ) π K Qx) dx, where ndk) is given y 5) Thus lower ound for λ 0 is otined The results given here my e lied to the eqution 57) x)y ) + qx)y =0, x, y chnge of vrile If y is solution of 57) stsfying y) =y) = 0, let Y t) =yx)wheret= ds/s) Then Y stisfies d Y + Rt)Y t) =0, Y0) = Y T )=0 dt where Rt) =x)qx)ndt= ds/s) Then Corollry tkes the form: if y is nontrivil solution of 57) stisfying y) =0=y), then there exist x,x such tht x ds/s) < qx) dx References P R Beesck nd KM Ds, Extensions of Oil s inequlity, Pcific J Mth 6 968), 5-3 MR 39:385 D W Boyd, Best constnts in clss of integrl inequlities, Pcific J Mth ), MR 0:80 3 B J Hrris nd Q Kong, On the oscilltion of differentil equtions with n oscilltory coefficient, Trns Amer Mth Soc ), MR 95h:3050 D S Mitrinović, JE Pěcrić, nd AM Fink, Inequlities involving functions nd their integrls nd derivtives, Kluwer Acdemic Pulishers, Dordrecht, 99 MR 93m:6036 Dertment of Mthemtics, University of Alm, Tuscloos, Alm 3587 E-mil ddress: drown@mthdetsuedu Dertment of Mthemtics, University of Tennessee, Knoxville, Tennessee E-mil ddress: hinton@novellmthutkedu x License or coyright restrictions my ly to redistriution; see htts://wwwmsorg/journl-terms-of-use