On the asymptotic behavior of solutions to nonlinear ordinary differential equations

Transcription

1 Asympoic Analysis 54 (2007) IOS Press On he asympoic behavior of soluions o nonlinear ordinary differenial equaions Ravi P. Agarwal a, Smail Djebali b, Toufik Moussaoui b,ocaviang.musafa c and Yuri V. Rogovchenko d a Deparmen of Mahemaical Sciences, Florida Insiue of Technology, Melbourne, FL 32901, USA agarwal@fi.edu b Deparmen of Mahemaics, E.N.S., P.O. Box 92, Kouba, Algiers, Algeria s: {djebali,moussaoui}@ens-kouba.dz c Deparmen of Mahemaics, Universiy of Craiova, Al. I. Cuza 13, Craiova, Romania ocaviangenghiz@yahoo.com d Deparmen of Mahemaics, Easern Medierranean Universiy, Famagusa, TRNC, Mersin 10, Turkey yuri.rogovchenko@emu.edu.r Absrac. We discuss a number of issues imporan for he asympoic inegraion of ordinary differenial equaions. Afer developing he ools required for applicaion of he fixed poin heory in he invesigaion, we presen some general resuls abou he long-ime behavior of soluions of n-h order nonlinear differenial equaions wih an emphasis on he exisence of polynomial-like soluions, he asympoic represenaion for he derivaives and he effec of perurbaions upon he asympoic behavior of soluions. 1. Inroducion In he recen years here has been a resurgence in one of he classical opics in he qualiaive heory of ordinary differenial equaions, he asympoic inegraion of nonlinear differenial equaions. Numerous sudies of he long-ime behavior of soluions, in many cases o srongly nonlinear equaions, have direc relaion o he analysis of he parial differenial equaions ha model physical, biological or social phenomena. There is an abundan lieraure concerning he asympoic inegraion of ordinary differenial equaions. The reader can find in [57,1] long liss of references. The paper [1] conains also a classificaion of known resuls according o he ype of nonlineariy in he equaions and he inegral condiions imposed on i. Relevan commens on he exising resuls and mehods will be made hroughou he paper. In his work, we conribue o he heory of asympoic inegraion of nonlinear differenial equaions discussing a number of imporan issues wihin a rigorous ye flexible framework of funcional analysis. In he firs par, we develop he ools for applicaion of several fixed poin mehods o problems of asympoic inegraion. In paricular, in Secion 2 four funcion spaces are inroduced, heir properies examined and relaive compacness crieria are derived. The coninuiy of general inegral operaors is discussed in Secion 3 and an inegral represenaion for he soluions is deailed in Secion /07/$ IOS Press and he auhors. All righs reserved

2 2 R.P. Agarwal e al. / On he asympoic behavior of soluions In he second par of he paper we presen a developmen of he general resuls in asympoic inegraion obained by Kusano and Trench [43,44,73] and Musafa and Rogovchenko [57,59]. We pay ribue o an excellen conribuion made o he heory of asympoic inegraion by Kusano and Trench and discuss relaed issues in Secion 5. In Secion 6, a heory of asympoic inegraion is consruced for he case of polynomial-like soluions ha is closely conneced o he analysis of disconjugacy (nonoscillaion, inerpolaion) of ordinary differenial equaions. Secions 7, 8 are devoed o he presenaion of an inegral represenaion for he derivaives and o a discussion abou he effec of perurbaions on he soluions in he relevan case of second order differenial equaions. 2. Funcion spaces and relaive compacness crieria The firs funcion space ha is needed here can be inroduced profiably wih an illusraion from an 1967 resul of Bihari [11, p. 2] regarding he inegral equaion x() = z() + k 1 (, s)f ( s, x(s) ) ds + k 2 (, s)f ( s, x(s) ) ds, 0, (1) 0 where i is assumed ha all he funcions involved are coninuous and lie in R n. Theorem 1 (Bihari). Suppose ha he funcions k 1, k 2, z are bounded, k i (, s) K i, i = 1, 2, z() γ, whereas he funcion f saisfies he inequaliy f(, x) G (, x ), 0, x R n, for a piecewise coninuous majorizing funcion G ha is nondecreasing in he second variable. Suppose furher ha here exiss a nonnegaive, coninuous funcion g ha verifies he inequaliy γ + K 1 G ( s, g(s) ) ds + K 2 G ( s, g(s) ) ds g(), 0. (2) 0 Then he inegral equaion (1) has a leas one coninuous soluion x() which exiss on he nonnegaive half-line and saisfies he inequaliy x() g() for all 0. This resul of Bihari has wo ineresing feaures. The firs one regards he bound for he soluion x() by a given posiive funcion g() while he second one is concerned wih he special form of inequaliy (2). Using a posiive funcion g() in Theorem 1 as a comparison funcion, one can inroduce he se X 1 ( 0 ; g) of all real-valued coninuous funcions (or, in a general framework, vecor funcions) u() ha are defined in [ 0, + ) forsome 0 R, and behave a infiniy as he given funcion g(), ha is, u() = O(g()) as +. This se, endowed wih he usual operaions on funcions, is a linear space [24, p. 2]. The consrucion of he norm in X 1 ( 0 ; g) akes ino accoun he fac ha, for any elemen u() of X 1 ( 0 ; g), he quaniy u()/g() is a bounded coninuous funcion on [ 0, + ). On he oher hand, he bounded, coninuous real-valued funcions defined on [ 0, + ) form a linear space C b ([ 0, + ), R) [39,

3 R.P. Agarwal e al. / On he asympoic behavior of soluions 3 p. 12] ha is successfully compleed by Chebyshev s sup norm. Now i is convenien o inroduce he norm in X 1 ( 0 ; g) by u = sup 0 u() g(). Le us noe ha he use of such a norm goes far beyond he inner needs of asympoic inegraion heory of differenial equaions [20, p. 250], [25, p. 2], [64, p. 1099]. The norm is helpful for Bernsein s C g - approximaion problem [3, pp. 315, 317]. This problem was saed and solved in 1924 for an imporan paricular case by Bernsein [10], cf. also [2, p. 95], [67]. As originally formulaed, C g is he se of all coninuous real-valued funcions u() ha exis on he enire line and verify he condiion u() lim ± g() = 0. C g is endowed wih he usual operaions on funcions and he norm u g = u() sup <<+ g(), see [2, p. 96], [3, p. 315]. Here, g is a posiive coninuous funcion defined on R such ha lim ± n = 0, n = 0, 1, 2,... g() The C g -approximaion problem asks for necessary and sufficien condiions o be imposed on g() in order o make he se of all polynomials dense in C g. Shor proofs of he original resuls of Bernsein are provided by Ahiezer and Babenko [3]. Furher developmens for he case when g is lower semiconinuous can be found in he paper by Carleson [16], whereas a complee soluion was obained in 1953 by Pollard [67]. In he nex saemen C 0 sands for he linear space of all real-valued coninuous funcions ha are defined in R and vanish a ±, equipped wih he Chebyshev norm. Theorem 2 (Pollard). In order ha ( n g() ) n 0 be dense in C 0 i is necessary and sufficien ha ln g() d = and ha here exiss a sequence of polynomials p n such ha p n () lim n + g() = 1 and p n () C, g() <<+. The conclusion of hese invesigaions is ha i is fuile o sudy he exisence of soluions of ordinary differenial equaions in X 1 ( 0 ; g) and similar spaces by applying he polynomial perurbaion echnique which works very well on finie inervals [21, p. 32], [52, p. 210], [53, p. 634] in virue of he Weiersrass approximaion heorem [23, p. 198].

4 4 R.P. Agarwal e al. / On he asympoic behavior of soluions Theorem 3. The space (X 1 ( 0 ; g), ) is a Banach space. The proof of his claim can be found, for insance, in [25, p. 2], [76, pp. 61, 349]. The nex funcion space is a subspace of X 1 ( 0 ; g) denoed X 2 ( 0 ; g) which is defined as he se of funcions u() from X 1 ( 0 ; g) such ha u() lim + g() = l u R. The space X 2 ( 0 ; g) inheris he linear srucure and he norm of X 1 ( 0 ; g). Theorem 4. The space (X 2 ( 0 ; g), ) is a Banach space. Proof. Le T > 0. Following he same lines as in [7, p. 149], [25, p. 3] for he case g = 1, i is no difficul o verify ha he operaor F : X 2 ( 0 ; g) C([ 0, T ], R) defined by he formula ( u (Fu)(τ) = 0 + τ )[ ( 0 g 0 + τ )] 1 0, 0 τ<t, T τ T τ l u, τ = T, (3) is an isomeric isomorphism beween (X 2 ( 0 ; g), )and(c([ 0, T ], R), ). We noe ha Akinson [6, p. 383] has esablished a similar isomeric isomorphism beween he subspace C 0 X 2 (0; 1) formed by all real-valued funcions u() forwhichl u = 0 and he subspace of C([0, 1], R) conaining all funcions v() such ha v(1) = 0. In order o use classical fixed poin heories for he asympoic inegraion of differenial equaions, besides having a hand an appropriae inegral operaor (see he discussion in [1]), one needs cerain crieria of relaive compacness. The keysone in his regard is he Arzelà Ascoli compacness crierion for he subses of C([ 0, T ], R), see [19, p. 81], [24, p. 30], [37, p. 21]. Theorem 5 (C. Arzelà, 1895). Le A be nonempy. Suppose ha for a family U = {u α : α A} from C([ 0, T ], R) he following holds: (i) U is uniformly bounded, ha is, here exiss an M>0such ha u(τ) M for all τ [ 0, T ] and all α A; (ii) (G. Ascoli, 1884) U is equiconinuous, ha is, for every ε>0 here exiss a δ = δ(ε) > 0 such ha u α (τ 1 ) u α (τ 2 ) <ε for all τ 1, τ 2 [ 0, T ] wih τ 1 τ 2 <δand all α A. Then, he family U is relaively compac.

5 R.P. Agarwal e al. / On he asympoic behavior of soluions 5 The inversion of he above heorem is he following resul, cf. [39, pp ]. Proposiion 1. Le A and U be as in Theorem 5. If U is relaively compac, hen condiions (i), (ii) of Theorem 5 are saisfied. An ineresing observaion abou condiion (ii) in Theorem 5 has been made by Harman [34] who esablished ha here exis uniformly bounded sequences (u n ) n of funcions from C([ 0, T ], R) having all he subsequences divergen almos everywhere. A relaive compacness crierion for C m ([ 0, T ], R), where m 1, can now be saed wihou difficuly as we inroduce he sup norm u = m see [39, p. 14]. sup [ 0,T ] u (i) (), (4) Theorem 6. Le A be nonempy and suppose ha for U = {u α : α A} C m ([ 0, T ], R) he following holds: (i) here exiss an M>0 such ha u (r) α (τ) M, 0 r m, for all τ [ 0, T ] and all α A; (ii) for every ε>0 here exiss a δ = δ(ε) > 0 such ha u (r) α (τ 1 ) u (r) α (τ 2 ) <ε, 0 r m, for all τ 1, τ 2 [ 0, T ] wih τ 1 τ 2 <δand all α A. Then, he family U is relaively compac. A relaive compacness crierion for he space X 2 ( 0 ; 1) can be formulaed by using he isomorphism (3), cf. [25, p. 3]. The resul has been published in 1969 by Avramescu [7, p. 149] and also by Akinson [6, p. 383]. A similar conclusion abou he srucure of compac ses of X 2 ( 0 ; 1) was presened laer by Philos in [63, p. 412], where he reader is referred o unpublished manuscrips by V. Saikos for a proof. Theorem 7. Le A be nonempy and suppose ha for U = {u α : α A} X 2 ( 0 ;1) he following holds: (i) here exiss an M>0 such ha u α () M for all 0 and all α A;

6 6 R.P. Agarwal e al. / On he asympoic behavior of soluions (ii) for every ε>0 here exiss a δ = δ(ε) > 0 such ha u α ( 1 ) u α ( 2 ) <ε for all 1, 2 0 wih 1 2 <δand all α A; (iii) U is equiconvergen, ha is, for every ε>0, here exiss a Q = Q(ε) > 0 such ha u α () l uα <ε (5) for all Q(ε) and all α A. Then, he family U is relaively compac. Le us give hins for a differen proof of Theorem 7. This can be achieved by adaping he echnique used in [15, Chaper 2], [39, pp ] o esablish he Arzelà Ascoli relaive compacness crierion (Theorem 5). I involves he noions of ε-ne and oal boundedness which we inroduce below for he convenience of he reader. Definiion 1 ([15, p. 8], [39, p. 22]). Le ε>0andm be a subse of he meric space (X, d). Ase M 1 X is said o be an ε-ne of M if for every poin x M here exiss an elemen y M 1 such ha d(x, y) <ε.thesem is called oally bounded if, for every ε>0, i has a finie ε-ne. I is well known ha in any complee meric space a se is relaively compac if and only if i is oally bounded, see [15, p. 10], [39, p. 23], [77, p. 13]. The consrucion of an ε-ne of U in Theorem 5 can be realized as follows [39, p. 24]. Fix an ε>0 andpickaδ = δ(ε) > 0 ha saisfies he hypohesis (ii) of Theorem 5. Denoe by he se { k : k = 1, 2,..., n, n = T, k k 1 <δ} and by P ([ 0, T ], [M 1, M 2 ], ) he se of all he coninuous funcions u() definedin[ 0, T ] wih values in [M 1, M 2 ], M i R, so ha he resricion of u() o[ k 1, k ]is a sraigh line for every k. Such funcions are usually called polygonal lines. Denoe now by P he se P ([ 0, T ], [ M, M], ), where M saisfies he hypohesis (i) of Theorem 5. I is no difficul o esablish he following: (1) u = max 0 k n u( k ), u P; (2) he se P is relaively compac in C([ 0, T ], R); (3) he se P is an ε-ne of U. Thus, P has a finie ε-ne in C([ 0, T ], R) which is a finie 2ε-ne of U. Reurning now o Theorem 7, denoe by P ([ 0, + ), [M 1, M 2 ],, T ) he se of all he coninuous funcions u() definedon[ 0, + ) wih values in [M 1, M 2 ], M i R, so ha he resricion of u() o [ 0, T ] is an elemen of P ([ 0, T ], [M 1, M 2 ], ) andu() = u(t )forall T. In oher words, he elemens of he new se are obained via coninuous exension o he righ over an infinie inerval of he elemens of P ([ 0, T ], [M 1, M 2 ], ) by assigning he consan value on [T, + ). Fix now an ε>0 and inroduce M, δ and Q as in Theorem 7. The se P ([ 0, + ), [ M, M],, Q) is a relaively compac 2ε-ne of U, see [56, p. 18]. Noe ha any finie ε-ne of he se P ([ 0, + ), [ M, M],, Q)inX 2 ( 0 ;1) is a finie 3ε-ne of U. An ineresing adapaion of hypohesis (iii) of Theorem 7 has been suggesed by Z. Yin [78, p. 393] who replaced he inequaliy (5) by a Bolzano Cauchy condiion for he exisence of a limi wihou specifying he value of he limi, ha is,

7 R.P. Agarwal e al. / On he asympoic behavior of soluions 7 (iii )foreveryε>0 here exiss a Q = Q(ε) > 0 such ha u α () u α (s) <ε for all, s Q(ε) andallα A. Our excursus abou he Arzelà Ascoli relaive compacness crierion canno be closed unless we menion he beauiful proof of he crierion realized in [35, p. 4] by means of he Canor selecion heorem. Noe ha i is sraighforward o obain a relaive compacness crierion for he subses of X 2 ( 0 ; g) once we have replaced in he saemen of Theorem 7 all he funcions u α ()byu α ()/g(). To inroduce he hird funcion space, le us fix an ineger n 1. Le also (ρ r ) 0 r n 1 be a se of posiive coninuous funcions defined on [ 0, + ). We denoe by X 3 ( 0 ; ρ r )heseofall(n 1)- imes coninuously differeniable real-valued funcions u() ha are defined in [ 0, + ) and saisfy u (r) () = O(ρ r ()) as +, 0 r n 1. Here, u (0) u. This se is endowed wih he usual funcion operaions whereas he norm on X 3 ( 0 ; ρ r ) may be defined as u = n 1 r=0 u (r) () sup 0 ρ r (). Theorem 8. The space (X 3 ( 0 ; ρ r ), ) is a Banach space. Proof. We follow an argumen from [25, p. 3]. I is clear ha for every u X 3 ( 0 ; ρ r ) one has u (r) X 1 ( 0 ; ρ r ), 0 r n 1. The quaniies below m T = u T = { [ min inf ρr () ]} { [, M T = max sup ρr () ]}, 0 r n 1 [ 0,T ] 0 r n 1 [ 0,T ] n 1 sup r=0 [ 0,T ] where T 0, verify he condiion u (r) () n 1, u T,ρ = 1 u T u T,ρ 1 u T. M T m T sup r=0 [ 0,T ] u (r) () ρ r (), The inequaliy shows ha all he Cauchy sequences from X 3 ( 0 ; ρ r ) are locally uniformly convergen o C n 1 -funcions. The srucure of he nex funcion space is a bi more complicaed. Le 0 1 in accordance wih [1, Secion 2] and fix an ineger q, where1 q n. We inroduce he se X q 4 ( 0)ofall(n 1)-imes coninuously differeniable real-valued funcions u() ha are defined on [ 0, + ) and saisfy lim + [ u (n 1 k) () k a u i (k + 1 i)! k+1 i ] = a u k+1, a u k R, a u 0 = 0,

8 8 R.P. Agarwal e al. / On he asympoic behavior of soluions for all 0 k q 1. As usual, we consider he ypical operaions on numerical funcions for he se X q 4 ( 0) and provide a norm by inroducing he quaniy u = n 1 { u (n 1 j) () sup j=1 0 j } q 1 + sup k=0 0 { u(n 1 k) () k a u } i (k + 1 i)! k+1 i. (6) I is no difficul o verify ha X q 4 ( 0) is a subspace of X 3 ( 0 ; n 1 r ). In fac, we noe ha one obains by repeaed applicaion of L Hospial s rule he esimaes u (n 2) () u (n 3) () u() lim + u(n 1) () = lim = 2 lim = = (n 1)! lim = a 1. n 1 As a by-produc, his jusifies he presence of he firs erm in he sum from (6). The reason for inroducing he second erm in he definiion of he norm relies on he fac ha u (n 1 k) () k a u i (k + 1 i)! k+1 i X 2 ( 0 ;1) for all 0 k q 1. To undersand beer he srucure of he elemens of X q 4 ( 0), we need he following echnical lemma. Lemma 1. Le u C n 1 ([ 0, + )R) and assume ha for a cerain 1 p n one has lim + [ u (n p) () p 1 ] a i (p i)! p i = a p, where a 0 = 0, a i R, 0 i p 1. Then, u() = p a i (n i)! n i + o ( n p) as +. The lemma may be proved by repeaed applicaion of L Hospial s rule. I can be also obained by performing an ieraed inegraion as in [9, p. 128], [56, p. 36]. Lemma 1 shows ha every elemen of X q 4 ( 0) can be wrien as u() = A 1 n A q n q + o ( n q) as +, where A i R. In oher words, he elemens of X q 4 ( 0)are(n 1)-imes coninuously differeniable funcions wih real values ha have polynomial asympoic expansions. Theorem 9. The space (X q 4 ( 0), ) is a Banach space.

9 R.P. Agarwal e al. / On he asympoic behavior of soluions 9 Proof. We shall explain he main idea of he proof. Le (u r ) r 1 be a Cauchy sequence in X q 4 ( 0)andle T> 0. For every γ>0, here exiss an R γ > 1 such ha u r1 u r2 < γ T n 1 for all r j R γ. As a consequence, we deduce ha u (n 1 j) r 1 () u (n 1 j) r 2 () < γ T n 1 j for all [ 0, T ], 0 j n 1, and r j R γ. Therefore, (u r ) r 1 is a Cauchy sequence in C n 1 ([ 0, T ], R) and i has he uniform limi u (n 1 j),whereu is a C n 1 -funcion. The firs erm in (6) allows us o deduce ha, for 0 j n 1, j u (n 1 j) r () converges uniformly o j u (n 1 j) () in [ 0, + ) asr +. Since he esimae u (n 1) r 1 () u (n 1) r 2 () < γ T n 1 remains valid on he infinie inerval [ 0, + ), we conclude ha a ur 1 1 a ur 2 1 which yields lim r + aur 1 = a u 1. γ T n 1, Consider now he sequence (u (n 2) r () a ur 1 ) r 1 which converges uniformly o a funcion v() on [ 0, + ). Furhermore, one has lim r + aur 2 = lim v() R. + This can be esablished easily noicing firs ha he sequence (u (n 2) r () a ur 1 ) r 1 is a Cauchy sequence in C([ 0, T ], R) aswehaveha u (n 2) r 1 () a ur 1 1 u (n 2) r 2 () a ur 2 1 [ 1 u (n 2) r 1 () u (n 2) r 2 () + a ur 1 1 a ur 2 1 ] T<2γ, r j R γ. Since he sequence (u (n 2) r () a ur 1 ) r 1 locally uniformly converges o u (n 2) () a u 1, we conclude ha he funcions u and v are relaed o each oher hrough he equaion u (n 2) () a u 1 = v() which holds for all 0.

10 10 R.P. Agarwal e al. / On he asympoic behavior of soluions As i can be seen from he proof of Theorem 9, he presence of he firs erm in (6) is moivaed by obvious compuaional advanages. Anoher explanaion is provided by a echnique due o Weiersrass used o esablish he uniform convergence in C n 1 ([ 0, T ], R). To his end, allow us o consider a sequence (f r ) r 1 of C 1 -funcions such ha he sequence of derivaives (f r) r 1 converges o g uniformly on [ 0, T ]. Assume also ha for a cerain 1 [ 0, T ] he sequence of real numbers (f r ( 1 )) r 1 converges o l R. Then, he sequence (f r ) r 1 converges uniformly o a C 1 -funcion f defined, for all [ 0, T ], by f() = l + g(s)ds, 1 see, for insance, [15, p. 16], [76, p. 152]. The key problem here is o link he convergence of he sequence of derivaives (f r) r 1 wih he convergence of he sequence of funcions (f r ) r 1, a ask achieved by inroducing he firs erm in he sum from (6). We noe ha some auhors have devised such norms in virue of he preceding echnique by using a sum like n 1 k=0 u (n 1 k) ( 0 ) k, 0 see [30, p. 39], [63, p. 413], whereas in [56, p. 35] are considered spaces of C n 1 -funcions wih u (k) ( 0 ) = 0for0 k n 1. In our opinion, he choice of he firs erm in (6) is exremely convenien for he purposes of asympoic inegraion heory no only since i follows direcly from he exisence of he limi lim + u(n 1) () bu also because of he special role played by such quaniies [1, Secion 2], [59]. We sae wihou proof a relaive compacness crierion for he subses of X q 4 ( 0) noing ha i can be esablished by ieraed applicaion of Theorem 7. Theorem 10. Le B be a se from X q 4 ( 0) such ha he following condiions are saisfied: (i) here exiss an L>0 such ha u (n 1 j) () j L, 1 j n 1, and k a u u(n 1 k) i () (k + 1 i)! k+1 i L, 0 k q 1, for all 0 and all u B;

11 R.P. Agarwal e al. / On he asympoic behavior of soluions 11 (ii) for every ε>0 here exiss a δ(ε) > 0 such ha u (n 1 j) ( 1 ) j 1 u(n 1 j) ( 2 ) j 2 <ε, 1 j n 1, and u(n 1 k) ( 1 ) u (n 1 k) ( 2 ) k a u i ( k+1 i 1 k+1 i ) 2 <ε, (k + 1 i)! 0 k q 1, for all s 0 wih 1 2 <δ(ε) and all u B; (iii) for every ε>0 here exiss a C(ε) > 0 such ha u (n 1 j) () j au 1 j! <ε, 1 j n 1, and k a u u(n 1 k) i () (k + 1 i)! k+1 i a u k+1 <ε, 0 k q 1, for all C(ε) and all u B. Then he se B is relaively compac in X q 4 ( 0). As in he case of Theorem 5, we have an inverse resul, similar o Proposiion 1. Proposiion 2. Le B be a relaively compac se from X q 4 ( 0). Then, he condiions (i) (iii) of Theorem 10 are saisfied. An imporan simplificaion of he compuaions required by Theorem 10 is given nex. Lemma 2. Le M be a subse of X q 4 ( 0) wih he following properies: (i) here exiss an L>0 such ha u (n 1 j) () j L, 0 j n 1, (7) for all 0 and all u M; (ii) for every ε>0 here exiss a C(ε) > 0 such ha u (n 1) () a u 1 <ε for all C(ε) and all u M.

12 12 R.P. Agarwal e al. / On he asympoic behavior of soluions Then, here exiss a C 1 (ε) C(ε) such ha u (n 1 j) () j au 1 j! ε, 1 j n 1, for all C 1 (ε) and all u M. Proof. Fix an ε>0. I is no hard o check ha lim γ + ε (L γ )[ ( L γ )n 1] γ ε + 1 ε 1 L γ ε + 1 = 0, and hus here exiss a γ>15 such ha ε (L γ )[ ( L γ )n 1] γ ε + 1 ε 1 L γ ε + 1 < ε 5. Now, le 1 l n 1and C(ε/γ). Inegraing l imes he inequaliy a u 1 ε γ <u(n 1) () <a u 1 + ε γ, we obain, for all u M, and l u (C (n 1 i) i=1 l u (C (n 1 i) i=1 ( )) [ ε C ( )] ε l i ( γ + a u 1 ε γ (l i)! γ ( )) [ ε C ( )] ε l i ( γ + a u 1 + ε γ (l i)! γ These inequaliies yield, for C(ε/γ), i=1 ) [ C ( )] ε l γ l! ) [ C ( )] ε l γ l! u (n 1 l) () u (n 1 l) (). l ( )) [ ( ε 1 C ε )] l i u (C (n 1 i) γ 1 (a γ (l i)! i + u 1 ε ) [ ( 1 C ε )] l γ u(n 1 l) () γ l! l and l i=1 ( )) ε 1 u (C (n 1 i) γ (l i)! [ C ( ε γ ) ] l i 1 i + (a u 1 + ε γ ) 1 l! [ C ( ε γ ) ] l u(n 1 l) () l. Noe ha, in accordance wih (7), we have a u 1 L.

13 Taking ino accoun ha R.P. Agarwal e al. / On he asympoic behavior of soluions 13 0 < C(ε/γ) 1, we conclude ha l ( )) [ ( ε 1 C ε )] l i u (C (n 1 i) γ 1 γ (l i)! i i=1 l 1 u (n 1 i)( C ( )) ε [ ( ) C ε ] l i [ ( γ C ε )] i [ ( (l i)! )] γ γ i=1 C ε i γ Thus, l ( )) [ ( ε 1 C ε )] l i u (C (n 1 i) γ 1 γ (l i)! i i=1 for all C 1 (ε) = (Lγ/ε + 1)C(ε/γ). Furhermore, ( a u 1 ± ε ) [ ( 1 C ε ) γ γ l! ( a u 1 + ε ){ 1 γ = ε γ ] l ( a u 1 ± ε γ ) [1 C( ε γ 3ε γ ) 1 l! a u 1 ± ε 1 γ l! ] n 1 } ( L + ε C 1 (ε) γ (L γ )[ ( L γ )n 1] ε + 1 ε 1 L γ ε + 1 < ε 5 )[ 1 { 1 l i=1 ( 1 ( ε ) 1 (l i)! LC γ [1 C( ε γ for all C 1 (ε)andallu M. Thus we conclude ha ( 1 1 l! au 1 ε γl! ) < u(n 1 l) () γ l < 1 ( 1 l! au 1 + ε γl! ) γ for all C 1 (ε)andallu M. Clearly, he choice of γ>15 implies ha 1 γl! + 3 γ < 1, )] n 1 } )n 1] 1 L γ ε + 1 3L C( ε ) γ. which complees he proof. 3. Coninuiy of he inegral operaors The posiive funcion g() from (2) plays a fundamenal role in Theorem 1 as i conrols he way in which he soluion x() of (1) evolves. In [11, p. 5], Bihari has presened an ingenious echnique for he

14 14 R.P. Agarwal e al. / On he asympoic behavior of soluions consrucion of such a funcion under reasonable condiions on G(, r). To explain his procedure, assume ha G(, r) = h()ω(r), where he funcions h() and ω(r) are coninuous, ω(r) > 0 for r > 0 and ω(r) is nondecreasing. Suppose also ha and 0+ dx + ω(x) = dx ω(x) =+ (8) 0 h()d<+. (9) We may assume wihou loss of generaliy (he complemenary case can be deal wih similarly) ha K 1 <K 2 in Theorem 1 and here exiss an a>0suchha I(a) = γ+a γ+ K 1 K 2 a dr ω(r) = (K 2 K 1 ) h()d. (10) 0 Then, ) g() = Ω (Ω(γ 1 + a) (K 2 K 1 ) h(s)ds, where, for all x>0andsomefixedu 0 > 0, he funcion Ω is defined by Ω(x) = x u 0 dr ω(r). An immediae example of a funcion ω(r) saisfying he preceding condiions is provided by ω(r) = r, r 0. A sraighforward compuaion yields a = γρ 1 e ρ 1 e ρ 2 ρ 1 e ρ 1 ρ2 e ρ 2, (11) where ρ i = K i 0 h()d, i = 1, 2. We noe ha a class of examples can be consruced simply by ieraion. In fac, le us inroduce he funcions (ω m ) m 1 hrough he formulas ω 1 (r) = ln(1 + r), ω m+1 (r) = ln ( 1 + r + ω m (r) ), r 0. I is easy o see ha 0 <ω m (r) <mr for all r>0. (12) Inequaliy (12) shows ha he funcions ω m (r) verify condiions (8) (9). Le now a m = γρ 1 e mρ 1 e mρ 2 ρ 1 e mρ 1 ρ2 e mρ 2.

15 R.P. Agarwal e al. / On he asympoic behavior of soluions 15 Then, defining γ+x I m (x) def dr =, x 0, γ+ K 1 x ω K m (r) 2 and aking ino accoun (12), one has I m (x) I m (a m ) > 1 ( m log sup x>0 γ + am γ + K 1 K 2 a m ) = (K 2 K 1 ) h()d. 0 Since he funcion I m (x) is coninuous and I m (0) = 0, we deduce ha here exiss an a>0suchha I m (a) = γ+a γ+ K 1 K 2 a dr ω m (r) = (K 2 K 1 ) h()d. 0 An analogous resul, in he case K 2 = 0, is discussed by Corduneanu [22, p. 362]. Inequaliies similar o (1) are used essenially in he sudy of global properies of nonlinear ordinary differenial equaions of arbirary order, cf. [72]. The inegral inequaliy of he form w(x) x a ( K x,, w() g() ) d, (13) where g() is a given coninuous posiive funcion, has been invesigaed by Saō [71, p. 281]. Finally, we noe ha resuls in his spiri can be designed wih he help of Kamke s funcion S, see [38, p. 289], [71, p. 284]. We shall esablish nex cerain deails allowing one o prove ha a case of inegral equaion (1) devised for use in he space X 3 ( 0 ; ρ r ) has soluion. Theorem 11. Le T> 0, K>0 and B be a se from X 3 ( 0 ; ρ r ) such ha b K for all b B. Le he funcions f :[ 0, + ) R n R, g, h :[ 0, + ) R be coninuous and consider ha h() does no vanish evenually. Assume also ha f ( s, u, u,..., u (n 1)) F ( s, u, u,..., u (n 1) ), where he comparison funcion F :[ 0, + ) R n + [0, + ) is coninuous and saisfies F ( s, Kρ0 (s),..., Kρ n 1 (s) ) ds<+. Then, he resricion of he funcion Q :[ 0, + ) B R, Q(, u) = g() + h() f ( s, u(s), u (s),..., u (n 1) (s) ) ds, 0, u B, o [ 0, T ] B is uniformly coninuous.

16 16 R.P. Agarwal e al. / On he asympoic behavior of soluions Proof. We inroduce he noaion and f[s, u] def = f ( s, u(s), u (s),..., u (n 1) (s) ) [u](s) def = ( u(s), u (s),..., u (n 1) (s) ). Then, Q( 2, u) Q( 1, v) g( 2 ) g( 1 ) + h( 2 ) 2 1 f[s, u] ds + h( 2 ) ( f[s, u] f[s, v] ) ds 1 + h( 2 ) h( 1 ) ( f[s, v] ) ds. Fix a β>0. Then, here exiss a T β >Tsuch ha T β F ( s, Kρ 0 (s),..., Kρ n 1 (s) ) ds< where def h = sup h(s). s [ 0,T ] 1 β 16 h, def Le us recall he noaions in he proof of Theorem 8. We inroduce he consan M β = M Tβ. Then, one has sup [ 0,T β ] u (r) () M β K, u B, 0 r n 1. Since he funcion f is uniformly coninuous in [ 0, T β ] [ M β K, M β K] n, here exiss a δ = δ(β) such ha f[s, u] f[s, v] < β 8(T β 0 ) h, for all s [ 0, T β ]andall[u](s), [v](s) [ M β K, M β K] n wih u (r) (s) v (r) (s) δ, where 0 r n 1.

17 Le u, v B saisfy u v Furher, u (r) (s) v (r) (s) f[s, u] f[s, v] ds R.P. Agarwal e al. / On he asympoic behavior of soluions 17 [ sup ρr () ] [ 0,T β ] δ M β. Then, for all s [ 0, T β ], one has δ M β δ. f[s, u] f[s, v] ds 1 0 Tβ 0 f[s, u] f[s, v] ds + Tβ 0 β 8(T β 0 ) h ds + 2 T β f[s, u] ds + β 16 h T β f[s, v] ds = β 4 h. (14) The idea behind his compuaion is ha, for any coninuous funcion f ha is also L 1 over an infinie inerval, here exiss an unbounded subinerval over which he remainder of he inegral of f can be made as small as desired. This allows one o repea on a compac subinerval of he domain of definiion of f he classical compuaion in he proof of Peano s exisence heorem, cf. [33, p. 15], [39, p. 40], and evaluae inegrals over non-compac inervals by small quaniies, see [57, pp ]. Finally, one has Q( 2, u) Q( 1, v) g( 2 ) g( 1 ) 2 + h F ( s, Kρ 0 (s),..., Kρ n 1 (s) ) ds + h 1 β 4 h + h( 2 ) h( 1 ) ( 0 F ( s, Kρ 0 (s),..., Kρ n 1 (s) ) ) ds g( 2 ) g( 1 ) + h F (s, Kρ 0,..., Kρ n 1 ) β 4 ( + F ( s, Kρ 0 (s),..., Kρ n 1 (s) ) ) ds h( 2 ) h( 1 ), 0 which yields he conclusion. I is no difficul o see ha Theorems 11 and 12 can be resaed, under appropriae hypoheses, for a funcion Q wih a more complicaed srucure defined by he formula Q(, u) = g() + + m i=l+1 l i=1 h i () h i () a i (s)f ( s, u(s), u (s),..., u (n 1) (s) ) ds 0 a i (s)f ( s, u(s), u (s),..., u (n 1) (s) ) ds,

18 18 R.P. Agarwal e al. / On he asympoic behavior of soluions as, for insance, he funcions ( s) n 1 Q 1 (, u) = g() 0 (n 1)! f( s, u(s), u (s),..., u (n 1) (s) ) ds and (s ) n 1 Q 2 (, u) = g() + (n 1)! f( s, u(s), u (s),..., u (n 1) (s) ) ds. Theorem 12. Le he hypoheses of Theorem 11 be verified, and assume in addiion ha g X 1 ( 0 ;1) and h() F ( s, Kρ 0 (s),..., Kρ n 1 (s) ) ds = o(1) as +. (15) Then, he funcion V : B X 1 ( 0 ;1) defined by V (u)() = Q(, u), 0, u B, is uniformly coninuous. Proof. Fix a β>0. I follows from (15) ha here exiss a T β > 0 such ha, for all T β, h() F ( s, Kρ 0 (s),..., Kρ n 1 (s) ) ds< β 2. Now, one has V (u)() V (v)() h() h() f[s, u] f[s, v] ds [ f[s, u] ds + f[s, v] ] ds β, T β, (16) where u, v B. On he oher hand, for [ 0, T β ], he resricion Q :[ 0, T β ] B R is uniformly coninuous. Then, here exiss a δ = δ(β) > 0 such ha V (u)() V (v)() <βfor all [ 0, T β ]andallu, v B saisfying u v < δ. Therefore, V (u) V (v) <β for all u, v B saisfying u v <δ. We would like o commen now on he echnique of (14), (16). Many auhors, perhaps in virue of he generaliy of hypoheses in he resuls, esablished he coninuiy of a cerain inegral operaor T by inroducing sequences of funcions (f n ) n and by he examinaion of he convergence of (T (f n )) n by means of he Lebesgue dominaed convergence heorem [14, p. 54], [70, p. 19]. See he deails in [1, Secion 4]. This approach is refleced, for insance, in he work of Bainov and Simeonov [8, pp. 94,

19 R.P. Agarwal e al. / On he asympoic behavior of soluions ], papers by Mâagli and Masmoudi [49, p. 301], Maaoug and Zribi [50, p. 725], Miller and Sell [51, p. 143], Granas and Guennoun [31, p. 706], Granas e al. [32, p. 156], Frigon and O Regan [29, pp. 39, 47], o cie a few. I originaes apparenly from he applicaion of he closed graph heorem [14, p. 20] o esablish cerain admissibiliy resuls for inegral operaors, cf. [22, p. 350]. In our opinion, he echnique refleced in (14) and (16) is more appropriae o sudy he asympoic behavior of soluions for general ordinary differenial equaions. A similar procedure has been devised by Przeradzki [68], who esablished he coninuiy of an inegral operaor using he local uniform coninuiy of he nonlineariy in quesion raher han Lebesgue s dominaed convergence heorem and hen compleed he proof by applying he degree heory for DC-mappings, cf. [42]. A serious difficuly when dealing wih he spaces X 1 ( 0 ; g), X 3 ( 0 ; ρ r ) is he lack of general relaive compacness crieria. We shall provide, however, in wha follows some relevan sufficien condiions for he relaive compacness of a subse of X 3 ( 0 ; ρ r ). To his end, le us follow he classical definiion due o Ascoli and call equiconinuous any se E X 3 ( 0 ; ρ r ) so ha for any ε>0 here exiss a δ = δ(ε) > 0 wih u (r) ( 2 ) ρ r ( 2 ) u(r) ( 1 ) ρ r ( 1 ) <ε for all 0 r n 1, all 1, 2 [ 0, + ) saisfying 2 1 <δand all u E. Similarly, he se E is considered equiconvergen if u (r) X 2 ( 0 ; ρ r )andforanyε>0 here exiss a Q = Q(ε) > 0 such ha u (r) () ρ r () lim s + u (r) (s) ρ r (s) <ε for all 0 r n 1andallu E. Proposiion 3. Le P be a bounded se from X 3 ( 0 ; ρ r ) and R k :[ 0, + ) X 3 ( 0 ; ρ r ) R, 0 k n 1, be coninuous funcions such ha for every T > 0 and every bounded B X 3 ( 0 ; ρ r ) he resricions R k :[ 0, T ] B R are uniformly coninuous. Furher, le V be he se of all he funcions v() in X 3 ( 0 ; ρ r ) for which here exiss an elemen p = p(v) Psuch ha v (k) () = R k (, p), 0,0 k n 1. Assume also ha R k (, p) ρ k () H(), 0, for all p P,0 k n 1, where H is a real-valued coninuous funcion on [ 0, + ) such ha lim H() = 0. + (17) Then, he se V is equiconinuous.

20 20 R.P. Agarwal e al. / On he asympoic behavior of soluions Proof. Noice ha, for all 0 r n 1, one has v (r) [ ( 2 ) ρ r ( 2 ) v(r) ( 1 ) ρ r ( 1 ) = 1 ρ r ( 2 ) 1 ] R r ( 2, p) + 1 [ Rr ( 2, p) R r ( 1, p) ]. ρ r ( 1 ) ρ r ( 1 ) Fix an ε>0. I follows from (17) ha here exiss a T ε > 0 such ha H() <ε/2forall T ε.the resricions R r :[ 0,2T ε ] P R are uniformly coninuous and bounded, R r (, p) H M 2Tε. Since he funcions (1/ρ r ()) 0 r n 1 are uniformly coninuous in [ 0,2T ε ], here exiss a δ = δ (ε) > 0 such ha 1 ρ r ( 2 ) 1 ρ r ( 1 ) ε 2 H M 2Tε for all 1, 2 [ 0,2T ε ] saisfying 2 1 <δ and all 0 r n 1. The resricions R r being uniformly coninuous here exiss a δ = δ (ε) > 0 such ha R r ( 2, p) R r ( 1, p) 1 2 ε m 2T ε for all 1, 2 [ 0,2T ε ] saisfying 2 1 <δ and all p P. Choosing δ = min{δ, δ, T ε }, for all 1, 2 0 such ha 2 1 <δone has eiher 1 2T ε and hus 2 T ε (he case 2 2T ε is reaed analogously) or 1, 2 2T ε. In he former case, i follows ha v (r) ( 2 ) ρ r ( 2 ) v(r) ( 1 ) ρ r ( 1 ) v(r) ( 2 ) ρ r ( 2 ) + v(r) ( 1 ) ρ r ( 1 ) H( 2 ) + H( 1 ) <ε for all 0 r n 1 whereas in he laer case we conclude ha v (r) ( 2 ) ρ r ( 2 ) v(r) ( 1 ) ρ r ( 1 ) ε 2sup 0 s 2T ε H(s) M 2Tε H( 2 )ρ r ( 2 ) + ε 2 m 2T ε 1 ρ r ( 1 ) ε for all 0 r n 1andallv V. Proposiion 4. Le g be a fixed elemen of X 3 ( 0 ; ρ r ). In he condiions of Proposiion 3, replace he se V by he se V g of all he funcions v() in X 3 ( 0 ; ρ r ) such ha v (k) () = g (k) () + R k (, p), 0,0 k n 1. Then, he se V g is relaively compac in X 3 ( 0 ; ρ r ).

21 R.P. Agarwal e al. / On he asympoic behavior of soluions 21 Proof. Using he fac ha, in any normed space, he propery of compacness is invarian o finie ranslaions, i suffices o esablish ha he se V g g is relaively compac in X 3 ( 0 ; ρ r ). According o Proposiion 3, he se V g g is bounded and equiconinuous. From (17) we deduce ha v (k) v (k) () X 2 ( 0 ; ρ k ) and l v (k) = lim = 0, k 0, n 1, + ρ k () uniformly wih respec o he elemens v of V g g which means ha V g g is also equiconvergen. Le us emphasize now an essenial feaure in our discussion. As he funcion space X 2 ( 0 ;1) is a opological subspace of X 1 ( 0 ;1), one can use Theorem 7 as a relaive compacness crierion for he subses of X 1 ( 0 ; 1) paying aenion o he fac is hypoheses are only sufficien and no necessary. According o his remark, an ieraed applicaion of Theorem 7 o he se of funcions { v (r) } () ρ r () : v V g for all 0 r n 1 yields he relaive compacness of V g. To give a hin abou our inenions, if we ake a funcion R in Proposiion 3 as R(, u) = hen he role of H will be played by H() = f ( s, u(s), u (s),..., u (n 1) (s) ) ds, F (s, Kρ 0 ( s, Kρ0 (s),..., Kρ n 1 (s) ) ds. 4. The formula of soluions We shall inroduce here and commen upon a fruiful inegral represenaion for he soluions of nonlinear ordinary differenial equaions. I is cusomary in he asympoic inegraion heory o regard he nonlineariy of he equaion as a perurbaion of he corresponding linear par. This approach has been presened in a well-developed form in he book by F. Tricomi [74, p. 142] and may be ermed he Fubini Tricomi procedure. The core idea of he procedure is o consider he nonlineariy simply as a nonhomogenous (forcing) erm of he remaining linear par of he equaion and hen obain an inegral represenaion for is soluions via Lagrange s mehod of variaion of consans. A furher generalizaion of his echnique, sepping ou from our inenions, leads o he Alekseev formula (he nonlinear variaion of consans mehod) [4] and is discussed in deail in [46, p. 78], [47, pp. 78, 83], [45, pp. 141, 151]. More general resuls of his ype for inegral and inegro-differenial equaions have been obained by Brauer [13]. We leave ou as well sophisicaed approaches in he asympoic inegraion heory given in [62,40]. Consider now he n-h order nonlinear differenial equaion u (n) + a 1 ()u (n 1) + + a n ()u + f (, u, u,..., u (n 1)) = 0, 0, (18)

22 22 R.P. Agarwal e al. / On he asympoic behavior of soluions along wih he associaed linear equaion z (n) + a 1 ()z (n 1) + + a n ()z = 0, 0. (19) Fix an ineger m 2. A funcion, denoed W [ϕ 1,..., ϕ m ] and named Wronskian, can be associaed naurally o every se of real-valued C m 1 -funcions (ϕ i ) 1 i m defined on [ 0, + ) bymeansofhe formula ϕ 1 () ϕ m () ϕ 1 W [ϕ 1,..., ϕ m ]() = () ϕ m()....,. ϕ (m 1) 1 () ϕ (m 1) m () where 0. If (z i ()) 1 i n is a se of fundamenal soluions of Eq. (19), le us inroduce he funcions (w i ) 1 i n by w i () = W [z 1,..., z i 1, z i+1,..., z n ](), 0,1 i n. (20) W [z 1,..., z n ]() In case i = 1 he firs argumen of W from (20) is z 2 whereas for i = n he las argumen of W is z n 1. Lemma 3. The following ideniies for he funcions w i hold: n i=1 ( 1) n i z (r) i ()w i () = δ r,n 1, 0 r n 1. Proof. For r = n 1 one has δ r,n 1 = 1. Since (z i ()) 1 i n is a se of fundamenal soluions, is Wronskian does no vanish. Therefore, expanding he deerminan W [z 1,..., z n ]() wih respec o he las row, we obain n W [z 1,..., z n ]() = ( 1) n+k z (n 1) k ()W [z 1,..., z k 1, z k+1,..., z n ]() k=1 n = ( 1) n k z (n 1) k ()W [z 1,..., z k 1, z k+1,..., z n ](). k=1 Dividing he ideniy by W [z 1,..., z n ](), we arrive a he conclusion. Suppose now ha r = 1. We have ha z 1 () z n () z 1 () z n()..... = 0, (21) 1 () z n (n 2) () z 1 () z n() z (n 2)

23 R.P. Agarwal e al. / On he asympoic behavior of soluions 23 where he deerminan in (21) is obained by replacing he las row in W [z 1,..., z n ]() wih is second row. Now, expanding he laer deerminan wih respec o he las row, we obain 0 = = n ( 1) n+k z k()w [z 1,..., z k 1, z k+1,..., z n ]() k=1 n k=1 ( 1) n k z k()w [z 1,..., z k 1, z k+1,..., z n ](). We ge he conclusion by dividing by W [z 1,..., z n ](). The inegral represenaion under consrucion here is needed o build an inegral operaor suiable for he applicaion of classical fixed poin heories in he fundamenal problem of he exisence of soluions of Eq. (18) ha behave a + like soluions of he linear equaion (19). We sar by ransforming Eq. (18) ino a perurbed sysem of firs-order linear differenial equaions and hen apply he variaion of consans formula. This gives d d u. u (n 1) = a n () a n 1 () a n 2 () a 1 () u. u (n 1) f[, u] Denoing by X() he principal fundamenal marix of he linear sysem, we search a soluion in he form u. = X() u (n 1) c 1 (). c n () where c i () are unknown C 1 -funcions, hus obaining., (22) c 1 () 0. = [X()] 1. c 0. (23) n() f[, u] Relaion (23) reads as c k() = f[, u]xkn 1 (), 1 k n, where X 1 kn () sands for he k-h elemen of las column of he marix [X()] 1. This means ha Xkn 1 () = W [z 1,..., z k 1, z k+1,..., z n ]() ( 1)n+k = ( 1) n k w k () W [z 1,..., z n ]()

24 24 R.P. Agarwal e al. / On he asympoic behavior of soluions and so c k() = ( 1) n k w k ()f[, u], 1 k n. (24) For oher deails in his maer see [24, pp ], [28, pp , 39], [36, p. 98], [76, p. 203]. Formula (22) becomes now u() = n c j ()X 1j () = j=1 n c j ()z j (). j=1 I is essenial o deduce from (24) convenien compuaional formulas for he coefficiens (c j ) 1 j n, i.e., expressions which agree wih he daa (iniial and erminal). A formal inegraion of Eqs (24) yields c i () = c i ( 0 ) f[s, u]( 1) n i w i (s)ds, 1 i p 1, 0 and c i () = c i ( ) + f[s, u]( 1) n i w i (s)ds, p i n, where p 1, n + 1isfixed.Ifp = 1 we inerpre p 1 as before whereas in he case p = n + 1we simply disregard he laer ideniy. The inegral represenaion of a soluion y() of Eq. (18) reads as p 1 y() = c i ( 0 )z i () + i=1 n + i=p ( 1) n i z i () n p 1 c i ( )z i () ( 1) n i z i () i=p i=1 w i (s)f[s, y]ds 0 w i (s)f[s, y]ds p 1 = ẑ() ( 1) n i z i () w i (s)f[s, y]ds (25) i=1 0 n + ( 1) n i z i () w i (s)f[s, y]ds. (26) i=p We remark ha ẑ() from (25) is a soluion of Eq. (19). Lemma 4. The following ideniies hold for all 0 r n 1: p 1 y (r) () = ẑ (r) () i=1 n + ( 1) n i z (r) i i=p ( 1) n i z (r) i () w i (s)f[s, y]ds 0 () w i (s)f[s, y]ds. (27)

25 R.P. Agarwal e al. / On he asympoic behavior of soluions 25 Proof. Clearly, for r = 0 he formula (27) reduces o (26). We shall esablish he ideniy only for he case r = 1. A differeniaion of (26) wih respec o yields p 1 y () = ẑ n () ( 1) n i z i() w i (s)f[s, y]ds + ( 1) n i z i() w i (s)f[s, y]ds i=1 0 i=p [ p 1 ] n f[, y] ( 1) n i z i ()w i () + ( 1) n i z i ()w i (). i=1 i=p The brackeed erm vanishes according o Lemma 3, which esablishes (27) for r = 1. To apply he above resuls o he general n-h order nonlinear differenial equaion u (n) + f (, u, u,..., u (n 1)) = 0, 0, (28) one may ake, cf. [43, p. 388], z i () = i 1 (i 1)! and w i () = n i, i 1, n. (29) (n i)! More deails on he inegral represenaion of soluions can be read in [27, pp. 161, 163]. We menion also he ineresing work of Vasilache [75]. We summarize he preceding discussion by inroducing he general inegral operaor below p 1 n Q(y)() = ẑ() ( 1) n i z i () w i (s)f[s, y]ds + ( 1) n i z i () w i (s)f[s, y]ds, (30) i=1 0 i=p where ẑ() is a soluion of Eq. (19). If p = n + 1 hen we simply disregard he las erm in (30). An imporan paricular case is provided by he inegral operaor Q(y)() = ẑ() 0 ( s) n 1 f[s, y]ds, (31) (n 1)! associaed wih (28), which can be wrien as (p = n + 1) n Q(y)() = ẑ() ( 1) n i z i () w i (s)f[s, y]ds i=1 0 n = ẑ() ( 1) i z n i () w n i (s)f[s, y]ds i=1 0 for z i (), w i () given by (29), in accordance wih Newon s binomial formula.

26 26 R.P. Agarwal e al. / On he asympoic behavior of soluions 5. The Kusano Trench heory of asympoic inegraion [43,44,73] 5.1. General resul Le ẑ() be a soluion of Eq. (19) and assume ha here exis he posiive and coninuous funcions (ρ r ) 0 r n 1 defined in [ 0, + ) which saisfy ẑ (r) () ρ r (), 0 r n 1. Suppose also ha here exiss a c>0 such ha he inequaliy p 1 i=1 z (r) i () w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds 0 n + z (r) i () w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds cρ r () (32) i=p holds for all 0 and 0 r n 1. Finally, assume ha z (r) i () w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds = o ( ρ r () ), 1 i p 1, 0 z (r) i () w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds = o ( ρ r () ), p i n, (33) as + for all 0 r n 1. Theorem 13. Assume ha he funcion f saisfies he inequaliy f ( s, u, u,..., u (n 1)) F ( s, u, u,..., u (n 1) ) (34) and (32), (33) hold. Then, Eq. (18) has a soluion y() defined in [ 0, + ) such ha y (r) () = ẑ (r) () + o ( ρ r () ), 0 r n 1, as +. Proof. In he space X 3 ( 0 ; ρ r ) we consider he closed ball of radius c cenered abou ẑ, hais, B = { y X 3 ( 0 ; ρ r ): y (r) () ẑ (r) () cρ r (), 0 r n 1 }. We claim ha he operaor Q : B B defined by (30) is well-defined and compleely coninuous. To prove his, le us noice ha

27 R.P. Agarwal e al. / On he asympoic behavior of soluions 27 y (r) () ẑ (r) () + cρ r () (1 + c)ρ r (), w i (s)f[s, y] ds w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds, 0 0 w i (s)f[s, y] ds w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds. Thus, [ Q(y) ](r) () ẑ (r) () p 1 z (r) i i=1 n + i=p cρ r () () w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds 0 z (r) i () w i (s) F ( s,(1+ c)ρ 0 (s),...,(1+ c)ρ n 1 (s) ) ds and so he operaor Q is well-defined. Nex, we inroduce he operaor V r : B X 1 ( 0 ; 1) wih he formula where ( Vr (y) ) () = [Q(y)](r) () ρ r () p 1 = g r () ( 1) g r () = ẑ (r) () ρ r (). i=1 n + ( 1) i=p n i z(r) i () ρ r () n i z(r) i () ρ r () 0 w i (s)f[s, y]ds w i (s)f[s, y]ds, Applying Theorem 12, wih K = 1 + c, we conclude ha all he funcions (V r ) r 0,n 1 are uniformly coninuous. Therefore, he operaor Q : B X 3 ( 0 ; ρ r ) is uniformly coninuous. Le us show furher ha he se Q(B) is relaively compac in X 3 ( 0 ; ρ r ). Indeed, he elemens of Q(B) can be wrien as where [ Q(y) ] (r) () = g (r) () + n Rr(, i y), g = ẑ, i=1 Rr(, i y) = ( 1) n i z (r) i () 0 w i (s)f[s, y]ds, 1 i p 1,

28 28 R.P. Agarwal e al. / On he asympoic behavior of soluions and Rr(, i y) = ( 1) n i z (r) i () w i (s)f[s, y]ds, p i n, for all r 0, n 1. The relaive compacness of Q(B) follows from Proposiion 4. The validiy of our claim is now esablished. Finally, by an applicaion of he Schauder Tikhonov fixed poin heorem we deduce ha he operaor Q has a fixed poin y in B Disconjugae equaions An accessible heory of disconjugacy for Eq. (19) can be read in [18, Chaper 3]. In his case he quaniies z i ()andw i () can be assumed posiive and such ha [ ] d wi () > 0, 0, d w j () and w i () z i () lim =+ and lim + w j () + z j () = 0 for all i<j.givenap 1, n, we inroduce [43, p. 386] he quaniies v rp () = 1 w p () n s=1 w s () z (r) s (), 0 r n 1. As we inend o use z p for he role of ẑ, he nex lemma is of essenial use. Lemma 5. Assume ha here exiss a K>0 such ha wp (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds<+. (35) Then, w i (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ( ) wi () ds = o, 1 i p 1, 0 w p () and w i (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ( ) wi () ds = o, p i n, w p () as +.

29 Proof. We sar wih he case 1 i p 1. Since R.P. Agarwal e al. / On he asympoic behavior of soluions 29 w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds = o(1) as +, i is obvious ha w i () w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ( ) wi () ds = o, w p () w p () where 1 i p 1. Furher, one has 0 w i (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds = w i() w p () According o (36), we obain ha 0 w i (s) w p (s) w p(s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds 0 w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds. (36) ( ) wi () 1 w i (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds w p () 0 0 w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds = O(1) as +. (37) If he inegral in (37) is bounded hen he proof is complee. Oherwise, le us add an o(1) o (37) and hen apply L Hospial s rule: lim + {( ) wi () 1 [ w i (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds w p () 0 + w i() w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ]} ds w p () d = lim + {{ d [ wi () + w p () = lim + [ ]} wi () 1 { w i ()F (, Kv 0p (),..., Kv n 1p () ) w p () ] [ w p ()F (, Kv 0p (),..., Kv n 1p () )] + d d }} w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds [ ] wi () w p () w p (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds = 0. (38) The rick (38) is adaped from a paper by Consanin [17, p. 133].

30 30 R.P. Agarwal e al. / On he asympoic behavior of soluions Consider now he case p i n. Thus, [ ] wi () 1 w i (s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds w p () w p (s) w i (s) w i(s)f ( s, Kv 0p (s),..., Kv n 1p (s) ) ds = o(1) as +, which complees he proof. Fix now p 1, n and assume ha for a cerain c>0 one has he nex varian of (32): p 1 i=1 z (r) i + () w i (s)f ( s,(1+ c)v 0p (s),...,(1+ c)v n 1p (s) ) ds 0 n i=p z (r) i () w i (s)f ( s,(1+ c)v 0p (s),...,(1+ c)v n 1p (s) ) ds cv rp (), 0 r n 1. In paricular, (35) holds for K = 1 + c. Now, Lemma 5 and he obvious inequaliy max 1 i n imply ha [ wi () w p () z (r) i () ] v rp (), 0, z (r) i () w i (s)f ( s,(1+ c)v 0p (s),...,(1+ c)v n 1p (s) ) ds = o ( v rp () ), 1 i p 1, 0 z (r) i () w i (s)f ( s,(1+ c)v 0p (s),...,(1+ c)v n 1p (s) ) ds = o ( v rp () ), p i n, which is a varian of condiion (33). We noe ha, in he case of Eq. (28), one may ake [43, p. 388] where v rp () = c rp p r 1, (39) c rp = (n p)! n i=r+1 The main resul of his secion is given nex. 1 (n i)!(i r 1)! = (n p)! 2n r 1 (n r 1)!.

31 R.P. Agarwal e al. / On he asympoic behavior of soluions 31 Theorem 14. Suppose ha he nonlineariy f in Eq. (28) saisfies he inequaliy (34). Assume furher ha for c, θ 0 > 0 one has p 1 i=1 z (r) i + () w i (s)f ( s,(θ 0 + c)v 0p (s),...,(θ 0 + c)v n 1p (s) ) ds 0 n i=p z (r) i () w i (s)f ( s,(θ 0 + c)v 0p (s),...,(θ 0 + c)v n 1p (s) ) ds cv rp (), 0 r n 1, where p 1, n is fixed. Then, for every θ R saisfying θ θ 0, Eq. (28) has a soluion y pθ () defined on [ 0, + ) such ha y (r) pθ () = θ z(r) p () + o ( v rp () ), 0 r n 1, as +. Proof. Le ẑ() = θ z p () and ρ r () = v rp (). Thus, for all y B we obain y (r) () ẑ (r) () + cv rp () ( θ + c ) v rp () (θ 0 + c)v rp (), and he proof follows he same lines as before. According o (39) he soluions of Eq. (28) have he asympoic developmens y (r) pθ () = θ p r 1 (p r 1)! + o( p r 1) as + (40) for 0 r p 1 and, respecively, y (r) pθ () = o( p r 1) as + (41) for p r n 1.

32 32 R.P. Agarwal e al. / On he asympoic behavior of soluions 6. An asympoic inegraion heory for he polynomial-like soluions Our ineres here is in he exisence of soluions of Eq. (28) wih he nex asympoic developmen y (n 1) () = a 1 + o(1), y (n 2) () = a 1 + a 2 + o(1),. q y (n q) a i () = (q i)! q i + o(1), (42) as +. In (42) we ake a 0 = 0and1 q n. We inroduce he comparison funcion F (, u, u,..., u (n 1) ) by he formula F (, u, u,..., u (n 1) ) n 1 ( u (i) ) = h() p i n 1 i, (43) where p i (x) are posiive, nondecreasing and coninuous funcions whereas h() is a nonnegaive coninuous funcion saisfying s q 1 h(s)ds<+. To moivae our choice for he comparison funcion (43), we noe ha, in order o esablish global exisence of soluions, T. Kusano and W. Trench were forced o impose quie resricive condiions. As seen in Secion 5, he convergence of a cerain sum of inegrals given in erms of he comparison funcion F was no enough and, in addiion, an upper bound for he sum was required (more resricions occur in he original formulaion [43,44,73]). Using he Bihari-like comparison funcion [1, Secion 2], we are able o formulae he principal hypoheses only in erms of convergence of a cerain inegral. For furher commens, see [59]. We spli he proof of he principal resul of his secion ino hree pars. We shall provide he proof of he firs wo pars before saing he resul on asympoic inegraion. Firs we shall examine he exisence of soluions for he problem (q 1, n 1) u (n) + f (, u, u,..., u (n 1)) = 0,, ] k lim + [u (n 1 k) a i () (k + 1 i)! k+1 i = a k+1, 0 k q 1, u (n q j) ( ) = u n q j, 1 j n q, where 1, a i and u n q j are real numbers saisfying cerain condiions which we inroduce laer. Noe ha he boundary value problem (44) has boh iniial and erminal daa. Le us explain how he inegral operaor associaed wih he problem (44) can be consruced. Alhough he conclusion of he compuaions will be in perfec agreemen wih Secion 4 a glance a he lieraure (44)

33 R.P. Agarwal e al. / On he asympoic behavior of soluions 33 shows ha i is useful o have a hand a direc, ieraion-based, echnique. Clearly, if y() is a soluion of his problem, i should saisfy he ideniies y (n p) () = p k=0 a k (p k)! p k + ( 1) p 1 s p 1 s 1 f[s 0, y]ds 0 ds q 1 def = A p [] + ( 1) p 1 B p [, y],, (45) where p 1, q. The formal inegraion in (45) can be verified by differeniaion wih respec o he variable.when p = 1 we have B 1 [, y] = f[s 0, y]ds 0. Furhermore, for 1 k p 1, i is easy o noice ha A (k) p [] = A p k [], B p (k) [, y] = ( 1) k B p k [, y]. Arriving o p = q means ha all he erminal daa have been used. Furher, we inegrae he ideniy (45) d + 1 imes, aking ino accoun he iniial daa ha have no been used ye, o obain d y (n q 1 d) ( ) b ( s) d () = u n q 1 d+b + A q [s]ds b! b=0 d! + ( 1) q 1 ( s) d B q [s, y]ds, (46) d! where d 0, n q 1. A sraighforward compuaion yields ( s) d d! A q [s]ds = q k=0 q k a k r=0 ( ) d+1+r (d r)! q k r (q k r)!. Since a 0 = 0, he resul of he inegraion is a polynomial, wih he degree less han d + q n 1, which depends on he erminal daa and he iniial value. The expression in he righ-hand side of (46) provides, for d = n q 1, he formula for he inegral operaor associaed wih he boundary value problem (44). The compuaion ha follows help us o provide a more convenien inegral represenaion for his operaor. Firs, we have ha ( s) d d! ( ( s) d B q [s, y]ds = d! ( s) d d! ) ( ) ds f[s 0, y]ds 1 ds q 1 s q 1 s 1 s f[s 0, y]ds 0 ds q 1 ds. s q 1 s 1