SOLUTIONS APPROACHING POLYNOMIALS AT INFINITY TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

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1 Elecronic Journal of Differenial Equaions, Vol. 2005(2005, No. 79, pp ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu (login: fp SOLUIONS APPROACHING POLYNOMIALS A INFINIY O NONLINEAR ORDINARY DIFFERENIAL EQUAIONS CHRISOS G. PHILOS, PANAGIOIS CH. SAMAOS Absrac. his paper concerns he soluions approaching polynomials a o n-h order (n > 1 nonlinear ordinary differenial equaions, in which he nonlinear erm depends on ime and on x, x,..., x (N, where x is he unknown funcion and N is an ineger wih 0 N n 1. For each given ineger m wih max{1, N} m n 1, condiions are given which guaranee ha, for any real polynomial of degree a mos m, here exiss a soluion ha is asympoic a o his polynomial. Sufficien condiions are also presened for every soluion o be asympoic a o a real polynomial of degree a mos n 1. he resuls obained exend hose by he auhors and by Purnaras [25 concerning he paricular case N = Inroducion Since is invenion by Isaac Newon around 1666, he heory of ordinary differenial equaions has occupied a cenral posiion in he developmen of mahemaics. One reason for his is is widespread applicabiliy in he sciences. Anoher is is naural conneciviy wih oher areas of mahemaics. In he heory of ordinary differenial equaions, he sudy of he asympoic behavior of he soluions is of grea imporance, especially in he case of nonlinear equaions. In applicaions of nonlinear ordinary differenial equaions, any informaion abou he asympoic behavior of he soluions is usually exremely valuable. hus, here is every reason for sudying he asympoic heory of nonlinear ordinary differenial equaions. Very recenly, he auhors and Purnaras [25 sudied soluions, which are asympoic a infiniy o real polynomials of degree a mos n 1, for he n-h order (n > 1 nonlinear ordinary differenial equaion x (n ( = f(, x(, > 0, (1.1 where f is a coninuous real-valued funcion on [, R. he work in [25 is essenially moivaed by he recen one by Lipovan [15 concerning he special case of he second order nonlinear ordinary differenial equaion x ( = f(, x(, > 0. ( Mahemaics Subjec Classificaion. 34E05, 34E10, 34D05. Key words and phrases. Nonlinear differenial equaion; asympoic properies; asympoic expansions; asympoic o polynomials soluions. c 2005 exas Sae Universiy - San Marcos. Submied February 13, Published July 11,

2 2 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 he applicaion of he main resuls in [25 o he second order nonlinear ordinary differenial equaion (1.2 leads o improved versions of he ones given in [15 (and of oher previous relaed resuls in he lieraure. Some closely relaed resuls for second order nonlinear differenial equaions involving he derivaive of he unknown funcion have been given by Rogovchenko and Rogovchenko [29 (see, also, Musafa and Rogovchenko [17. I is he purpose of he presen aricle o exend he resuls in [25 o he more general case of he n-h order (n > 1 nonlinear ordinary differenial equaion x (n ( = f(, x(, x (,..., x (N (, > 0, (1.3 where N is an ineger wih 0 N n 1, and f is a coninuous real-valued funcion on [, R N+1. Noe ha our houghs o exend he resuls in [25 for he differenial equaion (1.3, in some fuure ime, had been made known in his paper. hroughou he paper, we are ineresed in soluions of he differenial equaion (1.3 which are defined for all large, i.e., in soluions of (1.3 on an inerval [,,, where may depend on he soluion. For quesions abou he global exisence in he fuure of he soluions of (1.3, we refer o sandard classical heorems in he lieraure (see, for example, Corduneanu [6, Cronin [7, and Lakshmikanham and Leela [14. he paper is organized as follows. In Secion 2, for each given ineger m wih max{1, N} m n 1, sufficien condiions are presened in order ha, for any real polynomial of degree a mos m, he differenial equaion (1.3 has a soluion defined for all large, which is asympoic a o his polynomial and such ha he firs n 1 derivaives of he soluion are asympoic a o he corresponding firs n 1 derivaives of he given polynomial. Secion 3 is devoed o esablishing condiions, which are sufficien for every soluion defined for all large of he differenial equaion (1.3 o be asympoic a o a real polynomial of degree a mos n 1 (depending on he soluion and he firs n 1 derivaives of he soluion o be asympoic a o he corresponding firs n 1 derivaives of his polynomial. Moreover, in Secion 3, condiions are also given, which guaranee ha every soluion x defined for all large of (1.3 saisfies [x (j (/ n 1 j [c/(n 1 j! for (j = 0, 1,..., n 1, where c is some real number (depending on he soluion x. Secion 4 conains he applicaion of he resuls o he special case of second order nonlinear ordinary differenial equaions. For n = 2 and N = 0, (1.3 becomes (1.2. Moreover, in he special case where n = 2 and N = 1, (1.3 can be wrien as x ( = f(, x(, x (, > 0, (1.4 where f is a coninuous real-valued funcion on [, R 2. Some general examples are given in he las secion (Secion 5, which demonsrae he applicabiliy of he resuls (and, especially, of he main resul in Secion 2. he asympoic heory of n-h order (n > 1 nonlinear differenial equaions has a very long hisory. A cenral role in his heory plays he problem of he sudy of soluions which have a prescribed asympoic behavior via soluions of he equaion x (n = 0. In he special case of second order nonlinear differenial equaions, a large number of papers have appeared concerning his problem; see, for example, Cohen [3, Consanin [4, Dannan [8, Hallam [9, Kamo and Usami [10, Kusano, Naio and Usami [11, Lipovan [15, Musafa and Rogovchenko [17,

3 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 3 Naio [18, 19, 20, Philos and Purnaras [24, Rogovchenko and Rogovchenko [29, 30, Rogovchenko [31, Rogovchenko and Villari [32, Souple [34, ong [36, Walman [38, Yin [39, and Zhao [40. For higher order differenial equaions (ordinary or, more generally, funcional, he above menioned problem has also been invesigaed by several researchers; see, for example, Kusano and rench [12, 13, Meng [16, Philos [21, 22, 23, Philos, Sficas and Saikos [26, Philos and Saikos [27, and he references cied in hese papers. We also menion he paper by rench [37 concerning linear second order ordinary differenial equaions as well as he paper by Philos and samaos [28 abou he problem of he asympoic equilibrium for nonlinear differenial sysems wih reardaions. Before closing his secion, we noe ha i is especially ineresing o examine he possibiliy of generalizing he resuls of he presen paper in he case of he n-h order (n > 1 nonlinear delay differenial equaion x (n ( = f(, x( τ 0 (, x ( τ 1 (,..., x (N ( τ N (, > 0, where τ k (k = 0, 1,..., N are nonnegaive coninuous real-valued funcions on [, such ha lim [ τ k ( = (k = 0, 1,..., N. 2. Condiions for he Exisence of Soluions ha are Asympoic o Polynomials a Infiniy Our resuls in his secion are he heorem below and is corollary. heorem 2.1. Le m be an ineger wih max{1, N} m n 1, and assume ha f(, z 0, z 1,..., z N N k=0 p k (g k ( z k m k + q( for all (, z 0, z 1,..., z N [, R N+1, (2.1 where p k (k = 0, 1,..., N and q are nonnegaive coninuous real-valued funcions on [, such ha n 1 p k (d < (k = 0, 1,..., N, and n 1 q(d <, (2.2 and g k (k = 0, 1,..., N are nonnegaive coninuous real-valued funcions on [0, which are no idenically zero. Le c 0, c 1,..., c m be real numbers and be a poin wih, and suppose ha here exiss a posiive consan K so ha where max k=0,1,...,n + { N [ (s n 1 k (n 1 k! q(sds } K, (s n 1 k (n 1 k! p l(sds Θ l (c l, c l+1,..., c m ; ; K Θ 0 (c 0, c 1,..., c m ; ; K = sup { g 0 (z : 0 z K m + m and, provided ha N > 0, Θ l (c l, c l+1,..., c m ; ; K c i } m i (2.3 (2.4

4 4 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 = sup { g l (z : 0 z K m l + m i=l i(i 1...(i l + 1 c i m i } (l = 1,..., N. (2.5 hen he differenial equaion (1.3 has a soluion x on he inerval [,, which is asympoic o he polynomial c 0 + c c m m as ; i.e., x( = c 0 + c c m m + o(1 as, (2.6 and saisfies m x (j ( = i(i 1... (i j + 1c i i j + o(1 as (j = 1,..., m (2.7 i=j and, provided ha m < n 1, x (λ ( = o(1 as (λ = m + 1,..., n 1. (2.8 Corollary 2.2. Le m be an ineger wih max{1, N} m n 1, and assume ha (2.1 is saisfied, where p k (k = 0, 1,..., N and q, and g k (k = 0, 1,..., N are as in heorem 2.1. hen, for any real numbers c 0, c 1,..., c m, he differenial equaion (1.3 has a soluion x on an inerval [, (where max{, 1} depends on c 0, c 1,..., c m, which is asympoic o he polynomial c 0 + c c m m as ; i.e., (2.6 holds, and saisfies (2.7 and (2.8 (provided ha m < n 1. he mehod which will be applied in he proof of heorem 2.1 is based on he use of he well-known Schauder fixed poin heorem (Schauder [33. his heorem can be found in several books on funcional analysis (see, for example, Conway [5. heorem 2.3 (Schauder heorem. Le E be a Banach space and X be any nonempy convex and closed subse of E. If S is a coninuous mapping of X ino iself and SX is relaively compac, hen he mapping S has a leas one fixed poin (i.e., here exiss an x X wih x = Sx. We need o consider he Banach space BC([,, R of all bounded coninuous real-valued funcions on he given inerval [,, endowed wih he sup-norm : h = sup h( for h BC([,, R. In he proof of heorem 2.1, we will use he se (BC N ([,, R defined as follows: (BC 0 ([,, R coincides wih BC([,, R; for N > 0, (BC N ([,, R is he se of all bounded coninuous real-valued funcions on he inerval [,, which have bounded coninuous k-order derivaives on [, for each k = 1,..., N. Clearly, (BC N ([,, R is a Banach space endowed wih he norm N defined by h N = max k=0,1,...,n h(k for h (BC N ([,, R. o presen a compacness crierion for subses of he space (BC N ([,, R, we firs give some well-known definiions of noions referred o ses of real-valued funcions. Le U be a se of real-valued funcions defined on he inerval [,. he se U is called uniformly bounded if here exiss a posiive consan M such ha, for all funcions u in U, u( M for every.

5 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 5 Also, U is said o be equiconinuous if, for each ɛ > 0, here exiss a δ δ(ɛ > 0 such ha, for all funcions u in U, u( 1 u( 2 < ɛ for every 1, 2 wih 1 2 < δ. Moreover, U will be called equiconvergen a if all funcions in U are convergen in R a he poin and, for each ɛ > 0, here exiss a ɛ such ha, for all funcions u in U, u( lim s u(s < ɛ for every ɛ. We have he following compacness crierion for subses of (BC N ([,, R. Lemma 2.4 (Compacness crierion. Le H be a subse of he Banach space (BC N ([,, R endowed wih he norm N. Define H (0 = H and, provided ha N > 0, H (k = {h (k : h H} for k = 1,..., N. If H (k (k = 0, 1,..., N are uniformly bounded, equiconinuous and equiconvergen a, hen H is relaively compac. In he special case N = 0, i.e., in he case of he Banach space BC([,, R, he above compacness crierion is well-known (see Avramescu [1, Saikos [35. he mehod used in he proof of our compacness crierion is a generalizaion of he one applied in proving his crierion in he special case of he Banach space BC([,, R. Proof of Lemma 2.4. Firs, we noice ha he ses H (k (k = 0, 1,..., N are uniformly bounded if and only if he se H is uniformly bounded in (BC N ([,, R in he sense ha here exiss a posiive consan M such ha, for all funcions h in H, h (k ( M for every (k = 0, 1,..., N. Also, we observe ha H (k (k = 0, 1,..., N are equiconinuous if and only if H is equiconinuous in (BC N ([,, R, ha is, for each ɛ > 0, here exiss a δ δ(ɛ > 0 such ha, for all funcions h in H, h (k ( 1 h (k ( 2 < ɛ for every 1, 2 wih 1 2 < δ (k = 0, 1,..., N. Moreover, H (k (k = 0, 1,..., N are equiconvergen a if and only if H is equiconvergen a in (BC N ([,, R in he sense ha all funcions in H are convergen in R a he poin and, provided ha N > 0, he firs N derivaives of every funcion in H end o zero a, and, for each ɛ > 0, here exiss a ɛ such ha, for all funcions h in H, and, provided ha N > 0, h( lim s h(s < ɛ for every ɛ h (k ( < ɛ for every ɛ (k = 1,..., N. Le (BC N l ([,, R be he subspace of (BCN ([,, R consising of all funcions h in (BC N ([,, R such ha lim h( exiss in R and, provided ha N > 0, lim h(k ( = 0 (k = 1,..., N. Noe ha (BC N l ([,, R is a closed subspace of (BCN ([,, R.

6 6 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 Consider he Banach space C([0, 1, R of all coninuous real-valued funcions on he inerval [0, 1, endowed wih he sup-norm 0 : h 0 = sup h( 01 for h C([0, 1, R. Consider, also, he se C N ([0, 1, R defined as follows: C 0 ([0, 1, R coincides wih C([0, 1, R; for N > 0, C N ([0, 1, R is he se of all N-imes coninuously differeniable real-valued funcions on he inerval [0, 1. Clearly, C N ([0, 1, R is a Banach space endowed wih he norm 0 N defined by h 0 N = max k=0,1,...,n h(k 0 for h C N ([0, 1, R. By he Arzelà-Ascoli heorem, a subse of he Banach space C N ([0, 1, R is relaively compac if and only if i is uniformly bounded and equiconinuous. Noe ha a subse H 0 of C N ([0, 1, R is called uniformly bounded if here exiss a posiive consan M such ha, for all funcions h 0 in H 0, h (k 0 ( M for every [0, 1 (k = 0, 1,..., N. Also, a subse H 0 of C N ([0, 1, R is said o be equiconinuous if, for each ɛ > 0, here exiss a δ δ(ɛ > 0 such ha, for all funcions h 0 in H 0, h (k 0 ( 1 h (k 0 ( 2 < ɛ for every 1, 2 [0, 1 wih 1 2 < δ (k = 0, 1,..., N. Nex, we consider he funcion Φ : (BC N l ([,, R CN ([0, 1, R defined by he formula (Φx( = { x( + 1, if 0 < 1 lim s x(s, if = 1. I is no difficul o check ha Φ is a homeomorphism beween he Banach spaces (BC N l ([,, R and CN ([0, 1, R. So, i follows ha a subse of he space (BC N l ([,, R is relaively compac if and only if i is uniformly bounded, equiconinuous and equiconvergen a. Now, assume ha he ses H (k (k = 0, 1,..., N are uniformly bounded, equiconinuous and equiconvergen a. hen H is uniformly bounded, equiconinuous and equiconvergen a, in (BC N ([,, R. hus, H is a relaively compac subse of (BC N l ([,, R. Since (BCN l ([,, R is a closed subspace of (BC N ([,, R, we can be led o he conclusion ha H is also relaively compac in (BC N ([,, R. he proof of he lemma is complee. Proof of heorem 2.1. Se We have P (j m ( = P m ( = c 0 + c c m m m c i i for R. m i(i 1...(i j + 1c i i j for R (j = 1,..., m i=j and, provided ha m < n 1, P (λ m ( = 0 for R (λ = m + 1,..., n 1.

7 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 7 Furhermore, we see ha he subsiuion y = x P m ransforms he differenial equaion (1.3 ino he equaion y (n ( = f(, y( + P m (, y ( + P m(,..., y (N ( + P (N m (. (2.9 We observe ha y( = x( (c 0 + c c m m, m y (j ( = x (j ( i(i 1... (i j + 1c i i j (j = 1,..., m, i=j and, provided ha m < n 1, y (λ ( = x (λ ( (λ = m + 1,..., n 1. So, by aking ino accoun (2.6, (2.7 and (2.8, we can be led o he conclusion ha wha we have o prove is ha he differenial equaion (2.9 has a soluion y on he inerval [,, which saisfies lim y(ρ ( = 0 (ρ = 0, 1,..., n 1. (2.10 Le E denoe he Banach space (BC N ([,, R endowed wih he norm N, and le us define Y = {y E : y N K}. I is clear ha Y is a nonempy convex and closed subse of E. Consider, now, an arbirary funcion y in Y. hen y( K for every and, provided ha N > 0, hus, for every, we obain y (l ( K for every (l = 1,..., N. y( + P m ( m and, provided ha N > 0, y( m y (l ( + P m (l ( m l y(l ( m l + Hence, we have m i=l K m m l + i=l + m c i m i K m m + c i m i i(i 1... (i l + 1 c i m i i(i 1... (i l + 1 c i m i (l = 1,..., N. ( y( + P m ( g 0 Θ0 m (c 0, c 1,..., c m ; ; K for, where Θ 0 (c 0, c 1,..., c m ; ; K is defined by (2.4; moreover, provided ha N > 0, we have ( y (l ( + P m (l ( g l Θl m l (c l, c l+1,..., c m ; ; K for (l = 1,..., N, where Θ l (c l, c l+1,..., c m ; ; K are defined by (2.5. Bu, from (2.1 i follows ha ( f, y( + Pm (, y ( + P m(,..., y (N ( + P m (N p 0 (g 0 ( y( + P m ( m + p1 (g 1 ( y ( + P m( m 1 (

8 8 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 for all. So, we have ( y (N ( + P m (N + + p N (g N m N ( + q( f(, y( + P m (, y ( + P m(,..., y (N ( + P m (N ( N p l (Θ l (c l, c l+1,..., c m ; ; K for every. (2.11 his inequaliy, ogeher wih (2.2, guaranee ha (s n 1 (n 1! f(s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (sds exiss in R. More generally, for each ρ {0, 1,..., n 1}, (s n 1 ρ (n 1 ρ! f(s, y(s + P m(s, y (s + P m(s,..., y (N (s + P m (N (sds exiss in R. Nex, we use (2.11 o obain, for any k {0, 1,..., N} and for every, (s n 1 k ( (n 1 k! f s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (s ds (s n 1 k ( f s, y(s + P m (s, y (s + P (n 1 k! m(s,..., y (N (s + P m (s (N ds (s n 1 k ( f s, y(s + P m (s, y (s + P (n 1 k! m(s,..., y (N (s + P m (s (N ds N [ (s n 1 k (n 1 k! p l(sds Θ l (c l, c l+1,..., c m ; ; K + (s n 1 k (n 1 k! q(sds. Hence, by using (2.3, we have (s n 1 k (n 1 k! f( s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (s ds K for all (k = 0, 1,..., N. (2.12 We have hus proved ha every funcion y in Y is such ha (2.12 holds. So, i is no difficul o check ha he formula (Sy( = ( 1 n (s n 1 (n 1! f ( s, y(s + P m (s, y (s + P m(s,..., y (N (s + P (N m (s ds for defines a mapping S of Y ino iself. Our purpose is o apply he Schauder heorem for his mapping. We shall prove ha S saisfies he assumpions of he Schauder heorem. We will show ha SY is relaively compac. Define (SY (0 = SY and, provided ha N > 0, (SY (k = {(Sy (k : y Y } for k = 1,..., N. By he given compacness crierion, in order o show ha SY is relaively compac, i suffices o esablish

9 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 9 ha each one of he ses (SY (k ( k = 0, 1,..., N is uniformly bounded, equiconinuous, and equiconvergen a. Le k be an arbirary ineger in {0, 1,..., N}. Since SY Y, we obviously have (Sy (k K for all y Y, and consequenly (SY (k is uniformly bounded. Moreover, in view of (2.11, we obain, for any funcion y Y and every, (Sy (k ( 0 = ( f (s n 1 k (n 1 k! s, y(s + P m (s, y (s + P m(s,..., y (N (s + P (N m (s ds (s n 1 k (n 1 k! ( f s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (s (N ds N [ + (s n 1 k (n 1 k! p l(sds Θ l (c l, c l+1,..., c m ; ; K (s n 1 k (n 1 k! q(sds. hus, by (2.2, i follows easily ha (SY (k is equiconvergen a. Furhermore, by using again (2.11, for any y Y and for every 1, 2 wih 1 < 2, we have: (Sy (n 1 ( 2 (Sy (n 1 ( 1 ( = f = n f ( f s, y(s + P m (s, y (s + P m(s,..., y (n 1 (s + P (n 1 m s, y(s + P m (s, y (s + P m(s,..., y (n 1 (s + P (n 1 m ( f [ 2 s, y(s + P m (s, y (s + P m(s,..., y (n 1 (s + P (n 1 m (s ds (s ds (s ( s, y(s + P m (s, y (s + P m(s,..., y (n 1 (s + P (n 1 m (s ds 1 2 p l (sds Θ l (c l, c l+1,..., c m ; ; K + q(sds, 1 if k = n 1 (and so N = n 1; and (Sy (k ( 2 (Sy (k ( 1 (s 2 n 1 k ( = 2 (n 1 k! f s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (s 1 n 1 k ( 1 (n 1 k! f s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N [ (s r n 2 k = 2 r (n 2 k! f(s, y(s + P m(s, y (s + P m(s,..., y (N (s + P m (N (sds dr ds (s ds (s ds

10 10 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 [ (s r n 2 k 1 r (n 2 k! f(s, y(s + P m(s, y (s + P m(s,..., y (N (s + P m (N (sds dr 2 [ (s r n 2 k = 1 r (n 2 k! f( s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (s ds dr 2 [ (s r n 2 k f(s, y(s + P m (s, y (s + P 1 r (n 2 k! m(s,..., y (N (s + P m (N (s ds dr N { 2 [ (s r n 2 k } (n 2 k! p l(sds dr Θ l (c l, c l+1,..., c m ; ; K [ r r (s r n 2 k (n 2 k! q(sds dr, if k < n 1. Hence, i is no difficul o verify ha he se (SY (k is equiconinuous. We have hus proved ha SY is relaively compac. I remains o prove ha he mapping S is coninuous. o his end, le us consider a y Y and an arbirary sequence (y ν ν 1 in Y wih N lim ν y ν = y. hen we obviously have lim ν y ν = y and, provided ha N > 0, lim ν y(k ν = y (k (k = 1,..., N. On he oher hand, by (2.11, we have, for all ν 1, ( f, y ν ( + P m (, y ν( + P m(,..., y ν (N ( + P m ( (N N p l (Θ l (c l, c l+1,..., c m ; ; K + q( for every. So, because of (2.2, one can apply he Lebesgue dominaed convergence heorem o obain, for every, lim ν = (s n 1 (n 1! (s n 1 (n 1! f(s, y ν (s + P m (s, y ν(s + P m(s,..., y (N ν (s + P m (N (sds f(s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (sds. his ensures he poinwise convergence lim ν (Sy ν ( = (Sy( for. Nex, we esablish ha N lim ν Sy ν = Sy. (2.13 For his purpose, we consider an arbirary subsequence (Sy µν ν 1 of (Sy ν ν 1. Since SY is relaively compac, here exiss a subsequence (Sy µλν ν 1 of (Sy µν ν 1 and a u E so ha N lim ν Sy µ λν = u.

11 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 11 Since he N convergence implies he uniform convergence and, in paricular, he poinwise one o he same limi funcion, we mus have u = Sy. his means ha (2.13 holds rue. We have hus proved ha he mapping S is coninuous. Now, by applying he Schauder heorem, we conclude ha here exiss a y Y wih y = Sy. ha is, y( = ( 1 n ( f his yields (s n 1 (n 1! s, y(s + P m (s, y (s + P m(s,..., y (N (s + P (N m (s ds for. y (n ( = f(, y( + P m (, y ( + P m(,..., y (N ( + P (N m ( for and so y is a soluion on [, of he differenial equaion (2.9. Furhermore, for each ρ = 0, 1,..., n 1, we have ( 1 n ρ y (ρ ( (s n 1 ρ ( = (n 1 ρ! f s, y(s + P m (s, y (s + P m(s,..., y (N (s + P m (N (s ds for all. hus, i follows ha he soluion y saisfies (2.10. he proof of he heorem is now complee. Proof of Corollary 2.2. Le c 0, c 1,..., c m be given real numbers. By aking ino accoun he hypohesis ha g k (k = 0, 1,..., N are no idenically zero on [0,, we can consider a posiive consan K so ha Θ 0 0 sup { m g 0 (z : 0 z K + c i } > 0 and, provided ha N > 0, Θ 0 l sup { g l (z : 0 z K + m i(i 1... (i l + 1 c i } > 0 Furhermore, by (2.2, we can choose a poin max{, 1} such ha (s n 1 k (n 1 k! p K l(sds 2(N + 1Θ 0 (k, l = 0, 1,..., N l and (s n 1 k (n 1 k! q(sds K (k = 0, 1,..., N. 2 Since 1, we have K m m + c i m m i K + c i and, provided ha N > 0, m K m l + i=l for l = 1,..., N. Consequenly, i(i 1... (i l + 1 c i m i K + Θ l (c l, c l+1,..., c m ; ; K Θ 0 l (l = 1,..., N. m i(i 1... (i l + 1 c i i=l (l = 0, 1,..., N,

12 12 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 where Θ 0 (c 0, c 1,..., c m ; ; K is defined by (2.4 and, in he case where N > 0, Θ l (c l, c l+1,..., c m ; ; K (l = 1,..., N are defined by (2.5. Now, we obain max k=0,1,...,n + = N N { N [ (s n 1 k (n 1 k! q(sds } (s n 1 k (n 1 k! p l(sds Θ l (c l, c l+1,..., c m ; ; K K 2(N + 1Θ 0 l Θ l (c l, c l+1,..., c m ; ; K + K 2 K 2(N + 1 Θl(c l, c l+1,..., c m ; ; K Θ 0 + K l 2 K 2(N + 1 (N K 2 = K, which implies (2.3. Hence, he corollary follows from heorem Sufficien Condiions for all Soluions o be Asympoic o Polynomials a Infiniy Our resuls in his secion are formulaed as a proposiion and a heorem. Our proposiion is ineresing of is own as a new resul. Moreover, his proposiion will be used, in a basic way, in proving heorem 3.2. Proposiion 3.1. Assume ha f(, z 0, z 1,..., z N N k=0 p k (g k ( z k n 1 k + q( for all (, z 0, z 1,..., z N [, R N+1, (3.1 where p k (k = 0, 1,..., N and q are nonnegaive coninuous real-valued funcions on [, such ha p k (d < (k = 0, 1,..., N, and q(d < ; (3.2 and g k (k = 0, 1,..., N are coninuous real-valued funcions on [0,, which are posiive and increasing on (0, and such ha 1 dz N k=0 g =. (3.3 k(z hen every soluion x on an inerval [,,, of he differenial equaion (1.3 saisfies x (j ( = c (n 1 j! n 1 j + o( n 1 j as (j = 0, 1,..., n 1, (3.4 where c is some real number (depending on he soluion x.

13 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 13 heorem 3.2. Assume ha (3.1 is saisfied, where p k (k = 0, 1,..., N and q are as in heorem 2.1, i.e., nonnegaive coninuous real-valued funcions on [, such ha n 1 p k (d < (k = 0, 1,..., N, and n 1 q(d <, (3.5 and g k (k = 0, 1,..., N are as in Proposiion 3.1. hen every soluion x on an inerval [,,, of he differenial equaion (1.3 is asympoic o a polynomial c 0 + c c n 1 n 1 as ; i.e., and saisfies x( = c 0 + c c n 1 n 1 + o(1 as, (3.6 n 1 x (j ( = i(i 1... (i j + 1c i i j + o(1 as (j = 1,..., n 1, i=j (3.7 where c 0, c 1,..., c n 1 are real numbers (depending on he soluion x. More precisely, every soluion x on an inerval [,,, of (1.3 saisfies and x( = C 0 + C 1 ( + + C n 1 ( n 1 + o(1 as (3.8 n 1 x (j ( = i(i 1... (i j+1c i ( i j +o(1 as (j = 1,..., n 1, where C i = 1 i! i=j [x (i ( + ( 1 n 1 i (3.9 (s n 1 i (n 1 i! f(s, x(s, x (s,..., x (N (sds (i = 0, 1,..., n 1. (3.10 Combining Corollary 2.2 and heorem 3.2, we obain he following resul. Assume ha (3.1 is saisfied, where p k (k = 0, 1,..., N and q are nonnegaive coninuous real-valued funcions on [, such ha (3.5 holds, and g k (k = 0, 1,..., N are nonnegaive coninuous real-valued funcions on [0, which are no idenically zero. hen, for any real polynomial of degree a mos n 1, he differenial equaion (1.3 has a soluion defined for all large, which is asympoic a o his polynomial and such ha he firs n 1 derivaives of he soluion are asympoic a o he corresponding firs n 1 derivaives of he given polynomial. Moreover, if, in addiion, g k (k = 0, 1,..., N are posiive and increasing on (0, and such ha (3.3 holds, hen every soluion defined for all large of he differenial equaion (1.3 is asympoic a o a real polynomial of degree a mos n 1 (depending on he soluionand he firs n 1 derivaives of he soluion are asympoic a o he corresponding firs n 1 derivaives of his polynomial. he following lemma plays an imporan role in proving our proposiion. his lemma is he well-known Bihari s lemma (see Bihari [2; see, also, Corduneanu [6 in a simple form which suffices for our needs.

14 14 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 Lemma 3.3 (Bihari. Assume ha h( M + µ(sg(h(sds for 0, 0 where M is a posiive consan, h and µ are nonnegaive coninuous real-valued funcions on [ 0,, and g is a coninuous real-valued funcion on [0,, which is posiive and increasing on (0, and such ha hen 1 dz g(z =. h( G 1( G(M + µ(sds 0 for 0, where G is a primiive of 1/g on (0, and G 1 is he inverse funcion of G. Proof of Proposiion 3.1. Consider an arbirary soluion x on an inerval [,,, of he differenial equaion (1.3. From (1.3 i follows ha x (k ( = n 1 i=k ( i k x (i ( s n 1 k ( + (i k! (n 1 k! f( s, x(s, x (s,..., x (N (s ds (k = 0, 1,..., N for. herefore, in view of (3.1, for any k {0, 1,..., N} and every, we obain x (k ( n 1 i=k n 1 i=k n 1 i=k [ n 1 i=k ( i k x (i ( + (i k! i k (i k! x(i ( + n 1 k i k (i k! x(i ( + n 1 k i k (i k! x(i ( + n 1 k ( s n 1 k (n 1 k! f(s, x(s, x (s,..., x (N (s ds f(s, x(s, x (s,..., x (N (s ds [ N q(sds [ N ( x + n 1 k (l (s p l (sg l s n 1 l ds. hus, for any k {0, 1,..., N}, we have x (k ( n 1 k [ n 1 i=k 1 (i k! n 1 i x(i ( + ( x (l (s p l (sg l s n 1 l + q(s ds q(sds + [ N p l (sg l ( x (l (s s n 1 l ds

15 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 15 for every. So, by aking ino accoun (3.2, we immediaely conclude ha, for each k {0, 1,..., N}, here exiss a posiive consan M k such ha x (k ( n 1 k M k + [ N Hence, by seing M = max k=0,1,...,n M k x (k ( n 1 k M + ha is, where [ N p l (sg l ( x (l (s s n 1 l ds for. (M is a posiive consan, we obain p l (sg l ( x (l (s s n 1 l ds for (k = 0, 1,..., N. x (k ( h( for every (k = 0, 1,..., N, (3.11 n 1 k h( = M + [ N p l (sg l ( x (l (s s n 1 l ds for. Furhermore, by using (3.11 and he hypohesis ha g k (k = 0, 1,..., N are increasing on (0,, we obain for every Consequenly, h( M + M + M + h( M + [ N [ N p l (sg l ( x (l (s s n 1 l ds p l (sg l (h(s ds [ N [ N p l (s g l (h(s ds. [ N p l (s g(h(sds for, (3.12 where g = N g l. Clearly, g is a coninuous real-valued funcion on [0,, which is posiive and increasing on (0,. Moreover, because of (3.3, g is such ha Nex, we consider he funcion G(z = 1 z M dz =. (3.13 g(z du g(u for z M. We observe ha G is a primiive of he funcion 1/g on [M,. I is obvious ha G(M = 0 and ha G is sricly increasing on [M,. Also, by (3.13, we have G( =. So, i follows ha G([M, = [0,. hus, he inverse funcion G 1 of G is defined on [0,. Moreover, G 1 is also sricly increasing on [0,, and G 1 ([0, = [M,. Furhermore, we can ake ino accoun (3.12 and use he Bihari lemma o conclude ha h saisfies h( G 1( [ N N G(M + p l (s ds = G 1( p l (sds

16 16 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 for. herefore, in view of (3.2, i follows ha h( G 1( N p l (sds for every ; i.e., here exiss a posiive consan Λ such ha h( Λ for. So, (3.11 yields x (k ( Λ for all (k = 0, 1,..., N. (3.14 n 1 k Now, by aking ino accoun (3.1 and (3.14, we obain for f (, x(, x (,..., x (N ( and consequenly, in view of (3.2, N k=0 N k=0 ( x (k ( p k (g k n 1 k + q( p k ( [ sup g k (z + q( 0zΛ f ( s, x(s, x (s,..., x (N (s ds exiss in R. On he oher hand, from (1.3 i follows ha which gives x (n 1 ( = x (n 1 ( + lim x(n 1 ( = x (n 1 ( + f(s, x(s, x (s,..., x (N (sds for, f ( s, x(s, x (s,..., x (N (s ds c, where c is a real number (depending on he soluion x. Finally, by applying he L Hospial rule, we obain x (j ( lim n 1 j = 1 (n 1 j! lim c x(n 1 ( = (n 1 j! (j = 0, 1,..., n 1, which implies ha x saisfies (3.4. he proof of he proposiion is complee. Proof of heorem 3.2. Le x be an arbirary soluion on an inerval [,,, of he differenial equaion (1.3. Since (3.5 implies (3.2, as in he proof of Proposiion 3.1, we can be led o he conclusion ha (3.14 holds, where Λ is some posiive consan. his conclusion is also a consequence of Proposiion 3.1 iself; in fac, from his proposiion i follows ha, for each k = 0, 1,..., N, lim [x (k (/ n 1 k exiss (as a real number. By using (3.1 and (3.14, we obain f (, x(, x (,..., x (N ( N ( x (k ( p k (g k n 1 k + q( k=0 N k=0 p k ( [ sup g k (z + q( 0zΛ

17 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 17 for every. his, ogeher wih (3.5, guaranee ha L i (s n 1 i (n 1 i! f( s, x(s, x (s,..., x (N (s ds (i = 0, 1,..., n 1 are real numbers. Now, (1.3 gives, for, x( = n 1 ( i x (i ( + i! ( s n 1 f ( s, x(s, x (s,..., x (N (s ds. (3.15 (n 1! Following he same procedure as in he proof of he corresponding heorem in Philos, Purnaras and samaos [25, we can show ha n 1 ( s n 1 f ( s, x(s, x (s,..., x (N (s ds (n 1! ( i = ( 1 n 1 i L i + ( 1 n i! for all. So, (3.15 becomes x( = n 1 ( i [ x (i ( + ( 1 n 1 i L i i! + ( 1 n (s n 1 f ( s, x(s, x (s,..., x (N (s ds (n 1! (s n 1 f ( s, x(s, x (s,..., x (N (s ds for. (n 1! aking ino accoun he definiion of L i (i = 0, 1,..., n 1 as well as (3.10, we see ha he above equaion can be wrien as n 1 x( = C i ( i +( 1 n for all. We have lim (s n 1 f ( s, x(s, x (s,..., x (N (s ds (3.16 (n 1! (s n 1 f ( s, x(s, x (s,..., x (N (s ds = 0 (n 1! and hus (3.16 implies ha he soluion x saisfies (3.8. Furhermore, (3.16 gives Since n 1 x (j ( = i(i 1... (i j + 1C i ( i j lim i=j + ( 1 n j (s n 1 j (n 1 j! f(s, x(s, x (s,..., x (N (sds for (j = 1,..., n 1. (3.17 (s n 1 j (n 1 j! f(s, x(s, x (s,..., x (N (sds = 0 (j = 1,..., n 1, i follows from (3.17 ha he soluion x saisfies, in addiion, (3.9. Finally, we observe ha C 0 + C 1 ( + + C n 1 ( n 1 c 0 + c c n 1 n 1

18 18 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 for some real numbers c 0, c 1,..., c n 1. So, he soluion x saisfies (3.6 and (3.7. he proof is complee. 4. Applicaion of he Resuls o Second Order Nonlinear Ordinary Differenial Equaions his secion is devoed o he applicaion of he resuls o he special case of he second order nonlinear ordinary differenial equaions (1.2 and (1.4. In he case of he differenial equaion (1.2, heorem 2.1, Corollary 2.2, Proposiion 3.1, and heorem 3.2 are formulaed as follows: heorem 4.1. Assume ha f(, z p(g ( z + q( for all (, z [0, R, (4.1 where p and q are nonnegaive coninuous real-valued funcions on [, such ha p(d < and q(d < ; and g is a nonnegaive coninuous real-valued funcion on [0, which is no idenically zero. Le c 0, c 1 be real numbers and be a poin wih, and suppose ha here exiss a posiive consan K so ha [ (s p(sds K. sup { g(z : 0 z K + c 0 + c 1 } + (s q(sds hen he differenial equaion (1.2 has a soluion x on he inerval [,, which is asympoic o he line c 0 + c 1 as ; i.e., and saisfies x( = c 0 + c 1 + o(1 as, (4.2 x ( = c 1 + o(1 as. (4.3 Corollary 4.2. Assume ha (4.1 is saisfied, where p, q, and g are as in heorem 4.1. hen, for any real numbers c 0, c 1, he differenial equaion (1.2 has a soluion x on an inerval [, (where max{, 1} depends on c 0, c 1, which is asympoic o he line c 0 + c 1 as ; i.e., (4.2 holds, and saisfies (4.3. Proposiion 4.3. Assume ha (4.1 is saisfied, where p and q are nonnegaive coninuous real-valued funcions on [, such ha p(d < and q(d <, and g is a coninuous real-valued funcion on [0,, which is posiive and increasing on (0, and such ha dz 1 g(z =. hen every soluion x on an inerval [,,, of he differenial equaion (1.2 saisfies x( = c + o( and x ( = c + o(1, as, where c is some real number (depending on he soluion x.

19 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 19 heorem 4.4. Assume ha (4.1 is saisfied, where p and q are as in heorem 4.1, and g is as in Proposiion 4.3. hen every soluion x on an inerval [,,, of he differenial equaion (1.2 is asympoic o a line c 0 + c 1 as ; i.e., (4.2 holds, and saisfies (4.3, where c 0, c 1 are real numbers (depending on he soluion x. More precisely, every soluion x on an inerval [,,, of (1.2 saisfies where x( = C 0 + C 1 ( + o(1 and x ( = C 1 + o(1, as, C 0 = x( (s f(s, x(sds and C 1 = x ( + f(s, x(sds. he above resuls have also been obained in Philos, Purnaras and samaos [25 (as consequences of he main resuls given herein. Here, hese resuls are saed for he sake of compleeness. Now, we concenrae on he differenial equaion (1.4. By applying heorem 2.1, Corollary 2.2, Proposiion 3.1, and heorem 3.2 o he differenial equaion (1.4, we obain following resuls: heorem 4.5. Assume ha ( z 0 f(, z 0, z 1 p 0 (g 0 + p1 (g 1 ( z 1 + q( for all (, z 0, z 1 [, R 2, (4.4 where p 0, p 1, and q are nonnegaive coninuous real-valued funcions on [, such ha p 0 (d <, p 1 (d <, and q(d < ; and g 0 and g 1 are nonnegaive coninuous real-valued funcions on [0, which are no idenically zero. Le c 0, c 1 be real numbers and be a poin wih, and suppose ha here exiss a posiive consan K so ha [ (s p 0 (sds sup { g 0 (z : 0 z K + c 0 + c 1 } [ + (s p 1 (sds sup { g 1 (z : 0 z K + c 1 } + (s q(sds K and [ [ + p 0 (sds sup { g 0 (z : 0 z K + c 0 + c 1 } p 1 (sds sup { g 1 (z : 0 z K + c 1 } + q(sds K. hen he conclusion of heorem 4.1 holds for he differenial equaion (1.4. Corollary 4.6. Assume ha (4.4 is saisfied, where p 0, p 1, and q, and g 0 and g 1 are as in heorem 4.5. hen he conclusion of Corollary 4.2 holds for he differenial equaion (1.4.

20 20 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 Proposiion 4.7. Assume ha (4.4 is saisfied, where p 0, p 1, q are nonnegaive coninuous real-valued funcions on [, such ha p 0 (d <, p 1 (d <, and q(d < ; and g 0 and g 1 are coninuous real-valued funcions on [0,, which are posiive and increasing on (0, and such ha 1 dz g 0 (z + g 1 (z =. hen he conclusion of Proposiion 4.3 holds for he differenial equaion (1.4. heorem 4.8. Assume ha (4.4 is saisfied, where p 0, p 1, q are as in heorem 4.5, and g 0 and g 1 are as in Proposiion 4.7. hen he conclusion of heorem 4.4 holds for he differenial equaion (1.4 wih C 0 = x( (s f(s, x(s, x (sds, C 1 = x ( + 5. Examples f(s, x(s, x (sds. Example 5.1 (Philos, Purnaras, samaos [25. Consider he second order superlinear Emden-Fowler equaion x ( = a([x( 2 sgn x(, > 0, (5.1 where a is a coninuous real-valued funcion on [,. Applying heorem 2.1 (or, in paricular, heorem 4.1, we obain he following resul: Assume ha 3 a( d <. (5.2 Le c 0, c 1 be real numbers and be a poin wih, and suppose ha here exiss a posiive consan K so ha ( K A( + c c 1 K, (5.3 where A( = (s s 2 a(s ds. (5.4 hen (5.1 has a soluion x on he inerval [,, which is asympoic o he line c 0 + c 1 as ; i.e., x( = c 0 + c 1 + o(1 as, (5.5 and saisfies x ( = c 1 + o(1 as. (5.6 Now, assume ha (5.2 is saisfied, and le c 0, c 1 be given real numbers and be a fixed poin. Moreover, le A( be defined by (5.4. As i has been proved in [25, here exiss a posiive consan K so ha (5.3 holds if and only if hus, we have he following resul: A( ( c 0 + c (5.7

21 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 21 Assume ha (5.2 is saisfied, and le c 0, c 1 be real numbers and be a poin so ha (5.7 holds, where A( is defined by (5.4. hen (5.1 has a soluion x on he inerval [,, which saisfies (5.5and (5.6. In paricular, le us consider he differenial equaion (5.1 wih a( = σ µ( for, where σ is a real number and µ is a coninuous and bounded real-valued funcion on [,. In his case, here exiss a posiive consan θ so ha a( θ σ for every. We see ha (5.2 is saisfied if σ < 4. Furhermore, assume ha σ < 4 and le c 0, c 1 be real numbers and be a poin. hen (see [25 i follows ha (5.7 holds if σ+2 (σ + 3(σ + 4 ( c 0 + c 1. 4θ Example 5.2. Consider he n-h order (n > 1 sublinear Emden-Fowler equaion x (n ( = a( x( 1/2 sgn x(, > 0, (5.8 where a is a coninuous real-valued funcion on [,. For he differenial equaion (5.8, we have he following resul: Le m be an ineger wih 1 m n 1, and assume ha n 1+(m/2 a( d <. (5.9 hen, for any real numbers c 0, c 1,..., c m, he differenial equaion (5.8 has a soluion x on he (whole inerval [,, which is asympoic o he polynomial c 0 + c c m m as ; i.e., x( = c 0 + c c m m + o(1 as, and saisfies m x (j ( = i(i 1... (i j + 1c i i j + o(1 as (j = 1,..., m i=j and, provided ha m < n 1, x (λ ( = o(1 as (λ = m + 1,..., n 1. o prove he above resul, we assume ha (5.9 is saisfied and we consider arbirary real numbers c 0, c 1,..., c m. By heorem 2.1, i is sufficien o show ha here exiss a posiive consan K such ha where A( ( K m 0 A( = + m c i 1/2 K, (5.10 m i 0 (s n 1 s m/2 a(s ds. (n 1! In he rivial case A( = 0, (5.10 holds for any posiive consan K. So, in he sequel, we suppose ha A( > 0. We see ha (5.10 is equivalen o K 2 [A( 2 m K [A( 2 0 m c i 0 m i 0. (5.11

22 22 CH. G. PHILOS, P. CH. SAMAOS EJDE-2005/79 Le us consider he quadraic equaion Ω(ω ω 2 [A( 2 ω [A( 2 in he complex plane. he discriminan of his equaion is [ = [A( 2 2 [ m 4 [A( 2 m 0 m 0 m c i m i = 0 0 c i m i 0 We see ha > 0 and so he equaion Ω(ω = 0 has wo real roos: ω 1 = [A( m 0 2, ω 2 = [A( m wih ω 1 < ω 2. Clearly, ω 2 > 0. We have Ω(ω 0 for all ω ω 2. Hence, (5.11 (or, equivalenly, (5.10 is saisfied for every posiive consan K wih K ω 2 > 0. We have hus proved ha, in boh cases where A( = 0 or A( > 0, here exiss a posiive consan K so ha (5.10 holds. So, our resul has been proved. Example 5.3. Consider he second order Emden-Fowler equaion x ( = a( x( γ sgn x( + b( x ( δ sgn x (, > 0, (5.12 where a and b are coninuous real-valued funcions on [,, and γ and δ are posiive real numbers. By applying heorem 2.1 (or, in paricular, heorem 4.5 o he differenial equaion (5.12, we arrive a he nex resul: Assume ha 1+γ a( d < and b( d <. (5.13 Le c 0, c 1 be real numbers and be a poin wih, and suppose ha here exiss a posiive consan K so ha [ ( K (s s γ a(s ds + c 0 γ [ + c 1 + (s b(s ds (K + c 1 δ K and [ ( K s γ a(s ds + c 0 + c 1 γ + [. b(s ds (K + c 1 δ K. hen (5.12 has a soluion x on he inerval [,, which is asympoic o he line c 0 + c 1 as ; i.e., (5.5 holds, and saisfies (5.6. Moreover, an applicaion of Corollary 2.2 (or, in paricular, of Corollary 4.6 o he differenial equaion (5.12 leads o he following resul: Assume ha (5.13 is saisfied. hen, for any real numbers c 0, c 1, (5.12 has a soluion x on an inerval [, (where max{, 1} depends on c 0, c 1, which saisfies (5.5 and (5.6. Also, we can apply Proposiion 3.1 (or, in paricular, Proposiion 4.7 for he differenial equaion (5.12 o obain he resul: If γ a( d < and b( d <, (5.14

23 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 23 and γ 1 and δ 1, hen every soluion x on an inerval [,,, of (5.12 saisfies x( = c + o( and x ( = c + o(1, as, where c is some real number (depending on he soluion x. Furhermore, applying heorem 3.2 (or, in paricular, heorem 4.8 o he differenial equaion (5.12, we obain he following resul: Assume ha (5.13 is saisfied, and ha γ 1 and δ 1. hen every soluion x on an inerval [,,, of (5.12 is asympoic o a line c 0 + c 1 as ; i.e., (5.5 holds, and saisfies (5.6, where c 0, c 1 are real numbers (depending on he soluion x. More precisely, every soluion x on an inerval [,,, of (5.12 saisfies where C 0 = x( x( = C 0 + C 1 ( + o(1 and x ( = C 1 + o(1, as, C 1 = x ( + (s a(s x(s γ sgn x(sds a(s x(s γ sgn x(sds + (s b(s x (s δ sgn x (sds, b(s x (s δ sgn x (sds. Before ending his example, we concenrae on he Emden-Fowler equaion (5.12 wih a( = σ µ( for, and b( = τ ν( for, where σ and τ are real numbers, and µ and ν are coninuous and bounded realvalued funcions on [,. In his case, we have a( θ σ for, and b( = ξ τ for, where θ and ξ are posiive consans. We see ha (5.13 is saisfied if γ + σ < 2 and τ < 2. Moreover, we observe ha (5.14 holds if γ + σ < 1 and τ < 1. References [1 C. Avramescu; Sur l exisence des soluions convergenès de sysèmes d équaions différenielles non linéaires, Ann. Ma. Pura Appl. 81 (1969, [2 I. Bihari; A generalizaion of a lemma of Bellman and is applicaion o uniqueness problems of differenial equaions, Aca Mah. Acad. Sci. Hungar. 7 (1956, [3 D. S. Cohen; he asympoic behavior of a class of nonlinear differenial equaions, Proc. Amer. Mah. Soc. 18 (1967, [4 A. Consanin; On he asympoic behavior of second order nonlinear differenial equaions, Rend. Ma. Appl. 13 (1993, [5 J. B. Conway; A Course in Funcional Analysis, Springer, New York, [6 C. Corduneanu; Principles of Differenial and Inegral Equaions, Chelsea Publishing Company, he Bronx, New York, [7 J. Cronin; Differenial Equaions: Inroducion and Qualiaive heory, Second Ediion, Revised and Expanded, Marcel Dekker, Inc., New York, [8 F. M. Dannan; Inegral inequaliies of Gronwall-Bellman-Bihari ype and asympoic behavior of cerain second order nonlinear differenial equaions, J. Mah. Anal. Appl. 108 (1985, [9. G. Hallam; Asympoic inegraion of second order differenial equaions wih inegrable coefficiens, SIAM J. Appl. Mah. 19 (1970, [10 K. Kamo and H. Usami; Asympoic forms of posiive soluions of second-order quasilinear ordinary differenial equaions wih sub-homogeneiy, Hiroshima Mah. J. 31 (2001, [11. Kusano, M. Naio and H. Usami; Asympoic behavior of soluions of a class of second order nonlinear differenial equaions, Hiroshima Mah. J. 16 (1986,

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25 EJDE-2005/79 SOLUIONS APPROACHING POLYNOMIALS A INFINIY 25 [39 Z. Yin; Monoone posiive soluions of second-order nonlinear differenial equaions, Nonlinear Anal. 54 (2003, [40 Z. Zhao; Posiive soluions of nonlinear second order ordinary differenial equaions, Proc. Amer. Mah. Soc. 121 (1994, Chrisos G. Philos Deparmen of Mahemaics, Universiy of Ioannina, P. O. Box 1186, Ioannina, Greece address: cphilos@cc.uoi.gr Panagiois Ch. samaos Deparmen of Mahemaics, Universiy of Ioannina, P. O. Box 1186, Ioannina, Greece address: psamao@cc.uoi.gr