LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS. 1. Introduction

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1 ARCHIVUM MATHEMATICUM (BRNO) Tomus 43 (2007), LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS Mariella Cecchi, Zuzana Došlá and Mauro Marini Absrac. Some asympoic properies of principal soluions of he halflinear differenial equaion (*) (a()φ(x )) + b()φ(x) = 0, Φ(u) = u p 2 u, p > 1, is he p-laplacian operaor, are considered. I is shown ha principal soluions of (*) are, roughly speaking, he smalles soluions in a neighborhood of infiniy, like in he linear case. Some inegral characerizaions of principal soluions of (1), which complees previous resuls, are presened as well. Consider he half-linear equaion 1. Inroducion (1) ( a()φ(x ) ) + b()φ(x) = 0, where he funcions a, b are coninuous and posiive for 0, and Φ(u) = u p 2 u, p > 1. When (1) is nonoscillaory, he asympoic behavior of is soluions has been considered in many papers, see, e.g., [3, 4, 7, 9, 10, 11, 14, 15], he monographs [1, 8, 17] and references herein. In paricular, when (1) is nonoscillaory, he concep of a principal soluion has been formulaed for (1) in [11, 17], by exending he analogous one saed for he linear equaion (2) ( a()x ) + b()x = 0, 2000 Mahemaics Subjec Classificaion : 34C10, 34C11. Key words and phrases : half-linear equaion, principal soluion, limi characerizaion, inegral characerizaion. Suppored by he Research Projec MSMT of he Minisry of Educaion of he Czech Republic, and by he gran A of he Gran Agency of he Academy of Sciences of he Czech Republic. Received March 30, 2006.

2 76 M. CECCHI, Z. DOŠLÁ AND M. MARINI see, e.g., [13, Chaper 11]. More precisely, a nonrivial soluion u of (1) is called a principal soluion of (1) if for every nonrivial soluion x of (1) such ha x λu, λ R, we have (3) u () u() < x () x() for large. As in he linear case, he principal soluion u exiss and is unique up o a consan facor. Any nonrivial soluion x λu is called nonprincipal soluion. Denoe J a = 0 d Φ ( a() ), J b = 0 b()d, where Φ is he inverse of he map Φ, i.e. Φ (u) = u p 2 u, p = p/(p 1). The quesion concerning limi and inegral characerizaions of principal soluions, like in he linear case, has been posed in [7] and parially solved in [3] under any of he following assumpions (4) i) J a =, p 2, ii) J b =, 1 < p 2, iii) J a + J b <. In his paper we coninue such a sudy, by assuming (5) J a + J b =. We will characerize principal soluions of (1) by means of some limi or inegral properies, which exend our quoed resuls in [3]. The paper is organized as follows. In Secion 2 some preliminary resuls, concerning he classificaion of soluions of (1), are given. In Secion 3 principal soluions of (1) are characerized by showing ha hey are, roughly speaking, he smalles soluions in a neighborhood of infiniy, like in he linear case. Some inegral characerizaions of principal soluions of (1) are presened in Secion 4, compleing in such a way our previous resuls in [3]. Some open problems complee he paper. 2. Preliminaries We sar his secion by recalling some basic resuls, which will be useful in he sequel. I is easy o verify ha he quasi-derivaive y = x [1] of any soluion x of (1), where x [1] () = a()φ ( x () ), is a soluion of he so-called reciprocal equaion ( (6) Φ ( 1 ) ) ( Φ (y ) + Φ 1 ) Φ (y) = 0, b() a() which is obained from (1) by inerchanging he funcion a wih Φ (1/b) and b wih Φ (1/a). Conversely, he quasiderivaive y [1] () = Φ (1/b())Φ (y ()) of any soluion y of (6) is a soluion of (1). Observe ha J a [J b ] for (1) plays he same role as J b [J a ] for (6) and vice versa.

3 HALF-LINEAR DIFFERENTIAL EQUATIONS 77 In view of posiiveness of a and b, (1) and (6) have he same characer wih respec o he oscillaion, i.e. (1) is nonoscillaory if and only if (6) is nonoscillaory. When J a = J b =, hen (1) is oscillaory (see, e.g., [8, Th ]). If eiher J a =, J b < or J a <, J b =, hen boh oscillaion and nonoscillaion can occur (see, e.g., [8, 3.1]). Principal soluions of (1) and (6) are relaed, as he following resul, which can be proved by using he same argumen as in [3, Theorem 1], shows. Proposiion 1. Le (1) be nonoscillaory and assume (5). A soluion u of (1) is a principal soluion if and only if v = u [1] is a principal soluion of (6). When (1) is nonoscillaory, aking ino accoun ha (6) is nonoscillaory oo, we have ha any nonrivial soluion x of (1) belongs o one of he following wo classes: The following holds. M + = {x soluion of (1) : x 0 : x()x () > 0 for > x } M = {x soluion of (1) : x 0 : x()x () < 0 for > x }. Proposiion 2. Le (1) be nonoscillaory and assume (5). Le S be he se of nonrivial soluions of (1). Then J a = S M + ; J b = S M. Moreover, (1) does no have soluions x such ha (7) lim x() = c x, lim x [1] () = d x, 0 < c x <, 0 < d x <. Proof. The firs saemen follows by using a similar argumen as in [3, Lemma 1] (see also [8, Lemmas 4.1.3, 4.1.4]). Now le us prove (7). Assume J a = and le x be a soluion of (1) saisfying (7). Then x M + and, wihou loss of generaliy, suppose x() > 0, x () > 0 for large. From x [1] () = a()φ(x ()) we obain for large (8) x () 1 Φ (a()), where he symbol g 1 () g 2 () means ha g 1 ()/g 2 () has a finie nonzero limi, as. From (8) we obain ha x is unbounded (as ), which is a conradicion. The case J b = can be reaed by using a similar argumen. Noice ha if he assumpion (5) is no verified, hen boh saemens in Proposiion 2 fail, as i follows, for insance, from [12, Theorem 3] and applying his resul o he reciprocal equaion (6).

4 78 M. CECCHI, Z. DOŠLÁ AND M. MARINI In virue of he posiiveness of he funcions a, b, and Proposiion 2, boh classes M +, M can be divided, a-priori, ino he following subclasses: M + l,0 = { x M + : lim x() = c x, lim x [1] () = 0, 0 < c x < }, M +,0 = { x M + : lim x() =, lim x [1] () = 0 }, M +,l = { x M + : lim x() =, lim x [1] () = d x, 0 < d x < }, M 0,l = { x M : lim x() = 0, lim x [1] () = d x, 0 < d x < }, M 0, = { x M : lim x() = 0, lim x [1] () = }, M l, = { x M : lim x() = c x, lim x [1] () =, 0 < c x < }. The exisence of soluions in hese subclasses depends on he convergence or divergence of he following inegrals: T J 1 = lim Φ ( 1 )Φ ( ) b(s) ds d, T a() and 0 T J 2 = lim Φ ( 1 )Φ ( T ) b(s) ds d, T 0 a() T ( T Y 1 = lim b()φ Φ ( 1 ) ) ds d, T 0 a(s) T Y 2 = lim b()φ T 0 ( 0 0 Φ ( 1 ) ) ds d. a(s) Clearly, for he linear equaion (2) we have J 1 = Y 1, J 2 = Y 2. Observe ha he inegral J 1 for (1) plays he same role as Y 2 for (6) and vice versa; analogously J 2 for (1) plays he same role as Y 1 for (6) and vice versa. The following holds. Lemma A. Concerning he muual behavior of J 1, Y 1, he only possible cases are he following: J 1 = Y 1 = for 1 < p J 1 =, Y 1 < for 2 < p J 1 <, Y 1 = for 1 < p < 2 J 1 <, Y 1 < for 1 < p. Analogously for J 2, Y 2, he only possible cases are J 2 = Y 2 = for 1 < p J 2 =, Y 2 < for 2 < p J 2 <, Y 2 = for 1 < p < 2 J 2 <, Y 2 < for 1 < p. Moreover, if J 2 + Y 2 =, hen J a =, and, if J 1 + Y 1 =, hen J b =.

5 HALF-LINEAR DIFFERENTIAL EQUATIONS 79 Proof. The possible cases for J i, Y i (i = 1, 2) follow from [6, Corollary 1 and Examples 1, 2]. The relaions beween J i, Y i and J a, J b follow from [2, Lemma 2]. The following holds. Theorem A. i 1 ) Assume J a =. Then i 2 ) Assume J b =. Then M + l,0 J 2 <, M +,l Y 2 <. M 0,l Y 1 <, M l, J 1 <. Proof. Claim i 1 ) follows, for insance, from [14, Th.s 4.1 and 4.2 ] (see also [12, Secion 4], [16, Th. 4.3], in which a more general equaion is considered). Claim i 2 ) follows by applying i 1 ) o he reciprocal equaion (6). 3. Limi characerizaion When (1) is nonoscillaory, in [7] he quesion, wheher principal soluions are smalles soluions in a neighborhood of infiniy also in he half-linear case, has been posed. This problem has been solved in [3, Theorem 2] under any of assumpions in (4). To exend such a resul, he following uniqueness resul plays an imporan role. Theorem B. Le η 0 be a given consan. i 1 ) Assume J a =, J 2 <. Then here exiss a unique soluion x of (1) such ha x M + and lim x() = η. i 2 ) Assume J b =, Y 1 <. Then here exiss a unique soluion x of (1) such ha x M and lim x [1] () = η. Proof. Claim i 1 ) follows from [14, Theorem 4.3] (see also [8, Theorem 4.1.7]). Claim i 2 ) follows by applying i 1 ) o he reciprocal equaion (6). The following holds. Theorem 1. Le u be a soluion of (1) and assume eiher i 1 ) J a =, J 2 < or i 2 ) J b =, Y 1 <. Then u is a principal soluion if and only if for any nonrivial soluion x of (1) such ha x λu, λ R, we have u() (9) lim x() = 0. Proof. If (9) holds for any nonrivial soluion x of (1) such ha x λu, λ R, hen, by using he same argumen as in [3, Theorem 2], u is a principal soluion of (1). Conversely, suppose ha u is a principal soluion and le us show ha (9) holds for any nonrivial soluion x of (1) such ha x λu, λ R if eiher i 1 ) or i 2 ) holds. Assume case i 1 ). By Theorem A, we have M + l,0 and so (1) is nonoscillaory. Wihou loss of generaliy, suppose u evenually posiive. We claim ha u is

6 80 M. CECCHI, Z. DOŠLÁ AND M. MARINI bounded (as ). Assume ha u is unbounded and consider x M + l,0 such ha x is evenually posiive. From (3), he raio u/x is evenually posiive decreasing, which yields a conradicion because lim l [u()/x()] =. Then u is bounded and so u M + l,0. For any nonrivial soluion x of (1), such ha x λu, in view of Theorem B, we obain ha x is unbounded and so (9) holds. Now assume case i 2 ). Again by Theorem A, we have M 0,l and so (1) is nonoscillaory. Wihou loss of generaliy, suppose u and x evenually posiive. In view of Proposiion 2, we have u [1] () < 0, x [1] () < 0 for large. From (3), we obain for large u [1] () ( u() ) (10) x [1] () > Φ > 0. x() Applying Proposiion 1, u [1] is a principal soluion of (6). Since for (6) he case i 1 ) holds, we obain u [1] () lim x [1] () = 0 and so, from (10), he asserion follows. From Theorem B, Theorem 1 and Theorem 2 in [3], we obain he following. Corollary 1. The se of principal soluions of (1) is eiher M + l,0 or M 0,l according o eiher J a =, J 2 <, or J b =, Y 1 <, respecively. Remark 1. Summarizing Theorem 1 and [3, Theorem 2] (which holds under any of assumpions in (4)), and aking ino accoun Lemma A, we obain ha, if (1) is nonoscillaory, hen he limi characerizaion of principal soluions (9) holds in any case excep he following wo cases (11) J 2 = Y 2 =, 1 < p < 2 ; J 1 = Y 1 =, p > 2. When any of hese cases occurs (and (1) is nonoscillaory), we conjecure ha he limi characerizaion (9) coninues o hold, as he following example suggess. Example 1. Consider he Euler ype equaion ( 1) (12) ( Φ(x ) ) + ( γ ) pφ(x) = 0, where γ = (p 1)/p, 1 < p < 2. Obviously, J a = J 2 = and u() = γ is a soluion of (12). Moreover, any nonrivial soluion x λu, λ R, saisfies x() γ (log ) 2/p, and u() = γ is a principal soluion of (12) (see, e.g., [8, Example iii)]). Obviously, (9) is saisfied. 4. Inegral characerizaions I is well-known, see e.g. [13, Ch. XI, Theorem 6.4], ha, if he linear equaion (2) is nonoscillaory, hen principal soluions u of (2) can be equivalenly

7 HALF-LINEAR DIFFERENTIAL EQUATIONS 81 characerized by one of he following condiions (in which x denoes an arbirary nonrivial soluion of (2), linearly independen of u): (π 1) (π 2 ) (π 3) u () u() < x () x() u() lim x() = 0; for large ; d a()u 2 () =. The characerizaions (π 1 ), (π 2 ) depend on all he soluions of (2). Even if his is no a serious disadvanage in he linear case, because of he reducion of order formula, he characerizaion (π 3 ) seems prefereable, since i is, roughly speaking, self-conained. In his secion we sudy he possible exensions of he inegral characerizaion (π 3 ) o he half-linear case. In [7] principal soluions u of (1) have been characerized by means of he following inegral (13) Q u := u () u 2 ()u [1] () d. In paricular, when b may change is sign, he following holds. Theorem C [7, Theorem 3.1]. Suppose ha (1) is nonoscillaory and le 1 < p 2. If x is a nonprincipal soluion of (1), hen Q x <. When b() > 0, such a resul has been parially exended in [3] by he following way. Theorem D [3, Theorems 3, 4]. Le (1) be nonoscillaory and assume any of condiions i) J a =, p 2, ii) J b =, 1 < p 2. A soluion u of (1) is a principal soluion if and only if Q u =. In addiion in [3, Corrigendum] an example is given, illusraing ha he characerizaion (13) canno be exended o he case J a =, 1 < p < 2, wihou any addiional assumpions. Here we exend Theorems C, D by inroducing a new inegral characerizaion of principal soluions. Consider he inegral (14) R u := b()φ(u()) u()(u [1] ()) 2 d, which arises considering Q y, where y = u [1] is a soluion of he reciprocal equaion (6). Concerning he characerizaion of nonprincipal soluions, he following resul exends Theorem C.

8 82 M. CECCHI, Z. DOŠLÁ AND M. MARINI Theorem 2. Le (1) be nonoscillaory and assume (5). If x is a nonprincipal soluion of (1), hen Q x < and R x <. To prove his resul, he following lemma is useful. Lemma 1. Assume ha (1) is nonoscillaory and (5) holds. If x is a nonprincipal soluion of (1), hen lim sup x()x [1] () = or lim inf x()x[1] () =, according o J a = or J b =, respecively. Proof. Le J a =. Assume ha here exiss a consan h > 0 such ha for large x()x [1] () < h. Because x is a nonprincipal soluion, in view of Theorem A and Corollary 1, x is unbounded. Then x () Q x = x 2 ()x [1] () d 1 x () h x() d =, which conradics Theorem C or Theorem D, according o 1 < p 2 or p 2, respecively. Now le J b =. Consider he reciprocal equaion (6): applying he firs par of he proof and using Proposiion 1, we obain limsup y()y [1] () = for any nonprincipal soluion y of (6). Because y()y [1] () = x()x [1] (), he asserion follows. Proof of Theorem 2. Taking ino accoun Lemma 1 and using he ideniy T we obain x (s) x 2 (s)x [1] (s) ds = 1 x(t)x [1] (T) 1 x()x [1] () + Q x = 1 x(t)x [1] (T) + R x T b(s)φ(x(s)) x(s)(x [1] (s)) 2 ds, and so boh inegrals Q x, R x have he same behavior. Thus, if 1 < p 2, he asserion follows from Theorem C and if p > 2, he asserion follows applying again Theorem C o he reciprocal equaion (6). Concerning principal soluions, he following holds. Theorem 3. Le (1) be nonoscillaory and le u be a principal soluion of (1). i 1 ) Assume J a =. In addiion, when J 2 =, suppose p 2. Then R u =. i 2 ) Assume J b =. In addiion, when Y 1 =, suppose 1 < p 2. Then Q u =.

9 HALF-LINEAR DIFFERENTIAL EQUATIONS 83 Proof. Claim i 1 ). Since (1) is nonoscillaory, we have J b < (see, e.g., [8, Theorem 1.2.9]). By Proposiion 2 we have S M +. Wihou loss of generaliy, assume u() > 0, u [1] () > 0 for T 0. We have Then T u (s) u 2 (s)u [1] (s) ds = 1 u(t)u [1] (T) 1 u()u [1] () + < (15) Q u 1 u(t)u [1] (T) + T 1 u(t)u [1] (T) + R u. T b(s)φ(u(s)) u(s)(u [1] (s)) 2 ds. b(s)φ(u(s)) u(s)(u [1] (s)) 2 ds When p 2, from Theorem D we have Q u = and so (15) yields R u =. Now le 1 < p < 2. By assumpions and Lemma A we have J 2 < and so, in view of Corollary 1, u M + l,0. By using he l Hospial rule, we have (16) u [1] () b(s)ds. Thus, aking ino accoun ha J b < we obain R u b() (u [1] ()) 2 d = b() ( b(s)ds ) 2 d =. Claim i 2 ). The asserion follows by applying claim i 1 ) o he reciprocal equaion (6) and using Proposiion 1. From Theorems 2, 3 we obain he following. Corollary 2. Le (1) be nonoscillaory and assume (5). In addiion, when J 2 =, suppose p 2 and when Y 1 =, suppose 1 < p 2. A soluion u of (1) is a principal soluion if and only if Q u + R u =. Noice ha, when J a + J b <, he inegral characerizaion (13) fails, as, for insance, Example 2 in [3] shows. The same example illusraes ha also he inegral characerizaion (14) fails. We close his secion by sudying he behavior of inegrals Q u, R u, where u is a principal soluion of (1). The following holds. Theorem 4. Le u be a principal soluion of (1). i 1 ) Assume J a =, J 2 <. Then Q u = if and only if ( 1 ) 1/(p 1) ( (2 p)/(p 1) (17) b(s)ds) d =. a() 0 i 2 ) Assume J b =, Y 1 <. Then R u = if and only if (18) b()( ( 1 ) 1/(p 1)ds ) p 2d =. a(s)

10 84 M. CECCHI, Z. DOŠLÁ AND M. MARINI Proof. Wihou loss of generaliy, assume u() > 0 for large. Claim i 1 ). Inegraing (1) on (, ) and aking ino accoun ha, in view of Corollary 1, u M + l,0, (16) holds and so ( 1 ) (p 2)/(p 1) ( u () p 2 (p 2)/(p 1) b(s)ds). a() Thus (19) u () u [1] () = 1 a() from which he asserion follows. Claim i 2 ). Inegraing he equaliy 1 ( 1 ) 1/(p 1) ( (2 p)/(p 1) b(s)ds) u () p 2, a() u () = Φ ( u [1] () a() on (, ) and aking ino accoun ha, in view of Corollary 1, u M 0,l, we have u() Φ ( 1 ) ds, a(s) and herefore b()φ(u()) u() from which he asserion follows. ( b() ) ( 1 ) 1/(p 1)ds ) p 2, a(s) Remark 2. Using he previous resuls and inegral relaions saed in [5, Lemma 1], i is easy o show when he inegrals Q u, R u have he same behavior for any principal soluion u of (1). We sar by considering he case J a =. If p 2, from Theorems D and 3 we have Q u = R u =. Now consider he case J 2 <, 1 < p < 2 (and J a = ). By applying [5, Lemma 1] wih µ = (p 1)/(2 p) and λ = p 1 and aking ino accoun µ > λ, we obain ( 1 ) 1/(p 1) ( (2 p)/(p 1)d Y 2 = = b(s)ds) =. a() Thus, if Y 2 =, in virue of Theorems 3, 4, we have Q u = R u =. Observe ha when J a =, J 2 <, Y 2 <, 1 < p < 2, he condiion (17) can fail, as he example in [3, Corrigendum] shows. In such a circumsance, again from Theorems 3, 4, we have Q u <, R u = and so he inegrals Q u, R u have a differen behavior. In he case J b = he siuaion is similar. By applying he above argumen o he reciprocal equaion (6) we obain ha Q u = R u = when 1 < p 2. The same conclusion holds if J 1 <, Y 1 = and 1 < p < 2. Finally, when J b =, J 1 <, Y 1 <, p > 2, he condiion (18) can fail, and i is easy o produce an example in which Q u =, R u <. Remark 3. Analogously o he limi characerizaion, i remains an open problem o find an inegral characerizaion of principal soluions in boh cases (11).

11 HALF-LINEAR DIFFERENTIAL EQUATIONS 85 When b may change is sign, he limi and inegral characerizaion of he principal soluions have been parially solved in [4] provided J a <. These problems remain open in he opposie case J a = as well. References [1] Agarwal, R. P., Grace, S. R., O Regan, D., Oscillaion Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equaions, Kluwer Acad. Publ., Dordrech, The Neherlands, [2] Cecchi, M., Došlá, Z., Marini, M., On nonoscillaory soluions of differenial equaions wih p-laplacian, Adv. Mah. Sci. Appl (2001), [3] Cecchi M., Došlá Z., Marini M., Half-linear equaions and characerisic properies of he principal soluion, J. Differenial Equ. 208, 2005, ; Corrigendum, J. Differenial Equaions 221 (2006), [4] Cecchi, M., Došlá, Z., Marini, M., Half-linear differenial equaions wih oscillaing coefficien, Differenial Inegral Equaions (2005), [5] Cecchi, M., Došlá, Z., Marini, M., Vrkoč, I., Inegral condiions for nonoscillaion of second order nonlinear differenial equaions, Nonlinear Anal. 64 (2006), [6] Došlá, Z., Vrkoč, I., On exension of he Fubini heorem and is applicaion o he second order differenial equaions, Nonlinear Anal. 57 (2004), [7] Došlý, O., Elber, Á., Inegral characerizaion of he principal soluion of half-linear second order differenial equaions, Sudia Sci. Mah. Hungar. 36 (2000), [8] Došlý, O., Řehák, P., Half-linear Differenial Equaions, Norh-Holland, Mahemaics Sudies 202, Elsevier, Amserdam, [9] Došlý, O., Řezničková, J., Regular half-linear second order differenial equaions, Arch. Mah. (Brno) 39 (2003), [10] Elber, Á., On he half-linear second order differenial equaions, Aca Mah. Hungar. 49 (1987), [11] Elber, Á., Kusano, T.: Principal soluions of non-oscillaory half-linear differenial equaions, Adv. Mah. Sci. Appl. 8 2 (1998), [12] Fan, X., Li, W. T., Zhong, C., A classificaion scheme for posiive soluions of second order ieraive differenial equaions, Elecron. J. Differenial Equaions 25 (2000), [13] Harman, P., Ordinary Differenial Equaions, 2nd ed., Birkhäuser, Boson Basel Sugar, [14] Hoshino, H., Imabayashi, R., Kusano, T., Tanigawa, T., On second-order half-linear oscillaions, Adv. Mah. Sci. Appl. 8 1 (1998), [15] Jaroš, J., Kusano, T., Tanigawa, T., Nonoscillaion heory for second order half-linear differenial equaions in he framework of regular variaion, Resuls Mah. 43 (2003), [16] Jingfa, W., On second order quasilinear oscillaions, Funkcial. Ekvac. 41 (1998), [17] Mirzov, J. D., Asympoic Properies of Soluions of he Sysems of Nonlinear Nonauonomous Ordinary Differenial Equaions, (Russian), Maikop, Adygeja Publ., 1993; he english version: Folia Fac. Sci. Naur. Univ. Masaryk. Brun. Mah

12 86 M. CECCHI, Z. DOŠLÁ AND M. MARINI Deparmen of Elecronics and Telecommunicaions Universiy of Florence, Via S. Mara Florence, Ialy mariella.cecchi@unifi.i Deparmen of Mahemaics, Masaryk Universiy Janáčkovo nám. 2a, Brno, Czech Republic dosla@mah.muni.cz Deparmen of Elecronics and Telecommunicaion Universiy of Florence, Via S. Mara Florence, Ialy mauro.marini@unifi.i