MONOTONE SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR FUNCTIONAL DIFFERENTIAL SYSTEMS

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1 Dynamic Sysems and Applicaions MONOONE SOLUIONS OF WO-DIMENSIONAL NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS MARIELLA CECCHI, ZUZANA DOŠLÁ, AND MAURO MARINI Depar. of Elecronics and elecommunicaions, Universiy of Florence Via S. Mara 3, 5039 Firenze, Ialy Depar. of Mahemaics and Saisics, Masaryk Universiy, Janáčkovo nám. 2a, Brno, Czech Republic Depar. of Elecronics and elecommunicaions, Universiy of Florence Via S. Mara 3, 5039 Firenze, Ialy ABSRAC. he exisence of nonoscillaory soluions wih differen asympoic properies for a wo-dimensional nonlinear funcional differenial sysem is sudied. Some discrepancies in he coexisence of nonoscillaory soluions beween he general nonlinear sysem and he Emden-Fowler sysem or he half-linear equaion are poined ou. he roles of deviaing argumens are also discussed. AMS MOS Subjec Classificaion. Primary 34C0, Secondary 34C5.. INRODUCION Consider he nonlinear differenial sysem { x = afyr y = bgxs where a, b, r, s are posiive coninuous funcions on [,, r = s =, and f, g are nondecreasing coninuous funcions on R saisfying ufu > 0, ugu > 0 for u 0 and.2 fu = f u, gu = g u for u R. A coninuously differeniable vecor funcion x, y defined on [,,, is said o be a soluion of. on [, if here exis wo coninuous funcions x 0, y 0, defined on, ], such ha. is saisfied on [,, where x = x 0, y = y 0 for. hroughou his paper we shall consider only he soluions of. which exis on some ray [,, where may depend on he paricular soluion. For he coninuabiliy problem we refer o [5, Proposiion A]. As usually, a componen x [y] Received Sepember 4, $5.00 c Dynamic Publishers, Inc.

2 596 M. CECCHI, Z. DOSLA, AND M. MARINI of a soluion x, y of., defined on some neighborhood of infiniy, is said o be nonoscillaory if x 0 [y 0] for any large, and oscillaory oherwise. Clearly, x is nonoscillaory if and only if y is nonoscillaory oo. So, a soluion x, y of. is said o be oscillaory or nonoscillaory according o boh componens are oscillaory or nonoscillaory, respecively. Paricular cases of. are he Emden-Fowler sysem { x = a y /α sgn y.3 y = b x β sgn x where α > 0, β > 0, α β, he nonlinear equaion wih p-laplacian operaor.4 A x α sgn x + bgxs = 0, where A = a /α, and he half-linear equaion.5 A x α sgn x + b x α sgn x = 0. Sysem. wih s = r = and is paricular cases.3,.4,.5, have been widely invesigaed, see, e.g., he papers [2, 8, 0, 5] for oscillaion problems, [3, 5, 7,, 2, 4] for nonoscillaion ones and [6, 8] for boh. We refer also o he monographs [4, 3], in which a deailed sudy of.3 and.5, respecively, is presened and o [, 9], in which a deailed analysis on he above opics, joinly wih some ineresing open problems, are given. Pu I a = Here wo cases are considered, namely aτdτ, I b = bτdτ. I I a =, I b < ; II I a <, I b =. In boh cases, nonoscillaory soluions of. can be classified as subdominan, inermediae or dominan soluions, according o heir asympoic behavior see below for he definiion. As i is claimed in [, page 24], he exisence of inermediae soluions for. is a difficul problem, even in he special case where s = r =. Moreover, heir possible coexisence wih differen ypes of nonoscillaory soluions is a well-known problem see, e.g., [6, page 23], which has been compleely resolved for.5 in [3]. he aim of his paper is o sudy he exisence of inermediae, subdominan and dominan soluions of.. We show some discrepancies in he coexisence of hese soluions for. and.3, which are caused by he growh of nonlineariies f, g. Secions 3, 4 deal wih he case I. In Secion 5, by means of a dualiy propery, he obained resuls are exended o he case II. Our resuls improve or generalize analogous ones in [5, 7, 8, 2]. he role of he deviaing argumens r, s are also discussed and several examples illusrae he obained resuls.

3 NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS PRELIMINARIES When. is nonoscillaory, for sake of simpliciy, we will resric our aenion only o soluions x, y of. for which x is evenually posiive. We will denoe such soluions as soluions of class M + or M, according o y is evenually posiive or evenually negaive. he remaining cases can be easily reaed using analogous argumens. If x, y M +, hen x is posiive increasing and y is posiive decreasing for large ; if x, y M, hen x is posiive decreasing and y is negaive decreasing for large. I is easy o show ha, if I a =, hen M =. Similarly, if I b =, hen M + =. So, if x, y M + and x is bounded, hen lim y = 0. Similarly, if x, y M and lim x > 0, hen lim y =. hus, in case I soluions in M + can be a-priori divided ino he subclasses: M +,l = {x, y M+ : lim x =, lim y = l y, 0 < l y < }, M +,0 = {x, y M+ : lim x =, lim y = 0}, M + l,0 = {x, y M+ : lim x = l x, lim y = 0, 0 < l x < }, and in case II soluions in M ino he subclasses: M l, = {x, y M : lim x = l x, lim y =, 0 < l x < }, M 0, = {x, y M : lim x = 0, lim y = }, M 0, l = {x, y M : lim x = 0, lim y = l y, 0 < l y < }. Noice ha his classificaion is similar o he one given in [2], when r s. Following [6], soluions in M +,l, M+,0, M+ l,0 are called dominan soluions, inermediae soluions and subdominan soluions, respecively. his erminology is due o he fac ha, when g is unbounded and x, y M +,l, x 2, y 2 M +,0, x 3, y 3 M + l,0, hen x > x 2 > x 3 and y > y 2 > y 3 for any large. he same erminology of inermediae soluions is used for soluions in M 0,, for a similar reason. An imporan role in he exisence of nonoscillaory soluions, is played by he following inegrals depending on he parameers. In case I denoe J µ = aτf µ bσdσ dτ, K λ = bτg λ r τ and in case II W λ = aτf λ r τ bσdσ dτ, Z µ = bτg s τ aσdσ dτ, µ aσdσ dτ, s τ

4 598 M. CECCHI, Z. DOSLA, AND M. MARINI where r = max {, r}, s = max {, s}. 3. ON INERMEDIAE SOLUIONS heorem 3.. Assume I a =, I b <. If here exis wo posiive consans λ and µ, where µ < lim u gu, such ha hen M +,0. K λ <, J µ =, Proof. Le c be a posiive consan such ha 3. gc > µ and pu m = f λ/2. Choose large enough such ha r, s for and 3.2 gc 3.3 bσdσ m, c λ bτg λ sτ aσdσ 0 aσdσ, dτ m. Pu s = inf { : sτ τ } and le s, r be he funcions { for [, s 3.4 s =, r = max {r, }. s for s Clearly, if s >, hen s s = and so s is coninuous for. Denoe wih C[, he Fréche space of all coninuous funcions on [, endowed wih he opology of uniform convergence on compac subinervals of [, and consider he se Ω C[, given by { } Ω = v C[, : µ bσdσ v 2m,. Observe ha from 3. and 3.2 we have µ bσdσ < 2m for 0. Define in Ω he operaor given by v = bτg c + sτ aσf vrσ dσ dτ. From 3. we have v gc bτdτ µ bτdτ.

5 hus NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS 599 Moreover, in virue of 3.2, 3.3, we have s v = gc bτdτ + bτg c + s gc bτdτ + bτg c + λ 3.5 v gc s bτdτ + s bτg sτ sτ λ sτ aσfvrσdσ aσdσ aσdσ dτ, dτ dτ 2m, and so maps Ω ino iself. Le us show ha Ω is relaively compac, i.e. Ω consiss of funcions equibounded and equiconinuous on every compac inerval I of [,. Because Ω Ω, he elemens of Ω are equibounded wih firs derivaives equibounded on I and so he compacness follows. Now we show ha is coninuous in Ω C[,. Le {v n }, n N, be a sequence in Ω which uniformly converges on every compac inerval of [, o v Ω. Because Ω is relaively compac, he sequence {v n } admis a subsequence {v nj } converging, in he opology of C[,, o z v. In view of 3.3 and 3.5, by applying he Lebesgue dominaed convergence heorem, he sequence {v nj } poinwise converges o v. In view of he uniqueness of he limi, v = z v is he only cluser poin of he compac sequence {v n }, ha is he coninuiy of in he opology of C[,. Hence, by he ychonov fixed poin heorem here exiss a soluion of he inegral equaion 3.6 y = bτg I is easy o verify ha x, y, where 3.7 x = c + c + sτ aσf yrσ dσ aσfyrσdσ, is a soluion of. for large, say s. Since y Ω, we have for x x aτf µ bσdσ dτ and so, because J µ =, we obain x =. From 3.2 and 3.6 i resuls for y bτg c + λ sτ aσdσ and so, in view of K λ <, we ge y = 0. dτ rτ bτg λ dτ. sτ aσdσ dτ

6 600 M. CECCHI, Z. DOSLA, AND M. MARINI Remark 3.2. As already claimed, when s = r =, he exisence of inermediae soluions for. is considered in [2, heorem 2.4] and, for an equaion including.4, in [5, heorem 9], [8, heorem.3]. heorem 3. subsanially exends [5, heorem 9], [8, heorem.3], [2, heorem 2.4], because in hese resuls i is assumed ha J µ = for any µ > 0. he Example below illusraes his fac. Noice ha he proof of [2, heorem 2.4] is differen, since a differen operaor and a differen se Ω is considered. Moreover, i is no complee, because i remains o verify ha he se Ψ, considered in he proof, is convex, which seems difficul o prove. he following examples illusrae he role of he nonlineariy f and he condiion µ < lim u gu in heorem 3., respecively. Example 3.3. Consider he sysem 3.8 x = e fy, y = 2 3 gx, where f, g are nondecreasing coninuous funcions on R such ha ufu > 0, ugu > 0 for u 0 and We have fu = exp / u if 0 < u < gu = log u if u > e. K = 2 τ 3 ge τ edτ <, J = e τ e τ dτ =, and so, in view of heorem 3., his sysem has inermediae soluions. Neverheless he assumpion J µ = does no hold for any µ > 0. Indeed for µ = /4 we have J /4 = e τ dτ <. Example 3.4. Consider he sysem x = 2 e fy, y = 2 3 gx, where f, g are nondecreasing coninuous funcions on R such ha ufu > 0, ugu > 0 for u 0 and fu = exp / u if 0 < u <, gu = if u >. Such a sysem does no have inermediae soluion. Indeed, if here exiss x, y M +,0, we have y = 2 for large and so i resuls x = 2, which conradics he unboundedness of x. Noice ha K λ < for λ > 0, J µ < for 0 < µ and J µ = for µ > = lim u gu.

7 NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS COEXISENCE RESULS A naural quesion, which arises, is o sudy whenever he exisence of subdominan and dominan soluions depends on he limi value of heir firs and second componen, respecively. he following holds. heorem 4.. Assume I a =, I b <. i If here exiss λ > 0 such ha K λ =, hen. does no have soluions x, y saisfying x =, y = L, L > f λ. i 2 If here exiss λ > 0 such ha K λ <, hen. has soluions x, y saisfying x =, y = L, 0 < L < f λ. Proof. Claim i. By conradicion, assume here exiss a soluion x, y of. wih x =, y = L > f λ. Since y is evenually posiive decreasing, wihou loss of generaliy, we can assume x > 0, L < yr < 2L on [,,. From he firs equaion in. we ge x = x + aτfyrτdτ fl aτdτ. Le such ha s for. Hence, from he second equaion in. we obain for 4. y L bτg fl sτ aσdσ dτ. Since I a =, fixed ε wih λfl < ε <, i resuls for large τ, say τ 2, and so from 4. we ge sτ aσdσ > ε bτg εfl 2 sτ sτ aσdσ aσdσ dτ <. Since εfl > λ, we obain a conradicion wih K λ =. Claim i 2. Fixed c > 0, choose saisfying 3.2, 3.3 wih m = 2 f λ L and r, s for. Le r, s be he funcions defined in 3.4. Now consider he se Ω C[, given by Ω = { } v C[, : L v f λ for and define in Ω he operaor as follows v = L + bτg c + sτ aσfvrσdσ dτ

8 602 M. CECCHI, Z. DOSLA, AND M. MARINI Reasoning as in he proof of heorem 3., and applying he ychonov fixed poin heorem, we obain ha here exiss a soluion of he inegral equaion y = L bτg c + sτ aσfyrσdσ dτ. I is easy o verify ha x, y, where x is given by 3.7, is a soluion of. for large, say s. From 3.7, aking ino accoun ha y Ω, we obain x fl aσdσ for and so x =. Clearly y = L and he proof is complee. heorem 4.2. Assume I a =, I b <. i If here exiss µ > 0 such ha J µ =, hen. does no have soluions x, y saisfying x = L, L > g µ, y = 0. i 2 If here exiss µ > 0 such ha J µ <, hen. has soluions x, y saisfying x = L, 0 < L g µ, y = 0. Proof. Claim i. By conradicion, assume here exiss a soluion x, y of. wih x = L > g µ, y = 0. Le L ε such ha L > L ε > g µ. Since x, y M + l,0, we can suppose, wihou loss of generaliy, xr > L ε, y > 0 for any. From he second equaion in. we ge for σ 4.2 yσ = σ bτgxsτdτ. Le such ha r for. Using 4.2, from he firs equaion in. we obain for L x = aσf which is a conradicion. gl ε rσ aσf bτdτ bτgxrτdτ dσ rσ aσf µ bτdτ, rσ Claim i 2. he asserion follows by applying he ychonov fixed poin heorem o he operaor given by u = L aσf rσ bτgusτdτ dσ in he se Ω C[, { Ω = u C[, : } 2 L u L for.

9 NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS 603 where r = max {r, },s = max {s, } and is large so ha aτf gl bσdσ dτ L 2. he argumen is similar o he one given in [2, heorem 2.2], wih minor changes. rτ From heorems 4., 4.2 we obain he following. Corollary 4.3. Assume I a =, I b <. hen M +,l K λ < for some λ > 0. M + l,0 J µ < for some µ > 0. Remark 4.4. Corollary 4.3 can be proved direcly by using a similar argumen o he one given in [2, heorems 2.2, 2.3], wih minor changes. For he Emden-Fowler sysem.3 he convergence or divergence of inegrals K λ, J µ does no depend on he choice of he parameers λ, µ and so in case I, if K < [J < ], hen.3 has a soluion x, y M +,l [M+ l,0 ] such ha y = L [x = L] for any L > 0. From heorem 3. and Corollary 4.3, we obain he following coexisence resul. Corollary 4.5. Assume I a =, I b <. If here exis hree posiive consans λ, µ and ν, µ < ν < lim u gu, such ha K λ <, J µ <, J ν =, hen all subclasses in M + are nonempy, i.e. M + l,0, M+,0, M+,l. Remark 4.6. As already claimed, he coexisence of all ypes of nonoscillaory soluions is impossible for he half-linear equaion.5. Example 3.3 illusraes ha his is possible for sysem., since K <, J /4 < and J =. Anoher possible discrepancy beween. and.5, concerning he nonoscillaory soluions, is a consequence of he following resul. heorem 4.7. Assume I a =, I b <. If here exiss µ > 0 such ha s τ 4.3 bτg aσf µ bξdξ dσ dτ =, r σ for any, hen M +,0 = M +,l =.

10 604 M. CECCHI, Z. DOSLA, AND M. MARINI Proof. Firs, observe ha assumpion 4.3 yields g =. Le x, y M +,0. Le be such ha x is posiive increasing, y is posiive decreasing for and r, s for. Using he l Hopial rule, we obain ha here exiss a posiive consan d such ha y > d bξdξ for. hen evenually we have 4.4 yr > d r bξdξ. Wihou loss of generaliy, suppose ha 4.4 holds for any. Le such ha s on [,. From he firs equaion in. we obain for x aσf yrσ dσ aσf d bξdξ dσ and so for s g xs g aσf Hence s 4.5 y = bgxs bg aσf rσ d bξdξ dσ. rσ d bξdξ dσ. rσ Inegraing his inequaliy on,,, we obain a conradicion and so M +,0 =. Le us show ha M +,l =. Since I b <, from 4.3 we obain ha g is unbounded and J µ =. Since inermediae soluions do no exis, by heorem 3. we obain K λ = for any λ > 0. hus, applying Corollary 4.3 we ge M +,l =. he condiion 4.3 has o be verified for any. wo sronger condiions, which do no depend on he choice of, are given by he following: Corollary 4.8. Assume I a =, I b <. Le one of he following condiions hold: i here exis < λ < and µ > 0 such ha s τ 4.6 bτg λ aσf µ bξdξ dσ dτ = ; r σ i 2 guv gugv for any u >, v > and for some µ > 0 s τ 4.7 bτg aσf µ bξdξ dσ dτ =. hen M +,0 = M +,l =. Proof. Fixed, se J, = aσf r σ µ bξdξ dσ. r σ

11 NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS 605 Claim i. Since I b <, he funcion g is unbounded and J, =. Hence here exiss λ 0, such ha λj, s τ J, s τ for large τ. So, from he monooniciy of g, 4.3 holds and he asserion follows from heorem 4.7. Claim i 2. Choose τ large so ha J, [J, s τ] <. Hence he asserion follows from heorem 4.7 and J, J, s τ = J, s τ + 2J, s τ. J, s τ Remark 4.9. heorem 4.7 is no significan for he Emden-Fowler sysem.3 wih α β. Indeed for.3, he inegrals J µ, K λ reads as /α J = aτ µ bσdσ dτ, K = τ τ β bτ λ aσdσ dτ, respecively, and so, when he case I holds, 4.3 implies J = K =. Bu, in such a case, i is known ha all soluions of.3 are oscillaory see, e.g., [3, heorems.3,.4]. When α = β, he following example illusraes a possible discrepancy beween he coexisence of soluions of. wih r = s = and.5. Example 4.0. Consider he sysem x = e fy, y = 2 3 x x, where f is a nondecreasing coninuous funcion on R such ha ufu > 0 for u 0 and fu = exp / u if 0 < u <. hen 4.3 is saisfied for µ =. Since J /4 <, in view of Corollaries 4.3, 4.8 we have M + l,0, M,0 =, M,l =. Noe ha such siuaion never occurs for.5, because soluions in M + l,0 always coexis wih soluions eiher in M+,0 or in M+,l [3]. We close his secion by illusraing he role of he deviaing argumens o he asympoic properies of soluions of.. Example 4.. Consider he sysem 4.8 x = y α sgn y, y = 3gxs where α 0, /2] and g is a nondecreasing coninuous funcion on R such ha ugu > 0 for u 0 and { eu if u [0, gu = e u if u If s = + log, hen heorem 3. is applicable wih λ = µ =, and 4.8 has inermediae soluions. If s =, hen K λ = for any λ > 0 and all soluions of his sysem are oscillaory, as follows from [0] see also [, heorem 7..2].

12 606 M. CECCHI, Z. DOSLA, AND M. MARINI Example 4.2. Consider he sysem x = yr, y = 2x. If r = + log, assumpions in heorem 3. are verified wih λ = µ = and x M +,0, x M+,l, x M+ l,0 =. If r =, from Corollary 4.3 we have M +,l, M+ l,0 soluions do no exis, because he sysem is linear.. Moreover, inermediae 5. HE CASE II In his Secion we show how i is possible o exend all he above resuls o he case II, i.e. when I a <, I b =. In view of.2, if x, y is a soluion of., hen y, x is a soluion of he sysem { z = bgws 5. w = afzr and vice-versa, if z, w is a soluion of 5., hen w, z is a soluion of.. Observe ha 5. comes ou from. by inerchanging he roles of he funcions a, f, r wih b, g, s, respecively, and vice-versa. Moreover, he inegrals J µ, K λ become Z µ and W λ, respecively. So, such a dualiy propery permis us o exend all he above resuls o case I a <, I b = in a very simple way. Neverheless, we poin ou ha he assumpion.2 is no necessary for our resuls, because he same resuls can be also proved in a direc way by adding suiable assumpions on nonlineariies on, 0. We recall ha in case II, all nonoscillaory soluions of. belong o M. Here we repor only hose resuls which are significaive as regards he possible coexisence in he subclasses of M or he comparison wih known resuls. he remaining exensions are lef o he reader. heorem 5.. Assume I a <, I b =. If here exis wo posiive consans λ, µ, µ < lim u fu, such ha W λ <, Z µ =, hen M 0,. Proof. he asserion follows from heorem 3., by applying he dualiy propery. Remark 5.2. When s = r =, he exisence of soluions in M 0, is considered in [2, heorem 3.7]. heorem 5. subsanially exends such a resul, because heorem 3.7 requires Z µ = for any µ > 0 and, implicily, also he boundedness of f. Concerning he coexisence of nonoscillaory soluions, he following holds.

13 NONLINEAR FUNCIONAL DIFFERENIAL SYSEMS 607 Corollary 5.3. Assume I a <, I b =. If here exis hree posiive consans λ, µ and ν, ν < µ < lim u fu, such ha Z ν <, W λ <, Z µ =, hen all subclasses in M are nonempy, i.e. M 0, l, M 0,, M l,. Proof. he asserion follows from heorem 5., applying he dualiy propery o Corollary 4.3. Noice ha Corollary 5.3 gives a possibiliy of coexisence of all hree ypes of nonoscillaory soluions of., which is impossible for.5, as already observed in Remark 4.6. Applying he dualiy propery o heorem 4.7, we obain he following. heorem 5.4. Assume I a <, I b =. If here exiss λ > 0 such ha r τ 5.2 aτf bσg λ aξdξ dσ dτ = for any, hen M 0, = M l, =. Remark 5.5. heorem 5.4 exends [2, heorem 3.8], where a necessary condiion for he exisence of soluions in M 0,, when r = s =, is given. However such a resul is no well formulaed, because 3.8 in [2] should be 5.2 wih r τ = τ, s σ s σ = σ, as i follows from he proof of heorem 3.8. Acknowledgmen. he research of he second auhor was suppored by he Research Projec of he Minisery of Educaion of he Czech Republic and by he Gran 20/08/046 of he Gran Agency of he Czech Republic. REFERENCES [] R. P. Agarwal, S. R. Grace, D. O Regan, Oscillaion heory for Second Order Linear, Half- Linear, Superlinear and Sublinear Dynamic Equaions, Kluwer Acad. Publ. G, Dordrech, [2] M. Barušek, M. Cecchi, Z. Došlá, M. Marini, On oscillaory soluions of quasilinear differenial equaion, J. Mah. Anal. Appl., 320:08 20, [3] M. Cecchi, Z. Došlá, M. Marini, On inermediae soluions and he Wronskian for half-linear differenial equaions, J. Mah. Anal. Appl., 336:905 98, [4] O. Došlý, P. Řehák, Half-linear Differenial Equaions, Norh-Holland, Mahemaics Sudies 202, Elsevier Sci. B.V., Amserdam, [5] X. Fan, W.. Li, C. Zhong, A classificaion schemes for posiive soluions of second order nonlinear ieraive differenial equaions, Elec. J. Diff. Equa., 25: 4, [6] H. Hoshino, R. Imabayashi,. Kusano,. anigawa, On second-order half-linear oscillaions, Adv. Mah. Sci. Appl., 8:99 26, 998.

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