Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS YOUSSEF N. RAFFOUL Communicaed by K. Ciesielski Absrac. We give asympoic classificaion of he posiive soluions of a class of wo-dimensional nonlinear Volerra inegro-differenial equaions. Also, we furnish necessary sufficien condiions for he exisence of such posiive soluions. 1. Inroducion In his paper we consider he wo-dimensional nonlinear Volerra inegro-differen ial equaions { x = hx + 0 a, sfysds, y = py + 1.1 b, sgxsds, 0 s <. he funcions a, s b, s are posiive coninuous funcions for 0 s <. he funcions f g are real-valued coninuous in y x, respecively, increasing on he real line R. Also, he coefficiens h p are posiive coninuous funcions for 0 saisfy he condiion 0 hd < 0 pd <. 1.2 hroughou his paper we assume ha he funcions f g are well behaved so ha he soluions of sysem 1.1 exis on he inerval [0,. Also, we assume ha fx > 0, gx > 0 for x 0. Definiion 1.1. A pair of real-valued funcions x, y is said o be Dae: Received: 31 Ocober 2010; Revised: 13 May 2011; Acceped: 17 June 2011. 2010 Mahemaics Subjec Classificaion. Primary 34K20; Secondary 45J05, 45D05. Key words phrases. Volerra inegro-differenial equaions, posiive soluions, classificaion, Sysems, fixed poin heorems. 34
POSIIVE SOLUIONS OF NONLINEAR SYSEMS 35 1 a soluion of sysem 1.1 if i saisfies 1.1 for ; 2 evenually posiive if boh x y are evenually posiive; 3 nonoscillaory if boh x y are eiher evenually posiive or evenually negaive. here are many papers wrien on he subjec of oscillaory nonoscillaory behavior of soluions in differenial equaions. For such opics we refer he ineresed reader o [1], [2], [5]-[7] he reference herein. Recenly, Li [6] Li Cheng [7] sudied he class of wo-dimensional nonlinear differenial sysems of he form { x = afy, y = bgx, under similar assumpions. hey provided a classificaion scheme for posiive soluions of he above sysem. hey also provided condiions for he exisence of soluions wih designaed asympoic properies. However, no sudy has been devoed o sysems of Volerra inegro-differenial equaions. In [8] he auhor used he noion of Lyapunov funcional obain resuls concerning he boundedness of soluions of Volerra inegro-differenial equaions wih unbounded perurbaion. Sysems of he form of 1.1 are used for coninuous risk models where he risk process is a variaion/exension of he classical compound Poisson process, see for example Dickson dos Reis 1997 [2] Kluppelberg Sadmuller 1998 [4]. In his paper, we classify posiive soluions of 1.1 according o heir limiing behaviors hen provide sufficien /or necessary condiions for heir exisence. o simplify our noaion we le A = au, sds du, B = bu, sds du. Le A = lim A B = lim B. Noe ha if a, s = e s, hen A = 1 2. We will discuss each of he following cases: i A =, B = ; ii A =, B < ; iii A <, B = ; iv A <, B <. Le C be he se of all coninuous funcions define Ω = {x, y C[0,, R C[0,, R : x, y are evenually posiive }.
36 Y.N. RAFFOUL Le 0 wih x = x 0 y = y 0. By muliplying boh sides of he firs equaion in 1.1 by e R hsds 0 hen inegraing from u = o u = we arrive a he variaion of parameers formula R x = x 0 e hsds 0 + au, sfysds e R u hsds du. 1.3 In a similar fashion, we obain from he second equaion of 1.1, R y = y 0 e psds 0 + bu, sgxsds e R u psds du. 1.4 I is clear from 1.3 1.4 ha x y are posiive provided ha x 0, y 0 0. Moreover, since h, p > 0, hen from 1.1 we have x, y > 0. Now, for some posiive consans α β, we define he se Kα, β = {x, y Ω : lim x = α, lim y = β}. Noe ha α β maybe considered o be infinie. 2. Classificaion of Soluions Exisence In his secion, we should classify posiive soluions of 1.1 according o heir limiing behaviors hen provide necessary sufficien condiions for heir exisence in he cases ii, iii iv. Our resuls are based on he applicaion of Knaser s fixed poin heorem, which we sae below. Knaser s Fixed Poin heorem Le X be a parially ordered Banach space wih ordering. Le M be a subse of X wih he following properies: he infimum of M belongs o M every nonempy subse of M has a supermum which belongs o M. Le : M M be an increasing mapping, i.e., x y implies x y. hen has a fixed poin in M. heorem 2.1. Any soluion x, y Ω of 1.1 belongs o one of he following subses Kα, β, Kα,, K, β K,. Proof. Since x, y Ω, we have x, y > 0 for. hus x y are increasing. Hence, lim x = α > 0 or lim x =, lim y = β > 0 or lim y =. In he following we sae four heorems. Each heorem is relaed o one of he above menioned cases. heorem 2.2. Suppose 1.2, A = B = hold. soluion x, y Ω of 1.1 belongs o he se K,. hen any
POSIIVE SOLUIONS OF NONLINEAR SYSEMS 37 Proof. Le x, y Ω be a soluion of sysem 1.1. hen x, y > 0 for. hus x y are increasing. As a consequence of his he fac ha f is increasing, an inegraion of 1.1 yields x = x 0 + hsxsds + au, sfysds du fy 0 au, sds du, for, y = y 0 + psysds + bu, sgxsds du gx 0 bu, sds du, for. Showing ha x y, as. heorem 2.3. Suppose 1.2, A = B < hold. hen here exiss a soluion x, y Ω of 1.1 ha belongs o he se K, β if only if lim s bu, sg for some posiive consan c. v hτdτ v ] av, kfcdk dv ds du <, 2.1 Proof. Le x, y Ω be a soluion of sysem 1.1. hen x, y > 0 for. hus x y are increasing here exiss a posiive consan β > 0 such ha y 0 y β for. From 1.3 we have R x = x 0 e hsds 0 + au, sfysds e R u hsds du au, sfy 0 ds e R u hsds du, β y = y 0 + psysds + s bu, sg e R v s v hτdτ av, kfy 0 dk ] bu, sg xs ds du ] dv ds du. By aking he limi a infiniy in he above inequaliy we obain 2.1. Conversely, suppose ha 2.1 holds. Firs noice ha for, he second equaion of 1.1 can be wrien as y = y 0 + psysds + bu, sgxsds du.
38 Y.N. RAFFOUL Nex, we can choose a large enough so ha s bu, sg e R v s v hτdτ av, kdk psds 1 4. ] dv ds du c 4fc, Le X be he Banach space of all bounded real-valued funcions y on [, wih he norm y = sup y wih he usual poinwise ordering. Define a subse ω of X by ω = {y X : c 2 y c, }. I is clear ha for any subse B of ω, inf B ω sup B ω. Define he operaor E : ω X by Ey = c 2 + psysds s + bu, sg y ω. v hτdτ v We claim ha E maps ω ino ω. o see his we le y ω. hen c/2 Ey = c/2 + psysds s + bu, sg c/2 + c + fc 0 psds s bu, sg c/2 + c 4 + c 4 = c. v hτdτ v ] av, kfykdk dv ds du, v hτdτ v ] av, kfykdk dv ds du ] av, kdk dv ds du Since E is increasing, he mapping E saisfies he hypohesis of Knaser s fixed poin heorem hence we conclude ha here exiss y in ω such ha y = Ey. Se x = asfysds e R u hsds du, hen x = hx + asfysds,
POSIIVE SOLUIONS OF NONLINEAR SYSEMS 39 x = fy 0 In view of A =, we have On he oher h, y = c 2 + psysds s + bu, sg from which we obain, au, sfysds e R u hsds du au, sds du. lim x =. v hτdτ v lim y = β, ] av, kfykdk dv ds du, where β is a consan. Hence, x, y is a posiive soluion of 1.1 which belongs o K, β. he proof of he nex heorem follows along he lines of he proof of heorem 2.3 hence we omi. heorem 2.4. Suppose 1.2, A < B = hold. hen here exiss a soluion x, y Ω of 1.1 ha belongs o he se Kα, if only if s au, sf e R v ] s v pτdτ bv, kgcdk dv ds du <, lim for some posiive consan c. heorem 2.5. Suppose 1.2 hold. hen any soluion x, y Ω of 1.1 ha belongs o he se Kα, β if only A < B <. Proof. Le x, y be a soluion in Ω wih lim x α > 0 lim y β > 0. hen, here exiss a wo posiive consans; namely, c 1 c 2 such ha c 1 x α, c 2 y β for. From sysem 1.1 we have for ha x = x + hsxsds + au, sfysds du, 2.2 y = y + psysds + bu, sgxsds du. 2.3
40 Y.N. RAFFOUL hus c 1 x = x + x + α c 2 y = y + y + β hsxsds + hsds + psysds + psds + au, sfysds du, au, sfβds du <, bu, sgxsds du, bu, sgαds du <. Conversely, suppose ha A < B <. Firs noice ha for, he firs equaion of 1.1 can be wrien as x = x 0 + hsxsds + au, sfysds du. In a similar fashion, we obain from he second equaion of 1.1, y = y 0 + psysds + bu, sgxsds du. Nex, we can choose a large enough so ha ] au, sds du psds 1 4d. d 4fc, Le X be he Banach space of all bounded real-valued funcions x, y on [, wih he norm x, y = max{sup x, sup y } wih he usual poinwise ordering. Define a subse ω of X by ω = {x, y X : d 2 x d, c 2 y c, }. I is clear ha any subse B of ω, inf B ω sup B ω. Define he operaor E : ω X by x E y [ d = 2 c 2 ] + hsxsds + u au, sfysds du psysds + u, bu, sgxsds du
POSIIVE SOLUIONS OF NONLINEAR SYSEMS 41 x, y ω. We claim ha E maps ω ino ω. o see his we le x ω. hen d/2 Ex = d/2 + psxsds + au, sfysds du 0 d/2 + d psds + fc au, sds du d/2 + d 4 + d 4 = d. Showing ha, for y ω, c/2 Ey c is similar hence we omi i. Since E is increasing, he mapping E saisfies he hypohesis of Knaser s fixed poin heorem hence we conclude ha here exiss x, y in ω such ha x, y = Ex, y. ha is, x = d 2 + hsxsds + au, sfysds du, from which we obain, y = c 2 + psysds + lim x = α lim bu, sgxsds du y = β, where α β are posiive consans. Hence, x, y is a posiive soluion of 1.1 which belongs o Kα, β. Remark I is easy o exended he resuls of his paper o sysems of more han wo-dimensional Volerra inegro-differenial equaions. References 1. L.H. Erbe, Q.K. Kong B.G. Zhang, Oscillaion heory for Funcional Differenial Equaions, Marcel Dekker, New York, 1995. 2. D.C.M. Dickson A.D.E. dos Reis, he effec of ineres on negaive surplus, Insurance Mah. Econom. 21 1997, no. 1, 1 16. 3. I. Györi G. Ladas, Oscillaion heory of Delay Differenial Equaions wih Applicaions, he Clarendon Press, Oxford Universiy Press, New York, 1991. 4. C. Kluppelberg U. Smüller, Ruin probabiliies in he presence of heavy-ails ineres raes, Sc. Acuar. J. 1998, no. 1, 49 58. 5. G.S. Ladde, V. Lakshmikanham B.G. Zhang, Oscillaion heory of Differenial Equaions wih Deviaing Argumens, Pure Applied Mahemaics, 110, Marcel Dekker, New Yrok, 1987. 6. W.. Li C.K. Zhong, Unbounded posiive soluions of higher order nonlinear funcional differenial equaions, Appl. Mah. Le. 14 2001, no. 7, 825 830. 7. W.. Li, Classificaions exisence of nonoscillaory soluions of second order nonlinear differenial equaions, Ann. Polon. Mah. LXV 1997, no. 3, 283 302. 8. Y.N. Raffoul Exponenial analysis of soluions of funcional differenial equaions wih unbounded erms, Banach J. Mah. Anal. 3 2009, no. 2, 16 27. Deparmen of Mahemaics, Universiy of Dayon, Dayon, OH 45469-2316, USA. E-mail address: youssef.raffoul@noes.udayon.edu