The Existence, Uniqueness and Stability of Almost Periodic Solutions for Riccati Differential Equation

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1 ISSN (prin), (online) Inernaional Journal of Nonlinear Science Vol.5(2008) No.1,pp The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for Riccai Differenial Equaion Hua Ni Faculy of Science, Jiangsu Universiy, Zhenjiang, Jiangsu, , People s Repulic of China (Received 14 Novemer 2006, acceped 26 Decemer 2007) Asrac:In his paper, Riccai differenial equaion is sudied, y using he fixed poin heorem, he exisence and uniqueness of almos periodic soluion of his sysem are gained, y using second mehod of Liapunov, he uniformly asympoical sailiy and insailiy of he almos periodic soluions are oained. Key words: Riccai differenial equaion; almos periodic soluions; sailiy. AMS2000 sujec classificaion: 34C15; 34C25 1 Inroducion In research works, cerain equaions are ofen considered y scholars. Tian and Li[1] sudy he local well-posedness of he Cauchy prolem for he generalized Degasperis-Proces equaion; Ding and Tian[2] sudied dissipaive Camassa-Holm Equaion and oained he exisence and uniqueness of he saionary soluion elonging o asoring se; G.A.Afrouzi1 e al.[3] oained he exisence of soluions o a nonauonomous p-laplacian equaion; Fan e al.[4] invesigaed muliple compacons in a nonlinear aomic chain equaion. This paper deals wih he following Riccai differenial equaion dx d = a()x2 + ()x + c(), (1.1) where a(), (), c() are all coninuous funcions defined on R. Aou Riccai differenial equaion, many scholars have sudied he periodic soluions of he equaion (1.1), see [5-10], u aou is almos periodic soluion, ye relaed papers are no many, in [11], He Chongyou used he fixed poin heorem and oain he exisence and uniqueness of he almos periodic soluions for Riccai equaion (1.1), moreover, he used firs degree approximaion heory o discuss he sailiy of he almos periodic soluions and derived he sufficien condiions o guaranee he insailiy and asympoically sailiy of he almos periodic soluions for Riccai equaion (1.1). In his paper, we furher sudy he almos periodic soluions of he sysem (1.1), firs, y using variale change and he fixed poin heorem, we oain he exisence and uniqueness of he negaive and he posiive almos periodic soluions for Riccai equaion, hen y using second mehod of Liapunov, we discuss he sailiy of he almos periodic soluions and esalish he sufficien condiions which guaranee he insailiy and uniformly asympoically sailiy of he almos periodic soluions for he Riccai equaion (1.1), he resuls we oain are new. Corresponding auhor. address: nihua@ujs.edu.cn Copyrigh c World Academic Press, World Academic Union IJNS /126

2 H. Ni: The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for 59 2 Preliminary Definiion and Lemmas I is well known o us ha if f() is almos periodic, and f() 0, hen 1 f() is also almos periodic. To simplify our wriing, we inroduce he following noaions f, f, and we se f = sup 1 T m(f()) = lim T + T 0 f(s)ds. Consider he following sysem where a(), f() AP (E). dx d R f(), f = inf f() and R = a()x + f(), (2.1) Lemma 2.1 (Coppel[12]) If Re m(a()) 0, hen he equaion (2.1) exiss a unique almos periodic soluion ϕ(), and mod (ϕ) mod (a, f), ϕ() can e given as follows { ϕ() = e s a(τ)dτ f(s)ds, Re m(a()) > 0 e a(τ)dτ (2.2) f(s)ds, Re m(a()) < 0 Consider he following sysem dx = f(, x), (2.3) d assume f(, x) C(I Ω, R n ) can guaranee he uniqueness propery of soluion of he iniial value condiion for he sysem (2.3) and f(, 0) 0 for every I. Lemma 2.2 ([13]) Suppose ha here exis cerain ε > 0, 0 I; B ε Ω, and exis an open se ψ B ε, V : [ 0, + ) B ε R elonging o C 1 and is posiive definie and no decreasing, such ha when (, x) [ 0, + ) ψ, he following condiions hold: (1) 0 < V (, x) ( x ), where (r) is a coninuous posiive nondecreasing funcion; (II)V (, x) a( x ) where a(r) is a coninuous posiive nondecreasing funcion; (III) x = 0 is in ψ; (IV) V (, x) = 0, (, x) [ 0, + ) ( ψ B ε ); hen he null soluion of he sysem (2.3) is unsale. Lemma 2.3 ([14]) Suppose ha on he cerain region G H = {(, x) : 0, x < H}, here exis a funcion V (, x) and funcions a, where a(r), (r) are oh coninuous posiive nondecreasing funcions, such ha he following condiions hold: (I) 0 V (, x) a( x ); (II)V (, x) (2.3) ( x ); hen he null soluion of he sysem (2.3) is uniformly asympoically sale. 3 Main Resuls Theorem 3.1 Consider he sysem (1.1), suppose ha a(), (), c() are all almos periodic funcions in R, () < 0, a() < 0, 0 c() 2 (a()), where is a posiive consan which saisfies 2 < a 2 < +,hen here exiss a unique negaive almos periodic soluion ϕ () of he equaion (1.1), and mod (ϕ ) mod (a,, c); moreover, if 2ā a > 0, hen ϕ () is unsale. Proof. Taking x() = 1 u(), hen he sysem (1.1) is changed o he following form du d = ()u a() c()u2 (). (3.1) Since a() < 0, from (3.1), we know u() = 0 is no a soluion of he sysem (3.1), hus if u() is an almos periodic soluion of he sysem (3.1), i follows inf u() > 0, hen x() = 1 u() is an almos periodic soluion of he sysem (1.1). Following we show he proof of Theorem 3.1 in wo seps. R IJNS homepage:hp://

3 60 Inernaional Journal of Nonlinear Science,Vol.5(2008),No.1,pp (I) Show ha he equaion (1.1) exiss a unique almos periodic soluion. Define B = {ϕ(); ϕ is an almos periodic funcion, ā (1)ā ϕ(), and mod (ϕ) mod (a,, c)}. Take norm ϕ = sup ϕ() on B, hen (B, ) is a Banach space. R For any funcion ϕ() B, le us consider he following equaion du d = ()u a() c()ϕ2 (), (3.2) since () < 0, () > 0, oviously Re m(()) > 0, according o Lemma 2.1, he sysem (3.2) exiss a unique almos periodic soluion T ϕ(), and mod (T ϕ()) mod (a,, c), from (2.2), we have hence Also T ϕ() = T ϕ() = = 1 T ϕ() = ā e s (τ)dτ [a(s) c(s)ϕ 2 (s)]ds, e (τ)dτ [a(s) c(s)ϕ 2 (s)]ds e (s) [a(s) c(s)ϕ 2 (s)]ds e (s) [a(s) + a(s) 2 a 2 a 2 e (s) [ 1 a(s)]ds ā 2 ]ds e (τ)dτ [a(s) c(s)ϕ 2 (s)]ds e (s) [a(s)]ds Hence T ϕ() B, so T : B B, for any ϕ(), ψ() B, we have T ϕ() T ψ() = e (τ)dτ {c(s)[ϕ 2 (s) ψ 2 (s)]}ds e (τ)dτ c(s) ϕ(s) + ψ(s) ϕ(s) ψ(s) ds e (τ)dτ 2 a 2 (a(s)) 2ā ϕ(s) ψ(s) ds e (s) 2 a 2 a 2ā ϕ(s) ψ(s) ds e (s) 2 1 e (s) + = e (s) + = 2 ϕ ψ a 2ā ϕ(s) ψ(s) ds 2 a 2ā ϕ ψ 2 a 2 a ϕ ψ 2 IJNS for conriuion: edior@nonlinearscience.org.uk

4 H. Ni: The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for 61 hus T ϕ T ψ 2 ϕ ψ. Noe ha he condiion > 2, hus 2 < 1, herefore, T is a conracion mapping on B, T exiss a unique fixed poin on B, which is he unique negaive almos periodic soluion u () of he sysem (3.1), and mod (u ) mod (a,, c), noice ha x() = 1 u(), hus he sysem (1.1) exiss a unique negaive almos periodic soluion ϕ () = 1 u (), and mod (ϕ ) mod (a,, c). (II)we shall show he almos periodic soluion ϕ () of he sysem (1.1) is unsale. Define a Liapunov funcion as follows V (, x() ϕ ()) = (x() ϕ ()) 2, (3.3) where x() is he soluion of he sysem (1.1) wih iniial value ( 0, x 0 ), where ( 0, x 0 ) (R R), ϕ () is he unique almos periodic soluion, oviously V (, x() ϕ ()) saisfies he condiions (I),(III) and (IV) of Lemma 2.2. Differeniaing oh sides of (3.3) along he soluion of he sysem (1.1) gives V (, x() ϕ ()) (1.1) Noe ha hus herefore = 2(x() ϕ ())(x () ϕ ()) = 2(x() ϕ ()){a()x 2 () + ()x() + c() [a()ϕ () 2 + ()ϕ () + c()]} = 2(x() ϕ ()){a()[x 2 () ϕ () 2 ] + ()[x() ϕ ()]} = 2(x() ϕ ()){a()[x() ϕ () + 2ϕ ()][x() ϕ ()] + ()[x() ϕ ()]} = 2a()(x() ϕ ()) 3 + 4a()ϕ ()(x() ϕ ()) 2 + 2()(x() ϕ ()) 2 = 2a()(x() ϕ ()) 3 + [4a()ϕ () + 2()](x() ϕ ()) 2 ā 1)ā ϕ() (, ( 1)ā ϕ () a, V (, x() ϕ ()) (1.1) 2a()(x() ϕ ()) 3 + (4a()( a ) + 2())(x() ϕ ()) 2 2a()(x() ϕ ()) 3 + (4ā( a ) + 2)(x() ϕ ()) 2. Noice ha he condiion 2ā a > 0, hus here exiss a small posiive numer δ > 0 such ha 2ā a > δ > 0 holds, hence we have V (, x() ϕ ()) (1.1) 2a()(x() ϕ ()) 3 + δ(x() ϕ ()) 2, (3.4) noe ha [x() ϕ ()] 3 is infiniesimal of higher order of [x() ϕ ()] 2 as x() ϕ (), from (3.4), here exis a neighorhood D R n of ϕ () and a small posiive consan ε such ha V (, x() ϕ ()) (1.1) ε(x() ϕ ()) 2 holds when (, x) [ 0, + ) D, y Lemma 2.2, he almos periodic soluion ϕ () of he sysem (1.1) is unsale. Theorem 3.2 Consider he sysem (1.1), suppose ha a(), (), c() are all almos periodic funcions in, () > 0, a() < 0, 0 c() 2 (a()), where is a posiive consan which saisfies 2 < < +,hen a 2 here exiss a unique posiive almos periodic soluion ϕ () of he equaion (1.1), and mod (ϕ ) mod (a,, c); moreover, if 2ā + a < 0, hen ϕ () is uniformly asympoically sale. IJNS homepage:hp://

5 62 Inernaional Journal of Nonlinear Science,Vol.5(2008),No.1,pp Proof. Since a() < 0, from (3.1), we know u() = 0 is no a soluion of he sysem (3.1), hus if u() is an almos periodic soluion of he sysem (3.1), i follows inf u() > 0, hen x() = 1 u() is an almos periodic soluion of he sysem (1.1). Following we show he proof of Theorem 3.2 in wo seps. (I) Show ha he equaion (1.1) exiss a unique almos periodic soluion. Define B = {ϕ(); ϕ is an almos periodic funcion, 1 ( ā ) ϕ() a, and mod (ϕ) mod (a,, c)}. Take norm ϕ = sup ϕ() on B, hen (B, ) is a Banach space. R For any funcion ϕ() B, le us consider he following equaion R du d = ()u a() c()ϕ2 (), (3.5) since () > 0, () < 0, oviously Re m(()) < 0, according o Lemma 2.1, he sysem (3.5) exiss a unique almos periodic soluion T ϕ(), and mod (T ϕ()) mod (a,, c), from (2.2), we have hence T ϕ() = T ϕ() e s (τ)dτ [a(s) c(s)ϕ 2 (s)]ds, a e (s) [a(s)]ds Also T ϕ() = 1 e s (τ)dτ [a(s) c(s)ϕ 2 (s)]ds e (s) [a(s) 2 a 2 (a(s))(a )2 ]ds 1 ( ā ) e (s) [a(s)]ds Hence T ϕ() B, so T : B B, for any ϕ(), ψ() B, we have T ϕ() T ψ() = e s (τ)dτ {c(s)[ϕ 2 (s) ψ 2 (s)]}ds e s (τ)dτ c(s) ϕ(s) + ψ(s) ϕ(s) ψ(s) ds e (s) 2 a 2 (a(s)) 2(a ) ϕ(s) ψ(s) ds (s) 2 e a 2 a 2(a ) ϕ(s) ψ(s) ds (s) 2 e a 2 a ϕ(s) ψ(s) ds 1 e(s) a 2 a ϕ ψ = e (s) = 2 ϕ ψ 2 2 a 2 a 2 ϕ ψ IJNS for conriuion: edior@nonlinearscience.org.uk

6 H. Ni: The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for 63 hus T ϕ T ψ 2 ϕ ψ Noe ha he condiion > 2, hus 2 < 1, herefore, T is a conracion mapping on B, T exiss a unique fixed poin on B, which is he unique posiive almos periodic soluion u () of he sysem (3.1), and mod (u ) mod (a,, c), noice ha x() = 1 u(), hus he sysem (1.1) exiss a unique posiive almos periodic soluion ϕ () = 1 u (),and mod (ϕ ) mod (a,, c). (II)we shall show he almos periodic soluion ϕ () of he sysem (1.1) is uniformly asympoically sale Define a Liapunov funcion as follows V (, x() ϕ ()) = (x() ϕ ()) 2, (3.6) where x() is he soluion of he sysem (1.1) wih iniial value ( 0, x 0 ), where ( 0, x 0 ) (R R), ϕ () is he unique almos periodic soluion of he sysem (1.1), oviously V (, x() ϕ ()) saisfies he condiions (I) of Lemma 2.3. Differeniaing oh sides of (3.6) along he soluion of he sysem (1.1) gives V (, x() ϕ ()) (1.1) = 2(x() ϕ ())(x () ϕ ()) = 2(x() ϕ ()){a()x 2 () + ()x() + c() [a()ϕ () 2 + ()ϕ () + c()]} = 2(x() ϕ ()){a()[x 2 () ϕ () 2 ] + ()[x() ϕ ()]} = 2(x() ϕ ()){a()[x() ϕ () + 2ϕ ()][x() ϕ ()] + ()[x() ϕ ()]} = 2a()(x() ϕ ()) 3 + 4a()ϕ ()(x() ϕ ()) 2 + 2()(x() ϕ ()) 2 = 2a()(x() ϕ ()) 3 + [4a()ϕ () + 2()](x() ϕ ()) 2 Noe ha hus herefore 1 ( ā ) ϕ() a, a ϕ () ( 1 ā ), V (, x() ϕ ()) (1.1) 2a()(x() ϕ ()) 3 + (4a()( a ) + 2())(x() ϕ ()) 2 2a()(x() ϕ ()) 3 + (4ā( a ) + 2 )(x() ϕ ()) 2. Noice ha he condiion 2ā + a < 0, hus here exiss a posiive small numer δ > 0 such ha 2ā + a < δ < 0 holds, hence we have V (, x() ϕ ()) (1.1) 2a()(x() ϕ ()) 3 δ(x() ϕ ()) 2, (3.7) noe ha [x() ϕ ()] 3 is infiniesimal of higher order of [x() ϕ ()] 2 as x() ϕ (), from (3.7), here exis a neighorhood D R n of ϕ () and a small posiive consan ε such ha V (, x() ϕ ()) (1.1) ε(x() ϕ ()) 2 holds when (, x) [ 0, + ) D, hence V (, x() ϕ ()) also saisfies he condiions (II) of Lemma 2.3, y Lemma 2.3, he almos periodic soluion ϕ () of he sysem (1.1) is uniformly asympoically sale. IJNS homepage:hp://

7 64 Inernaional Journal of Nonlinear Science,Vol.5(2008),No.1,pp Conclusion Riccai Differenial Equaion is a famous differenial equaion, u he General Soluions of he Equaion can no e expressed y elemenary funcions or inegraions of elemenary funcions. In his paper, we oain he exisence and uniqueness of he negaive and he posiive almos periodic soluions for Riccai equaion y using variale change and he fixed poin heorem, and furher discuss he sailiy of he almos periodic soluions for he Riccai equaion, our resuls exend and improve some exising resuls. References [1] Tian Lixin, Li Xiuming: On he Well-posedness Prolem for he Generalized Degasperis-Procesi Equaion. Inernaional Journal of Nonlinear Science.2(2):67-76(2006) [2] Danping Ding, Lixin Tian: The Sudy of Soluion of Dissipaive Camassa-Holm Equaion on Toal Space. Inernaional Journal of Nonlinear Science. 1(1):37-42(2006) [3] G.A.Afrouzi1 e al: Exisence of Soluions o a Non-auonomous p-laplacian Equaion. Inernaional Journal of Nonlinear Science. 1(2): (2006) [4] Xinghua Fan, Lixin Tian, Lihong Ren: New Compacons in Nonlinear Aomic Chain Equaions wih firs-and second-neighour Ineracions. Inernaional Journal of Nonlinear Science. 1(2): (2006) [5] Weng Aizhi:The periodic soluions of Riccai equaions. Chinese journal of engineering mahemaics. 22(5): (2005) [6] Qing Yuanxuan: Periodic soluion of periodic coefficien Riccai differenial equaion. Chinese Science Bullein. 24(23): (1979) [7] Zhao Huaizhong: Periodic soluion of periodic coefficien Riccai differenial equaion. Chinese Science Bullein. 35(4): (1990) [8] Li Hongxiang: Periodic soluions of some cerain Riccai equaions and second-order differenial equaions. Applied mahemaics and mechanics. 3(2): (1982) [9] Wu Jingang: Crierion on exisence of periodic soluion of periodic coefficien Riccai equaion. Journal of sysems science and mahemaical sciences. 10(1): 24-30(1990) [10] Li Xiong:Periodic soluions of periodic coefficien higher Riccai equaion. Advances in Mahemaics. 28(4): (1999) [11] He Chongyou: Almos Periodic Differenial Equaions. Beijing: Higher Educaion Press(1992) [12] W, A. Coppel:Dichoomies in Sailiy heory. Lecure Noes in Mah. Vol Springer: Berlin(1978) [13] You Bingli: Implemen course of differenial equaions. Beijing: People educaion pulishing house(1981) [14] Liao Xiaoxin: Theories, mehods and applicaions of sailiy properies. Wuhan:Huazhong science and engineering universiy pulishing house(1999) IJNS for conriuion: edior@nonlinearscience.org.uk