Boundedness and Exponential Asymptotic Stability in Dynamical Systems with Applications to Nonlinear Differential Equations with Unbounded Terms

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1 Advances in Dynamical Sysems and Applicaions. ISSN Volume Number 1 007, pp Research India Publicaions hp:// Boundedness and Exponenial Asympoic Sabiliy in Dynamical Sysems wih Applicaions o Nonlinear Differenial Equaions wih Unbounded Terms Youssef N. Raffoul Deparmen of Mahemaics, Universiy of Dayon, Dayon, OH , USA youssef.raffoul@noes.udayon.edu Absrac Nonnegaive definie Lyapunov funcions are employed o obain sufficien condiions ha guaranee boundedness of soluions of a nonlinear differenial sysem. Also, sufficien condiions will be given o insure ha he zero soluion is exponenially and asympoically sable. Our heorems will make a significan conribuion o he heory of differenial equaions when dealing wih equaions ha migh conain unbounded erms. The heory is illusraed wih several examples. AMS subjec classificaion: 34C11, 34C35, 34K15. Keywords: Nonlinear differenial sysem, boundedness, exponenial sabiliy, Lyapunov funcions. 1. Inroducion For moivaional purpose, we consider he scalar linear iniial value problem ẋ = ax + g, 0, x = x 0, 0, 1.1 where a and g are coninuous in wih a 0, 0. By he variaional of parameers formula x = x 0 e asds 0 + gu e u asds du. 1. Received February 3, 007; Acceped April 8, 007

2 108 Youssef N. Raffoul Thus, if gu e u asds du M, 0 for some posiive consan M, hen expression 1. implies ha all soluions of Eqn. 1.1 are uniformly bounded. Then, i is reasonable o ask if he same can be done for nonlinear scalar or vecor equaions of he form ẋ = ax + g, x, 0, x = x 0, 0, 1.3 where a is coninuous in wih a 0, and g, x is coninuous in and x, 0. The answer is of course yes. Indeed, one ges he variaional of parameers formula x = x 0 e asds + gu, xu e u asds du. 1.4 Bu expression 1.4 hardly gives any informaion abou he boundedness of soluions of Eq. 1.3 unless we assume ha g, x λ, 0, 1.5 where λ is coninuous. Moreover, by making he assumpion ha λu e u asds du M for some posiive consan M, one concludes ha all soluions of Eqn. 1.3 are uniformly bounded. For more on his discussion, we refer he reader o [10, 1]. Condiion 1.5 is very resricive since he funcion g is bounded above by a funcion of. In his paper, we are ineresed in dynamical sysems ha are similar o Eqn. 1.3 wih he funcion g, x saisfying he less resricive growh condiion g, x λ x n + h, 0, 1.6 where λ, h are coninuous, unbounded and n is a posiive raional number. To beer illusrae our poin, we consider Eqn. 1.3 along wih 1.5 wih n = 1. Now, we consider he Lyapunov funcion Vx= x. Then along soluions of 1.3 we have V x ax + λ x + x + h = a + λ +1x + h, 1.7 where h may be unbounded. If a > 1 + λ and if we le α = a λ 1, hen inequaliy 1.7 is equivalen o V x αv x + h. 1.8

3 Boundedness and Exponenial Asympoical Sabiliy 109 From 1.8, we easily obain he variaion of parameers inequaliy x x 0 e αsds + Thus, he soluions are uniformly bounded provided ha h u e u du αsds M h u e u αsds du. 1.9 for some posiive consan M. We noe ha 1.9 is a major improvemen over 1.4 and consequenly 1.5. Inequaliy 1.9 was easily obained due o inequaliy 1.8. The quesion is, wha if insead of 1.8, we have he differenial inequaliy V x αwx + h, Wx = Vxfor x = Here α :[0, [0, is coninuous and W :[0, [0, is coninuous wih W0 = 0, Wr is sricly increasing, and Wr as r. Such a funcion is called a wedge. In he case ha h of 1.10 is bounded by some posiive consan, hen soluions of 1.3 are bounded. For a reference, we ask he reader o see [1, page 9 13], [4 6,8,1,13] and he references herein. In he las five years, fixed poin heory is being used o overcome some of he difficulies ha one may encouner using Lyapunov funcions or funcionals. One of he major hurdles when using he Lyapunov funcion mehod is he fac ha a may no be big enough in he negaive way so ha a condiion similar o 1.8 or 1.10 is saisfied. As we shall see in Secion 4, our resuls will offer a beer alernaive han he alernaive of fixed poin heory. Thus, his paper will offer new and powerful heorems ha will significanly advance he heory of exisence of soluions in differenial equaions and heir exponenial decay o zero in he case where he differenial equaion in consideraion has unbounded erms including unbounded forcing erms. Our moivaion is o use he negaive magniude of α o offse he unboundedness of he erm h, and hen arrive a new variaional of parameers inequaliies ha will serve as upper bounds on all he soluions of 1.3. For illusraion purpose, we wrie 1.3 as ẋ = ax + g, x + f, 0, 1.11 x = x 0, 0 where a,f and g are coninuous wih a 0, 0 and f is unbounded. By displaying a suiable Lyapunov funcion, one may arrive a he inequaliy V x αwx + M, 1.1 where M is a consan. Bu, his will pu so much weigh on he size of a, which makes i impossible for α 0 o hold for all 0.As we shall show in Example 3.1, he condiion α 0 may no hold for all 0, for he righ choice of fand a despie he fac ha a migh be unbounded. Currenly, he auhor is exending he conen of his paper o funcional differenial equaions wih bounded or unbounded delays.

4 110 Youssef N. Raffoul. Boundedness and Exponenial Asympoic Sabiliy In his secion we use nonnegaive Lyapunov ype funcions and esablish sufficien condiions o obain boundedness resuls on all soluions x of he dynamical sysem subjec o he iniial condiions ẋ = f,x, 0,.1 x = x 0, 0, x 0 R n,. where x R n, f : R + R n R n is a given nonlinear coninuous funcion in and x, where R +. Here R n is he n-dimensional Euclidean vecor space, R + is he se of all nonnegaive real numbers, x is he Euclidean norm of a vecor x R n. Also, in he case f,0 = 0, we obain condiions ha insure he zero soluion of.1 and. is exponenially asympoically sable. For more on boundedness and sabiliy, we refer he ineresed reader o [3, 7, 9, 10, 14]. In he spiri of he work in [11, 1], in his invesigaion, we esablish sufficien condiions ha yield all soluions of.1 and. are uniformly bounded. We achieve his by assuming he exisence of a Lyapunov funcion ha is bounded below and above and ha is derivaive along he rajecories of.1 o be bounded by a negaive definie funcion, plus a posiive coninuous funcion ha migh be unbounded. From his poin forward, if a funcion is wrien wihou is argumen, hen he argumen is assumed o be. Definiion.1. We say ha soluions of sysem.1 are bounded, if any soluion x,,x 0 of.1 and. saisfies x,,x 0 C x 0, for all, where C : R + R + R + is a consan ha depends on and x 0. We say ha soluions of sysem.1 are uniformly bounded if C is independen of. Definiion.. We say V : R n R + is a ype I Lyapunov funcion on R n provided Vx= n V i x i = V 1 x V n x n, i=1 where each V i : R + R + is coninuously differeniable and V i 0 = 0. If x is any soluion of sysem.1 and., hen for a coninuously differeniable funcion we define he derivaive V of V by V : R + R n R +, V, x = V, x + n i=1 V, x f i, x. x i

5 Boundedness and Exponenial Asympoical Sabiliy 111 In [1], he auhor proved he following heorem. Theorem.3. [7] Le D be a se in R n. Suppose here exiss a ype I Lyapunov funcion V : R + D R + ha saisfies λ 1 x p V,x λ x q.3 and V, x λ 3 x r + L.4 for some posiive consans λ 1,λ,λ 3,p,q,r and L. Moreover, if for some consan γ 0 he inequaliy V,x MV r/q, x γ holds for M = λ 3 /λ r/q, hen all soluions of.1 and. ha say in D saisfy { } 1 1/p [ x λ x 0 q + Mγ + L ] 1 p λ 1 M for all. In his paper we are ineresed in proving similar heorems, where he consan L in.4 is replaced by a coninuous funcion β, where β migh be unbounded. Thus, we have he following heorem. Theorem.4. Assume D R n and here exiss a ype I Lyapunov funcion V : D [0, such ha for all, x [0, D: W x Vx φ x,.5 V, x αψ x + β,.6 Vx ψφ 1 V x γ,.7 where W, φ, ψ are coninuous funcions such ha φ,ψ,w : [0, [0,, α, β :[0, [0, and coninuous in, ψ, φ and W are sricly increasing, γ is nonnegaive consan. Then all soluions of.1 and. ha say in D saisfy x W 1{ V,x 0 e αsds 0 + γ αu + βu e } u αsds du for all. Proof. Le x be a soluion o.1 and. ha says in D for all 0. Consider d [ ] e αsds 0 V,x = V, x + αv, x e αsds 0 d αψ x + β + αv, x e αsds 0, by.6 αψφ 1 V, x + β + αv, x e αsds 0 [ ] = α V,x ψφ 1 V, x + β e αsds 0, by.5 γ α + β e αsds 0, by.7.

6 11 Youssef N. Raffoul Inegraing boh sides from o wih x 0 = x, we obain, for [,, u V,xe αsds 0 V,x 0 + γ αu + βu e αsds 0 du. I follows ha V,x V,x 0 e αsds + for all [,. Thus by.5, we arrive a he formula x W 1{ V,x 0 e αsds + for all. This concludes he proof. γ αu + βu e u αsds du γ αu + βu e } u αsds du We now provide a special case of Theorem.4 for cerain funcions φ and ψ. Theorem.5. Assume D R n and here exiss a ype I Lyapunov funcion V : D [0, such ha for all, x [0, D: x p V,x x q,.8 V, x α x r + β,.9 V,x V r/q, x γ,.10 where α and β are nonnegaive and coninuous funcions, p, q, r are posiive consans, γ is nonnegaive consan. Then all soluions of.1 and. ha say in D saisfy x for all. {V,x 0 e αsds + γ αu + βu e } 1/p u αsds du.11 Proof. Le x be a soluion o.1 and. ha says in D for all 0. Consider d [ ] e αsds 0 V,x = V, x + αv, x e αsds 0 d α x r + β + αv, x e αsds 0. From condiion.5 we have x q V,x,and consequenly x r V r/q, x. Thus, by using.7 we ge d e αsds 0 V,x αv, x V r/q, x + β e αsds 0 d αγ + β e αsds 0.

7 Boundedness and Exponenial Asympoical Sabiliy 113 By inegraing boh sides from o, he res of he proof follows along he lines of he proof of Theorem.4. This concludes he proof. The nex heorem is an immediae consequence of Theorem.5 and hence we omi is proof. Theorem.6. Assume D R n and here exiss a ype I Lyapunov funcion V : D [0, such ha for all, x [0, D: x p V,x,.1 V, x αv q, x + β,.13 V,x V q, x γ,.14 where α and β are nonnegaive and coninuous funcions, p, q are posiive consans, γ is nonnegaive consan. Then all soluions of.1. ha say in D saisfy x {V,x 0 e αsds + γ αu + βu e } 1/p u αsds du.15 for all. From formula.11 we deduce a wealh of informaion regarding he qualiaive behavior of all soluions of.1 and.. Bu firs, we make he following definiion. Definiion.7. Suppose f,0 = 0. We say he zero soluion of.1,., 0, x 0 R n,isα-exponenially asympoically sable if here exiss a posiive coninuous funcion α such ha αsds as and consans d and C R + such ha for any soluion x,,x 0 of.1,. x,,x 0 C x 0, e d αsds 0 for all [,. The zero soluion of.1 is said o be α-uniformly exponenially asympoically sable if C is independen of. Corollary.8. Assume eiher of he hypohesis of Theorem.5 or Theorem.6 hold. i If γ αu + βu e u αsds du M, 0.16 for some posiive consan M, hen all soluions of.1 and. are uniformly bounded.

8 114 Youssef N. Raffoul ii If for some posiive consan M, and f,0 = 0, u γ αu + βu e αsds 0 du M.17 αsds as for all,.18 hen he zero soluion of.1 is α-exponenially asympoically wih d = 1/p. Proof. Le x be a soluion o.1,. ha says in D for all 0. Hence he proof of i is an immediae consequence of inequaliy.16. For he proof of ii, we consider he inequaliy from he proof of Theorem.4 V,xe αsds 0 V,x 0 + u γ αu + βu e αsds 0 du for all [,. This yields V,x [ u ] V,x 0 + γ αu + βu e αsds 0 du e αsds 0 for all [,. Using.5 we have x [ u ] V,x 0 + γ αu + βu e αsds 1/pe 0 du 1 p αsds 0. This concludes he proof. Remark.9. If f,0 = 0 and.17 and.18 hold, hen all soluions of.1 decay exponenially o zero. 3. Examples In his secion we give hree examples as applicaion o our heorems, where a imes, we consider β o be unbounded. Example 3.1. For a 0, we consider he scalar semi-linear differenial equaion x = a + 7 x + bx 1/3 + h, x = x 0,

9 Boundedness and Exponenial Asympoical Sabiliy 115 Le V,x= x. Then along soluions of 3.1 we have V, x = xx = a + 7 x + bx 4/3 + xh 6 a + 7 x + b x 4/3 + x + h To furher simplify 3., we make use of Young s inequaliy, which says for any wo nonnegaive real numbers w and z, wehave wz we e + zf f, wih 1 e + 1 f = 1. Thus, for e = 3/ and f = 3, we ge [ 1 b x 4/3 3 b 3 + x4/3 3/ ] 3/ = 4 3 x + 3 b 3. By subsiuing he above inequaliy ino 3., we arrive a V, x ax + 3 b 3 + h = αx + β, where α = a and β = 3 b 3 + h. One can easily check ha condiions.8.10 of Theorem.5 are saisfied wih p = q = r = and γ = 0. Le a = /, b = 1/3 and h = 1/. Then condiion.16 implies ha γ αu + βu e u αsds du = 5 ue u sds du 5 3 3, 0. Hence, condiion.16 is saisfied, which implies ha all soluions of x = + 7 x + 1/3 x 1/3 + 1/, x = x 0, are uniformly bounded. On he oher hand, if we le a = 1, b = e k 3 and h = 0, 0 and k>1, hen we have α = 1 and β = 3 e k. One can easily see ha condiions.17 and.18 are saisfied, and hence he zero soluion of x = 5 3 x + e k 3 x 1/3, x = x 0, is 1-exponenially asympoically sable.

10 116 Youssef N. Raffoul Nex, for he sake of comparison, we will use he same Lyapunov funcion and he same values for a, b and h, as in Example 3.1, and ry o ge a differenial inequaliy similar o 1.8 where he condiion α 0 will no hold for all 0. Suppose b p 1 q 1 and h p q for some posiive unbounded funcions p 1, p and posiive consans q 1 and q. Le V,xbe as in Example 3.1. Then along he soluions of 3.1 we have V, x x + p 1 q 1 x 4/3 + p q x a a Also, by making use of Young s inequaliy, we have x + p 1 q 1 x 4/3 + p x + q. p 1 q 1 x 4/3 4 3 p3/ 1 x + 3 q3 1. Wih his in mind, we have [ V, x a ] 3 p3/ 1 + p x + 3 q3 1 + q αx + M, where α = a p3/ 1 p, and M = 3 q3 1 + q is a posiive consan. Le a = /, b = 1/3 and h = 1/. Then α = a p3/ 1 p = / < for >. Tha is, α > 0 for >. Thus, condiion 1.1 canno hold. We conclude ha he curren available lieraure which makes use of condiions similar o 1.8 canno be applied o our example. Ye, according o our heorems, he soluions are uniformly bounded. In Example 3.1, condiion.10 did no come ino play, which was due o he fac ha r = q =. In he nex example, we consider a nonlinear sysem in which condiion.3 naurally comes ino play. Example 3.. Le D ={x R : x 1}. For a 5 1 consider he nonlinear differenial equaion and coninuous h, x = ax 3 + bx 1/3 + h, 0, x0 = 1.

11 Boundedness and Exponenial Asympoical Sabiliy 117 Consider he Lyapunov funcional V,x : R + D R + such ha V,x = x. Then along soluions of he differenial equaion we have V = xx = ax 4 + bx 4/3 + xh ax 4 + b x 4/3 + x h. 3.5 Using Young s inequaliy wih e = 3 and f = 3/, we ge x 4/3 b x b 3/. By a similar argumen we have Hence x h x4 + 3 h 4/3. V, x a + 5 x b 3/ + 3 h 4/3 = αx 4 + β, where α = a 5 6 and β = 3 b 3/ + 3 h 4/3. Hence, we have p = q = and r = 4. Thus, for x D V,x V r/q, x = x 1 x 0. Thus, condiion.3 is saisfied for γ = 0. Thus, by inequaliy.11 we have x {e as 5 6 ds + 3 bu 3/ + 3 hu 4/3 e } u as 5 1/p 6 ds du for all. We see ha if he righ side of he above inequaliy is uniformly bounded, hen all soluions ha are in D are bounded. As a maer of fac, we have ha every soluion x wih x D saisfies 1 x { e as ds + for all. 3 bu 3/ + 3 hu 4/3 e u as 5 6 ds du} 1/p, 3.6

12 118 Youssef N. Raffoul Le a = + 5 6, b = /3 and h = 3/4. Then inequaliy 3.5 implies ha all soluions of he nonlinear differenial equaion x = + 5 x + /3 x 1/3 + 3/4, x = 1, 0 6 saisfy { 1 x } 1/ for all. 6 As an applicaion of Theorem.6, we furnish he following example. Example 3.3. Le D ={y 1,y R : y1 + y following wo dimensional sysem }. For a > 0, consider he y 1 = y ay 1 y 1 + y 1h y 1 y = y 1 ay y + y h 1 + y y 1 = a 1, y = b 1 for 0 such ha a1 + b 1 =. Le us ake V,y 1,y = 1 y 1 + y. Then V, y 1,y = ay1 y 1 ay y + y 1 h y1 + y h 1 + y a y y 3 + h 1 + h [ y1 3 = a = a + y 3 [ y 1 3/ ] + h 1 + h + y 3/ ] + h 1 + h a y 1 + y 3/ 3/ + h 1 + h = av 3/, y 1,y + β, a + b l where we have used he inequaliy al + bl, a,b > 0, l>1. Thus, we have ha p = 1 and q = 3/. To verify.19, we noe ha for y = y 1,y D, we have Vy V q y = y 1 + y 1 y 1 + y 1/ 0.

13 Boundedness and Exponenial Asympoical Sabiliy 119 Hence, for γ = 0, α = a and β = h 1 + h, all he condiions of Theorem.6 are saisfied. We conclude ha all soluions of he above wo dimensional sysem ha are in D saisfy, y for all. {e asds + h 1 u + h u e } 1/p u asds du 4. Comparison Recenly, fixed poin heory has been revived o alleviae some of he resricions on he coefficien α, ora in When using Lyapunov funcion of funcionals mehod, he condiion ha a 0 for all is a poinwise condiion. In several papers, Buron used fixed poin heory o sudy he sabiliy of he rivial soluion and boundedness of soluions of funcional differenial equaions. The purpose was o relax he poinwise condiion on a and replace i wih a weaker condiion which is he average asds 0. In paricular, Buron and Furumochi, [], used he conracion mapping principle and showed ha under cerain condiions, all he soluions of he delay differenial equaion ẋ = ax + bgx h + f, 0, 4.1 are bounded. They make he remark ha regular Lyapunov funcional argumens will no work unless he funcions a, b, and f are bounded. For he sake of comparison, we consider he nonlinear differenial equaion ẋ = ax + bgx + f, 0, x = x 0, 0, 4. where all funcions are coninuous on heir respecive domains. We adjus [, Theorem 9.1], which is abou he soluions of 4.1 so ha i can be applied o 4. and hen compare i wih our resuls. Firs we make he following assumpions. Suppose here are posiive consans M,L,K,µ wih 0 0 e s audu fs ds M, 4.3 e s audu bs ds L, 4.4 gx gy µ x y, 4.5 µlk + M + 1 <K, 4.6

14 10 Youssef N. Raffoul and for each 1 > 0 and ε>0 here exiss > 1 such ha > implies e 0 audu <εand e 1 audu <ε. 4.7 Theorem 4.1. If hold, hen all soluions of 4. are bounded for large. The proof of Theorem 4.1 follows along he lines of he proof of [, Theorem 9.1] and hence we omi i. Insead, we make he following observaions. 1 In relaion o equaion 3.3 we have gx = x 1 3 and f = 1. One can easily see ha our gx canno saisfy condiion 4.5. Acually, gx = x 1 3 is no even differeniable a x = 0. Condiion 4.6 implies ha µl < 1 which may resric he ype of funcions g ha can be considered. 3 Our mehod will no work unless we ask ha a 0 for all, a condiion ha Buron replaces wih an averaging one audu. In conclusion, if we consider equaions ha are similar o 1.11, hen our mehods in his paper will handle cases when he funcion f is unbounded provided ha he coefficien a is large enough in he negaive way and when g is no Lipschiz. Thus, we have shown ha our resuls improve he exising resuls in he lieraure when using he radiional way by consrucing a suiable Lyapunov funcion or funcional. References [1] T. A. Buron. Sabiliy and periodic soluions of ordinary and funcional-differenial equaions, volume 178 of Mahemaics in Science and Engineering, Academic Press Inc., Orlando, FL, [] T. A. Buron and Tesuo Furumochi. Fixed poins and problems in sabiliy heory for ordinary and funcional differenial equaions, Dynam. Sysems Appl., 101:89 116, 001. [3] D. N. Cheban. Uniform exponenial sabiliy of linear periodic sysems in a Banach space, Elecron. J. Differenial Equaions, pages No. 3, 1 pp. elecronic, 001. [4] R. D. Driver. Ordinary and delay differenial equaions. Springer-Verlag, NewYork, Applied Mahemaical Sciences, Vol. 0. [5] Rodney D. Driver. Exisence and sabiliy of soluions of a delay-differenial sysem, Arch. Raional Mech. Anal., 10:401 46, 196. [6] John V. Erhar. Lyapunov heory and perurbaions of differenial equaions, SIAM J. Mah. Anal., 4:417 43, 1973.

15 Boundedness and Exponenial Asympoical Sabiliy 11 [7] Jack Hale. Theory of funcional differenial equaions. Springer-Verlag, New York, second ediion, 1977, Applied Mahemaical Sciences, Vol. 3. [8] Philip Harman. Ordinary differenial equaions, John Wiley & Sons Inc., New York, [9] Shigeo Kaō. Exisence, uniqueness, and coninuous dependence of soluions of delay-differenial equaions wih infinie delays in a Banach space, J. Mah. Anal. Appl., 1951:8 91, [10] N. M. Linh and V. N. Pha. Exponenial sabiliy of nonlinear ime-varying differenial equaions and applicaions, Elecron. J. Differenial Equaions, pages No. 34, 13 pp. elecronic, 001. [11] Allan C. Peerson and Youssef N. Raffoul. Exponenial sabiliy of dynamic equaions on ime scales, Adv. Difference Equ., : , 005. [1] Youssef N. Raffoul. Boundedness in nonlinear differenial equaions, Nonlinear Sud., 104: , 003. [13] T. Yoshizawa. Sabiliy heory and he exisence of periodic soluions and almos periodic soluions, Springer-Verlag, New York, 1975, Applied Mahemaical Sciences, Vol. 14. [14] Taro Yoshizawa. Sabiliy heory by Liapunov s second mehod, Publicaions of he Mahemaical Sociey of Japan, No. 9. The Mahemaical Sociey of Japan, Tokyo, 1966.