Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick Received 3 February 27 Absrac This paper discusses under wha condiions he soluions o a generalized Liénard equaion x + c()f(x)x + a()g(x) e() are bounded on [, ) wih specified condiions on c, f, a, g and e. Specifically, we shall show ha all soluions are bounded wheher e is bounded or absoluely inegrable. In he bounded case, however, we shall require ha e mus be of fixed sign along wih he condiion ha e ()a() e()a (). Finally, a brief discussion of L p -soluions is given under somewha more resricive condiions. In his noe we will sudy in deail some new resuls concerning he global properies of a generalized Liénard equaion of he form x + c()f(x)x + a()g(x) e(). () This equaion has been well-sudied and he resuls here exend he resuls of Fonda and Zanolin, Kroopnick, and Nkashama (see [2-6] as well as heir excellen liss of references) in which he auhors assumed a periodic forcing erm in [] and in [2] he forcing erm was or absoluely inegrable and he condiions on c and a were somewha less general. For our purposes, he forcing erm will eiher be absoluely inegrable or bounded. I is he bounded case which is new. For our firs resul we will assume ha: (a) c is a coninuous, non-negaive funcion for. (b) f is coninuous on R and non-negaive. (c) a is posiive on [, ) such ha a() >a > and a () on[, ). (d) g is coninuous on R such ha G(x) x gds as O ( x +δ) where δ>, and (e) e is coninuous on [, ) and x e ds <. Noice ha we do no require ha xg(x) > for all x. If he above condiions hold, all soluions as well as heir derivaives are bounded as. We now begin our analysis. Firs, using sandard exisence heory, we may conclude ha he soluions o () are local [, Chaper 6]. If we can show ha he soluions remain bounded hen we may conclude global exisence of all soluions [, pp.384-396]. Mahemaics Subjec Classificaions: 34C Universiy of Maryland Universiy College, 35 Universiy Boulevard, Adelphi, Maryland 2783, USA 4
A. J. Kroopnick 4 In order o see his, firs muliply equaion () by x () and hen inegrae from o where we inegrae by pars he hird erm on he LHS of () obaining 2 (x ()) 2 + cf(x) (x ) 2 ds + a()g(x() a G(x)ds ex ds + 2 (x ()) 2 + a()g(x()). (2) Using he fac ha e is absoluely inegrable we see ha 2 (x ()) 2 + cf(x) (x ) 2 ds + a()g(x() a G(x)ds e x ds + 2 (x ()) 2 + a()g(x()). (3) Applying he mean value heorem for inegrals o he erm e x ds ransforms equaion (3) ino 2 (x ()) 2 + x ( ) cf(x) (x ) 2 ds + a()g(x()) a G(x)ds e ds + 2 (x ()) 2 + a()g(x()) (4) where < <.Should ( x or x become unbounded, hen, by our hypoheses, he LHS approaches as O x +δ) ( and O x 2) while he RHS approaches infiniy as O ( x ). Since his is impossible, we have ha boh x and x mus say bounded. This firs resul we will call Theorem I. Nex we sae Theorem II. The only difference in our hypohesis is a (). Also, he derivaives are no guaraneed o be bounded. THEOREM II. The hypoheses are idenical o Theorem I excep ha condiion (c) changes o a (). Under hose condiions, all soluions o () are bounded. If a() is bounded from above by a posiive consan A, hen he derivaives, oo, are bounded. PROOF. In his case we muliply () by x ()/a() and inegrae from o where we inegrae by pars only he firs erm on he LHS of () and proceeding as before we obain (x ()) 2 + x 2 a() 2 (a ) 2 ds + cf(x) (x ) 2 ds + G(x()) (x ()) 2 + G(x()) + x ( ) 2 a() e ds. (5) Again, noice ha should eiher x or x becomes infinie, he LHS of (5) would approach faser han he RHS of (5), so he soluions mus remain bounded. Should a() be bounded from above by some consan A, he derivaives, oo, are bounded. We now urn our aenion o he case when e is bounded wih a derivaive of fixed sign. Saemen and proof now follow.
42 Generalized Liénard Equaion Wih Forcing Term THEOREM III. They hypoheses are he same as Theorem I excep for condiion (e). e is a bounded funcion wih a derivaive of fixed sign. In such cases, all soluions are bounded along wih heir derivaives. PROOF. We firs proceed as we did in Theorem I obaining equaion (2). However, we hen inegrae by pars he erm ex ds obaining 2 (x ()) 2 + cf(x) (x ) 2 ds + a()g(x()) a G(x)ds e()x() e()x() e xds + 2 (x ()) 2 + a()g(x()). (6) We now apply he mean value heorem for inegrals o he erm e xds which ransform (6) ino 2 (x ()) 2 + cf(x) (x ) 2 ds + a()g(x() a G(x)ds x()e() x()e() x( )(e() e()) + 2 (x ()) 2 + a()g(x()), (7) where < <. Arguing as before, boh x and x mus remain bounded. Oherwise, he LHS of (7) again becomes infinie faser han he RHS which is impossible. We now sae our final heorem. THEOREM IV. The hypohesis are he same as Theorem III excep ha a () on [, ) and a()e () a ()e() on[, ), hen all soluions are bounded as. Furher if a () is bounded from above by some consan A, hen he derivaives, oo, are bounded. PROOF. We proceed as in Theorem II o obain, (x ()) 2 + x 2 a() 2 (a ) 2 ds + cf(x) (x ) 2 ds + G(x()) (x ()) 2 + G(x()) + 2 a() x e ds. (8) Inegraing by pars he las erm of (8) we see ha (x ()) 2 + x 2 a() 2 (a ) 2 ds + cf(x) (x ) 2 ds + G(x()) (x ()) 2 + G(x()) + x()e() x()e() 2 a() a() a() ( x e ) ds. (9) As before, we apply he mean value heorem for inegrals o he las erm of (9) obaining, (x ()) 2 + x 2 a() 2 (a ) 2 ds + cf(x) (x ) 2 ds + G(x()) (x ()) 2 + G(x()) + x()e() x()e() ( ) e x( ) ds. () 2 a() a() a()
A. J. Kroopnick 43 where < <. Simplifying () by inegraing he las erm yields finally, (x ()) 2 + x 2 a() 2 (a ) 2 ds + x () 2 x()e() + G(x()) + x()e() x( ) 2 a() a() a() cf(x) (x ) 2 ds + G(x()) ( e() a() e() ). () a() Equaion () clearly show ha x remains bounded. Oherwise, he LHS of () approaches faser han he RHS which is a conradicion. Moreover, if A a(), hen he derivaives, oo, say bounded for he same reason. Under somewha more resricive condiion, we can show ha all soluions o () are in L p [, ). Specifically, we assume e is an elemen of L [, ), c() >c >, c () and f(x) >f >, xg(x) > for x, and xg(x) K x p for p 2 and K a posiive consan. We immediaely see ha x is square inegrable from (2). Nex, muliply () by x and inegrae he firs erm by pars geing x()x () x 2 ds + cf(x)xx ds + g(x)xds exds + x()x (). (2) Nex, define F (x) x f(u)udu and inegrae he hird erm of (2) by pars so ha (2) becomes, x()x () x 2 ds + c()f (x()) + g(x)xds c F (x)ds exds + x()x () + c()f (x()). (3) From (3), since all erms on he LHS are posiive and all erms on he RHS are bounded, we may conclude ha x is indeed an elemen of L p [, ) As an example, consider he following equaion x + kx + p(x) C (4) where p(x) is a polynomial of odd degree 2n + and k and C are consans. From our above remarks, all soluions are bounded. Nex, consider he equaion, x +( + sin())x + p(x) C (5) where he consan has been replaced by + sin. All soluions o (5) are bounded.
44 Generalized Liénard Equaion Wih Forcing Term References [] F. Brauer and J. A. Nohel, Inroducion o Ordinary Differenial Equaions wih Applicaions, New York, Harper and Row, 985. [2] A. Fonda and F. Zanolin, Bounded soluions of second order ordinary differenial equaions, Discree and Coninuous Dynamical Sysems, 4(998), 9 98. [3] A. Kroopnick, Noe on bounded L p -soluions of a generalized Liénard equaion, Pacific J. Mah, 94(98), 7 75. [4] A. Kroopnick, Bounded and L p -soluions o a second order nonlinear differenial equaion wih inegrable forcing erm, Iner. Jour. Mah. and Mah Sci., 33(999), 569 57. [5] A. Kroopnick, Bounded and L p -soluions o a second order nonlinear differenial equaion wih inegrable forcing erm, Missouri Journal of Mahemaical Sciences, (998), 5 9. [6] M. N. Nkashama, Periodically perurbed nonconservaive sysems of Liénard ype, Proc. Amer. Mah. Soc., (99), 677 682.