A Note on the Qualitative Behavior of Some Second Order Nonlinear Equation

Transcription

1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol. 8, Issue (December 3), pp Applicaions and Applied Mahemaics: An Inernaional Journal (AAM) A Noe on he Qualiaive Behavior of Some Second Order Nonlinear Equaion Juan E. Nápoles Valdes UNNE, FaCENA, Maemáicas Av. Liberad 545, (34) Corrienes, Argenina UTN-FRRE, French 44 Maerias Básicas, (35) Resisencia, Chaco, Argenina jnapoles@exa.unne.edu.ar, jnapoles@frre.un.edu.ar Received: Augus 7, 3; Acceped: November 6, 3 Absrac In his paper, we presen wo qualiaive resuls concerning he soluions of some second order nonlinear equaions, under suiable assumpions. The firs resul ceners on he boundedness of he soluions while he second discusses he square inegrabiliy of he soluions. These resuls are obained by exending and improving he curren lieraure hrough sound mahemaical analysis. Keywords: Bounded, L -soluions, square-inegrable, asympoic behavior AMS-MSC No: 34C. Inroducion We consider here he equaion (p()x ) +f(,x,x )x +a()g(x)=q(,x,x ), () under he following condiions: 767

2 768 Juan E. Nápoles Valdes i) p and q are coninuous funcions on I:=[,+ ) such ha < p p() <+ and < a a() a <+. ii) f is a coninuous funcion on IxR saisfying < f f(,x,x ). iii) g is also a coninuous funcion for all x such ha xg(x)> for x and g ( x) dx. iv) q(, x, x') e( ), where e() is a non-negaive and coninuous funcion of saisfying e ( ) d M and M is a consan. We shall deermine sufficien condiions for he boundedness and he L properies of he soluions of equaion (). Our approach differs from hose of he earlier research as all hey consruced energy or Liapunov Funcions; so, our resuls differ significanly from hose obained previously, see some aemps in ha sense in Kroopnick (8) and Tunc (a), and references cied herein. The soluions of equaion () are bounded if here exiss a consan K> such ha x()< K for all T > for some T. By an L -soluion, we mean a soluion of equaion () such ha x ( ) d. In he las four decades, many auhors have invesigaed he Liénard equaion x +f(x)x +g(x)=. () They examined some qualiaive properies of he soluions. The book of Sansone and Coni (964) conains an almos complee lis of papers dealing wih hese equaions as well as a summary of he resuls published up o 96. The book of Reissig, Sansone and Coni (963) updaes his lis and summary up o 96. The lis of he papers which appeared beween 96 and 97 is presened in he paper of Graef (97). Among he papers published in he las years are Buron (965); Buron and Townsend (968); Hara and Yoneyama (985); Hricisakova (993); Kao (988); Nápoles (); Omari, Villari and Zanolin (987); Tunc (a); Villari (983); and Villari (987). If in () we make p(), q(,x,x ), f(,x)=f(x) and a(), hen equaion () becomes equaion () so, every qualiaive resul for he equaion () produces a qualiaive resul for (). The paper is organized as follows: in we sae and prove our resuls on boundedness and L properies of soluions of (); in he 3 we presen some applicaions of our resuls o show he advanages over hose repored in he lieraure and in 4 we reflec on a paricular case of Theorem, an open problem, a simple example of he necessary crierion for posiiviy of he funcion f and some relaions wih resuls obained recenly.

3 AAM: Inern. J., Vol. 8, Issue (December 3) 769. Resuls We now sae and prove a general boundedness heorem. Wihou loss of generaliy, we shall assume. Theorem. Assuming ha condiions i)-iv) above holds, hen any soluion x() of (), as well as is derivaive, is bounded as and x ' ( ) d Proof: By sandard exisence heory, here is a soluion of () which exiss on [,T) for some T>. Muliply he equaion () by x and inegrae from o and apply he assumpions i) and iv) we obain x( s) x'() (, (), '()) '() () () '(). (3) x'( ) p f s x s x s x s dsa g u du p e s x s ds x() Now if x() becomes unbounded hen i follows ha all erms on he lef hand side of () are posiive from our hypoheses. By he Cauchy-Schwarz inequaliy for inegrals on he righ hand side of (3), we ge ( ) x s x'() x'( ) p f (, s x(), s x '()) s x '() s ds a g() u du p e () s ds x '() s ds. x() Now, le X ( ) x' ( s) ds. Dividing boh sides by X() yields x( s) x'( ) X () p f(, s x(), s x'()) s x'() s ds a g() u du x() x '() X () p e ( s) ds. (4) Taking ino accoun he posiiviy of he lef hand side of (4) if x() increase wihou bound and he erm X ( ) f x' ( s) ds f x s ds ' ( )

4 77 Juan E. Nápoles Valdes is bounded by he righ hand side of equaion (4) we obain ha x is square inegrable and is also bounded afer we examine he firs erm of he lef hand side of (4). However, he above implies ha x() mus be bounded. Oherwise, he lef hand side of (4) becomes infinie which is impossible. A sandard argumen now permis he soluion o be exended o all of I, see for example Boudonov (nodaed); Reissig, Sansone and Coni (963) and Sansone and Coni (964). The proof is hus complee. By imposing more sringen condiions on g(,x) and p(), all soluions become L -soluions. This case is covered by he following resul. Theorem. Under hypoheses of Theorem, we suppose ha g(x)x >g x for some posiive consan g >, and < p < p() < P <+, hen all he soluions of equaion () are L -soluions. Proof: In order o see ha xl [,) we mus firs muliply equaion () by x, he inegraion from o yields ( ( ) ') ( ) ' ( ) (,, ') ( ) '( ) ( ) ( ) ( ( )) x px psx sds f xx xsx sds xsasgxs ds x( ) Nex, le zf ( x where x() x() p() x'() q(( s, x( s, x'( s)) x( s) ds. ( z), z, z') dz F( x). So, he above equaion may be rewrien as px ( ) x'( ) P x' ( s) ds F( x) a g x ( s) ds K, (5) K P x( ) x'() e( s) x( s) ds. Noice ha he las erm is bounded by e ( s) ds x ( s) ds by using he Cauchy-Scwharz inequaliy. Dividing he lef hand side of (9) by

5 AAM: Inern. J., Vol. 8, Issue (December 3) 77 M ( ) e( s) x( s) ds and using he hypoheses of Theorem we obain M () px() x '() Px '() s ds F() x agx () s ds px() x'() e M () () s ds. (6) Since he righ hand side of () is bounded and all he erms of he lef hand side are eiher bounded or posiive, he resul follows because he lef hand side canno be unbounded. Here, we need ha x is square inegrable. 3. Some Applicaions In his secion our resuls are applied o some repored in he lieraure. If in () he funcions involved saisfies p(), e(), f(,x,x ) f and g(x)=g x, from assumpions ii) and iii) of Theorem we obain ii ) f >, iii ) g >. Then, hese assumpions amoun o he usual Rouh-Hurwiz crierion (see Boudonov). In Nápoles (999) he auhor proved for he generalized Liénard equaion () wih resoring erm h(), he following resul: Theorem. We assume ha gc(r), wih limi a infiniy and g(-)<g(x)<g(+), xr. In addiion, eiher pv, g(-)<p()<g(+), or pl (I), g(-)=-, g(+)=g(+), where V T { h L ( I) : hm Lim h( ) d T uniformly in }; denoing by h m he medium value of h, hen () has a soluion in W, (R). Also, >, > such ha for any soluion x() of () wih x( )+ x ( ), for some I, hen x()+x (),.

6 77 Juan E. Nápoles Valdes This resul is easily obained from our Theorem. In Repilado and Ruiz (985) and Repilado and Ruiz (986) he auhors sudied he asympoic behaviour of he soluions of he equaion x +f(x)x +a()g(x)= (7) under he following condiions: a) f is a coninuous and nonnegaive funcion for all xr, b) g is also a coninuous funcion wih xg(x)>, for x, and c) a() >, for all I and ac. In paricular in Repilado and Ruiz (986), he following resul is proved: Theorem. Under condiions. a ( ) d. a'( ). d, a () - = max{-a (),}. a( ) 3. There exiss a posiive consan N such ha G(x) N for x(-,), where x G( x) g( s) ds, all soluions of equaion (6) are bounded if and only if a ( ) f [ k( )] d, (8) for all k > and some. The firs resul of his naure was obained by Buron and Grimmer (97) when hey showed ha all coninuable soluions of equaions x +a()f(x)= under condiion b) and c) are oscillaory (and bounded) if and only if he condiion (8) is fulfilled, wih f insead of g. I is easy o obain he sufficiency of he above resul from our Theorem. If in () we ake f(,x), e() and g(,x)=g()x, our resul becomes Theorem of Nápoles and Negrón (996), referen o boundedness of x() and p()x (), for all a wih a some posiive consan. Casro and Alonso (987) considered he special case x +h()x +x=, (9)

7 AAM: Inern. J., Vol. 8, Issue (December 3) 773 of equaion () under condiion hc (I) and h() b >. Furher, hey required ha he condiion ah ()+h() 4a be fulfilled, and obained various resuls on he sabiliy of he rivial soluion of (8). I is clear ha all assumpions of Theorem are saisfied. Thus, we obain a consisen resul under milder condiions. This resul complees hose of Ignaiev referred o in equaion (5), see Ignaiev (997), wih resoring erm x +f()x +g()x=h(), () Taking h() coninuous on I (in Ignaiev s resuls h ) such ha h ( ) d and f()>f >, g()>g > wih coninuous nonposiive derivaives we have ha all he soluions of (), as well as heir derivaives, are bounded and in L (I). Our resuls conain and improve hose of Ruiz (988), obained wih h, referring o he boundedness of he soluions of equaion x +f()x +a()g(x)=h(), because he auhor used regulariy assumpions on funcion a(), which are no used here. In Kroopnick (8) he auhor discussed he boundedness and L characer of equaion () wih f(,x,x )=c()f(x) and p(). Thus, our resuls conain hose of Kroopnick. Tunc sudied he boundedness of he soluions of equaion () under derivabiliy assumpions on p() and a(), see Tunc (a). Taking ino accoun he resuls obained above, hese complee and improve Tunc s work. The resuls obained in his paper are consisen wih hose of Shao and Song () where he auhors sudy he sublinear equaion (a()y ) +b()y k =, wih regulariy assumpions on r(). 4. Conclusions A he end of his paper and o compare our resuls wih several of he references, we give in his secion a paricular case of Theorem, an open problem, a simple example of he necessary characer of posiiviy of funcion f and some concluding remarks. A paricular case. Ignaiev (997) considered an oscillaor described by he following equaion x +f()x +g()x=, ()

8 774 Juan E. Nápoles Valdes where he damping and rigidiy coefficiens f() and g() are coninuous and bounded funcions. If in equaion () we pu p(), e(), f(,x)=f() and g(,x)=g()x, hen we improve he Theorem of Ignaiev, since he assumpion g ( ) f ( ), g( ) is no necessary, and f()< M, g()< M, g ()< M 3, is dropped. Under he above remarks, he Ignaiev s Corollary is obvious. An open problem. Taking ino accoun he Applicaion relaed o Theorem of Repilado and Ruiz (986), and Theorem of same reference, raises he following open problem Under which addiional hypoheses, he assumpion g( x) dx condiion for boundedness of he soluions of equaion ()? is a necessary and sufficien This is no a rivial problem. The resoluion implies obaining a necessary and sufficien condiion for compleing he sudy of asympoic naure of soluions of (). On he posiiviy of f. Under assumpions f(,x,x ) f > for some posiive consan f, he class of equaion () is no very large, bu if his condiion is no fulfilled, we can exhibi equaions ha have unbounded soluions. For example e ' x' ( )e 3 3 3, has he unbounded soluion x()=e and f(,x,x ). Final Remarks. Tunc () esablished some new sufficien condiions which guaranee he boundedness of soluions of non-linear differenial equaion: x +f(,x,x )g(,x,x )+b()h(x)=e(,x,x ), ()

9 AAM: Inern. J., Vol. 8, Issue (December 3) 775 where b, f, g, h and e are coninuous funcions in heir respecive domains and b () exiss and is coninuous. Under hese assumpions he proved he following resul. Theorem. In addiion o he basic assumpions imposed on he funcions b, f, g, h and e ha appear in equaion (), we assume ha here exiss a posiive consan such ha he following assumpions hold: i) f(,x,x )g(,x,x ), for all R + :=[,+) and x,yr, b(), b (), for all R +, ii) e(, x, y) p( ), wih p( s) ds. Then all soluions of equaion () are bounded. h ( x) x, for all x. Clearly his heorem and our resuls are consisen and complemen each oher and hey have differen applicaions. Equaion () conains a general funcion p() while in equaion () p(), he forcing erm of boh are similar, he funcion a() in equaion () is wider han ha p() in (), he consideraion of he funcion h(x) in () is more resricive han ours on g(x), alhough he erm resorer of equaion () is more general han ours. The same auhor exends and complees several known resuls on boundedness and sabiliy, o he case of funcional differenial equaions of various ypes, which illusraes a work address very promising (see Tunc (b); Tunc (a) and Tunc (b)). Acknowledgmens The auhor wish o hank he referees and edior for he valuable suggesions and commens which have resuled in a grea improvemen of his paper. REFERENCES Boudonov, N. (no daed). Qualiaive heory of ordinary differenial equaions, Universidad de la Habana, Cuba (In Spanish). Buron, T. A. (965). The generalized Liénard equaion, SIAM J. Conrol SerA#, 3-3. Buron, T. A. and Grimmer, R. (97). On he asympoic behaviour of soluions of x +a()f(x)=, Proc. Camb. Phil. Soc. 7, Buron, T. A. and Townsend, C.C. (968). On he generalized Liénard equaion wih forcing erm, J. Differenial Equaions 4, Casro, A. and Alonso, R. (987). Varians of wo Salvadori s resuls on asympoic sabiliy, Revisa Ciencias Maemáicas VIII, (In Spanish). Graef, J. R. (97). On he generalized Liénard equaion wih negaive damping, J. Differenial Equaions, 34-6.

10 776 Juan E. Nápoles Valdes Hara, T. and Yoneyama, T. (985). On he global cener of generalized Liénard equaion and is applicaion o sabiliy problems, Funkcial. Ekvac., 8, 7-9. Hricisakova, D. (993). Exisence of posiive soluions of he Liénard differenial equaion, J. Mah. Anal. And Appl. 76, Ignaiev, A. O. (997). Sabiliy of a linear oscillaor wih variable parameers, Elecronic J. Differenial Equaions 7, -6. Kao, J. (988). Boundedness heorems on Liénard ype differenial equaions wih damping, J. Norheas Normal Universiy, -35. Kroopnick, A. (8). Properies of soluions o a generalized Liénard equaion wih forcing erm, Applied Mah., E-Noes, 8, Nápoles, J. E. (999). On he global sabiliy of non-auonomous sysems, Revisa Colombiana de Maemáicas, 33, -8. Nápoles, J. E. (). On he boundedness and global asympoic sabiliy of he Liénard equaion wih resoring erm, Revisa de la UMA, 4, Nápoles, J. E. and Negrón, C. (996). On behaviour of soluions of second order linear differenial equaions, Revisa Inegración, 4, Omari, P.; Villari, C. and Zanolín, F. (987). Periodic soluions of he Liénard equaion wih one-side growh resricions, J. Differenial Equaions 67, Reissig, R., Sansone, C. and Coni, R. (963). Qualiaive heorie nichlinearer differenialgleichungen, Edizioni Cremonese, Rome. Repilado, J. A. and Ruiz, A.I. (985). On he behaviour of soluions of differenial equaion x +g(x)x +a()f(x)=(i), Revisa Ciencias Maemáicas, VI, 65-7 (In Spanish). Repilado, J. A. and Ruiz, A.I. (986). On he behaviour of soluions of differenial equaion x +g(x)x +a()f(x)= (II), Revisa Ciencias Maemáicas, VII, (In Spanish). Ruiz, A. I. (988). Behaviour of rajecories of nonauonomous sysems of differenial equaions, Universidad de Oriene, Cuba (In Spanish). Shao, Jing and Song, Wei (). Limi circle/limi poin crieria for second order sublinear differenial equaions wih damping erm, Absrac and Applied Analysis, Aricle ID Sansone, C. and Coni, R. (964). Nonlinear differenial equaions, MacMillan, New York. Tunc, C. (). Boundedness resuls for soluions of cerain nonlinear differenial equaions of second order, J. Indones. Mah. Soc. Vol. 6 No., 5-6. Tunc, C. (a). On he boundedness of soluions of a non-auonomous differenial equaion of second order, Sarajevo J. Mah., 7(9), 9-9. Tunc, C. (b). Uniformly sabiliy and boundedness of soluions of second order nonlinear delay differenial equaions, Appl. Compu. Mah.,, no.3, Tunc, C. (a). On he sabiliy and boundedness of soluions of a class of nonauonomous differenial equaions of second order wih muliple deviaing argumens, Afrika Maemaika 3, no., Tunc, C. (b). On he inegrabiliy of soluions of non-auonomous differenial equaions of second order wih muliple variables deviaing argumens, J. Compu. Anal. Appl., 4, no. 5, Villari, C. (983). On he exisence of periodic soluions for Liénard s equaion, Nonlin. Anal., 7, Villari, C. (987). On he qualiaive behavior of soluions of Liénard equaion, J. Differenial Equaion 67,