Fonda, Alessandro

Zanolin, Fabio

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeAlessandro Fonda; Fabio ZanolinPeriodic solutions to second order differential equations of Liénard type withjumping nonlinearitiesCommentationes Mathematicae Universitatis Carolinae, Vol. 28 (1987), No. 1, 33--41Persistent URL: http://dml.cz/dmlcz/106507Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1987Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 28.1 (1987) PERIODIC SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS OF LIENARD TYPE WITH JUMPING NONLINEARITIES Alessandro FONDA, Fabio ZANOLIN Abstract; An existence theorem of periodic solutions for the first order differential system x» = y - F(x) + Ej.(t), y' * - g(x) + E2(t) is proved and applications are given to the.scalar Lienard equation x" + f(x)x' + g(x) » ett), under suitable assumptions on the limits lim P(x)/x and lim g(x)/x. In , x-+±°° x-+±" this way, a theorem of S. Pucik, concerning the second order equation x" + g(x) = e(t), is extended to the Lienard equation. Key words: Periodic solutions, Lienard equation, jumping nonlinearities, homogeneous equations, conditions for the existence of a global center. Classification: 34C25, 34 B15 Introduction. In [8], S. Fucfk considered the problem of the existence of periodic solutions to the second order scalar differential equation x" + g(x) -* e(t), assuming that the limits lim g(x)/x « gx , lim g(x)/x » g2 X *+--» X-++00 exist finite. More precisely, he found a critical set K in the (g-,^) -plane such that the existence of periodic solutions (having the same period of •(•) ) is ensured provided that (gi,g2) ff K. For related results, see also 121. Fucik's theorem was further generalized by several authors (see, e.g. (3], [6J, [91» [12], [13] ) either considering conditions on the lim inf and lim sup of g(x)/x, or dealing with more general equations. He refer to [9] and [5] for more details and a more complete bibliography. - 33 -In the present paper, we extend Fuc'ik's result to the differential equation of Lienard ty£e x" + f(x)x* + g(x) * e(t) allowing suitable "jumping" behaviour not only on g but also on f. To this end we prove an existence theorem for a more general first order differential 2 system in » . Different results in this direction were obtained by E.N. Dancer in [4] who also studied the Rayleigh equation x" + F(x') + g(x) = e(t) and, more recently, by P. Drabek and S. Invernizzi in [6] who considered the Duffing equation xw + ex* •-g(t,x) « e(t). Our main argument, which combines topologi-cal degree tools with phase plane analysis of homogeneous equations, provides a new proof of Fuc'ik's theorem in the case f = 0 as well. For the significance of the periodic problem to the Lienard equation and for an exhaustive collection of the most classical results, see, e.g. f j l 7 ] A 1 8 ] . • M a i n results. In tnis paper we deal with the existence of T - periodic solutions to,the following differential system: (1)' x' • y - F(x) + Ex(t) , y» = - g(x) + E2(t) where F, g : K •• H are continuous functions and Ex , E2 : K -+• M are continuous and T-periodic. System (1) is usually considered in the. study of some second order scalar differential equations like, for instance, Lienard or Rayleigh equations. We assume that the limits F-t:-- lim - F(x)/x , F2:= lim F(x)/x X-*-""* X-**+«> gxi~ lim g(x)/x , g2:=- lim g(x)/x x->-*o X->-+<» exist and are finite. Throughout the paper we also suppose that gx and g2 are both positive. The case in which gx and g2 are both negative was already examined in {16] where it was proved that existence of T - periodic solutions to (1) is achieved without any other assumption on P. ISie case in which gx and g2 have opposite sign is contained in [7]. In this case, for any choice of F, thexe always exist forcing terms (E1,E2) such that (1) has no periodic solutions. Finally, if g * 0 for some i - 1,2 , further informations on the behaviour of the function g are needed in order to get results of existence of periodic solutions. Accordingly, this situation is left out in the present discussion. Our main theorem is a generalization of some results obtained by Fuclk et al. (see e.g. [9] ) in the case F = 0. In order to prove our result we need to study the existence of non-trivial T - periodic solutions to the homogeneous system x + + (2) x* = y - Fxx - F2x , y* « gxx - g2x where we have set x t= max{0,-x} , x :• max{0,x} , x = x - x . Lemma 1. System (2) has nontrivial periodic solutions if and only.if tihe following conditions hold: (fe?) Ai := gj, - (Fi/2)2>0 (i - 1, 2) (fc2) Fi/Vgi - F2//g2 . Moreover, system (2) has nontrivial solutions of period T if and only if (fef), (fc2) and (fe3) 2/A1/A2/(I>(/A1 + /A2) e J* hold, with-U) :« 21T/T . Proof. First we observe that a nontrivial periodic solution to system (2) exists if and only if the origin is a global center. lliis is due to the fact that the nonlinearities in the vector field of (2) are positively homogeneous functions. Accordingly, we can use some standard facts about the theory of plane dynamical systems (see [18],[11]). Assume {k1) and {&?}; then, Theorem 4.2 in [11] assures that (2) has a global center at the origin. If (fe2) holds but (fel) does not, Corollary 4.1 in the same paper guarantees the nonexistence of nontrivial periodic orbits. Finally, let us consider the case when (12) does not hold. By Lemma 3.1 in [11], there is a homeomorphism between the prbits of (2) and those of (3) u' - v - (F^^u""" - (FiAg-.iu** , v* * - u - 35 and it is easily seen that if F1//gi ? F2//g2 , (3) does not have any nontrivial closed orbit. The minimal positive period T* of a nontrivial periodic orbit of (2) can be computed by examining separately the cases x(t) SO and x(t) £0 (observe that all the nontrivial periodic solutions of (2) have the same minimal period). (tore precisely, given a closed orbit which intersects the y-axis at the points pi * COjyj) and P2 =- (0,y2) .with yj. > 0 > y2 , by simple calculations we find that the minimal time T x (resp. T2 ) needed to reach Px (resp. P2 ) from P2 (resp. Px ) is 1T//A-, (resp. TT//A2 ). Hence we have T* * Tx + T 2 = 7T(/A1 + /AiJ/ZAi/Az and (fe.3) easily follows. The set of all 4-tuples (F1-F2r9ir92) such that [k1), [k2) and (fe3) hold true will be denoted by K(T). The proof of our main theorem is based on the following abstract existence result (cf. [10 1, [14 ] for the terminology). Lemma 2. Let X, Z be real Banach spaces. We consider the operators: L : dom(L)c x -+ Z is a linear (not necessarily contiguous) Fredholm mapping of index zero, with a continuous generalized inverse K ; P,Q T : X •+ Z is anL - completely continuous operator such that [tl) lim ITXI/IXI - 0? |x|-i-t» S* : [0,l]xx-+Z is at\L - completely continuous operator such that . U?) S*(0,») is linemr, UZ) S*(X,Ux) - US*(X,x) for every Xe[o,l], x e x , U > 0 , (43) for each Xe[0,l], Lx - S*(X,x) if and only if x » 0. Set S « S*(l,«). *nieh, for each z e Z , the operator equation (4) Lx « Sx + Tx + z has at least one solution in dom(L). Proof. It is sufficient to find an a-priori bound for the solutions of eq. - 56 -(4 ) Lx • S*(X,x) + XTx + Xz ( X€[0,U ). In fact, if this is the case, the coincidence degree of the homotopy H(X,x) - Lx - S*(X,x) - XTx - Xz ( Xe[Orl] ) is well defined and nonzero - by {&1), (.43) - on balls with sufficiently large radius (see [ 14, Prop. 11.16, Th. IV.ll ). Suppose, by contradiction, that no a-priori bound exists. Then for each new, there are X e [0,1] and x e &om(L) such that lx I -+» and x is a n n n n solution to (4, ). Then, dividing (4, ) by lx I , we have, by (-62), X X n n n (5A ) L(x / l x | ) » S*(X ,x / l x I ) + X (Tx ) / l x I + X z / l x I :*- y . X n n n n n n n n n n n n Equat ion (5 ) i s e q u i v a l e n t , (cf . [14] ) # t o • A n (6y ) x / | x | « Px / | x | + J Q y + K y , X n n n n n P,Q n n where P: X -+ ker(L), Q: Z •+ coker(L) . are continuous projectors, J: Im(Q) -+ ker(L) is a linear isomorphism and K is the (continuous) generalized P*Q inverse of L associated to the pair (P,Q). Assumption {£1} and the L - complete continuity of the operators imply, by standard arguments, that the set {x /lx I , n n new } is relatively compact in X. Therefore, passing if necessary to subsequen-ces, we can assume X -+ X* e [0,1] , x /lx I •+ x* e x and Ix*I -» 1. Taking the n n n limits in (5 ) - (6 ) as n •+ °o and taking into account (-£?), we obtain n n x* * Px* + JQS*(X*,x*) + K S*(X*,x*), that is x* edom(L) and L&* «- S*CX*,x*). P#Q By (43), x* = 0 an$| the contradiction is achieved. Theorem 1. System (1) has at least one T - periodic solution provided that ih1) (FirF2,glrg2) ¥ K(T) * holds. Proof. In order to use Lemma 2 we set r 0 2 . X - Z - l h - C (l,B);h is T-periodic } , equipped w i t h t h e sup norm | h | « | h | « s u p { | h ( t ) | : O S t S T } , where | • | i s t h e °° - 3 7 -2 e u c l i d e a n norm i n R . 1 2 dom(L) » XHC ( m, B ) . L : ( x ( * ) , y ( ' » ) ) ^ + ( x ' ( * ) , y ' ( 0 ) , T : ( x O , y O ) H^-> ( F - x O + F 2 x O - F ( x O ) , - g -xC)"" + g 2 x O+ - g ( x O ) ) S*(X,«) : ( x O , y ( * ) ) H - ^ ( y ( ' ) - F x ( X ) x ( - ) " " - F 2 ( X ) x O + ^ g 1 ( X ) x C ) " - g 2 ( X ) x O+ ) S * ( l , « ) = S : ( x C ) , y ( ' ) ) N * + ( y ( « ) - F 1 x ( « ) " * - F 2 x ( « ) , g 1 x ( » ) ~ - g 2 x O+ ) z : - ( E X C ) , E 2 C ) ) , where ,\x I 2X<3i + (1 * 2X)a i f O S X S l / 2 en (X) * i 9 1 l g i i f 1/2 <X S I f 2Xg2 + (1 ~ 2X)a i f 0 S X S 1 / 2 9 2 t g 2 i f 1/2 <X SI r vc i f 0 < X < l l / 2 lK l (2X - 1)FX + 2(1 - X)vc i f 1/2 < X ^ 1 x r - v c i f OS XS 1/2 2^ = t (2X - 1)F 2 - 2(1 - X)vc i f 1 /2<X<.1 with a > 0, c > 0 and v e {- 1 , 1 } to be chosen in the following way: if Fx//gx £F2//g2 otherwise . (.', It is clear that w(t) * (x(t),y(t)) is a T-periodic solution of system (I) if and only if wedom(L) fulfils the operator equation Lw * Sw + Tw + z. It can be easily seen that, by the choice of a, c and v, Fx(X)//gx(X) 4 F2(X)//g2(X), for every Xe [0,1[, so that, by definition of K(T) and using hypothesis (ft?), we have that, for each Xe[0,l] , (FX(X) ,F2 (X) ,gx (X) ,g2(X)) 4 K(T). Hence, by 9 Lemma 1, (43) of Lemma 2 holds. All the other assumptions of Lemma 2 are easily verified and the result follows. Remark 1. Theorem 1 generalizes a previous result of S. Fu2j(k in [ 8] , since in the case Fx =- F2 »- 0 we have A » g . However the proof is carried - 38 -out in a different way even in this case. Corollary 1. Consider the equation of Lienard type (7) x" + f(x)x* • g(x) * e(t) with f, g, e : B -*• jR continuous and' e(*),- T-periodic. Suppose the limits fx :« lim -fix) , • f2 :* lim f(x) X-*-«» K^+« gx x» lim g(x)/x r g2 :« lim g(x)/x x'•--»• x + +» exist and are finite, with g^ and g2 positive. Then equation (7) has a T-periodic solution if (fx ,f2*9i/92^ ^ K{T). Proof. For the proof it is sufficient to observe that equation (7) is equivalent to the system x* * y - F(x) + E(t) , y* « - g(x) + e , where P(x) * /* f(u)du, e » (1/T)/ e(s)dsr E(t) «- /-<•<•) «- * )d« , and then apply Theorem 1 observing that F * f (i • I, 2), As already remarked, the case in which gx and g2 are both negative is contained in [16] where the existence of T - periodic solutions to (7) is proved for £ arbitrary. Remark 2. If, in the situation of Corollary i, we have £|»£2 <0, with g1 and g2 positive, then the existence of a T aperiodic solution to (7) is •WMJMMI, since [k.2) fails and sor (tl ,t2 ,<3\t<32) t K(T). Shis fact, which could also be obtained as a consequence of more classical results (like, e.g. (15) ) was also covered by [6, Th. 3.2 ] in the case of the Duffing equation, that is, with f2 * *" f x * c f* ° <se* aiso I - J ) • Remark 3. Theorem 1 also applies to equations of the form (8) x" + b|x»| • ax * e(t) * e(t + T) (a><3). - 39 -We have: (8) has a T - periodic solution whenever * a - (b2/4) f- k V (k - 0, 1, ... ). -This result shows that our theorem is not contained in those by E.N. Dancer in [ 4}. Remark 4. The regularity assumptions on the functions Ej and E2 could be weakened by standard arguments provided that solutions were considered in a suitable generalized sense. In particular, Corollary 1 can be extended to x" + f(x)x' + g(x) • e(t), e e L (0,T; K ) , with boundary conditions x(T) - x(0) - x'(T) - x'(0) = 0. In this situation, however, it is convenient to produce the proof directly on equation (7) and use the homotopy on system (1) only for getting the needed a-priori bounds. In the, case of 2-points BVP we are able to obtain similar results, which # will be discussed elsewhere. R e f e r e n c e s { [ 1] J. BEBERNES, M. MARTELLI: Periodic solutions for Lienard systems. Boll. Un. Mat. Ital. (5) 16 -A (1979), 398 - 405. [ 2] E.N. DANCER: Boundary-value problems for weakly nonlinear ordinary differen-tial equations. Bull. Austral. Math. Soc. 15 (1976), 321 - 328. [ 3] E.N. DANCER: On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. Roy. Soc. Edinburgh (Ser. A) 76 (1977), 283 - 300. [ 4] E.N. DANCER: On the ranges of certain damped nonlinear differential equations. Ann. Mat. Pura Appl. (4) 119 (1979), 281 - 295. I 5l P. DRABEK: Remarks on nonlinear noncoercive problems with jumping nonlinear-ities. Comment. Math. Univ. Carolinae 25 (1984), 373 - 3.99. [ 6] P. DRABEK, S. INVERNIZZI: On the periodic BVP for the forced Duffing equation with jumping nonlinearity. Nonlinear Analysis 10 (1986), 643 - 650. [ 7] C. FABRY, J. MAWHIN, M.N. NKASHAMA: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations.-Bull. London Math. Soc. 18 (1986), 173 - 180. [ 8] S. FUCIK: Boundary value problems with jumping nonlinearities. Casopis Pest. Mat. 101 (1976), 69 - 87. - 40 -I 9] S. FUC*IK: "Solvability of Nonlinear Equations and Boundary Value Problems? Reidel P. Company, Dordrecht - Boston, 1980. [10] R. GAINES, J. MAWHIN:"Coincidence degree and nonlinear differential equations? Lecture Notes in Mathematics 568, Springer -Verlag, Berlin - New York, 1977. [11] T. HARA, T. YONEYAMA: On the global center of generalized Lienard equation and its application to stability problems. Funkcialaj Ekvacioj 28 (1985), 171 - 192. [12] S. INVERNIZZI: A note on nonuniform nonresonance for jumping nonlinearitles. Comment. Math. Univ. Carolinae 27 (t$86), 285 - 291. [13] J. MAWHIN: Periodic solutions of nonlinear telegraph equation. In "Dynamical Systems" (A.R. Bednarek and L. Cesari, ed.), Academic Press, New York 1977, 193 - 210. [14] J. MAWHIN:"Topological degree methods in nonlinear boundary value problems? CBMS n. 40, Reg. Conf. Ser. in Math., Amer. Math. Soc, Providence, R.I., 1979. [15] S. MIZOHATA, S. YAMAGUTI: On the existence of periodic solutions of the nonlinear differential equation x" + a(x)x* + <|>(x) * p(t). Mem. Coll. Sci. Univ. Kyoto (Sec. A) 27 (1952), 109 - 113. [16] R. REISSIG: Extensions of some results concerning the generalized Lienard equation. Ann. Mat. Pura Appl. (4) 104 (1975), 269 - 281. [17] R. REISSIG, G. SANSONE, R. CONTI:"Qualitative Theorie nichtlinearer Differentialgleichungen1.' Cremonese, Roma, 1963. I18] G. SANSONE, R. CONTI: "Equazionj. differenziali non linear!? Cremonese, Roma, 1956. A. Fonda: International School for Advanced Studies, Strada Costlera 11, 34Ó14 Trieste, Italy F. Zanolin: Dipartimento di Scienze Matematiché, Universita, P.le Europa 1, 34100 Trieste, Italy (Oblátům 6.8. 1986) - 41 -

Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities

http://dml.cz/bitstream/handle/10338.dmlcz/106507/CommentatMathUnivCarol_028-1987-1_5.pdf

Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.