Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T12:59:47.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions*

Published online by Cambridge University Press:  14 November 2011

R. Iannacci ,
M.N. Nkashama ,
P. Omari  and
F. Zanolin
R. Iannacci
Affiliation:
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università “La Sapienza”, Via A. Scarpa 10, 00161 Roma, Italy
M.N. Nkashama
Affiliation:
Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152, U.S.A.
P. Omari
Affiliation:
Dipartimento di Scienze Matematiche, Università, P.le Europa 1, 34100 Trieste, Italy
F. Zanolin
Affiliation:
Dipartimento di Matematica ed Informatica, Università, Via Zanon 6, 33100 Udine, Italy
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equation

The key assumptions relate the asymptotic behaviour as x →± ∞of g(t; x)/x to the “critical values” of the positively 1-homogeneous problem

No condition on f, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 110 , Issue 3-4 , 1988 , pp. 183 - 198
Copyright
Copyright © Royal Society of Edinburgh 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ambrosetti, A. and Prodi, G.. On the inversion of some mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93 (1973), 321–347.Google Scholar
2Ambrosetti, A. and Prodi, G.. Analisi non lineare. Quaderno 1 (1973).Google Scholar
3Cesari, L. and Kannan, R.. Periodic solutions in the large of Liénard systems with forcing terms. Boll. Un. Mat. ltal. A (6) 1 (1982), 217–224.Google Scholar
4Dancer, E. N.. Boundary-value problems for weakly nonlinear ordinary differential equations. Bull. Austral. Math. Soc. 15 (1976), 321–328.Google Scholar
5Dancer, E. N.. On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 283–300.CrossRefGoogle Scholar
6Dancer, E. N.. On the ranges of certain damped nonlinear differential equations. Ann. Mat. Pura Appl. (4) 119 (1979), 281–295.Google Scholar
7Drabek, P.. Remarks on nonlinear noncoercitive problems with jumping nonlinearities. Comment. Math. Univ. Carolin. 25 (1984), 373–399.Google Scholar
8Drabek, P. and Invernizzi, S.. On the periodic BVP for the forced Duffing equation with jumping nonlinearity. Nonlinear Anal. 10 (1986), 643–650.Google Scholar
9Fabry, C., Mawhin, J. and Nkashama, M. N.. A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. London Math. Soc. 18 (1986), 173–180.Google Scholar
10Fonda, A. and Zanolin, F.. Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities. Comment. Math. Univ. Carolin. 28 (1987), 33–41.Google Scholar
11Fučik, S.. Boundary value problems with jumping nonlinearities. Časopis. Pěst. Mat. 101 (1976), 69–87.Google Scholar
12Fučik, S.. Solvability of nonlinear equations and boundary value problems (Dordrecht: Reidel, 1980).Google Scholar
13Gupta, C. P. and Mawhin, J.. Asymptotic conditions at the two first eigenvalues for the periodic solutions of Liénard differential equations and an inequality of E. Schmidt. Z. Anal. Anwendungen 3 (1984), 33–42.Google Scholar
14Hale, J. K.. Ordinary differential equations (New York: Wiley, 1969).Google Scholar
15Invernizzi, S.. A note on nonuniform nonresonance for jumping nonlinearities. Comment. Math. Univ. Carolin. 27 (1986), 285–291.Google Scholar
16Lazer, A. C. and McKenna, P. J.. Large scale oscillatory behaviour in loaded asymmetric systems. Ann. Inst. H. Poincaré (4) 3 (1987), 243–274.Google Scholar
17Martelli, M.. On forced nonlinear oscillations. J. Math. Anal. Appl. 69 (1979), 496–504.CrossRefGoogle Scholar
18Mawhin, J.. An extension of a theorem of A. C. Lazer on forced nonlinear oscillations. J. Math. Anal. Appl. 40 (1972), 20–29.Google Scholar
19Mawhin, J.. Remarks on the preceding paper of Ahmad and Lazer on periodic solutions. Boll. Un. Mat. ltal. A(6) 3 (1984), 229–236.Google Scholar
20Mawhin, J. and Jr, J. R. Ward. Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations. Rocky Mountain J. Math. 12 (1982), 643–654.Google Scholar
21Mawhin, J. and Jr, J R. Ward. Periodic solutions of some forced Liénard differential equations at resonance. Arch. Math. (Basel) 41 (1983), 337–351.Google Scholar
22Omari, P., Villari, G. and Zanolin, F.. Periodic solutions of the Liénard equation with one-sided growth restrictions. J. Differential Equations 67 (1987), 278–293.Google Scholar
23Omari, P. and Zanolin, F.. On the existence of periodic solutions of forced Liénard differential equations. Nonlinear Anal. 11 (1987), 275–284.CrossRefGoogle Scholar
24Reissig, R.. Extensions of some results concerning the generalized Liénard equation. Ann. Mat. Pura Appl. (4) 104 (1975), 269–281.Google Scholar
25Reissig, R.. Schwingungssätze für die verallgemeinerte Liénardsche Differentialgleichung. Abh. Math. Sent. Univ. Hamburg 44 (1975), 45–51.Google Scholar
26Reissig, R., Sansone, G. and Conti, R.. Qualitative theorie nichtlinearer differentialgleichungen (Rome: Cremonese, 1963).Google Scholar
27Schmitt, K.. Boundary value problems with jumping nonlinearities. Rocky Mountain J. Math. 16 (1986), 481–496.CrossRefGoogle Scholar