References
L. Amaral andM. P. Pera, A note on periodic solutions at resonance. Boll. Un. Mat. Ital. C(5)18, 107–117 (1981).
L. Cesari andR. Kannan, Solutions in the large of Liénard equations with forcing terms. Ann. Mat. Pura Appl. (4)111, 101–124 (1976).
C. P. Gupta, On functional equations of Fredholm and Hammerstein type with applications to existence of periodic solutions of certain ordinary differential equations. J. Integral Equations (1)3, 21–42 (1981).
C. P. Gupta andJ. Mawhin, Asymptotic conditions at the two first eigenvalues for the periodic solutions of Liénard differential equations and an inequality of E. Schmidt. Z. Anal. Anwendungen (1)3, 33–42 (1984).
P. Habets andM. N. Nkashama, On periodic solutions of non-linear second order vector differential equations. Sém. Math. (Nouv. Sér.) Univ. Catholique Louvain68, IX-1–24 (1985).
R. Iannacci andM. N. Nkashama, Periodic solutions for some forced second order Liénard and Duffing systems. Sém. Math. (Nouv. Sér.) Univ. Catholique Louvain30, XIV-1–11 (1983).
S. Invernizzi andF. Zanolin, Periodic solutions of functional-differential systems with sublinear non-linearities. J. Math. Anal. Appl.101, 588–597 (1984).
M. A.Krasnosel'skii et al., Integral operators in spaces of summable functions. Leyden 1982.
A. C. Lazer, On Schauder's fixed point theorem and forced second order non-linear oscillations. J. Math. Anal. Appl.21, 421–425 (1968).
M. Martelli, On forced non-linear oscillations. J. Math. Anal. Appl.69, 496–504 (1979).
M. Martelli andJ. D. Schuur, Periodic solutions of Liénard type second-order ordinary differential equations. Tohoku Math. J.32, 201–207 (1980).
J. Mawhin, An extension of a theorem of A. C. Lazer on forced non-linear oscillations. J. Math. Anal. Appl.40, 20–29 (1972).
J. Mawhin, Boundary value problems at resonance for vector second order non-linear ordinary differential equations. Proc. Equadiff. IV, Prague 1977, LNM703, 241–249, Berlin-Heidelberg-New York 1979.
J. Mawhin, Topological degree methods in non-linear boundary value problems. CBMS Conf. in Math.40, Amer. Math. Soc., Providence, R.I. 1979.
J. Mawhin andJ. R. Ward, Jr., Non-uniform non-resonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations. Rocky Mountain J. Math. (4)12, 643–654 (1982).
J. Mawhin andJ. R. Ward, Jr., Periodic solutions of some forced Liénard differential equations at resonance. Arch. Math.41, 337–351 (1983).
J. Mawhin, Remarks on the preceding paper of Ahmad and Lazer on periodic solutions. Boll. Un. Mat. Ital. A(6)3, 229–238 (1984).
P. Omari andF. Zanolin, On forced non-linear oscillations inn-th order differential systems with geometric conditions. J. Non-linear Anal., TMA (7)8, 723–748 (1984).
P. Omari andF. Zanolin, Existence results for forced non-linear periodicBVPs at resonance. Ann. Mat. Pura Appl. (4)141, 127–157 (1985).
P.Omari and F.Zanolin, Sharp non-resonance conditions for periodic boundary value problems. In: Non-linear oscillations for conservative systems (A. Ambrosetti ed.), 73–88, Bologna 1985.
W. V. Petryshyn andS. Z. Yu, Periodic solutions of non-linear second-order differential equations which are not solvable for the highest derivative. J. Math. Anal. Appl.89, 462–488 (1982).
R. Reissig, Über einen allgemeinen Typ erzwungener nicht-linearer Schwingungen zweiter Ordnung. Rend. Accad. Naz. Lincei, Cl. Sci. fis. mat. natur. (8)56, 297–302 (1974).
R. Reissig, Extensions of some results concerning the generalized Liénard equation. Ann. Mat. Pura Appl. (4)104, 269–281 (1975).
R. Reissig, Schwingungssätze für die verallgemeinerte Liénardsche Differentialgleichung. Abh. Math. Sem. Univ. Hamburg44, 45–51 (1975).
R.Reissig, On a general type of second-order forced non-linear oscillations. In: Dynamical Systems, an International Symposium, (Vol. II) (Cesari, Hale and LaSalle eds.), 223–226, New York 1976.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Omari, P., Zanolin, F. Sharp non-resonance conditions for periodically perturbed Liénard systems. Arch. Math 46, 330–342 (1986). https://doi.org/10.1007/BF01200464
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01200464