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**Resonant and nonresonant oscillations for some third order nonlinear ordinary differential equations.**
*(English)*
Zbl 0676.34021

The subjects of the paper are three third-order ordinary differential equations, one of which is typically \(x'''+ax''+bx'+g(t,x)=p(t,x,x',x''),\) where a and b are constants. Similar equations were the subject of an earlier paper by J. O. C. Ezeilo and J. Onyia [J. Niger Math. Soc. 3, 83-96 (1984; Zbl 0599.34055)] in which existence and uniqueness of \(2\pi\)-periodic solutions were established subject to certain nonresonant conditions. Here the conditions are weakened in that p is assumed only to be a Caratheodory function. Also some new existence results for two of the equations are proved including the existence of a \(2\pi\)-periodic solution at resonance for one of the equations.

Reviewer: P.Smith

### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

### Citations:

Zbl 0599.34055
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\textit{J. O. C. Ezeilo} and \textit{M. N. Nkashama}, Nonlinear Anal., Theory Methods Appl. 12, No. 10, 1029--1046 (1988; Zbl 0676.34021)

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### References:

[1] | Dunford, N.; Schwartz, J. T., Linear Operators, Vol. 1 (1964), Interscience: Interscience New York |

[2] | Ezeilo, J. O.C.; Onyia, J., Non-resolant oscillations for some third order differential equations I, J. Nigerian Math. Soc., 3, 83-96 (1984) · Zbl 0599.34055 |

[3] | Ezeilo, J. O.C.; Nkashama, M. N., Non-uniform non-resonance and existence of periodic solutions of some third order nonlinear differential equations (1985), ICTP: ICTP Trieste, preprint IC/85/83 · Zbl 0676.34021 |

[4] | Mawhin, J., Topological degree methods in nonlinear boundary value problems, (CBMS Regional Conf. Ser. Math. No. 40 (1979), Am. Math. Soc: Am. Math. Soc Providence, RI), (second printing 1981). · Zbl 0414.34025 |

[5] | Mawhin, J., Compacite, monotonie et convexite dans l’etude de problemes aux limites semi-lineaires, (Semin. Anal. Moderne (1981), Universite de Sherbrooke: Universite de Sherbrooke Quebec, Canada), No. 19 · Zbl 0497.47033 |

[6] | Mawhin, J.; Ward, J. R., Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions of forced Lienard and Duffing equations, Rocky Mount. J. Math., 12, 643-653 (1982) · Zbl 0536.34022 |

[7] | Mawhin, J.; Ward, J. R., Periodic solutions of some forced Lienard differential equations at resonance, Arch. Math., 41, 337-351 (1983) · Zbl 0537.34037 |

[8] | Reissig, R., Extension of some results concerning the generalized Lienard equations, Annali Mat. pura appl., 104, 269-281 (1975) · Zbl 0313.34037 |

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