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


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


Zbl 0599.34055
Full Text: DOI


[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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