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**Periodic solutions for coupled first order nonlinear differential systems of Hamiltonian type.**
*(English)*
Zbl 0562.34030

The author shows there are periodic solutions of periodic systems \(- x'+f(t,x,y)=p(t)\), \(y'+g(t,x,y)=q(t)\) under various sign and growth conditions. These results follow from analysis of nonlinear equations in Banach spaces. Abstract theorems use techniques of alternative methods and Leray-Schauder-Mawhin continuation: the latter involves a priori estimates derived from hypothesized operator estimates. It seems possible to alter the results to allow the removal of the assumption that p, q have mean value zero. In the proofs found in the first section, the author implicitly assumes that (\(\bullet\),\(\bullet)\) is positive definite on \(X_ 2\); this causes no difficulty in the second section. It seems to the reviewer that for Corollary 2.4 to imply Corollary 2.5 one must assume \(\int^{T}_{0}\Gamma (t)dt<-T+1/4\) in the latter. The paper is written well and almost completely self-contained.

Reviewer: L.Turyn

### MSC:

34C25 | Periodic solutions to ordinary differential equations |

47J05 | Equations involving nonlinear operators (general) |

55M20 | Fixed points and coincidences in algebraic topology |

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

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\textit{C. P. Gupta}, Nonlinear Anal., Theory Methods Appl. 8, 1271--1285 (1984; Zbl 0562.34030)

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### References:

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[10] | Mawhin, J.; Ward, J. R., Nonuniform non-resonance conditions at the two first eigenvalues for periodic solutions of forced Lienard and Duffing equations, Rocky Mountain J. Math., 12, 643-654 (1982) · Zbl 0536.34022 |

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