## Summary

The equation x′=A(t)x+f(t, x, ε) is investigated for periodic solutions in the critical case. Explicit estimates for the existence region in ε for these solutions are obtained by employing implicit function theorem techniques.

## References

- [1]
R. F. Arenstorf,

*New periodic solutions of the plane three body problem corresponding to elliptic motion in lunar theory*, J. Differential Equations, 4 (1968), pp. 202–256. - [2]
R. Berlman,

*Stability Theory of Differential Equations*, McGraw Hill (1953). - [3]
G. R. Clements,

*Implicit functions defined by equations with vanishing Jacobian*, Trans. Amer. Math. Soc., 14 (1913), pp. 325–342. - [4]
E. A. Coddington andN. Levinson,

*Theory of ordinary Differential Equations*, McGraw Hill (1955). - [5]
H. I. Freedman,

*Estimates on the existence region for periodic solutions of equations involving a small parameter*, I: The noncritical case, SIAM J. Appl. Math., 16 (1968), pp. 1341–1349. - [6]
—— ——,

*An explicit estimate on the norm of the inverse of a matrix*, SIAM Review, 11 (1969), pp. 254–256. - [7]
-- --,

*Estimates on the existence region for solutions of equations involving a small parameter*, Ph. D. thesis, (1967), University of Minnesota. - [8]
E. Goursat,

*Cours d’Analyse Mathematique*, Vol. 1, Gauthier-Villars (1910). - [9]
D. C. Lewis,

*Periodic solutions of differential equations containing a small parameter*, Duke Math. J., 22 (1955), pp. 39–56. - [10]
—— ——,

*On the perturbation of a periodic solution when the variational system has nontrivial periodic solutions*, J. Rational Mech. Anal., 4 (1955), pp. 795–815. - [11]
—— ——,

*On the role of first integrals in the perturbation of periodic solutions*, Ann. of Math., 63 (1956), pp. 535–548.

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## Additional information

This paper is based on a Ph. D. thesis rubmitted by the authour to the faculty of the Graduate School of the University of Minnesota. The research was paid for in part by U. S. Army Research Contract No. DA-ARO-D-31-124G737.

Entrata in Redazione il 15 febbraio 1971.

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### Cite this article

Freedman, H.I. Estimates on the existence region for periodic solutions of equations involving a small parameter. Il: Critical cases.
*Annali di Matematica* **90, **259–279 (1971) doi:10.1007/BF02415051

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### Keywords

- Periodic Solution
- Small Parameter
- Implicit Function Theorem
- Critical Case
- Explicit Estimate