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Analytical theory of non-linear oscillations

I. An existence theorem for periodic oscillations of equations of the second order

Published online by Cambridge University Press:  24 October 2008

Chike Obi
Chike Obi
Affiliation:
Mathematics Department, University of Lagos, Nigeria
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Extract

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equation

where g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 285 - 296
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Friedricks, K. O. and Stoker, J. J.Forced vibrations of systems with non-linear restoring force. Quart. Appl. Math. 1 (1943), 97–115.CrossRefGoogle Scholar
(2)Langenhop, C. E. and Farnell, A. B. The existence of forced periodic solutions of second order differential equations. Contributions to Non-Linear Oscillations 1, 291–292, section 1 (Princeton, 1950).Google Scholar
(3)Lefschetz, S.Differential Equation: Geometric Theory, pp. 301–306, section 7 (New York, 1963).Google Scholar
(4)Loud, W. S.Periodic solutions of . Memoirs of the American Math. Soc. no. 31 (1959).CrossRefGoogle Scholar
(5)Morris, G. R.A differential equation for undamped forced non-linear oscillations I. Proc. Cambridge Philos. Soc. 51 (1955), 297–312.CrossRefGoogle Scholar
(6)Morris, G. R.A differential equation for undamped forced non-linear oscillations II. Proc. Cambridge Philos. Soc. 54 (1958), 426–438.CrossRefGoogle Scholar
(7)Morris, G. R.A differential equation for undamped forced non-linear oscillations III. Proc. Cambridge Philos. Soc. 61 (1963), 133–155.CrossRefGoogle Scholar
(8)Obi, Chike. Periodic solutions of non-linear differential equations of the second order IV. Proc. Cambridge Philos. Soc. 47 (1951), 741–751.CrossRefGoogle Scholar
(9)Obi, Chike. Periodic solutions of the equation Proc. Peking Symposium Natural Sciences A.1 (1964), 21–45.Google Scholar
(10)Obi, Chike. Sub-harmonic solutions of non-linear differential equations of the second order. J. Math. London Soc. 25 (1950), 220–1, section 2.1.Google Scholar
(11)Shimuzu, Tatsujoro. On differential equations for non-linear oscillations I. Math. Jap. 2 (1951), 86–96.Google Scholar
(12)Stoker, J. J.Non-linear Vibration, p. 83, section 2 (New York, 1950).Google Scholar