## Asymptotic conditions for periodic solutions of ordinary differential equations

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- by James R. Ward PDF
- Proc. Amer. Math. Soc.
**81**(1981), 415-420 Request permission

## Abstract:

We obtain sufficient conditions for the existence of periodic solutions to differential equations of the form \[ {x^{(m)}} + {a_{m - 1}}{x^{(m - 1)}} + \cdots + {a_1}x’ + g(t,x) = f(t)\quad (m \geqslant 1).\] The conditions are necessary for some classes of functions, and require no growth condition on $g(t,x)$ for $x \geqslant 0(x \leqslant 0)$.## References

- E. N. Dancer,
*Boundary-value problems for weakly nonlinear ordinary differential equations*, Bull. Austral. Math. Soc.**15**(1976), no. 3, 321–328. MR**430384**, DOI 10.1017/S0004972700022747 - Svatopluk Fučík and Vladimír Lovicar,
*Periodic solutions of the equation $x^{^{\prime \prime }}(t)+g(x(t))=p(t)$*, Časopis Pěst. Mat.**100**(1975), no. 2, 160–175. MR**0385239** - Chaitan P. Gupta,
*On functional equations of Fredholm and Hammerstein type with applications to existence of periodic solutions of certain ordinary differential equations*, J. Integral Equations**3**(1981), no. 1, 21–41. MR**604314** - A. C. Lazer,
*On Schauder’s fixed point theorem and forced second-order nonlinear oscillations*, J. Math. Anal. Appl.**21**(1968), 421–425. MR**221026**, DOI 10.1016/0022-247X(68)90225-4 - J. Mawhin,
*Periodic solutions of some vector retarded functional differential equations*, J. Math. Anal. Appl.**45**(1974), 588–603. MR**333400**, DOI 10.1016/0022-247X(74)90053-5 - J. Mawhin,
*Topological degree methods in nonlinear boundary value problems*, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR**525202** - J. Mawhin and K. Schmitt,
*Rothe and Altman type coincidence theorems and applications to differential equations*, Nonlinear Anal.**1**(1976/77), no. 2, 151–160. MR**637077**, DOI 10.1016/0362-546X(77)90006-2 - R. Reissig,
*Periodic solutions of certain higher order differential equations*, Nonlinear Anal.**2**(1978), no. 5, 635–642. MR**512159**, DOI 10.1016/0362-546X(78)90011-1

## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**81**(1981), 415-420 - MSC: Primary 34C25
- DOI: https://doi.org/10.1090/S0002-9939-1981-0597653-2
- MathSciNet review: 597653