Existence of periodic solutions of nonlinear differential equations
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- by R. Kannan PDF
- Trans. Amer. Math. Soc. 217 (1976), 225-236 Request permission
Abstract:
The nonlinear differential equation $x'' = f(t,x(t))$, f being $2\pi$-periodic in t, is considered for the existence of $2\pi$-periodic solutions. The equation is reduced to an equivalent system of two Hammerstein equations. The case of nonlinear perturbation at resonance is also discussed.References
- H. Brezis, M. G. Crandall, and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970), 123–144. MR 257805, DOI 10.1002/cpa.3160230107 L. Cesari, Nonlinear functional analysis, Lecture notes, C.I.M.E., 1972.
- L. Cesari and R. Kannan, Functional analysis and nonlinear differential equations, Bull. Amer. Math. Soc. 79 (1973), 1216–1219. MR 333861, DOI 10.1090/S0002-9904-1973-13385-3 J. K. Hale, Applications of alternative problems, Brown University Lecture Notes, 1971.
- Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 0419901
- R. Kannan, Periodically perturbed conservative systems, J. Differential Equations 16 (1974), no. 3, 506–514. MR 417493, DOI 10.1016/0022-0396(74)90006-0
- A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl. (4) 82 (1969), 49–68. MR 249731, DOI 10.1007/BF02410787
- A. C. Lazer and D. A. Sánchez, On periodically perturbed conservative systems, Michigan Math. J. 16 (1969), 193–200. MR 245906, DOI 10.1307/mmj/1029000261
- D. E. Leach, On Poincaré’s perturbation theorem and a theorem of W. S. Loud, J. Differential Equations 7 (1970), 34–53. MR 251308, DOI 10.1016/0022-0396(70)90122-1
- W. S. Loud, Periodic solutions of nonlinear differential equations of Duffing type, Proc. U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967) Benjamin, New York, 1967, pp. 199–224. MR 0223656
- George J. Minty, on a “monotonicity” method for the solution of non-linear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1038–1041. MR 162159, DOI 10.1073/pnas.50.6.1038
- Mitio Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), 497–511. MR 42697, DOI 10.2307/2372304
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 217 (1976), 225-236
- MSC: Primary 34C25
- DOI: https://doi.org/10.1090/S0002-9947-1976-0397091-4
- MathSciNet review: 0397091