Autonomous Noether Boundary-Value Problems not Solved with Respect to the Derivative
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In monographs of N. V. Azbelev, A. M. Samoilenko, and A. A. Boichuk, constructive methods of study of Noether boundary-value problems have been developed. These methods continue the investigation of periodic problems stated by H. Poincaré, A. M. Lyapunov, N. M. Krylov, N. N. Bogolyubov, I. G. Malkin, and O. Veivoda by the methods of small parameter. We propose an improved scheme of study of autonomous Noether boundary-value problems for nonlinear systems in critical cases. In the case of multiple roots of the equation for generating constants, we obtain sufficient conditions of existence of solutions to an autonomous boundary-value problem not solved with respect to the derivative. The effectiveness of the scheme proposed is illustrated by an example of the periodic problem for the Liénard equation.
Keywords and phrasesautonomous boundary-value problem ordinary differential equation Liénard equation
AMS Subject Classification34B15
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- 1.N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations, World Federation Publ., Atlanta (1995).Google Scholar
- 3.A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht, Boston (2004).Google Scholar
- 8.S. M. Chuiko, O. V. Starkova, and O. E. Pirus, “Nonlinear Noether boundary-value problems not solved with respect to the derivative,” Dinam. Sist., 30, Nos. 1-2, 169–186 (2012).Google Scholar
- 10.O. Vejvoda, “On perturbed nonlinear boundary-value problems,” Czech. Math. J., No. 11, 323-364 (1961).Google Scholar
- 11.V. Volterra, Le¸cons sur la Th´eorie Mathematique de la Lutte pour la Vie, Gauthiers-Villars, Paris (1931).Google Scholar
- 12.V. F. Zaitsev and A. D. Polyanin, Handbook of Ordinary Differential Equations [in Russian], Fizmatlit, Moscow (2001).Google Scholar