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Periodic Solutions of Periodic Retarded Functional Differential Equations

Marcin Pawłowski (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

The paper presents a geometric method of finding periodic solutions of retarded functional differential equations (RFDE) x ' ( t ) = f ( t , x t ) , where f is T-periodic in t. We construct a pair of subsets of ℝ × ℝⁿ called a T-periodic block and compute its Lefschetz number. If it is nonzero, then there exists a T-periodic solution.

Periodic Solutions of Second Order Nonautonomous Systems with the Potentials Changing Sign

Mario Girardi, Michele Matzeu (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Some existence and multiplicity results for periodic solutions of second order nonautonomous systems with the potentials changing sign are presented. The proofs of the existence results rely on the use of a linking theorem and the Mountain Pass theorem by Ambrosetti and Rabinowitz [2]. The multiplicity results are deduced by the study of constrained critical points of minimum or Mountain Pass type.

Periodic solutions of the Rayleigh equation with damping of definite sign

Pierpaolo Omari, Gabriele Villari (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The existence of a non-trivial periodic solution for the autonomous Rayleigh equation x ¨ + F x ˙ + g x = 0 is proved, assuming conditions which do not imply that F x x has a definite sign for x large. A similar result is obtained for the periodically forced equation x ¨ + F x ˙ + g x = e t .

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