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Existence of forced oscillations for some nonlinear differential equations. (English) Zbl 0588.34028

This article studies the existence of T-periodic solutions for systems of nonlinear second order ordinary differential equations of the type ẍ\(+V'(x)=f(t)\). Here, \(x: R\to R^ N\), \(V\in C^ 2(R^ N,R)\) and \(f: R\to R^ N\) is a given T-periodic forcing term \((T>0\) is given). Assuming V to be superquadratic, it is shown that this system possesses infinitely many T-periodic solutions. The proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial. Some more general results concerning systems of the type ẍ\(+\hat V'\!_ x(t,x)=0\) are also presented here.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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