Resonance in Undamped Second-order Nonlinear Equations with Periodic Forcing
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چکیده
where g and p are real valued functions continuous on the reals R , p(t+2n) = p(t), and the solutions of (1) are uniquely determined by their initial conditions. If g is nonlinear, the question of whether all solutions of (1) are bounded on R has long been recognized as nontrivial and challenging. For the special case of g(x) = 2x Morris [1] was able to show that this question has an affirmative answer. Later Dieckerhoff and Zehnder [2] were able to prove it for the case g(jc) = x2"+1 + Pin^x2" + • • • + P\{t)x where the pk(t) are 27i-periodic and sufficiently smooth. A more recent and considerably more general result is due to Ding [3] and asserts that if g is continuously differentiate and g(x)/x —> oo as |x[ —> oo, the answer is in the affirmative. On the other hand, Littlewood [4] has given an example of an equation like (1) with g(x)/x —+ oo as |x —► oo where g(.v) and p(t) are not continuous which has an unbounded solution on R. As Morris has pointed out in [1], it is easy to modify the g(x) in Littlewood's example so that it is continuously differentiable, in fact, C°° , and still have an unbounded solution for (1); whether the function p(t), which in Littlewood's example is piecewise constant-valued, can be modified to be continuous seems not entirely obvious. However, in Ding's proof in [3], only the fact that the Poincare map associated with (1) is an area-preserving twist homeomorphism is used, and since this is also true if g(x) is continuously differentiable and p(t) only piecewise continuous, a question arises as to the validity of Ding's proof or Littlewood's example. In this note we consider the problem of giving conditions on g for which all solutions of (1) are unbounded on R\ specifically on [0, oo), primarily for cases where g is nonlinear. Our main result states that if for some integer n, \g(x)-n~x\ is bounded on R, and if a suitable Fourier coefficient of p(t) has sufficiently large absolute value, then all solutions of (1) are unbounded on [0, oo).
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تاریخ انتشار 2016