Abstract
We obtain a new variant of Moser’s small twist theorem and apply this new version to investigate the boundedness of solutions for the following semilinear Duffing equation x ̈ + n 2 x + g ( x ) = p ( t ) , where p is a 2 π -periodic smooth function and lim | x | → ∞ x − 1 g ( x ) = 0 . We obtain some sharp sufficient conditions for the boundedness of all solutions to the above equation at resonance. Unlike many existing results in the literature where the function g is required to be a bounded function with asymptotic limits, our main results here allow g be unbounded or oscillatory without asymptotic limits.
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