It is shown that all solutions of x + 2x = pit) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p .The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almostperiodic solutions and, for each positive integer m , an infinity of periodic solutions of least period 2im (2TT being taken as the least period of p ).It is suggested that for a large class of equations the same attack would show all solutions of x + g(x) = pit) bounded.However, in order to show the method clearly, no generalisation is attempted here.

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A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.