Uniform boundedness theorem for a nonlinear Mathieu equation
Bolotin [1, Ch. 3-Ch. 6] has derived the equation /"(<) + 2e/'(0 + G2(l 2M cos 6t)f{t) + /'(«), f (0) = 0 (1) to describe the motion of a parametrically excited pin-ended elastic column. The function fit) is related to the displacement at time t of the column from its undisturbed position as follows. Let I be the length of the column and x the displacement at height h. Then x = fit) sin (vh/l). In a recent paper Genin and Maybee [2], using an energy function technique, have shown that all solutions are bounded when (1) is of the form f" = (2e/' + 02(1 2p. cos 8t)f + yf + 2e;/'f + 2k/'2/)/(1 + 2K/2) (2) and certain restrictions are placed on the parameters e, 0, n, 8, y, , and k. In this note a theorem is presented which shows that every solution of (2) has a bound which is independent of the solution's initial values. This result is obtained under less restrictive conditions than those used in [2]. Theorem. There is a bounded region A in the xy-plcme such that if f(t) satisfies (2), /(<0) = /o , and f'(t0) = /o , then (/(<), /'(<)) G A for all t > T where T is a finite value which depends on t0 , f0 , and f'a . The conditions placed on the parameters of (2) are: €, 7 > 0; €; , k > 0; and e; > 0 if k > 0. Proof. Let