On boundedness of solutions to conservative second-order differential equations (Q2761381)

The author considers the differential equation NEWLINE\[NEWLINE \ddot x+ x^{2m-1} = f(x,t),\tag{1} NEWLINE\]NEWLINE where \(m>1 \) is an integer, the function \(f(x,t)\) is decomposed into the Laurent series convergent for \(|x|>x^*>0\) NEWLINE\[NEWLINE f(x,t) = \sum_{k=-\infty}^{2m-2} a_k(t)x^k, NEWLINE\]NEWLINE whose coefficients \(a_k(t)\) are real analytical \(2\pi\)-periodic functions of \(t\) for \(|\text{Im} t|<p_0\). It is proved that under the assumptions made, the solutions to equation (1) are bounded.