Results on non-resonant oscillations for some nonlinear vector fourth order differential systems (Q2847089)

The authors consider the following nonlinear vector differential equations of fourth order: NEWLINE\[NEWLINE X^{(4)} + A{X}''' + B{X}'' + \frac{d}{dt}g(X) + DX = P(t) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE X^{(4)} + A{X}''' + B{X}'' + G(t,X) + DX = P(t) NEWLINE\]NEWLINE subject to the periodic boundary conditions NEWLINE\[NEWLINE D^{(r)}X(0) = D^{(r)}X(T), (D = \frac{d}{dt}),\quad r = 0,1,2,3, NEWLINE\]NEWLINE on \([0,T]\) with \(T > 0.\)NEWLINENEWLINEHere, \(X \equiv (x_i )_{1 \leq i \leq n} :[0,T] \to \mathbb R^n\). They prove two existence results. They extend and improve some related results from scalar cases to vectorial cases.