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Results on non-resonant oscillations for some nonlinear vector fourth order differential systems. (English) Zbl 1283.34043

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

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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