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New ultimate boundedness and periodicity results for certain third-order nonlinear vector differential equations. (English) Zbl 1138.34322

Consider the vector differential equation
\[ \dddot x+ F(x,\dot x, \ddot x)\ddot x+ g(\dot x)+ h(x)= p(t,x,\dot x,\ddot x)\tag{\(*\)} \] with \(x\in\mathbb{R}^n\), \(F\) is a continuous symmetric \(n\times n\)-matrix, \(g: \mathbb{R}^n\to \mathbb{R}^n\), \(h: \mathbb{R}^n\to \mathbb{R}^n\), \(p: \mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n\) are continuous functions with \(h(0)= g(0)= 0\). The authors give conditions such that all solutions of \((*)\) are ultimately bounded and that there is a periodic solution with the same period as \(p\).

MSC:

34C25 Periodic solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
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