On the stability and boundedness of solutions of nonlinear vector differential equations of third order. (English) Zbl 1162.34043

Tunç, Cemil

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 6, 2232-2236 (2009).

Consider the vector differential equation \[ {d^3 x\over dt^3}+ \Psi\Biggl({dx\over dt}\Biggr) {d^2 x\over dt^2}+ B{dx\over dt}+ cx= p(t),\tag{\(*\)} \] where \(B\) is a constant symmetric \(n\times n\)-matrix, \(c\) is a positive constant, \(\Psi\) is a continuous symmetric \(n\times n\)-matrix function. In case \(p\equiv 0\), the author proves a theorem on global asymptotic stability of the equilibrium \(x= 0\). His second theorem is concerned with boundedness of all solutions of \((*)\) under certain assumptions on \(p\).
The proofs are based on the construction of suitable Lyapunov functions.

Reviewer: Klaus R. Schneider (Berlin)

MSC:

34D20	Stability of solutions to ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations

Keywords:

boundedness; stability; Lyapunov function; differential equations of third order

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