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

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.

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
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References:

[1] Chukwu, E.N., On boundedness of solutions of third order differential equations, Ann. math. pura. appl., 104, 123-149, (1975) · Zbl 0319.34027
[2] Ezeilo, J.O.C., A generalization of a boundedness theorem for a certain third-order differential equation, Proc. Cambridge philos. soc., 63, 735-742, (1967) · Zbl 0163.10507
[3] Liapunov, A.M., Stability of motion, (1966), Academic Press London, p. 203
[4] Mehri, B.; Shadman, D., Boundedness of solutions of certain third order differential equation, Math. inequal. appl., 2, 4, 545-549, (1999) · Zbl 0943.34022
[5] Rao, M.R.M., Ordinary differential equations, (1980), Affiliated East-West Private Limited London · Zbl 0173.34702
[6] Reissig, R.; Sansone, G.; Conti, R., Nonlinear differential equations of higher order, (1974), Noordhoff Groningen · Zbl 0275.34001
[7] Tejumola, H.O., A note on the boundedness and the stability of solutions of certain third-order differential equations, Ann. mat. pura appl., 92, 4, 65-75, (1972) · Zbl 0242.34046
[8] Tunç, C., Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait J. sci. eng., 32, 1, 39-48, (2005) · Zbl 1207.34043
[9] Tunç, C.; Ateş, M., Stability and boundedness results for solutions of certain third order nonlinear vector differential equations, Nonlinear dynam., 45, 3-4, 273-281, (2006) · Zbl 1132.34328
[10] Wall, Edward T.; Moe, Maynard L., An energy metric algorithm for the generation of Liapunov functions, IEEE trans. automat. control, 13, 1, 121-122, (1968)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.