Boundedness of Certain System of Second Order Differential Equations

Abstract

This work is concerned with the ultimate boundedness of solutions of the system of vector differential equations X˙ = H (Y ), Y˙ = − F (X, Y )Y − G (X ) + P (t,X, Y ), where t ∈ ℝ+, X = X(t), Y = Y (t) ∈ ℝn, F : ℝn × ℝn → ℝn×n, G,H : ℝn → ℝn and P : ℝ+ × ℝn × ℝn → ℝn. By using a Lyapunov function as a basic technique, we prove that the solutions of the system of equations are ultimately bounded. In addition, result obtained includes and improves some related results in literature.