Boundedness and Asymptotic Behaviour of Solutions of a Second‐Order Nonlinear System

Consider the second‐order nonlinear system{x ˙ = 1 a ( x )[ c ( y ) ‐ b ( x ) ] ,y ˙ = ‐ a ( x ) [ h ( x ) ‐ e ( t ) ] ,where a is a positive and continuous function on R:(− ∞, ∞), b , c and h are continuous functions on R, and e is a continuous function on I :[0, ∞). We obtain a sufficient condition for all solutions of (*) to be bounded, and obtain a sufficient condition for all solutions of (*) to tend to zero. Our results can be applied to the well‐known equationx ¨ + ( f ( x ) + g ( x ) x ˙ ) x ˙ + h ( x ) = e ( t ) , which substantially improves and extends several known results in the literature.