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Boundedness and Asymptotic Behaviour of Solutions of a Second‐Order Nonlinear System
Author(s) -
Qian Chuanxi
Publication year - 1992
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/24.3.281
Subject(s) - mathematics , bounded function , continuous function (set theory) , order (exchange) , nonlinear system , zero (linguistics) , function (biology) , mathematical analysis , pure mathematics , combinatorics , physics , finance , quantum mechanics , economics , linguistics , philosophy , evolutionary biology , biology
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.