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Necessary and Sufficient Conditions for Oscillations of the Generalized Liénard Systems
College of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou, China
College of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou, China
College of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou, China
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
- 1 College of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou, China
- 2 College of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou, China
- 3 College of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou, China
- 4 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
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Abstract
In this paper, we study the second-order nonlinear differential systems of Liénard-type x ˙ = 1 a ( x ) [ h ( y ) − F ( x ) ] , y ˙ = − a ( x ) g ( x ) . Necessary and sufficient conditions to ensure that all nontrivial solutions are oscillatory are established by using a new nonlinear integral inequality. Our results substantially extend and improve previous results known in the literature.
KeywordsGeneralized Liénard SystemNonlinear Integral InequalityOscillation
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