Abstract and Applied Analysis

Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance

Zaihong Wang

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Abstract

We study the existence of periodic solutions of Liénard equation with a deviating argument x ′′ + f ( x ) x ' + n 2 x + g ( x ( t - τ ) ) = p ( t ) , where f , g , p : R R are continuous and p is 2 π -periodic, 0 τ < 2 π is a constant, and n is a positive integer. Assume that the limits l i m x ± g ( x ) = g ( ± ) and l i m x ± F ( x ) = F ( ± ) exist and are finite, where F ( x ) = 0 x f ( u ) d u . We prove that the given equation has at least one 2 π -periodic solution provided that one of the following conditions holds: 2 c o s ( n τ ) [ g ( + ) - g ( - ) ] 0 2 π p ( t ) s i n ( θ + n t ) d t , for all θ [ 0,2 π ] , 2 n c o s ( n τ ) [ F ( + ) - F ( - ) ] 0 2 π p ( t ) s i n ( θ + n t ) d t , for all θ [ 0,2 π ] , 2 [ g ( + ) - g ( - ) ] - 2 n s i n ( n τ ) [ F ( + ) - F ( - ) ] 0 2 π p ( t ) s i n ( θ + n t ) d t , for all θ [ 0,2 π ] , 2 n [ F ( + ) - F ( - ) ] - 2 s i n ( n τ ) [ g ( + ) - g ( - ) ] 0 2 π p ( t ) s i n ( θ + n t ) d t , for all θ [ 0,2 π ] .

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 906972, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511880

Digital Object Identifier
doi:10.1155/2013/906972

Mathematical Reviews number (MathSciNet)
MR3055862

Zentralblatt MATH identifier
1277.34102

Citation

Wang, Zaihong. Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance. Abstr. Appl. Anal. 2013 (2013), Article ID 906972, 10 pages. doi:10.1155/2013/906972. https://projecteuclid.org/euclid.aaa/1393511880


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