Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance. (8th May 2013) - Wang, Zaihong
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 ] .