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

and Applied Analysis 3 (3) The Brouwer degree deg{JQN,Ω∩Ker L, 0} ̸ = 0, where J : ImQ → Ker L is an isomorphism. Then equation Lx = Nx has at least one solution onD(L)∩Ω. 3. Main Results In this section, we will use the continuation theorem introduced in Section 2 to prove the existence of periodic solutions of (1). To this end, we first quote some notations and definitions. Let X and Y be two Banach spaces defined by the following: X = {x ∈ C 1 (R,R) : x (t + 2π) = x (t) , ∀t ∈ R} , Y = {y ∈ C (R,R) : y (t + 2π) = y (t) , ∀t ∈ R} (14) with the following norms ‖x‖ X = max {‖x‖ ∞ , 󵄩 󵄩 󵄩 󵄩 󵄩 x 󸀠󵄩 󵄩 󵄩 󵄩∞ } , 󵄩 󵄩 󵄩 󵄩 y 󵄩 󵄩 󵄩 󵄩Y = 󵄩 󵄩 󵄩 󵄩 y 󵄩 󵄩 󵄩 󵄩∞ . (15) Define a linear operator L : D (L) ⊂ X 󳨀→ Y, Lx = x 󸀠󸀠 + n 2 x, (16) where D(L) = {x ∈ X : x󸀠󸀠 ∈ C(R,R)}, and a nonlinear operator N : X 󳨀→ Y, (Nx) (t) = −f (x (t)) x 󸀠 (t) − g (x (t − τ)) + p (t) . (17) It is easy to see that Ker L = Span {sin nt, cos nt} , Im L = {y ∈ Y : ∫ 2π 0 y (t) sin ntdt = 0,