Antiperiodic Solutions for a Generalized High-Order (p, q)-Laplacian Neutral Differential System with Delays in the Critical Case

and Applied Analysis 3 (d) If p = q, x = y, |c| = |d| = 1, k = l = 1, m = n = 2, θ0(t) = θ1(t) ≡ 0, θ0(t) = δ(t), F = F (t, xθ0(t), x 󸀠 θ1(t) , xθ0(t)) = f (x (t)) x 󸀠 (t) + g (t, x (t − δ (t))) + e (t) , (15) then system (10) reduces to (7). The main purpose of this paper is to establish sufficient conditions for the existence of π-antiperiodic solutions to system (10) by using the method of coincidence degree. The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, by using the method of coincidence degree, we establish sufficient conditions for the existence of π-antiperiodic solutions to system (10). An illustrative example is given in Section 4. 2. Preliminaries The following continuation theorem of coincidence degree is crucial in the arguments of our main results. Lemma 1 (see [48]). Let X,Y be two Banach spaces; let Ω ⊂ X be open bounded and symmetric with 0 ∈ Ω. Suppose that L : D(L) ⊂ X → Y is a linear Fredholm operator of index zero withD(L)∩Ω ̸ = 0 andN : Ω → Y is L-compact. Further, one also assumes that (H) Lx−Nx ̸ = λ(−Lx−N(−x)), for all x ∈ D(L)∩∂Ω, λ ∈ (0, 1]. Then equation Lx = Nx has at least one solution onD(L)∩ Ω. Definition 2. Let u(t) : R → R be continuous. u(t) is said to be T/2-antiperiodic on R, if u (t + T) = u (t) , u (t + T 2 ) = −u (t) , ∀t ∈ R. (16) We will adopt the following notations: C k 2π := {u ∈ C (R,R) : u is 2π-periodic} , k ∈ N, |u|∞ = max t∈[0,2π] |u (t)| , (17) where u is a 2π-periodic function. For the sake of convenience, we introduce the following assumptions. (H1) There exist nonnegative constants α1, α2, . . . , αk+l+2, β1, β2, . . . , βk+l+2, such that 󵄨󵄨󵄨F (t, s1, s2, . . . , sk+l+2) − F (t, z1, z2, . . . , zk+l+2) 󵄨󵄨󵄨 ≤ k+l+2 ∑ i=1 αi 󵄨󵄨󵄨si − zi 󵄨󵄨󵄨 , 󵄨󵄨󵄨G (t, s1, s2, . . . , sk+l+2) − G (t, z1, z2, . . . , zk+l+2) 󵄨󵄨󵄨 ≤ k+l+2 ∑ i=1 βi 󵄨󵄨󵄨si − zi 󵄨󵄨󵄨 (18) for any (t, s1, s2, . . . , sk+l+2), (t, z1, z2, . . . , zk+l+2) ∈ R. (H2) For all (t, s1, s2, . . . , sk+l+2) ∈ R , F (t + π, −s1, −s2, . . . , −sk+l+2) = −F (t, s1, s2, . . . , sk+l+2) , G (t + π, −s1, −s2, . . . , −sk+l+2) = −G (t, s1, s2, . . . , sk+l+2) . (19) In order to apply Lemma 1 to study the existence of antiperiodic solutions for system (10), we set X = {x = (x1 (t) , x2 (t) , y1 (t) , y2 (t)) T ∈ C k 2π × C m−k−1 2π ×C l 2π × C n−l−1 2π : x (t + π) = −x (t) } , Y = {x = (x1 (t) , x2 (t) , y1 (t) , y2 (t)) T ∈ C 0 2π × C 0 2π ×C 0 2π × C 0 2π : x (t + π) = −x (t) } (20) are two Banach spaces with the norms ‖x‖X = k ∑ i=0 󵄨󵄨󵄨󵄨 x (i) 1 󵄨󵄨󵄨󵄨∞ + m−k−1 ∑ i=0 󵄨󵄨󵄨󵄨 x (i) 2 󵄨󵄨󵄨󵄨∞ + l ∑ i=0 󵄨󵄨󵄨󵄨 y (i) 1 󵄨󵄨󵄨󵄨∞ + n−l−1 ∑ i=0 󵄨󵄨󵄨󵄨 y (i) 2 󵄨󵄨󵄨󵄨∞ , ‖x‖Y = 2 ∑ j=1 ( 󵄨󵄨󵄨󵄨 xj 󵄨󵄨󵄨󵄨∞ + 󵄨󵄨󵄨󵄨 yj 󵄨󵄨󵄨󵄨∞ ) , (21) respectively. Define D = {x = (x1 (t) , x2 (t) , y1 (t) , y2 (t)) T ∈ C k 2π × C m−k 2π ×C l 2π × C n−l 2π : x (t + π) = −x (t) }