... Tris completely continuous in Ω. Therefore, Trhas a fixed point x ∈ Ω by Lemma3.2, which is a ω-periodic solution of (1.1). The proof is complete. Proof of Theorem 1.2 Since g ∈ C1[m, M], ... dic solutions for a class of higher-order difference equationsHuantao Zhu1and Weibing Wang∗21Hunan College of Information, Changsha, Hunan 410200, P.R. China2Department of Mathematics, ... (i − 1)k), x ∈ X.Clearly, the fixed point of Tpin X is a periodic solution of (1.1).Proof of Theorem 1.1 Let p = r and Ω = {x ∈ X : m ≤ x(n) ≤ M for n ∈ Z},then Ω is a closed and convex set....