RENDICONTI DEL SEMINARIO MATEMATICO

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 1 (2007)Subalpine Rhapsody in DynamicsC. Bereanu - J. MawhinPERIODIC SOLUTIONS OF FIRST ORDER NONLINEARDIFFERENCE EQUATIONSAbstract. This paper surveys some recent results on the existence and multiplicity of periodicsolutions of nonlinear difference equations of the first order under Ambrosetti-Prodi orLandesman-Lazer type conditions.1. IntroductionPeriodic solutions of first and second order nonlinear difference equations have beenwidely studied, and the reader can consult [1, 9] for references. In some recent workwith C. Bereanu, we have adapted the topological approach to the upper and lower solutionsmethod to this class of problems and used it, together with Brouwer degree, toobtain new existence and multiplicity results of the Ambrosetti-Prodi and Landesman-Lazer type [2, 3]. In [4], we have used the same methodology to prove similar resultsfor second order nonlinear difference equations with Dirichlet boundary conditions.The present paper surveys some of those results and is restricted, for the sake of simplicity,to the case of periodic solutions of first order difference equations. Some of thearguments of [2, 3] are simplified, and some of the conclusions are sharpened.2. Periodic solutionsLet n ≥ 2 be a fixed integer. For (x 1 ,..., x n ) ∈ R n , define the first order differenceoperator (Dx 1 ,..., Dx n−1 ) ∈ R n−1 byDx m := x m+1 − x m (1 ≤ m ≤ n − 1).Let f m : R n → R (1 ≤ m ≤ n − 1) be continuous functions. We study the existenceof solutions for the periodic boundary value problem(1)Dx m + f m (x 1 ,..., x n ) = 0 (1 ≤ m ≤ n − 1), x 1 = x n .Let(2)U n−1 = {x ∈ R n : x 1 = x n },so that U n−1 ≃ R n−1 because an element of U n−1 can be characterized by the coordinatesx 1 ,..., x n−1 . The restriction L : R n−1 → R n−1 of D to R n−1 is givenby(3)(Lx) m = x m+1 − x m (1 ≤ m ≤ n − 2), (Lx) n−1 = x 1 − x n−1 ,17