RENDICONTI DEL SEMINARIO MATEMATICO

18 C. Bereanu - J. Mawhinor, in matrix form, by the circulant matrix [6]⎛⎞−1 1 0 ··· ··· 00 −1 1 0 ··· 0··· ··· ··· ··· ··· ···(4)⎜ ··· ··· ··· ··· ··· ···.⎟⎝ 0 ··· ··· 0 −1 1 ⎠1 0 ··· ··· 0 −1If we define F : R n−1 → R n−1 byF m (x 1 ,··· , x n−1 ) = f m (x 1 , x 2 ,..., x n−1 , x 1 ) (1 ≤ m ≤ n − 1),problem (1) is equivalent to study the zeros of the continuous mapping H : R n−1 →R n−1 defined by(5)H m (x) = (Lx) m + F m (x) (1 ≤ m ≤ n − 1).3. Bounded nonlinearitiesLet us first consider the linear periodic problem(6)Dx m + αx m = 0 (1 ≤ m ≤ n − 1), x 1 = x n ,where α ∈ R. The solutions of the corresponding difference system are given byx m = (1 − α) m−1 x 1(1 ≤ m ≤ n),and hence (6) has a solution if and only ifx 1 = (1 − α) n−1 x 1 .This immediately implies the followingLEMMA 1. Problem (6) has only the trivial solution if n is even and α ̸= 0 or ifn is odd and α ̸∈ {0, 2}. When α = 0, the solutions are of the form x m = c (1 ≤ m ≤n), and when n is odd and α = 2, they have the form x m = (−1) m−1 c (1 ≤ m ≤ n),with c ∈ R arbitrary.Let(7)b m : R n → R, (x 1 ,..., x n ) ↦→ b m (x 1 ,..., x n ) (1 ≤ m ≤ n − 1)be continuous and bounded, and consider the semilinear periodic problem(8)Dx m + αx m + b m (x 1 ,..., x n ) = 0 (1 ≤ m ≤ n − 1), x 1 = x n .If L is defined like above and B : R n−1 → R n−1 byB m (x 1 ,..., x n−1 ) = b m (x 1 , x 2 ,..., x n−1 , x 1 ) (1 ≤ m ≤ n − 1),