RENDICONTI DEL SEMINARIO MATEMATICO

Periodic solutions of difference equations 19then problem (8) is equivalent to the semilinear problem in R n−1(9)Lx + αx + B(x) = 0.We have the following existence result.THEOREM 1. Assume n odd and α ̸= 0 or n even and α ̸∈ {0, 2} and assumethat the functions b m in (7) are continuous and bounded. Then problem (8) has at leastone solution and, for all sufficiently large R,d B [L + αI + B, B(R), 0] = ±1.Proof. Let M > 0 is such that ‖B(v)‖ ≤ M for all v ∈ R n−1 . For each λ ∈ [0, 1],each possible zero u of L + αI + λB is such that, using Lemma 1,‖u‖ = λ‖(L + αI) −1 B(u)‖ ≤ ‖(L + αI) −1 ‖M.Hence, if we take any R > ‖(L + αI) −1 ‖M, and denote the Brouwer degree by d B(see [7]), the homotopy invariance of the degree implies thatd B [L + αI + B, B(R), 0] = d B [L + αI, B(R), 0] = ±1,and the existence follows from the existence property of Brouwer degree [7].4. Upper and lower solutionsLet f m : R → R (1 ≤ m ≤ n − 1) be continuous functions, and let us consider theperiodic problem(10)Dx m + f m (x m ) = 0 (1 ≤ m ≤ n − 1), x 1 = x n .DEFINITION 1. α = (α 1 ,...,α n ) (resp. β = (β 1 ,...,β n )) is called a lowersolution (resp. upper solution) for (10) ifand the inequalitiesα 1 ≥ α n (resp. β 1 ≤ β n ),(11)Dα m + f m (α m ) ≥ 0 (resp. Dβ m + f m (β m ) ≤ 0)hold for all 1 ≤ m ≤ n − 1. Such a lower or upper solution will be called strict if theinequality (11) is strict for all 1 ≤ m ≤ n − 1.The basic theorem for the method of upper and lower solutions goes as follows.The proof given here is a simplification of that given in [2], which is modeled on thecorresponding one for differential equations in [11].