RENDICONTI DEL SEMINARIO MATEMATICO

24 C. Bereanu - J. Mawhinhas the same nontrivial solution ϕ. As b m = δ nm (1 ≤ m ≤ n) (Kronecker symbol) isnot orthogonal to the kernel of the adjoint system (25), the problemx 1 − x n = 0x 2 + x 1 = 0... ... ...x n−1 + x n−2 = 0x n + x n−1 = 1has no solution, or, equivalently the problem(26)Dx m + 2x m = 0 (1 ≤ m ≤ n − 2), Dx n−1 + 2x n−1 = 1, x 1 = x nhas no solution. However, α = (1,...,1) is a lower solution and β = (0,...,0) is anupper solution of (26) such that β m ≤ α m (1 ≤ m ≤ n).If now n > 2 is even, the problem(27)Dx m + 2x m = 0 (1 ≤ m ≤ n − 3), Dx n−2 + 2x n−2 = 1,Dx n−1 = 0, x 1 = x nis of course equivalent to the problemDx m + 2x m = 0 (1 ≤ m ≤ n − 3), Dx n−2 + 2x n−2 = 1, x 1 = x n−1 .As n − 1 is odd, it follows from the counter-example (26) that problem (27) has nosolution. However α = (1,...,1) is a lower solution and β = (0,...,0) is an uppersolution of (27) such that β m ≤ α m (1 ≤ m ≤ n). Those counter-examples were firstgiven in [3].For n = 2, problem (10) is equivalent to the unique scalar equationf 1 (x 1 ) = 0and, in this case, the validity of the method of upper and lower solutions, independentlyof their order, follows from its equivalence with Bolzano’s theorem applied to the realfunction f 1 .REMARK 2. Notice that, in contrast to the periodic problem for differenceequations, whose eigenvalues are in the left half-plane, all the eigenvalues λ k = 2kπiT(k ∈ Z) of the differential operatordt d with periodic boundary conditions on [0, T]are on the imaginary axis. This explains that the method of upper and lower solutionsworks irrespectively to the order of the lower and the upper solution.7. Ambrosetti-Prodi type multiplicity resultLet f 1 ,..., f n−1 : R → R be continuous functions, s ∈ R. Consider the problem,with n ≥ 2,(28)Dx m + f m (x m ) = s (1 ≤ m ≤ n − 1), x 1 = x n ,