RENDICONTI DEL SEMINARIO MATEMATICO

Periodic solutions of difference equations 27that|d B [G(s 2 ,·), B(ρ ′ ) \ R̂x , 0]| = |d B [G(s 2 ,·), B(ρ ′ ), 0] − d B [G(s 2 ,·), R̂x , 0]|= |d B [G(s 2 ,·), R̂x , 0]| = 1,and (28) with s = s 2 has a second solution in B(ρ ′ ) \ R̂x .(e) s 1 ∈ S 1 .Taking a decreasing sequence (σ k ) k∈N in ]s 1 ,∞[ converging to s 1 , a correspondingsequence (x k 1 ,..., xk n ) of solutions of (28) with s = σ k and using Lemma 3, we obtaina subsequence (x j k1 ,..., x j kn ) which converges to a solution (x 1 ,..., x n ) of (28) withs = s 1 .COROLLARY 4. If p > 0, a m > 0 and b m ∈ R (1 ≤ m ≤ n − 1), there existss 1 ∈ R such that the periodic problemDx m + a m |x m | p = s + b m (1 ≤ m ≤ n − 1), x 1 = x nhas no solution if s < s 1 , at least one solution if s = s 1 and at least two solutions ifs > s 1 .Similar arguments allow to prove the following result.THEOREM 4. If the functions f m satisfy condition(34)f m (x) → −∞ as |x| → ∞ (1 ≤ m ≤ n − 1).then there is s 1 ∈ R such that (28) has zero, at least one or at least two solutionsaccording to s > s 1 , s = s 1 or s < s 1 .COROLLARY 5. If p > 0, a m > 0 and b m ∈ R (1 ≤ m ≤ n − 1), there existss 1 ∈ R such that the periodic problemDx m − a m |x m | p = s + b m (1 ≤ m ≤ n − 1), x 1 = x nhas no solution if s > s 1 , at least one solution if s = s 1 and at least two solutions ifs < s 1 .8. One-side bounded nonlinearitiesThe nonlinearity in Ambrosetti-Prodi type problems is bounded from below and coerciveor bounded from above and anticoercive. In this section, we consider nonlinearitieswhich are bounded from below or above but have different limits at +∞ and −∞.Let n ≥ 2 be an integer and f m : R → R continuous functions (1 ≤ m ≤ n−1).Consider the problem(35)Dx m + f m (x m ) = 0 (1 ≤ m ≤ n − 1), x 1 = x n .