RENDICONTI DEL SEMINARIO MATEMATICO

32 C. Bereanu - J. MawhinWhen a m < 0 and b m ∈ R (1 ≤ m ≤ n − 1), problem (48) has at least one solution ifand only ifn−1∑b m ≤ 0.m=1Proof. For the necessity, if problem (48) has a solution x, thenn−1∑ ∑b m = a m (x m + )p ≥ 0.m=1n−1m=1For the sufficiency, each function f m (x m ) = a m (x + m )p − b m is bounded from below by−b m . Furthermore, ifR ≥(∑ n−1m=1 b ) 1/pm∑ n−1m=1 a ,mthen ∑ n−1m=1 f m(x m ) ≥ 0 when min 1≤m≤n−1 x m ≥ R. On the other hand, ∑ n−1m=1 f m(x m )= − ∑ n−1m=1 b m ≤ 0 when max 1≤m≤n−1 x m ≤ 0. Hence the result follows from Theorem5. The proof of the other case is similar.References[1] AGARWAL R.P., Difference equations and inequalities, 2nd ed., Marcel Dekker, New York 2000.[2] BEREANU C. AND MAWHIN J., Existence and multiplicity results for periodic solutions of nonlineardifference equations, J. Difference Equations Applic. 12 (2006), 677–695.[3] BEREANU C. AND MAWHIN J., Upper and lower solutions for periodic problems : first order differencevs first order differential equations, in: “Proceed. Intern. Conf. Analysis Applic. (Craiova 2005)”,(Eds. Niculescu C. and Radulescu V.), Amer. Inst. Physics, Melville, NY, vol. 835, 2006, 30–36.[4] BEREANU C. AND MAWHIN J., Existence and multiplicity results for nonlinear second order differenceequations with Dirichlet boundary conditions, Math. Bohemica 131 (2006), 145–160.[5] CHIAPPINELLI R., MAWHIN J. ANS NUGARI R., Generalized Ambrosetti-Prodi conditions for nonlineartwo-point boundary value problems, J. Differential Equations 69 (1987), 422–434.[6] DAVIS PH.J., Circulant matrices, Wiley, New York 1979.[7] DEIMLING K., Nonlinear functional analysis, Springer, Berlin 1985.[8] FABRY C., MAWHIN J. AND NKASHAMA M., A multiplicity result for periodic solutions of forcednonlinear second order ordinary differential equations, Bull. London Math. Soc. 18 (1986), 173–180.[9] KELLEY W.G. AND PETERSON A.C., Difference equations, 2nd ed., Harcourt, San Diego 2001.[10] MAWHIN J., Topological degree methods in nonlinear boundary value problems, CBMS Reg. Conf. inMath. 40, Amer. Math. Soc., Providence 1979.[11] MAWHIN J., Points fixes, points critiques et problèmes aux limites, Sémin. Math. Sup. 92, PressesUniv. Montréal, Montréal 1985.[12] MAWHIN J., Ambrosetti-Prodi type results in nonlinear boundary value problems, Lecture Notes Math.1285, Springer, Berlin 1986, 390–413.