Foliations, associated reductions and lower and upper solutions

In this work, we answer by the affirmative the question of J. Mawhin [14] whether the boundary value problem u + g(t, u) = 0, u(0) = u(T), u(0) = u(T), has a solution, provided the nonlinearity is aymptotically linear, satisfy a nonresonance condition to the left of the eigenvalue (2pi/T)(2) (see condition (2)) as well as an Ahmad-Lazer-Paul condition to the right of the eigenvalue 0 (see condition (3)). More generally, we generalize condition (2) considering the relation with the Fucik spectrum. Our approach is mixed as it combines variational reduction arguments and lower and upper solutions method in that miming [20,21]. In our opinion this approach is of independent interest, since we believe it is applicable in a number of different situations. The idea of this mixed approach can be resumed in the following way: a real function phi of a single real variable is associated to the functional J which describes some of its more relevant features and a pair of lower and upper solutions can be found in case phi is non-monotone. This is done without reference to any kind of Palais-Smale condition.