A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle

where is an –Caratheodory function, that is, a function such that is measurable for every , is continuous a.e. and for any , there exists a function such that for all and a.e. . The method of lower and upper solutions for this problem is a classical topic with a wide literature (see the monograph [3] and the references therein). The most popular result is the following: If and are a couple of lower and upper solutions (see definition in Section 3) with then problem (1.1) has a solution. In this sense we speak about “usual” ordering. On the contrary, if this order is reversed ( ), it is necessary to include additional conditions in order to get solvability of problem (1.1), as it is shown by the simple example , where constant upper and lower solutions in the reversed order appear and however there is resonance. Several authors have succeed in the search of such a condition by using several strategies, which includes variational techniques [1], topological degree [11] and monotone methods [3, 13, 2]. In this paper we are interested in this last option. In few words, the strategy is to exploit an anti–maximum principle for the linear equation in order to construct a monotone approximation scheme converging to the solution. Our aim is to prove a new anti–maximum principle based on an –norm criterion (Section 2) which generalizes the known –anti–maximum principle (see [3, Lemma 4.11]). A less known anti–maximum principle appears in [12], by using an –norm criterion. Our result can be seen as a link between these two results. Such a principle enables us to obtain a new existence result (Section 3) which generalizes in some sense the previous ones. Finally, in Section 4 some applications will be developed illustrating the advantages of our result, specially when the nonlinearity does not admit the decomposition .