In this paper, we present an ongoing work to discover maximal frequent itemsets in a transactional database. We propose an algorithm called ABS for Adaptive Borders Search, which is in the spirit of algorithms based on the concept of dualization. From an abstract point of view, our contribution can be seen as an improvement of the basic APRIORI algorithm for mining maximal frequent itemsets. The key point is to decide dynamically at which iteration, if any, the dualization has to be made to avoid the enumeration of all subsets of large maximal itemsets. Once the first dualization has been done from the current negative border, APRIORI is no longer used and instead, another dualization is carried out from the positive border known so far. The process is repeated until no change occurs anymore in the positive border in construction. Experiments have been done on FIMI datasets from which tradeoffs on adaptive behavior have been proposed to guess the best iteration for the first dualization. Far from being the best implementation wrt FIMI'03 contributions, performance evaluations of ABS exhibit better performance than IBE, the only public implementation based on the concept of dualization.