New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities

New fixed-point theorems for two maps defined on product spaces are obtained. These new results only require one of them to satisfy a noncompactness condition. Previous results required each map to satisfy a noncompactness condition. Applications of our results are given to intersection problems for transfer-closed maps, to problems on sets with convex sections, and to minimax inequalities of Sion-type. Our results generalize many well-known results.