On characterizations of preinvex fuzzy mappingsI

Several characterizations about preinvex fuzzy mapping are obtained in this paper. Firstly, an equivalent condition of preinvex fuzzy mapping is established under certain conditions. Furthermore, the necessary and sufficient conditions for differentiable and twice differ- entiable preinvex fuzzy mapping are provided by using the given equivalent condition of preinvex fuzzy mapping. Finally, a new proof of some known important conclusions is of- fered. These results generalize and improve some known results.