On some characterizations of preinvex fuzzy mappings

Jeyakumar (Methods Oper. Res. 55:109-125, 1985) and Weir and Mond (J. Math. Anal. Appl. 136:29-38, 1988) introduced the concept of preinvex function. The preinvex functions have some interesting properties. For example, every local minimum of a preinvex function is a global minimum and nonnegative linear combinations of preinvex functions are preinvex. Invex functions were introduced by Hanson (J. Math. Anal. Appl. 80:545-550, 1981) as a generalization of differentiable convex functions. These functions are more general than the convex and pseudo convex ones. The type of invex function is equivalent to the type of function whose stationary points are global minima. Under some conditions, an invex function is also a preinvex function. Syau (Fuzzy Sets Syst. 115:455-461, 2000) introduced the concepts of pseudoconvexity, invexity, and pseudoinvexity for fuzzy mappings of one variable by using the notion of differentiability and the results proposed by Goestschel and Voxman (Fuzzy Sets Syst. 18:31-43, 1986). Wu and Xu (Fuzzy Sets Syst 159:2090-2103, 2008) introduced the concepts of fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex, and fuzzy preinvex mapping from ℝⁿ to the set of fuzzy numbers based on the concept of differentiability of fuzzy mapping defined by Wang and Wu (Fuzzy Sets Syst. 138:559-591, 2003). In this paper, we present some characterizations of preinvex fuzzy mappings. The necessary and sufficient conditions for differentiable and twice differentiable preinvex fuzzy mapping are provided. These characterizations correct and improve previous results given by other authors. This fact is shown with examples. Moreover, we introduce additional conditions under which these results are valid.