Study On The GH-differentiable Problems Of Fuzzy Mapping

In 1972,S.S.L.Chang and L.A.Zadeh introduced the definition of fuzzy number.Since then,many scholars have carried out the research on the fuzzy numerical function(i.e.,fuzzy mapping).The differentiability of fuzzy mapping is one of the important concepts in fuzzy analysis,which plays a critical role in the study of fuzzy optimization problems and fuzzy differential equation theories.In general,there are two kinds of differentiability concepts in terms of the differentiability of fuzzy mapping and its applications.One is the concept of differentiability or H-differentiability given by H-difference;the other is the concept of generalized differentiability or gH-differentiability given by gH-difference.In the study of the gH-differentiable problem of fuzzy mapping,the concept of gH-differentiability merely considers the rate of change in the direction of coordinate axis,but ignores the rate of change in other specific directions.On the other hand,similar to classical mapping,fuzzy mapping is not necessarily gH-differentiable,so it is necessary to study and analyze fuzzy mapping whose properties are slightly weaker than gH-differentiability.That is,the problem of gH-subdifferentiability.Based on this idea,the research work done by this article is following:1.The concept of gH-directional differentiability of fuzzy mapping and its related properties are clarified,and it is proved that both gH-derivative and gH-partial derivative are the gH-directional derivatives of fuzzy mapping along coordinate axis;At the same time,the relationship of gH-directional differentiability between fuzzy mapping and interval-valued mapping is discussed,and the sufficient conditions for the gH-directional differentiability of fuzzy mapping are given by using a family of interval-valued mapping and two families of endpoint functions;Moreover,the gH-directional differentiability of fuzzy mapping is equivalent to the D-directional differentiability under certain conditions.2.The concept of LgH-directional differentiability of fuzzy mapping and its related properties are given,and the definition of LgH-partial derivative is explained by making use of the concept of LgH-derivative.Then by analysis,both LgH-derivative and LgH-partial derivative of fuzzy mapping are LgH-directional derivatives along the direction of coordinate axis;Meanwhile,the relationship between gH-directional differentiability and LgH-directional differentiability of fuzzy mapping is also discussed,and the fact that if gH-directional differentiability is certain then LgH-directional differentiability must be certain is proved.3.The concepts and properties of gH-differentiability/gH-subdifferentiability of fuzzy mapping are clarified;The relationship between differentiability/subdifferentiability and gH-differentiability/gH-subdifferentiability of fuzzy mapping is discussed.What's more,several sufficient conditions for gH-differentiable and gH-subdifferentiable are proposed;It is proved that under certain circumustance,the gH-subdifferentiability of the fuzzy mapping is equivalent to its corresponding two families of endpoint functions and they are all subdifferentiable.Finally,the problem of the gH-subdifferentiability of convex fuzzy mapping is discussed,and it is proved that the gH-subdifferentiability of fuzzy mapping at a point is equivalent to the gH-subdifferentiability of convex fuzzy mapping at that point.