Averaging Method For Some Set Differential Systems

As one of the generalized forms of ordinary differential systems,set differential systems have many applications in the fields of physics,astronomy and engineering.Due to the complexity of the set differential systems,it is difficult to find the exact solutions.So,it is important to study the qualitative and stability problems of set differential systems.There are some results on the existence and stability of solutions for set differential systems,but the theory of approximate solutions is few.As a method to study the theory of approximate solution,the averaging method can simplify the complex system effectively and find the approximate solution.In this paper,some set differential systems are studied by using averaging method,and the relationship between the solutions of the set differential systems and the average systems is proved.The main contents include:In the first part,the basic concepts of set differential system and fuzzy differential system are introduced.In the second part,the functional differential equations with small parameter are discussed.By using averaging method,the standard functional differential equations,the functional differential equations with two sum forms at the right hand and the functional differential equations with two product forms at the right hand are studied,the full averaging method,the partial addition averaging method and partial multiplication averaging method are proved respectively.Finally,two examples are given to verify the effectiveness of the obtained results.In the third part,the impulsive differential equations with small parameter and with CancyNicoletti type multipoint boundary value problems are studied.By using averaging method,the full averaging method and the partial averaging method are proved.In the fourth part,the fuzzy control differential inclusions with small parameter are considered.By using averaging method,the full averaging method is proved,then we consider the case when the average of right-hand is absent,and the averaging method in this case is obtained.