Research On Some Nonlinear Problems In Modular Space

Since the introduction of the concepts of modular space from Nakano in 1950,the researches on the related nonlinear problems in this framework have attracted the attentions of many scholars both at home and abroad.For example,the researches on the fixed point theorem,KKM theorem and minimax inequality have made some meaningful results.But the researches on the common fixed point,the equilibrium problem and the Ekeland variational principle are still rare in modular space.This paper is intended to explore the above issues under the new space framework.The main contents are as followed:1.The research on common fixed point theorems in modular spaces.Firstly the concepts of the common fixed points of four mappings and infinite groups of self mappings in modular spaces are given.And the related common fixed point theorems are obtained.Lastly the stability of the iterative procedure is discussed.2.The research on the Ekeland’s variational principle and its application in modular space.The Ekeland’s variational principle in modular space is given and proved,and the Caristi’s fixed point theorem is proved by the above proved result.3.The research on equilibrium problems in modular space.The concepts of general equilibrium and the equilibrium problem with lower and upper bounds are given in modular space.And by using the results of KKM theorem of Shabanian S et al^[14],the existence’s proof of the solution of the above equilibrium problems are given.4.The research on vector variational inequality in modular spaces.We propose a class of vector variational inequality in modular space.And by using the results of KKM theorem above,the existence of the solution of such a vector variational inequality is proved under certain conditions.