Research Of Some Nonlinear Questions

In chapter 1, we have discussed the equivalence property of Caristi fixed point theorem and Banach contracting mapping principle. We know Caristi fixed point theorem and Ekeland variation principle is epuivalence. In the same time, the two principles and the completeness of space is epuivalence. Moreover, under certain conditions, Banach contracting mapping principle and the completeness of space is epuivalence. So I will think over the equivalence property of Caristi fixed point theorem and Banach contracting mapping principle. We proved, under certainconditions, there exists a equivalent metric d~*, such that the mapping F which satisfies the condition of caristi fixed point theorem, is a Banach contraction mapping respect to d~*. So under certain conditions, caristi fixed point theorem and Banachcontraction map principle is equivalent.In chapter 2, we mainly discussed the problems related to Ekeland variation principle. It is well known that Ekeland variation principle plays an important role in nonlinear analysis and geometry of Banach spaces, optimization theory, and multivalued differential calculus. Just because of the importance of the Ekeland variation principle, it is necessary to study the stability of the principle. In this paper, we give an example and point out that e - solutions of Ekeland variation principle are not always lower semicontinuous in Banach spaces with tepect to uniform metric. To provide some results of stability, we prove that e - solutions is almost lower semicontinuous in general metric spaces.We discussed the properties of set valued mapping in the Caristi fixed point theorem. We know Caristi fixed point theorem and Ekeland variation principle is epuivalence and there are good properties of e - solutions in the Ekeland variation principle. So we discussed the properties of the set of fixed points in the Caristi fixed point theorem. On the other hand, we try to generalize F in the Ekeland variation principle to vector valued function.In chapter 3, we discussed the properties of continuity of set valued mapping in the vector optimization problem. Vector optimization theory have been already applied in many fields. In the world, there are many results. These results mainly discussed the best solutions of the existence and stability in the vector optimization. These solutions include effective solution, weak effective solution and added weight solution. So this text considered the properties of this solutions of set valued mapping...