Set-valued Mapping Cone Continuity And Cone Continuous Set-valued Stability Of The Optimal Solution

The vector optimization theories have been widely applied to many domains, like engineering design, economical and management, military and political, production and plan, resources reasonable use as well as ecological environment protection and so on. But when we solve the actual problem, we should consider many factors, for example,we not only consider the commodity's price,quality and the use value but also the different brand,different factory and different packing in similary commodity.Most of the optimized model of these problem is set-valued vector optimization problem.Suppose that X is a topological space, Y is a topological linear space, K is the internal non-empty point closed convex cone, consider the mapping F:Xâ†'P₀(Y), where P₀(Y) is the collection of all non-empty subsets in 7.This paper consists of two chapters, the first chapter research the cone continuous properties of set-valued mapping.In the main research the cone convergincy of the set net in topological linear space, the cone continuity of the set-valued mapping, the continuity of semi-continuous maps, the continuous selection of the semi-continuous maps.First section has introduced in the topological linear space the concept of the cone topology, the cone limit point and the cone cluster point of the net and of the set net, and then we give the definition of the cone lower limit point and the cone upper limit point of the set net and proved some preliminary properties. Second section we quoted about set-valued mapping some basic concepts and some basic properties which has, in which some has been able to use in the following various chapters research. Third section proposes on the set-valued mapping cone upper semi- continuity, cone lower semi-continuity with the cone continuity concepts, proposing cone lower semi-continuity overcomes the fault of which the original cone upper semi-continuity and cone lower semi-continuity for set-valued map can not distinguish between the upper semi-continuity and lower semi-continuity for single-valued map. And then gives and proves on their equal description and the close properties of the weak lower level set for upper semi- continuous set-valued maps and of the strong upper level set for lower semi-continuous set-valued maps; then has studied their some fundamental operation properties. Fourth section gives the generic continuity conclusions of the semi-continuous single-valued map and the generic cone continuity of the cone semi-continuous set-valued map by building the first category set .Fifth section gives the continuous cone selection concept, several conclusions on the continuous cone selection and approximate continuous cone selection.Second chapter gives on the upper semi-continuity of the weakly efficient solution of vector optimization and an equivalence condition of the upper semi-continuity for the cone upper semi-continuous maps, the equivalence description of the lower semi-continuity properties of the weakly efficient solution and the efficient solution for the cone upper semi-continuous vector optimization problem.