One Theorem On Set-valued Mapping And Its Derivatives

Nonlinear analysis is a flourishing family, whose power of solving practice problem become more prominent. The set-valued analysis is main portion of nonlinear analysis, which is widely applied not only in much area such as game theory, control theory and biologic mathematics, but also in many mathematic disciplines like topology, functional analysis, variational calculus, approximation theory, convex analysis and non-smooth analysis, and differential equation and differential inclusion. Its idea and method infiltrate into lots of research of social science, natural science and technology field. The survey of theory and application of set-valued analysis is in the ascendant.Continuous selection and continuous approximation theory and set-valued mapping differential theory constitute major part of set-valued analysis. The former has become primary instrument in many area of modern mathematics, the latter hold the balance in game theory and optimization theory. In this paper, a theorem is studied thoroughly, which was originally published on the Journal of MATHEMATICAL ANALYSIS AND APPLICATIONS and modified later. The search in the paper includes as follows:Firstly, the absence of condition in original theorem is found. An example is put forward to overthrow the theorem. A new existence theorem is arrived after the condition is recomposed.Secondly, another choice and existence theory is achieved under weaker condition, which has the same result as the former. A rigorous proof is provided, in which a set of prove method is established.