The Research Of Convex Functional And Its Duality In The Non-reflexive Space

The good properties of convex function are widely used in variational calculus, optimization theory, optimal control, and so on. It has been a long time, the convex function is the study object of domestic and foreign scholars. What’s more, a lot of meaningful results are obtained. In this paper, we study Lagrangian convex functional and its properties of duality in the non-reflexive Banach space. And Two generalized sub-differentials are introduced, the relationship between them is also discussed. Last but not least, a meaningful conclusion is that a Lagrangian convex functional is also B-self-dual. This paper is divide into three chapters.In the first chapter, we mainly introduce the research background and development of convex functions, duality theory, and the sub-differentials of convex functions.In the second chapter, a type of weak* lower semi-continuity convex functional’s Legendre duality has been discussed. We found another convenient calculation method for Legendre duality. An calculation example have been given by using two calculation methods. We found that it is more convenience to calculate with the conclusion of theorem.In the third chapter, we introduce two concepts of generalized sub-differentials of Lagrangian convex functions, the relation between these two concepts were studied. An example is given to verify the relationship. Importantly, we prove a Lagrangian convex functional is also B-self-dual in the non-reflexive Banach space.