Characterizations Of Topical And Non-topical Functions And Applications Of Optimization Problems

In this dissertation, we consider a function l: X ×X â†'R on a normed linear space X. For every x, y ∈X, we define(x, y) =sup{λ: xl λ-∈ 1yR ≤}.This function is a topical function in the first argument and a non-topical function in the second argument. We independently investigate these functions about several notions of abstract analysis such as support set, subdifferential, and some related optimization problems. The main work is as follows:In chapter 2, we present some characterizations for topical and non-topical functions, and also give specific relationships between these functions and abstract convexity. Furthermore, some relative results are also obtained about the function u(x, y) =inf{λ ∈R: x-y ≤λ1}.In chapter 3, we introduce the properties for support set and subdifferential of topical and non-topical functions with respect to downward and upward sets.In chapter 4, we propose maximal elements of the set involving strictly non-topical functions. As an application, we give the necessary and sufficient conditions for global minimum of the difference of two strictly non-topical functions.