The Extention of Mean Value Theorem in Asplund Spaces

The First Mean Value Theorem for Integrals

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

The Mean Value Theorem and Its Consequences

The point (M,f(M)) is called an absolute maximum of f if f(x) ≤ f(M) for every x in the domain of f . The point (m, f(m)) is called an absolute minimum of f if f(x) ≥ f(m) for every x in the domain of f . More than one absolute maximum or minimum may exist. For example, if f(x) = |x| for x ∈ [−1, 1] then f(x) ≤ 1 and there are absolute maxima at (1, 1) and at (−1, 1), but only one absolute mini...

φ - regular functions in Asplund spaces

We introduce in the context of Asplund spaces, a new class of φ-regular functions. This new concept generalizes the one of prox-regularity introduced by Poliquin & Rockafellar (2000) in R and extended to Banach spaces by Bernard & Thibault (2004). In particular, the class of φ-regular functions includes all lower semicontinuous convex functions, all lower-C functions, and convexly C−composite f...

The Convex Point of Continuity Property in Asplund Spaces

Given a property of Banach spaces that is hereditary, it is natural to ask whether a Banach space has the property if each of its subspaces with a w Ž . Ž . particular structure such as a Schauder basis or a Schauder finite-dix mensional decomposition has the property. The motivation for such questions is that it is much easier to deal with Banach spaces with such an additional structure. A sub...

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition