In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).
In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken...
We shall prove the following statements: Given a sequence in a Banach space enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) of the sequence such that whenever belongs to the closed convex hull of the set of weak limit points of . In case has the Banach-Saks property and is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the...