Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRepresentation of measures of noncompactness and its applications related to an initial-value problem in Banach spaces
96
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, defined on Banach spaces and its applications. With the aid of a three-time order preserving embedding theorem, we show that for every Banach space $X$, there exist a Banach function space $C(K)$ for some compact Hausdorff space $K$, and an order-preserving affine mapping $mathbb T$ from the super space $mathscr B$ of all nonempty bounded subsets of $X$ endowed with the Hausdorff metric to the positive cone $C(K)^+$ of $C(K)$ such that for every convex measure, in particular, regular measure, homogeneous measure, sublinear measure of non generalized compactness $mu$ on $X$, there is a convex function $digamma$ on the cone $V=mathbb T(mathscr B)$ which is Lipschitzian on each bounded set of $V$ such that [digamma(mathbb T(B))=mu(B),;;forall;Binmathscr B.] As its applications, we show a class of basic integral inequalities related to an initial-value problem in Banach spaces, and prove a solvability result of the initial-value problem, which is an extension of some classical results due to Goebel, Rzymowski, and Bana{s}.
rate research
Read More
An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].
In this paper, we give the definability of bilinear singular and fractional integral operators on Morrey-Banach space, as well as their commutators and we prove the boundedness of such operators on Morrey-Banach spaces. Moreover, the necessary condition for BMO via the bounedness of bilinear commutators on Morrey-Banach space is also given. As a application of our main results, we get the necessary conditions for BMO via the bounedness of bilinear integral operators on weighted Morrey space and Morrey space with variable exponents. Finally, we obtain the boundedness of bilinear C-Z operator on Morrey space with variable exponents.
We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces and it is shown that these functionals are Lipschitz continuous. The risk functionals cannot be defined on strictly larger spaces of random variables which creates a particular interest for the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
