scholarly journals Datko criteria for uniform instability in Banach spaces

2021 ◽  
Vol 66 (1) ◽  
pp. 115-122
Author(s):  
Rovana Boruga Toma ◽  
Mihail Megan

The main objective of this paper is to present some necessary and sufficient conditions of Datko type for the uniform exponential and uniform polynomial instability concepts for evolution operators in Banach spaces.

2015 ◽  
Vol 93 (3) ◽  
pp. 473-485 ◽  
Author(s):  
JIAN-ZE LI

In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.


1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.


Author(s):  
Andrea Bejenaru ◽  
Mihai Postolache

This paper provides a new, symmetric, nonexpansiveness condition to extend the classical Suzuki mappings. The newly introduced property is proved to be equivalent to condition (E) on Banach spaces, while it leads to an entirely new class of mappings when going to modular vector spaces; anyhow, it still provides an extension for the modular version of condition (C). In connection with the newly defined nonexpansiveness, some necessary and sufficient conditions for the existence of fixed points are stated and proved. They are based on Mann and Ishikawa iteration procedures, convenient uniform convexities and properly selected minimizing sequences.


2009 ◽  
Vol 52 (1) ◽  
pp. 28-38 ◽  
Author(s):  
Changsun Choi ◽  
Ju Myung Kim ◽  
Keun Young Lee

AbstractNew necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.


2014 ◽  
Vol 12 (02) ◽  
pp. 195-208 ◽  
Author(s):  
STEVAN PILIPOVIĆ ◽  
DIANA T. STOEVA

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, ‖ ⋅ ‖1 ≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XF and its dual [Formula: see text], as well of some of the generating spaces of XF with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations.


Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 235
Author(s):  
Rovana Boruga(Toma) ◽  
Mihail Megan ◽  
Daniela Maria-Magdalena Toth

The aim of this paper is to present some integral characterizations for the concept of uniform stability with growth rates in Banach spaces. In this sense, we prove necessary and sufficient conditions (of Barbashin and Datko type) for an evolution operator to be uniform h- stable. As particular cases of this notion, we obtain four characterizations for uniform exponential stability and two characterizations for uniform polynomial stability.


Author(s):  
Larisa Elena Biriş ◽  
Claudia Luminiţa Mihiţ ◽  
Traian Ceauşu ◽  
Ioan-Lucian Popa

AbstractThe aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.


1990 ◽  
Vol 41 (3) ◽  
pp. 441-450
Author(s):  
Robert H. Lohman ◽  
Thaddeus J. Shura

This paper considers direct sums of normed spaces with respect to a Banach space with a normalised, unconditionally strictly monotone basis. Necessary and sufficient conditions are given for such direct sums to have the λ-property. These results are used to construct examples of reflexive Banach spaces U and V such that U has the uniform λ-property but U* fails to have the λ-property, while V and V* fail to have the λ-property.


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