Necessary conditions for the optimality of control in Banach spaces

1966 ◽  
pp. 63-85 ◽  
Author(s):  
Ju. V. Egorov
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions

scholarly journals Approximative K-atomic decompositions and frames in Banach spaces

2018 ◽  
Vol 26 (1/2) ◽  
pp. 153-166
Author(s):  
Shah Jahan
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic Decomposition ◽  
Necessary Conditions ◽  
Bounded Operator ◽  
Special Kind ◽  
Frame Theory ◽  
Bessel Sequence ◽  
Atomic Decompositions ◽  
Counter Example

L. Gǎvruţa (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K-atomic decompositions in Banach spaces. Also some results on the existence of approximative K-atomic decompositions are obtained. We discuss several methods to construct approximative K-atomic decomposition for Banach Spaces. Further, approximative d-frame and approximative d-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative d-Bessel sequence and approximative d-frame give rise to a bounded operator with respect to which there is an approximative K-atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.

scholarly journals Strongly Extreme Points and Middle Point Locally Uniformly Convex in Orlicz Spaces Equipped with s-Norm

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Yunan Cui ◽  
Yujia Zhan
Keyword(s):  
Banach Spaces ◽  
Orlicz Space ◽  
Necessary Conditions ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Uniformly Convex ◽  
Middle Point ◽  
Sufficient And Necessary Conditions

As is well known, the extreme points and strongly extreme points play important roles in Banach spaces. In this paper, the criterion for strongly extreme points in Orlicz spaces equipped with s-norm is given. We complete solved criterion-Orlicz space that generated by Orlicz function. And the sufficient and necessary conditions for middle point locally uniformly convex in Orlicz spaces equipped with s-norm are obtained.

scholarly journals On the range of the derivative of a smooth mapping between Banach spaces

2005 ◽  
Vol 2005 (5) ◽  
pp. 499-507
Author(s):  
Robert Deville
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differentiable Mapping ◽  
Smooth Mapping ◽  
Lipschitz Continuous ◽  
Fréchet Differentiable ◽  
New Phenomena

We survey recent results on the structure of the range of the derivative of a smooth mappingfbetween two Banach spacesXandY. We recall some necessary conditions and some sufficient conditions on a subsetAofℒ(X,Y)for the existence of a Fréchet differentiable mappingffromXintoYso thatf′(X)=A. Wheneverfis only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mappingffromℓ1(ℕ)intoℝ2, which is bounded, Lipschitz-continuous, and so that for allx,y∈ℓ1(ℕ), ifx≠y, then‖f′(x)−f′(y)‖>1.

scholarly journals On projection constant problems and the existence of metric projections in normed spaces

2001 ◽  
Vol 6 (7) ◽  
pp. 401-411 ◽  
Author(s):  
Entisarat El-Shobaky ◽  
Sahar Mohammed Ali ◽  
Wataru Takahashi
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Metric Projection ◽  
Infinite Matrix ◽  
Normed Spaces ◽  
Reflexive Banach Spaces ◽  
Linear Spaces ◽  
Sufficient And Necessary Conditions ◽  
Projection Constant

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spacesl p,1≤p<∞andc 0. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator froml p,1≤p<∞orc 0onto anyone of their maximal proper subspaces.

scholarly journals On Existence and uniqueness to solutions nonlocal integrodifferential equations

2018 ◽  
Vol 1 (25) ◽  
pp. 493-508
Author(s):  
Fawzi Mutter Ismaael
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Necessary Conditions ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Nonlinear Functional ◽  
The Fixed Point Theorem

The Study aims in this paper to give and investigate the existence and uniqueness of mild solutions to nonlinear functional integrodifferential equations in Banach Spaces. the fixed point theorem, according to Sadovskii and sutible necessary conditions, are concepts consulted to obtain the results in the work

scholarly journals INTERPOLATION METHODS DEFINED BY MEANS OF POLYGONS AND COMPACT OPERATORS

2007 ◽  
Vol 50 (3) ◽  
pp. 653-671 ◽  
Author(s):  
Luz M. Fernández-Cabrera ◽  
Antón Martínez
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Closed Operator ◽  
Compact Operators ◽  
Operator Ideals ◽  
Interpolation Methods

AbstractWe work with interpolation methods for $N$-tuples of Banach spaces associated with polygons. We compare necessary conditions for interpolating closed operator ideals with conditions required to interpolate compactness. We also establish a formula for the measure of non-compactness of interpolated operators.

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