scholarly journals Adding a Nonreflecting Weakly Compact Set

2019 ◽  
Vol 60 (3) ◽  
pp. 503-521 ◽  
Author(s):  
Brent Cody
Keyword(s):  
Compact Set ◽  
Weakly Compact ◽  
Weakly Compact Set

Rearrangements of functions on unbounded domains

1994 ◽  
Vol 124 (4) ◽  
pp. 621-644 ◽  
Author(s):  
R. J. Douglas
Keyword(s):  
Variational Problem ◽  
Unbounded Domain ◽  
Extreme Points ◽  
Unbounded Domains ◽  
Compact Set ◽  
Weakly Compact ◽  
Weak Closure ◽  
Weakly Compact Set ◽  
Steady Vortices

A characterisation is provided for the weak closure of the set of rearrangements of a function on an unbounded domain. The extreme points of this convex, weakly compact set are classified. This result is used to study the maximising sequences of a variational problem for steady vortices.

scholarly journals Some fixed point theorems on non-convex sets

2017 ◽  
Vol 18 (2) ◽  
pp. 377
Author(s):  
Mohanasundaram Radhakrishnan ◽  
S. Rajesh ◽  
Sushama Agrawal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Compact Set ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Weakly Compact Set

<span style="color: #000000;">In this paper, we prove that if </span><span style="color: #008000;">$K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonempty</span><span style="color: #000000;"> weakly compact set in a </span><span style="text-decoration: underline; color: #000000;">Banach</span><span style="color: #000000;"> space </span><span style="color: #008000;">$X$</span><span style="color: #000000;">, </span><span style="color: #008000;">$T:K\to K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonexpansive</span><span style="color: #000000;"> map satisfying </span><span style="color: #008000;">$\frac{x+Tx}{2}\in K$</span><span style="color: #000000;"> for all </span><span style="color: #008000;">$x\in K$</span><span style="color: #000000;"> and if </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> is </span><span style="color: #008000;">$3-$</span><span style="color: #000000;">uniformly convex or </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> has the </span><span style="text-decoration: underline; color: #000000;">Opial</span><span style="color: #000000;"> property, then </span><span style="color: #008000;">$T$</span><span style="color: #000000;"> has a fixed point in </span><span style="color: #008000;">$K.$ <br /></span>

Every weakly compact set can be uniformly embedded into a reflexive Banach space

2009 ◽  
Vol 25 (7) ◽  
pp. 1109-1112 ◽  
Author(s):  
Li Xin Cheng ◽  
Qing Jin Cheng ◽  
Zheng Hua Luo ◽  
Wen Zhang
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Compact Set ◽  
Weakly Compact ◽  
Weakly Compact Set

Existence of solutions of Hammerstein equations of compact type

1981 ◽  
Vol 89 (1) ◽  
pp. 149-158 ◽  
Author(s):  
Ronald I. Becker
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Existence Of Solutions ◽  
Compact Set ◽  
Resonance Case ◽  
Compact Type ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Image Position ◽  
Uniformly Bounded

AbstractThe existence of solutions is considered for equations of the formfor x ∈ H (a Hilbert space), P a compact linear operator on H; Q(x) a bounded linear operator on H and continuous in x and uniformly bounded; g(x) a continuous uniformly bounded map with range in H. Two situations are considered: Q(x) lies in a weakly compact set of operators for which (a) (I − PQ(x)) is invertible (non-resonance case) or (b) (I − P(Q(x) + λI)) is invertible for 0 < λ ≤ α (resonance case).

A ξ-weak Grothendieck compactness principle

2021 ◽  
pp. 1-16
Author(s):  
KEVIN BEANLAND ◽  
RYAN M. CAUSEY
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Compact Set ◽  
Closed Convex Hull ◽  
Compact Sets ◽  
Weakly Compact ◽  
Schur Property ◽  
Compactness Principle ◽  
Weakly Compact Sets

Abstract For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

Common Fixed Points in Modular Spaces

2018 ◽  
pp. 502 ◽  
Author(s):  
Salwa Salman Abed ◽  
Karrar Emad Abdul Sada
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Fixed Point Theorems ◽  
Active Role ◽  
Common Fixed Points ◽  
Weakly Compact ◽  
Expansive Mapping ◽  
Modular Spaces ◽  
Opial’S Property ◽  
Common Fixed Point Theorems

     In this paper,there are   new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a -non-expansive mapping defined on a star-shaped weakly compact subset are proved,  Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results.  

scholarly journals A Robust Observer—Based Adaptive Control of Second—Order Systems with Input Saturation via Dead-Zone Lyapunov Functions

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 82
Author(s):  
Alejandro Rincón ◽  
Gloria M. Restrepo ◽  
Fredy E. Hoyos
Keyword(s):  
Lyapunov Functions ◽  
Dead Zone ◽  
Tracking Error ◽  
Input Saturation ◽  
Chemical Reactor ◽  
Second Order ◽  
Control Law ◽  
Compact Set ◽  
State Variables ◽  
Robust Observer

In this study, a novel robust observer-based adaptive controller was formulated for systems represented by second-order input–output dynamics with unknown second state, and it was applied to concentration tracking in a chemical reactor. By using dead-zone Lyapunov functions and adaptive backstepping method, an improved control law was derived, exhibiting faster response to changes in the output tracking error while avoiding input chattering and providing robustness to uncertain model terms. Moreover, a state observer was formulated for estimating the unknown state. The main contributions with respect to closely related designs are (i) the control law, the update law and the observer equations involve no discontinuous signals; (ii) it is guaranteed that the developed controller leads to the convergence of the tracking error to a compact set whose width is user-defined, and it does not depend on upper bounds of model terms, state variables or disturbances; and (iii) the control law exhibits a fast response to changes in the tracking error, whereas the control effort can be reduced through the controller parameters. Finally, the effectiveness of the developed controller is illustrated by the simulation of concentration tracking in a stirred chemical reactor.

scholarly journals Application of Psychometric Approach for ASD Evaluation in Russian 3–4-Year-Olds

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1608
Author(s):  
Andrey Nasledov ◽  
Sergey Miroshnikov ◽  
Liubov Tkacheva ◽  
Kirill Miroshnik ◽  
Meriam Uld Semeta
Keyword(s):  
Autistic Spectrum Disorder ◽  
Online Survey ◽  
Total Sample ◽  
Compact Set ◽  
Healthy Controls ◽  
Screening System ◽  
Children With Asd ◽  
Discriminant Analyses ◽  
Biological Problem ◽  
Choice Tasks

Background: Autistic spectrum disorder (ASD) is a significant socio-biological problem due to its wide prevalence and negative outcomes. In the current study, we aimed to develop an autism scale for early and accurate differentiation of 3- to 4-year-olds at risk for ASD since there is no systematic monitoring of young children in Russia yet. Methods: The total sample (N = 324) included 116 children with ASD, 131 children without ASD (healthy controls), and 77 children with developmental delay (DD). An online survey of specialists working with children was conducted based on a specially designed autism questionnaire consisting of 85 multiple-choice tasks distributed across 12 domains. Initially, each child was assessed by 434 items using a dichotomous scale (0 = no, 1 = yes). Factor and discriminant analyses were performed to identify a compact set of subscales that most accurately and with sufficient reliability predicted whether a child belongs to the ASD group. Results: As a result, four subscales were obtained: Sensorics, Emotions, Hyperactivity, and Communication. The high discriminability of the subscales in distinguishing the ASD group from the non-ASD group was revealed (accuracy 85.5–87.0%). Overall, the obtained subscales meet psychometric requirements and allow for creating an online screening system for wide application.

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