The Lipschitz Property of the Metric Projection in the Hilbert Space

2020 ◽  
Vol 250 (3) ◽  
pp. 391-403
Author(s):  
M. V. Balashov
Keyword(s):  
Hilbert Space ◽  
Metric Projection ◽  
Lipschitz Property

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Approximation properties of sets with bounded complements

1981 ◽  
Vol 89 (1-2) ◽  
pp. 75-86 ◽  
Author(s):  
C. Franchetti ◽  
P. L. Papini
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Projection ◽  
Special Role ◽  
Approximation Properties ◽  
Chebyshev Sets

SynopsisGiven a Banach space X, we investigate the behaviour of the metric projection PF onto a subset F with a bounded complement.We highlight the special role of points at which d(x, F) attains a maximum. In particular, we consider the case of X as a Hilbert space: this case is related to the famous problem of the convexity of Chebyshev sets.

scholarly journals Functor of extension in Hilbert cube and Hilbert space

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Uniform Continuity ◽  
Hilbert Cube ◽  
Compact Sets ◽  
Lipschitz Property ◽  
Michigan Math ◽  
Extension Of Maps ◽  
Convergence Of Sequences

AbstractIt is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.

scholarly journals New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 466
Author(s):  
Li Wei ◽  
Yingzi Shang ◽  
Ravi P. Agarwal
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Special Cases ◽  
Accretive Mappings ◽  
Number Sequences ◽  
Zero Points

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified.

scholarly journals General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Nopparat Wairojjana ◽  
Poom Kumam
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Bounded Operator ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Linear Bounded Operator ◽  
New Methods

This paper deals with new methods for approximating a solution to the fixed point problem; findx̃∈F(T), whereHis a Hilbert space,Cis a closed convex subset ofH,fis aρ-contraction fromCintoH,0<ρ<1,Ais a strongly positive linear-bounded operator with coefficientγ̅>0,0<γ<γ̅/ρ,Tis a nonexpansive mapping onC,andPF(T)denotes the metric projection on the set of fixed point ofT. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality〈(A-γf)x̃+τ(I-S)x̃,x-x̃〉≥0forx∈F(T), whereτ∈[0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

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