On ℒ2, the set of Lipschitz continuous operators is a set of first category in the set of uniformly continuous operators

2019 ◽  
Author(s):  
Vincent Fromion ◽  
Gerard Scorletti
Keyword(s):  
Lipschitz Continuous ◽  
First Category ◽  
Continuous Operators ◽  
Uniformly Continuous ◽  
Set Of First Category

scholarly journals POROSITY OF CERTAIN SUBSETS OF LEBESGUE SPACES ON LOCALLY COMPACT GROUPS

2012 ◽  
Vol 88 (1) ◽  
pp. 113-122 ◽  
Author(s):  
I. AKBARBAGLU ◽  
S. MAGHSOUDI
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Young’S Inequality ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
First Category ◽  
Young's Inequality ◽  
Set Of First Category

AbstractLet $G$ be a locally compact group. In this paper, we show that if $G$ is a nondiscrete locally compact group, $p\in (0, 1)$ and $q\in (0, + \infty ] $, then $\{ (f, g)\in {L}^{p} (G)\times {L}^{q} (G): f\ast g\text{ is finite } \lambda \text{-a.e.} \} $ is a set of first category in ${L}^{p} (G)\times {L}^{q} (G)$. We also show that if $G$ is a nondiscrete locally compact group and $p, q, r\in [1, + \infty ] $ such that $1/ p+ 1/ q\gt 1+ 1/ r$, then $\{ (f, g)\in {L}^{p} (G)\times {L}^{q} (G): f\ast g\in {L}^{r} (G)\} $, is a set of first category in ${L}^{p} (G)\times {L}^{q} (G)$. Consequently, for $p, q\in [1+ \infty )$ and $r\in [1, + \infty ] $ with $1/ p+ 1/ q\gt 1+ 1/ r$, $G$ is discrete if and only if ${L}^{p} (G)\ast {L}^{q} (G)\subseteq {L}^{r} (G)$; this answers a question raised by Saeki [‘The ${L}^{p} $-conjecture and Young’s inequality’, Illinois J. Math. 34 (1990), 615–627].

scholarly journals On compositions of special cases of Lipschitz continuous operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pontus Giselsson ◽  
Walaa M. Moursi
Keyword(s):  
Optimization Problems ◽  
Nonexpansive Mappings ◽  
Structure And Properties ◽  
Monotone Inclusion ◽  
Lipschitz Continuous ◽  
Iterative Optimization ◽  
Special Cases ◽  
Firmly Nonexpansive ◽  
Nonexpansive Operators ◽  
Continuous Operators

AbstractMany iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas–Rachford and forward–backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.

scholarly journals On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

2020 ◽  
Author(s):  
Hendrik Ranocha
Keyword(s):  
Differential Equations ◽  
Hyperbolic Conservation Laws ◽  
Strong Stability ◽  
Nonlinear Operators ◽  
Dissipative Operators ◽  
Hyperbolic Conservation ◽  
Lipschitz Continuous ◽  
Fully Discrete ◽  
Runge Kutta ◽  
Continuous Operators

Abstract Explicit Runge–Kutta methods are classical and widespread techniques in the numerical solution of ordinary differential equations (ODEs). Considering partial differential equations, spatial semidiscretizations can be used to obtain systems of ODEs that are solved subsequently, resulting in fully discrete schemes. However, certain stability investigations of high-order methods for hyperbolic conservation laws are often conducted only for the semidiscrete versions. Here, strong stability (also known as monotonicity) of explicit Runge–Kutta methods for ODEs with nonlinear and semibounded (also known as dissipative) operators is investigated. Contrary to the linear case it is proven that many strong-stability-preserving (SSP) schemes of order 2 or greater are not strongly stable for general smooth and semibounded nonlinear operators. Additionally, it is shown that there are first-order-accurate explicit SSP Runge–Kutta methods that are strongly stable (monotone) for semibounded (dissipative) and Lipschitz continuous operators.

scholarly journals ALGORITHM FOR VARIATIONAL INEQUALITY PROBLEM OVER THE SET OF SOLUTIONS THE EQUILIBRIUM PROBLEMS

2020 ◽  
pp. 18-30
Author(s):  
Ya. I. Vedel ◽  
S. V. Denisov ◽  
V. V. Semenov
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Proximal Method ◽  
Iterative Regularization ◽  
Inequality Problem ◽  
Lipschitz Continuous ◽  
Convergence Of Sequences ◽  
Continuous Operators ◽  
Monotone Bifunctions

In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithm is proved.

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