Description

• Title:
  On the Kuratowski measure of noncompactness for duality mappings
• Creator:
  Dinca, George
• Description:
  Let $(X,\Vert \cdot\Vert ) $ be an infinite dimensional real Banach space having a Fréchet differentiable norm and $\varphi\colon \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a gauge function. Denote by $J_{\varphi}\colon X\rightarrow X^{\ast}$ the duality mapping on $X$ corresponding to $\varphi$. Then, for the Kuratowski measure of noncompactness of $J_{\varphi}$, the following estimate holds: $$ \alpha( J_{\varphi}) \geq \sup\bigg\{ \frac{\varphi(r) }{r}\ \bigg|\ r> 0\bigg\} . $$ In particular, for $-\Delta_{p}\colon W_{0}^{1,p}( \Omega)\rightarrow W^{-1,p^{\prime}}( \Omega) $, $1< p< \infty$, ${1}/{p}+{1}/{p^{\prime}} = 1$, viewed as duality mapping on $W_{0}^{1,p}(\Omega)$, corresponding to the gauge function $\varphi(t)=t^{p-1}$, one has $$ \alpha( -\Delta_{p}) =\cases 1 & \text{for }p=2,\\ \infty & \text{for }p\in( 1,2) \cup( 2,\infty). \endcases $$
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2012.04.23
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 40, No 1 (September 2012); 181 - 187;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2012.027/8376
• Subject:
  Kuratowski measure of noncompactness;
  smooth Banach spaces;
  duality mappings;
  p-Laplacian;
  47H08;
  35J70
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2012.027
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other