Topological Methods in Nonlinear Analysis

On the Kuratowski measure of noncompactness for duality mappings

George Dinca

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Abstract

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'}( \Omega) $, $1< p< \infty$, $l/p+l/p' = 1$, viewed as duality mapping on $W_{0}^{1,p}(\Omega)$, corresponding to the gauge function $\varphi(t)=t^{p-1}$, one has $$ \alpha( -\Delta_{p}) =\begin{cases} 1 & \text{for }p=2,\\ \infty & \text{for }p\in( 1,2) \cup( 2,\infty). \end{cases} $$

Article information

Source
Topol. Methods Nonlinear Anal., Volume 40, Number 1 (2012), 181-187.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461243458

Mathematical Reviews number (MathSciNet)
MR3026107

Zentralblatt MATH identifier
1290.47052

Citation

Dinca, George. On the Kuratowski measure of noncompactness for duality mappings. Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 181--187. https://projecteuclid.org/euclid.tmna/1461243458


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