Topological Methods in Nonlinear Analysis

On the Kuratowski measure of noncompactness for duality mappings

George Dinca

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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 $$ \alpha( -\Delta_{p}) =\begin{cases} 1 & \text{for }p=2,\\ \infty & \text{for }p\in( 1,2) \cup( 2,\infty). \end{cases} $$

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

First available in Project Euclid: 21 April 2016

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Dinca, George. On the Kuratowski measure of noncompactness for duality mappings. Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 181--187.

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