On the Kuratowski measure of noncompactness for duality mappings

George Dinca

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

Keywords


Kuratowski measure of noncompactness; smooth Banach spaces; duality mappings; p-Laplacian

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