On the Kuratowski measure of noncompactness for duality mappings

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