  Let $X$ be a real locally uniformly convex Banach space and $X^*$ be the dual
space of $X$. Let $\varphi:\mathbf R_+\to \mathbf R_+$ be a strictly increasing
and continuous function such that $\varphi(0) = 0$, $\varphi(r) \to \infty$ as
$r\to\infty$, and let $J_\varphi$ be the duality mapping corresponding to
$\varphi$. We will prove that for every $R>0$ and every $x_0\in X$ there exists
a nondecreasing function $\psi = \psi (R, x_0) :\mathbf R_+\to \mathbf R_+$
such that $\psi(0) = 0$, $\psi(r)>0$ for $r>0$, and $\langle x^*- x_0^*,
x-x_0\rangle \ge \psi(\|x-x_0\|) \|x-x_0\|$ for all $x$ satisfying
$\|x-x_0\|\le R$ and all $x^*\in J_\varphi x$ and $x_0^*\in J_\varphi x_0.$
This result extends the previous results of Pr\"{u}ss and Kartsatos who studied
the normalized duality mapping $J$ (with $\varphi(r)=r$) for uniformly convex
and locally uniformly Banach spaces, respectively. As an application of the
above result, we give a concise proof of the continuity of the Yosida
approximants $A_\lambda^\varphi$ and resolvents $J_\lambda^\varphi$ of a
maximal monotone operator $A:X\supset X\to 2^{X^*}$ on $(0, \infty) \times X$
for an arbitrary $\varphi$ when $X$ is reflexive and both $X$ and $X^*$ are
locally uniformly convex. In addition, we discuss pseudomonotone homotopy of
the Yosida approximants $A_\lambda^\varphi$ with reference to the Browder
degree.
