Continuity of the Yosida Approximants Corresponding to General Duality Mappings

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.