Continuity of the Yosida Approximants Corresponding to General Duality Mappings

Let $X$ be a real locally uniformly convex Banach space and $X^{\ast }$ be the dual space of $X$. Let $\phi :{\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ be a strictly increasing and continuous function such that $\phi (0)=0$, $\phi (r)\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$, and let $J_{\phi }$ be the duality mapping corresponding to $\phi$. 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 $⟨x^{\ast }-x_0^{\ast },x-x_0⟩\ge \psi (\mathrm{\parallel }x-x_0\mathrm{\parallel })\mathrm{\parallel }x-x_0\mathrm{\parallel }$ for all $x$ satisfying $\mathrm{\parallel }x-x_0\mathrm{\parallel }\le R$ and all $x^{\ast }\in J_{\phi }x$ and $x_0^{\ast }\in J_{\phi }{x}_0\mathrm{.}$ This result extends the previous results of Prüss and Kartsatos who studied the normalized duality mapping $J$ (with $\phi (r)=r$) 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 $A_{\lambda }^{\phi }$ and resolvents $J_{\lambda }^{\phi }$ of a maximal monotone operator $A:X\supset X\to 2^{X^{\ast }}$ on $(0,\mathrm{\infty })\times X$ for an arbitrary $\phi$ when $X$ is reflexive and both $X$ and $X^{\ast }$ are locally uniformly convex. In addition, we discuss pseudomonotone homotopy of the Yosida approximants $A_{\lambda }^{\phi }$ with reference to the Browder degree.