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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 $φ:\mathbf R_+\to \mathbf R_+$ be a strictly increasing and continuous function such that $φ(0) = 0$, $φ(r) \to \infty$ as $r\to\infty$, and let $J_φ$ be the duality mapping corresponding to $φ$. We will prove that for every $R>0$ and every $x_0\in X$ there exists a nondecreasing function $ψ= ψ(R, x_0) :\mathbf R_+\to \mathbf R_+$ such that $ψ(0) = 0$, $ψ(r)>0$ for $r>0$, and $\langle x^*- x_0^*, x-x_0\rangle \ge ψ(\|x-x_0\|) \|x-x_0\|$ for all $x$ satisfying $\|x-x_0\|\le R$ and all $x^*\in J_φx$ and $x_0^*\in J_φx_0.$ This result extends the previous results of Prüss and Kartsatos who studied the normalized duality mapping $J$ (with $φ(r)=r$) 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 $A_λ^φ$ and resolvents $J_λ^φ$ of a maximal monotone operator $A:X\supset X\to 2^{X^*}$ on $(0, \infty) \times X$ for an arbitrary $φ$ when $X$ is reflexive and both $X$ and $X^*$ are locally uniformly convex. In addition, we discuss pseudomonotone homotopy of the Yosida approximants $A_λ^φ$ with reference to the Browder degree.