2022/08/23 by Dhruba R. Adhikari, Adhikari, Dhruba R.
Computer Science · Mathematics · #47H05 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2208.10689
openalex publication_date 2022/08/23 · openalex created_date 2023/02/14 · openalex updated_date 2026/07/28
Let X be a real locally uniformly convex Banach space and X^* be the dual space of X. Let φ:\mathbf R+→ \mathbf R+ be a strictly increasing and continuous function such that φ(0) = 0, φ(r) → ∞ as r→∞, and let Jφ be the duality mapping corresponding to φ. We will prove that for every R>0 and every x0∈ X there exists a nondecreasing function ψ= ψ(R, x0) :\mathbf R+→ \mathbf R+ such that ψ(0) = 0, ψ(r)>0 for r>0, and ⟨ x^*- x0^*, x-x0⟩ ≥ ψ(‖x-x0‖) ‖x-x0‖ for all x satisfying ‖x-x0‖≤ R and all x^*∈ Jφx and x0^*∈ Jφx0. 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 Aλφ and resolvents Jλφ of a maximal monotone operator A:X⊃ X→ 2X^* on (0, ∞) × 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 Aλφ with reference to the Browder degree.