Abstract In this paper we introduce a gyro-analogue of L p {L^{p}} spaces; we achieve this goal by constructing normed gyrolinear structures on the set of functions from an arbitrary measure space to the Möbius gyrovector space ℳ s {\mathcal{M}_{s}} and the Einstein gyrovector space ℰ s {\mathcal{E}_{s}} . This gives rise to two major classes of gyro- L p {L^{p}} spaces: Möbius s - L p {L^{p}} gyrospaces and Einstein s - L p {L^{p}} gyrospaces. These two spaces will form a normed gyrolinear space with respect to the ∥ ⋅ ∥ p ℳ s {\|\cdot\|_{p}^{\mathcal{M}_{s}}} gyronorm and the ∥ ⋅ ∥ p ℰ s {\|\cdot\|_{p}^{\mathcal{E}_{s}}} gyronorm, respectively. We show that the Einstein L p {L^{p}} -gyrospaces and Möbius L p {L^{p}} -gyrospaces are isomorphic, and the isomorphism between them also preserves the L p {L^{p}} gyronorm, thus giving rise to a gyroisometry. Furthermore, we show that the Möbius and Einstein L p {L^{p}} gyronorms induce a complete metric space structure on the Möbius L p {L^{p}} gyrospace and the Einstein L p {L^{p}} gyrospace, respectively.
Abstract Let { ω k } {\{\omega_{k}\}} be a basis of polynomial solutions to the equation E u = 0 {Eu=0} , where E is a higher-order elliptic operator with real constant coefficients and no lower-order terms, in any number of variables. The aim of the paper is to prove that the system { ( ω k , ∂ ν ω k , … , ∂ ν m - 1 ω k ) } {\{(\omega_{k},\partial_{\nu}\omega_{k},\dots,\partial^{m-1}_{\nu}\omega_{k})\}} is complete in the Besov space [ B p , p 1 - 1 / p ( ∂ Ω ) ] m {[B^{1-{1}/{p}}_{p,p}(\partial\Omega)]^{m}} ( 1 ≤ p < ∞ ) {(1\leq p<\infty)} , Ω being a C ∞ {C^{\infty}} bounded domain in ℝ n {\mathbb{R}^{n}} such that ℝ n ∖ Ω ¯ {\mathbb{R}^{n}\setminus\overline{\Omega}} is connected.
Abstract In this paper, we introduce a new four-step fixed point iteration scheme in the framework of hyperbolic spaces as defined by Kohlenbach. We establish strong convergence theorems for the proposed iteration scheme for contractive-like mappings having a fixed point. Furthermore, we present a detailed rate analysis and prove that explicit error majorants for our iteration decay faster than those for the Abbas–Nazir iteration under suitable control-parameter assumptions. For positive affine contractions, this comparison also applies to actual errors. We also establish a data dependence result for the new scheme. Finally, we illustrate the theoretical findings through numerical examples and an application to a fractional differential equation. The examples show fewer iteration cycles at stated tolerances; iteration counts, operator-evaluation costs, and discretization errors are distinguished.
