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The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method. [ABSTRACT FROM PUBLISHER]

Dorff et al. ['Convolutions of harmonic convex mappings', Complex Var. Elliptic Equ. 57 (5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results. [ABSTRACT FROM AUTHOR]

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra, $L(V)$ , is the Lie algebra of derivations of the moduli algebra of $V$. It is a finite-dimensional solvable algebra and its dimension $\unicode[STIX]{x1D706}(V)$ is the Yau number. Fewnomial singularities are those which can be defined by an $n$ -nomial in $n$ indeterminates. Yau and Zuo [‘A sharp upper estimate conjecture for the Yau number of weighted homogeneous isolated hypersurface singularity’, Pure Appl. Math. Q.12(1) (2016), 165–181] conjectured a bound for the Yau number and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper, we verify this conjecture for weighted homogeneous fewnomial surface singularities. [ABSTRACT FROM AUTHOR]

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying $$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$ for some $\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$, along with their rotations, play an important role among those functions that are harmonic and orientation-preserving and map the unit disc onto a convex domain. The main theorem in this paper generalises results in recent literature by showing that convex combinations of functions in ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ are convex. [ABSTRACT FROM AUTHOR]

Clunie and Sheil-Small ['Harmonic univalent functions', Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 3-25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction. [ABSTRACT FROM AUTHOR]

In this paper we prove the following result: let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)|m-n|$-torsion free semiprime ring and let $D:R\rightarrow R$ be an $(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation $(m+n)D(x^{2})=2mD(x)x+2nxD(x)$ for $x\in R$. Then $D$ is a derivation which maps $R$ into its centre. [ABSTRACT FROM PUBLISHER]

In this paper, we establish a convergence theorem for fixed points of generalised weak contractions in complete metric spaces under some new control conditions on the functions. An illustrative example of a generalised weak contraction is discussed to show how the new conditions extend known results. [ABSTRACT FROM PUBLISHER]

We study linear mappings which preserve vectors at a specific angle. We introduce the concept of $(\unicode[STIX]{x1D700},c)$ -angle preserving mappings and define $\widehat{\unicode[STIX]{x1D700}}\,(T,c)$ as the 'smallest' number $\unicode[STIX]{x1D700}$ for which $T$ is an $(\unicode[STIX]{x1D700},c)$ -angle preserving mapping. We derive an exact formula for $\widehat{\unicode[STIX]{x1D700}}\,(T,c)$ in terms of the norm $\Vert T\Vert$ and the minimum modulus $[T]$ of $T$. Finally, we characterise approximately angle preserving mappings. [ABSTRACT FROM AUTHOR]

Varieties of the form $G\times S_{\!\text{reg}}$, where $G$ is a complex semisimple group and $S_{\!\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’, Math. Res. Let., to appear] use a Hamiltonian $G$-action to endow $G\times S_{\!\text{reg}}$ with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian $G$-actions, we consider a holomorphic symplectic variety $X$ carrying an abstract integrable system induced by a Hamiltonian $G$-action. Under certain hypotheses, we show that there must exist a $G$-equivariant variety isomorphism $X\cong G\times S_{\!\text{reg}}$. [ABSTRACT FROM AUTHOR]

Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$. To characterise Lie $n$-higher derivations on ${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie $n$-higher derivations into the same problems concerning Lie $n$-derivations. We prove that: (1) if every Lie $n$-derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$-higher derivation on ${\mathcal{A}}$; (2) if every linear mapping Lie $n$-derivable at several points is a Lie $n$-derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points; (3) if every linear mapping Lie $n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$th commutators of these points. We also give several applications of these results. [ABSTRACT FROM AUTHOR]

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family. [ABSTRACT FROM PUBLISHER]

We show that any mapping between two real $p$-normed spaces, which preserves the unit distance and the midpoint of segments with distance $2^{p}$, is an isometry. Making use of it, we provide an alternative proof of some known results on the Aleksandrov question in normed spaces and also generalise these known results to $p$-normed spaces. [ABSTRACT FROM PUBLISHER]

It is proved that any surjective morphism ƒ: ℤk → K onto a locally compact group K is open for every cardinal k. This answers a question posed by Hofmann and the second author. [ABSTRACT FROM AUTHOR]

We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings. [ABSTRACT FROM PUBLISHER]

We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations. [ABSTRACT FROM AUTHOR]