On involutive division on monoids

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

On compositions of special cases of Lipschitz continuous operators

Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas–Rachford and forward–backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.

On a power-type coupled system with mean curvature operator in Minkowski space

We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u|_{\partial B}=v|_{\partial B}=0, \end{cases} $$ { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , where $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$ M ( w ) = div ( ∇ w 1 − | ∇ w | 2 ) and B is a unit ball in $\mathbb{R}^{N} (N\geq 2)$ R N ( N ≥ 2 ) . We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.

Order continuity from a topological perspective

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups, and partially ordered vector spaces, respectively. An order topology is introduced such that in the latter two settings under mild conditions order continuity is a topological property. We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.

Factorisation of orthogonal projectors

We study the problem of a special factorisation of an orthogonal projector~$P$ acting in the Hilbert space $L_2(\mathbb R)$ with $\dim\ker P<\infty$. In particular, we prove that the orthogonal projector~$P$ admits a special factorisation in the form$P=VV^*$, where $V$ is an isometric upper-triangular operator in the Banach algebra of all linear continuous operators in $L_2(\mathbb R)$. Moreover, wegive an explicit formula for the operator $V$.