On disjointness, bands and projections in partially ordered vector spaces
Jochen Glück
备注：
23 pages. This is version 3. Compared to v2, a referee's comments have been implemented, several references have been added, and several further minor improvements have been made
Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If $X$ is such a lattice, those notions seem - at first glance - intimately related to the lattice operations on $X$. The last fifteen years, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces.
In fact if $X$ is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give rise to a non-trivial structure theory.
The purpose of this note is twofold: (i) We show that, on any pre-Riesz space, the structure of the space of all band projections is remarkably close to what we have in the case of vector lattices. In particular, this space is a Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already be a vector lattice. These criteria are coined in terms of disjointness and closely related concepts, and they mark how lattice-like the order structure of pre-Riesz spaces can get before the theory collapses to the vector lattice case.
We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extens…
In this paper, we study the connections between the normality, regularity, full regularity, and chain-complete property in partially ordered Banach spaces. Then, by applying these properties, we prove some fixed point theorems on partially ordered Banach spaces. As applications of these fixed point theorems, we prove the existence of solutions of some integral equations, such as Hammerstein integral equations, in Banach spaces.
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.
We obtain coupled coincidence and coupled common fixed point theorems for mixed $g$-monotone nonlinear operators $F:X \times X \rightarrow X$ in partially ordered metric spaces. Our results are generalizations of recent coincidence point theorems due to Lakshmikantham and \' Ciri\' c [Lakshmikantham, V., \' Ciri\' c, L., \textit{Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces}, Nonlinear Anal. \textbf{70} (2009), 4341-4349], of coupled fixed point theorems established by Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point …
In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article will investigate these notions and their relations. In particular we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin,Wolk and Vulikh.
Consider an Archimedean partially ordered vector space $X$ with generating cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear projection on $X$ such that both $P$ and its complementary projection $I - P$ are positive; we prove that the range of $P$ is a band. This shows that the well-known concept of band projections on vector lattices can, to a certain extent, be transferred to the framework of ordered vector spaces.
In this article, we present some fixed point theorems in partially ordered G-metric space using the concept of $(ψ,ϕ)$- weak contraction which extend many existing fixed point theorems in such space. We also give some examples to show that if we transform a metric space into a G-metric space our results are not equivalent to the existing results in metric space.
In this paper we are concerned with existence results for a coupled system of quadratic functional differential equations. This system is reduced to a fixed point problem for a block operator matrix with nonlinear inputs. To prove the existence we are established some fixed point theorem of Dhage's type for the block matrix operator acting in partially ordered Banach algebras.
In an Archimedean directed partially ordered vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(Ω)$, where $Ω$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $Ω$. We also analyze two methods to extend bands in $X$ to $C(Ω)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered vector space with…
In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. \textbf{65} (2006) 1379-1393] and [N.V. Luong and N.X. Thuan, \textit{Coupled fixed points in partially ordered metric spaces and application}, Nonlinear Anal. \textbf{74} (2011) 983-992], by weakening the involved contractive condition. An example as well an application to nonlinear Fredholm integral equations are also given in order to …
We introduce the metric space valued in partially ordered groups, and define the convergence of sequences and the multi-valued weak contractions, etc., on the space. We then establish endpoint theorems for the defined maps. Our contributions extend the theory of cone metric space constructed by Huang and Zhang (2007) and some recent results on the fixed point and endpoint theory, such as the endpoint theorem given by Amini-Harandi (2010).
The aim of this paper is to present some fixed point theorems for generalized contractions by altering distance functions in a complete cone metric spaces endowed with a partial order. We also generalize fixed point theorems of J. Harjani, K. Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis 72 (2010) 1188-1197] from metric spaces to cone metric spaces.
In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. TMA \textbf{65} (2006) 1379-1393] by significantly weakening the involved contractive condition. Our technique of proof is essentially different and more natural. An example as well an application to periodic BVP are also given in order to illustrate the effectiveness of our generalizations.
The concept of nonlinear split ordered variational inequality problems on partially ordered vector spaces is a natural extension of linear split vector variational inequality problems on Banach spaces. The results about nonlinear split ordered variational inequality problems are immediately applied to solving nonlinear split vector optimization problems. In this paper, we prove the solvability of some nonlinear split ordered variational inequality problems by using some fixed point theorems on partially ordered spaces, in which the considered mappings may not be required to have any type of co…
In this paper we introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ in partially ordered metric spaces. The obtained results extend, complement and unify some recent coupled fixed point theorems due to Samet [B. Samet, \textit{Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces}, Nonlinear Anal. \textbf{72} (2010), 4508-4517], Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordere…
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.
In this paper, we prove several fixed point theorems on both of normal partially ordered Banach spaces and regular partially ordered Banach spaces by using the normality, regularity, full regularity, and chain -complete property. Then, by applying these theorems, we provide some existence and uniqueness of solutions to some integral equations. We also prove the solvability of some equilibrium problems in Banach spaces.
In this paper, we investigate the existence and uniqueness of fixed point for partially ordered contraction type operators in Banach Space. We also present applications to integral and differential equations.
We use the method of monotone iterations to obtain fixed point and coupled fixed point results for mixed monotone operators in the setting of partially ordered sets, with no additional assumptions on the partial order and with no convergence structure. We define the concept of attractive fixed point with respect to the partial order and obtain several criteria for the existence, uniqueness and order-attractiveness of the fixed points, both in the presence and in the absence of a coupled lower-upper fixed point. As an application, we present a fixed point result for a class of mixed monotone op…
Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of \ $C^*$-algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of $C^*$-algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those induc…
In this paper, we generalize the existence result in [14] and prove convergence theorems of the iterative scheme in [12, 16] for monotone generalized alpa-nonexpansive mappings in uniformly convex partially ordered hyperbolic metric spaces. And we also give a numerical example to show that this scheme converges faster than the scheme in [14] and apply the result to the integral equation.
The paper deals with $C^*$-algebras generated by a net of Hilbert spaces over a partially ordered set. The family of those algebras constitutes a net of $C^*$-algebras over the same set. It is shown that every such an algebra is graded by the first homotopy group of the partially ordered set. We consider inductive systems of $C^*$-algebras and their limits over maximal directed subsets. We also study properties of morphisms for nets of Hilbert spaces as well as nets of $C^*$-algebras.
