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 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.
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 continue with the study of greedy-type bases in quasi-Banach spaces started in [3]. In this paper, we study partially-greedy bases focusing our attention in two main results: -Characterization of partially-greedy bases in quasi-Banach spaces in terms of quasi-greediness and different conservative-like properties. -Given a $C$-partially-greedy basis in a quasi-Banach space, there exists a "renorming" such that the basis is $1$-partially-greedy.
We give an example of a positive element $a$ in some ordered Banach algebra $A$ such that its spectrum is equal to $\{1\}$ and it is not greater than or equal to the unit element of $A$.
We work in an ordered Banach space with closed generating positive cone. We show that a positive compact operator has zero spectral radius or a positive eigenvector with the corresponding eigenvalue equal to the spectral radius.
In this paper, we provide a sublinear function $p$ on ordered Banach spaces, which depends on the order structure of the space. With respect to this $p$, we study the relation between $p$-contractivity of positive semigroups and the $p$-dissipativity of its generators. The positive off-diagonal property of generators is also studied in ordered vector spaces.
Free Banach lattices over pre-ordered Banach spaces
Mathematical Institute, Leiden University（莱顿大学数学研究所）；Department of Mathematics and Applied Mathematics, University of Pretoria（比勒陀利亚大学数学与应用数学系）；College of Mathematics, Sichuan University（四川大学数学学院）
We consider a monotone increasing operator in an ordered Banach space having $u_-$ and $u_+$ as a strong super- and subsolution, respectively. In contrast with the well studied case $u_+ < u_-$, we suppose that $u_- < u_+$. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the ordered interval $[u_-,u_+].$
It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on ordered Banach space with a base have been studied. In this paper, we consider uniformly mean ergodic and asymptotically stable Markov operators on ordered Banach spaces. In terms of the ergodicity coefficient, we prove uniform mean ergodicity criterion in terms of the ergodicity coefficient. Moreover, we develop the perturbation theory for uniformly asympto…
The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with th…
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…
We introduce a notion of p-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.
In this note, we consider several notions related to the Grothendieck property. Among them, we introduce the notion "unbounded Grothendieck property" in a Banach lattice as an unbounded version of the known Grothedieck property in the Banach space theory. Beside other results, surprisingly, we show that spaces with the unbounded Grothendieck property are exactly the reflexive Banach lattices.
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.
We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that are not Banach lattice…
In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,α)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper we adapt the theory to the ordered context. We construct a pre-ordered crossed product of a Banach algebra from a pre-ordered Banach algebra dynamical system $(A,G,α)$ and a given uniformly bounded class $\mathcal{R}$ of continuous covariant representations of $(A,G,α)$. If $A$ has a positive bounded approximate left identity and $\mathcal{R}$ consists of n…
We characterize $M$-ideals in order smooth $\infty$-normed spaces by extending the notion of split faces of the state space to those of the quasi-state space. We also characterize approximate order unit spaces as those order smooth $\infty$-normed spaces $V$ that are $M$-ideals in $\tilde{V}.$ Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V.$ To prove these results, we develop an order theoretic version of the "Alfsen-Efffros' cone decomposition theorem" for order smooth $1$-normed spaces. (As a quick application of this result, we sharpen a result on the ext…
We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a Valdivia compact and every separable subspace of $C(K)$ is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.
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.
Let $A$ be a Banach algebra with a bounded left approximate identity $\{e_λ\}_{λ\inΛ}$, let $π$ be a continuous representation of $A$ on a Banach space $X$, and let $S$ be a non-empty subset of $X$ such that $\lim_λπ(e_λ)s=s$ uniformly on $S$. If $S$ is bounded, or if $\{e_λ\}_{λ\inΛ}$ is commutative, then we show that there exist $a\in A$ and maps $x_n: S\to X$ for $n\geq 1$ such that $s=π(a^n)x_n(s)$ for all $n\geq 1$ and $s\in S$. The properties of $a\in A$ and the maps $x_n$, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous fac…
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.
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…
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.
Results of a previous paper [Commun. Contemp. Math., 09 (2007) 217-251] on the existence of solutions to a nonlinear evolution equation in an abstract Lebesgue space, arising from kinetic theory, are re-obtained in the more general setting of a real ordered Banach space, with additive norm on the positive cone, which is not necessarily a (Banach) lattice. In addition, an easily correctable technical error in the aforementioned paper is pointed out, and repaired.
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 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).
We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach spa…
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 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. 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.
In this paper, we extend the concept of split variational inequality problems from Hilbert spaces to Banach spaces. Then we apply the Fan-KKM theorem to prove the existence of solutions to some split variational inequality problems and some split minimization problems in Banach spaces. We also apply fixed point theorems on Chain complete posets to show the solvability of some split variational inequality problems in partially ordered Banach space.
We consider fractional powers of non-densely defined non-negative operators in Banach spaces defined by means of the Balakrishnan operator. Under mild assumptions on the operator we show that the fractional powers can partially be obtained by a generalised Dirichlet-to-Neumann operator for a Bessel-type differential equation.
In this paper we show that by renorming an ordered Banach space, every cone P can be converted to a normal cone with constant K = 1 and consequently due to this approach every cone metric space is really a metric one and every theorem in metric space valid for cone metric space automatically.
We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to Bénilan and Crandall.
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.
It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base (OBSB), and study its properties. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous Markov chains. This extends earlier results obtained in case of von Neumann algebras. Such a result allowed to establish a category theorem for uniformly ergodic Markov operators. We find necessary …
The general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first. Motivated by recent work on barycentric maps on spaces of probability measures on ordered Banach spaces, we introduce and study a stochastic order on the space of probability measures $\mathcal{P}(X)$, where $X$ is a metric space equipped with a closed partial order, and derive several useful equivalent versions of the definition. We establish the antisymmetry and closedness of the stochastic order (and hence that…
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.
We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of our investigation is the notion of connection on an infinite-dimensional bundle, and we survey some equivalent ocurrences of such a notion in the literature.
