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In this paper, we introduce the class of demi-strongly order bounded operators on a Riesz space generalization of strongly order bounded operators. Let M be a Riesz space, an operator H from M into M is said to be a demi-strongly order bounded operator if for every net {uα} in M+ whenever 0 ≤ uα ↑ ≤ u ′′, u ′′ in M∼∼ and {uα −H(uα)} is order bounded in M, then {uα} is order bounded in M. We obtain a characterization of the b-property by the term of demi-strongly order bounded operators. In addition, we study the relationship between strongly order bounded operators and demi-strongly order bounded operators. Finally, we also investigate some properties of the class of demi-strongly order bounded operators. [ABSTRACT FROM AUTHOR]

The contraction condition in the Banach contraction principle forces a function to be continuous. Many authors overcome this obligation and weaken the hypotheses via metric spaces endowed with a partial order. In this paper, we present some coupled fixed point theorems for the functions having mixed monotone properties on ordered vector metric spaces, which are more general spaces than partially ordered metric spaces. We also define the double monotone property and investigate the previous results with this property. In the last section, we prove the uniqueness of a coupled fixed point for non-monotone functions. In addition, we present some illustrative examples to emphasize that our results are more general than the ones in the literature. [ABSTRACT FROM AUTHOR]

We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation T=Ta+Tc$T = T_a + T_c$, where Ta$T_a$ is the sum of an absolutely order summable family of disjointness preserving operators and Tc$T_c$ is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators. [ABSTRACT FROM AUTHOR]

The statistically unbounded p-convergence is an abstraction of the statistical order, unbounded order, and p-convergences. We investigate the concept of the statistically unbounded convergence on lattice-normed Riesz spaces with respect to statistical p-decreasing sequences. Also, we get some relations between this concept and the other kinds of statistical convergences on Riesz spaces. [ABSTRACT FROM AUTHOR]

A new class of the Itô integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is a motivitation to construct near-martingales in Riesz spaces. Furthermore, we state Doob–Meyer decomposition theorem for near-submartingales in Riesz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate that under which conditions the inverse of an invertible ideal operator becomes an ideal operator. To determine conditions, in the first case, we consider the center of a Riesz space and achieve the solution by using some properties of it. In the second case, we first give an example of an ideal operator which is not order bounded. Then, we research when an ideal operator is order bounded. Thus, we conclude that if an invertible operator is order bounded then being an ideal operator, a band operator, and a disjointness preserving operator are equivalent and the same situation holds for the inverses of these three operators. [ABSTRACT FROM AUTHOR]

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) [ Measure Theory , Torres Fremlin, Colchester, 2004] concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions. [ABSTRACT FROM AUTHOR]

In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the same null part is itself positive. This result allows us to study the domination problem in the class of positive relations. [ABSTRACT FROM AUTHOR]

Let U and V be two Archimedean Riesz spaces. An operator S : U → V is said to be unbounded order continuous (uo-continuous), if rα uo → 0 in U implies Srα uo → 0 in V . In this paper, we give some properties of the uo-continuous dual Uuo~ of U. We show that a nonzero linear functional f on U is uo-continuous if and only if f is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the uo-continuous dual Uuo~. In general, by giving an example that the uo-continuous dual U~ uo is not a band in U~, we obtain the conditions for the uo-continuous dual of a Banach lattice U to be a band in U~. Then, we examine the properties of uo-continuous operators. We show that S is an order continuous operator if and only if S is an unbounded order continuous operator when S is a lattice homomorphism between two Riesz spaces U and V . Finally, we proved that if an order bounded operator S : U → V between Archimedean Riesz space U and atomic Dedekind complete Riesz space V is uo-continuous, then /S/ is uo-continuous. [ABSTRACT FROM AUTHOR]

The Generalized Relative Entropy inequality is a ubiquitous property in mathematical models applied in physics or biology. In spite of its importance, it is currently proved on a case-by-case basis in the literature. Here, we show that GRE is actually a generic consequence of Łoskot-Rudnicki’s inequality that is reminiscent of Jensen’s inequality. [ABSTRACT FROM AUTHOR]

In this work we review some common fixed point theorems for four non-continuous mappings in ε-metric spaces, where the metric is Riesz space valued. These results cover famous comparable results in the existing literature by considering fewer conditions. [ABSTRACT FROM AUTHOR]

Let be a unital -module over an -algebra . With the help of Arens extension theory, a module structure on can be defined. The paper deals mainly with properties of the Arens homomorphism , which is defined by the module structure on . Necessary and sufficient conditions for an submodule of to be an order ideal are obtained. [ABSTRACT FROM AUTHOR]

In this paper we present a fixed point theorem for order-preserving Prešić type operators on ordered vector metric spaces. This result extends many results in the literature obtained for Prešić type operators both on metric spaces and partially ordered metric spaces. We also emphasize the relationships between our work and the previous ones in the literature. Finally we give an example showing the fact that neither results for Prešić type contractions on metric spaces nor the results for ordered Prešić type contractions on ordered metric space is applicable to it. [ABSTRACT FROM AUTHOR]

We present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow's theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e., multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories. [ABSTRACT FROM AUTHOR]

Statistical convergence is an active area, and it appears in a wide variety of topics. However, it has not been studied extensively in Riesz spaces. There are a few studies about the statistical convergence on Riesz spaces, but they only focus on the relationship between statistical and order convergences of sequences in Riesz spaces. In this paper, we introduce the notion of (V, λ)-order summable by using the concept of λ-statistical monotone and the λ-statistical order convergent sequences in Riesz spaces. Moreover, we give some relations between (V, λ)-order summable and λ-statistical order convergence. [ABSTRACT FROM AUTHOR]

In this paper we introduce two new classes of operators which are called strongly order continuous and order-to-unbounded order continuous operators and we study their lattice properties. With under some conditions, we show that order continuity is equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all strongly continuous linear functionals on a Riesz space E is a band of E~. [ABSTRACT FROM AUTHOR]

In this paper, we define ordered vectorial quasi contractions. We show that ordered quasi contractions are ordered vectorial quasi contractions, but the reverse is not true. We also define ordered vectorial almost contractions and present fixed point theorems for this type of contractions. Hence, we disclose many results in the literature. With the help of examples, we illustrate the relationship between these two types of contractions and some others in the literature. [ABSTRACT FROM AUTHOR]