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In this paper we investigate the polynomial-like iterative equation on Riesz spaces. Since a Riesz space does not need to have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using the Knaster–Tarski fixed point theorem, we first obtain the existence and uniqueness of order-preserving solutions on convex complete sublattices of Riesz spaces. Then, restricting to R and R n , special cases of Riesz space, we obtain semi-continuous solutions and integrable solutions, respectively. Finally, we present more special cases of Riesz space in which solutions to the iterative equation can be discussed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the existence of a marginal pricing economic equilibrium in presence of increasing returns and externalities in a commodity space general enough as to encompass the vast majority of economic situations. This extends the existing literature on competitive equilibria in vector lattices by incorporating market failures, and it also generalises several non-competitive existence results to a larger class of commodity spaces. The key features are a suitable definition for the marginal pricing rule and an adaptation of the properness condition. [ABSTRACT FROM AUTHOR]

We define the product integral for two Riesz space-valued Daniell integrals and prove Fubini's theorem establishing the relation between product integrals and iterated integrals. The properties of the Fremlin tensor product of two Archimedean Riesz spaces enable one to do this in an elegant way. [ABSTRACT FROM AUTHOR]

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 Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each 0 < e ∈ A , the Yosida space for (e dd , e) , denoted Y e , has a base of clopen sets. This yields a short “Yosida based" proof of FST. On the other hand, SMP implies that each Y e has a π -base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation). [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]

Let E and F be two Archimedean Riesz spaces. An operator T : E → F is said to be unbounded order continuous (uo-continuous), if u α → uo 0 in E implies T u α → uo 0 in F. In this study, our main aim is to give the solution to two open problems which are posed by Bahramnezhad and Azar. Using this, we obtain that the space L uo (E , F) of order bounded unbounded order continuous operators is an ideal in L b (E , F) for Dedekind complete Riesz space F. In general, by giving an example that the space L uo (E , F) of order bounded unbounded order continuous operators is not a band in L b (E , F) , we obtain the conditions on E or F for the space L uo (E , F) to be a band in L b (E , F) . Then, we give the extension theorem for uo-continuous operators similar to Veksler’s theorem for order continuous operators. [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]