Maximal Theorem Research Articles

Scalarized system of nonsmooth vector quasi-variational inequalities with applications to Debreu type vector equilibrium problems

In this work, we utilize a scalarization method to introduce a system of nonsmooth vector quasi-variational inequalities. We also study their relationship to Debreu type vector equilibrium problems. Then we establish some existence results for solutions of these systems by using maximal element theorems for a family of set-valued maps.

Maximal Manipulation of Envy-Free Solutions in Economies with Indivisible Goods and Money

We consider the problem of the fair allocation of indivisible goods and money with non-quasi-linear preferences. The purpose of the present study is to examine strategic manipulation under envy-free solutions. We show that under a certain domain-richness condition, each individual obtains the welfare level of his “optimal” envy-free allocation by maximally manipulating the solutions. This maximal manipulation theorem is helpful in analyzing the set of Nash equilibrium allocations in the direct revelation games associated with a given envy-free solution: if an envy-free solution satisfies a mild condition, the set of Nash equilibrium allocations in its associated direct revelation game coincides with that of envy-free allocations.

On Equilibrium for Abstract Economies in GFC-Spaces

In this paper, the GFC-KKM mapping is introduced and GFC-KKM theorems are established in GFC-spaces. As applications, a fixed point theorem and maximal element theorem are obtained. Our results unify, improve and generalize some known results in recent reference. Finally, equilibrium existence theorems for qualitative games and abstract economies are yielded in GFC-spaces.

System of Operator Quasi Equilibrium Problems

We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive some existence theorems for solutions to these problems with and without involving Φ-condensing mappings.

The Equilibrium Existence Theorem for Systems of General Quasiequilibrium Problems in GFC-Spaces

In this paper, the GFC-KKM mapping is introduced and a GFC-KKM theorem is established in GFC-spaces. As applications, a matching theorem and a maximal element theorem are obtained. Our results unify, improve and generalize some known results in recent reference. Finally, an equilibrium existence theorem for systems of general quasiequilibrium problems is yielded in GFC-spaces.

A maximal element theorem in FWC-spaces and its applications.

A maximal element theorem is proved in finite weakly convex spaces (FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature.

New Results for Multipoint Singular Boundary Value Problems on a Measure Chain

We study the existence and uniqueness of positive solutions for a class of singularm-point boundary value problems of second order differential equations on a measure chain. A sharper sufficient condition for the existence and uniqueness ofCrd⁡1[0,T]positive solutions as well asCrd⁡1[0,T]positive solutions is obtained by the technique of lower and upper solutions and the maximal principle theorem.

A remark on maximal functions for noncommutative martingales

Let \({\mathcal{M}}\) be a finite von Neumann algebra equipped with a normal tracial state τ. It is shown that if \({\{x_n\}_{n\geq1}}\) is a sequence of positive marginales that is bounded in \({L^1(\mathcal{M},\mathcal{T})}\), then for every 0 < p < 1, there exists \({y \in L^p(\mathcal{M},\mathcal{T})}\) satisfying the property that \({x_n \leq y}\) for all \({n\geq 1}\). Thus we obtain a noncommutative analogue of a maximal function theorem from classical martingale theory.

An RS-KKM Theorem in FC-Metric Spaces with the Application to Fixed Points

In this paper, RS-KKM mappings are introduced and an RS-KKM theorem is established in noncompact complete FC-metric spaces. As applications, a Ky Fan section theorem, a maximal element theorem and a Fan-Browder type fixed point theorem are obtained in noncompact complete FC-metric spaces. These results unify, improve and generalize some known results in recent literature.

Some Properties of Quasinearly Subharmonic Functions and Maximal Theorem for Bergman Type Spaces

Let denote the class of quasinearly subharmonic functions in unit ball . We provide, following result: if and if , then , where is the radial maximal function and , and . Also, we prove a maximal theorem for Bergman type spaces.

A New Maximal Theorem in Product GFC-Spaces with Application to Systems of Generalized Mixed Vector Quasiequilibrium Problems

In this paper, a new maximal element theorem is established in product GFC-spaces. As application, a new existence theorem of solutions for systems of generalized mixed vector quasi-equilibrium problems is obtained.

Generalizations of a theorem of Paul Erdős with application

Let (Xk:k=1,2,…) be a sequence of random variables. It is not assumed that the Xkʼs are mutually independent or that they are identically distributed. Set(1)S(b,n):=∑k=b+1b+nXkandM(b,n):=max1⩽k⩽n|S(b,k)|, where b⩾0 and n⩾1 are integers. We provide bounds on the expectation EMγ(b,n) in terms of given bounds of E|S(b,n)|γ, where γ>1 is real.This problem goes back to a theorem of Erdős [P. Erdős, On the convergence of trigonometric series, J. Math. Phys. (Massachusetts Institute of Technology), 22 (1943), 37–39] on the almost everywhere convergence of such trigonometric series that the indices of the nonzero coefficients satisfy condition (B2) (see in Section 1). Our maximal Theorem 4 is a generalization of the Erdős-Stechkin maximal inequality (see both in Section 2).Relying on Theorem 4, we prove the upper part of the law of iterated logarithm for uniformly bounded, equinormed, strongly multiplicative systems (ϕk(t):k=1,2,…;t∈[0,1]). We also state the central limit theorem for uniformly bounded multiplicative (ϕk(t)) systems.

VECTOR-VALUED THEOREM FOR THE UNCENTRED MAXIMAL OPERATOR ON BESSEL–KINGMAN HYPERGROUPS

AbstractIn this paper we introduce a vector-valued uncentred maximal operator in the setting of one-dimensional Bessel–Kingman hypergoups, and prove a maximal theorem for it.

Existence Results for Nonsmooth Vector Quasi-Variational-Like Inequalities

We introduce nonsmooth vector quasi-variational-like inequalities (NVQVLI) by means of a bifunction. We establish some existence results for solutions of these inequalities by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in Al-Homidan et al. (2012), one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

A New Fixed Point Theorem in FC-Metric Spaces with the Application to Saddle Points

In this paper,with the help of the KKM technique,a new fixed point theorem for transfer compactly open valued mappings is established in FC-metric spaces.As applications,a maximal element theorem,an intersection theorem,a minimax inequality and a saddle point theorem are obtained in FC-metric spaces.Our results unify,improve and generalize some known results in recent references.

On the Littlewood–Paley [formula omitted]-function and Calderón’s area theorem

This paper contains some known and some new properties of the Littlewood–Paley g-function. By using the Fefferman–Stein vector maximal theorem and the area theorem of Calderón, we prove the equivalence of the following four conditions, for all p>0: (a)f∈Hp;(b)∫02π(∫01(1−r)|f′(reiθ)|2dr)p/2dθ<∞;(c)∫02π(∫01(1−r)supρ<r|f′(ρeiθ)|2dr)p/2dθ<∞,(d)∫02π(∑n=0∞2−2n|f′(rneiθ)|2)p/2dθ<∞, where rn=1−2−n. As a consequence of (a)⇔(b) we have: f∈Hp⇒limr→1−(1−r)|f′(reiθ)|=0 for almost all θ. The rest of the paper is mainly devoted to the presentation of Oswald’s results Oswald (1983) [14] on a generalized Littlewood–Paley g-function as well as to the proof of Calderón’s theorem on a generalized area function.

Maximal inequality and ergodic theorems for Markov groups

The paper shows that for actions of Markov semigroups, in particular, of finitely generated word hyperbolic groups, the Cesaro means of spherical averages converge almost everywhere for any function from the class Lp, p > 1.

Maximal Manipulation in Fair Allocation

We herein consider the problems of fairly allocating indivisible objects and money. The objective of the present study is to examine strategic manipulation under envy-free solutions. We therefore investigate the consequence of individual manipulation. Each individual is observed to obtain the welfare level of his “optimal” envy-free allocation by maximally manipulating the solutions. This maximal manipulation theorem leads to several fruitful results on the manipulability of envy-free solutions: (i) we present a characterization of non-manipulable preference profiles under a given envy-free solution; (ii) we analyze the set of Nash equilibrium allocations in the direct revelation games associated with a given envy-free solution; (iii) we formulate envy-free solutions that are strategy-proof for at least one individual; and (iv) we identify the functional form of the least manipulable envy-free solutions on the quasi-linear domain.

Generalized vector mixed general quasi-variational-like inequalities in Hausdorff topological vector spaces

In this paper, we consider a generalized vector mixed general quasi-variational-like inequality in Hausdorff topological vector spaces. By using maximal element theorem, we prove existence theorems for two types of generalized vector mixed general quasi-variational-like inequalities without monotonicity and compactness.

Existence Theorem for Vector Quasi-Variational-like Inequalities

In this paper, using the maximal element theorem, we prove some existence theorem for vector quasi-variational-like inequalities without monotonicity and without compactness.