Fixed point theorem in subsets of topological vector spaces

We prove new existence results of fixed points for upper semicontinuous multi-valued maps with not necessarily convex values. The definition domains are assumed to have the simplicial approximation property. To cite this article: Y. Askoura, C. Godet-Thobie, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Fixed point theorem in ordered Banach spaces and applications to matrix equations

In this paper, we give an important generalization of Lakshmikantham’s theorem (Theorem 2.1 in Nonlinear Anal Theory Methods Appl 3448–3458, 2009). Our main result asserts the existence of fixed point for a class of nonlinear operators defined in an ordered Banach space, and gives a new monotone Newton-like method to compute this fixed point. We also apply our new result to some important matrix equations.     

The applications of the Fan–Browder fixed point theorem in topological ordered spaces

In this work, for topological ordered spaces, by using the Fan–Browder fixed point theorem, we obtain existence results for solutions for some generalized quasi-Ky Fan inequalities and Nash equilibrium points for a game system.     

Full characterizations of minimax inequality,fixed point theorem,saddle point theorem,and KKM principle in arbitrary topological spaces

This paper provides necessary and sufficient conditions for the existence of solutions for some important problems from optimization and non-linear analysis by replacing two typical conditions—continuity and quasiconcavity with a unique condition, weakening topological vector spaces to arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, \(\gamma \)-recursive transfer lower semicontinuity, which fully characterizes the existence of \(\gamma \)-equilibrium of minimax inequality without imposing any restrictions on topological space. The result is then used to provide full characterizations of fixed point theorem, saddle point theorem, and KKM principle.     

A fixed point theorem for contractions of rational type in partially ordered metric spaces

The purpose of this paper is to present a fixed point theorem due to Dass and Gupta (Indian J Pure Appl Math 6:1455–1458, 1975) in the context of partially ordered metric spaces.     

Bochner integrals in ordered vector spaces

We present a natural way to cover an Archimedean directed ordered vector space E by Banach spaces and extend the notion of Bochner integrability to functions with values in E. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.     

Orthomodular lattices in ordered vector spaces

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments , a < b, is a base for the Euclidean topology). Received January 7, 2005; accepted in final form November 26, 2005.     

The Schauder fixed point theorem in geodesic spaces

We give a direct proof of Schauder's fixed point theorem in the setting of geodesic metric spaces, generalizing the classical Schauder's theorem and improving a recent version of this theorem in CAT(κ)$\mathrm{CAT}(\kappa )$ spaces. As an application we prove an existence result for a variational inequality in the setting of CAT(κ)$\mathrm{CAT}(\kappa )$ spaces.     

Nonstandard hulls of ordered vector spaces

This paper undertakes the investigation of ordered vector spaces by applying nonstandard analysis. We introduce and study two types of nonstandard hulls of ordered vector spaces. Norm-nonstandard hulls of ordered Banach spaces are also investigated.     

Topological concepts in partially ordered vector spaces

Positivity - In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations....     

Directed ideals in partially ordered vector spaces

For a linear subspace I$I$ of a Riesz space there are various well-known properties that are equivalent to I$I$ being an ideal, such as I$I$ is a full Riesz subspace, I$I$ is a solid subspace, I$I$ is a Riesz subspace and the kernel of a positive linear map, I$I$ is the kernel of a Riesz homomorphism. Generalizations of these properties to partially ordered vector spaces are considered and their relations are investigated. It is shown that for directed subspaces all these generalizations are equivalent, just as in the case of Riesz spaces.     

Shape preserving approximation in vector ordered spaces

The aim of this note is to extend some classical results on the shape preserving approximation of real functions (of real variables) to functions with values in ordered vector spaces.     

Fixed point theorems in ordered abstract spaces

We extend some fixed point theorems in -spaces, obtaining extensions of the Banach fixed point theorem to partially ordered sets.

     

Extension of Caristi’s fixed point theorem to vector valued metric spaces

The paper deals with the classical Caristi fixed point theorem in vector valued metric spaces. The results obtained seem to be new in this setting.