Some properties of the mapping TμT_{\mu} introduced by a representation in Banach and locally convex spaces

Let $\mathcal{S}=\{T_{s}:s\in S\}$ be a representation of a semigroup $S$. We show that the mapping $T_{\mu}$ introduced by a mean on a subspace of $l^{\infty}(S)$ inherits some properties of $\mathcal{S}$ in Banach spaces and locally convex spaces. The notions of $Q$-$G$-nonexpansive mapping and $Q$-$G$-attractive point in locally convex spaces are introduced. We prove that $T_{\mu}$ is a $Q$-$G$-nonexpansive mapping when $T_{s}$ is $Q$-$G$-nonexpansive mapping for each $s\in S$ and a point in a locally convex space is $Q$-$G$-attractive point of $T_{\mu}$ if it is a $Q$-$G$-attractive point of $\mathcal{S}$

New kinds of generalized variational-like inequality problems in topological vector spaces

AbstractIn this work, we consider a generalized nonlinear variational-like inequality problem, in topological vector spaces, and, by using the KKM technique, we prove an existence theorem. Our result extends a theorem of Ahmad and Irfan [R. Ahmad, S.S. Irfan, On the generalized nonlinear variational-like inequality problems, Appl. Math. Lett. 19 (2006) 294–297]

Vector F-implicit complementarity problems in topological vector spaces

AbstractRecently, Huang and Li [J. Li, N.J. Huang, Vector F-implicit complementarity problems in Banach spaces, Appl. Math. Lett. 19 (2006) 464–471] introduced and studied a new class of vector F-implicit complementarity problems and vector F-implicit variational inequality problems in Banach spaces. In this work, we study this class in topological vector spaces and drive some existence theorems for the vector F-implicit variational inequality and vector F-implicit complementarity problem. Also, their equivalence is presented under certain conditions

Strong Vector Equilibrium Problems in Topological Vector Spaces Via KKM Maps

In this paper, we establish some existence results for strong vector equilibrium problems (for short, SVEP) in topological vector spaces. The solvability of the SVEP is presented using the Fan-KKM lemma. These results give a positive answer to an open problem proposed by Chen and Hou and generalize many important results in the recent literature.En este artículo, establecemos algunos resultados de existencia para problemas de equilibrio strong vector en espacios vectoriales topológicos (abreviadamente, SVEP). La salubilidad del SVEP es presentada usando el lema de Fan-KKM. Estos resultados dan una respuesta positiva a problemas abiertos propuestos por Chen y Hon y generalizan varios resultados importantes en la literatura reciente

An intersection theorem for set-valued mappings

summary:Given a nonempty convex set $X$ in a locally convex Hausdorff topological vector space, a nonempty set $Y$ and two set-valued mappings $T\colon X\rightrightarrows X$, $S\colon Y\rightrightarrows X$ we prove that under suitable conditions one can find an $x\in X$ which is simultaneously a fixed point for $T$ and a common point for the family of values of $S$. Applying our intersection theorem we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems