International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 390720, 9 pages
http://dx.doi.org/10.1155/2011/390720
Research Article

Fixed-Point Theory on a Frechet Topological Vector Space

Afif Ben Amar,1 Mohamed Amine Cherif,2 and Maher Mnif2

1Departement de Mathématiques, Faculté des Sciences de Gafsa, Université de Gafsa, Cite Universitaire Zarrouk, Gafsa 2112, Tunisia
2Departement de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, Route de Soukra Km 3.5, B.P.1171, Sfax 3000, Tunisia

Received 6 December 2010; Revised 14 February 2011; Accepted 15 February 2011

Academic Editor: Genaro Lopez

Copyright © 2011 Afif Ben Amar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish some versions of fixed-point theorem in a Frechet topological vector space 𝐸 . The main result is that every map 𝐴 = 𝐵 𝐶 (where 𝐵 is a continuous map and 𝐶 is a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem for U-contractions and weakly compact mappings, while the second one, by assuming that the family { 𝑇 ( ⋅ , 𝑦 ) ∶ 𝑦 ∈ 𝐶 ( 𝑀 ) where 𝑀 ⊂ 𝐸 and 𝐶 ∶ 𝑀 → 𝐸 a compact o p e r a t o r } is nonlinear 𝜑 equicontractive, we give a fixed-point theorem for the operator of the form 𝐸 𝑥 ∶ = 𝑇 ( 𝑥 , 𝐶 ( 𝑥 ) ) .