Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces
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Research Article Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces F. Akbar1 and A. R. Khan2 1 2
Department of Mathematics, University of Sargodha, Sargodha, Pakistan Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
Correspondence should be addressed to A. R. Khan, [email protected] Received 21 February 2009; Accepted 14 April 2009 Recommended by Anthony Lau Common fixed point results for some new classes of nonlinear noncommuting maps on a locally convex space are proved. As applications, related invariant approximation results are obtained. Our work includes improvements and extension of several recent developments of the existing literature on common fixed points. We also provide illustrative examples to demonstrate the generality of our results over the known ones. Copyright q 2009 F. Akbar and A. R. Khan. 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.
1. Introduction and Preliminaries In the sequel, E, τ will be a Hausdorff locally convex topological vector space. A family {pα : α ∈ I} of seminorms defined on E is said to be an associated family of seminorms for τ if the family {γU : γ > 0}, where U ni1 Uαi and Uαi {x : pαi x < 1}, forms a base of neighborhoods of zero for τ. A family {pα : α ∈ I} of seminorms defined on E is called an augmented associated family for τ if {pα : α ∈ I} is an associated family with property that the seminorm max{pα , pβ } ∈ {pα : α ∈ I} for any α, β ∈ I. The associated and augmented associated families of seminorms will be denoted by Aτ and A∗ τ, respectively. It is well known that given a locally convex space E, τ, there always exists a family {pα : α ∈ I} of seminorms defined on E such that {pα : α ∈ I} A∗ τ see 1, page 203. The following construction will be crucial. Suppose that M is a τ-bounded subset of E. For this set M we can select a number λα > 0 for each α ∈ I such that M ⊂ λα Uα , where Uα {x : pα x ≤ 1}. Clearly, B α λα Uα is τ-bounded, τ-closed, absolutely convex and contains M. The linear span EB of B in E is ∞ n1 nB. The Minkowski functional of B is a norm · B on EB . Thus EB , · B is a normed space with B as its closed unit ball and supα pα x/λα xB for each x ∈ EB for details see 1–3.
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Fixed Point Theory and Applications
Let M be a subset of a locally convex space E, τ. Let I, J : M → M be mappings. A mapping T : M → M is called I, J-Lipschitz if there exists k ≥ 0 such that pα T x − T y ≤ kpα Ix − Jy for any x, y ∈ M and for all pα ∈ A∗ τ. If k < 1 resp., k 1, then T is called an I, J-contraction resp., I, J-nonexpansive. A point x ∈ M is a common fixed coincidence point of I and T if x Ix T xIx T x. The set of coincidence points of I and T is denoted by CI, T , and the set of fixed points of T
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