Fixed Point Theorems for Middle Point Linear Operators in
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Research Article Fixed Point Theorems for Middle Point Linear Operators in L1 Milena Chermisi1 and Anna Martellotti2 1 2
Fachbereich Mathematik, Universit¨at Duisburg-Essen, Lotharstraße 65, 47057 Duisburg, Germany Dipartimento di Matematica e Informatica, Universit`a di Perugia, via Vanvitelli 1, 06123 Perugia, Italy
Correspondence should be addressed to Milena Chermisi, [email protected] Received 3 February 2008; Revised 11 August 2008; Accepted 8 September 2008 Recommended by Klaus Schmitt We introduce the notion of middle point linear operators. We prove a fixed point result for middle point linear operators in L1 . We then present some examples and, as an application, we derive a Markov-Kakutani type fixed point result for commuting family of α-nonexpansive and middle point linear operators in L1 . Copyright q 2008 M. Chermisi and A. Martellotti. 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 Furi and Vignoli 1 proved that any α-nonexpansive map T : K → K on a nonempty, bounded, closed, convex subset K of a Banach space X satisfies inf T x − x 0,
x∈K
1.1
where α is the Kuratowski measure of noncompactness on X. It is of great importance to obtain the existence of fixed points for such mappings in many applications such as eigenvalue problems as well as boundary value problems, including approximation theory, variational inequalities, and complementarity problems. Such results are used in applied mathematics, engineering, and economics. In this paper, we give optimal sufficient conditions for T to have a fixed point on K in case that X L1 μ, where μ is an σ-finite measure, and as a minor application, in case that X is a reflexive Banach space. The study of fixed point theory has been pursued by many authors and many results are known in literature. In order to have an overview of the problem, we present a brief survey of most relevant fixed point theorems. Darbo 2 showed that any α-contraction T : K → K has at least one fixed point on every nonempty, bounded, closed, convex subset K of
2
Fixed Point Theory and Applications
a Banach space. Later, Sadovski˘ı 3 extended the Darbo’s result for α-condensing mappings. Belluce and Kirk 4 obtained fixed point results for nonlinear mappings T , defined on a convex and weakly compact subset K of a Banach space, for which V : I − T satisfies V x y ≤ 1 V x V y, 2 2
for any x, y ∈ K.
1.2
Lennard 5 proved that any nonexpansive map T : K → K has at least one fixed point on every nonempty, ·L1 -bounded, ρ-compact, convex subset of L1 μ, where ρ is the metric of the convergence locally in measure. For other results, we refer to 6–11. The purpose of this paper is two-fold: i to introduce the notion of middle point linear operator, which extends the notion of convexity in the sense of 1.2; ii to show that any cont
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