Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces
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Research Article Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces A. Abkar and M. Eslamian Department of Mathematics, Imam Khomeini International University, P.O.Box 288, Qazvin 34149, Iran Correspondence should be addressed to M. Eslamian, [email protected] Received 1 March 2010; Accepted 17 June 2010 Academic Editor: Tom´as Dominguez Benavides Copyright q 2010 A. Abkar and M. Eslamian. 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. In the first part of this paper, we prove the existence of common fixed points for a commuting pair consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition in a uniformly convex Banach space. In this way, we generalize the result of Dhompongsa et al. 2006. In the second part of this paper, we prove a fixed point theorem for upper semicontinuous mappings satisfying the Suzuki condition in strictly Lτ spaces; our result generalizes a recent result of Dom´ınguez-Benavides et al. 2009.
1. Introduction A mapping T on a subset E of a Banach space X is said to be nonexpansive if T x − T y ≤ x − y ,
x, y ∈ E.
1.1
In 2008, Suzuki 1 introduced a condition which is weaker than nonexpansiveness. Suzuki’s condition which was named by him the condition C reads as follows: a mapping T is said to satisfy the condition C on E if 1 x − T x ≤ x − y ⇒ T x − T y ≤ x − y, 2
x, y ∈ E.
1.2
He then proved some fixed point and convergence theorems for such mappings. We shall at times refer to this concept by saying that T is a generalized nonexpansive mapping in
2
Fixed Point Theory and Applications
the sense of Suzuki. Very recently, the current authors used a modified Suzuki condition for multivalued mappings and proved a fixed point theorem for multivalued mappings satisfying this condition in uniformly convex Banach spaces see 2. In this paper, we first present a common fixed point theorem for commuting pairs consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition. This result extends a result of Dhompongsa et al. 3. In the next part, we shall consider a recent result of Dom´ınguez-Benavides et al. 4 on the existence of fixed points in an important class of spaces which are usually called strictly Lτ spaces. These spaces contain all Lebesgue function spaces Lp Ω for p ≥ 1. In this paper, we also generalize results of Dom´ınguez-Benavides et al. 4 to upper semicontinuous mappings satisfying the Suzuki condition.
2. Preliminaries Given a mapping T on a subset E of a Banach space X, the set of its fixed points will be denoted by FixT {x ∈ E : T x x}.
2.1
We start by the following definition due to Suzuki. Definition 2.1 see 1. Let T be a mapping on a subset E of a Banach space X. The mapping T is said to satisfy the Suzuki condition
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