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Fixed point theorems for a class of generalized nonexpansive mappings
- Fatemeh Lael^{1}Email author and
- Zohre Heidarpour^{2}
https://doi.org/10.1186/s13663-016-0571-y
© Lael and Heidarpour 2016
- Received: 13 February 2016
- Accepted: 20 July 2016
- Published: 1 August 2016
Abstract
In this paper, we introduce a new class of generalized nonexpansive mappings. Some new fixed point theorems for these mappings are obtained.
Keywords
- monotone mapping
- nonexpansive mapping
- fixed point
- \(L_{p}\)
MSC
- 47H10
1 Introduction and preliminaries
A nonexpansive mapping has a Lipschitz constant equal to 1. The fixed point theory for such mappings is very rich [1–5] and has many applications in nonlinear functional analysis [6].
We first commence some basic concepts about generalization of nonexpansive mappings as formulated by Suzuki et al. [7, 8].
Definition 1
[8]
Let C be a nonempty subset of a Banach space X. We say that a mapping \(T:C \rightarrow C\) satisfies condition \((C)\) on C if \(\frac{1}{2}\|x-T(x)\| \leq\|x-y\|\) implies \(\|T(x)-T(y)\| \leq\|x-y\|\), for \(x,y\in C\).
Of course, every nonexpansive mapping satisfies condition \((C)\) but the converse is not correct and you can find some counterexamples for it in [8]. So the class of mappings which has condition \((C)\) is broader than the class of nonexpansive mappings.
In [7], condition \((C)\) is generalized as follows.
Definition 2
[7]
Let C be a nonempty subset of a Banach space X and \(\lambda\in(0,1)\). We say that a mapping \(T:C \rightarrow X\) satisfies (\(C_{\lambda}\))-condition on C if \(\lambda\|x-T(x)\| \leq\|x-y\|\) implies \(\|T(x)-T(y)\| \leq\|x-y\|\), for \(x,y\in C\).
So if \(\lambda=\frac{1}{2}\), we will have condition \((C)\). There are examples that show the converse is false; see [7].
In [9], monotone nonexpansive mappings are defined in \(L_{1}[0,1]\).
We next review some notions in \(L_{p}[0,1]\). All of them can be found in [10].
In this paper, we redefine Definition 2 on a subset of Banach space \(L_{p}\) and those theorems which are proved in [9] generalize to a wider class of monotone (\(C_{\lambda}\))-condition with preserving their fixed point property.
2 Main results
Let C be a nonempty subset of \(L_{p}\) which is equipped with a vector order relation ⪯. A map \(T:C\rightarrow C\) is called monotone if for all \(f\preceq g\) we have \(T(f)\preceq T(g)\).
We generalize the (\(C_{\lambda}\))-condition as follows.
Definition 3
Let C be a nonempty subset of a Banach space \(L_{p}\). For \(\lambda\in(0,1)\), we say that a mapping T monotone (\(C_{\lambda}\))-condition on C if T is monotone and for all \(f\preceq g\), \(\lambda\|f-T(f)\| \leq\|g-f\|\) implies \(\|T(g)-T(f)\| \leq\|g-f\|\).
Note Definition 3 is a generalization of the monotone nonexpansive mapping which is defined in [9] as follows.
A map T is said to be monotone nonexpansive if T is monotone and for \(f\preceq g\), we have \(\|T(g)-T(f)\| \leq\|g-f\|\).
The next example is a direct generalization of monotone nonexpansive mapping.
Example 1
Let \(f=2.9\) and \(g=3\). Then \(f\preceq g\) while \(\|T(f)-T(g)\|\nleq \|f-g\|\). Thus, T is not monotone nonexpansive.
The following lemmas will be crucial to prove the main result of this paper.
Lemma 1
Proof
A sequence \(\{f_{n}\}\) in C is called an almost fixed point sequence for T, if \(\|f_{n}-T(f_{n})\|\rightarrow0\) (a.f.p.s. in short).
Lemma 2
Proof
Lemma 3
[11]
In the following, let C be a nonempty, convex, and bounded set and \(T:C\rightarrow C\) be a monotone \((C_{\lambda})\)-condition, for some \(\lambda\in(0,1)\).
Theorem 1
Let \(f_{1}\in C\) such that \(f_{1}\preceq T(f_{1})\). Then \(f_{n}\) defined in (⋆) is an a.f.p.s.
Proof
Example 2
We show that T, which is defined in Example 1, has an a.f.p.s. It is easy to see that C is a nonempty, convex, and bounded subset of \(L_{p}\). Also, we proved T obeys the monotone \((C_{\frac{1}{2}})\)-condition. Moreover, \(0\preceq T(0)\). Thus, by Theorem 1, T has an a.f.p.s.
Theorem 2
Let C be compact. Assume there exists \(f_{1}\in C\) such that \(f_{1}\) and \(T(f_{1})\) are comparable. Then T has a fixed point.
Proof
By Theorem 2, we can see that T in Example 1, has a fixed point.
The following example shows that monotone \((C_{\lambda})\)-condition is a direct generalization of \((C_{\lambda})\)-condition.
Example 3
Note, for \(\lambda\in(0,1)\), T does not obey the \((C_{\lambda})\)-condition. Because, for \(f=x\) and \(g=\frac{x}{2}+\frac{1}{2}\sin(x)\), we have \(\lambda\|f-T(f)\|\leq\|f-g\|\), but \(\|T(g)-T(f)\|\nleq\|f-g\|\).
Theorem 3
Let C be a weakly compact subset of \(L_{2}\). Assume, there is \(f_{1}\in C\) such that \(f_{1}\preceq T(f_{1})\). Then T has a fixed point.
Proof
This result is a generalization of the original existence theorem in [7, 9] form monotone nonexpansive to monotone \((C_{\lambda})\)-condition. Therefore this class is bigger and is used to answer the question asked by T Benavides [12]: Does X also satisfy the fixed point property for Suzuki-type mappings?
Declarations
Acknowledgements
The first author acknowledges Buein Zahra Technical University for supporting this research.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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