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# Fixed point theorems for a class of generalized nonexpansive mappings

*Fixed Point Theory and Applications*
**volumeÂ 2016**, ArticleÂ number:Â 82 (2016)

## Abstract

In this paper, we introduce a new class of generalized nonexpansive mappings. Some new fixed point theorems for these mappings are obtained.

## 1 Introduction and preliminaries

A nonexpansive mapping has a Lipschitz constant equal to 1. The fixed point theory for such mappings is very rich [1â€“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].

Consider the Riesz Banach space \(L_{p}[0,1]\), where \(\int_{0}^{1}|f(x)|^{p} \,dx<+\infty\) and \(p\in(0,+\infty)\). Also, we have \(f=0\) when the set

has Lebesgue measure zero. In this case, we say \(f=0\) almost everywhere. An element of \(L_{p}[0,1]\) is therefore seen as a class of functions. The norm of any \(f\in L_{p}[0,1]\) is given by \(\|f\|_{p}=(\int_{0}^{1}|f(x)|^{p} \,dx)^{\frac{1}{p}}\). Throughout this paper, we will write \(L_{p}\) instead of \(L_{p}[a,b]\), \(a,b\in\mathbb{R}\) and \(\|\cdot\|\) instead of \(\|\cdot\|_{p}\).

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 âª¯. 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 \(C=\{f\in L_{p}[0,3]: f(x)=a\}\), where \(a\in[0,3]\). For \(f,g\in C\), consider the partial order relation

Let \(T:C\rightarrow C\) be defined by

Then the mapping *T* satisfies the monotone (\(C_{\frac{1}{2}}\))-condition but it fails monotone nonexpansiveness. Indeed, whenever \(f\preceq g\), if \(0\leq f(x)\leq g(x)<3\), then \(\|T(f)-T(g)\|\leq\|f-g\|\). On the other hand, \(0\leq f(x)<3\) and \(g=3\), so if \(0\leq f(x)\leq2\) and \(g=3\), then we have again \(\|T(f)-T(g)\|\leq\|f-g\|\), but if \(2< f(x)< 3\) and \(g=3\), then \(\frac{1}{2}\|f\|\nleq\|f-3\|\). Thus, the mapping *T* satisfying monotone (\(C_{\frac{1}{2}}\))-condition on \([0,3]\).

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

*Let*
*C*
*be convex and*
*T*
*monotone*. *Assume that for some*
\(f_{1}\in C\), \(f_{1}\preceq T(f_{1})\). *Then the sequence*
\(f_{n}\)
*defined by*

\(\lambda\in(0,1)\), *satisfies*

*for*
\(n\geq1\).

### Proof

First, we prove that \(f_{n}\preceq T(f_{n})\). By assumption, we have \(f_{1}\preceq T(f_{1})\). Assume that \(f_{n}\preceq T(f_{n})\), for \(n\geq1\). Then we have

*i.e.*
\(f_{n}\preceq f_{n+1}\). Since *T* is monotone, \(T(f_{n})\preceq T(f_{n+1})\). We have

Thus

for \(n\geq1\). The proof is closely modeled on Lemma 3.1 of [9].â€ƒâ–¡

Note that under the assumption of Lemma 1, if we assume \(T(f_{1})\preceq f_{1}\), then we have

for any \(n\geq1\).

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

*Let*
\(T:C\rightarrow L_{p}\)
*be a monotone*
\((C_{\lambda })\)-*condition mapping and*
\(f_{n}\)
*be a bounded a*.*f*.*p*.*s*. *for*
*T*. *Then*

*for*
\(f\in C\)
*which*
\(f_{n}\preceq f\)
*and*
\(\liminf_{n}\|f_{n}-f\|>0\), *for all*
\(n\geq1\).

### Proof

Fix \(f\in C\) such that \(f_{n}\preceq f\). Since \(f_{n}\) is an a.f.p.s., for \(\epsilon=\frac{1}{2}\liminf_{n}\|f_{n}-f\|\), there is \(n_{0}\) such that \(\|f_{n}-T(f_{n})\|<\epsilon\), for all \(n\geq n_{0}\). This implies that

for all \(n\geq n_{0}\). Since *T* satisfies the monotone (\(C_{\lambda}\))-condition, we have

for all \(n\geq n_{0}\). So by the triangle inequality and (1), we have

Thus \(\liminf_{n}\|f_{n}-T(f)\|\leq \liminf_{n}\|f_{n}-f\|\). The proof is closely modeled on Lemma 1 of [7].â€ƒâ–¡

### Lemma 3

[11]

*If*
\(\{f_{n}\}\)
*is a sequence of*
\(L_{p}\)-*uniformly bounded functions on a measure space*, *and*
\(f_{n} \rightarrow f\)
*almost everywhere*, *then*

*for all*
\(p\in(0,\infty)\).

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* (â‹†) *is an a*.*f*.*p*.*s*.

### Proof

Since \(f_{n+1}=\lambda T(f_{n})+(1-\lambda)f_{n}\), for \(n\geq1\), we have

By Lemma 1, we have \(f_{n}\preceq f_{n+1}\). Therefore, monotone (\(C_{\lambda}\))-condition implies that \(\|T(f_{n})-T(f_{n+1})\|\leq\|f_{n}-f_{n+1}\|\). Now, we can apply Lemma 3 of [1] to conclude that \(\lim_{n}\|f_{n}-T(f_{n})\|=0\).â€ƒâ–¡

### 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.

Now, we construct an a.f.p.s. according (â‹†). Let \(f_{1}=0\). So \(f_{n}=0\). Therefore

Thus \(f_{n}\) is 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

Let \(f_{n}\) be a sequence which is defined in (â‹†). By Theorem 1, \(f_{n}\) is an a.f.p.s. Since *C* is compact, \(f_{n}\) has a convergent subsequence \(f_{n_{k}}\) to *f*. By triangle inequality, we get

Since \(f_{n}\) is an a.f.p.s., we have

Again, by triangle inequality, we have

Therefore,

From equations (2) and (3), we have

By using the partially order and convergent properties \(f_{n_{k}}\preceq f\). Lemma 1 implies \(f_{n_{k}}\preceq f_{n_{k}+1}\preceq f\). So \(\Vert f_{n_{k}+1}-f_{n_{k}}\Vert \leq \Vert f-f_{n_{k}}\Vert \). Since \(f_{n_{k}+1}-f_{n _{k}}=\lambda(f_{n_{k}}-T(f_{n_{k}}))\), we get

Therefore

Thus the monotone (\(C_{\lambda}\))-condition implies

Since \(f_{n_{k}}\) is bounded, Lemma 3 implies

From equation (4), we get

From equation (5), we get

This implies that \(T(f)=f\).â€ƒâ–¡

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

Let \(C=co\{x,\sin(x)\}\), where \(x\in[-\frac{\pi }{2},\frac{\pi}{2}]\). Define a partial order on *C* as follows:

Let \(T:C\rightarrow C\) be

Since *C* is convex hull of a compact set \(\{x,\sin(x)\}\), so it is a nonempty, convex and compact subset of \(L_{p}\). Put \(f=x\). Then *f* and \(T(f)\) are comparable. Also, *T* obeys the monotone \((C_{\lambda})\)-condition. Thus, by Theorem 2, *T* has a fixed point.

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

By Theorem 1, *T* has an a.f.p.s. \(f_{n}\). Since *C* is weakly compact, there is a weakly convergent subsequence \(f_{n_{k}}\) to some \(f\in C\). If \(\liminf_{n_{k}}\|f_{n_{k}}-f\|=0\), then \(f_{n_{k}}\) is convergent and we will have the same proof of Theorem 2. On the other hand, if \(\liminf_{n_{k}}\|f_{n_{k}}-f\|>0\), then by Lemma 2,

We claim that \(f= T(f)\). Because if \(f\neq T(f)\), since \(L_{2}\) satisfies Opial condition, we have

which is a contradiction with inequality (6).â€ƒâ–¡

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?

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## Acknowledgements

The first author acknowledges Buein Zahra Technical University for supporting this research.

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Lael, F., Heidarpour, Z. Fixed point theorems for a class of generalized nonexpansive mappings.
*Fixed Point Theory Appl* **2016**, 82 (2016). https://doi.org/10.1186/s13663-016-0571-y

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DOI: https://doi.org/10.1186/s13663-016-0571-y