Reach Us
+32-10-28-02-25

**Naderi F ^{*}**

Department of Mathematics, Tarbiat Modares University, Iran

- *Corresponding Author:
- Naderi F

Department of Mathematics

Tarbiat Modares University, Iran

**Tel:**00989127309428

**E-mail:**[email protected]

**Received Date:** April 12, 2015 **Accepted Date:** May 14, 2015 **Published Date:** May 22, 2015

**Citation:** Naderi F (2015) A Fixed Point Theorem for Left Amenable Semi-Topological Semi Groups. J Phys Math 6:138. doi:10.4172/2090-0902.1000138

**Copyright:** © 2015 Naderi F. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Physical Mathematics

In this note, we extend and improve the corresponding result of Takahashi. Explanation of DeMarr’s theorem is

further generalized for some semi groups of non-expansive self- maps on K by the following considerations which are explained in the paper. The application of Zorn’s lemma and its application are explained. An application of Zorn’s lemma shows that there exists a minimal non-empty compact convex and S-invariant subset.

Non-expansive mappings; Semi-topological semi groups; Amenable; Left reversible

Let K be a subset of a Banach space E. A self-**mapping **T on K is said to be non- expansive if ||T (x) -T (y)|| ≤ ||x- y|| for all x, y ∈ K. In [1] DeMarr proved the following theorem:

**Theorem 1.1: **For any non-empty compact convex subset K of a Banach space E, each commuting family of non-expansive selfmappings on K has a common fixed point in K.

DeMarr’s theorem can be further generalized for some semigroups of non-expansive self- maps on K by the following considerations.

Let S be a * semi-topological semigrou*p, i.e. S is a semigroup with a Hausdorff topology such that for each a ∈ S, the mappings and from S into S are continuous. S is called left reversible if any two closed right ideals of S have non-void intersection. Let l

An action of S on a topological space E is a mapping from S × E into E such that (st)(x)=s(t(x)) for s, t ∈ S, x ∈ E. The action is separately continuous if it is continuous in each variable when the other is kept fixed. Every action of S on E induces a representation of S as a semigroup of self-mappings on E denoted by S, and the two semigroups are usually identified. When the action is separately continuous, each member of S is a continuous mapping on E. A subset K ⊆ E is called S-invariant if sK ⊆ K for each s ∈ S. We say that S has a common fixed point in E, if there exists a singleton S-invariant subset of E. When E is a normed space the action of S on E is called nonexpansive if ||s(x) − s(y)|| ≤ ||x – y|| for all s ∈ S and x, y ∈ E.

Takahashi [6] proved a generalization of DeMarr’s fixed point theorem as follows:

**Theorem 1.2: **Let K be a non-empty compact convex subset of a Banach space E and S be an amenable **discrete semigroup **which acts on K separately continuous and non-expansive. Then S has a common fixed point in K. It is well-known that every left amenable discrete semigroup is left reversible [4], so Mitchell [7] proved the following theorem:

**Theorem 1.3: **Let K be a non-empty compact convex subset of a Banach space E and S be a left reversible discrete semigroup which acts on K separately continuous and non- expansive. Then S has a common fixed point in K. But it is not the case that all left amenable semitopological semigroups are left reversible as the following example shows [4]:

**Example 1.4:** Let S be a topological space which is regular and Hausdorff. Then C_{b}(S) consists of constant functions only. Define on S the multiplication st=s for all s, t ∈ S. Let a ∈ S be fixed. Define μ(f)=f (a) for all a ∈ S. Then μ is a left invariant mean on C (S), but S is not reversible.

Now the question naturally arises as to whether this is true if one considers a left amenable semi-topological semigroup in Takahashi’s theorem.

In this paper, we show that the answer is affirmative. Our theorem is new and is not a result of any previous work.

The space of almost periodic functions is the space of all f ∈ C (S) such that {l_{s}f: s ∈ S} is relatively compact in the sup-norm topology of C(S) and is denoted by AP(S). For any semi-topological semigroup S we have the following theorem [1].

**Theorem 2.1:** (a)f ∈ AP(S) if and only if {r_{s}f: s ∈ S} is relatively compact in the sup-norm topology of C (S).

(b) AP(S) ⊆ LUC (S) ∩ RUC (S).

The following lemma is important in proving our main theorem and lets one replace the discrete semigroup in Takahashi’s theorem by a general semi-topological semigroup.

**Lemma 2.2: **Let S be a semi-topological semigroup which acts separately continuous and non-expansive on a compact subset M of a Banach space E. Then for each m ∈ M and each f ∈ C (M) we have f_{m} ∈ LU C (S) where f_{m}(s)=f (sm) (s ∈ S).

**Proof:** For f ∈ C (M) define a new function A: M → C (S) by A(m)=f_{m} so A(m)(s)=f (sm) for all s ∈ S. Put sup-norm topology on C(S). We show that A is continuous. Given m ∈ M, ε>0 we must find a suitable neighborhood for m such that for all m' in it the inequality holds. By continuity of f and compactness of M the function f is uniformly continuous, so there is a positive number δ such that if u, v ∈ M and ||u − v || < δ, then . By Archimedean property of numbers, there is a natural number k for which . For each in the ball and each s ∈ S we have

because the action is non-expansive. Now use uniform continuous property of f to get . Hence corresponds to ε>0 we found the ball so that if , then

for all s ∈ S. Consequently

which shows that A is continuous. On the other hand for each right translate of f_{m}=A(m) we have

that is raA(m)=A(am) hence {r_{a}f_{m} : a ∈ S}=A(Sm). The set Sm is relatively compact in M and A is continuous, so A(Sm) is relatively compact in the sup-norm topology of C(S). Therefore by theorem 2.1 part (a) we see that f_{m}= A(m) ∈ AP(S) and from part (b) f_{m} ∈ LU C(S).

Now we use the above lemma to modify Takahashi’s proof [7] for left amenable semi- topological semigroups which are not necessarily discrete.

**Theorem 2.3: **Let K be a non-empty compact convex subset of a Banach space E and S be a left amenable semi-topological semigroup which acts on K separately continuous and non-expansive. Then S has a common fixed point in K.

**Proof: **An application of Zorn’s lemma shows that there exists a minimal non-empty compact convex and S-invariant subset X ⊆ K. If X is a singleton we are done, otherwise apply Zorn’s lemma for the second time to get a minimal non-empty compact and S- invariant subset M ⊆ X.

We claim that M is S-preserved, i.e. M=sM for all s ∈ S. Let ν be a left invariant mean on LUC (S) and define μ(f)=ν(f_{m}), where f_{m} is defined as in lemma 2.2. Then by Riesz representation theorem, μ induces a regular probability measure on M (still denoted by μ) such that μ(sB)=μ(B) for all Borel sets B ⊆ M and s ∈ S. Let F be the support of μ. Each s ∈ S defines a measurable continuous function from M into M, so by basic properties of support F ⊆ sM, μ(sM)=μ(M)=1 [7]. Assume that χ_{F} is the characteristic function of F. For each s ∈ S,

(s^{−1}F means the pre-image of F under s) again by the definition and properties of support we see that F ⊆s^{−1}F , meaning that F is S-invariant. Hence F=M by the minimality of M. Consequently M=F ⊆ ⊆ sM for each s ∈ S. But M was already S-invariant, so sM=M for each s in S.

Now if M is singleton we are done, otherwise if δ(M)=diam(M) >0, we get a contradiction by DeMarr’s lemma [1] which implies that

such that r_{0}=sup{||m – u||: m ∈ M } < δ(M).

Define then X_{0} is a non-empty (indeed u ∈ X_{0}) compact convex proper subset of X such that sX_{0} ⊆ X_{0} for each s in S (the inclusion follows from the fact that M is S-preserved). But this contradicts the minimality of X. Therefore M contains only one point which is a common fixed point for the action of S.

Obviously every amenable discrete semigroup is a left amenable semi-topological semi- group, so we can deduce Takahashi’s theorem from our theorem:

**Corollary 2.4: **Let K is a non-empty compact convex subset of a Banach space E and S is an amenable discrete semigroup which acts on K separately continuous and non-expansive. Then S has a common fixed point in K [6].

- DeMarr R (1963) Common fixed points for commuting contraction mappings. Pacific J Math 13:1139-1141.
- Berglund JF, Junghen HD, Milnes P (1989) Analysis on semi groups. John Wiley & Sons Inc., New York, USA.
- Day MM (1957) Amenable semi groups. Illinois J Math 1:509-544.
- Lau ATM (2010) Normal structure and common fixed point properties for semi groups of non-expansive Mappings in Banach spaces. Fixed Point Theory and Application.
- Paterson AL (1988) Amenability. American Mathematical Society, Providence
- Takahashi W (1969) Fixed point theorem for amenable semi group of non-expansive mappings. Kodai MathSem Rep 21:383-386.
- Mitchell T (1970) Fixed points of reversible semi groups of non-expansive mappings. Kodai Math Sem Rep 22:322-323.

Select your language of interest to view the total content in your interested language

- Algebraic Geometry
- Analytical Geometry
- Axioms
- Complex Analysis
- Differential Equations
- Fourier Analysis
- Hamilton Mechanics
- Integration
- Noether's theorem
- Physical Mathematics
- Quantum Mechanics
- Quantum electrodynamics
- Relativity
- Riemannian Geometry
- Theoretical Physics
- Theory of Mathematical Modeling
- Topology
- mirror symmetry
- vector bundle

- Total views:
**12456** - [From(publication date):

November-2015 - Jan 24, 2020] - Breakdown by view type
- HTML page views :
**8550** - PDF downloads :
**3906**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2020-21