- Open Access
Some geometric properties of a new modular space defined by Zweier operator
© Et et al.; licensee Springer. 2013
- Received: 14 December 2012
- Accepted: 5 June 2013
- Published: 25 June 2013
In this paper, we define the modular space by using the Zweier operator and a modular. Then, we consider it equipped with the Luxemburg norm and also examine the uniform Opial property and property β. Finally, we show that this space has the fixed point property.
MSC:40A05, 46A45, 46B20.
- Zweier operator
- Luxemburg norm
- modular space
- uniform Opial property
Dedicated to Professor Hari M Srivastava
In literature, there are many papers about the geometrical properties of different sequence spaces such as [1–9]. Opial  introduced the Opial property and proved that the sequence spaces () have this property but (, ) does not have it. Franchetti  showed that any infinite dimensional Banach space has an equivalent norm that satisfies the Opial property. Later, Prus  introduced and investigated the uniform Opial property for Banach spaces. The Opial property is important because Banach spaces with this property have the weak fixed point property.
Let be a real Banach space and let (resp. ) be the unit sphere (resp. the unit ball) of X. A Banach space X has the Opial property if for any weakly null sequence in X and any x in , the inequality holds. We say that X has the uniform Opial property if for any there exists such that for any with and any weakly null sequence in the unit sphere of X, the inequality holds.
For a bounded set , the ball-measure of noncompactness was defined by . The function Δ defined by is called the modulus of noncompact convexity. A Banach space X is said to have property , if . This property is an important concept in the fixed point theory and a Banach space X possesses property if and only if it is reflexive and has the uniform Opial property.
A Banach space X is called nearly uniformly convex (NUC) if for every , there exists such that for every sequence with , we have . Huff  proved that every (NUC) Banach space is reflexive and has property . A Banach space X has property if and only if for each , there exists such that for each element and each sequence in with , there is an index k for which .
if and only if ,
for all scalar α with ,
for all and all with .
for all and all with .
if and , then .
If the modular ρ is convex, then the equality defines a norm which is called the Luxemburg norm.
A modular ρ is said to satisfy the -condition if for any , there exist constants , such that for all with . If ρ provides the -condition for any with dependent on a, then ρ provides the strong -condition (briefly ).
for and .
where ℱ is the field of all complex or real numbers. The Zweier operator was studied by Şengönül and Kayaduman .
It is easy to see that the space is a Banach space with respect to the Luxemburg norm.
where and with .
Since is reflexive and convex, -type spaces have many useful applications, and it is natural to consider a geometric structure of these spaces. From this point of view, we generalized the space by using the Zweier operator and then obtained the equality , that is, it was seen that the structure of the space was preserved. In this section, our goal is to investigate a geometric structure of the modular space related to the fixed point theory. For this, we will examine property and the uniform Opial property for . Finally, we will give some fixed point results. To do this, we need some results which are important in our opinion.
Lemma 3.1 
where with and .
Lemma 3.2 
If , convergence in norm and in modular are equivalent in .
Lemma 3.3 
If , then for any , there exists such that implies .
Now we give the following two lemmas without proof.
Lemma 3.4 If for any , then .
Lemma 3.5 For any , if and only if .
for all with , where and .
for each and . □
Lemma 3.7 If , then for any , there exists such that implies .
This is a contradiction. So, the proof is complete. □
Theorem 3.8 The space has property .
for any by Lemma 3.7.
So, the inequality (3.4) implies that . Consequently, the space possesses property . □
Since property implies NUC, NUC implies property and property implies reflexivity, we can give the following result from Theorem 3.8.
Corollary 3.9 The space is nearly uniform convex, reflexive and also it has property .
Theorem 3.10 The space has the uniform Opial property.
Since , it follows from Lemma 3.3 that there is τ depending on ζ only such that . □
Corollary 3.11 The space has property and the fixed point property.
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