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Some geometric properties of a new modular space defined by Zweier operator
Fixed Point Theory and Applications volume 2013, Article number: 165 (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.
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.
2 Definition and preliminaries
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 said to satisfy the weak fixed point property if every nonempty weakly compact convex subset C and every nonexpansive mapping have a fixed point, that is, there exists such that . Property and the fixed point property were also studied by Goebel and Kirk , Toledano et al. , Benavides , Benavides and Phothi . A Banach space X is said to have property if every weakly convergent sequence on the unit sphere is convergent in norm. Clarkson  introduced the uniform convexity, and it is known that the uniform convexity implies the reflexivity of Banach spaces. Huff  introduced the concept of nearly uniform convexity of Banach spaces. A Banach space X is called uniformly convex (UC) if for each , there is such that for , the inequality implies that . For any , the drop determined by x is the set . A Banach space X has the drop property if for every closed set C disjoint with , there exists an element such that . Rolewicz  showed that the Banach space X is reflexive if X has the drop property. Later, Montesinos  extended this result and proved that X has the drop property if and only if X is reflexive and has property . A sequence is said to be ε-separated sequence for some if
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 .
For a real vector space X, a function is called a modular if it satisfies the following conditions:
if and only if ,
for all scalar α with ,
for all and all with .
The modular ρ is called convex if
for all and all with .
For any modular ρ on X, the space
is called a modular space. In general, the modular is not subadditive and thus it does not behave as a norm or a distance. But we can associate the modular with an F-norm. A functional defines an F-norm if and only if
if and , then .
F-norm defines a distance on X by . If the linear metric space is complete, then it is called an F-space. The modular space can be equipped with the following F-norm:
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 ).
Let us denote by the space of all real sequences. The Cesàro sequence spaces
were introduced by Shiue . Jagers  determined the Köthe duals of the sequence space (). It can be shown that the inclusion is strict for although it does not hold for . Also, Suantai  defined the generalized Cesàro sequence space by
where . If is bounded, then
The sequence space was defined by Bilgin  as follows:
for with , where denotes a sum over the ranges . The special case of for is the space
which was introduced by Lim . Also, the inclusion holds. A paranorm on is given by
for and .
The Z-transform of a sequence is defined by by using the Zweier operator
where ℱ is the field of all complex or real numbers. The Zweier operator was studied by Şengönül and Kayaduman .
Now we introduce a new modular sequence space by
where and . If we take , then ; if and , then . It can be easily seen that is a modular on . We define the Luxemburg norm on the sequence space as follows:
It is easy to see that the space is a Banach space with respect to the Luxemburg norm.
Throughout the paper, suppose that is bounded with for all and
for and . In addition, we will require the following inequalities:
where and with .
3 Main results
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 
If , then for any and , there exists such that
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 .
Lemma 3.6 If , then for any , there exist and such that
for all with , where and .
Proof Let be fixed. Then there exists such that . Let γ be a real number , and so there exists such that for all . Choose such that . Therefore, we have
for each and . □
Lemma 3.7 If , then for any , there exists such that implies .
Proof Suppose that lemma does not hold. So, there exist and such that and . Take , and so as . Let . There exists such that
for every with , since . We have
for all by (3.1). Therefore, and from Lemma 3.5 we have
This is a contradiction. So, the proof is complete. □
Theorem 3.8 The space has property .
Proof Let and with and . For each , we can find such that . Let
Since for each , is bounded, by using the diagonal method, we can find a subsequence of such that converges for each with . Therefore, there exists an increasing sequence of positive integers such that . Thus, there exists a sequence of positive integers with such that for all . Since , there is such that
from Lemma 3.3. However, there exist and such that
for all and by Lemma 3.6. There exists such that
for any by Lemma 3.7.
By Lemma 3.1, there exists such that
where and . Hence, we get that since . Then there exists such that . Let and . Then
We obtain from (3.3) and (3.5) that
Choose . By the inequalities (3.2), (3.3), (3.6) and the convexity of the function , we have
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.
Proof Let and be such that and be a weakly null sequence in . By , there exists independent of x such that by Lemma 3.2. Also since , by Lemma 3.1, there is such that
whenever and . Take such that
Hence, we have
and this implies that
Since , by the inequality (3.10), there exists such that
Again, by , there is such that for all
where for all . Therefore, we obtain that
by the triangle inequality of the norm. It follows from the definition of the Luxemburg norm that
and this implies that
By (3.7), (3.8), (3.11), (3.15) and since (coordinatewise), we have for any that
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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The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Et, M., Karakaş, M. & Çınar, M. Some geometric properties of a new modular space defined by Zweier operator. Fixed Point Theory Appl 2013, 165 (2013). https://doi.org/10.1186/1687-1812-2013-165
- Zweier operator
- Luxemburg norm
- modular space
- uniform Opial property