In this paper, we define a generalized modular sequence space by using the generalized de la Vallée-Poussin mean with a generalized Riesz transformation. Moreover, we investigate the property and the uniform Opial property which is equipped with the Luxemburg norm. Finally, we show that this space has the fixed point property.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
1 Introduction
A number of mathematicians are studying the geometric properties of Banach spaces, because such properties were identified as an important characteristic of the Banach spaces. For example, if Banach spaces have some geometric properties such as the uniformly rotund, P-convexity, Q-convexity, Banach-Saks property, then they are reflexive spaces. The investigations of the metric geometry of Banach spaces date back to 1913, when Radon [1] introduced the Kadec-Klee property (sometimes called the Radon-Riesz property, or property ), and later, when Riesz [2, 3] showed that the classical -spaces, , have the Kadec-Klee property. Although the space (with Lebesgue measure) fails to have the Kadec-Klee property. In 1936, Clarkson [4] introduced the notion of the uniform convexity property or the uniform rotund property of Banach spaces, and it was shown that with are examples of such space. In 1967, Opial [5] introduced a new property which was called the Opial property and proved that the sequence spaces () have this property but (, ) do not have it. In 1980, Huff [6] introduced the nearly uniform convexity for Banach spaces and he also proved that every nearly uniformly convex Banach space is reflexive and it has the uniformly Kadec-Klee property . In 1987, Rolewicz [7] defined the drop property and property and the characterization of property , which is proved in [8]. In 1991, Kutzarova [8] defined and studied k-nearly uniformly Banach spaces. In 1992, Prus [9] introduced the notion of the uniform Opial property. There are many papers about the geometrical properties of sequence spaces. In 2003, Suantai [10, 11] defined the generalized Cesàro sequence space with a bounded sequence of positive real numbers. In 2010, Şimşek et al. [12] introduced a new modular sequence space which is more general than the Cesàro sequence space defined by Shiue [13] and the generalized Cesàro sequence space defined by Suantai. In 2013, Mongkolkeha and Kumam [14] defined the generalized Cesàro sequence space for a bounded sequence with for all and of positive real numbers. Recently, Şimşek et al. [15, 16] defined it by the modular sequence space with de la Vallée-Poussin’s mean and studied some geometric properties in these spaces. Some examples of the geometry of sequence spaces and their generalizations have been extensively studied in [17‐20].
The main purpose of this paper is to investigate the property and the uniform Opial property equipped with the Luxemburg norm of the new modular sequence space, which is defined by using the generalized de la Vallée-Poussin mean with generalized Riesz transformation. Furthermore, we show that this space has the fixed point property.
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2 Preliminaries and notations
Let be the space of all real sequences. For , the Cesàro sequence space (, for short) of Shue is defined by
equipped with the norm
(2.1)
The generalized Cesàro sequence space for a bounded sequence of positive real numbers with for all of Suantai [10, 11] is defined by
where
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equipped with the Luxemburg norm
In the case when , for all , the generalized Cesàro sequence space is nothing but the Cesàro sequence space and the Luxemburg norm is expressed by (2.1).
Let be a nondecreasing sequence of positive real numbers tending to infinity and let and . The generalized de la Vallée-Poussin means of a sequence is defined as follows:
The modular sequence space of Şimşek et al. [15, 16] is defined by de la Vallée-Poussin’s mean, namely
where
equipped with the Luxemburg norm
Let be a sequence of positive real numbers and . Then the Riesz transformation of is defined as
(2.2)
In 2012, Mursaleen et al. [21] has modified the definition of weighted statistical convergence due to Karakaya and Chishti [22], they showed that the definition must be as follows: A sequence is weighted statistically convergent (or -convergent) to L if, for every ,
where as . In the same year, Mongkolkeha and Kumam [14] defined the generalized Cesàro sequence space for a bounded sequence with for all and of positive real numbers by
where
and with as . Thus, we see that , for all ; then reduces to defined by Khan [23]. Recently, Belen and Mohiuddine [24] generalized the concept of weighted statistical convergence due to Mursaleen et al. for a nondecreasing sequence of positive real numbers tending to infinity and let and . That is, let a sequence of nonnegative numbers be such that and as and
(2.3)
where .
A sequence is weighted λ-statistically convergent (or -convergent) to L if for every ,
Now, we define the new generalized modular sequence space for a bounded sequence of positive real numbers with for all and is a sequence of positive real numbers such that . Let as by
(2.4)
where
equipped with the Luxemburg norm
when and for .
By applying the reasoning of Remark 2.4 in [24], if we take for all , then the weighted generalized modular sequence space becomes the space . If we take for all , then the weighted generalized modular sequence space becomes the space . Also, if we take and for all , then the weighted generalized modular sequence space becomes the space .
Let be a real Banach space and let (resp., ) be a closed unit ball (resp., the unit sphere) of X. A point is an H-point of if for any sequence in X such that as , the weak convergence of to x implies that as . If every point in is an H-point of , then X is said to have the property . A Banach space X has the property if for each there exists such that implies , where denotes the Kuratowski measure noncompactness of a subset A of X defined as the infimum of all such that A can be covered by a finite union of sets of diameter less than ε. The following characterization of the property is very useful (see [25]): A Banach space X has the property if and only if for each there exists such that for each element and for each sequence in with there is an index k for which where . A Banach space X is nearly uniformly convex if for each and every sequence in with , there exists such that . A Banach space X is said to have the Opial property (see [5]) if every sequence weakly convergent to satisfies
for every . Opial proved in [5] that the sequence spaces () have this property but (, ) do not have it. A Banach space X is said to have the uniform Opial property (see [9]), if for each there exists such that for any weakly null sequence in and with the following holds:
For example, the spaces in [19, 20] have the uniform Opial property.
The ball-measure of noncompactness was defined in [26, 27] by
A Banach space X is said to have property if , where . The function Δ is called the modulus of noncompact convexity (see [26]). It has been proved in [9] that property is a useful tool in fixed point theory and that a Banach space X has property if and only if it is reflexive and has the uniform Opial property.
Throughout this paper, we assume that and and for , , we denote
In addition, we put for all .
First, we start with a brief recollection of basic concepts and facts in modular space. For a real vector space X, a function is called a modular if it satisfies the following conditions:
(i)
if and only if ;
(ii)
for all scalar α with ;
(iii)
, for all and all with .
The modular ρ is called convex if
(iv)
, for all and all with .
For modular ρ on X, the space
is called the modular space.
A sequence in is called modular convergent to if there exists a such that as .
A modular ρ is said to satisfy the -condition () if for any there exist constants and such that
for all with .
If ρ satisfies the -condition for any with dependent on a, we say that ρ has the strong -condition ().
In this section, we prove the property and uniform Opial property in a generalized modular sequence space . Finally, we show that this space has the fixed point property. First we shall give some results which are very important for our consideration.
Proposition 3.1The functionalϱis a convex modular on .
Proof Let . It is obvious that if and only if and for scalar α with . Let , with . By the convexity of the function , for all , we have
□
Proposition 3.2For , the modularϱonsatisfies the following properties:
(i)
if , thenand ;
(ii)
if , then ;
(iii)
if , then .
Proof (i) Let . Then we have
By the convexity of modular ϱ, we have , so (i) is obtained.
(ii) Let . Then
Hence (ii) is satisfied. (iii) follows from the convexity of ϱ. □
Following the line of the proof in [10, 11, 17], we get the following results.
Proposition 3.3For any , we have
(i)
if , then ;
(ii)
if , then ;
(iii)
if and only if ;
(iv)
if and only if ;
(v)
if and only if .
Proposition 3.4For any , we have
(i)
ifand , then ;
(ii)
ifand , then .
Proposition 3.5Letbe a sequence in .
(i)
Ifas , thenas .
(ii)
Ifas , thenas .
Lemma 3.6For any , there existandsuch thatfor allwith , where
andcorresponding tofor .
Proof Let be fixed. So there exist such that . Let α be a real number such that , then there exists such that for all . Choose to be a real such that . Then for each and , we have
□
Lemma 3.7For anyandthere existssuch thatimplies .
Proof For a proof of this lemma, we apply and follow the line of the proof of Theorem 1.39(4) in [29]. Suppose that the lemma does not hold, then there exist and such that and . Let . Then as . Let . By , i.e., , there exists such that
(3.1)
for every with . By (3.1), we have for all . Hence . By Proposition 3.1 and Proposition 3.2(iii), we have
which is a contradiction. □
Theorem 3.8The spaceis a Banach space with respect to the Luxemburg norm.
Proof Let be a Cauchy sequence in and . Thus there exists such that for all . By Proposition 3.3(i), we have
(3.2)
That is,
(3.3)
For fixed k, we see that
Thus, let be a Cauchy sequence in ℝ for all . Since ℝ is complete, for each , there exists such that as . Thus for fixed k and for each , we have
This implies that
(3.4)
That is,
(3.5)
as . By (3.3), we have
and hence as . So we have . Since and the linearity of the sequence space , we get . Therefore the sequence space is a Banach space, with respect to the Luxemburg norm, and the proof is complete. □
Theorem 3.9The spacehas property .
Proof Let and with . For each , there exist such that . Let
where corresponds to for . This is so since for each , is bounded. By using the diagonal method, we see that for each we can find a subsequence of such that converges for each . Therefore, for any there exists an increasing sequence such that . Hence for each there exists a sequence of positive integers with such that and, since , by Lemma 2.2 we may assume that there exists such that for all , that is,
(3.6)
for all . On the other hand, by Lemma 3.6, there exist and such that
(3.7)
for all and . From Lemma 3.7, there exists such that for any
(3.8)
Since again , by Lemma 2.1, there exists such that
(3.9)
whenever and . Since , we have . Then there exists such that . We put and ,
From (3.7) and (3.9), we have
(3.10)
By (3.6), (3.9), (3.10), and convexity of the function , for all , we have
So it follows from (3.8) that
Therefore, the space has property . □
By the facts that property implies , and implies property , property , and reflexivity (see [29‐31]). The following results are obtained directly from Theorem 3.9.
Corollary 3.10The spacehas property .
Corollary 3.11The spaceis nearly uniform convexity and reflexive.
Corollary 3.12The spacehas propertyand property .
Corollary 3.13The spaceis nearly uniform convexity and reflexive.
Corollary 3.14The spacehas propertyand .
Next, we will prove the uniform Opial property for the space .
Theorem 3.15The spacehas the uniform Opial property.
Proof Take any and with . Let be a weakly null sequence in . By , hence by Lemma 2.2 there exists independent of x such that . Also, by and Lemma 2.1, one may assert that there exists such that
(3.11)
whenever and . Choose such that
(3.12)
So, we have
(3.13)
which implies that
(3.14)
Since , there exists such that
(3.15)
for all , since weak convergence implies coordinatewise convergence. Again, by , there exists such that
(3.16)
for all . Hence, by the triangle inequality of the norm, we get
(3.17)
It follows from Proposition 3.3(ii) that
(3.18)
implies that
(3.19)
for all . By inequality (3.11), (3.12), (3.15), and (3.19), we have for any
Since and by Lemma 2.3 there exists τ depending on δ only such that , which implies that , hence the proof is complete. □
Corollary 3.16The spacehas the uniform Opial property.
For any , the spacehas the uniform Opial property.
Corollary 3.19The spacehas propertyand the fixed point property.
Corollary 3.20The spacehas propertyand the fixed point property.
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.