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.
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 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 .
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 
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 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.
- Cui Y, Hudzik H: Some geometric properties related to fixed point theory in Cesàro spaces. Collect. Math. 1999, 50(3):277–288.MathSciNetGoogle Scholar
- Cui Y, Hudzik H: On the uniform Opial property in some modular sequence spaces. Funct. Approx. Comment. Math. 1998, XXVI: 93–102.MathSciNetGoogle Scholar
- Karakaya V: Some geometric properties of sequence spaces involving lacunary sequence. J. Inequal. Appl. 2007., 2007: Article ID 81028Google Scholar
- Savaş E, Karakaya V, Şimşek N:Some -type new sequence spaces and their geometric properties. Abstr. Appl. Anal. 2009., 2009: Article ID 696971Google Scholar
- Şimşek N, Savaş E, Karakaya V: Some geometric and topological properties of a new sequence space defined by de la Vallée-Poussin mean. J. Comput. Anal. Appl. 2010, 12(4):768–779.MathSciNetGoogle Scholar
- Maligranda L, Petrot N, Suantai S: On the James constant and B -convexity of Cesàro and Cesàro-Orlicz sequences spaces. J. Math. Anal. Appl. 2007, 326(1):312–331. 10.1016/j.jmaa.2006.02.085MathSciNetView ArticleGoogle Scholar
- Mursaleen M, Çolak R, Et M: Some geometric inequalities in a new Banach sequence space. J. Inequal. Appl. 2007., 2007: Article ID 86757Google Scholar
- Petrot N, Suantai S: On uniform Kadec-Klee properties and rotundity in generalized Cesàro sequence spaces. Int. J. Math. Sci. 2004, 2: 91–97.MathSciNetView ArticleGoogle Scholar
- Petrot N, Suantai S: Uniform Opial properties in generalized Cesàro sequence spaces. Nonlinear Anal. 2005, 63(8):1116–1125. 10.1016/j.na.2005.05.032MathSciNetView ArticleGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar
- Franchetti C: Duality mapping and homeomorphisms in Banach theory. In Proceedings of Research Workshop on Banach Spaces Theory. University of Iowa Press, Iowa City; 1981.Google Scholar
- Prus S: Banach spaces with uniform Opial property. Nonlinear Anal. 1992, 8: 697–704.MathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk W: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
- Toledano JMA, Benavides TD, Acedo GL: Measureness of noncompactness. 99. In Metric Fixed Point Theory, Operator Theory: Advances and Applications. Birkhäuser, Basel; 1997.Google Scholar
- Benavides TD: Weak uniform normal structure in direct sum spaces. Stud. Math. 1992, 103(37):293–299.Google Scholar
- Benavides TD, Phothi S: The fixed point property under renorming in some classes of Banach spaces. Nonlinear Anal. 2010, 72(3):1409–1416.MathSciNetView ArticleGoogle Scholar
- Clarkson JA: Uniformly convex spaces. Trans. Am. Math. Soc. 1936, 40: 396–414. 10.1090/S0002-9947-1936-1501880-4MathSciNetView ArticleGoogle Scholar
- Huff R: Banach spaces which are nearly uniformly convex. Rocky Mt. J. Math. 1980, 10(4):743–749. 10.1216/RMJ-1980-10-4-743MathSciNetView ArticleGoogle Scholar
- Rolewicz S: On Δ-uniform convexity and drop property. Stud. Math. 1987, 87(2):181–191.MathSciNetGoogle Scholar
- Montesinos V: Drop property equals reflexivity. Stud. Math. 1987, 87(1):93–100.MathSciNetGoogle Scholar
- Shiue JS: On the Cesàro sequence space. Tamkang J. Math. 1970, 2: 19–25.MathSciNetGoogle Scholar
- Jagers AA: A note on Cesàro sequence spaces. Nieuw Arch. Wiskd. 1974, 22(3):113–124.MathSciNetGoogle Scholar
- Suantai S: On the H -property of some Banach sequence spaces. Arch. Math. 2003, 39: 309–316.MathSciNetGoogle Scholar
- Bilgin T:The sequence space and related matrix transformations. Punjab Univ. J. Math. 1997, 30: 67–77.MathSciNetGoogle Scholar
- Lim KP: Matrix transformation in the Cesàro sequence spaces. Kyungpook Math. J. 1974, 14: 221–227.MathSciNetGoogle Scholar
- Şengönül M, Kayaduman K:On the -Nakano sequence space. Int. J. Math. Anal. 2010, 4(25–28):1363–1375.Google Scholar
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