Some geometric properties of the metric space
© Çinar et al.; licensee Springer 2013
Received: 18 September 2012
Accepted: 26 December 2012
Published: 22 January 2013
In this study, we consider the space with an invariant metric. Then, we examine some geometric properties of the linear metric space such as property (β), property (H) and k-NUC property.
MSC:40A05, 46A45, 46B20.
Let be a linear metric space and (resp., ) be a closed unit ball (resp., the unit sphere) of X. A linear metric space has property (β) if and only if for each and , there exists such that for each element and each sequence in with , there is an index k for which , where . If for each and , implies , a linear metric space is said to have property (H). Let be an integer. A linear metric space is said to be k-nearly uniform convex (k-NUC) if for every and , there exists such that for any sequence with , there are such that . These properties have been studied by Mongkolkeha and Pumam , Sanhan and Suantai , Cui et al.  and Cui and Hudzik .
Ahuja et al.  introduced the notions of strict convexity and U.C.I (uniform convexity) in linear metric spaces which are generalizations of the corresponding concepts in linear normed spaces. Later, Sastry and Naidu  introduced the notions of U.C.II and U.C.III in linear metric spaces and showed that these three forms are not always equivalent. Further, Junde et al. [9, 10] showed that if a linear metric space is complete and U.C.I, then it is reflexive.
In summability theory, de la Vallée-Poussin mean was first used to define the -summability by Leindler . -summable sequences have been studied by many authors including Et et al. [12, 13], Savas [14–18], Savas and Malkowsky  and Şimsek et al. [20, 21]. Let be a nondecreasing sequence of positive real numbers tending to infinity and let and . The generalized de la Vallée-Poussin mean is defined by , where for . A sequence is said to be -summable to a number ℓ if as . If , then -summability is reduced to Cesàro summability.
2 Main results
In this part of the paper, our main purpose is to define a metric on and show that possesses property (β), property (H) and k-NUC property. Let be a bounded sequence of real numbers with for all . The mapping is a metric on the space , where and since the function is convex for . First, we will show that the space has property (β) under the above metric. To do this, we need the following two lemmas. To prove these lemmas, we use the technique given in Sanhan and Mongkolkeha .
where and .
From (2.1) and (2.2), we obtain that . □
Theorem 2.3 The space has property (β).
Therefore, we have whenever . Consequently, the space possesses property (β). □
Now, we will show that the space has k-NUC property.
Theorem 2.4 The space is k-NUC for any integer .
Thus, we have for . Hence, is k-NUC. □
Since k-NUC implies NUC and NUC implies property (H), by using the previous theorem, we can give the following result.
Corollary 2.5 The space has property (H).
- Sanhan W, Mongkolkeha C: A locally uniform rotund and property ( β ) of generalized Cesàro metric space. Int. J. Math. Anal. 2011, 5(9–12):549–558.MATHMathSciNetGoogle Scholar
- Junde W, Narang TD: H -property, normal structure and fixed points of nonexpansive mappings in metric linear spaces. Acta Math. Vietnam. 2000, 25(1):13–18.MATHMathSciNetGoogle Scholar
- Mongkolkeha C, Kumam P: Some geometric properties of lacunary sequence spaces related to fixed point property. Abstr. Appl. Anal. 2011., 2011: Article ID 903736Google Scholar
- Sanhan W, Suantai S: On k -nearly uniform convex property in generalized Cesàro sequence spaces. Int. J. Math. Math. Sci. 2003, 57: 3599–3607.MathSciNetView ArticleGoogle Scholar
- Cui Y, Hudzik H, Ping W: On k -nearly uniform convexity in Orlicz spaces. Perspectives in mathematical analysis. Rev. R. Acad. Cienc. Exactas Fís. Nat. 2000, 94(4):461–466. (Spanish)MATHMathSciNetGoogle Scholar
- Cui Y, Hudzik H: Some geometric properties related to fixed point theory in Cesàro spaces. Collect. Math. 1999, 50(3):277–288.MATHMathSciNetGoogle Scholar
- Ahuja GC, Narang TD, Trehan S: Best approximation on convex sets in metric linear spaces. Math. Nachr. 1977, 78: 125–130. 10.1002/mana.19770780110MATHMathSciNetView ArticleGoogle Scholar
- Sastry KPR, Naidu SVR: Convexity conditions in metric linear spaces. Math. Semin. Notes 1979, 7: 235–251.MATHMathSciNetGoogle Scholar
- Junde W, Lianchang C: Reflexivity of uniform convexity in metric linear spaces and its applications. Adv. Math. (China) 1994, 23: 439–444.Google Scholar
- Junde W, Dehai Y, Wenbo Q: The uniform convexity and reflexivity in metric linear spaces. Math. Appl., Chin. Ser. 1995, 8: 322–324.MATHGoogle Scholar
- Leindler L: Über die verallgemeinerte de la Vallee-Poussinsche summierbarkeit allgemeiner Orthogonalreihen. Acta Math. Acad. Sci. Hung. 1965, 16: 375–387. 10.1007/BF01904844MATHMathSciNetView ArticleGoogle Scholar
- Et M: Spaces of Cesàro difference sequences of order r defined by a modulus function in a locally convex space. Taiwan. J. Math. 2006, 10: 865–879.MATHMathSciNetGoogle Scholar
- Güngör M, Et M, Altin Y:Strongly -summable sequences defined by Orlicz functions. Appl. Math. Comput. 2004, 157: 561–571. 10.1016/j.amc.2003.08.051MATHMathSciNetView ArticleGoogle Scholar
- Savas E, Savas R: Some λ -sequence spaces defined by Orlicz functions. Indian J. Pure Appl. Math. 2003, 34: 1673–1680.MATHMathSciNetGoogle Scholar
- Savas E: Strong almost convergence and almost λ -statistical convergence. Hokkaido Math. J. 2000, 29(3):531–536.MATHMathSciNetView ArticleGoogle Scholar
- Savas E: On asymptotically λ -statistical equivalent sequences of fuzzy numbers. New Math. Nat. Comput. 2007, 3(3):301–306. 10.1142/S1793005707000781MATHMathSciNetView ArticleGoogle Scholar
- Savas E:On -statistically convergent double sequences of fuzzy numbers. J. Inequal. Appl. 2008., 2008: Article ID 147827Google Scholar
- Savas E:-double sequence spaces of fuzzy real numbers defined by Orlicz function. Math. Commun. 2009, 14(2):287–297.MATHMathSciNetGoogle Scholar
- Savas E, Malkowsky E: Some λ -sequence spaces defined by a modulus. Arch. Math. 2000, 36(3):219–228.MATHMathSciNetGoogle Scholar
- Simsek N, Savas 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: 768–779.MATHMathSciNetGoogle Scholar
- Simsek N: On some geometric properties of sequence space defined by de la Vallée-Poussin mean. J. Comput. Anal. Appl. 2011, 13: 565–573.MATHMathSciNetGoogle Scholar
- Suantai S: On the H -property of some Banach sequence spaces. Arch. Math. 2003, 39: 309–316.MATHMathSciNetGoogle Scholar
- Shiue JS: On the Cesàro sequence space. Tamkang J. Math. 1970, 2: 19–25.MathSciNetGoogle Scholar
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