- Research Article
- Open Access
A New Class of Sequences Related to the Spaces Defined by Sequences of Orlicz Functions
© Naim L. Braha. 2011
- Received: 5 November 2010
- Accepted: 18 February 2011
- Published: 10 March 2011
We introduce new sequence space defined by combining an Orlicz function, seminorms, and -sequences. We study its different properties and obtain some inclusion relation involving the space Inclusion relation between statistical convergent sequence spaces and Cesaro statistical convergent sequence spaces is also given.
- Banach Space
- Natural Number
- Topological Space
- Sequence Space
- Positive Real Number
By , we denote the space of all real or complex valued sequences. If , then we simply write instead of . Also, we will use the conventions that . Any vector subspace of is called a sequence space. We will write , , and for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by , we denote the sequence space of all -absolutely convergent series, that is, for . Throughout the article, , , and denote, respectively, the spaces of all, bounded, and -absolutely summable sequences with the elements in , where is a seminormed space. By , we denote the zero element in . denotes the set of all subsets of , that do not contain more than elements. With , we will denote a nondecreasing sequence of positive real numbers such that and , as . The class of all the sequences satisfying this property is denoted by .
which yields that and hence is -convergent to . We therefore deduce that the ordinary convergence implies the -convergence to the same limit.
The space is closely related to the space which is an Orlicz sequence space with . An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as . It is well known that if is a convex function and , then for all with .
An Orlicz function is said to satisfy the -condition for all values of , if there exists a constant such that (see, Krasnoselskii and Rutitsky ). In the later stage, different Orlicz sequence spaces were introduced and studied by Bhardwaj and Singh , Güngör et al. , Tripathy and Mahanta , Esi , Esi and Et , Parashar and Choudhary , and many others.
Since the proofs of the following theorems are not hard we omit them.
The sequence spaces are linear spaces over the complex field .
In what follows, we will show inclusion theorems between spaces .
Therefore, , which is contradiction.
Let , , be Orlicz functions which satisfy the -condition and , and seminorms. Then
(4)If is stronger than , then , and
(5)If is equivalent to , then .
Proof is similar to [14, Theorem 2.5].
Let be an Orlicz function which satisfy the -condition. Then .
Let be a sequence of Orlicz functions. Then the sequence space is solid and monotone.
which proves that is solid space and monotone.
for some .
If is any Orlicz function, strictly increasing sequence, then .
Taking the limit as , it follows that .
If is any Orlicz bounded function, strictly increasing sequence, then , for every .
where the summation is over and the summation is over . Taking the limit as and , it follows that .
If is an Orlicz function. Then .
Let . Then for any Orlicz function, , .
from this, it follows that .
Let . Then for any Orlicz function, , .
- Mursaleen M, Noman AK: On the spaces of λ -convergent and bounded sequences. Thai Journal of Mathematics 2010,8(2):311–329.MathSciNetMATHGoogle Scholar
- Sargent WLC: Some sequence spaces related to the spaces. Journal of the London Mathematical Society 1960, 35: 161–171. 10.1112/jlms/s1-35.2.161MathSciNetView ArticleMATHGoogle Scholar
- Rath D: Spaces of -convex sequences and matrix transformations. Indian Journal of Mathematics 1999,41(2):265–280.MathSciNetMATHGoogle Scholar
- Rath D, Tripathy BC: Characterization of certain matrix operators. Journal of Orissa Mathematical Society 1989, 8: 121–134.Google Scholar
- Tripathy BC, Sen M: On a new class of sequences related to the space . Tamkang Journal of Mathematics 2002,33(2):167–171.MathSciNetMATHGoogle Scholar
- Tripathy BC, Mahanta S: On a class of sequences related to the space defined by Orlicz functions. Soochow Journal of Mathematics 2003,29(4):379–391.MathSciNetMATHGoogle Scholar
- Lindenstrauss J, Tzafriri L: On Orlicz sequence spaces. Israel Journal of Mathematics 1971, 10: 379–390. 10.1007/BF02771656MathSciNetView ArticleMATHGoogle Scholar
- Krasnoselskii MA, Rutitsky YB: Convex Function and Orlicz Spaces. P.Noordhoff, Groningen, The Netherlands; 1961.Google Scholar
- Bhardwaj VK, Singh N: Some sequence spaces defined by Orlicz functions. Demonstratio Mathematica 2000,33(3):571–582.MathSciNetMATHGoogle Scholar
- Güngör M, Et M, Altin Y: Strongly -summable sequences defined by Orlicz functions. Applied Mathematics and Computation 2004,157(2):561–571. 10.1016/j.amc.2003.08.051MathSciNetView ArticleMATHGoogle Scholar
- Esi A: On a class of new type difference sequence spaces related to the space . Far East Journal of Mathematical Sciences 2004,13(2):167–172.MathSciNetMATHGoogle Scholar
- Esi A, Et M: Some new sequence spaces defined by a sequence of Orlicz functions. Indian Journal of Pure and Applied Mathematics 2000,31(8):967–973.MathSciNetMATHGoogle Scholar
- Parashar SD, Choudhary B: Sequence spaces defined by Orlicz functions. Indian Journal of Pure and Applied Mathematics 1994,25(4):419–428.MathSciNetMATHGoogle Scholar
- Altun Y, Bilgin T: On a new class of sequences related to the space defined by Orlicz function. Taiwanese Journal of Mathematics 2009,13(4):1189–1196.MathSciNetMATHGoogle Scholar
- Fast H: Sur la convergence statistique. Colloquium Mathematicum 1951, 2: 241–244.MathSciNetMATHGoogle Scholar
- Connor JS: The statistical and strong -Cesàro convergence of sequences. Analysis 1988,8(1–2):47–63.MathSciNetView ArticleMATHGoogle Scholar
- Freedman AR, Sember JJ: Densities and summability. Pacific Journal of Mathematics 1981,95(2):293–305.MathSciNetView ArticleMATHGoogle Scholar
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