SciELO - Scientific Electronic Library Online

 
vol.12 número3Self-Dual and Anti-Self-Dual Solutions of Discrete Yang-Mills Equations on a Double ComplexSome Generalizations of Mulit-Valued Version of Schauder’s Fixed Point Theorem with Applications índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.3 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000300008 

CUBO A Mathematical Journal Vol.12, N°03, (121-138). October 2010

 

Calculations in New Sequence Spaces and Application to Statistical Convergence

 

BRUNO DE MALAFOSSE AND VLADIMIR RAKOČEVIC1

LMAH Université du Havre, BP 4006 IUT Le Havre, 76610 Le Havre. France email: bdemalaf@wanadoo.fr

Department of Mathematics, University of Niš, Videgradska 33, 18000 Niš, Serbia email: vrakoc@bankerinter.net


ABSTRACT

In this paper we recall recent results that are direct consequences of the fact that (w(λ) ,w(λ)) is a Banach algebra. Then we define the set Wτ = Dτw and characterize the sets Wτ (A) where A is either of the operators Δ, ∑, Δ(λ), or C(λ). Afterwardswe consider the sets [A1,A2]Wτ of all sequences X such that A1 (λ)(|A2(μ) X|) Wτ where A1 and A2 are of the form C(ξ), C+ (ξ), Δ(ξ), or Δ+ (ξ) and it is given necessary conditions to get |A1 (λ),A2(μ)| Wτ in the form Wξ. Finally we apply the previous results to statistical convergence. So we have conditions to have xk L(S(A)) where A is either of the infinite matrices D1/τC(λ)C(μ), D1/τΔ(λ)Δ(μ), D1/τΔ(λ)C(μ). We also give conditions to have xk 0(S(A)) where A is either of the operators D1/τC+ (λ)Δ(μ), D1/τC(λ)C(μ), D1/τC+ (λ)C+(μ), or D1/τΔ(λ)C+(μ).

Key words and phrases: Banach algebra, statistical convergence, A−statistical convergence, infinite matrix.


RESUMEN

Recordamos resultados recientes que son consecuencia directa del hecho de que (w(λ), w(λ)) es una algebra de Banach. Entonces nosotros definimos el conjunto = Dτwy caracterizamos los conjuntos (A) donde A es uno de los siguientes operadores Δ, ∑, Δ(λ), o C(λ). Después consideramos los conjuntos[A1,A2] de todas las sucesiones X tal que A1 (λ)(|A2(μ) X|) dondeA1 y A2 son de la forma C(ξ), C+ (ξ), Δ(ξ), or Δ+ (ξ) y son dadas condiciones necesarias para obtener |A1 (λ),A2(μ)| en la forma Wξ. Finalmente, aplicamos los resultados previos para tener xk L(S(A)) donde A es una de las matrices infinitas D1/τC(λ)C(μ), D1/τΔ(λ)Δ(μ), D1/τΔ(λ)C(μ) . Nosotros también damos condiciones para tener xk 0(S(A)) donde A es uno de los operadores D1/τC+ (λ)Δ(μ), D1/τC(λ)C(μ), D1/τC+ (λ)C+(μ), o D1/τΔ(λ)C+(μ).

Math. Subj. Class.: 40C05, 40F05, 40J05, 46A15.



Notas

1Supported by Grant No. 144003 of the Ministry of Science, Technology and Development, Republic of Serbia

References

[1] ÇOLAK, R., Lacunary strong convergence of difference sequences with respect to a modulus, Filomat, 17 (2003), 9-14.        [ Links ]

[2] DE MALAFOSSE, B., On some BK space, Int. J. of Math. and Math. Sc., 28 (2003), 1783- 1801.        [ Links ]

[3] DE MALAFOSSE, B., On the set of sequences that are strongly α-bounded and α- convergent to naught with index p, Seminario Matematico dell’Università e del Politecnico di Torino, 61 (2003), 13-32.        [ Links ]

[4] DE MALAFOSSE, B., Calculations on some sequence spaces, Int. J. of Math. and Math. Sc., 31 (2004), 1653-1670.        [ Links ]

[5] DE MALAFOSSE, B. AND MALKOWSKY, E., The Banach algebra (w(λ) ,w(λ)), in press Far East Journal Math.        [ Links ]

[6] DE MALAFOSSE, B. AND RAKOCEVIC, V., Matrix Transformations and Statistical convergence, Linear Algebra and its Applications, 420 (2007), 377-387.        [ Links ]

[7] FAST, H., Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.        [ Links ]

[8] FRIDY, J.A., On statistical convergence, Analysis, 5 (1985), 301-313.        [ Links ]

[9] FRIDY, J.A., Statistical limit points, Proc. Amer. Math. Soc., 118 (1993), 1187-1192.        [ Links ]

[10] FRIDY, J.A. AND ORHAN, C., Lacunary statistical convergence, Pacific J. Math., 160 (1993), 43-51.        [ Links ]

[11] FRIDY, J.A. AND ORHAN, C., Statistical core theorems, J. Math. Anal. Appl., 208 (1997), 520-527.        [ Links ]

[12] MADDOX, I.J., On Kuttner’s theorem, J. London Math. Soc., 43 (1968), 285-290.        [ Links ]

[13] MADDOX, I.J., Elements of Functionnal Analysis, Cambridge University Press, London and New York, 1970.        [ Links ]

[14] MALKOWSKY, E., The continuous duals of the spaces c0 (?) and c (?) for exponentially bounded sequences ?, Acta Sci. Math (Szeged), 61, (1995), 241-250.        [ Links ]

[15] MALKOWSKY, E. AND RAKOCEVIC, V., An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik radova, Matematˇcki institut SANU, 9 (17) (2000), 143-243.        [ Links ]

[16] STEINHAUS, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.        [ Links ]

[17] WILANSKY, A., Summability through Functional Analysis, North-Holland Mathematics Studies, 85, 1984.        [ Links ]