International Journal of Nonlinear Analysis and Applications
Sequences of Cesaro type using lacunary notion
Document Type : Research Paper
Author
Basic Science Department, College of Science and Theoretical Studies, Saudi Electronic University-Abha Male, 61421, Kingdom of Saudi Arabia
Abstract
The scenario of this article is to introduce the space $\mathfrak{R}^{t}_s(p, \Delta)$ based on a general Riesz sequence space. Its completeness property is derived and its linear isomorphism property with $\ell(p)$ is proved. The K\"{o}the-dual property of the space $\mathfrak{R}^{t}_s(p, \Delta)$ is also derived. Furthermore, its basis is constructed and some characterization of infinite matrices are given.
Keywords
[1] A.A. Albanese, J. Bonet and W.J. Ricker, Multiplier and averaging operators in the Banach spaces ces(p), 1 <
p < ∞, Positivity 23 (2018), no. 1, 177–193.
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(1991), 14–19.
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[7] G. P. Curbera and W. J. Ricker, Solid Extensions of the Ces`aro Operator on ℓp and c0, Integral Equations and
Operator Theory 80 (2014), no. 1, 61–77.[8] E. D¨undar, N. Pancaroˇglu Akın, and U. Ulusu, Lacunary II-Invariant convergence of sequences of sets, Proc. Nat.
Acad. Sci. India, Sect. A Phys. Sci. 91 (2021), 517–522.
[9] M. Et, On some generalised Ces`aro difference sequence spaces, Istanbul Univ. Fen Fak. Mat. Derg. 55 (1996),
221–229.
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[11] A. R. Freedman, J.J. Sember and M. Raphael, Some Ces`aro type summability spaces, Proc. Lond. Math. Soc.
37 (1978), 508–520.
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[13] A.H. Ganie, Sigma bounded sequence and some matrix transformations, Algebra Lett. 3 (2013), 1–7.
[14] A.H. Ganie and N.A. Sheikh, Matrix transformation into a new sequence space related to invariant means, Chamchuri J. Math., 4 (2012), 71–72.
[15] D. Ghosh and P. D. Srivastava, On some vector valued sequence spaces defined using a modulus function, Indian
J. Pure Appl. Math. 30 (1999), no. 8, 819–826.
[16] A.A. Jagers, A note on Ces`aro sequence spaces, Nieuw. Arch. Wisk., 22 (1974), 113–124.
[17] E. Kolk, Inclusion theorems for some sequence spaces defined by a sequence of modulli, Acta Comment. Univ.
Tartu 970 (1994), 65–72.
[18] G.M. Leibowitz, A note on the Ces`aro sequence spaces, Tamkang J. Math. 2 (1971), no. 2, 151–157.
[19] W.A.J. Luxemburg and A.C. Zaanen, Some examples of normed K¨othe spaces, Math. Ann. 162 (1966), 337-–350.
[20] I.J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Cam. Philos. Soc., 100 (1986), 161–166.
[21] P.-N. Ng and P.-Y. Lee, Ces`aro sequences spaces of non-absolute type, Comment. Math. Prace Mat., 20 (1978),
429–433.
[22] Programma van Jaarlijkse Prijsvragen (Annual Problem Section), Nieuw Arch. Wiskd. 16 (1968), 47–51.
[23] W.H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25 (1973),
1973–1978.
[24] E. Sava¸s, On some generalized sequence spaces defined by a modulus, Indian J. pure and Appl. Math. 30 (1999),
459–464.
[25] J.S. Shiue, A note on the Ces`aro sequence space, Tamkang J. Math. 1 (1970), no. 2, 91–95.
[26] H. Seng¨u, Some Ces`aro-type summability spaces defined by modulus function of order (α, β), Commun. Fac. Sci.
Univ. Ank. Ser. A1 66 (2017), no. 2, 80–90.
[27] U. Ulusu and F. Nuray, On the strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinf. 3
(2013), no. 3, 75–88.
p < ∞, Positivity 23 (2018), no. 1, 177–193.
[2] A. Alexiewicz, On Cauchy’s condensation theorem, Studia Math. 16 (1957), 80—85
[3] A. Arhangelskii, Some types of factor mappings and the relations between classes of topological spaces, Soviet
Math. Dokl. 4 (1963), 1726–1729.
[4] I. Bala, On Ces`aro sequence space defined by a modulus function, Demonstr. Math. 45 (2012), no. 4, 157–163.
[5] M. Ba¸sarir, Paranormed Ces`aro difference sequence space and related matrix transformation, Turk. J. Math. 15
(1991), 14–19.
[6] G. Bennett, Factorizing the classical inequalities, Mem. Amer. Math. Soc. 576 (1996).
[7] G. P. Curbera and W. J. Ricker, Solid Extensions of the Ces`aro Operator on ℓp and c0, Integral Equations and
Operator Theory 80 (2014), no. 1, 61–77.[8] E. D¨undar, N. Pancaroˇglu Akın, and U. Ulusu, Lacunary II-Invariant convergence of sequences of sets, Proc. Nat.
Acad. Sci. India, Sect. A Phys. Sci. 91 (2021), 517–522.
[9] M. Et, On some generalised Ces`aro difference sequence spaces, Istanbul Univ. Fen Fak. Mat. Derg. 55 (1996),
221–229.
[10] D. Fathima and A.H. Ganie, On some new scenario of ∆- spaces, J. Nonlinear Sci. Appl. 14 (2021), 163–167.
[11] A. R. Freedman, J.J. Sember and M. Raphael, Some Ces`aro type summability spaces, Proc. Lond. Math. Soc.
37 (1978), 508–520.
[12] A.H. Ganie, New spaces over modulus function, Bol. Soc. Paranaense Mat. (in press), 1–6.
[13] A.H. Ganie, Sigma bounded sequence and some matrix transformations, Algebra Lett. 3 (2013), 1–7.
[14] A.H. Ganie and N.A. Sheikh, Matrix transformation into a new sequence space related to invariant means, Chamchuri J. Math., 4 (2012), 71–72.
[15] D. Ghosh and P. D. Srivastava, On some vector valued sequence spaces defined using a modulus function, Indian
J. Pure Appl. Math. 30 (1999), no. 8, 819–826.
[16] A.A. Jagers, A note on Ces`aro sequence spaces, Nieuw. Arch. Wisk., 22 (1974), 113–124.
[17] E. Kolk, Inclusion theorems for some sequence spaces defined by a sequence of modulli, Acta Comment. Univ.
Tartu 970 (1994), 65–72.
[18] G.M. Leibowitz, A note on the Ces`aro sequence spaces, Tamkang J. Math. 2 (1971), no. 2, 151–157.
[19] W.A.J. Luxemburg and A.C. Zaanen, Some examples of normed K¨othe spaces, Math. Ann. 162 (1966), 337-–350.
[20] I.J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Cam. Philos. Soc., 100 (1986), 161–166.
[21] P.-N. Ng and P.-Y. Lee, Ces`aro sequences spaces of non-absolute type, Comment. Math. Prace Mat., 20 (1978),
429–433.
[22] Programma van Jaarlijkse Prijsvragen (Annual Problem Section), Nieuw Arch. Wiskd. 16 (1968), 47–51.
[23] W.H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25 (1973),
1973–1978.
[24] E. Sava¸s, On some generalized sequence spaces defined by a modulus, Indian J. pure and Appl. Math. 30 (1999),
459–464.
[25] J.S. Shiue, A note on the Ces`aro sequence space, Tamkang J. Math. 1 (1970), no. 2, 91–95.
[26] H. Seng¨u, Some Ces`aro-type summability spaces defined by modulus function of order (α, β), Commun. Fac. Sci.
Univ. Ank. Ser. A1 66 (2017), no. 2, 80–90.
[27] U. Ulusu and F. Nuray, On the strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinf. 3
(2013), no. 3, 75–88.
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Volume 13, Issue 2
July 2022Pages 1399-1405
- Receive Date: 31 August 2020
- Revise Date: 27 August 2021
- Accept Date: 04 September 2021