Paranormed Riesz Difference Sequence Spaces of Fractional Order
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Authors: T. YAYING
DOI: 10.46793/KgJMat2202.175Y
Abstract:
In this article we introduce paranormed Riesz difference sequence spaces of fractional order α, r0t
, rct
and r∞t
defined by
the composition of fractional difference operator Δ(α), defined by
(Δ(α)x)k = ∑
i=0∞(−1)i
xk−i, and Riesz mean matrix Rt. We give
some topological properties, obtain the Schauder basis and determine the
α-, β- and γ- duals of the new spaces. Finally, we characterize certain
matrix classes related to these new spaces.
Keywords:
Riesz difference sequence spaces, difference operator Δ(α), Schauder basis, α-, β-, γ- duals, matrix transformation.
References:
[1] B. Altay and F. Başar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26 (2002), 701–715.
[2] B. Altay and F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 30 (2006), 591–608.
[3] E. Malkowsky, Recent results in the theory of matrix transformations in sequence spaces, Mat. Vesnik 49 (1997), 187–196.
[4] M. Başarir and M. Öztürk, On the Riesz difference sequence space, Rend. Circ. Mat. Palermo (2) 57 (2008), 377–389.
[5] M. Başarir, On the generalized Riesz B-difference sequence spaces, Filomat 24(4) (2010), 35–52.
[6] M. Başarir and M. Kayikçi, On the generalized Bm-Riesz difference sequence space and β-property, J. Inequal. Appl. 2009 (2009), Article ID 385029, 18 pages.
[7] M. Başarir and M. Öztürk, On some generalized Bm-difference Riesz sequence spaces and uniform opial property, J. Inequal. Appl. 2011 (2011), Article ID 485730, 17 pages.
[8] M. Yeşilkayagil and F. Başar, On the paranormed Nörlund sequence space of nonabsolute type, Abstr. Appl. Anal. 2014 (2014), Article ID 858704, 9 pages.
[9] M. Yeşilkayagil and F. Başar, Domain of the Nörlund matrix in some of Maddox’s spaces, Proceedings of the National Academy of Sciences, India - Section A 87(3) (2017), 363–371.
[10] P. Baliarsingh and S. Dutta, A unifying approach to the difference operators and their applications, Bol. Soc. Parana. Mat. 33 (2015), 49–57.
[11] P. Baliarsingh and S. Dutta, On the classes of fractional order difference sequence spaces and their matrix transformations, Appl. Math. Comput. 250 (2015), 665–674.
[12] P. Baliarsingh, Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput. 219 (2013), 9737–9742.
[13] T. Yaying, A. Das, B. Hazarika and P. Baliarsingh, Compactness of binomial difference sequence spaces of fractional order and sequence spaces, Rend. Circ. Mat. Palermo (2) (2018), DOI 10.1007/s12215-018-0372-8.
[14] T. Yaying and B. Hazarika, On sequence spaces generated by binomial difference operator of fractional order, Math. Slovaca 69(4) (2019), 901–918.
[15] C. Bektas, M. Et and R. Çolak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl. 292 (2004), 423–432.
[16] Z. U. Ahmad and M. Mursaleen, Köthe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd) (N.S.) 42 (1987), 57–61.
[17] M. Et, On some difference sequence spaces, Dǒga Tr. J. Math. 17 (1993), 18–24.
[18] M. Mursaleen, Generalized spaces of difference sequences, J. Anal. Math. Appl. 203 (1996), 738–745.
[19] Z. U. Ahmad, M. Mursaleen and A. H. A. Bataineh, On some generalized spaces of difference sequences, Southeast Asian Bull. Math. 29(4), 635–649.
[20] S. Dutta and P. Baliarsingh, A note on paranormed difference sequence spaces of fractional order and their matrix transformations, J. Egyptian Math. Soc. 22 (2014), 249–253.
[21] M. Et and R. Çolak, On generalized difference sequence spaces, Soochow J. Math. 21 (1995), 377–386.
[22] M. Et and A. Esi, On Köthe-Toeplitz duals of generalized difference sequence spaces, Bull. Malays. Math. Sci. Soc. 231 (2000), 25–32.
[23] M. Et and M. Basarir, On some new generalized difference sequence spaces, Period. Math. Hungar. 35 (1997), 169–175.
[24] A. Karaisa and F. Özger, Almost difference sequence spaces derived by using a generalized weighted mean, Journal of Computational Analysis and Applications 19(1) (2015), 27–38.
[25] U. Kadak and P. Baliarsingh, On certain Euler difference sequence spaces of fractional order and related dual properties, J. Nonlinear Sci. Appl. 8 (2015), 997–1004.
[26] P. Baliarsingh, U. Kadak and M. Mursaleen, On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems, Quaest. Math. 41(8) (2018), 1117–1133.
[27] U. Kadak, Generalized statistical convergence based on fractional order difference operator and its applications to approximation theorems, Iran. J. Sci. Technol. Trans. A Sci. 43(1)(2019), 225–237.
[28] U. Kadak, Generalized lacunary statistical difference sequence spaces of fractional order, Int. J. Math. Math. Sci. 2015 (2015), Article ID 984283, 6 pages.
[29] U. Kadak, Generalized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variables, Results Math. 72(3) (2017), 1181–1202.
[30] H. Furkan, On some λ-difference sequence spaces of fractional order, J. Egypt Math. Soc. 25(1) (2017), 37–42.
[31] F. Özger, Characterisations of compact operators on ℓp-type fractional sets of sequences, Demonstr. Math. 52 (2019), 105–115.
[32] F. Özger, Some geometric characterisations of a fractional Banach set, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 68(1) (2019), 546–558.
[33] F. Özger, Some general results on fractional Banach sets, Turkish J. Math. 43(2) (2019), 783–794.
[34] F. Özger, Compact operators on the sets of fractional difference sequences, Sakarya University Journal of Science 23(3) (2019), 425–434.
[35] E. Malkowsky, F. Özger and V. Veličkovi
, Some mixed
paranorm spaces, Filomat 31(4) (2017), 1079–1098.
[36] E. Malkowsky, F. Özger and V. Veličković, Matrix transformations on mixed paranorm spaces, Filomat 31(10) (2017), 2957–2966.
[37] V. Veličković, E. Malkowsky and F. Özger, Visualization of the spaces W(u,v; ℓp) and their duals, AIP Conf. Proc. 1759 (2016), DOI 10.1603/1.4959634.
[38] F.Özger and F. Başar, Domain of the double sequential band
matrix B(
,
) on some Maddox’s spaces, Acta Math. Sci. Ser. B
Engl. Ed. 34(2) (2014), 394–408.
[39] E. Malkowsky, F. Özger and A. Alotaibi, Some notes on matrix mappings and their Hausdorff measure of noncompactness, Filomat 28(5) (2014), 1059–1072.
[40] H. Kızmaz, On certain sequence spaces, Can. Math. Bull. 24 (1981), 169–176.
[41] K. G. Grosse-Erdmann, Matrix transformation between the sequence spaces of Maddox, J. Math. Anal. Appl. 180 (1993), 223–238.
[42] C. G. Lascarides and I. J. Maddox, Matrix transformations between some classes of sequences, Mathematical Proceedings of the Cambridge Philosophical Society 68(1) (1970), 99–104.
[43] I. J. Maddox, Spaces of strongly summable sequences, Q. J. Math. 18 (1967), 345–355.
[44] I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Mathematical Proceedings of the Cambridge Philosophical Society 64 (1968), 335–340.
[45] I. J. Maddox, Elements of Functional Analysis, 2nd ed., Cambridge University Press, Cambridge, 1988.
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