On a Generalized Difference Sequence Spaces of Fractional Order associated with Multiplier Sequence Defined by A Modulus Function

Year 2020, , 1105 - 1114, 01.10.2020

Taja Yayıng

https://doi.org/10.16984/saufenbilder.744881

Abstract

Let Γ(m) denotes the gamma function of a real number m∉{0,-1,-2,…}. Then the difference matrix Δ^α of a fractional order α is defined as
(Δ^α v)_k=∑_i〖(-1)^i (Γ(α+1))/(i!Γ(α-i+1)) v_(k+i) 〗.
Using the difference operator Δ^α, we introduce paranormed difference sequence spaces N_θ (Δ^α,f,Λ,p) and S_θ (Δ^α,f,Λ,p) of fractional orders involving lacunary sequence, θ; modulus function, f and multiplier sequence, Λ=(λ_k). We investigate topological structures of these spaces and examine various inclusion relations.

Keywords

Difference operator $\Delta^{\alpha}$, Paranormed sequence space, Lacunary sequence, Modulus function, Multiplier sequence

References

• H. Kızmaz, On certain sequence spaces, Canadian Mathematical Bulletin, Vol. 24, no. 2, pp. 169-176, 1981.
• M. Et, R. Ҫolak, On some generalized difference sequence spaces, Soochow Journal of Mathematics, Vol. 21, no. 4, pp. 377-386, 1995.
• M. Et, A. Esi, On Köthe-Toeplitz duals of generalized difference sequence spaces, Bulletin of Malaysian Mathematical Science Society, Vol. 23, pp. 25-32, 2000.
• M. Et, M. Başarır, On some new generalized difference sequence spaces, Periodica Mathematica Hungarica, Vol. 35, no. 3, pp. 169-175, 1997.
• E. Malkowsky, S.D. Parashar, Matrix transformations in spaces of bounded and convergent difference sequences of order m, Analysis, Vol. 17, pp. 87-97, 1997.
• R. Ҫolak, Lacunary strong convergence of difference sequences with respect to a modulus function, Filomat, Vol. 17, pp. 9-14, 2003.
• C. Aydin, F. Başar, Some new difference sequence spaces, Applied Mathematics and Computation, Vol. 157, no. 3, pp. 677-693, 2004.
• M. Mursaleen, Generalized spaces of difference sequences, Journal of Mathematical Analysis and Application, Vol. 203, no. 3, pp. 738-745, 1996.
• Ç. A. Bektaş, M. Et, R. Çolak, Generalized difference sequence spaces and their dual spaces, Journal of Mathematical Analysis and Application, Vol. 292, pp. 423-432, 2004.
• P. Baliarsingh, S. Dutta, A unifying approach to the difference operators and their applications, Boletim da Sociedade Paranaense de Matematica, Vol. 33, pp. 49-57, 2015.
• P. Baliarsingh, S. Dutta, On the classes of fractional order difference sequence spaces and their matrix transformations, Applied. Mathematics and Computation, Vol. 250, pp. 665-674, 2015.
• P. Baliarsingh, Some new difference sequence spaces of fractional order and their dual spaces, Applied Mathematics and Computation, Vol. 219, pp. 9737-9742, 2013.
• P. Baliarsingh, U. Kadak, On matrix transformations and Hausdorff measure of noncompactness of Euler difference sequence spaces of fractional order, Quaestiones Mathematica, DOI: 10.2989/16073606.2019.1648325.
• J. Meng, L. Mei, Binomial difference sequence spaces of fractional order, Journal of Inequalities and Application, vol. 2018, 274, 2018.
• T. Yaying, A. Das, B. Hazarika, P. Baliarsingh, Compactness of binomial difference sequence spaces of fractional order and sequence spaces, Rendiconti del Circolo Matematico di Palermo series II, vol. 68, 459-476, 2019.
• T. Yaying, B. Hazarika, On sequence spaces generated by binomial difference operator of fractional order, Mathematica Slovaca, vol. 69, no. 4, pp. 901-918, 2019.
• T. Yaying, Paranormed Riesz difference sequence spaces of fractional order, Kragujevac Journal of Mathematics, vol. 46, no. 2, pp. 175-191, 2022.
• S. Dutta, P. Baliarsingh, A note on paranormed difference sequence spaces of fractional order and their matrix transformations, Journal of Egyptian Mathematical Society, vol. 22, pp. 249-253, 2014.
• U. Kadak, P. Baliarsingh, On certain Euler difference sequence spaces of fractional order and related dual properties, Journal of Nonlinear Science and Application, vol. 8, pp. 997-1004, 2015.
• P. Baliarsingh, U. Kadak, M. Mursaleen, On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems, Quaestiones Mathematica, vol. 41, no. 8, pp. 1117-1133, 2018.
• U. Kadak, Generalized statistical convergence based on fractional order difference operator and its applications to approximation theorems, Iran Journal of Science and Technology Transactions A Science, vol. 43, no. 1, 225-237, 2019.
• U. Kadak, Generalized lacunary statistical difference sequence spaces of fractional order, International Journal of Mathematics and Mathematical Science, vol. 2015, Article ID 984283, 6 pages, 2015.
• U. Kadak, Generalized Lacunary Statistical Difference Sequence Space of Fractional Order, International Journal of Mathematics and Mathematical Sciences, vol. 2015, Article ID 984283, 6 pages, 2015.
• U. Kadak, Generalized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variables, Results Mathematics, vol. 72, no. 3, pp. 1181-1202, 2017.
• U. Kadak, Weighted statistical convergence based on generalized difference operator involving (p,q)-Gamma function and its application to approximation theorems, Journal of Mathematical Analysis and Applications, vol. 448, no. 2, pp. 1633-1650, 2017.
• L. Nayak, M. Et, P. Baliarsingh, On certain generalized weighted mean fractional difference sequence spaces, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 89, pp. 163-170, 2019.
• T. Yaying, On a new class of generalized difference sequence spaces of fractional order defined by modulus function, Proyecciones, vol. 38, no. 3, 485-497, 2018.
• H. Furkan, On some λ-difference sequence spaces of fractional order, Journal of Egyptian Mathematical Society, vol. 25, no. 1, pp. 37-42, 2017.
• F. Özger, Characterisations of compact operators on l_p-type fractional sets of sequences, Demonstrio Mathematica, vol. 52, pp. 105-115, 2019.
• F. Özger, Some geometric characterisations of a fractional Banach set, Communications Faculty of Sciences University of Ankara Series A1, Mathematics and Statistics, vol. 68, no. 1, 546-558, 2019.
• H. Fast, Sur la convergence statistique, Colloquium Mathematicum, vol. 2, pp. 241-244, 1951.
• I.J. Schoenberg, The integrability of certain functions and related summability methods, American Mathematical Monthly, vol. 66, pp. 361-375, 1959.
• I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proceedings of Cambridge Philosophical Society, vol. 64, pp. 335-340, 1968.
• H. Nakano, Concave Modulars, Journal of Mathematical Society of Japan, vol. 5, pp. 29-49, 1959.
• I.J. Maddox, Spaces of strongly summable sequences, The Quarterly Journal of Mathematics, Oxford, vol. 18, no. 2, pp. 345-355, 1967.
• W.H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canadian Journal of Mathematics, vol. 25, pp. 973-978, 1973.
• I.J. Maddox, Elements of Functional Analysis, Second edition., University Press, Cambridge, 1988.
• B.C. Tripathy, M. Et, On Generalized Difference Lacunary Statistical Convergence, Studia Univ. "Babeş-Bolyai", Mathematica, L(1), 2015.
• M. Et, Generalized Cesàro difference sequence spaces of non absolute type involving lacunary sequence, Applied Mathematics and Computation, vol. 219, pp. 9372-9376, 2013.
• G. Goes, S. Goes, Sequences of bounded variation and sequences of Fourier coefficients, Mathematische Zeitschrift, vol. 118, pp. 93-102, 1917.
• A. Esi, B.C. Tripathy, Strongly almost convergent generalized difference sequences associated with multiplier sequences, Mathematica Slovaca, vol. 57, no. 4, pp. 339-348, 2017.
• J.A. Fridy, C. Orhan, Lacunary Statistical Convergence, Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43-51, 1993.
• S. Pehlivan, B. Fisher, On Some Sequence Spaces, Indian Journal of Pure and Applied Mathematics, vol. 25, no. 10, pp. 1067-1071, 1994.
• A.R. Freedman, J.J. Sember, M. Raphael, Some Cesàro-type Summability Spaces, Proceedings of London Mathematical Society, vol. 37, no. 3, 508-520, 1978.