Some general results on fractional Banach sets

The gamma function which is expressed by an improper integral is used to establish the fractional differenceoperators and fractional Banach sets. In this study, we achieve some comprehensive and complementary results relatedto characterizations of the matrix classes of fractional Banach sets. We also obtain some identities or inequalities for theHausdorff measure of noncompactness of the corresponding matrix operators, and finally find the necessary and sufficientconditions for those matrix operators to be compact.

PDF

___

[1] Altay B, Başar F. The fine spectrum and the matrix domain of the difference operator Δ on the sequence space ℓp , (0 < p < 1). Communications in Mathematical Analysis 2007; 2 (2): 1-11.
[2] Aydın C, Başar F. Some new difference sequence spaces. Applied Mathematics and Computation 2004; 157: 677-693.
[3] Baliarsingh P. Some new difference sequence spaces of fractional order and their dual spaces. Applied Mathematics and Computation 2013; 219: 9737-9742.
[4] Baliarsingh P, Dutta S. On the classes of fractional order of difference sequence spaces and matrix transformations. Applied Mathematics and Computation 2015; 250: 665-674.
[5] Başar F. Summability Theory and its Applications. İstanbul, Turkey:Bentham Science Publishers e-books, Monographs, 2012. ISBN: 978-1-60805-420-6.
[6] Başar F, Altay, B. On the space of sequences of p-bounded variation and related matrix mappings. Ukrainian Mathematical Journal 2003; 55 (1): 136-147.
[7] Başar F, Malkowsky E. The characterisation of compact operators on spaces of strongly summable and bounded sequences. Applied Mathematics and Computation 2011; 217 (12): 5199-5207.
[8] Djolović I, Malkowsky E. A note on compact operators on matrix domains. Journal of Mathematical Analysis and Applications 2008; 340: 291-303.
[9] Djolović I, Malkowsky E. Characterization of some classes of compact operators between certain matrix domains of triangles. Filomat 2016; 30: 1327-1337.
[10] Djolović I, Malkowsky E. A note on Fredholm operators on (c0)T . Applied Mathematics Letters 2009; 22: 1734-1739.
[11] Et M, Çolak R. On some generalized difference sequence spaces. Soochow Journal of Mathematics 1995; 21 (4): 377–386.
[12] Karaisa A, Özger F. Almost difference sequence space derived by using a generalized weighted mean. Journal of Computational Analysis and Applications 2015; 19: 27-38.
[13] Kızmaz H. On certain sequence spaces. Canadian Mathematical Bulletin 1981; 24: 169-176.
[14] Malkowsky E, Özger F, Veličković V. Some mixed paranorm spaces. Filomat 2017; 31: 1079-1098.
[15] Malkowsky E, Özger F. A note on some sequence spaces of weighted means. Filomat 2012; 26: 511-518.
[16] Malkowsky E, Özger F. Compact operators on spaces of sequences of weighted means. American Institute of Physics Conference Proceedings 2012; 1470: 179-182.
[17] Malkowsky E, Özger F, Alotatibi A. Some notes on matrix mappings and their Hausdorff measure of noncompactness. Filomat 2014; 28: 1059-1072.
[18] Malkowsky E, Özger F, Veličković V. Some spaces related to Cesaro sequence spaces and an application to crystallography. MATCH Communications in Mathematical and Computer Chemistry 2013; 70: 867-884.
[19] Malkowsky E, Özger F., Veličković V. Matrix transformations on mixed paranorm spaces. Filomat 2017; 31: 2957- 2966.
[20] Mohiuddine SA, Mursaleen M, Alotaibi A. The Hausdorff measure of noncompactness for some matrix operators. Nonlinear Analysis 2013; 92: 119-129.
[21] Mohiuddine SA, Mursaleen M, Alotaibi A. Erratum: The Hausdorff measure of noncompactness for some matrix operators. Nonlinear Analysis 2015; 117: 221.
[22] Malkowsky E, Rakočević V. On matrix domains of triangles. Applied Mathematics and Computation 2007; 189: 1148-1163.
[23] Malkowsky E, Rakočević V. An introduction into the theory of sequence spaces and measures of noncompactness. Zb. Rad. (Beogr.) 2000; 9: 143-234.
[24] Özger F. Some geometric characterizations of a fractional Banach set. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 2019; 68 (1): 546-558
[25] Özger F. Compact operators on the sets of fractional difference sequences. Sakarya University Journal of Science 2019; 23(3): 425-434.
[26] Özger F. Characterizations of compact operators on ℓp -type fractional sets of sequences. Demonstratio Mathematica 2019; 52(2). doi:10.1515/dema-2019-0015
[27] Özger F, Başar F. Domain of the double sequential band matrix B(er; es) on some Maddox’s spaces. Acta Mathematica Scienta 2014; 34: 394-408.
[28] Özger F, Başar F. Domain of the double sequential band matrix B(er; es) on some Maddox’s spaces. American Institute of Physics Conference Proceedings 2012; 1470: 152-155.
[29] Stieglitz M, Tietz H. Matrixtransformationen von Folgenräumeneine Ergebnisübersicht. Mathematische Zeitschrift 1977; 154: 1-16.
[30] Wilansky A. Summability through Functional Analysis. New York, NY, USA: North–Holland Mathematics Studies 85, 1984.
[31] Veličković V, Malkowsky E, Özger F. Visualization of the spaces W(u; v; ℓp) and their duals. American Institute of Physics Conference Proceedings 2016, 1759: doi: 10.1063/1.4959634
[32] Yeşilkayagil M, Başar F. Some topological properties of almost null and almost convergent sequences. Turkish Journal of Mathematics 2016; 40: 624-630.