Compact Operators on the Sets of Fractional Difference Sequences

Year 2019, Volume: 23 Issue: 3, 425 - 434, 01.06.2019

Faruk Özger

https://doi.org/10.16984/saufenbilder.463368

Cited By: 3

Abstract

The sets of fractional difference sequences have been studied in the literature recently. In this work, some identities or estimates for the operator norms and the measure of noncompactness of certain operators on difference sets of sequences of fractional orders are established. Some classes of compact operators on those spaces are characterized.

Keywords

measure of noncompactness, fractional operators, compact operator

References

• A.M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat \textbf{17}, (2003), 59--78.
• A. Karaisa, F. \"{O}zger, Almost difference sequence space derived by using a generalized weighted mean. J. Comput. Anal. Appl. \textbf{19}(1), (2015), 27--38.
• A. Karaisa, F. \"{O}zger, On almost convergence and difference sequence spaces of order $m$ with core theorems, Gen. Math. Notes, \textbf{26}(1), (2015), 102--125.
• A. Wilansky, Summability through Functional Analysis, North--Holland Mathematics Studies 85, Amsterdam, New York, Oxford, 1984.
• C. Ayd\i n, B. Altay, Domain of generalized difference matrix $ B(r, s) $ on some Maddox's spaces, Thai J. Math. \textbf{11}(1), (2012), 87-–102.
• E. Malkowsky, V. Rako\v{c}evi\'{c}, An introduction into the theory of sequence spaces and measures of noncompactness, Zb. Rad. (Beogr.) \textbf{9}(17) (2000), 143--234.
• E. Malkowsky, F. \"Ozger, V. Veli\v{c}kovi\'{c}, Some mixed paranorm spaces, Filomat, \textbf{31}(4), (2017), 1079--1098.
• E. Malkowsky, F. \"{O}zger, A note on some sequence spaces of weighted means, Filomat \textbf{26}(3), (2012), 511–-518.
• E. Malkowsky, F. \"{O}zger, Compact operators on spaces of sequences of weighted means, AIP Conf. Proc. \textbf{1470}, (2012), 179--182.
• E. Malkowsky, F. \"{O}zger, A. Alotatibi, Some notes on matrix mappings and their Hausdorff measure of noncompactness, Filomat \textbf{28}(5), (2014), 1059–-1072.
• E. Malkowsky, F. \"Ozger, V. Veli\v{c}kovi\'{c}, Some spaces related to Cesaro sequence spaces and an application to crystallography, MATCH Commun. Math. Comput. Chem. \textbf{70}(3), (2013), 867--884.
• E. Malkowsky, F. \"Ozger, V. Veli\v{c}kovi\'{c}, Matrix transformations on mixed paranorm spaces, Filomat \textbf{31}(10), (2017), 2957--2966.
• E. Malkowsky, V. Rako\v{c}evi\'{c}, On matrix domains of triangles, Appl. Math. Comput. \textbf{189}, (2007), 1148--1163.
• F. Ba\c{s}ar, Summability theory and its applications, Bentham Science Publishers. e-books, Monographs, Istanbul, (2012) ISBN: 978-1-60805-420-6.
• F. Nuray, U. Ulusu, E. D\"{u}ndar, Lacunary statistical convergence of double sequences of sets, Soft Comput (2016) \textbf{20}(7), (2016), 2883-–2888.
• F. \"Ozger, Some geometric characterizations of a fractional Banach set, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. (2019), to be appear.
• F. \"Ozger, F. Ba\c{s}ar, Domain of the double sequential band matrix $B(\widetilde{r},\widetilde{s})$ on some Maddox's spaces, AIP Conf. Proc. \textbf{1470} (2012), 152--155.
• F. \"Ozger, F. Ba\c{s}ar, Domain of the double sequential band matrix $B(\widetilde{r},\widetilde{s})$ on some Maddox's spaces, Acta Math. Sci. Ser. B Engl. Ed. \textbf{34}(2) (2014), 394-408.
• G. Kılınç, M. Candan, A different approach for almost sequence spaces defined by a generalized weighted means, Sakarya University Journal of Science, 21(6), (2017)1529--1536.
• H. Furkan, On some $\lambda$ difference sequence spaces of fractional order, J. Egypt. Math. Soc. \textbf{25}, (2017), 37--42.
• H. K\i zmaz, On certain sequence spaces, Canad. Math. Bull. \textbf{24}(2) (1981), 169--176.
• I. Djolovi\'{c}, E. Malkowsky, A note on compact operators on matrix domains, J. Math. Anal. Appl. \textbf{340} (2008), 291--303.
• M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl. \textbf{2012}(281), (2012),doi:10.1186/1029-242X-2012-281.
• M. Kiri\c{s}\c{c}i, F. Ba\c{s}ar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput.Math. Appl. \textbf{60}(5) (2010), 1299--1309.
• M. Mursaleen, V. Karakaya, H. Polat, N. Sim\c{s}ek, Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means, Comput. Math. Appl. \textbf{62}(2), (2011), 814--820.
• M. Mursaleen, S.A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $ \ell_{p} $ spaces, Nonlinear Anal. \textbf{75}(4), (2012), 2111--2115.
• M. Stieglitz, H. Tietz, Matrixtransformationen von Folgenr\"{a}umeneine Ergebnis\"{u}bersicht, Math. Z. \textbf{154}, (1977), 1--16.
• M. Ye\c{s}ilkayagil, F. Ba\c{s}ar, Spaces of $ A_\lambda $-almost null and $ A_\lambda $-almost convergent sequences, J. Egypt. Math. Soc. \textbf{23}(1), (2015), 119-126.
• N.L. Braha, F. Ba\c{s}ar, On the domain of the triangle $ A(\lambda) $ on the Spaces of null, convergent, and bounded sequences, Abstr. Appl. Anal. (2013), doi:10.1155/2013/476363.
• N.A. Sheikh, A.H. Ganie, On some new sequence spaces of non-absolute type and matrix transformations, J. Egypt. Math. Soc. \textbf{21}(2), (2013), 108--114.
• O. Duyar, S. Demiriz, On vector--valued operator Riesz sequence spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. \textbf{68}(1), (2019), 236--247.
• P. Baliarsingh, S. Dutta, On the classes of fractional order of difference sequence spaces and matrix transformations, Appl. Math. Comput. \textbf{250}, (2015), 665--674.
• S. Ercan, \c{C}. Bekta\c{s}, Some topological and geometric properties of a new BK space derived by using regular matrix of Fibonacci numbers, Linear Multilinear Algebra, \textbf{65}(5), (2017), 909--921.
• U. Kadak, P. Baliarsingh, On certain Euler difference sequence spaces of fractional order and related dual properties, J. Nonlinear Sci. Appl. \textbf{8}, (2015), 997--1004.
• U. Kadak, N.L. Braha, H.M. Srivastava, Statistical weighted $ B $--summability and its applications to approximation theorems, Appl. Math. Comput. \textbf{302}, (2017), 80--96.
• V. Veli\v{c}kovi\'{c}, E. Malkowsky, F. \"Ozger, Visualization of the spaces $ W (u, v; \ell_{p}) $ and their duals, AIP Conf. Proc. \textbf{1759}, (2016), doi:10.1063/1.4959634.