On composition operators of Fibonacci matrix and applications of Hausdorff measure of noncompactness
DOI:
https://doi.org/10.5269/bspm.39960Abstract
The aim of the paper is introduced the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right).$ Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of new spaces and also construct the basis for the space $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$
Finally we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators by applying the Hausdorff measure of noncompactness, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\},$ and $1\leq p<\infty.$
References
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10. A. Das, B. Hazarika, Some new Fibonacci difference spaces of non-absolute type and compact operators, Linear Multilinear Algebra, 65(12)(2017), 2551-2573.
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31. M. Mursaleen, A.K. Gaur, A.H. Saifi, Some new sequence spaces and their duals and matrix transformations, Bull. Calcutta Math. Soc. 88 (3) (1996), 207-212.
32. M. Mursaleen, A.K. Noman, Compactness by the Hausdorff measure of non-compactness, Nonlinear Anal. TMA 73 (8)(2010), 2541-2557.
33. M. Mursaleen, A.K. Noman, On the spaces of λ-convergent and bounded sequences, Thai J. Math. 8 (2) (2010), 311-329.
34. V. Rakocevic, Measures of non-compactness and some applications, Filomat 12 (2) (1998), 87-120.
35. M. Stieglitz, H. Tietz, Matrix transformationen von Folgenraumen Eine Ergebnisubersicht, Math. Z. 154 (1977), 1-16.
36. B. C. Tripathy, Matrix transformation between some classes of sequences, J. Math. Anal. Appl. 206(1997) 448-450.
37. O. Tug, F. Basar, On the domain of Norlund mean in the spaces of null and convergent sequences, TWMS J. Pure Appl. Math. 7 (1) (2016), 76–87.
38. O. Tug, F. Basar, On the spaces of Norlund almost null and Norlund almost convergent sequences, Filomat 30 (3) (2016), 773–783.
39. A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies 85, Amsterdam · New York · Oxford, 1984.
2. B. Altay, F. Basar, Certain topological properties and duals of the domain of a triangle matrix in a sequence space, J. Math. Anal. Appl. 336 (2007), 632-645.
3. J. Banas, K. Goebel, Measure of non-compactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York · Basel, 1980.
4. F. Basar, B. Altay, M. Mursaleen, Some generalizations of the space bvp of p-bounded variation sequences, Nonlinear Anal. TMA 68 (2008), 273-287.
5. F. Basar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1) (2003), 136-147.
6. F. Basar, E. Malkowsky, The characterization of compact operators on spaces of strongly summable and bounded sequences, Appl. Math. Comput. 217 (12) (2011), 5199-5207.
7. H. Capan, F. Basar, Domain of the double band matrix defined by Fibonacci numbers in the Maddox’s space â„“(p), Electron. J. Math. Anal. Appl. 3 (2) (2015), 31–45.
8. A. Das, B. Hazarika, Some properties of generalized Fibonacci difference bounded and p-absolutely convergent sequences, Bol. Soc. Parana. Mat. 36(1)(2018), 37-50.
9. A. Das, B. Hazarika, Matrix transformation of Fibonacci band matrix on generalized bv-space and its dual spaces, Bol. Soc. Parana. Mat. 36(3)(2018), 41-52.
10. A. Das, B. Hazarika, Some new Fibonacci difference spaces of non-absolute type and compact operators, Linear Multilinear Algebra, 65(12)(2017), 2551-2573.
11. J. Diestel, Sequences and Series in Banach Spaces, vol. 92 (1984), Springer, New York, NY, USA.
12. I. Djolovic, Compact operators on the spaces a r0(∆) and a rc(∆), J. Math. Anal. Appl. 318 (2) (2006), 658–666.
13. I. Djolovic, E. Malkowsky, A note on compact operators on matrix domains, J. Math. Anal. Appl. 340(1) (2008), 291–303.
14. J. Garcıa-Falset, Stability and fixed points for nonexpansive mapping, Houst. J. Math. 20 (3) (1994), 495-506.
15. J. Garcıa-Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (2) (1997), 532-542.
16. A. M. Jarrah, E. Malkowsky, BK spaces, bases and linear operators, Rendiconti Circ. Mat. Palermo II 52 (1990) 177–191.
17. A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat 17 (2003), 59-78.
18. E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl. 2013, 2013:38.
19. E. E. Kara and Merve Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra, 64(11)(2016) 2208-2223.
20. E. E. Kara, M. Basarır, M. Mursaleen, Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers, Kragujevac J. Math. 39(2)(2015) 217-230.
21. E. E. Kara, S. Demiriz, Some new paranormed difference sequence spaces derived by Fibonacci numbers, Miskolc Math. Notes 16(2)(2015) 907-923.
22. M. Kirisci, F. Basar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl. 60 (2010), 1229-1309.
23. H. Knaust, Orlicz sequence spaces of Banach-Saks type, Arch. Math. 59 (6) (1992), 562-565.
24. T. Koshy, Fibonacci and Lucas Numbers with applications, Wiley, 2001.
25. E. Malkowsky, Klassen von Matrixabbildungen in paranormierten FK-R¨aumen, Analysis (Munich) 7 (1987), 275-292.
26. E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of non-compactness, in: Zb. Rad. (Beogr.), vol. 9 (17), Matematicki institut SANU, Belgrade, 2000, pp. 143-234.
27. E. Malkowsky, V. Rakocevic, S. Zivkovic, Matrix transformations between the sequence spaces w p0(Λ), vp0(Λ), cp0(Λ), 1 < p < ∞, and BK spaces, Appl. Math. Comput. 147 (2004), 377-396.
28. M. Mursaleen, Generalized spaces of difference sequences, J. Math. Anal. Appl. 203 (3) (1996), 738-745.
29. M. Mursaleen, V. Karakaya, H. Polat, N. S¸imsek, Measure of non-compactness of matrix operators on some difference sequence spaces of weighted means, Comput. Math. Appl. 62 (2011), 814-820.
30. M. Mursaleen, A.K. Noman, On some new sequence spaces of non-absolute type related to the spaces ℓp and ℓ∞ I, Filomat 25 (2) (2011), 33-51.
31. M. Mursaleen, A.K. Gaur, A.H. Saifi, Some new sequence spaces and their duals and matrix transformations, Bull. Calcutta Math. Soc. 88 (3) (1996), 207-212.
32. M. Mursaleen, A.K. Noman, Compactness by the Hausdorff measure of non-compactness, Nonlinear Anal. TMA 73 (8)(2010), 2541-2557.
33. M. Mursaleen, A.K. Noman, On the spaces of λ-convergent and bounded sequences, Thai J. Math. 8 (2) (2010), 311-329.
34. V. Rakocevic, Measures of non-compactness and some applications, Filomat 12 (2) (1998), 87-120.
35. M. Stieglitz, H. Tietz, Matrix transformationen von Folgenraumen Eine Ergebnisubersicht, Math. Z. 154 (1977), 1-16.
36. B. C. Tripathy, Matrix transformation between some classes of sequences, J. Math. Anal. Appl. 206(1997) 448-450.
37. O. Tug, F. Basar, On the domain of Norlund mean in the spaces of null and convergent sequences, TWMS J. Pure Appl. Math. 7 (1) (2016), 76–87.
38. O. Tug, F. Basar, On the spaces of Norlund almost null and Norlund almost convergent sequences, Filomat 30 (3) (2016), 773–783.
39. A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies 85, Amsterdam · New York · Oxford, 1984.
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