On the domain of Cesàro matrix defined by weighted means in \(\ell_{t(.)}\), and its pre-quasi ideal with some applications

Volume 26, Issue 1, pp 41--66 http://dx.doi.org/10.22436/jmcs.026.01.05

Publication Date: October 14, 2021 Submission Date: May 08, 2021 Revision Date: June 14, 2021 Accteptance Date: August 10, 2021

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Authors

A. A. Bakery - Department of Mathematics, College of Science and Arts at Khulis, University of Jeddah, Jeddah, Saudi Arabia. - JeddahDepartment of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt. E. A. E. Mohamed - Department of Mathematics, College of Science and Arts at Khulis, University of Jeddah, , Jeddah, Saudi Arabia . - Department of Mathematics, Faculty of Education, Alzaeim Alazhari University, Khartoum, Sudan. O. K. S. K. Mohamed - Department of Mathematics, College of Science and Arts at Khulis, University of Jeddah, Jeddah, Saudi Arabia. - Academy of Engineering and Medical Sciences, Department of Mathematics, Khartoum, Sudan.

Abstract

In this article, we have constructed the sequence space \(\left(\Xi(p,r,t)\right)_{\upsilon}\) by the domain of Cesàro matrix defined by weighted means in Nakano sequence space \(\ell_{(t_{l})}\), where \(t\!=\!(t_{l})\) and \(r\!=\!(r_{l})\) are sequences of positive reals, and \(\upsilon(f)\!=\!\displaystyle\sum^{\infty}_{l=0}\left(p_{l}\left|\sum^{l}_{z=0}r_{z}f_{z}\right|\right)^{t_{l}}\), with \(f=(f_{z})\in \Xi(p,r,t)\). Some geometric and topological actions of \(\left(\Xi(p,r,t)\right)_{\upsilon}\), the multiplication maps stand-in on \(\left(\Xi(p,r,t)\right)_{\upsilon}\), and the eigenvalues distribution of operator ideal formed by \(\left(\Xi(p,r,t)\right)_{\upsilon}\) and \(s\)-numbers are discussed. We offer the existence of a fixed point of Kannan contraction operator improvised on these spaces. It is curious that various numerical experiments are introduced to present our results. Moreover, a few gilded applications to the existence of solutions of non-linear difference equations are examined.

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Keywords

• Cesàro matrix
• weighted means
• \(s\)-numbers
• multiplication operator
• minimum space
• Kannan contraction operator

MSC

•  46B10
•  46B15
•  47B10
•  46C05
•  46E05
•  46E15
•  46E30

References

• [1] R. P. Agarwal, N. Hussain, M.-A. Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr. Appl. Anal., 2012 (2012), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [2] B. Altay, F. Başar, Generalization of the sequence space $\ell(p)$ derived by weighted means, J. Math. Anal. Appl., 330 (2007), 147--185
  • View Article
  • MathSciNet
  • Google Scholar


• [3] Y. Altin, M. Et, B. C. Tripathy, The sequence space $|\overline{N}_{p}| (M, r, q, s)$ on seminormed spaces, Appl. Math. Comput., 154 (2004), 423--430
  • View Article
  • MathSciNet
  • Google Scholar


• [4] A. A. Bakery, A. R. A. Elmatty, A note on Nakano generalized difference sequence space, Adv. Difference Equ., 2020 (2020), 17 pages
  • View Article
  • Google Scholar


• [5] A. A. Bakery, O. M. K. S. K. Mohamed, Kannan Prequasi Contraction Maps on Nakano Sequence Spaces, J. Funct. Spaces, 2020 (2020), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [6] A. A. Bakery, O. M. K. S. K. Mohamed, $(r_{1},r_{2})$-Cesaro summable sequence space of non-absolute type and the involved pre-quasi ideal, J. Inequal. Appl., 2021 (2021), 20 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [7] A. A. Bakery, M. M. Mohammed, Some properties of pre-quasi operator ideal of type generalized Cesaro sequence space defined by weighted means, Open Math., 17 (2019), 1703--1715
  • View Article
  • MathSciNet
  • Google Scholar


• [8] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3 (1922), 133--181
  • View Article
  • Google Scholar


• [9] M. Başarir, E. E. Kara, On compact operators on the Riesz $B(m)$-difference sequence space, Iran. J. Sci. Technol. Trans. A Sci., 35 (2011), 279--285
  • View Article
  • MathSciNet
  • Google Scholar


• [10] M. Başarir, E. E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal., 2 (2011), 114--129
  • View Article
  • MathSciNet
  • Google Scholar


• [11] M. Başarir, E. E. Kara, On the $B$-difference sequence space derived by weighted mean and compact operators, J. Math. Anal. Appl., 391 (2012), 67--81
  • View Article
  • MathSciNet
  • Google Scholar


• [12] M. Başarir, E. E. Kara, On the $m^{th}$ order Difference Sequence Space of Generalized Weighted Mean and Compact Operators, Acta Math. Sci. Ser. B (Engl. Ed.), 33 (2013), 797--813
  • View Article
  • MathSciNet
  • Google Scholar


• [13] N. Faried, A. A. Bakery, Small operator ideals formed by $s$ numbers on generalized Cesaro and Orlicz sequence spaces, J. Inequal. Appl., 2018 (2018), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [14] S. J. H. Ghoncheh, Some Fixed point theorems for Kannan mapping in the modular spaces, Ciencia e Natura, 37 (2015), 462--466
  • View Article
  • Google Scholar


• [15] B. Hazarika, A. Esi, Lacunary Ideal quasi Cauchy sequences, An. Univ. Craiova Ser. Mat. Inform., 45 (2018), 220--231
  • View Article
  • MathSciNet
  • Google Scholar


• [16] N. Hussain, A. R. Khan, R. P. Agarwal, Krasnoselkii and Ky Fan type fixed point theorems in ordered Banach spaces, J. Nonlinear Convex Anal., 11 (2010), 475--789
  • Google Scholar


• [17] M. İlkhan, S. Demiriz, E. E. Kara, Multiplication operators on Cesaro second order function spaces, Positivity, 24 (2020), 605--614
  • View Article
  • MathSciNet
  • Google Scholar


• [18] N. J. Kalton, Spaces of compact operators, Math. Ann., 208 (1974), 267--278
  • View Article
  • MathSciNet
  • Google Scholar


• [19] R. Kannan, Some results on fixed points--II, Amer. Math. Monthly, 76 (1969), 405--408
  • View Article
  • MathSciNet
  • Google Scholar


• [20] E. E. Kara, M. Başarir, On compact operators and some Euler $B(m)$-difference sequence spaces, J. Math. Anal. Appl., 379 (2011), 499--511
  • View Article
  • MathSciNet
  • Google Scholar


• [21] V. A. Khan, R. K. A. Rababah, A. Esi, S. A. A. Abdullah, K. M. A. S. Aslhlool, Some new spaces of ideal convergent double sequences by using compact operator, J. Appl. Sci., 9 (2017), 467--474
  • View Article
  • Google Scholar


• [22] B. S. Komal, S. Pandoh, K. Raj, Multiplication operators on Cesaro sequence spaces, Demonstr. Math., 49 (2016), 430--463
  • View Article
  • MathSciNet
  • Google Scholar


• [23] B. M. Makarov, N. Faried, Some properties of operator ideals constructed by $s$ numbers, (In Russian), Academy of Science, Siberian section, Novosibirsk, Russia, 1977 (1977), 206--211
  • Google Scholar


• [24] T. Mrowka, A Brief Introduction to Linear Analysis: Fredholm Operators, Geometry of Manifolds, Massachusetts Institute of Technology: MIT Open Couse Ware, (Fall 2004) (2004)
  • View Article


• [25] M. Mursaleen, F. Başar, Domain of Cesaro mean of order one in some spaces of double sequences, Studia Sci. Math. Hungar., 51 (2014), 335--356
  • View Article
  • MathSciNet
  • Google Scholar


• [26] M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73 (2010), 2541--2557
  • View Article
  • MathSciNet
  • Google Scholar


• [27] M. Mursaleen, A. K. Noman, On some new sequence spaces of non\textendash absolute type related to the spaces $\ell_{p}$ and $\ell_{\infty}$ I, Filomat, 25 (2011), 33--51
  • View Article
  • Google Scholar


• [28] M. Mursaleen, A. K. Noman, On some new sequence spaces of non-absolute type related to the spaces $\ell_{p}$ and $\ell_{\infty}$ II, Math. Commun., 16 (2011), 383--398
  • View Article
  • MathSciNet
  • Google Scholar


• [29] M. Mursaleen, A. K. Noman, Compactness of matrix operators on some new difference sequence spaces, Linear Algebra Appl., 436 (2012), 41--52
  • View Article
  • MathSciNet
  • Google Scholar


• [30] P.-N. Ng, P.-Y. Lee, Cesaro sequences spaces of non-absolute type, Comment Math. Prace Math., 20 (1978), 429--433
  • View Article
  • Google Scholar


• [31] A. Pietsch, $s$-numbers of operators in Banach spaces, Studia Math., 51 (1974), 201--223
  • View Article
  • MathSciNet


• [32] A. Pietsch, Small ideals of operators, Studia Math., 51 (1974), 265--267
  • View Article
  • MathSciNet


• [33] A. Pietsch, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin (1978)
  • MathSciNet
  • Google Scholar


• [34] A. Pietsch, Operator Ideals, North-Holland Publishing Co., Amsterdam (1980)
  • Google Scholar


• [35] A. Pietsch, Eigenvalues and $s$-numbers, Cambridge University Press, New York (1986)
  • View Article
  • Google Scholar


• [36] K. Raj, A. Esi, S. Jamwal, Orlicz-Quasi Cauchy double sequence spaces for RH-Regular Matrix, J. Class. Anal., 10 (2017), 67--76
  • View Article
  • MathSciNet
  • Google Scholar


• [37] B. E. Rhoades, Operators of $A-p$ type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 59 (1975), 238--241
  • View Article
  • MathSciNet


• [38] P. Salimi, A. Latif, N. Hussain, Modified $\alpha$-$\psi$-contractive mappings with applications, Fixed Point Theory Appl., 2013 (2013), 19 pages
  • View Article
  • Google Scholar


• [39] B. C. Tripathy, On a class of difference sequences related to the $p$-normed space $l^p$, Demonstratio Math., 36 (2003), 867--872
  • View Article
  • MathSciNet
  • Google Scholar


• [40] B. C. Tripathy, On a new class of sequences, Demonstratio Math., 37 (2004), 377--381
  • View Article
  • MathSciNet
  • Google Scholar


• [41] B. C. Tripathy, A. Esi, B. Tripathy, On a new type of generalized difference Cesaro sequence spaces, Soochow J. Math., 31 (2005), 333--340
  • View Article
  • Google Scholar


• [42] B. C. Tripathy, P. Sudipta, R. D. Nanda, Banach and Kannan fixed point results in fuzzy $2$-metric spaces, Proyecciones, 32 (2013), 359--375
  • View Article
  • MathSciNet
  • Google Scholar


• [43] T. Yaying, B. Hazarika, S. A. Mohiuddine, M. Mursaleen, Estimation of upper bounds of certain matrix operators on Binomial weighted sequence spaces, Adv. Oper. Theory, 5 (2020), 1376--1389
  • View Article
  • MathSciNet
  • Google Scholar


• [44] T. Yaying, B. Hazarika, M. Mursaleen, Norm of matrix operator on Orlicz-binomial spaces and related operator ideal, J. Math. Comput. Sci., 23 (2021), 341--353
  • View Article
  • Google Scholar


• [45] T. Yaying, B. Hazarika, M. Mursaleen, On sequence space derived by the domain of $q$-Cesaro matrix in $\ell_p$ space and the associated operator ideal, J. Math. Anal. Appl., 493 (2021), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar