Fixed point property for nonexpansive mappings on large classes in Köthe-Toeplitz duals of certain difference sequence spaces

Veysel Nezir

doi:10.3934/mfc.2024026

Article Contents

2025, Volume 8, Issue 4: 529-539. Doi: 10.3934/mfc.2024026

This issue Previous Article Second-Order Hermitian–Toeplitz Determinants for the classes of Sakaguchi Convex and Starlike functions Next Article Extended dissipativity-based sampled-data controller design for fuzzy distributed parameter systems

Fixed point property for nonexpansive mappings on large classes in Köthe-Toeplitz duals of certain difference sequence spaces

Veysel Nezir

Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Received: November 2023

Revised: May 2024

Early access: June 2024

Published: August 2025

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Abstract

In 1970, Cesàro sequence spaces were introduced by Shiue. In 1981, Kızmaz defined difference sequence spaces for $ {\ell }^{\infty } $, $ {\mathrm{c}}_0 $, and $ \mathrm{c} $. In 1989, Çolak obtained new types of sequence spaces by generalizing Kızmaz's idea. Using Çolak's structure, Et and Esi, in 2000, obtained generalized difference sequences. In fact, they found the corresponding Köthe-Toeplitz duals and examined geometric properties for those spaces. We will be interested in their generalizations and we study the Goebel and Kuczumow analogy for the spaces they introduced. We recall that in 1979, Goebel and Kuczumow found that there exists a large class of closed, bounded, convex subsets in $ {\ell }^{\mathrm{1}} $ with the fixed point property for nonexpansive mappings. Their study has become a pioneer for researchers investigating if nonreflexive spaces can be renormed to have the fixed point property and even the first example was given by Lin, in 2008. Lin's study was on $ {\ell }^{\mathrm{1}} $ and there are, indeed, traces of impression from Goebel and Kuczumow. In the present study, we aim to discuss Goebel and Kuczumow analogous results for Köthe-Toeplitz duals of certain generalized difference sequence spaces studied by Et and Esi. We show that there exists a very large class of closed, bounded, convex subsets in those spaces with the fixed point property for nonexpansive mappings.

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Mathematics Subject Classification: Primary: 46B45, 47H09; Secondary: 46B10.

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[1]	A. Aasma, H. Dutta and P. N. Natarajan, An Introductory Course in Summability Theory, John Wiley & Sons, Inc., Hoboken, USA, 2017. doi: 10.1002/9781119397786.
[2]	D. E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc., 82 (1981), 423-424. doi: 10.1090/S0002-9939-1981-0612733-0.
[3]	F. Başar and H. Dutta, Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties, Chapman and Hall/CRC, Taylor & Francis Group, Boca Raton, London, New York, 2020.
[4]	Ç. A. Bektaş, M. Et and R. Çolak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292 (2004), 423-432. doi: 10.1016/j.jmaa.2003.12.006.
[5]	F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272-1276. doi: 10.1073/pnas.53.6.1272.
[6]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041-1044. doi: 10.1073/pnas.54.4.1041.
[7]	S. Chen, Y. Cui, H. Hudzik and B. Sims, Geometric properties related to fixed point theory in some Banach function lattices, Handbook of Metric Fixed Point Theory, Springer, Dordrecht, (2001), 339-389.
[8]	Y. Cui and H. Hudzik, Some geometric properties related to fixed point theory in Cesàro spaces, Collect. Math., 50 (1999), 277-288.
[9]	Y. Cui, H. Hudzik and Y. Li, On the Garcfa-Falset coefficient in some Banach sequence spaces, Function Spaces, Lecture Notes in Pure and Appl. Math., Marcel Dekker, Inc., New York, 213 (2000), 163-170.
[10]	Y. Cui, C. Meng and R. Płuciennik, Banach—Saks property and property ($\alpha$) in Cesàro sequence spaces, Southeast Asian Bull. Math., 24 (2000), 201-210. doi: 10.1007/s100120070003.
[11]	R. Çolak, On some generalized sequence spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 38 (1989), 35-46.
[12]	P. N. Dowling, C. J. Lennard and B. Turett, Some fixed point results in $\ell^{1}$ and $c_{0}$, Nonlinear Anal., 39 (2000), 929-936. doi: 10.1016/S0362-546X(98)00259-4.
[13]	H. Dutta, Characterization of certain matrix classes involving generalized difference summability spaces, Appl. Sci. APPS., 11 (2009), 60-67.
[14]	H. Dutta and T. Bilgin, Strongly $(V^\lambda, A, \Delta_{(vm)}^n, p)$-summable sequence spaces defined by an Orlicz function, Appl. Math. Lett., 24 (2011), 1057-1062. doi: 10.1016/j.aml.2011.01.022.
[15]	H. Dutta and B. E. Rhoades, Current Topics in Summability Theory and Applications, Springer, Singapore, 2016. doi: 10.1007/978-981-10-0913-6.
[16]	M. Et, On some generalized Cesàro difference sequence spaces, İstanb. Univ. Sci. Fac. J. Math. Phys. Astronom., 55 (1996), 221-229.
[17]	M. Et and R. Çolak, On some generalized difference sequence spaces, Soochow J. Math., 21 (1995), 377-386.
[18]	M. Et and A. Esi, On Köthe-Toeplitz duals of generalized difference sequence spaces, Bull. Malays. Math. Sci. Soc., 23 (2000), 25-32.
[19]	T. M. Everest, Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in $\ell^{1}$, Ph.D thesis, University of Pittsburgh, Pittsburgh, 2013.
[20]	J. G. Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl., 215 (1997), 532-542. doi: 10.1006/jmaa.1997.5657.
[21]	K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math., 47 (1973), 135-140. doi: 10.4064/sm-47-2-134-140.
[22]	K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., 28, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511526152.
[23]	K. Goebel and T. Kuczumow, Irregular convex sets with fixed-point property for nonexpansive mappings, Colloq. Math., 2 (1979), 259-264. doi: 10.4064/cm-40-2-259-264.
[24]	D. Göhde, Zum prinzip der kontraktiven abbildung, Math. Nachr., 30 (1965), 251-258. doi: 10.1002/mana.19650300312.
[25]	W. Kaczor and S. Prus, Fixed point properties of some sets in $\ell^1$, Proceedings of the International Conference on Fixed Point Theory and Applications, (2004), 11 pp.
[26]	H. Kızmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169-176. doi: 10.4153/CMB-1981-027-5.
[27]	W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004-1006. doi: 10.2307/2313345.
[28]	T. C. Lim, Fixed point theorems for uniformly Lipschitzian mappings in Lp spaces, Nonlinear Anal., 7 (1983), 555-563. doi: 10.1016/0362-546X(83)90044-5.
[29]	P. K. Lin, There is an esuivalent norm on $\ell^{1}$ that has the fixed point property, Nonlinear Anal., 68 (2008), 2303-2308. doi: 10.1016/j.na.2007.01.050.
[30]	C. A. Hernández-Linares and M. A. Japón, Renormings and fixed point property in non-commutative L1-spaces Ⅱ: Affine mappings, Nonlinear Anal., 75 (2012), 5357-5361. doi: 10.1016/j.na.2012.04.050.
[31]	E. Malkowsky, M. Mursaleen and S. Suantai, The dual spaces of sets of difference sequences of order m and matrix transformations, Acta Math. Sin. (Engl. Ser.), 23 (2007), 521-532. doi: 10.1007/s10114-005-0719-x.
[32]	M. Mursaleen, A. H. Ganie and N. A. Sheikh, New type of generalized difference sequence space of non-absolute type and some matrix transformations, Filomat, 28 (2014), 1381-1392. doi: 10.2298/FIL1407381M.
[33]	N. N. NgPeng-Nung and L. Y. LeePeng-Yee, Cesàro sequence spaces of nonabsolute type, Comment. Math., 20 (1978), 429-433.
[34]	C. Orhan, Casáro differance sequence spaces and related matrix transformations, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 32 (1983), 55-63.
[35]	J. Schauder, Der fixpunktsatz in funktionalraumen, Studia. Math., 2 (1930), 171-180.
[36]	J. S. Shiue, On the Cesáro sequence spaces, Tamkang J. Math., 1 (1970), 19-25.
[37]	B. C. Tripathy, A. Esi and B. Tripathy, On new types of generalized difference Cesáro sequence spaces, Soochow J. Math., 31 (2005), 333-340.