The degree of nondensifiability of linear bounded operators and its applications

In the present paper we define the degree of nondensifiability (DND for short) of a bounded linear operator T on a Banach space and analyze its properties and relations with the Hausdorff measure of non-compactness (MNC for short) of T. As an application of our results, we have obtained a formula to find the essential spectral radius of a bounded operator T on a Banach space as well as we have provided the best possible lower bound for the Hyers-Ulam stability constant of T in terms of the aforementioned DND.

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References:

Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics vol. 50, American Math. Soc., Providence, 2002. https://doi.org/10.1090/gsm/050

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measure of noncompactness and condensing operators,Birkhäuser Verlag Basel, 1992. https://doi.org/10.1007/978-3-0348-5727-7

A .G. Aksoy and J. M. Almira, On approximation schemes and compactness, Proceedings of the first conference on classical and functional analysis, Azuga-Romania (2014), 5-24.

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064

J. Appell, Measure of noncompactness and some applications, Fixed Point Theory 6 (2005), 157-229.

J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser Verlag, 1997. https://doi.org/10.1007/978-3-0348-8920-9

D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7

J. Brzdęk, D. Popa, I. Raşa and B. Xu, Ulam Stability of Operators, Academic Press 2018. https://doi.org/10.1007/978-3-030-28972-0

C. Buşe, D. O'Regan and O. Saierli, Hyers-Ulam stability for linear differences with time dependent and periodic coefficients, Symmetry 11 (2019). https://doi.org/10.3390/sym11040512

E. Chafai and M. Boumazgour, Some perturbation results for ascent and descent via measure of non-compactness, Filomat 32 (2018), 3495-3504. https://doi.org/10.2298/FIL1810495C

Y. Cherruault and G. Mora, Optimisation Globale. Théorie des Courbes α-denses, Económica, Paris, 2005.

D. E. Edmuns and W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, 1987.

G. García, A quantitative version of the Arzelá-Ascoli theorem based on the degree of nondensifiability and applications, Appl. Gen. Topol. 10 (2019), 265-279. https://doi.org/10.4995/agt.2019.10930

G. García, Existence of solutions for infinite systems of ordinary differential equations by densifiability techniques, Filomat 32 (2018), 3419-3428. https://doi.org/10.2298/FIL1810419G

G. García, Solvability of initial value problems with fractional order differential equations in Banach spaces by alpha-dense curves, Fract. Calc. Appl. Anal. 20 (2017), 646-661. https://doi.org/10.1515/fca-2017-0034

G. García and G. Mora, A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral equations, J. Math. Anal. Appl. 472, no. (2019), 1220-1235. https://doi.org/10.1016/j.jmaa.2018.11.073

G. García and G. Mora, The degree of convex nondensifiability in Banach spaces, J. Convex Anal. 22 (2015), 871-888.

G. García and G. Mora, On the Ulam-Hyers stability of the complex functional equation F(z)+F(2z)+...+F(nz)=0, Aequat. Math. 94 (2020), 899-911. https://doi.org/10.1007/s00010-019-00693-2

G. García and G. Mora, Projective limits of generalized scales of Banach spaces and applications, Ann. Funct. Anal. 13 (2022), 76. https://doi.org/10.1007/s43034-022-00224-2

O. Hatori, K. Kobayashi, T. Miura, H. Takagi and S. E. Takahasi, On the best constant of Hyers-Ulam stability, J. Nonlinear Convex Anal. 5 (2004), 387-393.

D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. US 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222

J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, 1961.

S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optim. Appl., vol. 48, Springer, New York (2011). https://doi.org/10.1007/978-1-4419-9637-4

J. L. Kelley, General Topology, D. van Nostrand, New York, 1955.

Y. Li and Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010), 306-309. https://doi.org/10.1016/j.aml.2009.09.020

T. Miura, S. Miyajima and S. I. Takahasi, Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (2003), 90-96. https://doi.org/10.1002/mana.200310088

G. Mora, The Peano curves as limit of α-dense curves, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 9 (2005), 23-28.

G. Mora, On the Ulam stability of F(z)+F(2z)=0, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 114 (2020), 108. https://doi.org/10.1007/s13398-020-00846-y

G. Mora and Y. Cherruault, Characterization and generation of α-dense curves, Computers Math. Applic. 33 (1997), 83-91. https://doi.org/10.1016/S0898-1221(97)00067-9

G. Mora and J. A. Mira, Alpha-dense curves in infinite dimensional spaces, Int. J. Pure Appl. Math. 5 (2003), 437-449.

G. Mora and D. A. Redtwitz, Densifiable metric spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105 (2011), 71-83. https://doi.org/10.1007/s13398-011-0005-y

M. Mursaleen and K. J. Ansari, On the stability of some positive linear operators from approximation theory, Bulletin of Mathematical Sciences 5 (2015), 147-157. https://doi.org/10.1007/s13373-015-0064-z

D. Popa, Hyers-Ulam stability of the linear recurrence with constant coefficients, Adv. Difference Equ. 2005 (2005), Article ID 407076. https://doi.org/10.1155/ADE.2005.101

D. Popa and I. Raşa, On the best constant in Hyers-Ulam stability of some positive linear operators, J. Math. Anal. Appl. 412 (2014), 103-108. https://doi.org/10.1016/j.jmaa.2013.10.039

T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1

V. Rakočević, Measure of noncompactness and some applications, Filomat 12 (1998), 87-120.

W. Rudin, Functional Analysis 2nd ed., MacGraw-Hill, New York, 1991.

H. Takagi, T. Miura and S. E. Takahasi, Essential norm and stability constant of weighted composition operators on C(X), Bull. Korean Math. Soc. 40 (2003), 583-591. https://doi.org/10.4134/BKMS.2003.40.4.583

S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York (1960).

S. Willard, General Topology, Dover Inc. Pubs., New York 1970.

A. Zada, U. Riaz and F. Khan, Hyers-Ulam stability of impulsive integral equations, Bollettino dell Unione Matematica Italiana 12 (2019), 453-467. https://doi.org/10.1007/s40574-018-0180-2

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