Some hyperstability results of a p-radical functional equation related to Drygas mappings in non-Archimedean Banach spaces
Keywords:Hyperstability, non-Archimedean Banach spaces, Radical functional equations, Drygas functional equations
The aim of this paper is to introduce and solve the following p-radical functional equation related to Drygas mappings
f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzdȩk’sfixed point theorem , we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to Drygas mappings
L. Aiemsomboon and W. Sintunavarat, “On a new type of stability of a radical quadratic functional equation using Brzdȩk’s fixed point theorem”, Acta mathematica hungarica, vol. 151, no. 1, pp. 35–46, Nov. 2016, doi: 10.1007/s10474-016-0666-2
L. Aiemsomboon and W. Sintunavarat, “On generalized hyperstability of a general linear equation”, Acta mathematica hungarica, vol. 149, no. 2, pp. 413–422, May 2016, doi: 10.1007/s10474-016-0621-2
C. Alabiso and I. Weiss, A primer on Hilbert space theory: linear spaces, topological spaces, metric spaces, normed spaces, and topological groups. Cham: Springer, 2015, doi: 10.1007/978-3-319-03713-4
Z. Alizadeh and A. G. Ghazanfari, “On the stability of a radical cubic functional equation in quasi-β-spaces,” Journal of fixed point theory and applications, vol. 18, no. 4, pp. 843–853, Aug. 2016, doi: 10.1007/s11784-016-0317-9
M. Almahalebi, A. Charifi and S. Kabbaj, “Hyperstability of a Cauchy functional equation”, Journal of nonlinear analysis and optimization: theory applications, vol. 6, no. 2, pp. 127-137, 2015. [On line]. Available: https://bit.ly/3ny4yMN
M. Almahalebi, “On the hyperstability of σ-Drygas functional equation on semigroups”, Aequationes mathematicae, vol. 90, no. 4, pp. 849-857, Aug. 2016, doi: 10.1007/s00010-016-0422-2
M. Almahalebi and C. Park, “On the hyperstability of a functional equation in commutative groups”, Journal of computational analysis applications, vol. 20, no. 1, pp. 826-833, 2016.
T. Aoki, “On the stability of the linear transformation in Banach spaces”, Journal of the Mathematical Society of Japan, vol. 2, no. 1-2, pp. 64–66, Sep. 1950, doi: 10.2969/jmsj/00210064
D. G. Bourgin, “Classes of transformations and bordering transformations”, Bulletin of the American mathematical society, vol. 57, no. 4, pp. 223-237, 1951, doi: 10.1090/S0002-9904-1951-09511-7
N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliński, “On some recent developments in Ulams type stability”, Abstract and applied analysis, vol. 2012, pp. Art ID. 716936, Dec. 2012, doi: 10.1155/2012/716936
J. Brzdȩk, “A note on stability of additive mappings”, in Stability of mappings of Hyers-Ulam type, T. M. Rassias and J. Tabor, Eds. Palm Harbor, FL: Hadronic Press, 1994, pp. 19–22.
J. Brzdȩk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations”, Nonlinear analysis: theory, methods & applications, vol. 74, no. 17, pp. 6728–6732, Dec. 2011, doi: 10.1016/j.na.2011.06.052
J. Brzdȩk and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces”, Nonlinear analysis: theory, methods & applications, vol. 74, no. 18, pp. 6861–6867, Dec. 2011, doi: 10.1016/j.na.2011.06.050
J. Brzdęk, “Stability of additivity and fixed point methods”, Fixed point theory and applications, vol. 2013, no. 1, Art ID. 285, Nov. 2013, doi: 10.1186/1687-1812-2013-285
J. Brzdȩk “Hyperstability of the Cauchy equation on restricted domains,” Acta mathematica hungarica, vol. 141, no. 1-2, pp. 58–67, Feb. 2013, doi: 10.1007/s10474-013-0302-3
J. Brzdęk, L. Cădariu, and K. Ciepliński, “Fixed point theory and the Ulam stability”, Journal of function spaces, vol. 2014, Art ID. 829419, 2014, doi: 10.1155/2014/829419
J. Brzdęk, W. Fechner, M. S. Moslehian, and J. Sikorska, “Recent developments of the conditional stability of the homomorphism equation”, Banach journal of mathematical analysis, vol. 9, no. 3, pp. 278–326, 2015, doi:10.15352/bjma/09-3-20
J. Brzdęk, “Remarks on solutions to the functional equations of the radical type”, Advances in the theory of nonlinear analysis and its application, vol. 1, no. 2, pp. 125-135, Dec. 2017, doi: 10.31197/atnaa.379095
H. Drygas, “Quasi-inner products and their applications”, in Advances in multivariate statistical analysis, A. K. Gupta, Ed. Dordrecht: Springer, 1987, pp. 13–30, doi: 10.1007/978-94-017-0653-7_2
B. R. Ebanks, P. L. Kannappan, and P. K. Sahoo, “A common generalization of functional equations characterizing normed and quasi-inner-product spaces”, Canada mathematical bulletin, vol. 35, no. 3, pp. 321- 327, Sep. 1992, doi: 10.4153/CMB-1992-044-6
M. E. Gordji, H. Khodaei, A. Ebadian, and G. H. Kim, “Nearly radical quadratic functional equations in p-2-normed spaces”, Abstract and applied analysis, vol. 2012 Art ID. 896032 , 2012, doi: 10.1155/2012/896032
M. E. Gordji and M. Parviz, “On the Hyers Ulam stability of the functional equation”, Nonlinear functions analysis applications, vol. 14, no. 3, pp. 413-420, 2009. [On line]. Available: https://bit.ly/3pVAIDN
P. Găvruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings”, Journal mathematics analysis applications, vol. 184, no. 3, pp. 431-436, Jun. 1994, doi: 10.1006/jmaa.1994.1211
D. H. Hyers, “On the stability of the linear functional equation”, Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941, doi: 10.1073/pnas.27.4.222
S.-M. Jung, and P. K. Sahoo, “Stability of a functional equation of Drygas”, Aequationes mathematicae, vol. 64, no. 3, pp. 263-273, Dec. 2002, doi: 10.1007/PL00012407
H. Khodaei, M. E. Gordji, S. Kim, and Y. Cho, “Approximation of radical functional equations related to quadratic and quartic mappings”, Journal of mathematical analysis and applications, vol. 395, no. 1, pp. 284–297, Nov. 2012, doi: 10.1016/j.jmaa.2012.04.086
S. Kim, Y. Cho, and M. E. Gordji, “On the generalized Hyers-Ulam-Rassias stability problem of radical functional equations”, Journal of inequalities and applications, Art ID. 186, Aug. 2012, doi: 10.1186/1029-242X-2012-186
G. Maksa and Z. Pales, “Hyperstability of a class of linear functional equations”, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, vol. 17, no. 2, pp. 107-112, 2001. [on line]. Available: https://bit.ly/3nzQLFv
M. Piszczek and J. Szczawińska, “Hyperstability of the Drygas functional equation”, Journal function spaces and applied, vol. 2013, Art ID 912718, Jul. 2013, doi: 10.1155/2013/912718
University of Cracow and AGH University of Science and Technology “16th International Conference on Functional Equations and Inequalities, Będlewo, Poland, May 17-23, 2015”, Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, vol. 14, no. 1, pp. 163-202, Dec. 2015, doi: 10.1515/aupcsm-2015-0012
Th. M. Rassias, “On the stability of the linear mapping in Banach spaces”, Proceedings of the American mathematical society, vol. 72, pp. 297-300, 1978, doi: 10.1090/S0002-9939-1978-0507327-1
Th. M. Rassias, “Problem 16, 2°. The Twenty-seventh International Symposium on Functional Equations, August 14–24, 1989, Bielsko-BiałKatowice—Kraków, Poland”, Aequationes mathematicae, vol. 39, no. 2-3, pp. 292-293, Apr. 1990, doi: 10.1007/BF01833155
Th. M. Rassias, “On a modified Hyers-Ulam sequence”, Journal of mathematical analysis and applications, vol. 158, no. 1, pp. 106-113, Jun. 1991, doi: 10.1016/0022-247X(91)90270-A
P. K. Sahoo and P. Kannappan, Introduction to functional equations. Boca Raton, FL: CRC Press, 2011.
M. Sirouni and S. Kabbaj, “A fixed point approach to the hyperstability of Drygas functional equation in metric spaces”, Journal of mathematical and computational science, vol. 4, no. 4, 2014, 705-715. [On line]. Available: https://bit.ly/2MXuvJb
W. Smajdor, “On set-valued solutions of a functional equation of Drygas”, Aequationes mathematicae, vol. 77, pp. 89-97, Mar. 2009, doi: 10.1007/s00010-008-2935-9
S. M. Ulam, Problems in modern mathematics. New York, NY: J. Wiley, 1964.
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