ON NEW APPROXIMATIONS FOR GENERALIZED CAUCHY FUNCTIONAL EQUATIONS USING BRZDȨK AND CIEPLIŃSKI'S FIXED POINT THEOREMS IN 2-BANACH SPACES

doi:10.1007/s10473-020-0316-1

Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (3): 824-834.doi: 10.1007/s10473-020-0316-1

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ON NEW APPROXIMATIONS FOR GENERALIZED CAUCHY FUNCTIONAL EQUATIONS USING BRZDȨK AND CIEPLIŃSKI'S FIXED POINT THEOREMS IN 2-BANACH SPACES

Laddawan AIEMSOMBOON, Wutiphol SINTUNAVARAT

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand

Received:2019-02-11 Revised:2019-08-29 Online:2020-06-25 Published:2020-07-17
Contact: Wutiphol SINTUNAVARAT E-mail:wutiphol@mathstat.sci.tu.ac.th
Supported by:
This work was supported by Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and Thammasat University Research Fund, Contract No. TUGG 33/2562. The second author would like to thank the Thailand Research Fund and Office of the Higher Education Commission under grant no. MRG6180283 for financial support during the preparation of this manuscript.

Abstract

Abstract: In this work, we apply the Brzdȩk and Ciepliński's fixed point theorem to investigate new stability results for the generalized Cauchy functional equation of the form
f(ax + by)=af(x) + bf(y),
where a,b ∈ N and f is a mapping from a commutative group (G, +) to a 2-Banach space (Y,||·,·||). Our results are generalizations of main results of Brzdȩk and Ciepliński[J Brzdȩk, K Ciepliński. On a fixed point theorem in 2-normed spaces and some of its applications. Acta Mathematica Scientia, 2018, 38B(2):377-390].

Key words: Fixed point theorem, generalized Cauchy functional equation, 2-normed space, stability

CLC Number:

47H10

Laddawan AIEMSOMBOON, Wutiphol SINTUNAVARAT. ON NEW APPROXIMATIONS FOR GENERALIZED CAUCHY FUNCTIONAL EQUATIONS USING BRZDȨK AND CIEPLIŃSKI'S FIXED POINT THEOREMS IN 2-BANACH SPACES[J].Acta mathematica scientia,Series B, 2020, 40(3): 824-834.

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References

[1] Aiemsomboon L, Sintunavarat W. A note on the generalised hyperstability of the general linear equation. Bull Aust Math Soc, 2017, 96:263-273
[2] Aiemsomboon L, Sintunavarat W. On a new type of stability of a radical quadratic functional equation using Brzdȩk's fixed point theorem. Acta Math Hungar, 2017, 151(1):35-46
[3] Aiemsomboon L, Sintunavarat W. On generalized hyperstability of a general linear equation. Acta Math Hungar, 2016, 149(1):413-422
[4] Aoki T. On the stability of the linear transformation in Banach spaces. J Math Soc Jpn, 1950, 2:64-66
[5] Brzdȩk J, Pietrzyk A. A note on stability of the general linear equation. Aequationes Math, 2008, 75:267-270
[6] Brzdȩk J. Stability of additivity and fixed point methods. J Fixed Point Theory Appl, 2013, 2013:285
[7] Brzdȩk J, Ciepliński K. Hyperstability and Superstability. Abstr Appl Anal, Vol 2013. Article ID 401756, 13 pages. http://dx.doi.org/10.1155/2013/401756
[8] Brzdȩk J, Ciepliński K. On a fixed point theorem in 2-Banach spaces and some of its applications. Acta Mathematica Scientia, 2018, 38B(2):377-390
[9] Gajda Z. On stability of additive mappings. Int J Math Math Sci, 1991, 14(3):431-434
[10] Gähler S. Lineare 2-normierte Räume. Math Nachr, 1964, 28:1-43(in German)
[11] Gähler S. Ber 2-normed-Räume. Math Nachr, 1969, 42:335-347(in German)
[12] Gǎvruţa P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mapping. J Math Anal Appl, 1994, 184:431-436
[13] Hyers D H. On the stability of the linear functional equation. Proc Natl Acad Sci USA, 1941, 27:222-224
[14] Hyers D H, Isac G, Rassias Th M. Stability of Functional Equations in Several Variables. Berlin:Birkhäuser, 1998
[15] Piszczek M. Remark on hyperstability of the general linear equation. Aequat Math, 2014, 88:163-168
[16] Piszczek M. Hyperstability of the general linear functional equation. Bull Korean Math Soc, 2015, 52:1827-1838
[17] Rassias T M. On the stability of the linear mapping in Banach spaces. Proc Am Math Soc, 1978, 72:297-300
[18] Rassias T M. On a modified Hyers-Ulam sequence. J Math Anal Appl, 1991, 158:106-113
[19] Ulam S M. A collection of mathematical problems. New York:Interscience, 1960
[20] White A. 2-normed spaces. Math Nachr, 1969, 42:43-60

ON NEW APPROXIMATIONS FOR GENERALIZED CAUCHY FUNCTIONAL EQUATIONS USING BRZDȨK AND CIEPLIŃSKI'S FIXED POINT THEOREMS IN 2-BANACH SPACES

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