Articles

ON A FIXED POINT THEOREM IN 2-BANACH SPACES AND SOME OF ITS APPLICATIONS

  • Janusz BRZDȨ ,
  • K ,
  • Krzysztof CIEPLINSKI
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  • 1. Pedagogical University, Department of Mathematics, Podchor??ych 2, 30-084 Kraków, Poland;
    2. AGH University of Science and Technology, Faculty of Applied Mathematics, Mickiewicza 30, 30-059 Kraków, Poland

Received date: 2016-04-12

  Revised date: 2017-06-06

  Online published: 2018-04-25

Abstract

The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

Cite this article

Janusz BRZDȨ , K , Krzysztof CIEPLINSKI . ON A FIXED POINT THEOREM IN 2-BANACH SPACES AND SOME OF ITS APPLICATIONS[J]. Acta mathematica scientia, Series B, 2018 , 38(2) : 377 -390 . DOI: 10.1016/S0252-9602(18)30755-0

References

[1] Hyers D H, Isac G, Rassias Th M. Stability of Functional Equations in Several Variables. Boston:Birkhäuser, 1998
[2] Agarwal R P, Xu B, Zhang W. Stability of functional equations in single variable. J Math Anal Appl, 2003, 288(2):852-869
[3] Brillouët-Belluot N, Brzdȩk J, Ciepliński K. On some recent developments in Ulam's type stability. Abstr Appl Anal, 2012, Art ID 716936
[4] Brzdȩk J, Ciepliński K. Hyperstability and superstability. Abstr Appl Anal, 2013, Art ID 401756
[5] Brzdȩk J, Ciepliński K, Lésniak Z. On Ulam's type stability of the linear equation and related issues. Discrete Dyn Nat Soc, 2014, Art ID 536791
[6] Forti G L. Hyers-Ulam stability of functional equations in several variables. Aequationes Math, 1995, 50(1/2):143-190
[7] Jung S M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. New York:Springer, 2011
[8] Miura T, Takahasi S, Choda H. On the Hyers-Ulam stability of real continuous function valued differentiable map. Tokyo J Math, 2001, 24(2):467-476
[9] Gähler S. Lineare 2-normierte Räume. Math Nachr, 1964, 28:1-43
[10] Gao J. On the stability of the linear mapping in 2-normed spaces. Nonlinear Funct Anal Appl, 2009, 14(5):801-807
[11] Cho Y J, Park C, Eshaghi Gordji M. Approximate additive and quadratic mappings in 2-Banach spaces and related topics. Int J Nonlinear Anal Appl, 2012, 3(2):75-81
[12] Chung S C, Park W G. Hyers-Ulam stability of functional equations in 2-Banach spaces. Int J Math Anal (Ruse), 2012, 6(17/20):951-961
[13] Ciepliński K. Approximate multi-additive mappings in 2-Banach spaces. Bull Iranian Math Soc, 2015, 41(3):785-792
[14] Ciepliński K, Surowczyk A. On the Hyers-Ulam stability of an equation characterizing multi-quadratic mappings. Acta Mathematica Scientia, 2015, 35B(3):690-702
[15] Ciepliński K, Xu T Z. Approximate multi-Jensen and multi-quadratic mappings in 2-Banach spaces. Carpathian J Math, 2013, 29(2):159-166
[16] Park W G. Approximate additive mappings in 2-Banach spaces and related topics. J Math Anal Appl, 2011, 376(1):193-202
[17] Brzdȩk J, Čadariu L, Ciepliński K. Fixed point theory and the Ulam stability. J Funct Spaces, 2014, Art ID 829419
[18] Ciepliński K. Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey. Ann Funct Anal, 2012, 3(1):151-164
[19] Xu B, Brzdȩk J, Zhang W. Fixed-point results and the Hyers-Ulam stability of linear equations of higher orders. Pacific J Math, 2015, 273(2):483-498
[20] Brzdȩk J, Chudziak J, Páles Zs. A fixed point approach to stability of functional equations. Nonlinear Anal, 2011, 74(17):6728-6732
[21] Brzdȩk J, Ciepliński K. A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal, 2011, 74(18):6861-6867
[22] Čadariu L, Gǎvrutą L, Gǎvrutą P. Fixed points and generalized Hyers-Ulam stability. Abstr Appl Anal, 2012, Art ID 712743
[23] Bahyrycz A, Brzdȩk J, Jab lónska E, Olko J. On functions that are approximate fixed points almost everywhere and Ulam's type stability. J Fixed Point Theory Appl, 2015, 17(4):659-668
[24] Kannappan Pl. Functional Equations and Inequalities with Applications. New York:Springer, 2009
[25] Kuczma M. An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality. Basel:Birkhäuser Verlag, 2009
[26] Sahoo P K, Kannappan Pl. Introduction to Functional Equations. Boca Raton:CRC Press, 2011
[27] Borelli Forti C. Solutions of a nonhomogeneous Cauchy equation. Rad Mat, 1989, 5(2):213-222
[28] Ebanks B R. Generalized Cauchy difference functional equations. Aequationes Math, 2005, 70(1-2):154-176
[29] Ebanks B R. Generalized Cauchy difference equations. Ⅱ. Proc Amer Math Soc, 2008, 136(11):3911-3919
[30] Ebanks B R, Kannappan Pl, Sahoo P K. Cauchy differences that depend on the product of arguments. Glasnik Mat Ser Ⅲ, 1992, 27(47)(2):251-261
[31] Fenyö I, Forti G-L. On the inhomogeneous Cauchy functional equation. Stochastica, 1981, 5(2):71-77
[32] Járai A, Maksa Gy, Páles Zs. On Cauchy-differences that are also quasisums. Publ Math Debrecen, 2004, 65(3-4):381-398
[33] Brzdȩk J. Hyperstability of the Cauchy equation on restricted domains. Acta Math Hungar, 2013, 141(1-2):58-67
[34] Brzdȩk J. Remarks on hyperstability of the Cauchy functional equation. Aequationes Math, 2013, 86(3):255-267
[35] Brzdȩk J. A hyperstability result for the Cauchy equation. Bull Aust Math Soc, 2014, 89(1):33-40
[36] Isac G, Rassias Th M. Functional inequalities for approximately additive mappings//Rassias Th M, Tabor, J Stability of Mappings of Hyers-Ulam Type. Palm Harbor:Hadronic Press, 1994:117-125
[37] Piszczek M. Remark on hyperstability of the general linear equation. Aequationes Math, 2014, 88(1-2):163-168
[38] Freese R W, Cho Y J. Geometry of Linear 2-normed Spaces. Hauppauge, NY:Nova Science Publishers, Inc, 2001
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