×

On a fixed point theorem in 2-Banach spaces and some of its applications. (English) Zbl 1399.39063

Summary: 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.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hyers, D. H.; Isac, G.; Rassias Th, M., Stability of Functional Equations in Several Variables (1998), Birkhäuser: Birkhäuser Boston · Zbl 0907.39025
[2] Agarwal, R. P.; Xu, B.; Zhang, W., Stability of functional equations in single variable, J Math Anal Appl, 288, 2, 852-869 (2003) · Zbl 1053.39042
[3] Brillouët-Belluot, N.; Brzdęk, J.; Ciepliński, K., On some recent developments in Ulam’s type stability, Abstr Appl Anal (2012), Art ID 716936 · Zbl 1259.39019
[4] Brzdęk, J.; Ciepliński, K., Hyperstability and superstability, Abstr Appl Anal (2013), Art ID 401756 · Zbl 1293.39013
[5] Brzdęk, J.; Ciepliński, K.; Leśniak, Z., On Ulam’s type stability of the linear equation and related issues, Discrete Dyn Nat Soc (2014), Art ID 536791 · Zbl 1419.39053
[6] Forti, G. L., Hyers-Ulam stability of functional equations in several variables, Aequationes Math, 50, 1/2, 143-190 (1995) · Zbl 0836.39007
[7] Jung, S. M., Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis (2011), Springer: Springer New York · Zbl 1221.39038
[8] Miura, T.; Takahasi, S.; Choda, H., On the Hyers-Ulam stability of real continuous function valued differentiable map, Tokyo J Math, 24, 2, 467-476 (2001) · Zbl 1002.39039
[9] Gähler, S., Lineare 2-normierte Räume, Math Nachr, 28, 1-43 (1964) · Zbl 0142.39803
[10] Gao, J., On the stability of the linear mapping in 2-normed spaces, Nonlinear Funct Anal Appl, 14, 5, 801-807 (2009) · Zbl 1383.39027
[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, 3, 2, 75-81 (2012) · Zbl 1281.39035
[12] Chung, S. C.; Park, W. G., Hyers-Ulam stability of functional equations in 2-Banach spaces, Int J Math Anal (Ruse), 6, 17/20, 951-961 (2012) · Zbl 1311.39041
[13] Ciepliński, K., Approximate multi-additive mappings in 2-Banach spaces, Bull Iranian Math Soc, 41, 3, 785-792 (2015) · Zbl 1373.39025
[14] Ciepliński, K.; Surowczyk, A., On the Hyers-Ulam stability of an equation characterizing multi-quadratic mappings, Acta Mathematica Scientia, 35B, 3, 690-702 (2015) · Zbl 1340.39038
[15] Ciepliński, K.; Xu, T. Z., Approximate multi-Jensen and multi-quadratic mappings in 2-Banach spaces, Carpathian J Math, 29, 2, 159-166 (2013) · Zbl 1299.39023
[16] Park, W. G., Approximate additive mappings in 2-Banach spaces and related topics, J Math Anal Appl, 376, 1, 193-202 (2011) · Zbl 1213.39028
[17] Brzdęk, J.; Cădariu, L.; Ciepliński, K., Fixed point theory and the Ulam stability, J Funct Spaces (2014), Art ID 829419 · Zbl 1314.39029
[18] Ciepliński, K., Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey, Ann Funct Anal, 3, 1, 151-164 (2012) · Zbl 1252.39032
[19] Xu, B.; Brzdęk, J.; Zhang, W., Fixed-point results and the Hyers-Ulam stability of linear equations of higher orders, Pacific J Math, 273, 2, 483-498 (2015) · Zbl 1319.39018
[20] Brzdęk, J.; Chudziak, J.; Páles, Zs, A fixed point approach to stability of functional equations, Nonlinear Anal, 74, 17, 6728-6732 (2011) · Zbl 1236.39022
[21] Brzdęk, J.; Ciepliński, K., A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal, 74, 18, 6861-6867 (2011) · Zbl 1237.39022
[22] Cădariu, L.; Găvruţa, L.; Găvruţa, P., Fixed points and generalized Hyers-Ulam stability, Abstr Appl Anal (2012), Art ID 712743 · Zbl 1252.39030
[23] Bahyrycz, A.; Brzdęk, J.; Jabłońska, E.; Olko, J., On functions that are approximate fixed points almost everywhere and Ulam’s type stability, J Fixed Point Theory Appl, 17, 4, 659-668 (2015) · Zbl 1328.39041
[24] Kannappan, Pl, Functional Equations and Inequalities with Applications (2009), Springer: Springer New York · Zbl 1178.39032
[25] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality (2009), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1221.39041
[26] Sahoo, P. K.; Kannappan, Pl, Introduction to Functional Equations (2011), CRC Press: CRC Press Boca Raton · Zbl 1223.39012
[27] Borelli Forti, C., Solutions of a nonhomogeneous Cauchy equation, Rad Mat, 5, 2, 213-222 (1989) · Zbl 0697.39009
[28] Ebanks, B. R., Generalized Cauchy difference functional equations, Aequationes Math, 70, 1-2, 154-176 (2005) · Zbl 1079.39017
[29] Ebanks, B. R., Generalized Cauchy difference equations. II, Proc Amer Math Soc, 136, 11, 3911-3919 (2008) · Zbl 1206.39022
[30] Ebanks, B. R.; Kannappan, Pl; Sahoo, P. K., Cauchy differences that depend on the product of arguments, Glasnik Mat Ser III, 27, 47, 251-261 (1992), 2 · Zbl 0780.39007
[31] Fenyö, I.; Forti, G-L, On the inhomogeneous Cauchy functional equation, Stochastica, 5, 2, 71-77 (1981) · Zbl 0492.39002
[32] Járai, A.; Maksa, Gy; Páles, Zs, On Cauchy-differences that are also quasisums, Publ Math Debrecen, 65, 3-4, 381-398 (2004) · Zbl 1071.39026
[33] Brzdęk, J., Hyperstability of the Cauchy equation on restricted domains, Acta Math Hungar, 141, 1-2, 58-67 (2013) · Zbl 1313.39037
[34] Brzdęk, J., Remarks on hyperstability of the Cauchy functional equation, Aequationes Math, 86, 3, 255-267 (2013) · Zbl 1303.39016
[35] Brzdęk, J., A hyperstability result for the Cauchy equation, Bull Aust Math Soc, 89, 1, 33-40 (2014) · Zbl 1290.39016
[36] Isac, G.; Rassias Th, M., Functional inequalities for approximately additive mappings, (Rassias, Th M.; Tabor, J., Stability of Mappings of Hyers-Ulam Type (1994), Hadronic Press: Hadronic Press Palm Harbor), 117-125 · Zbl 0844.39015
[37] Piszczek, M., Remark on hyperstability of the general linear equation, Aequationes Math, 88, 1-2, 163-168 (2014) · Zbl 1304.39033
[38] Freese, R. W.; Cho, Y. J., Geometry of Linear 2-normed Spaces (2001), Nova Science Publishers, Inc: Nova Science Publishers, Inc Hauppauge, NY
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.