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Some new results on hyperstability of the general linear equation in $$(2,\beta)$$-Banach spaces. (English) Zbl 1396.39021
Summary: In this paper, we first introduce the notions of $$(2,\beta)$$-Banach spaces and we will reformulate the fixed point theorem [J. Brzdęk et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6728–6732 (2011; Zbl 1236.39022), Theorem 1] in this space. We also show that this theorem is a very efficient and convenient tool for proving the new hyperstability results of the general linear equation in $$(2,\beta)$$-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. Our results are improvements and generalizations of the main results of M. Piszczek [Aequationes Math. 88, No. 1–2, 163–168 (2014; Zbl 1304.39033)], J. Brzdęk [Acta Math. Hung. 141, No. 1–2, 58–67 (2013; Zbl 1313.39037); Bull. Aust. Math. Soc. 89, No. 1, 33–40 (2014; Zbl 1290.39016)] and A. Bahyrycz and M. Piszczek [Acta Math. Hung. 142, No. 2, 353–365 (2014; Zbl 1299.39022)] in $$(2,\beta)$$-Banach spaces.
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B62 Functional inequalities, including subadditivity, convexity, etc. 47H14 Perturbations of nonlinear operators 47H10 Fixed-point theorems
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##### References:
 [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. · Zbl 0040.35501 [2] A. Bahyrycz and M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), no. 2, 353–365. · Zbl 1299.39022 [3] D. G. Bourgin, Approximately isometric and multiplicative transformations on contin- uous function rings, Duke Math. J. 16 (1949), 385–397. · Zbl 0033.37702 [4] , Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237. · Zbl 0043.32902 [5] J. Brzdęk, Remarks on hyperstability of the Cauchy functional equation, Aequationes Math. 86 (2013), no. 3, 255–267. [6] , Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), no. 1-2, 58–67. · Zbl 1313.39037 [7] , A hyperstability result for the Cauchy equation, Bull. Aust. Math. Soc. 89 (2014), no. 1, 33–40. · Zbl 1290.39016 [8] , Remarks on stability of some inhomogeneous functional equations, Aequationes Math. 89 (2015), no. 1, 83–96. · Zbl 1316.39011 [9] J. Brzdęk and K. Ciepliński, Hyperstability and superstability, Abstr. Appl. Anal. 2013 (2013), Art. ID 401756, 13 pp. [10] J. Brzdęk, J. Chudziak, and Z. P´ales, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011), no. 17, 6728–6732. · Zbl 1236.39022 [11] J. Brzdęk and A. Pietrzyk, A note on stability of the general linear equation, Aequationes Math. 75 (2008), no. 3, 267–270. · Zbl 1149.39018 [12] Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak, Theory of 2-inner product spaces, Nova Science Publishers, Inc., Huntington, NY, 2001. · Zbl 1016.46002 [13] Iz. EL-Fassi, Hyperstability of an n-dimensional Jensen type functional equation, Afr. Mat. 27 (2016), no. 7-8, 1377–1389. · Zbl 1381.39027 [14] Iz. EL-Fassi and S. Kabbaj, On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces, Proyecciones J. Math 34 (2015), no. 4, 359–375. · Zbl 1345.39017 [15] Iz. EL-Fassi, S. Kabbaj, and A. Charifi, Hyperstability of Cauchy-Jensen functional equations, Indag. Math. (N.S.) 27 (2016), no. 3, 855–867. · Zbl 1382.39037 [16] S. Elumalai, Y. J. Cho, and S. S. Kim, Best approximation sets in linear 2-normed spaces, Commun. Korean Math. Soc. 12 (1997), no. 3, 619–629. · Zbl 0944.46003 [17] R. W. Freese and Y. J. Cho, Geometry of linear 2-normed spaces, Nova Science Publishers, Inc., Hauppauge, NY, 2001. · Zbl 1051.46001 [18] S. G¨ahler, Lineare 2-normierte R¨aume, Math. Nachr. 28 (1964), 1–43. [19] , ¨Uber 2-Banach-R¨aume, Math. Nachr. 42 (1969), 335–347. 916IZ. EL-FASSI [20] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431–434. · Zbl 0739.39013 [21] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately ad- ditive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431–436. [22] E. Gselmann, Hyperstability of a functional equation, Acta Math. Hungar. 124 (2009), no. 1-2, 179–188. · Zbl 1212.39044 [23] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222–224. · JFM 67.0424.01 [24] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of functional equations in sev- eral variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkh¨auser Boston, Inc., Boston, MA, 1998. [25] S.-M. Jung, Hyers-Ulam Stability of Functional Equation in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. [26] S.-M. Jung and M. Th. Rassias, A linear functional equation of third order associated with the Fibonacci numbers, Abstr. Appl. Anal. 2014 (2014), Art. ID 137468, 7 pp. [27] S.-M. Jung, M. Th. Rassias, and C. Mortici, On a functional equation of trigonometric type, Appl. Math. Comput. 252 (2015), 294–303. · Zbl 1338.39041 [28] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, Springer, New York, 2009. · Zbl 1178.39032 [29] M. Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śl¸askiego w Katowicach, 489, Uniwersytet Śl¸aski, Katowice, 1985. · Zbl 0555.39004 [30] Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), no. 2, 397–403. · Zbl 1152.39023 [31] Y.-H. Lee, S.-M. Jung, and M. Th. Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput. 228 (2014), 13–16. · Zbl 1364.39023 [32] G. Maksa and Z. P´ales, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyh´azi. (N.S.) 17 (2001), no. 2, 107–112. · Zbl 1004.39022 [33] G.V. Milovanović and M.Th. Rassias (eds.), Analytic Number Theory, Approximation Theory and Special Functions, Springer, New York, 2014. [34] M. Piszczek, Remark on hyperstability of the general linear equation, Aequationes Math. 88 (2014), no. 1-2, 163–168. · Zbl 1304.39033 [35] D. Popa, Hyers-Ulam-Rassias stability of the general linear equation, Nonlinear Funct. Anal. Appl. 7 (2002), no. 4, 581–588. · Zbl 1031.39021 [36] , On the stability of the general linear equation, Results Math. 53 (2009), no. 3-4, 383–389. · Zbl 1181.39020 [37] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300. · Zbl 0398.47040 [38] , On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), no. 1, 106–113. · Zbl 0746.46038 [39] T. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers- Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989–993. · Zbl 0761.47004 [40] S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. · Zbl 0137.24201 [41] A. G. White, Jr., 2-Banach spaces, Math. Nachr. 42 (1969), 43–60. · Zbl 0185.20003 [42] D. Zhang, On hyperstability of generalised linear functional equations in several vari- ables, Bull. Aust. Math. Soc. 92 (2015), no. 2, 259–267. · Zbl 1330.39032
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