Fixed Point Theory, 18(2017), No. 1, 69-84 http://www.math.ubbcluj.ro/ ∼ nodeacj/sfptcj.html APPROXIMATELY p-WRIGHT AFFINE FUNCTIONS, INNER PRODUCT SPACES AND DERIVATIONS ANNA BAHYRYCZ ** , JANUSZ BRZDĘK * AND MAGDALENA PISZCZEK * * Department of Mathematics, Pedagogical University Podchorążych 2, 30-084 Kraków, Poland E-mail: bahyrycz@agh.edu.pl, jbrzdek@up.krakow.pl, magdap@up.krakow.pl ** AGH University of Science and Technology, Faculty of Applied Mathematics Mickiewicza 30, 30-059 Krakow, Poland Abstract. We prove a result on hyperstability (in normed spaces) of the equation that defines the p-Wright affine functions and show that it yields a simple characterization of complex inner product spaces. We also obtain in this way some inequalities describing derivations, Lie derivations and Lie homomorphisms. Key Words and Phrases: Hyperstability, p-Wright affine function, inner product space, derivation, Lie derivation, Lie homomorphism, fixed point theorem. 2010 Mathematics Subject Classification: 16W25, 39B52, 39B62, 39B82, 46C99, 47J99. References [1] J. Acz´el, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989. [2] M. Albert, J.A. Baker, Functions with bounded m-th differences, Ann. Polon. Math., 43(1983), 93-103. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950), 64-66. [4] A. Bahyrycz, J. Brzdęk, M. Piszczek, J. Sikorska, Hyperstability of the Frechet equation and a characterization of inner product spaces, J. Funct. Spaces Appl., 2013(2013), Article ID 496361, 6 pages. [5] A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar., 142(2014), no. 2, 353-365. [6] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous func- tion rings, Duke Math. J., 16(1949), 385-397. [7] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57(1951), 223-237. [8] N. Brillou¨et-Belluot, J. Brzdęk, K. Ciepliński, On some recent developments in Ulam’s type stability, Abstr. Appl. Anal., 2012(2012), Article ID 716936, 41 pages. [9] J. Brzdęk, Stability of the equation of the p-Wright affine functions, Aequationes Math., 85(2013), no. 3, 497-503. [10] J. Brzdęk, Remarks on hyperstability of the Cauchy functional equation, Aequationes Math., 86(2013), no. 3, 255-267. [11] J. Brzdęk, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar., 141(2013), no. 1-2, 58-67. 69