Abstract
Recently, functions of several variables satisfying, with respect to each variable, some functional equation (among them Cauchy’s, Jensen’s, quadratic and other ones) have been studied. We give a new characterization of multi-Cauchy–Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy–Jensen. Next, using a fixed point theorem, it is proved that a function which approximately satisfies (on restricted domain) the equation characterizing such functions is close (in some sense) to the solution of the equation. This result is a tool for obtaining a generalized Hyers–Ulam stability or hyperstability of this equation for particular control functions, which is presented in several examples.
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References
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)
Bae, J.-H., Park, W.-G.: On the solution of a bi-Jensen functional equation and its stability. Bull. Korean Math. Soc. 43, 499–507 (2006)
Bae, J.-H., Park, W.-G.: On the solution of a a multi-additive functional equation and its stability. J. Appl. Math. Comput. 22, 517–522 (2006)
Bae, J.-H., Park, W.-G.: Solution of a vector variable bi-additive functional equation. Commun. Korean Math. Soc. 23, 191–199 (2008)
Baker, J.A.: The stability of certain functional equations. Proc. Am. Math. Soc. 112, 729–732 (1991)
Bae, J.-H., Park, W.-G.: Stability of a Cauchy–Jensen functional equation in quasi-Banach spaces. J. Inequal. Appl. Art. ID 151547, 9 pp (2010)
Bae, J.-H., Park, W.-G.: A fixed point approach to the stability of a Cauchy–Jensen functional equation. Abstr. Appl. Anal. Art. ID 205160, 10 pp (2012)
Bahyrycz, A.: On stability and hyperstability of an equation characterizing multi-additive mappings. Fixed Point Theory 18, 445–456 (2017)
Bahyrycz, A., Ciepliński, K., Olko, J.: On an equation characterizing multi-Cauchy–Jensen mappings and its Hyers–Ulam stability, Acta Math. Sci. Ser. B Engl. Ed. 35B, 1349–1358 (2015)
Bahyrycz, A., Ciepliński, K., Olko, J.: On an equation characterizing multi-additive-quadratic mappings and its Hyers–Ulam stability. Appl. Math. Comput. 265, 448–455 (2015)
Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951)
Brillouët-Belluot, N., Brzdȩk, J., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. Art. ID 716936, 41 pp (2012)
Brzdȩk, J., Cǎdariu, L., Ciepliński, K.: Fixed point theory and the Ulam stability. J. Funct. Spaces. Art. ID 829419, 16 pp (2014)
Brzdȩk, J., Chudziak, J., Páles, Zs: A fixed point approach to stability of functional equations. Nonlinear Anal. 74, 6728–6732 (2011)
Brzdȩk, J., Ciepliński, K.: Remarks on the Hyers–Ulam stability of some systems of functional equations. Appl. Math. Comput. 219, 4096–4105 (2012)
Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. Art. ID 401756, 13 pp (2013)
Brzdȩk, J., Fechner, W., Moslehian, M.S., Sikorska, J.: Recent developments of the conditional stability of the homomorphism equation. Banach J. Math. Anal. 9(3), 278–326 (2015)
Ciepliński, K.: On multi-Jensen functions and Jensen difference. Bull. Korean Math. Soc. 45, 729–737 (2008)
Ciepliński, K.: Stability of the multi-Jensen equation. J. Math. Anal. Appl. 363, 249–254 (2010)
Ciepliński, K.: Generalized stability of multi-additive mappings. Appl. Math. Lett. 23, 1291–1294 (2010)
Ciepliński, K.: Applications of fixed point theorems to the Hyers–Ulam stability of functional equations—a survey. Ann. Funct. Anal. 3, 151–164 (2012)
Găvruţa, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)
Jun, K.-W., Lee, Y.-H.: On the stability of a Cauchy–Jensen functional equation. II. Dyn. Syst. Appl. 18, 407–421 (2009)
Jun, K.-W., Lee, J.-R., Lee, Y.-H.: On the Hyers–Ulam–Rassias stability of a Cauchy–Jensen functional equation II. J. Chungcheong Math. Soc. 21, 197–208 (2008)
Jun, K.-W., Lee, Y.-H., Cho, Y.-S.: On the generalized Hyers–Ulam stability of a Cauchy-Jensen functional equation. Abstr. Appl. Anal. Art. ID 35151, 15 pp (2007)
Jun, K.-W., Lee, Y.-H., Cho, Y.-S.: On the stability of a Cauchy–Jensen functional equation. Commun. Korean Math. Soc. 23, 377–386 (2008)
Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)
Kannappan, Pl: Functional Equations and Inequalities with Applications. Springer, New York (2009)
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Birkhäuser Verlag, Basel, Cauchy’s equation and Jensen’s inequality (2009)
Lee, Y.-H.: On the Hyers–Ulam–Rassias stability of a Cauchy–Jensen functional equation. J. Chungcheong Math. Soc. 20, 163–172 (2007)
Park, W.-G., Bae, J.-H.: On a Cauchy–Jensen functional equation and its stability. J. Math. Anal. Appl. 323, 634–643 (2006)
Prager, W., Schwaiger, J.: Multi-affine and multi-Jensen functions and their connection with generalized polynomials. Aequ. Math. 69, 41–57 (2005)
Prager, W., Schwaiger, J.: Stability of the multi-Jensen equation. Bull. Korean Math. Soc. 45, 133–142 (2008)
Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Schwaiger, J.: Some remarks on the stability of the multi-Jensen equation. Cent. Eur. J. Math. 11, 966–971 (2013)
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This work was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks and dean grant within subsidy of Ministry of Science and Higher Education.
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Bahyrycz, A., Olko, J. On Stability and Hyperstability of an Equation Characterizing Multi-Cauchy–Jensen Mappings. Results Math 73, 55 (2018). https://doi.org/10.1007/s00025-018-0815-8
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DOI: https://doi.org/10.1007/s00025-018-0815-8