# zbMATH — the first resource for mathematics

Remarks on stability of some inhomogeneous functional equations. (English) Zbl 1316.39011
A general result is stated, showing an equivalence between stability of a given homogeneous functional equation and stability of its inhomogeneous counterpart. Applying this result the author proves several stability results concerning the inhomogeneous Cauchy functional equation $f(x+y)=f(x)+f(y)+d(x,y),$ as well as similar orthogonally additive, Jensen and linear functional equations.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B55 Orthogonal additivity and other conditional functional equations
Full Text:
##### References:
 [1] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66, (1950) · Zbl 0040.35501 [2] Bahyrycz, A.; Piszczek, M., Hyperstability of the Jensen functional equation, Acta Math. Hungarica, 142, 353-365, (2014) · Zbl 1299.39022 [3] Borelli Forti, C., Solutions of a non-homogeneous Cauchy equation, Radovi. Mat., 5, 213-222, (1989) · Zbl 0697.39009 [4] Brillouët-Belluot, N., Brzdęk, J., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal., Art. ID 716936 (2012) · Zbl 0398.47040 [5] Brzdęk, J.; Rassias, Th.M. (ed.); Tabor, J. (ed.), A note on stability of additive mappings, 19-22, (1994), Palm Harbor, FL · Zbl 0844.39012 [6] Brzdęk, J., On a generalization of the Cauchy functional equation, Aequ. Math., 46, 56-75, (1993) · Zbl 0788.39004 [7] Brzdęk, J., On approximately additive functions, J. Math. Anal. Appl., 381, 299-307, (2011) · Zbl 1235.39017 [8] Brzdęk, J., Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar., 141, 58-67, (2013) · Zbl 1313.39037 [9] Brzdęk, J., Remarks on hyperstability of the Cauchy functional equation, Aequ. Math., 86, 255-267, (2013) · Zbl 1303.39016 [10] Brzdęk, J., A hyperstability result for the Cauchy equation, Bull. Austral. Math. Soc., 89, 33-40, (2014) · Zbl 1290.39016 [11] Brzdęk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. 2013, Article ID 401756, 13 pp (2013) · Zbl 0844.39015 [12] Davison, T.M.K.; Ebanks, B., Cocycles on cancellative semigroups, Publ. Math. Debrecen, 46, 137-147, (1995) · Zbl 0860.39037 [13] Ebanks, B., Generalized Cauchy difference functional equations, Aequ. Math., 70, 154-176, (2005) · Zbl 1079.39017 [14] Ebanks, B., Generalized Cauchy difference equations. II, Proc. Amer. Math. Soc., 136, 3911-3919, (2008) · Zbl 1206.39022 [15] Ebanks, B.; Kannappan, P.; Sahoo, P.K., Cauchy differences that depend on the product of arguments, Glasnik Mat., 27, 251-261, (1992) · Zbl 0780.39007 [16] Ebanks B., Sahoo P., Sander W.: Characterizations of Information Measures. World Scientific, Singapore, New Jersey, London, Hong Kong (1998) · Zbl 0923.94002 [17] Erdös, J., A remark on the paper “on some functional equations” by S. kurepa, Glasnik Mat. Fiz. Astronom. (2), 14, 3-5, (1959) · Zbl 0085.32903 [18] Fechner, W.; Sikorska, J., On the stability of orthogonal additivity, Bull. Pol. Acad. Sci. Math., 58, 23-30, (2010) · Zbl 1197.39016 [19] Fenyö, I.; Forti, G.-L., On the inhomogeneous Cauchy functional equation, Stochastica, 5, 71-77, (1981) · Zbl 0492.39002 [20] Gajda, Z., On stability of additive mappings, Int. J. Math. Math. Sci., 14, 431-434, (1991) · Zbl 0739.39013 [21] Hyers, D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224, (1941) · Zbl 0061.26403 [22] Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston, Basel, Berlin (1998) · Zbl 0907.39025 [23] Járai, A.; Maksa, Gy.; Páles, Zs., On Cauchy-differences that are also quasisums, Publ. Math. Debrecen, 65, 381-398, (2004) · Zbl 1071.39026 [24] Jessen, B.; Karpf, J.; Thorup, A., Some functional equations in groups and rings, Math. Scand., 22, 257-265, (1968) · Zbl 0183.04004 [25] Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer Optimization and its Applications 48, Springer, New York (2011) · Zbl 1221.39038 [26] Moszner, Z., On the stability of functional equations, Aequ. Math., 77, 33-88, (2009) · Zbl 1207.39044 [27] Moszner, Z., On stability of some functional equations and topology of their target spaces, Ann. Univ. Paedagog. Crac. Stud. Math., 11, 69-94, (2012) · Zbl 1292.39027 [28] Piszczek, M.: Remark on hyperstability of the general linear equation. Aequ. Math., in press. doi:10.1007/s00010-013-0214-x · Zbl 1304.39033 [29] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300, (1978) · Zbl 0398.47040 [30] Rassias, Th.M.; Isac, G.; Rassias, Th.M. (ed.); Tabor, J. (ed.), Functional inequalities for approximately additive mappings, 117-125, (1994), Palm Harbor · Zbl 0844.39015 [31] Stetkær H.: Functional Equations on Groups. World Scientific, Singapore (2013) · Zbl 1298.39018
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.