On a Pexiderized conditional exponential functional equation. (English) Zbl 1212.39039

Let \(f:X\to Y\) be a function where \((X, +)\) is a groupoid and \((Y, \cdot)\) a semigroup, both with neutral elements. The stability of the conditional exponential equation \[ f(x+y)=f(x)f(y)\qquad((x,y)\in D\subset X\times X) \] and its Pexiderized version is studied in a quite general setting. The results include some known theorems concerning orthogonal and some other conditional exponential equations.


39B55 Orthogonal additivity and other conditional functional equations
39B82 Stability, separation, extension, and related topics for functional equations
Full Text: DOI


[1] J. Aczél, Lectures on Functional Equations and their Applications, Academic Press (New York – London, 1966).
[2] J. Aczél and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications, Cambridge University Press (Cambridge, 1989).
[3] K. Baron and G.-L. Forti, Orthogonality and additivity modulo Z, Results Math., 26 (1994), 205–210
[4] K. Baron, F. Halter-Koch and P. Volkmann, On orthogonally exponential functions, Arch. Math. (Basel), 64 (1995), 410–414. · Zbl 0821.39006
[5] J. Brzdęk, On functionals which are orthogonally additive modulo Z, Results Math., 30 (1996), 25–38. · Zbl 0862.39012
[6] J. Brzdęk, On orthogonally exponential and orthogonally additive mappings, Proc. Amer. Math. Soc., 125 (1997), 2127–2132. · Zbl 0870.39011
[7] J. Brzdęk, On orthogonally exponential functionals, Paci-c J. Math., 181 (1997), 247–267. · Zbl 1010.39011
[8] J. Brzdęk, On the isosceles orthogonally exponential mappings, Acta Math. Hungar., 87 (2000), 147–152. · Zbl 0963.46015
[9] J. Brzdęk, On the quotient stability of a family of functional equations, Nonlinear Analysis (2009), doi:10.1016/j.na.2009.02.123.
[10] J. Brzdęk and J. Sikorska, A conditional exponential functional equation and its stability (submitted).
[11] W. Fechner and J. Sikorska, A note on the stability of the orthogonal additivity (submitted). · Zbl 1197.39016
[12] R. Ger, Superstabitity is not natural, Rocznik Naukowo-Dydaktyczny WSP w Krakowie, Prace Mat., 159 (1993), 109–123.
[13] R. Ger and P. Šemrl, The stability of the exponential function, Proc. Amer. Math. Soc., 124 (1996), 779–787. · Zbl 0846.39013
[14] R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math., 43 (1995), 143–151. · Zbl 0833.39007
[15] R. Ger and J. Sikorska, On the Cauchy equation on spheres, Ann. Math. Sil., 11 (1997), 89–99. · Zbl 0894.39009
[16] R. C. James, Orthogonality in normed linear spaces, Duke Math. J., 12 (1945), 291–302. · Zbl 0060.26202
[17] J. Rätz, On orthogonally additive mappings, Aequationes Math., 28 (1985), 35–49. · Zbl 0569.39006
[18] J. Sikorska, Orthogonal stability of some functional equations (in Polish), PhD Thesis, Silesian University (Katowice, 1997).
[19] J. Sikorska, Generalized orthogonal stability of some functional equations, J. Inequal. Appl. (2006), Art. ID 12404, 23 pp. · Zbl 1133.39023
[20] J. Sikorska, Generalized stability of the Cauchy and Jensen functional equations on spheres, J. Math. Anal. Appl., 345 (2008), 650–660. · Zbl 1157.39019
[21] Gy. Szabó, A conditional Cauchy equation on normed spaces, Publ. Math. Debrecen, 42 (1993), 256–271. · Zbl 0807.39010
[22] Gy. Szabó, Isosceles orthogonally additive mappings and inner product spaces, Publ. Math. Debrecen, 46 (1995), 373–384. · Zbl 0865.46012
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.