## 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.

### MSC:

 39B55 Orthogonal additivity and other conditional functional equations 39B82 Stability, separation, extension, and related topics for functional equations
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### References:

 [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
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