Exponential functional equation on spheres. (English) Zbl 1205.39032

Let \(X\) be a real, at least 2-dimensional, normed space and let \(Y\) be a uniquely 2-divisible Abelian semigroup with a neutral element. For \(f_1,f_2,f_3: X\to Y\) the conditional functional equation
\[ f_1(x+y)=f_2(x)f_3(y)\qquad \text{whenever}\;\|x\|=\|y\| \]
is considered. For this equation both the solutions are given and, for \(Y=\mathbb{K}\), the stability is proved. Namely, assuming that the quotient \(f_1(x+y)/f_2(x)+f_3(y)\) is close to 1 whenever \(\|x\|=\|y\|\), there exist \(g_1,g_2,g_3\) satisfying the considered exponential equation on spheres and such that \(f_i/g_i\) are close to 1 on \(X\).


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI


[1] Alsina, C.; Garcia-Roig, J.-L., On a conditional Cauchy equation on rhombuses, () · Zbl 0877.39015
[2] Szabó, Gy., A conditional Cauchy equation on normed spaces, Publ. math. debrecen, 42, 256-271, (1993) · Zbl 0807.39010
[3] Ger, R.; Sikorska, J., On the Cauchy equation on spheres, Ann. math. sil., 11, 89-99, (1997) · Zbl 0894.39009
[4] Ziółkowski, M., On conditional Jensen equation, Demonstratio math., 34, 809-818, (2001) · Zbl 0995.39007
[5] Sikorska, J., On two conditional Pexider functional equations and their stabilities, Nonlinear anal., 70, 2673-2684, (2009) · Zbl 1162.39018
[6] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403
[7] Ulam, S.M., A collection of the mathematical problems, (1960), Interscience Publ. New York · Zbl 0086.24101
[8] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Boston · Zbl 0894.39012
[9] Moszner, Z., On the stability of functional equations, Aequationes math., 77, 33-88, (2009) · Zbl 1207.39044
[10] Brzdęk, J., On the Cauchy difference on normed spaces, Abh. math. sem. univ. Hamburg, 66, 143-150, (1996) · Zbl 0864.39009
[11] Brzdęk, J., On the isosceles orthogonally exponential mappings, Acta math. hungar., 87, 1-2, 147-152, (2000) · Zbl 0963.46015
[12] Ger, R., Superstabitity is not natural, Rocznik nauk.-dydakt. prace mat., 159, 109-123, (1993) · Zbl 0964.39503
[13] Chudziak, J., Approximate solutions of the gołąb – schinzel functional equation, J. approx. theory, 136, 21-25, (2005) · Zbl 1083.39025
[14] Sikorska, J., On a pexiderized conditional exponential functional equation, Acta math. hungar., (2009) · Zbl 1212.39039
[15] Ger, R.; Šemrl, P., The stability of the exponential function, Proc. amer. math. soc., 124, 779-787, (1996) · Zbl 0846.39013
[16] Sikorska, J., Generalized stability of the Cauchy and Jensen functional equations on spheres, J. math. anal. appl., 345, 650-660, (2008) · Zbl 1157.39019
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.