Abstract
Let R denote the set of reals, J a real interval and X a real linear space. We determine the functions g : X →J, M : J → R and H : J 2 → R satisfying the equationg(x+M(g(x))y)=H(g(x),g(y)),under the assumptions that g is continuous on rays, M is continuous, and H is symmetric. As a consequence we obtain characterizations of some groups and semigroups.
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References
J. Aczél, Lectures on Functional Equations and their Applications, Academic Press (1966).
J. Aczél and J. Schwaiger, Continuous solutions of the Goł\(a\)b-Schinzel equation on the nonnegative reals and on related domains, Sitzungsber. Österr. Akad. Wiss., Math.-naturwiss. Kl., Abt. II, 208 (1999), 171-177.
K. Baron, On the continuous solutions of the Goł\(a\)b-Schinzel equation, Aequationes Math., 38 (1989), 155-162.
K. Baron, On the continuity of additive-like functions and Jensen convex functions which are Borel on a sphere, in: Functional Equations — Results and Advances, (eds. Z. Daróczy and Z. Páles), Kluwer Academic Publishers (2002), pp. 17-20.
N. Brillouët-Belluot, On some functional equations of Goł\(a\)b-Schinzel type, Aequationes Math., 42 (1991), 239-270.
J. Brzd\(e\)k, Subgroups of the group Z n and a generalization of the Goł\(a\)b-Schinzel functional equation, Aequationes Math., 43 (1992), 59-71.
J. Brzd\(e\)k, Some remarks on solutions of the functional equation f(x + f(x)n y) = tf(x)f(y), Publ. Math. Debrecen, 43 (1993), 147-160.
J. Brzd\(e\)k, On continuous solutions of a conditional Goł\(a\)b-Schinzel equation, Anzeiger Österr. Akad. Wiss., Math.-naturwiss. Kl., Abt. II, 138 (2001), 3-6.
J. Chudziak, Continuous solutions of a generalization of the Goł\(a\)b-Schinzel equation, Aequationes Math., 61 (2001), 63-78.
K. Domańska and R. Ger, Addition formulae with singularities, to appear.
P. Javor, Continuous solutions of the functional equation f(x + yf(x)) = f(x)f(y), in: Proc. Internat. Sympos. on Topology and its Applications (Herceg-Novi, 1968), Savez Drŭstava Mat. Fiz. i Astronom. (Belgrade, 1969), pp. 206-209.
P. Kahlig and J. Matkowski, On some extensions of the Goł\(a\)b-Schinzel functional equation, Ann. Math. Siles., 8 (1994), 13-31.
P. Kahlig and J. Matkowski, A modified Goł\(a\)b-Schinzel equation on a restricted domain (with applications to meteorology and fluid mechanics), Sitzungsber. Österr. Akad. Wiss., Math.-naturwiss. Kl., Abt. II, in print.
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Państwowe Wydawnictwo Naukowe & Uniwersytet Śl\(a\)ski (1985).
A. Mureńko, On solutions of the Goł\(a\)b-Schinzel equation, Internat. J. Math. Math. Sci., 27 (2001), 541-546.
P. Plaumann and S. Strambach, Zweidimensionale Quasialgebren mit Nullteilern, Aequationes Math., 15 (1977), 249-264.
L. Reich, Über die stetigen Lösungen der Goł\(a\)b-Schinzel-Gleichung auf R ≧0, Sitzungsber. Österr. Akad. Wiss., Math.-naturwiss. Kl., Abt. II, 208 (1999), 165-170.
W. Ring, P. Schöpf and J. Schwaiger, On functions continuous on certain classes of ‘thin’ sets, Publ. Math. Debrecen, 51 (1997), 205-224.
M. Sablik, A conditional Goł\(a\)b-Schinzel equation, Anzeiger Österr. Akad. Wiss., Math.-naturwiss. Kl., Abt. II, 137 (2000), 11-15.
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Brzdęk, J. A generalization of addition formulae. Acta Mathematica Hungarica 101, 281–291 (2003). https://doi.org/10.1023/B:AMHU.0000004940.94722.b4
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DOI: https://doi.org/10.1023/B:AMHU.0000004940.94722.b4