Fischer-Muszély additivity on Abelian groups

• Autorzy: Ger R.
• Języki publikacji: EN

Abstrakty

EN

A description of a general solution f : X -> Y mapping a commutative group (X, +) into a real normed linear space (Y, || o ||) of the functional equation [formula] is given in terms of isometrics and additive mappings. Several results describing the solutions of this equation that were obtained earlier under some alternative assumptions regarding the domains, ranges and//or by imposing some regularity upon the map f become special cases of our main result. To gain a proper proof tool we have also established an improvement of E. Berz's [4] representation theorem for sublinear functionals on Abelian groups.

Słowa kluczowe

EN

functional equations; additivity; isometry; sublinear functionals

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
• Rocznik: 2004
• Tom: Vol. 44, nr spec.
• Strony: 83--96
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Ger R.

• Instytut Matematyki, Uniwersytet Śląski, ul. Bankowa 14, 40-007 Katowice, Poland,

Bibliografia

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• [4] E. Berz, Sublinear functions on R, Aequationes Math. 12 (1975), 200-206.
• [5] J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979.
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• [7] P. Fischer and G. Muszély, On some new generalizations of the functional equation of Cauchy,, Canad. Math. Bull. 10 (1967), 197-205.
• [8] R. Ger, On a characterization of strictly convex spaces, Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 127 (1993), 131-138.
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• [11] R. Ger and B. Koclęga, Isometries and a generalized Cauchy equation, Aequationes Math. 60 (2000), 72-79.
• [12] P. Kranz, Additive functionals on abelian semigroups, Comment. Math. Prace Mat. 16 (1972), 239-246.
• [13] M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish Scientific Publishers & Silesian University, Warszawa-Kraków-Katowice, 1985.
• [14] P. Schöpf, Solutions of || f(ξ + η) || = || f (ξ) + f(η) ||, Math. Pannon. 8 (1997), 117-127.
• [15] F. Skof, On the functional equation || f (x + y) - f (x) || = || f (y) ||„ Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 127 (1993), 229-237.
• [16] J. Tabor, Jr., Stability of the Fischer-Muszély functional equation, Publ. Math. Debrecen 62 (2003), 205-211.
• [17] J. Tabor, Jr., Isometries from R to a Banach space, (manuscript).

Uwagi

Dedicated to Professor Julian Musielak on his 75th birthday.