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Abstrakty
We study the stability of some functional equations postulated for orthogonal vectors in a ball centered at the origin. The maps considered are denned on a finite-dimensional normed linear space with Birkhoff-James orthogonality and take their values in a real sequentially complete linear topological space. The main results establish the stability of the corresponding conditional Cauchy functional equation on a half-ball and in uniformly convex spaces on a whole ball. The methods used in the first part of the paper are similar to those from [10]. Since, however, now in a general structure, some additional problems arise, we need several new tools.
Wydawca
Czasopismo
Rocznik
Tom
Strony
579--596
Opis fizyczny
Bibliogr. 15 poz.
Twórcy
autor
- Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
Bibliografia
- [1] D. Amir, Characterization of Inner Product Spaces, Birkhäuser Verlag, Basel-Boston, 1986.
- [2] R. Ger, J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43, No.2 (1995), 143-151.
- [3] S. Gudder, D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), 427-436.
- [4] R.C. James, Inner products in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947), 559-566.
- [5] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292.
- [6] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22, No.2 (1989), 499-507.
- [7] J. Lawrence, Orthogonality and additive functions on normed linear spaces, Colloq. Math. 49 (1985), 253-255.
- [8] J. Rätz, On approximately additive mappings, General Inequalities 2, Internat. Ser. Numer. Math. 47, Birkhäuser Verlag, Basel, 1980, 233-251.
- [9] J. Rätz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35-49.
- [10] J. Sikorska, Orthogonal stability of the Cauchy equation on balls, Demonstratio Math. 33, No.3 (2000), 527-546.
- [11] F. Skof, Proprietà locali e approssimazione di operatori, Rend. Semin. Mat. Fis. Milano 53 (1983), 113-129.
- [12] F. Skof, Sull'approssimazione delle applicazioni localmente ð-additive, Atti Accad. Sci. Torino CI. Sci. Fis. Mat. Natur. 117 (1983), 377-389.
- [13] F. Skof, S. Terracini, On the stability of the quadratic functional equation on a restricted domain (Italian), Atti Accad. Sci. Torino CI. Sci. Fis. Mat. Natur. 121 (1987), 153-167.
- [14] K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187-190.
- [15] Gy. Szabó, On mappings orthogonally additive in the Birkhoff-James sense, Aequationes Math. 30 (1986), 93-105.
Typ dokumentu
Bibliografia
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bwmeta1.element.baztech-article-PWA3-0010-0031