Quasi-additive functions |
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Authors: | J Tabor |
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Institution: | (1) Department of Mathematics, Pedagogical University of Cracow, Podchorazych, Cracow, Poland |
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Abstract: | Summary Let 0 ![les](/content/hm2n8m217n78m5g7/xxlarge10877.gif) < 1 and letX, Y be real normed spaces. In this paper we consider the following functional inequality: f(x + y) – f(x) – f(y) min{ f(x + y) , f(x) + f(y) } forx, y R, wheref: X Y. Mainly continuous solutions are investigated. In the case whereY = R some necessary and some sufficient conditions for this inequality are given.Let 0 ![les](/content/hm2n8m217n78m5g7/xxlarge10877.gif) <1. The following functional inequality has been considered in 5]: f(x + y) – f(x) – f(y) min{ f(x + y) , f(x) + f(y) } forx, y R, wheref: R R. It appeared that the solutions of this inequality have properties very similar to those of additive functions (cf. 1], 2], 3]). The inequality under consideration seems to be interesting also because of its physical interpretation (cf. 5]). In this paper we shall consider this inequality in a more general case, wheref is defined on a real normed space and takes its values in another real normed space.The first part of the paper concerns the general case; in the second part we assume that the range off is inR. |
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Keywords: | Primary 51G05 Secondary 11E81 12K05 12K10 |
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