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  Volume 8, Issue 2, Article 35
A Simultaneous System of Functional Inequalities and Mappings which are Weakly of a Constant Sign

    Authors: Dorota Krassowska, Janusz Matkowski,  
    Keywords: Linear functional inequalities, Kronecker's theorem, Mappings of weakly constant sign.  
    Date Received: 17/10/06  
    Date Accepted: 07/06/07  
    Subject Codes:

Primary 39B72; Secondary 26D05.

    Editors: Zsolt Pales,  

It is shown that, under some algebraic conditions on fixed reals $ alpha _{1},alpha _{2},dots ,alpha _{n+1}$ and vectors $ mathbf{a}_{1}mathbf{,a} _{2},mathbf{dots ,a}_{n+1}in mathbb{R}^{n}$, every continuous at a point function $ f:mathbb{R}^{n}rightarrow mathbb{R}$ satisfying the simultaneous system of inequalities

$displaystyle f(mathbf{x+a}_{i})leq alpha _{i}+f(mathbf{x}),quad mathbf{x}in mathbb{R}^{n},i=1,2,dots ,n+1,$

has to be of the form $ f(mathbf{x})=mathbf{pcdot x}+f(mathbf{0}),$ $ mathbf{x}in mathbb{R}^{n}$, with uniquely determined $ mathbf{p}in mathbb{R}^{n}$. For mappings with values in a Banach space which are weakly of a constant sign, a counterpart of this result is given.

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