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Functional equations stemming from numerical analysis

Tomasz Szostok — 2015

Always when a numerical method gives exact results an interesting functional equation arises. And, since no regularity is assumed, some unexpected solutions may appear. Here we deal with equations constructed in this spirit. The vast majority of this paper is devoted to the equation i = 0 l ( y - x ) i [ f 1 , i ( α 1 , i x + β 1 , i y ) + + f k i , i ( α k i , i x + β k i , i y ) ] = 0 ( 1 ) and its particular cases. We use Sablik’s lemma to prove that all solutions of (1) are polynomial functions. Since a continuous polynomial function is an ordinary polynomial, the crucial problem throughout the whole...

On ω-convex functions

Tomasz Szostok — 2011

Banach Center Publications

In Orlicz spaces theory some strengthened version of the Jensen inequality is often used to obtain nice geometrical properties of the Orlicz space generated by the Orlicz function satisfying this inequality. Continuous functions satisfying the classical Jensen inequality are just convex which means that such functions may be described geometrically in the following way: a segment joining every pair of points of the graph lies above the graph of such a function. In the current paper we try to obtain...

On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms

Michał BaczyńskiTomasz SzostokWanda Niemyska — 2014

Kybernetika

Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) f ( min ( x + y , a ) ) = min ( f ( x ) + f ( y ) , b ) , where a , b > 0 and f : [ 0 , a ] [ 0 , b ] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation f ( m 1 ( x + y ) ) = m 2 ( f ( x ) + f ( y ) ) , where m 1 , m 2 are functions...

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