Hosszúova funkcijska jednadžba

Abstract

Hosszúova funkcijska jednadžba je jednadžba oblika $$f(x+y-xy)+f(xy)=f(x)+f(y), \qquad x,y \in \mathbb{R}$$ Glavninu ovog rada predstavlja rješenje Hosszúove funkcijske jednadžbe i dokaz njezine stabilnosti, te rješenje i stabilnost njezine poopćene varijante $$f(x+y-\alpha xy)+g(xy)=h(x)+k(y), \qquad x,y \in \mathbb{R}.$$ Dokazano je da je svako rjesenje Hosszúove funkcijske jednadžbe oblika $$f(x)=A(x)+ a,$$ pri čemu je $A$ aditivno preslikavanje i $a$ realna konstanta. Pored toga prikazana je i veza Hosszúove funkcijske jednadžbe s nekim drugim poznatim funkcijskim jednadžbama. Konkretno, dan je dokaz ekvivalentnosti Hosszúove i Jensenove funkcijske jednadžbe te Hosszúove i tzv. Hosszú-Jensenove funkcijske jednadžbe.Hosszú functional equation is an equation of the form $$f(x+y-xy)+f(xy)=f(x)+f(y), \qquad x,y \in \mathbb{R}$$ The main goal of this thesis is to present the solution of the Hosszú functional equation and the proof of its stability, as well as the solution and stability of its generalization, the functional equation $$f(x+y-\alpha xy)+g(xy)=h(x)+k(y), \qquad x,y \in \mathbb{R}.$$ It is proved that every solution of the Hosszú functional equation is of the form $$f(x)=A(x)+ a,$$ where $A$ is an additive function and $a$ is an arbitrary constant. Furthermore, relations to some other well-known functional equations have also been mentioned. Specifically, a proof of the equivalency of the Hosszú and the Jensen functional equation is also presented, as well as the equivalency of the Hosszú and the so-called Hosszú-Jensen functional equation