# Functions with Bounded nth Differences

• Donald H. Hyers
• George Isac
• Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

## Abstract

Let S be a commutative semigroup with a zero element and Y a real Banach space. For functions f: S → Y, we define the difference operator △ h f (x)by △ h f (x)= f (x + h) - f (x). Similarly, we define$$\Delta _{{h_1}{h_2}}^2f\left( x \right) = {\Delta _{{h_2}}}\left( {{\Delta _{{h_1}}}f\left( x \right)} \right)and\Delta _{{h_1} \cdots {h_{n + 1}}}^{n + 1}f\left( x \right) = {\Delta _{{h_{n + 1}}}}\left( {\Delta _{{h_1} \cdots {h_n}}^nf\left( x \right)} \right),n = 1,2,....$$Note that the nth difference is symmetric in the increments h1,, h n . When all the increments are equal to h we write$$\Delta _h^nf\left( x \right)$$instead of$$\Delta _{h \cdots h}^nf\left( x \right)$$.

## Authors and Affiliations

• Donald H. Hyers
• George Isac
• 1
• Themistocles M. Rassias
• 2