Proof of Bernhard Riemann’s Functional Equation using Gamma Function

A functional equation proof of the distributive-triples theorem

with domains M,V , and S. The substances are assumed to be homogeneous. In this case, mass has two decompositions. The first is the conjoint one 〈V × S,%M 〉 underlying the usual multiplicative relation, where %M is a qualitative weak ordering determined by placing masses on the two scales of an equal-arm balance. A standard monotonicity assumption induces orderings %V on volumes and %S on homog...

On the Stability of Generalized Gamma Functional Equation

We obtain the Hyers-Ulam stability and modified Hyers-Ulam stability for the equations of the formg(x+p)=φ(x)g(x) in the following settings: |g(x+p)−φ(x)g(x)| ≤ δ, |g(x+p)−φ(x)g(x)| ≤φ(x), |(g(x+p)/φ(x)g(x))−1| ≤ψ(x). As a consequence we obtain the stability theorems for the gamma functional equation.

Notes on Stability of the Generalized Gamma Functional Equation

for some positive constant ε depending only on δ. Sometimes we call f a δ-approximate solution of (1.1) and g ε-close to f . Such an idea of stability was given by Ulam [13] for Cauchy equation f(x+y) = f(x)+f(y) and his problem was solved by Hyers [4]. Later, the Hyers-Ulam stability was studied extensively (see, e.g., [6, 8, 10, 11]). Moreover, such a concept is also generalized in [2, 3, 12]...

Hyers-Ulam Stability of Fibonacci Functional Equation

Insertion of a gamma-continuous function

A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.