D’Alembert’s and Wilson’s functional equations on step 2 nilpotent groups

Summary. We describe the set of solutions of Wilsons functional equation on any step 2 nilpotent group and how the set of classical solutions in certain cases must be supplemented by 4-dimensional spaces of solutions.     

On an alternative d’Alembert’s equation

Aequationes mathematicae - Roger Cuculière [Problem 11998, The American Mathematical Monthly 124 no. 7 (2017)] has posed the following problem: Find all continuous functions $$f: mathbb R...     

The Kac–Bernstein functional equation on Abelian groups

Aequationes mathematicae - We consider the Kac–Bernstein functional equation $$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$ on an arbitrary Abelian group...     

A link between Wilson’s and d’Alembert’s functional equations

If \({f, g : G \to \mathbb{C}}\), f ≠ 0, is a solution of Wilson’s functional equation on a group G, then g is a d’Alembert function.     

A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent

According to Schmidt’s Theorem a finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is solvable. In this paper we prove that every finite group with less than 22 non-nilpotent subgroups is solvable and that this estimate is sharp.     

A note on Wilson’s functional equation

Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\). If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \)-functional equation such that \(f \ne 0\), then g satisfies d’Alembert’s \(\mu \)-functional equation.     

Reduction of the multidimensional d’Alembert equation to two-dimensional equations

We give a classification of maximal subalgebras of rankn−1 for the extended Poincaré algebra , which is realized on the set of solutions of the d'Alembert equation . These subalgebras are used for constructing anzatses that reduce this equation to differential equations with two invariant variables. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 651–662, June, 1994.     

Kannappan’s functional equation on semigroups with involution

We express the complex-valued solutions of Kannappan’s functional equation on semigroups with involution in terms of solutions of d’Alembert’s functional equation.     

Solving Chisini’s functional equation

We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x), . . . , G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the existence and uniqueness of solutions. When F is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results.