Funktionalgleichungen für konstante Funktionen
Conditions are given for the solutions f of the functional equation \[ (1)\quad f(x)=p(x)f(a(x))+q(x)f(b(x)) \] (on a proper interval J of the reals) to be constant functions. The result obtained for (1) implies the following: Take p, \(q>0\), \(p+q=1\) and consider the equation \[ (2)\quad f(x)=p f(px-1)+q f(qx+1). \] A bounded function f:(-1/q,1/p)\(\to {\mathbb{R}}\) which is continuous at a point and satisfies (2) is necessarily a constant one. Due to the result for (1) conditions are established for the Cauchy functional equation on a curve, \(f(x)=f(a(x))+f(b(x))\) on J, to have \(f(x)=cx\), \(x\in J\) as the only solutions. The general solution of (2) is also determined in the class of functions which are meromorphic in the complex plane and regular on the interval (-1/q,1/p).
scientific article; zbMATH DE number 3865874
scientific article