A conjecture of Gy. Petruska
The paper deals with the solution of the integral equation \(f(x)=(1/\phi (x))\int^{\phi (x)}_{0}f(t)dt\quad (0<x<1).\) A conjecture by \textit{Gy. Petruska} [Aequationes Math. 38, 66-72 (1989; Zbl 0679.39007)] states that if \(0<\phi (x)<1\) is a continuous and increasing function in (0,1) with \(\lim_{x\to 0}\phi (x)/x=0\), then the equation admits a nonconstant bounded solution. In the paper the conjecture is proved not to be consistent; it is shown that every bounded solution of the given equation must be constant, provided \(1\leq f(x)\leq 2\) for all \(0<x<1\).
scientific article; zbMATH DE number 4161335