Fixed points of an integral operator
The author studies the equation \[ g(x)=(1-x)^{\gamma}\frac{\gamma}{2}\int_{0}^{1} \frac{1+tx}{(1-tx)^{\gamma+1}}g(t)t^{\gamma/2-1}\,dt, \qquad \gamma \geq 2, \] arising naturally in the study of the invariant-mean-value properties of hyperbolically-harmonic function. The main result establishes necessary and sufficient conditions (in terms of \(\gamma\)) of the following property: constants are the only \(L^{1}([0,1])\)-solutions of this equation.
scientific article; zbMATH DE number 2103491
scientific article