A refinement of the Browder-Göhde-Kirk fixed point theorem and some applications (Q2087428)

Let \(X\) be a uniformly convex Banach space and \(C\) a bounded closed and convex subset of \(X\). In the paper under review it is proved that if \(T:C\rightarrow C\) is a continuous mapping satisfying condition: \(\left\Vert Tx-Ty\right\Vert \leq \beta (\left\Vert x-y\right\Vert ),\ x\neq y\in C,\) for a function \(\beta :(0,\infty )\rightarrow \lbrack 0,\infty )\) with \(\lim_{t\rightarrow 0^{+}}\frac{\beta (t)}{t}=1\), then \(T\) has a fixed point. This generalizes the Browder-Göhde-Kirk fixed point theorem concerning nonexpansive maps. An application of this theorem in the theory of iterative functional equations is given, and a refinement of Radamacher's theorem concerning differentiability of functions is proposed.