A refinement of the Browder–Göhde–Kirk fixed point theorem and some applications

Abstract

AbstractThe following generalization of the Browder–Göhde–Kirk fixed point theorem is proved: ifCis a nonempty bounded closed and convex subset of a uniformly convex normed spaceXandTis a self-mapping ofCsuch that$$\left\| Tx-Ty\right\| \le \beta \left( \left\| x-y\right\| \right) $$ T x - T y ≤ β x - y for all $$x,y\in C,$$ x , y ∈ C , $$x\ne y,$$ x ≠ y , where a function$$\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) $$ β : 0 , ∞ → 0 , ∞ is such that$$ \lim _{t\rightarrow 0+}\frac{\beta \left( t\right) }{t}=1,$$ lim t → 0 + β t t = 1 , thenThas a fixed point. Two modifications of this theorem as well as some accompanying results on Lipschitz-type mappings are given. An application in the theory of $$L^{p}$$ L p -solutions of an iterative functional equation, and some refinements of the Radamacher theorem are proposed.