Fixed point theorems for certain class of mappings. (English) Zbl 0613.47048

A fixed point theorem is proved in a uniformly convex Banach space for mappings T satisfying: for each x,y in the domain and for \(n=1,2,..\). \[ \| T^ nx-T^ ny\| \leq a_ n\| x-y\| +b_ n(\| x-T^ nx\| +\| y-T^ ny\|)+c_ n(\| x-T^ ny\| +\| y-T^ nx\|) \] where \(a_ n,b_ n,c_ n\) are nonnegative, \(b_ n+c_ n<1\) for large n and \(\lim_{n\to \infty}(a_ n+3b_ n+c_ n)/(1-b_ n-c_ n)=1\). This result generalises a fixed point theorem of Goebel and Kirk for asymptotically non-expansive mapping. The result is then extended to common fixed point theorems.


47H10 Fixed-point theorems