Fixed points of a certain class of mappings in uniformly convex Banach spaces

摘要

In this paper, we prove in p-uniformly convex space a fixed point theorem for a class of mappings $T$ satisfying: for each $x, y $ in the domain and for $n = 1, 2, 3, cdots,$ $$ligned Vert T^n x - T^n y Vert le a cdot Vert x - y Vert &+ b(Vert x - T^nx Vert + Vert y - T^ny Vert) &qquad + c(Vert x - T^ny Vert + Vert y - T^nx Vert), endaligned $$ where $a, b, c $ are nonnegative constants satisfying certain conditions. Further we establish some fixed point theorems for these mappings in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{p,k} $ for $1 < p < infty $ and $k ge 0$. As a consequence of our main result, we also extend the results of Goebel and Kirk [7], Lim [8], Lifshitz [12], Xu [20] and others.