Bulletin of the
Korean Mathematical Society BKMS

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 $n = 1,\ 2,\ 3,\ \cdots,$ $$\aligned \Vert T^n x - T^n y \Vert \le a \cdot \Vert x - y \Vert &+ b(\Vert x - T^nx \Vert + \Vert y - T^ny \Vert)\\ &\qquad + c(\Vert x - T^ny \Vert + \Vert y - T^nx \Vert), \endaligned $$ 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.

Keywords: $p$-uniformly convex Banach space, normal structure, asymptotic center, fixed points

MSC numbers: 47H10