Comparison of Harder stability and Rus stability of Mann iteration procedure and their equivalence

Abstract

In this paper, we study the stability of Mann iteration procedure in two directions, namely one due to Harder and the second one due to Rus with respect to a map $T:Kto K$ where $K$ is a nonempty closed convex subset of a normed linear space $X$ and there exist $deltain(0,1)$ and $Lgeq 0$ such that $||Tx-Ty||leqdelta||x-y||+L||x-Tx||$ for $x,yin K$. Also, we show that the Mann iteration procedure is stable in the sense of Rus may not imply that it is stable in the sense of Harder for weak contraction maps. Further, we compare and study the equivalence of these two stabilities and provide examples to illustrate our results.