comparison of harder stability and rus stability of mann iteration procedure and their equivalence

In this paper, we study the stability of mann iteration procedure in two directions, namely one due to harder and the other one due to rus with respect to a map t : k → k where k is a nonempty closed convex subset of a normed linear space x and that satisfies the property: there exist δ ∈ (0, 1) and l ≥ 0 such that ||t x − t y|| ≤ δ||x − y|| + l||x − t x|| for x, y ∈ k. also, we show that the stability of the mann iteration procedure in the sense of rus may not imply that of harder for weak contraction maps. further, we compare and study the equivalence of these two stabilities under certain hypotheses and provide examples to illustrate the phenomena.