REMARKS ABOUT A PAPER DEALING WITH THE EQUIVALENCE OF MANN AND ISHIKAWA ITERATIONS

. We give an aﬃrmative answer to the following question: are Mann and Ishikawa iterations equivalent under the assumptions that lim n →∞ α n 6 = 0 and lim n →∞ β n 6 = 0? MSC 2000. 47H10.

Definition 1.Let X be a real Banach space.Let B be a nonempty subset.A map T : B → B is called strongly pseudocontractive if there exists k ∈ (0, 1) and a j(x − y) ∈ J(x − y) such that A map S : X → X is called strongly accretive if there exists k ∈ (0, 1) and a j(x − y) ∈ J(x − y) such that Remark 1.The operator T is strongly pseudocontractive map if and only if (I − T ) is strongly accretive.
Let X be a real Banach space, B be a nonempty, convex subset of X, and T : B → B be an operator.Let u 1 , x 1 ∈ B. The following iteration is known as Mann iteration, see [3]: where the sequence satisfies 0 ≤ α n ≤ 1, ∀n ∈ N. Ishikawa iteration is given by, see [1]: where the sequences satisfy 0 ≤ α n , β n ≤ 1, ∀n ∈ N.
(ii) For the above T, B, X, we have shown in [4] that Mann and Ishikawa iterations are equivalent when (6) 0 (iii) In [5] and [6], the authors have additional assumptions on {α n } and {β n }, without changing the most important conditions from ( 6) : All convergence results from [5] and [6] are dealing with It seems that it is impossible to apply Theorem 4 from [4] in the case (8) .This is only an appearance.
Theorem 2. Let B be a closed convex subset of an arbitrary Banach space X and let T be a Lipschitzian strongly pseudocontractive selfmap of B, with Lipschitz constant L ≥ 1.Let x 0 = u 0 ∈ B and {x n } and {u n } be the Ishikawa and Mann iterations (5) and (4) , with {α n } and {β n } satisfying (7) and (10)

Then the following are equivalent:
(i) the Mann iteration converges to x * , (ii) the Ishikawa iteration converges to x * .
Proof.The proof is exactly the same as the proof of Theorem 4 from [4].The difference now is that we cannot apply formula (30) from page 457, from [4].However, this formula is satisfied now by the use of (10) .Note that (9) can be replaced successfully with (10) .
Let S be a strongly accretive operator.Let us consider when the equation Sx = f has a solution for a given f ∈ X.It easy to see that (11)

Theorem 3 .
is a strongly pseudocontractive operator.A fixed point for T is the solution of Sx = f, and conversely.It is well known that if S is bounded (I − S) could be unbounded, for example take S : R → B := [−1, 1] with S(x) = (1/2) cos x.Remark 1, Definition 1 and Theorem 2 lead us to the following result.Let X be an arbitrary Banach space and let T be a Lipschitzian strongly accretive selfmap of X, with S (X) bounded and Lipschitz constant L ≥ 1.Let x 0 = u 0 ∈ B and {x n } and {u n } be the Ishikawa and Mann iterations (5) and (4) , with {α n } and {β n } satisfying (7) and (10) .Then the following are equivalent:(i) the Mann iteration with T x = f + (I − S) x converges to x * (ii) the Ishikawa iteration with T x = f + (I − S) x converges to x * .