Let $X$ be a real Banach space and let $B$ be a nonempty, convex subset. Let $u_{1},x_{1}\in B$ be two arbitrary fixed points. Let $T:B\rightarrow B$ be a map.

Definition 1. The map $T$ is said to be asymptotically nonexpansive if there exists a sequence $\left(k_{n}\right)_{n},k_{n}\in[1,\infty),\forall n\in\mathbb{N},\lim_{n\rightarrow\infty}k_{n}=1$, such that

The following remark will be useful.

Remark 2. An asymptotically nonexpansive map is uniformly Lipschitzian for some $L\geqslant 1$, i.e., $\left(\exists L\geqslant 1:\left\|T^{n}x-T^{n}y\right\|\leqslant L\|x-y\|,\forall x,y\in B,\forall n\in\mathbb{N}\right.$ ).

Proof. Let $L:=\sup_{n\in\mathbb{N}}k_{n}$. Because $\lim_{n\rightarrow\infty}k_{n}=1$ and $k_{n}\geqslant 1,\forall n\in\mathbb{N}$, one can deduce that $L\in[1,\infty)$. From Definition 1,

In [5] the following class of maps was introduced.
Definition 3. A map $T$ is said to be asymptotically pseudocontractive if there exists a sequence $\left(k_{n}\right)_{n},k_{n}\in[1,\infty),\forall n\in\mathbb{N},\lim_{n\rightarrow\infty}k_{n}=1$, and there exists $j(x-y)\in J(x-y)$ such that

When $n=1$ in (2) we get the usual definition of a strongly pseudocontractive map. The following remark is Remark 1 from [5].

Remark 4 [5]. An asymptotically nonexpansive map is asymptotically pseudocontractive. The converse is not true.

We consider the following iteration (see [3]):

The sequence $\left(\alpha_{n}\right)_{n}\subset(0,1)$ is convergent, such that $\lim_{n\rightarrow\infty}\alpha_{n}=0$ and $\sum_{n=1}^{\infty}\alpha_{n}=\infty$. This iteration is known as Mann type iteration. We consider the following iteration, known as Ishikawa type iteration (see [1]):

The sequences $\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n}\subset(0,1)$ are such that

The sequence $\left(\alpha_{n}\right)_{n}$ remains the same in both iterations. For $\beta_{n}=0,\forall n\in\mathbb{N}$, from (5) we get (4). We denote by $F(T)=\left\{x^{*}\in B:F\left(x^{*}\right)=x^{*}\right\}$. Replacing $T^{n}$ by $T$ in (3) and (5) gives the Mann and Ishikawa iteration, respectively.

The aim of this paper is to prove an equivalence between the convergences of the above two iterations when $T$ is an asymptotically nonexpansive respective asymptotically pseudocontractive map.

The following lemma appears in [2].
Lemma 5 [2]. Let $X$ be a Banach space and $x,y\in X$. Then $\|x\|\leqslant\|x+ry\|$ for all $r>0$ if and only if there exists $j(x)\in J(x)$ such that $\langle y,j(x)\rangle\geqslant 0$.

Using this lemma we are able to prove the following result.
Lemma 6. Let $B$ be a nonempty subset of a Banach space $X$ and let $T:B\rightarrow B$ be a map. Then the following conditions are equivalent:
(i) $T$ is asymptotically pseudocontractive map;
(ii) There exists $k_{n}\in[1,\infty)$ such that

Proof. Lemma 5 assures that relation (6) is $\left\langle\left(k_{n}I-T^{n}\right)x-\left(k_{n}I-T^{n}\right)y,j(x-y)\right\rangle\geqslant 0$, $\forall n\geqslant n_{0}$, which is equivalent with $\left\langle T^{n}x-T^{n}y,j(x-y)\right\rangle\leqslant k_{n}\langle x-y,j(x-y)\rangle=k_{n}\|x-y\|^{2},\forall x,y\in B$, that is (2).

The following lemma is Lemma 4 from [6].
Lemma 7 [6]. Let $\left(a_{n}\right)_{n}$ be a nonnegative sequence which satisfies the following inequality:

where $\lambda_{n}\in(0,1),\forall n\in\mathbb{N},\sum_{n=1}^{\infty}\lambda_{n}=\infty$, and $\sigma_{n}=o\left(\lambda_{n}\right)$. Then $\lim_{n\rightarrow\infty}a_{n}=0$.

We are now able to give the following result.
Theorem 8. Let $B$ be a closed convex subset of an arbitrary Banach space $X$ and $\left(x_{n}\right)_{n}$ and $\left(u_{n}\right)_{n}$ defined by (3) and (4) with $\left(\alpha_{n}\right)_{n}$ and $\left(\beta_{n}\right)_{n}$ satisfying (5). Let $T$ be an asymptotically pseudocontractive and Lipschitzian with $L\geqslant 1$ self-map of $B$. Let $x^{*}$ be the fixed point of $T$. If $u_{0}=x_{0}\in B$, then the following two assertions are equivalent:
(i) Mann type iteration (3) converges to $x^{*}\in F(T)$;
(ii) Ishikawa type iteration (4) converges to $x^{*}\in F(T)$.

Proof. If the Ishikawa iteration (4) converges then setting $\beta_{n}=0,\forall n\in\mathbb{N}$, the convergence of Mann iteration (3). Conversely, we shall prove that (i) $\Rightarrow$ (ii). The proof is similar to the proof of Theorem 4 from [4]. We have

Also

From (8) and (9) we get

The norm of the sum of the first two terms on the right-hand side of (10) is equal to

Using (6) with $x:=x_{n+1},y:=u_{n+1}$, we obtain

From (10) it follows that

Thus, we have

But

and

because $1\leqslant L\Rightarrow 1-\beta_{n}+\beta_{n}L\leqslant L$.
Substituting (14) into (13) we obtain

Using (14),

From (4) we have

Substituting (18) and (17) into (16), we obtain

Using (14) we have

Substituting (19) and (20) into (12), and using the facts that $\left(1+\alpha_{n}^{2}\right)^{-1}\leqslant 1$, we get

We may write

where

We have

The last inequality is true because $L\geqslant 1$. From (5) it follows that for all $n$ sufficiently large we have

thus