Let $X$ be an arbitrary real Banach space and $J:X\rightarrow 2^{X^{*}}$ the normalized duality mapping given by

In [13] the following class of maps was introduced:
Definition 1.1. Let $X$ be a normed space and $B$ a subset of $X$. A map $T$ is said to be asymptotically pseudocontractive if there exists a sequence $\left\{K_{n}\right\},K_{n}\in[1,\infty),\forall n\in\mathbb{N},\lim_{n\rightarrow\infty}K_{n}=$ 1 , and there exists $j(x-y)\in J(x-y)$ such that

If there exists $x^{*}$ such that $Tx^{*}=x^{*}$, by setting $y:=x^{*}$ in (1.2) we get

such a map is called asymptotically hemicontractive.

The modified Mann iteration, (see [8]), is defined by

The modified Ishikawa iteration is defined, (see [6]), by

The sequences $\left\{\alpha_{n}\right\}\subset(0,1),\left\{\beta_{n}\right\}\subset[0,1)$ satisfy

We shall give here the most general result concerning the convergence of Mann and Ishikawa iterations dealing with a uniformly Lipschitzian and asymptotically pseudocontractive map. Our result generalizes the main results from [3], [4], [10] and [12]. We also give an affirmative answer to the open question from [4] if the Mann or Ishikawa iteration converges when applied to an asymptotically pseudocontractive (respectively an asymptotically hemicontractive map), in more general spaces than Hilbert spaces.

We recall the following auxiliary results.
Lemma 2.1. [7] Let $X$ be a Banach space and $x,y\in X$. Then

for all $r>0$ if and only if there exists $j(x)\in J(x)$ such that $\langle y,j(x)\rangle\geq 0$.

Lemma 2.2 [11] 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 an asymptotically pseudocontractive map,
(ii) for $k_{n}\in[1,\infty),\forall n\in\mathbb{N}$, we have

Definition 2.1. Let $X$ be a normed space and $B$ a subset of $X$, then the map $T:B\rightarrow B$ is a uniformly Lipschitzian map if for some $L\geq 1$, we have $\left\|T^{n}x-T^{n}y\right\|\leq L\|x-y\|,\forall x,y\in B,\forall n\in\mathbb{N}$.

Lemma 2.3. [14] Let $\left\{\Psi_{n}\right\}$ be a nonnegative sequence satisfying

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

Theorem 3.1. Let $B$ be a closed convex subset of an arbitrary Banach space $X$ and $\left(u_{n}\right)_{n}$ defined by (1.4) with $\left(\alpha_{n}\right)_{n}$ and $\left(\beta_{n}\right)_{n}$ satisfying (1.6). Let $T$ be an asymptotically pseudocontractive (or asymptotically hemicontractive) and uniformly Lipschitzian map with $L\geq 1$ self-map of $B$. If $u_{0}\in B$, then the modified Mann iteration (1.4) strongly converges to the nearest $x^{*}$ fixed point of $T$.

Proof. From (1.4) we obtain

By using $T^{n}x^{*}=x^{*}$ we observe that

From (3.1) and (3.2) we get

The norm of the sum of the first two terms on the right-hand side of 3.3) is equal to

Using (2.1) with

we obtain

From (3.3) and (3.6) it follows that

We shall prove later the first inequality from 3.7). Supposing that 3.7) holds, we obtain

Also, we know that

Using (1.4), 3.9 and the fact that $T$ is a uniformly Lipschitzian map, we obtain

From (3.8), (3.9) and (3.10), by using $\left(1+\alpha_{n}^{2}\right)^{-1}\leq 1,\forall n\in\mathbb{N}$, we get

The condition $\lim_{n\rightarrow\infty}\alpha_{n}=0$ implies the existence of $n_{0}\in\mathbb{N}$, such that

Condition (3.12) assures the following inequalities, $\forall n\geq n_{0}$,

Using (3.11) and (3.13) we observe that

Relations (3.11), (3.14), lead us to

Setting in (2.3) from lemma 2.3

we get

We prove now the first inequality from 3.7 . Set in 3.7

to obtain

We shall now prove (3.19) using the following relations

The last inequality from (3.20) is given by (2.1), that is $\left\|a^{\prime}\right\|\geq\|a\|$. We further prove that

By using

we obtain

The last equality is true because $T^{n}x^{*}=x^{*}$. Finally, we have

The last equality is true because we already know that

Remark 3.1. If $\lim_{n\rightarrow\infty}\alpha_{n}\neq 0$, then our Theorem 3.1 holds supposing condition (3.12) is satisfied.

The modified Ishikawa iteration also converges, being equivalent to the modified Mann iteration.

Theorem 3.2. [11] Let $B$ be a closed convex subset of an arbitrary Banach space $X,\left(x_{n}\right)_{n}$ and $\left(u_{n}\right)_{n}$ defined by (1.5) and (1.4) with $\left(\alpha_{n}\right)_{n}$ and $\left(\beta_{n}\right)_{n}$ satisfying (1.6). Let $T$ be an asymptotically pseudocontractive and uniformly Lipschitzian with $L\geq 1$ self-map of $B$. Let $x^{*}$ be a fixed point of $T$. If $u_{0}=x_{0}\in B$, then the following two assertions are equivalent:
(i) the modified Mann iteration (1.4) strongly converges to $x^{*}$,
(ii) the modified Ishikawa iteration (1.5) strongly converges to $x^{*}$.

Remark 3.2. Each fixed point has its own basin of attraction. The map $T$ has no unique fixed point. The starting point is crucial for the convergence of Mann or Ishikawa iteration. For example, take $T=I$, the identity map on $B$, with $k_{n}=1,\forall n\in\mathbb{N}$. Each point of $B$ becomes a fixed point and the starting point is directly a fixed point.

Theorem 3.1 generalizes the Theorem from [3] because in [3] the set $B$ is bounded, the space $X$ is uniformly convex and $\left(\alpha_{n}\right)_{n}$ and $\left(\beta_{n}\right)_{n}$ satisfy some additional conditions. We also generalize Theorem 1 from [12], because the space is smooth and the following conditions are required: $\sum\left(k_{n}-1\right)<+\infty,\sum\alpha_{n}^{2}<+\infty$ and $\sum\beta_{n}<+\infty$. Our Theorem 3.1 generalizes the main results from [2] and [10] because the map $T$ satisfies the following restrictive condition:

where $\left(x_{n}\right)_{n}$ is the modified Mann (respectively modified Ishikawa) iterations, $x^{*}$ is a fixed point and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a strictly increasing function with $\phi(0)=0$.

In [1] and [5] the convergence of (1.4) and (1.5) is shown, dealing with an asymptotically pseudocontractive map without being uniformly Lipschitzian. However, in [1] and [5] the assumptions are more restrictive than those from our Theorem 3.1; the Banach space is uniformly smooth, the set $B$ is bounded, respectively $T(B)$ is bounded and the map $T$ satisfies condition (3.26).

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