Let E be a real Banach space and let J denote the normalized duality mapping from E into {2}^{{E}^{\ast}} defined by
J(x)=\{f\in {E}^{\ast}:\u3008x,f\u3009={\parallel x\parallel}^{2}={\parallel f\parallel}^{2}\}
for all x\in E, where {E}^{\ast} denotes the dual space of E and \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing, respectively. The normalized duality mapping J has the following properties:

(1)
J is an odd mapping, i.e., J(x)=J(x).

(2)
J is positive homogeneous, i.e., for any \lambda >0, J(\lambda x)=\lambda J(x).

(3)
J is bounded, i.e., for any bounded subset A of E, J(A) is a bounded subset of {E}^{\ast}.

(4)
If E is smooth (or {E}^{\ast} is strictly convex), then J is singlevalued.
In the sequel, we denote the singlevalued normalized duality mapping by j. In a Hilbert space H, j is the identity mapping.
Let D be a nonempty closed convex subset of E. A mapping T:D\to D is said to be asymptotically nonexpansive with a sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}) and {lim}_{n\to \mathrm{\infty}}{k}_{n}=1 if, for all x,y\in D,
\parallel {T}^{n}x{T}^{n}y\parallel \le {k}_{n}\parallel xy\parallel
(1.1)
for all n\ge 1. The mapping T is said to be asymptotically pseudocontractive with a sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}) and {lim}_{n\to \mathrm{\infty}}{k}_{n}=1 if, for any x,y\in D, there exists j(xy)\in J(xy) such that
\u3008{T}^{n}x{T}^{n}y,j(xy)\u3009\le {k}_{n}{\parallel xy\parallel}^{2}
(1.2)
for all n\ge 1. Furthermore, the mapping T is said to be uniformly LLipschitzian if, for any x,y\in D, there exists a constant L>0 such that
\parallel {T}^{n}x{T}^{n}y\parallel \le L\parallel xy\parallel
(1.3)
for all n\ge 1.
It is easy to see that if T is an asymptotically nonexpansive mapping, then it is both asymptotically pseudocontractive and uniformly LLipschitzian. The converse is not true in general. Therefore, it is interesting to study these mappings in fixed point theory and its applications. In fact, the asymptotically nonexpansive and asymptotically pseudocontractive mappings were first introduced by GoebelKirk [1] and Schu [2], respectively. Since then, some authors have studied several iterative sequences for asymptotically nonexpansive and asymptotically pseudocontractive mappings in Hilbert spaces and Banach spaces (see [3–11]).
In [2], Schu proved the following theorem.
Theorem 1.1 [2]
Let H be a Hilbert space, K be a nonempty bounded closed convex subset of H and T:K\to K be a completely continuous, uniformly LLipschitzian and asymptotically pseudocontractive mapping with a sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}) satisfying the following conditions:
(a1) {k}_{n}\to 1 as n\to \mathrm{\infty};
(a2) {\sum}_{n=1}^{\mathrm{\infty}}({q}_{n}^{2}1)<\mathrm{\infty}, where {q}_{n}=2{k}_{n}1.
Suppose further that \{{\alpha}_{n}\} and \{{\beta}_{n}\} are two sequences in [0,1] such that \u03f5<{\alpha}_{n}<b for all n\ge 1, where \u03f5>0 and b\in (0,{L}^{2}[{(1+{L}^{2})}^{1/2}1]). For any {x}_{1}\in K, let \{{x}_{n}\} be an iterative sequence defined by
{x}_{n+1}=(1{\alpha}_{n}){x}_{n}+{\alpha}_{n}{T}^{n}{x}_{n}
for all n\ge 1. Then \{{x}_{n}\} converges strongly to a fixed point of T in K.
In [12], Chang extended above Theorem 1.1 to the setting of real uniformly smooth Banach spaces and proved the following.
Theorem 1.2 [12]
Let E be a uniformly smooth Banach space, K be a nonempty bounded closed convex subset of E and T:K\to K be an asymptotically pseudocontractive mapping with a sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}), {lim}_{n\to \mathrm{\infty}}{k}_{n}=1 and F(T)\ne \mathrm{\varnothing}, where F(T) is the set of fixed points of T in K. Let \{{\alpha}_{n}\} be a sequence in [0,1] satisfying the following conditions:
(a1) {\alpha}_{n}\to 0 as n\to \mathrm{\infty};
(a2) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}.
For any {x}_{0}\in K, let \{{x}_{n}\} be an iterative sequence defined by
{x}_{n+1}=(1{\alpha}_{n}){x}_{n}+{\alpha}_{n}{T}^{n}{x}_{n}
for all n\ge 0. If there exists a strictly increasing function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
\u3008{T}^{n}x{x}^{\ast},j(x{x}^{\ast})\u3009\le {k}_{n}{\parallel x{x}^{\ast}\parallel}^{2}\mathrm{\Phi}\left(\parallel x{x}^{\ast}\parallel \right)
for all x\in K and n\ge 0, where {x}^{\ast}\in F(T), then {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}.
In [13], Ofoedu extended Theorem 1.2 in a uniformly smooth Banach space to the setting of arbitrary real Banach spaces and dropped the boundedness assumption in Theorem 1.2.
Theorem 1.3 [13]
Let E be a real Banach space, K be a nonempty closed convex subset of E and T:K\to K be a uniformly LLipschitzian asymptotically pseudocontractive mapping with a sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}), {lim}_{n\to \mathrm{\infty}}{k}_{n}=1 and {x}^{\ast}\in F(T). Let \{{\alpha}_{n}\} be a sequence in [0,1] satisfying the following conditions:
(a1) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
(a2) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}<\mathrm{\infty};
(a3) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}({k}_{n}1)<\mathrm{\infty}.
For any {x}_{0}\in K, let \{{x}_{n}\} be an iterative sequence defined by
{x}_{n+1}=(1{\alpha}_{n}){x}_{n}+{\alpha}_{n}{T}^{n}{x}_{n}
for all n\ge 0. If there exists a strictly increasing function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
\u3008{T}^{n}x{x}^{\ast},j(x{x}^{\ast})\u3009\le {k}_{n}{\parallel x{x}^{\ast}\parallel}^{2}\mathrm{\Phi}\left(\parallel x{x}^{\ast}\parallel \right)
for all x\in K and n\ge 0. Then

(1)
\{{x}_{n}\} is bounded;

(2)
\{{x}_{n}\} converges strongly to {x}^{\ast}\in F(T).
Theorem 1.4 [13]
Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, T:K\to K be a uniformly LLipschitzian asymptotically pseudocontractive mapping with a sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}) with {lim}_{n\to \mathrm{\infty}}{k}_{n}=1 and {x}^{\ast}\in F(T). Let \{{a}_{n}\}, \{{b}_{n}\} and \{{c}_{n}\} be real sequences in [0,1] satisfying the following conditions:
(a1) {a}_{n}+{b}_{n}+{c}_{n}=1;
(a2) {\sum}_{n\ge 0}({b}_{n}+{c}_{n})=\mathrm{\infty};
(a3) {\sum}_{n\ge 0}{({b}_{n}+{c}_{n})}^{2}<\mathrm{\infty};
(a4) {\sum}_{n\ge 0}({b}_{n}+{c}_{n})({k}_{n}1)<\mathrm{\infty};
(a5) {\sum}_{n\ge 0}{c}_{n}<\mathrm{\infty}.
For arbitrary {x}_{0}\in K, let \{{x}_{n}\} be a sequence in K iteratively defined by
{x}_{n+1}={a}_{n}{x}_{n}+{b}_{n}{T}^{n}{x}_{n}+{c}_{n}{u}_{n}
for all n\ge 0, where \{{u}_{n}\} is a bounded sequence in K. Suppose that there exists a strictly increasing continuous function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
\u3008{T}^{n}x{x}^{\ast},j(x{x}^{\ast})\u3009\le {k}_{n}{\parallel x{x}^{\ast}\parallel}^{2}\mathrm{\Phi}\left(\parallel x{x}^{\ast}\parallel \right)
for all x\in K. Then {\{{x}_{n}\}}_{n\ge 0} converges strongly to {x}^{\ast}\in F(T).
Very recently, in [14], Chang et al. proved the following theorem.
Theorem 1.5 [14]
Let E be a real Banach space. Let K be a nonempty closed convex subset of E, {T}_{i}:K\to K be two uniformly {L}_{i}Lipschitzian mappings with F({T}_{1})\cap F({T}_{2})\ne \mathrm{\varnothing} and {x}^{\ast}\in F({T}_{1})\cap F({T}_{2}). Let \{{k}_{n}\}\subset [1,+\mathrm{\infty}) be a sequence with {lim}_{n\to \mathrm{\infty}}{k}_{n}=1. Let \{{\alpha}_{n}\} and \{{\beta}_{n}\} be two sequences in [0,1] satisfying the following conditions:
(a1) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
(a2) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}<\mathrm{\infty};
(a3) {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}<\mathrm{\infty};
(a4) {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}({k}_{n}1)<\mathrm{\infty}.
For any {x}_{0}\in K, let \{{x}_{n}\} be an iterative sequence in K defined by
\{\begin{array}{c}{x}_{n+1}=(1{\alpha}_{n}){x}_{n}+{\alpha}_{n}{T}_{1}^{n}{y}_{n},\hfill \\ {y}_{n}=(1{\beta}_{n}){x}_{n}+{\beta}_{n}{T}_{2}^{n}{x}_{n}\hfill \end{array}
for all n\ge 0. If there exists a strictly increasing function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
\u3008{T}_{i}^{n}x{x}^{\ast},j(x{x}^{\ast})\u3009\le {k}_{n}{\parallel x{x}^{\ast}\parallel}^{2}\mathrm{\Phi}\left(\parallel x{x}^{\ast}\parallel \right)
for all j(x{x}^{\ast})\in J(x{x}^{\ast}) and x\in K, i=1,2, then \{{x}_{n}\} converges strongly to {x}^{\ast}.
Also, some authors have studied the modified Halpern, Mann and Ishikawa iterative sequences for nonlinear mappings in Hilbert spaces and Banach spaces (see [15, 16]).
The aim of this paper is to give some strong convergence theorems for uniformly LLipschitzian and asymptotically pseudo contractive mappings in Banach spaces. Our results not only include the past ones known in [3–11], but also provide quite a different proof method.
For our main purpose, we recall some concepts and lemmas.
Definition 1.6 [17]
For arbitrary {x}_{1}\in D, the sequence \{{x}_{n}\} in D defined by
\{\begin{array}{c}{y}_{n}=(1{b}_{n}{d}_{n}){x}_{n}+{b}_{n}{T}^{n}{x}_{n}+{d}_{n}{v}_{n},\hfill \\ {x}_{n+1}=(1{a}_{n}{c}_{n}){x}_{n}+{a}_{n}{T}^{n}{y}_{n}+{c}_{n}{u}_{n}\hfill \end{array}
(1.4)
for all n\ge 1 is called the modified Ishikawa iteration with errors, where \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{d}_{n}\} are four real sequences in [0,1] satisfying {a}_{n}+{c}_{n}\le 1, {b}_{n}+{d}_{n}\le 1 and \{{u}_{n}\}, \{{v}_{n}\} are any bounded sequences in D.
In particular, if {b}_{n}={d}_{n}=0 in (1.4), then the sequence \{{x}_{n}\} defined by
{x}_{n+1}=(1{a}_{n}{c}_{n}){x}_{n}+{a}_{n}{T}^{n}{x}_{n}+{c}_{n}{u}_{n}
(1.5)
for all n\ge 1 is called the modified Mann iteration with errors.
If {c}_{n}={d}_{n}=0 in (1.4) and (1.5), then the sequence \{{x}_{n}\} defined by
\{\begin{array}{c}{y}_{n}=(1{b}_{n}){x}_{n}+{b}_{n}{T}^{n}{x}_{n},\hfill \\ {x}_{n+1}=(1{a}_{n}){x}_{n}+{a}_{n}{T}^{n}{y}_{n}\hfill \end{array}
(1.6)
and
{x}_{n+1}=(1{a}_{n}){x}_{n}+{a}_{n}{T}^{n}{x}_{n}
(1.7)
for all n\ge 1 is called the modified Ishikawa iteration and the modified Mann iteration, respectively.
Lemma 1.7 [18]
Let E be a real Banach space and J:E\to {2}^{{E}^{\ast}} be a normalized duality mapping. Then
{\parallel x+y\parallel}^{2}\le {\parallel x\parallel}^{2}+2\u3008y,j(x+y)\u3009
for all x,y\in E and j(x+y)\in J(x+y).
Lemma 1.8 [19]
Let \{{\delta}_{n}\}, \{{\lambda}_{n}\} and \{{\gamma}_{n}\} be three nonnegative real sequences and \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) be a strictly increasing continuous function with \mathrm{\Phi}(0)=0 satisfying the following inequality:
{\delta}_{n+1}^{2}\le {\delta}_{n}^{2}{\lambda}_{n}\mathrm{\Phi}({\delta}_{n+1})+{\gamma}_{n}
for all n\ge 0, where {\lambda}_{n}\in [0,1] with {\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty} and {\gamma}_{n}=o({\lambda}_{n}). Then {\delta}_{n}\to 0 as n\to \mathrm{\infty}.