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Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 247 (2013)
Abstract
The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 6179, 2007), associated with nonexpansive and ϕhemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.
Dedication
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
1 Preliminaries
Let K be a nonempty subset of an arbitrary Banach space X, and let ${X}^{\ast}$ be its dual space. Let $T:X\to X$ be an operator. The symbols $D(T)$ and $R(T)$ stand for the domain and the range of T, respectively. We denote $F(T)$ by the set of fixed points of a singlevalued mapping $T:K\to K$. We denote by J the normalized duality mapping from X to ${2}^{{X}^{\ast}}$ defined by
Let $T:D(T)\subseteq X\to X$ be an operator.
Definition 1 T is called LLipschitzian if there exists $L\ge 0$ such that
for all $x,y\in D(T)$. If $L=1$, then T is called nonexpansive, and if $0\u2a7dL<1$, T is called contraction.

(i)
T is said to be strongly pseudocontractive if there exists a $t>1$ such that for each $x,y\in D(T)$, there exists $j(xy)\in J(xy)$ satisfying
$$Re\u3008TxTy,j(xy)\u3009\le \frac{1}{t}{\parallel xy\parallel}^{2}.$$ 
(ii)
T is said to be strictly hemicontractive if $F(T)\ne \mathrm{\varnothing}$ and if there exists a $t>1$ such that for each $x\in D(T)$ and $q\in F(T)$, there exists $j(xy)\in J(xy)$ satisfying
$$Re\u3008Txq,j(xq)\u3009\le \frac{1}{t}{\parallel xq\parallel}^{2}.$$ 
(iii)
T is said to be ϕstrongly pseudocontractive if there exists a strictly increasing function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\varphi (0)=0$ such that for each $x,y\in D(T)$, there exists $j(xy)\in J(xy)$ satisfying
$$Re\u3008TxTy,j(xy)\u3009\le {\parallel xy\parallel}^{2}\varphi (\parallel xy\parallel )\parallel xy\parallel .$$ 
(iv)
T is said to be ϕhemicontractive if $F(T)\ne \mathrm{\varnothing}$ and if there exists a strictly increasing function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\varphi (0)=0$ such that for each $x\in D(T)$ and $q\in F(T)$, there exists $j(xy)\in J(xy)$ satisfying
$$Re\u3008Txq,j(xq)\u3009\le {\parallel xq\parallel}^{2}\varphi (\parallel xq\parallel )\parallel xq\parallel .$$
Clearly, each strictly hemicontractive operator is ϕhemicontractive.
For a nonempty convex subset K of a normed space $X,S:K\to K$ and $T:K\to K$,

(a)
the Mann iteration scheme [4] is defined by the following sequence $\{{x}_{n}\}$:
where $\{{b}_{n}\}$ is a sequence in $[0,1]$;

(b)
the sequence $\{{x}_{n}\}$ defined by
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ are sequences in $[0,1]$ is known as the Ishikawa [2] iteration scheme;

(c)
the sequence $\{{x}_{n}\}$ defined by
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ are sequences in $[0,1]$, is known as the AgarwalO’ReganSahu [5] iteration scheme;

(d)
the sequence $\{{x}_{n}\}$ defined by
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ are sequences in $[0,1]$, is known as the modified AgarwalO’ReganSahu iteration scheme.
Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudocontractive mapping from a bounded closed convex subset of ${L}_{p}$ (or ${l}_{p}$) into itself. Afterwards, several authors generalized this result of Chidume in various directions [3, 6–12].
The purpose of this paper is to characterize conditions for the convergence of the iterative scheme in the sense of Agarwal et al. [5] associated with nonexpansive and ϕhemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our results improve and generalize most results in recent literature [1, 3, 5, 6, 8, 9, 11, 12].
2 Main result
The following result is now well known.
Lemma 3 [13]
For all $x,y\in X$ and $j(x+y)\in J(x+y)$,
Now, we prove our main result.
Theorem 4 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, let $S:K\to K$ be nonexpansive, and let $T:K\to K$ be a uniformly continuous ϕhemicontractive mapping such that S and T have the common fixed point. Suppose that ${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$ are sequences in $[0,1]$ satisfying conditions

(i)
${lim}_{n\to \mathrm{\infty}}(1{b}_{n})={lim}_{n\to \mathrm{\infty}}{b}_{n}^{\prime}=0$,

(ii)
${\sum}_{n=1}^{\mathrm{\infty}}(1{b}_{n})=\mathrm{\infty}$.
For any ${x}_{1}\in K$, define the sequence ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ inductively as follows:
Then the following conditions are equivalent:

(a)
${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to the common fixed point q of S and T.

(b)
${\{S{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{T{y}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are bounded.
Proof First, we prove that (a) implies (b).
Since T is ϕhemicontractive, it follows that $F(T)$ is a singleton. Let $F(S)\cap F(T)=\{q\}$ for some $q\in K$.
Suppose that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=q$. Then the continuity of S and T yields that
and
Thus, ${lim}_{n\to \mathrm{\infty}}T{y}_{n}=q$. Therefore, ${\{S{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{T{y}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are bounded.
Second, we need to show that (b) implies (a). Suppose that ${\{S{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{T{y}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are bounded.
Put
It is clear that $\parallel {x}_{1}q\parallel \le {M}_{1}$. Let $\parallel {x}_{n}q\parallel \le {M}_{1}$. Next, we will prove that $\parallel {x}_{n+1}q\parallel \le {M}_{1}$.
Note that
Thus, we can conclude that the sequence ${\{{x}_{n}q\}}_{n\ge 1}$ is bounded, and hence, there is a constant $M>0$ satisfying
Let ${w}_{n}=\parallel T{y}_{n}T{x}_{n+1}\parallel $ for each $n\ge 1$. The uniform continuity of T ensures that
because
By virtue of Lemma 3 and (2.1), we infer that
The real function $f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, $f(t)={t}^{2}$ is increasing and convex. For all $a\in [0,1]$ and ${t}_{1},{t}_{2}>0$, we have
Hence,
where the second inequality holds by the convexity of ${\parallel \cdot \parallel}^{2}$.
By substituting (2.5) in (2.4), we get
where
as $n\to \mathrm{\infty}$.
Let $\delta =inf\{\parallel {x}_{n+1}q\parallel :n\ge 0\}$. We claim that $\delta =0$. Otherwise, $\delta >0$. Thus, (2.7) implies that there exists a positive integer ${N}_{1}$ such that ${l}_{n}<\varphi (\delta )\delta $ for each $n\ge {N}_{1}$. In view of (2.6), we conclude that
which implies that
which contradicts (ii). Therefore, $\delta =0$. Thus, there exists a subsequence ${\{{x}_{{n}_{i}+1}\}}_{n=1}^{\mathrm{\infty}}$ of ${\{{x}_{n+1}\}}_{n=1}^{\mathrm{\infty}}$ such that
Let $\u03f5>0$ be a fixed number. By virtue of (2.7) and (2.9), we can select a positive integer ${i}_{0}>{N}_{1}$ such that
Let $p={n}_{{i}_{0}}$. By induction, we show that
Observe that (2.6) means that (2.11) is true for $m=1$. Suppose that (2.11) is true for some $m\ge 1$. If $\parallel {x}_{p+m+1}q\parallel \ge \u03f5$, by (2.6) and (2.10), we know that
which is impossible. Hence, $\parallel {x}_{p+m+1}q\parallel <\u03f5$. That is, (2.11) holds for all $m\ge 1$. Thus, (2.11) ensures that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=q$. This completes the proof. □
Taking $S=I$ in Theorem 4, we get the following.
Corollary 5 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, and let $T:K\to K$ be a uniformly continuous ϕhemicontractive mapping. Suppose that ${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$ are sequences in $[0,1]$ satisfying conditions (i)(ii) of Theorem 4. For any ${x}_{1}\in K$, define the sequence ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ inductively as follows:
Then the following conditions are equivalent:

(a)
${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to the unique fixed point q of T.

(b)
${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is bounded.
Remark 6

1.
All the results can also be proved for the same iterative scheme with error terms.

2.
The known results for strongly pseudocontractive mappings are weakened by the ϕhemicontractive mappings.

3.
Our results hold in arbitrary Banach spaces, where as other known results are restricted for ${L}_{p}$ (or ${l}_{p}$) spaces and quniformly smooth Banach spaces.

4.
Theorem 4 is more general in comparison to the results of Agarwal et al. [5] in the context of the class of ϕhemicontractive mappings. Theorem 4 extends convergence results coercing ϕhemicontractive mappings in the literature in the framework of AgarwalO’ReganSahu iteration process (see also [14–21]).
3 Applications
Theorem 7 Let X be an arbitrary real Banach space, $S:X\to X$ be nonexpansive, and let $T:X\to X$ be uniformly continuous ϕstrongly accretive operators, respectively. Suppose that ${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$ are sequences in $[0,1]$ satisfying conditions (i)(ii) of Theorem 4. For any ${x}_{1}\in X$, define the sequence ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ inductively as follows:
where $f\in X$, and I is the identity operator. Then the following conditions are equivalent:

(a)
${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to the solution of the system $Sx=f=Tx$.

(b)
${\{(IS){x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{(IT){x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{(IT){y}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are bounded.
Proof Suppose that ${x}^{\ast}$ is the solution of the system $Sx=f=Tx$. Define $G,{G}^{\prime}:X\to X$ by $Gx=f+(IS)x$ and ${G}^{\prime}x=f+(IT)x$, respectively. Since S and T are nonexpansive and uniformly continuous ϕstrongly accretive operators, respectively, so are G and ${G}^{\prime}$, then ${x}^{\ast}$ is the common fixed point of G and ${G}^{\prime}$. Thus, Theorem 7 follows from Theorem 4. □
Example 8 Let $X=\mathbb{R}$ be the reals with the usual norm and $K=[0,1]$. Define $S:K\to K$ by
and $T:K\to K$ by
By the mean value theorem, we have
Noticing that ${T}^{\prime}(c)=1{sec}^{2}(c)$ and $1<{sup}_{c\in (0,1)}{T}^{\prime}(c)=2.4255$. Hence,
where $L=2.4255$. It is easy to verify that T is ϕhemicontractive mapping with $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by $\varphi (t)=tan(t)$ for all $t\in [0,\mathrm{\infty})$. Moreover, 0 is the common fixed point of S and T. Let ${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$ be sequences in $[0,1]$ defined by
Then ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ defined by (2.1) in Theorem 4 converges to 0, which is the common fixed point of S and T.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.
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Hussain, N., Sahu, D. & Rafiq, A. Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces. Fixed Point Theory Appl 2013, 247 (2013). https://doi.org/10.1186/168718122013247
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Keywords
 modified iterative scheme
 nonexpansive mappings
 ϕhemicontractive mappings
 Banach spaces