Convergence Theorems for a Hybrid Pair of Generalized Nonexpansive Mappings in Banach Spaces

Abstract

In this paper, we study convergence of modified Ishikawa iteration process involving a pair of hybrid mappings satisfying a generalized nonexpansive condition on Banach spaces. In process, several relevant results are generalized and improved.

1 Introduction

Let $X$ be a Banach space and $K$ be a nonempty subset of $X$. Let $CB(K)$ be the family of nonempty closed bounded subsets of $K$, while $KC(K)$ be the family of nonempty compact convex subsets of $K$. A subset $K$ of $X$ is called proximinal if for each $x \in X$, there exists an element $k \in K$ such that

$$\begin{aligned} d(x, k) = dist(x, K) = \inf \{\Vert x - y\Vert : y \in K\}. \end{aligned}$$

It is well known that every closed convex subset of a uniformly convex Banach space is proximinal. We shall denote by $PB(K)$, the family of nonempty bounded proximinal subsets of $K$. The Hausdorff metric $H$ on $CB(K)$ is defined as

$$\begin{aligned} H(A, B)= max \bigg \{\sup _{x\in A} d(x, B), \sup _{y\in B} d(y, A) \bigg \} ~\text{ for }~ A,~ B \in CB(K). \end{aligned}$$

A mapping $f: K\rightarrow K$ is said to be nonexpansive if

$$\begin{aligned} \Vert fx-fy\Vert \le \Vert x-y\Vert , ~\text{ for } \text{ all }~ x,~ y \in K, \end{aligned}$$

while a multi-valued mapping $T: K \rightarrow CB(K)$ is said to be nonexpansive if

$$\begin{aligned} H(T(x), T(y)) \le \Vert x-y\Vert , ~\text{ for } \text{ all }~ x,~ y \in K. \end{aligned}$$

We use the notation $Fix(T)$ for the set of fixed points of the mapping $T$, while $Fix(f)\bigcap Fix(T)$ denotes the set of common fixed points of $f$ and $T$, i.e. a point $x$ is said to be a common fixed point of $f$ and $T$ if $x=fx\in Tx.$

In 2008, Suzuki [1] introduced a new class of mappings, which is larger than the class of nonexpansive mappings and name the defining condition as condition $(C)$ which runs follows: A mapping $T$ defined on a subset $K$ of a Banach space $X$ is said to satisfy condition $(C)$ if

$$\begin{aligned} {\frac{1}{2}}\Vert x-Tx\Vert \le \Vert x-y\Vert \Rightarrow \Vert Tx-Ty\Vert \le \Vert x-y\Vert ,\,x,y\in K. \end{aligned}$$

He then proved some fixed point as well as convergence theorems for such mappings. Inspired from this, recently García-Falset et al. [2] defined two new generalizations of condition $(C)$ and term their conditions as condition $(E)$ and Condition $(C_\lambda )$ and studied the existence of fixed point besides their asymptotic behaviour for both the classes of mappings in Banach spaces.

In 2010, Sokhuma and Kaewkhao [3] introduced a modified Ishikawa iterative process involving a pair of single-valued and multi-valued nonexpansive mappings in Banach spaces and proved strong convergence theorems.

The purpose of this paper is to study modified Ishikawa iterative method for generalized nonexpansive mappings introduced by García-Falset et al. [2].

2 Preliminaries

With a view to make, our presentation self contained, we collect some basic definitions, needed results and some iterative methods which will be used frequently in the text later. We start with the following definitions:

Definition 2.1

[2] Let $K$ be a nonempty subset of a Banach space $X$ and $S : K \rightarrow X$ be a single-valued mapping. Then $S$ is said to satisfy the condition $(E_{\mu })$ for some $\mu \ge 1$ if

$$\begin{aligned} \Vert x-Sy\Vert \le \mu \Vert x-Sx\Vert +\Vert x-y\Vert , ~{\text{ for }} ~x,~y\in K. \end{aligned}$$

We say that $S$ satisfies condition $(E)$ whenever $S$ satisfies the condition $(E_{\mu })$ for some $\mu \ge 1$.

Definition 2.2

[2] Let $K$ be a nonempty subset of a Banach space $X$ and $S : K \rightarrow X$ be a single-valued mapping. Then $S$ is said to satisfy the condition $(C_ \lambda )$ for some $\lambda \in (0,1)$ if

$$\begin{aligned} \lambda \Vert x-Sx\Vert \le \Vert x-y\Vert \Rightarrow \Vert Sx-Sy\Vert \le \Vert x-y\Vert , ~{\text{ for }}~x,~y\in K. \end{aligned}$$

From the foregoing definitions, one can infer the following observations:

Lemma 2.1

[2] If $S: K\rightarrow X$ is a nonexpansive mapping, then $S$ satisfies the condition $(E_1)$. But the converse statement is not true in general.

Lemma 2.2

[2] From Lemma 1 in [1], one can see that if $S:K\rightarrow K$ satisfies the condition $(C)$, then $S$ satisfies condition $(E_3)$. But the converse statement is not true in general.

The following example supports two preceding facts.

Example 2.1

[2] Consider $X= C[0,1]$ and the nonempty subset $K$ as follows:

$$\begin{aligned} K=\{f\in C[0,1]: 0=f(0)\le f(x)\le f(1)=1\}. \end{aligned}$$

For any $g\in K$, we construct the function $F_{g}: K\rightarrow K$

$$\begin{aligned} F_{g}h(t)=(g\circ h)(t)=g(h(t)). \end{aligned}$$

It is easy to verify that $F_g$ satisfies the condition $(E_1)$ and but does not satisfy the condition $(C)$.

By choosing $\lambda = \frac{1}{2}$, we can retrieve the class of Suzuki’s generalized nonexpansive mappings. From the definition of condition $(C_\lambda )$, one can easily see that condition $(C_{\lambda _{1}})$ implies the condition $(C_{\lambda _{2}})$ for $0<\lambda _1<\lambda _2<1.$ The following example shows that the converse fails.

Example 2.2

[2] For a given $\lambda \in (0,1)$ define a self-mapping $S$ on $[0,1]$ by

$$\begin{aligned} S(x)= {{\left\{ \begin{array}{ll} \frac{x}{2}\quad \mathrm{{if}}~ x\ne 1 \\ \\ \frac{1+\lambda }{2+\lambda }\quad \mathrm{{ if}}~ x =1.\\ \end{array}\right. }} \end{aligned}$$

Then the mapping $S$ satisfies condition $(C_{\lambda })$ but it fails condition $(C_{\lambda _1})$ whenever $0< \lambda _1<\lambda .$ Moreover $S$ satisfies condition $(E_\mu )$ for $\mu =\frac{2+\lambda }{2}$.

In the same paper, García-Falset et al. used the concept of strongly demiclosed to prove a weaker version of famous principle of demiclosedness for generalized nonexpansive mappings in which weak convergence has been replaced by strong convergence. Definition runs as follows:

Definition 2.3

[2] Given a mapping $S:K\rightarrow X$, we say that $(I-S)$ is strongly demiclosed at $0$ if for every sequence $\{x_n\}$ in $K$ strongly convergent to $z\in K$ and such that $x_n-Sx_n \rightarrow 0$ we have that $z=Sz$.

Lemma 2.3

[2] Let $K$ be a nonempty subset of a Banach space $X$. If $S:K\rightarrow X$ satisfies condition$(E)$ on $K$, then $(I-S)$ is strongly demiclosed at 0.

Kaewcharoen and Panyanak [4] gave the multi-valued version of Condition $(C_\lambda )$.

Definition 2.4

[4] Let $T:K\rightarrow CB(X)$ be a multi-valued mapping. Then $T$ is said to satisfy condition $(C_{\lambda })$ for some $\lambda \in (0,1)$ if each $x,~y \in K$,

$$\begin{aligned} \lambda ~dist (x, Tx) \le \Vert x-y\Vert ~{\text{ implies }}~H(Tx, Ty) \le \Vert x-y\Vert . \end{aligned}$$

As the single-valued case, one can easy verify that condition for $0<\lambda _{1}< \lambda _{2}<1$, $(C_{\lambda _{1}})$ implies condition $(C_{\lambda _{2}})$.

Lemma 2.4

[4] Let $K$ be a nonempty subset of a Banach space $X$ and $T:K\rightarrow CB(X)$ be a mapping. If T is a nonexpansive, then $T$ satisfies condition $(C_{\lambda })$.

The following example exhibits that the converse statement need not be true.

Example 2.3

[4] Define a mapping $T: [0,3]\rightarrow CB([0,3])$ by

$$\begin{aligned} T(x)= \left\{ \begin{array}{l} \left\{ 0\right\} ,\quad {\mathrm{if}}~ x\ne 3,\\ {[}0.5, 1],\quad {\mathrm{if}}~ x =3. \\ \end{array}\right. \end{aligned}$$

$T$ satisfies condition $(C_\lambda )$ for $\lambda =\frac{1}{2}$ but $T$ is not nonexpansive.

Now we list the following important property of a uniformly convex Banach space due to Schu [5] and an important lemma due to Sokhuma and Kaewkhao  [3].

Lemma 2.5

Let $X$ be a uniformly convex Banach space, let $\{u_n\}$ be a sequence of real numbers such that $0 < b\le u_n \le c < 1$ for all $n \ge 1$, and let $\{x_n\}$ and $\{y_n\}$ be sequences in $X$ such that $\limsup \limits _{n\rightarrow \infty } \Vert x_n\Vert \le a$ , $\limsup \limits _{n\rightarrow \infty } \Vert y_n\Vert \le a$, and $\lim \limits _{n\rightarrow \infty } \Vert u_n x_n + (1-u_n)y_n\Vert = a$ for some $a>0$. Then, $\lim \limits _{n\rightarrow \infty } \Vert x_n-y_n\Vert = 0$.

Lemma 2.6

Let $X$ be a Banach space, and let $K$ be a nonempty closed convex subset of $X$. Then,

$$\begin{aligned} dist(y, Ty) \le \Vert y - x\Vert + dist(x, Tx) + H(Tx,Ty), \end{aligned}$$

where $x, y\in K$ and $T$ is a multi-valued nonexpansive mapping from $K$ into $CB(K)$.

In 1974, Ishikawa [6] introduced an iterative procedure for approximating fixed point as follows: For $x_{0} \in K$, the sequence $\{x_n\}$ of Ishikawa iteration defined by

$$\begin{aligned} y_n&= (1-\beta _{n})x_{n}+ \beta _{n} S x_{n}, \\ x_{n+1}&= (1-\alpha _{n}) x_{n} + \alpha _{n} S y_{n},\,n\ge 0, \end{aligned}$$

where $\{\alpha _{n}\}$ and $\{\beta _{n}\}$ are real sequences in [0, 1].

In 2005, Sastry and Babu [7] defined Ishikawa iteration scheme for multi-valued mappings. Let $T: K\rightarrow PB(K)$ a multi-valued mapping and fix $p\in F(T)$. Then the sequence of Ishikawa iteration is defined as follows:

Choose $x_0 \in K$,

$$\begin{aligned} y_{n}= \beta _{n} z_{n} +(1-\beta _{n}) x_{n},\,\beta _n\in [0, 1],~~n\ge 0, \end{aligned}$$

where $z_{n} \in T(x_{n})$ such that $\Vert z_{n}-p\Vert =\mathrm{dist}(p, T x_{n})$ and

$$\begin{aligned} x_{n+1}= \alpha _{n} z_{n}^{\prime } +(1-\alpha _{n}) x_{n}, ~~\alpha _n\in [0,1],~ n\ge 0, \end{aligned}$$

where ${z_{n}^{\prime }}\in T(y_{n})$ such that $\Vert {z_{n}^{\prime }}-p\Vert = \mathrm{dist}(p, T y_{n}).$

Sastry and Babu [7] proved that Ishikawa iteration scheme for a multi-valued nonexpansive mapping $T$ with a fixed point $p$ converges to a fixed point $q$ of $T$ under certain conditions. In 2007, Panyanak [8] extended the results of Sastry and Babu to uniformly convex Banach space for multi-valued nonexpansive mappings. Panyanak also modified the iteration scheme of Sastry and Babu and imposed the question of convergence of this scheme. He introduced the following modified Ishikawa iteration method,

Choose $x_{0} \in K$, then

$$\begin{aligned} y_{n}= \beta _{n} z_{n} +(1-\beta _{n}) x_{n},\,\beta _{n} \in [a,b], ~0<a<b<1,~n\ge 0, \end{aligned}$$

where $z_{n}\in Tx_{n}$ is such that $\Vert z_n-u_n\Vert ={\text{ dist }}(u_n, Tx_n),$ and $u_n \in F(T)$ such that $\Vert x_n-u_n\Vert ={\text{ dist }}(x_n, F(T)),$ and

$$\begin{aligned} x_{n+1}= \alpha _{n}{z_{n}^{\prime }}+(1-\alpha _{n}) x_{n}\,\alpha _n\in [a, b], \end{aligned}$$

where ${z_{n}^{\prime }} \in T(y_{n})$ such that $\Vert {z_{n}^{\prime }}-v_n\Vert ={\text{ dist }}(v_n, Ty_n),$ and $v_n \in F(T)$ such that $\Vert y_n-v_n\Vert ={\text{ dist }}(y_n, F(T)).$

In 2009, Song and Wang [9] pointed out the gap in the result of Panyanak [8]. They solved/revised the gap and gave the partial answer to the question raised by Panyanak using the following iteration scheme.

Let $\alpha _{n},~\beta _n\in [0,1]$ and $\gamma _{n}\in (0, \infty )$ such that $\lim \limits _{n\rightarrow \infty } \gamma _{n}=0$. Choose $x_0\in K,$ then

$$\begin{aligned} y_{n}&= \beta _{n} z_{n} +(1-\beta _{n}) x_{n},\\ x_{n+1}&= \alpha _{n}{z_{n}^{\prime }}+(1-\alpha _{n}) x_{n}, \end{aligned}$$

where $\Vert z_n- {z_{n}^{\prime }}\Vert \le H(Tx_n, Ty_n)$ + $\gamma _{n}$ and $\Vert z_{n+1}- {z_{n}^{\prime }}\Vert \le H(Tx_{n+1}, Ty_n)$ + $\gamma _{n}$ for $z_n\in Tx_n$ and ${z_{n}^{\prime }}\in Ty_n.$

Simultaneously, Shahzad and Zegeye [11] extended the results of Sastry and Babu, Panyanak, and Song and Wang to quasi nonexpansive multi-valued mappings and also relaxed the end point condition and compactness of the domain using the following modified iteration scheme and gave the affirmative answer to the Panyanak question in a more general setting.

$$\begin{aligned} y_{n}&= \beta _{n} z_{n} +(1-\beta _{n}) x_{n},~~\beta _n\in [0, 1],~~n\ge 0,\\ x_{n+1}&= \alpha _{n}{z_{n}^{\prime }}+(1-\alpha _{n}) x_{n},~~\alpha _n\in [0,1],~~n\ge 0, \end{aligned}$$

where $z_n\in Tx_n$ and ${z_{n}^{\prime }} \in Ty_n$. Recently, Sokhuma and Kaewkhao [3] introduced the following modified Ishikawa iteration scheme for a pair of single-valued and multi-valued mapping.

Let $K$ be a nonempty closed and bounded convex subset of Banach space $X$, let $S: K\rightarrow K$ be a single-valued nonexpansive mapping and let $T: K\rightarrow CB(K)$ be a multi-valued nonexpansive mapping. The sequence $\{x_n\}$ of the modified Ishikawa iteration is defined by

$$\begin{aligned} {{\left\{ \begin{array}{ll} y_{n}= \beta _{n} z_{n} +(1-\beta _{n}) x_{n},\\ x_{n+1}= \alpha _{n}Sy_n+(1-\alpha _{n}) x_{n},\\ \end{array}\right. }} \end{aligned}$$

(1)

where $x_0\in K$, $z_n\in Tx_n$ and $0<a\le \alpha _n,~\beta _n\le b < 1$.

Furthermore, they proved the following strong convergence theorem:

Theorem 2.7

Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space $X$, and let $S : K\rightarrow K$ and $T : K \rightarrow CB(K)$ be a single-valued and a multi-valued nonexpansive mapping, respectively, and $Fix(S) \bigcap Fix(T)\not = \emptyset$ satisfying $Tw = \{w\}$ for all $w \in Fix(S) \bigcap Fix(T)$. Let $\{x_n\}$ be the sequence of the modified Ishikawa iteration defined by (1) with $0 < a \le \alpha _n,~\beta _n \le b < 1$. Then $\{x_n\}$ converges strongly to a common fixed point of $S$ and $T$.

3 Main Results

Before establishing our main results we prove following lemmas. These results generalize the results of [3, 10] and will be used to accomplish our main Theorem 3.6.

Lemma 3.1

Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space $X$, and let $S : K \rightarrow K$ and $T : K \rightarrow CB(K)$ be a single-valued and a multi-valued mapping such that both satisfy the condition $(C_{\lambda })$, respectively, and $Fix(S)\cap Fix(T) \not =\emptyset$ satisfying $Tw = \{w\}$ for all $w \in Fix(S)\cap Fix(T)$. Let $\{x_{n}\}$ be the sequence of the modified Ishikawa iteration defined by (1). Then, $\lim \limits _{n\rightarrow \infty }\Vert x_{n}-w\Vert$ exists for all $w \in Fix(S)\cap Fix(T)$.

Proof

Letting $w \in Fix(S)\cap Fix(T).$ Since for $\lambda \in (0,~1),$

$$\begin{aligned} \lambda \Vert w-Sw\Vert =0 \le \Vert ((1-\beta _{n})x_{n}+ \beta _{n}z_{n})-w\Vert , \end{aligned}$$

using definition of Condition $(C_{\lambda })$, we get

$$\begin{aligned} \Vert S((1-\beta _{n})x_{n}+ \beta _{n}z_{n})-Sw\Vert \le \Vert (1-\beta _{n})x_{n} +\beta _{n}z_{n}-w\Vert . \end{aligned}$$

Similarly, since $\lambda ~{\text{ dist }} (w,Tw)=0 \le \Vert x_{n}-w\Vert$, then we get, $H(Tx_{n},Tw)\le \Vert x_{n}-w\Vert$. Consider

$$\begin{aligned} \Vert x_{n+1}-w\Vert&= \Vert (1-\alpha _{n})x_{n}+\alpha _{n}S((1-\beta _{n})x_{n}+\beta _{n}z_{n})-w\Vert \\&= \Vert (1-\alpha _{n})x_{n}+\alpha _{n}S((1-\beta _{n})x_{n}+\beta _{n}z_{n})-(1-\alpha _{n})w-\alpha _{n}w\Vert \\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert S((1-\beta _{n})x_{n}+\beta _{n}z_{n})-w\Vert \\&= (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert S((1-\beta _{n})x_{n}+\beta _{n}z_{n})-Sw\Vert \\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert (1-\beta _{n})x_{n}+\beta _{n}z_{n}-w\Vert \\&= (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert (1-\beta _{n})x_{n}+\beta _{n}z_{n}-(1-\beta _{n})w-\beta _{n}w\Vert \\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\{(1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}\Vert z_{n}-w\Vert \}\\&= (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\{(1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}dist(z_{n},Tw)\}\\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\{(1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}H(Tx_{n},Tw)\}\\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\{(1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}\Vert x_{n}-w\Vert \}\\&= \Vert x_{n}-w\Vert . \end{aligned}$$

Since $\{\Vert x_{n}-w\Vert \}$ is a bounded and decreasing sequence, hence the limit of $\{\Vert x_{n}-w\Vert \}$ exists.

Using Lemma 2.5 we get the following results: $\square$

Lemma 3.2

Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space $X$, and let $S: K\rightarrow K$ and $T:K \rightarrow CB(K)$ be a single-valued and a multi-valued mapping such that both satisfy the condition $(C_\lambda )$, respectively, and $Fix(S)\cap Fix(T)\not =\emptyset$ satisfying $Tw=\{w\}$ for all $w \in Fix(S)\cap Fix(T)$. Let $\{x_{n}\}$ be the sequence of the modified Ishikawa iteration defined by (1). If $0<a \le \alpha _{n}, \beta _n\le b<1$ for some $a,b \in \mathbb R$, then $\lim \limits _{n\rightarrow \infty }\Vert Sy_{n}-x_{n}\Vert =0.$

Proof

Let $w \in Fix(S)\cap Fix(T)$.

Since for any $\lambda \in (0,1)$, $\lambda \Vert w -Sw\Vert =0\le \Vert y_{n}-w\Vert$, therefore using the definition of condition $(C_\lambda )$, we get $\Vert Sy_{n}-Sw\Vert \le \Vert y_{n}-w\Vert$. Similarly, since $\lambda {\text{ dist }} (w, Tw)=0 \le \Vert x_{n}-w\Vert$ then we get $H(Tx_{n},Tw)\le \Vert x_{n}-w\Vert$.

By Lemma 3.1, $\lim \limits _{n\rightarrow \infty }\Vert x_{n}-w\Vert$ exists, say c, and consider,

$$\begin{aligned} \Vert Sy_{n}-w\Vert&= \Vert Sy_{n}-Sw\Vert \\&\le \Vert y_{n}-w\Vert \\&= \Vert (1-\beta _{n})x_{n}+\beta _{n}z_{n}-w\Vert \\&\le (1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}{\text{ dist }}(z_{n},Tw)\\&\le (1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}H(Tx_{n},Tw)\\&\le (1-\beta _{n})\Vert x_{n}-w\Vert +\beta _{n}\Vert x_{n}-w\Vert \\&= \Vert x_{n}-w\Vert . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty }\Vert Sy_{n}-w\Vert \le \limsup \limits _{n\rightarrow \infty } \Vert y_{n}-w\Vert \le \limsup \limits _{n\rightarrow \infty }\Vert x_{n}-w\Vert =c. \end{aligned}$$

(2)

Further, we have

$$\begin{aligned} c&= \lim \limits _{n\rightarrow \infty }\Vert x_{n+1}-w\Vert \\&= \lim \limits _{n\rightarrow \infty }\Vert (1-\alpha _{n})x_{n}+\alpha _{n}Sy_{n}-w\Vert \\&= \lim \limits _{n\rightarrow \infty }\Vert \alpha _{n}Sy_{n}-\alpha _{n}w+x_{n}-\alpha _{n}x_{n}+\alpha _{n}w-w\Vert \\&= \lim \limits _{n\rightarrow \infty }\Vert \alpha _{n}(Sy_{n}-w)+(1-\alpha _{n})(x_{n}-w)\Vert . \end{aligned}$$

By Lemma 2.5, we obtain $\lim \limits _{n\rightarrow \infty }\Vert (Sy_{n}-w)-(x_{n}-w)\Vert =\lim \limits _{n\rightarrow \infty }\Vert Sy_{n}-x_{n}\Vert =0.$

$\square$

Lemma 3.3

Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space $X$, and let $S: K\rightarrow K$ and $T:K\rightarrow CB(K)$ be a single-valued and a multi-valued mapping such that both satisfy the condition $(C_\lambda )$, respectively, and $Fix(S)\cap Fix(T)\not =\emptyset$ satisfying $Tw=\{w\}$ for all $w \in Fix(S)\cap Fix(T)$. Let $\{x_n\}$ be the sequence of the modified Ishikawa iteration defined by (1). If $0 < a \le \alpha _{n},~\beta _{n}\le b<1$ for some $a,~b \in \mathbb R$, then $\lim \limits _{n\rightarrow \infty } \Vert x_{n}-z_{n}\Vert =0.$

Proof

Let $w \in Fix(S)\cap Fix(T).$

Since $\lambda \Vert w -Sw\Vert =0\le \Vert y_{n}-w\Vert$, we get $\Vert Sy_{n}-Sw\Vert \le \Vert y_{n}-w\Vert$. Similarly, since $\lambda dist (w,Tw)=0 \le \Vert x_{n}-w\Vert$ then we get $H(Tx_{n},Tw)\le \Vert x_{n}-w\Vert$. We put $\lim \limits _{n\rightarrow \infty }\Vert x_{n}-w\Vert =c.$ For $n\ge 0$, we have

$$\begin{aligned} \Vert x_{n+1}-w\Vert&= \Vert (1-\alpha _{n})x_{n}+\alpha _{n}Sy_{n}-w\Vert \\&= \Vert (1-\alpha _{n})x_{n}+\alpha _{n}Sy_{n}-(1-\alpha _{n})w-\alpha _{n}w\Vert \\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert Sy_{n}-w\Vert \\&= (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert Sy_{n}-Sw\Vert \\&\le (1-\alpha _{n})\Vert x_{n}-w\Vert +\alpha _{n}\Vert y_{n}-w\Vert \end{aligned}$$

and therefore

$$\begin{aligned} \Vert x_{n+1}-w\Vert -\Vert x_{n}-w\Vert \le \alpha _{n} (\Vert y_{n}-w\Vert -\Vert x_{n}-w\Vert ),\\ \frac{\Vert x_{n+1}-w\Vert -\Vert x_{n}-w\Vert }{\alpha _{n}} \le \Vert y_{n}-w\Vert -\Vert x_{n}-w\Vert . \end{aligned}$$

Since $0< a \le \alpha _n \le b< 1$, we have

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty } \Big \{\Big (\frac{\Vert x_{n+1}-w\Vert -\Vert x_{n}-w\Vert }{\alpha _{n}}\Big ) +\Vert x_{n}-w\Vert \Big \}\le \liminf \limits _{n\rightarrow \infty } \Vert y_{n}-w\Vert . \end{aligned}$$

It follows that

$$\begin{aligned} c\le \liminf \limits _{n\rightarrow \infty } \Vert y_{n}-w\Vert . \end{aligned}$$

Since, from (2) $\limsup \limits _{n\rightarrow \infty } \Vert y_n-w\Vert \le c$, hence we have

$$\begin{aligned} c&= \lim \limits _{n\rightarrow \infty } \Vert y_n - w\Vert \nonumber \\&= \lim \limits _{n\rightarrow \infty } \Vert (1-\beta _n)x_n + \beta _n z_n - w\Vert \\&= \lim \limits _{n\rightarrow \infty } \Vert (1 - \beta _n)(x_n - w) + \beta _n(z_n-w)\Vert .\nonumber \end{aligned}$$

(3)

Recall that $\Vert z_n-w\Vert = {\text{ dist }}(z_n, Tw) \le H(Tx_n, Tw) \le \Vert x_n -w\Vert .$ Thus, we have

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty } \Vert z_n - w\Vert \le \limsup \limits _{n\rightarrow \infty } \Vert x_n- w\Vert = c. \end{aligned}$$

Since $0 < a \le \beta _n\le b < 1$ and by Lemma 2.5 and (3), we obtain $\lim \limits _{n\rightarrow \infty } \Vert x_n-z_n\Vert = 0$.

$\square$

Lemma 3.4

Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space X. Let $S : K \rightarrow K$ be a single-valued mapping which satisfies conditions $(C_\lambda )$ and $(E)$ while $T : K \rightarrow CB(K)$ be a multi-valued mapping which satisfies condition $(C_{\lambda })$. Also, $Fix(S) \bigcap Fix(T) \not = \emptyset$ and $Tw=\{w\}$ for $w\in Fix(S) \bigcap Fix(T)$. Let $\{x_n\}$ be the sequence of the modified Ishikawa iteration defined by (1). If $0 < a \le \alpha _n, \beta _n \le b < 1$, then $\lim \limits _{n\rightarrow \infty } \Vert Sx_n-x_n\Vert = 0$.

Proof

As $S$ satisfies the condition $(E)$, there exists $\mu \ge 1,$ such that

$$\begin{aligned} \Vert Sx_n-x_n\Vert&= \Vert Sx_n- y_n + y_n -x_n\Vert \\&\le \Vert Sx_n - y_n\Vert + \Vert y_n-x_n\Vert \\&\le \mu \Vert Sy_n - y_n\Vert + 2 \Vert y_n - x_n\Vert \\&\le \mu \Vert Sy_n -x_n\Vert +(\mu + 2) \Vert y_n-x_n\Vert \\&= \mu \Vert Sy_n -x_n\Vert +(\mu + 2) \Vert (1-\beta _{n}) x_{n}+\beta _{n} z_{n} - x_n\Vert \\&= \mu \Vert Sy_n -x_n\Vert +(\mu + 2)\beta _{n}\Vert x_n-z_n\Vert , \end{aligned}$$

therefore,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert Sx_n-x_n\Vert \le \lim \limits _{n\rightarrow \infty } \mu \Vert Sy_n -x_n\Vert + \lim \limits _{n\rightarrow \infty } (\mu + 2)\beta _{n}\Vert x_n-z_n\Vert . \end{aligned}$$

Thus, by Lemma 3.2 and Lemma 3.3,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert Sx_n-x_n\Vert =0. \end{aligned}$$

Now, we are equipped to prove the our main results of this section. $\square$

Theorem 3.5. Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space $X$. Let $S : K\rightarrow K$ be a single-valued mapping which satisfies conditions $(C_\lambda )$ and $(E)$ while $T : K \rightarrow CB(K)$ be a multi-valued mapping which satisfies condition $(C_{\lambda })$. Also, $Fix(S) \bigcap Fix(T) \not = \emptyset$ and $Tw=\{w\}$ for $w\in Fix(S) \bigcap Fix(T)$. Let $\{x_n\}$ be the sequence of the modified Ishikawa iteration defined by (1). If $0 < a \le \alpha _n,~\beta _n\le b < 1$, then $\{x_{n_i}\} \rightarrow y$ for some subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $\{x_{n_i} \} \in Tx_{n_i}$ for all $n_i$ implies $y \in Fix(S) \bigcap Fix(T)$.

Proof

Assume that $\lim \limits _{i\rightarrow \infty } \Vert x_{n_i}- y\Vert = 0$. By Lemma 3.4, we obtain $0 = \lim \limits _{i\rightarrow \infty } \Vert Sx_{n_i}- x_{n_i}\Vert = \lim \limits _{i\rightarrow \infty } \Vert (I-S)(x_{n_i})\Vert$. Since $I-S$ is strongly demiclosed at 0, we have $(I - S)(y) = 0$. Thus $y = Sy$, i.e. $y\in Fix(S)$. Since $\lambda dist(x_{n_i}, Tx_{n_i}) = 0 \le \Vert x_{n_i} -y\Vert$, then we have $H(Tx_{n_i}, T y) \le \Vert x_{n_i} -y\Vert$. By Lemma 2.6, we have

$$\begin{aligned} dist(y, Ty)&\le \Vert y-x_{n_i}\Vert +dist(x_{n_i}, Tx_{n_i}) + H(Tx_{n_i}, Ty)\\&\le \Vert y -x_{n_i} \Vert + dist(x_{n_i}, Tx_{n_i}) + \Vert x_{n_i}- y\Vert ~\rightarrow 0 \end{aligned}$$

as $i\rightarrow \infty$. It follows that $y \in Fix(T)$. Thus $y \in Fix(S) \bigcap Fix(T)$. $\square$

Theorem 3.5

Let $K$ be a nonempty compact convex subset of a uniformly convex Banach space X. Let $S: K \rightarrow K$ be a single-valued mapping which satisfies conditions $(C_\lambda )$ and $(E)$ while $T : K \rightarrow CB(K)$ be a multi-valued mapping which satisfies condition $(C_{\lambda })$. Also, $Fix(S) \bigcap Fix(T) \not = \emptyset$ and $Tw=\{w\}$ for $w\in Fix(S) \bigcap Fix(T)$. Let $\{x_n\}$ be the sequence of the modified Ishikawa iteration defined by (1) with $0 < a \le \alpha _n,~\beta _n \le b < 1$. Then $\{x_n\}$ converges strongly to a common fixed point of $S$ and $T$.

Proof

Since $\{x_n\}$ is contained in $K$ which is compact, there exists a subsequence $\{x_{n_i} \}$ of $\{x_n\}$ such that $\{ x_{n_i} \}$ converges strongly to some point $y \in K$, that is, $\lim \limits _{i\rightarrow \infty } \Vert x_{n_i} - y\Vert = 0.$ By Theorem 2.7, we get $y \in Fix(S) \bigcap Fix(T)$, and by Lemma 3.1, we have that $\lim \limits _{ n\rightarrow \infty } \Vert x_n -y\Vert$ exists. It must be the case in which $\lim \limits _{ n\rightarrow \infty } \Vert x_n - y\Vert = \lim \limits _{i\rightarrow \infty } \Vert x_{n_i} - y\Vert = 0$. Thus, $\{x_n\}$ converges strongly to $y \in$ $Fix(S) \bigcap Fix(T)$. $\square$