Convergence theorems for finite family of a general class of multi-valued strictly pseudo-contractive map**s

Abstract

Let H be a real Hilbert space, K a nonempty subset of H, and $T:K\rightarrow \mathit{CB}(K)$ a multi-valued map**. Then T is called a generalized k-strictly pseudo-contractive multi-valued map** if there exists $k\in[0,1)$ such that, for all $x,y\in D(T)$, we have $D^{2}(Tx,Ty)\leq\|x-y\|^{2}+kD^{2}(Ax,Ay)$, where $A:=I-T$, and I is the identity operator on K. A Krasnoselskii-type algorithm is constructed and proved to be an approximate fixed point sequence for a common fixed point of a finite family of this class of maps. Furthermore, assuming existence, strong convergence to a common fixed point of the family is proved under appropriate additional assumptions.

1 Introduction

Let H be a real Hilbert space, $\mathit{CB}(H)$ denote the collection of nonempty, closed, and bounded subsets of H. For $A, B\in \mathit{CB}(H)$, the Hausdorff metric is defined by

$$D(A,B) = \max \Bigl\{ \sup_{a\in A} d(a,B), \sup _{b\in B} d(b,A) \Bigr\} . $$

Let $T : D(T) \subseteq H \rightarrow \mathit{CB}(H)$ be a multi-valued map** on H. A point $x\in D(T)$ is called a fixed point of T if $x \in Tx$. The fixed point set of T is denoted by $F(T) := \{x \in D(T) : x \in Tx\}$.

The study of fixed points for multi-valued nonexpansive map**s using Hausdorff metric was introduced by Markin [1], and studied extensively by Nadler [2]. Since then, many results have appeared in the literature (see, e.g., Nadler [2] and Panyanak [3], and the references contained in them). Many of these results have found nontrivial applications in pure and applied sciences. Examples of such applications are, in control theory, convex optimization, differential inclusions, and economics (especially in game theory and market economy). For early results involving fixed points of multi-valued map**s and their applications see, for example, Brouwer [4], Daffer and Kaneko [5], Downing and Kirk [6], Geanakoplos [7], Kakutani [8], Nash [9, 10]. For details on the applications of this type of map**s in nonsmooth differential equations, one may consult Chang [11], Chidume et al. [12], Deimling [13], Khan et al. [14, 15], Reich et al. [16–18], Song and Wnag [19] and the references therein.

In studying the equation $Au=0$, where A is a monotone operator defined on a real Hilbert space, Browder [20], introduced an operator T defined by $T:=I-A$, where I is the identity map** on H. He called such an operator a pseudo-contractive map**. It is easily seen that the zeros of A are precisely the fixed points of the pseudo-contractive map** T. It is well known that every nonexpansive map** is pseudo-contractive but the converse is not true. In fact, in general, pseudo-contractive map**s are not necessarily continuous.

Moreover, Browder and Petryshyn [21] introduced the subclass of single-valued pseudo-contractive maps given below.

Definition 1.1

Let K be a nonempty subset of a real Hilbert space H. A map $T:K\rightarrow H$ is called strictly pseudo-contractive if there exists $k\in[0,1)$ such that

$$ \|Tx-Ty\|^{2}\leq\|x-y\|^{2}+k\bigl\Vert (x-Tx)-(y-Ty)\bigr\Vert ^{2}, \quad \forall x,y\in K. $$

(1.1)

Since then, several extensions of this class of map**s to multi-valued cases have been defined and studied. For results on the approximation of common fixed points of families of multi-valued nonexpansive map**s (see, for example, Abbas et al. [22]), and for multi-valued strictly pseudo-contractive map**s see, Chidume et al. [12], Chidume and Ezeora [23], Ofoedu and Zegeye [24], Panyanak [3], Shahzad and Zegeye [25] and the references therein.

The class of multi-valued pseudo-contractive map**s introduced by Chidume et al. [12] is as follows.

Definition 1.2

Let H be a real Hilbert space and let K be a nonempty, closed, and convex subset of H. Let $T:K\rightarrow \mathit{CB}(K )$ be a map**. Then T is called a multi-valued k-strictly pseudo-contractive map** if there exists $k\in[0,1)$ such that, for all $x,y\in D(T)$, we have

$$ D^{2}(Tx,Ty)\leq\|x-y\|^{2}+k\bigl\Vert (x-u)-(y-v)\bigr\Vert ^{2} $$

(1.2)

for all $u\in Tx$, $v\in Ty$.

Remark 1.3

It is easy to see that inequality (1.2) is equivalent to

$$ D^{2}(Tx,Ty)\leq\|x-y\|^{2}+k \inf_{(u,v) } \bigl\Vert (x-u)-(y-v)\bigr\Vert ^{2} . $$

Using the above definition, Chidume et al. [12] proved the following theorem, which extends the result of Browder and Petryshyn [21].

Theorem 1.4

([12])

Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Suppose that $T:K\rightarrow \mathit{CB}(K)$ is a multi-valued k-strictly pseudo-contractive map** such that $F(T)\neq\emptyset$. Assume that $Tp=\{p\}$ for all $p\in F(T)$. Let $\{x_{n}\}$ be a sequence defined by $x_{0}\in K$,

$$x_{n+1}=(1-\lambda)x_{n}+\lambda y_{n}, $$

where $y_{n}\in Tx_{n}$ and $\lambda\in (0, 1-k)$. Then $\lim_{n\rightarrow\infty}d(x_{n}, Tx_{n})=0$.

Chidume and Ezeora [23] extended the theorems obtained in Chidume et al. [12] to a finite family $\{T_{i}, i=1,2,\ldots ,m\}$ of multi-valued $k_{i}$-strictly pseudo-contractive map**s using a Krasnoselskii-type algorithm.

More precisely, they obtained the following theorems.

Theorem 1.5

([23])

Let K be a nonempty, closed, and convex subset of a real Hilbert space H and $T_{i}:K \rightarrow \mathit{CB}(K)$ be a finite family of multi-valued $k_{i}$-strictly pseudo-contractive map**s, $k_{i} \in (0,1)$, $i = 1, \ldots ,m$, such that $\bigcap_{i=1}^{m}F(T_{i})\neq\emptyset$. Assume that, for $p\in\bigcap_{i=1}^{m}F(T_{i})$, $T_{i}p=\{p\}$.

Let $\{x_{n}\}$ be a sequence defined by $x_{0}\in K$,

$$x_{n+1} = \lambda_{0}x_{n} + \lambda_{1}y_{n}^{1}+ \cdots+\lambda_{m}y_{n}^{m}, $$

where $y_{n}^{i}\in T_{i}x_{n}$, $n\geq1$ and $\lambda_{i}\in(k,1)$, $i=0,1,\ldots ,m$, such that $\sum_{i=0}^{m}\lambda_{i}=1$ and $k:=\max\{k_{i}, i=1,\ldots ,m\}$. Then $\lim_{n\rightarrow\infty}d(x_{n}, T_{i}x_{n})=0$, $\forall i=1,\ldots,m$.

Theorem 1.6

([23])

Let K be a nonempty, closed, and convex subset of a real Hilbert space H and $T_{i}:K\rightarrow \mathit{CB}(K)$ be a finite family of multi-valued $k_{i}$-strictly pseudo-contractive map**s, $k_{i}\in(0,1)$, $i=1, \ldots ,m$, such that $\bigcap_{i=1}^{m}F(T_{i})\neq \emptyset$. Assume that, for $p\in \bigcap_{i=1}^{m}F(T_{i})$, $T_{i}p=\{p\}$ and $T_{i}$, $i=1,\ldots,m$, is hemicompact and continuous. Let $\{ x_{n}\}$ be a sequence defined by $x_{0}\in K$,

$$x_{n+1} = \lambda_{0}x_{n} + \lambda_{1}y_{n}^{1}+ \cdots+\lambda_{m}y_{n}^{m}, $$

where $y_{n}^{i}\in T_{i}x_{n}$, $n\geq1$ and $\lambda_{i}\in(k,1)$, $i=0,1,\ldots ,m$ such that $\sum_{i=0}^{m}\lambda_{i}=1$ with $k:=\max\{k_{i}, i=1,\ldots ,m\}$. Then the sequence $\{x_{n}\}$ converges strongly to an element of $\bigcap_{i=1}^{m}F(T_{i})$.

Chidume and Okpala [26] introduced the following definition for multi-valued k-strictly pseudo-contractive map**s.

Definition 1.7

([26])

Let H be a real Hilbert space and let K be a nonempty subset of H. Let $T:K\rightarrow \mathit{CB}(K )$ be a map**. Then T is called a generalized k-strictly pseudo-contractive multi-valued map** if there exists $k\in[0,1)$ such that, for all $x,y\in D(T)$, we have

$$ D^{2}(Tx,Ty)\leq\|x-y\|^{2}+kD^{2}(Ax,Ay), \quad \text{where } A:=I-T, $$

(1.3)

and I is the identity operator on K.

Remark 1.8

It is shown in [26] that every multi-valued k-strictly pseudo-contractive map as defined in [12] is a generalized k-strictly pseudo-contractive multi-valued map**. An example is given in [26] of a generalized k-strictly pseudo-contractive map** that is not a multi-valued k-strictly pseudo-copseudo-contractive map**. The definition given here appears to be more natural than that given in [12].

It is our purpose in this paper to prove that a Krasnoselskii-type algorithm, under appropriate conditions, converges strongly to a common fixed point of a finite family $\{T_{i}, i=1,2,\ldots,m\}$ of generalized $k_{i}$-strictly pseudo-contractive multi-valued map**s in a real Hilbert space. In the setting where the algorithms agree, our theorems generalize the results of Chidume et al. [12], and Chidume and Ezeora [23]. Moreover, they improve and extend to a finite family the results of Chidume and Okpala [26]. Also, several assumptions in the results of Chidume and Ezeora [23] (e.g., $\lambda_{i}\in(k,1)$, for all i and $T_{i}$ is continuous (see, e.g., [23], Theorem 2.4) and hemicompact for each i), are significantly weakened.

The rest of this paper is organized as follows. Some known results and useful lemmas are listed in Section 2. In Section 3, we state and prove our main theorem and the corollaries that follow from the theorem. In the last section, we show an illustrative example where our theorem is applicable.

2 Preliminaries

We first recall some definitions, notations, and results which will be needed in proving our main results:

• (i) $x_{n}\rightarrow x$: $\{x_{n}\}$ converges strongly to x as $n\rightarrow\infty$.

• (ii) H: a real Hilbert space with an induced norm $\|\cdot\|$.

• (iii) $F(T):=\{x\in K: x\in Tx\}$.

• (iv) $\mathit{CB}(K)$, is the collection of nonempty, closed, and bounded subsets of K.

To simplify notation, we shall denote $(D(A,B))^{2}$ by $D^{2}(A,B)$ for all A, B elements of $\mathit{CB}(H)$.

Definition 2.1

A multi-valued map** $T:K\subseteq H\rightarrow \mathit{CB}( K)$ is called Lipschitzian if there exists $L>0$ such that

$$ D(Tx,Ty)\leq L \|x-y\| $$

(2.1)

for each $x,y\in K$. If $L<1$ in inequality (2.1), the map** T is called a contraction, and if $L=1$, it is called nonexpansive.

We recall the following proposition.

Proposition 2.2

([26])

Let K be a nonempty subset of a real Hilbert space H and $T:K\rightarrow \mathit{CB}(K)$ be a generalized k-strictly pseudo-contractive multi-valued map**. Then T is Lipschitzian.

Remark 2.3

Since every Lipschitz map is continuous, we would not make any continuity assumption on our map** T throughout this paper.

Definition 2.4

A map $T:K\rightarrow \mathit{CB}(K)$ is said to be hemicompact if, for any sequence $\{x_{n}\}$ such that $\lim_{n\rightarrow\infty}d(x_{n}, Tx_{n})=0$, there exists a subsequence, say, $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $x_{n_{k}}\rightarrow p\in K$.

Remark 2.5

If K is compact, then every multi-valued map** $T:K\rightarrow \mathit{CB}(K)$ is hemicompact.

The following lemma will also be used in the sequel.

Lemma 2.6

([27])

Let $\{a_{n}\}$ be a sequence of nonnegative real numbers satisfying the following condition:

$$a_{n+1}\leq a_{n}+\sigma_{n},\quad n\geq0, $$

such that $\sum_{n=1}^{\infty}\sigma_{n}<\infty$. Then $\lim a_{n}$ exists. If, in addition, $\{a_{n}\}$ has a subsequence that converges to 0, then $a_{n}$ converges to 0 as $n\rightarrow\infty$.

Lemma 2.7

([23])

Let H be a real Hilbert space and let $\{x_{i}, i=1,2,\ldots,m\}\subseteq H$. For $\alpha_{i}\in(0,1)$, $i=1,2,\ldots,m$ such that $\sum_{i=1}^{m}\alpha_{i}=1$, the following identity holds:

$$\Biggl\Vert \sum_{i=1}^{m} \alpha_{i}x_{i}\Biggr\Vert ^{2}=\sum _{i=1}^{m}\alpha_{i}\|x_{i}\| ^{2}-\sum_{1\leq i< j\leq m}\alpha_{i} \alpha_{j}\|x_{i}-x_{j}\|^{2}. $$

The following properties of the Hausdorf distance were established in [26].

Lemma 2.8

([26])

Let E be a normed linear space, $B_{1}, B_{2}\in \mathit{CB}(E)$, and $x, y\in E$ arbitrary. The following hold:

• (a) $D(B_{1}, B_{2})=D(x+B_{1}, x+B_{2})$,

• (b) $D(B_{1}, B_{2})=D(-B_{1},-B_{2})$,

• (c) $D(x+B_{1}, y+B_{2})\leq\|x-y\|+D(B_{1},B_{2})$,

• (d) $D(\{x\},B_{1})=\sup_{b_{1}\in B_{1}}\|x-b_{1}\|$,

• (e) $D(\{x\}, B_{1})=D(0,x-B_{1})$.

3 Main results

In this section, we prove strong convergence theorems for a common fixed point of a finite family of generalized k-strictly pseudo-contractive multi-valued map**s in a real Hilbert space.

Henceforth, for any given finite family $\{T_{i}, i=1,\ldots,m\}$ of generalized $k_{i}$-strictly pseudo-contractive multi-valued map**s and arbitrary sequence $\{x_{n}\}\subseteq K$, let

$$S_{n}^{i}:= \biggl\{ y_{n}^{i}\in T_{i}x_{n}:D^{2}\bigl(\{x_{n}\}, T_{i}x_{n}\bigr)\leq\bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2}+\frac{1}{n^{2}} \biggr\} . $$

Certainly, $S_{n}^{i}$ is not empty for each $n\geq1$ since by Lemma 2.8(d), we have

$$D\bigl(\{x_{n}\}, T_{i}x_{n}\bigr)=\sup _{y_{n}\in T_{i}x_{n}}\|x_{n}-y_{n}\|. $$

We now state and prove our main theorem.

Theorem 3.1

Let K be a nonempty, closed, and convex subset of a real Hilbert space H. For $i=1,2,\ldots,m$, let $T_{i}:K\rightarrow \mathit{CB}(K)$ be a family of generalized $k_{i}$-strictly pseudo-contractive multi-valued map**s with $k_{i}\in(0,1)$. Suppose that $\bigcap_{i=1}^{m} F(T_{i})\neq \emptyset$ and assume that, for $p\in\bigcap_{i=1}^{m} F(T_{i})$, $T_{i}p=\{p\}$. Define a sequence $\{x_{n}\}$ by $x_{0}\in K$ arbitrary and

$$ x_{n+1}=(\lambda_{0})x_{n}+\sum _{i=1}^{m}\lambda_{i} y_{n}^{i}, $$

(3.1)

where $y_{n}^{i} \in S_{n}^{i}$, $\lambda_{0} \in(k,1)$, $\sum_{i=0}^{m}\lambda _{i}=1$, and $k:=\max\{k_{i}, i=1,2,\ldots,m\}$. Then, for each $i=1,2,\ldots,m$, $\lim_{n\rightarrow\infty}d(x_{n}, T_{i}x_{n})= 0$.

Proof

Let $p\in\bigcap_{i=1}^{m} F(T_{i})$. Then, using Lemma 2.7 together with Lemma 2.8(d) and (e), we have

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^{2} =& \Biggl\Vert \lambda_{0}(x_{n}-p)+\sum_{i=1}^{m} \lambda_{i} \bigl(y_{n}^{i}-p\bigr)\Biggr\Vert ^{2} \\ =&\lambda_{0}\Vert x_{n}-p\Vert ^{2}+ \sum _{i=1}^{m}\lambda_{i} \bigl\Vert y_{n}^{i}- p\bigr\Vert ^{2}-\sum _{i=1}^{m}\lambda_{0}\lambda_{i} \bigl\Vert x_{n}-y_{n}^{i}\bigr\Vert ^{2} \\ &{}-\sum_{1\leq i\leq j\leq m}\lambda_{i} \lambda_{j}\bigl\Vert y_{n}^{i}-y_{n}^{j} \bigr\Vert ^{2} \\ \leq&\lambda_{0}\Vert x_{n}-p\Vert ^{2}+\sum _{i=1}^{m}\lambda_{i} D^{2}(T_{i}x_{n}, T_{i}p)-\sum _{i=1}^{m}\lambda_{0}\lambda_{i} \bigl\Vert x_{n}-y_{n}^{i}\bigr\Vert ^{2} \\ \leq&\lambda_{0}\Vert x_{n}-p\Vert ^{2}+\sum _{i=1}^{m}\lambda_{i} \bigl(\Vert x_{n}-p\Vert ^{2}+k_{i}D^{2} \bigl(x_{n}-T_{i}x_{n},\{0\}\bigr)\bigr) \\ &{}-\sum _{i=1}^{m}\lambda_{0} \lambda_{i}\bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2} \\ =&\sum_{i=0}^{m}\lambda_{i} \Vert x_{n}-p\Vert ^{2}+\sum_{i=1}^{m} \lambda_{i} k_{i}D^{2}\bigl(\{ x_{n} \},T_{i}x_{n}\bigr)-\sum_{i=1}^{m} \lambda_{0}\lambda_{i}\bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2} \\ \leq&\sum_{i=0}^{m}\lambda_{i} \Vert x_{n}-p\Vert ^{2}+\sum_{i=1}^{m} \lambda_{i} k \biggl(\bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2}+\frac{1}{n^{2}}\biggr) \\ &{}-\sum _{i=1}^{m}\lambda_{0}\lambda_{i} \bigl\Vert x_{n}-y_{n}^{i}\bigr\Vert ^{2}, \quad \text{since }y_{n}^{i}\in S_{n}^{i} \\ \leq&\Vert x_{n}-p\Vert ^{2}+\frac{k}{n^{2}}-\sum _{i=1}^{m}\lambda_{i}( \lambda_{0}-k) \bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2}. \end{aligned}$$

Therefore,

$$ \|x_{n+1}-p\|^{2} \leq\|x_{n}-p\|^{2}+ \frac{k}{n^{2}}-\sum_{i=1}^{m}\lambda _{i}(\lambda_{0}-k) \bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2}, $$

(3.2)

and then

$$ \|x_{n+1}-p\|^{2}\leq\|x_{n}-p\|^{2}+ \frac{k}{n^{2}}. $$

(3.3)

Inequality (3.3) and Lemma 2.6 then show that the sequence $\{\|x_{n}-p\| \}$ has a limit and, therefore, $\{ x_{n}\}$ is bounded. Moreover, we have from inequality (3.2) that

$$ \sum_{i=1}^{m}\lambda_{i}( \lambda_{0}-k) \bigl\Vert x_{n}-y_{n}^{i} \bigr\Vert ^{2}\leq\frac{k}{n^{2}} +\| x_{n}-p \|^{2}-\|x_{n+1}-p\|^{2}, $$

and so

$$ \lambda_{i}(\lambda_{0}-k) \bigl\Vert x_{n}-y_{n}^{i}\bigr\Vert ^{2}\leq \frac{k}{n^{2}}+ \|x_{n}-p\|^{2}-\| x_{n+1}-p \|^{2}\rightarrow0 \quad (\text{as } n\rightarrow\infty) $$

for each $i=1,2,\ldots,m$. Thus, for each $i=1,2,\ldots,m$, $\lim_{n\rightarrow\infty}\|x_{n}-y_{n}^{i}\|=0$ and using the fact $d(x_{n},T_{i}x_{n})\leq\|x_{n}-y_{n}^{i}\|$, it follows that $\lim_{n\rightarrow \infty}d(x_{n}, T_{i}x_{n})=0$. □

Corollary 3.2

Let K be nonempty, closed, and convex subset of a real Hilbert space H. For $i=1,2,\ldots,m$, let $T_{i}:K\rightarrow \mathit{CB}(K)$ be a family of generalized $k_{i}$-strictly pseudo-contractive multi-valued map** with $\bigcap_{i=1}^{m} F(T_{i})\neq\emptyset$. Assume that, for $p\in\bigcap_{i=1}^{m} F(T_{i})$, $T_{i}p=\{p\}$, and that $T_{i_{0}}$ is hemicompact for some $i_{0}$. Then the sequence $\{x_{n}\}$ defined by (3.1) converges strongly to a common fixed point of $\{T_{i}, i=1,2,\ldots,m\}$.

Proof

By Theorem 3.1, $\lim_{n\rightarrow\infty}d(x_{n}, T_{i}x_{n})=0$ for each i and in particular $\lim_{n\rightarrow\infty }d(x_{n}, T_{i_{0}}x_{n})=0$. Since $T_{i_{0}}$ is hemicompact, let $\{x_{n_{j}}\}$ be a subsequence of $\{x_{n}\}$ such that $x_{n_{j}}\rightarrow q$ as $j\rightarrow\infty$. For each $i=1,2,\ldots,m$, choose $y_{n_{j}}^{i}\in T_{i}x_{n_{j}}$ such that $\|x_{n_{j}}-y_{n_{j}}^{i}\|\leq d(x_{n_{j}}, T_{i}x_{n_{j}})+\frac{1}{j}$. Then

$$\begin{aligned} d(q,T_{i}q)&\leq\|q-x_{n_{j}}\|+\bigl\Vert x_{n_{j}}-y_{n_{j}}^{i}\bigr\Vert +d \bigl(y_{n_{j}}^{i},T_{i}q\bigr) \\ &\leq \|q-x_{n_{j}}\|+d(x_{n_{j}}, T_{i}x_{n_{j}})+ \frac{1}{j}+D(T_{i}x_{n_{j}}, T_{i}q) \\ &\leq \|q-x_{n_{j}}\|+d(x_{n_{j}}, T_{i}x_{n_{j}})+ \frac{1}{j}+\frac{1+\sqrt {k}}{1-\sqrt{k}} \|x_{n_{j}}-q \|. \end{aligned}$$

Thus, taking the limits on the right-hand side as $j\rightarrow\infty$, we have $d(q,T_{i}q)=0$. Since $T_{i}q$ is closed, $q\in T_{i}q$ for each i and therefore $q\in\bigcap_{i=1}^{m}T_{i}q$. Moreover, $x_{n_{j}}\rightarrow q$ as $j\rightarrow\infty$ gives $\| x_{n_{j}}- q\| \rightarrow0$ as $j\rightarrow\infty$. Thus, using inequality (3.3) and Lemma 2.6, $\lim_{n\rightarrow\infty}\| x_{n}-q\|=0$. Therefore $\{x_{n}\}$ converges strongly to a common fixed point q of the maps $T_{i}$, as claimed. □

Remark 3.3

Corollary 3.2 is a significant improvement and generalization of Theorem 2.4 of [23] in the following sense:

• (i) The theorem is proved for the much larger class of generalized k-strictly pseudo-contractive multi-valued map**s.

• (ii) No continuity assumption is imposed on our maps.

• (iii) Only one arbitrary map is required to be hemicompact.

• (iv) The condition $\lambda_{i}\in(k,1)$, for all i is replaced by the weaker condition $\lambda_{0}\in(k,1)$.

Furthermore, the condition $y_{n}^{i}\in S_{n}^{i}$ is more readily applicable than requiring that Tx is proximinal and weakly closed for each x, and then finding $y_{n}\in Tx_{n}$ such that $\|y_{n}-x_{n}\|=d(x_{n}, Tx_{n})$ at each iterative step, as it is in [3] and in many other results.

Corollary 3.4

Let K be nonempty, compact and convex subset of a real Hilbert space H, and for $i=1,\ldots,m$, let $T_{i}:K\rightarrow \mathit{CB}(K)$ be a family of generalized $k_{i}$-strictly pseudo-contractive multi-valued map**s with $\bigcap_{i=1}^{m} F(T_{i})\neq\emptyset$. Assume that, for $p\in\bigcap_{i=1}^{m} F(T_{i})$, $T_{i}p=\{p\}$. Then the sequence $\{x_{n}\}$ defined by (3.1) converges strongly to a common fixed point of the maps $T_{i}$.

Proof

Since every multi-valued map defined on a compact set is necessarily hemicompact, $T_{i}:K\rightarrow \mathit{CB}(K)$ is hemicompact for each i. Thus, by Corollary 3.2, $\{x_{n}\}$ converges strongly to some $p\in\bigcap_{i=1}^{m} F(T_{i})$. □

Remark 3.5

Our theorem and corollaries improve convergence theorems for multi-valued nonexpansive map**s in [3, 12, 14, 19, 22, 23, 26, 28], in the following sense:

• (i) The class of map**s considered in this paper contains the class of multi-valued k-strictly pseudo-contractive map**s as special case, which itself properly contains the class of multi-valued nonexpansive maps.

• (ii) The algorithm here is of Krasnoselskii-type, which is well known to have a geometric order of convergence.

• (iii) The method of proof used here is of independent interest as it does not assume that Tx is weakly closed for each $x\in K$, or proximinal subset of K, as imposed in [12] and [23].

• (iv) In the case where we have only one map, $m=1$, we recover all the results of Chidume and Okpala [26].

4 An example

We finally give an example where the theorems are applicable. For the example, we shall need the following lemma.

Lemma 4.1

Let a, b, c be real numbers such that $0\leq a\leq bc$, $c>0$. Then

$$ (a-b)^{2}\leq b^{2}+ \biggl(\frac{c-2}{c} \biggr)a^{2}. $$

(4.1)

Proof

The proof is trivially established as follows:

$$\begin{aligned}& 0\leq a\leq bc,\quad c>0 \\& \quad \Rightarrow\quad a^{2}\leq abc \\& \quad \Rightarrow\quad \frac{a^{2}}{c}\leq ab \\& \quad \Rightarrow\quad -2ab\leq-\frac{2a^{2}}{c} \\& \quad \Rightarrow\quad a^{2}-2ab+b^{2}\leq a^{2} - \frac{2a^{2}}{c}+b^{2} \\& \quad \Rightarrow\quad (a-b)^{2}\leq b^{2}+ \biggl( \frac{c-2}{c} \biggr)a^{2}. \end{aligned}$$

 □

Remark 4.2

If we take $c=4$ in this lemma, we recover Lemma 3.5 of [26].

Example 4.3

Let $T_{i}:\mathbb{R}\rightarrow \mathit{CB}(\mathbb{R})$ be defined by

$$ T_{i}x:= \textstyle\begin{cases} {[(-1-\alpha_{i})x, (-1+\alpha_{i})x]}, &x>0, \\ \{0\}, & x=0, \\ {[(-1+\alpha_{i})x, (-1-\alpha_{i})x]}, &x< 0, \end{cases} $$

(4.2)

where $\alpha_{i}=\frac{i}{5}$, $i=1,2,3,4,5$. We have

$$ x-T_{i}x:= \textstyle\begin{cases} {[(2-\alpha_{i})x, (2+\alpha_{i})x]}, &x>0, \\ \{0\}, & x=0, \\ {[(2+\alpha_{i})x, (2-\alpha_{i})x]}, &x< 0. \end{cases} $$

Then, for nonzero $x,y\in\mathbb{R}$,

$$D(T_{i}x,T_{i}y)=|x-y|+\alpha_{i}\bigl\vert \vert x\vert -\vert y\vert \bigr\vert $$

and

$$D(x-T_{i}x,y-T_{i}y)=2|x-y|+\alpha_{i}\bigl\vert \vert x\vert -\vert y\vert \bigr\vert . $$

Now, set

$$a:=D(x-T_{i}x, y-T_{i}y);\qquad b:=|x-y|. $$

Then $a-b=D(T_{i}x,T_{i}y)$ and

$$\begin{aligned} a&=2|x-y|+\alpha_{i}\bigl\vert \vert x\vert -\vert y\vert \bigr\vert \\ &\leq(2+\alpha_{i})|x-y|. \end{aligned}$$

Therefore, taking $c=c_{i}:=2+\alpha_{i}$, $i=1,2,3,4,5$, we have $\frac {c_{i}-2}{c_{i}}=\frac{\alpha_{i}}{2+\alpha_{i}}$, and by Lemma 4.1, we obtain

$$ D^{2}(T_{i}x,T_{i}y)\leq|x-y|^{2}+ \frac{\alpha_{i}}{2+\alpha_{i}}D(x-T_{i}x,y-T_{i}y). $$

Thus, each $T_{i}$, $i=1,\ldots,5$, is a generalized $k_{i}$-strictly pseudo-contractive multi-valued map** with $k_{i}=\frac{\alpha _{i}}{2+\alpha_{i}}\in(0,1)$. Moreover, $p\in T_{i}p$ if and only if $p=0$. Thus, for $p\in\bigcap_{i=1}^{5}F(T_{i}p)$, $T_{i}p=\{p\}$.

It is also interesting to note that this family of map**s does not belong to the class discussed in [23]. We prove this by contradiction. For $i=5$, $\alpha_{5}=1$ we have

$$ T_{5}x:= \textstyle\begin{cases} {[-2x, 0]}, &x>0, \\ \{0\}, & x=0, \\ {[0, -2x]}, &x< 0. \end{cases} $$

(4.3)

Observe that $0\in T_{5}x$ for all $x\in\mathbb{R}$. Now, suppose for contradiction that $T_{5}$ is a multi-valued k-strictly pseudo-contractive map** in the sense of Definition 1.2, i.e., $T_{5}$ satisfies inequality (1.2) for some $k\in (0,1)$. Let $x=1$, $y=2$, $u=v=0$. Then $u\in[-1,0]= T_{5}x$ and $v\in [-2,0]= T_{5}y$, and

$$\begin{aligned}& |x-y|=1=\bigl\vert \vert x\vert -\vert y\vert \bigr\vert , \qquad D(T_{5}1, T_{5}2)=2 , \\& \bigl\vert (x-u)-(y-v)\bigr\vert =|x-y|=1. \end{aligned}$$

Thus, we have

$$ 4=D^{2}(Tx,Ty)\leq|x-y|^{2}+k\bigl\vert (x-u)-(y-v)\bigr\vert \leq2. $$

Since this is impossible, $T_{5}$ is not a k-strictly pseudo-contractive multi-valued map**, in the sense of Definition 1.2, for any $k\in(0,1)$.