Convergence theorems for new classes of multivalued hemicontractive-type mappings

Abstract

Weak and strong convergence theorems are proved in Hilbert spaces for new classes of multivalued demicontractive-type and hemicontractive-type mappings which are related to the class of multivalued pseudocontractive-type mappings studied by Isiogugu (Fixed Point Theory Appl. 2013:61, 2013). Thus our results extend and improve several corresponding results in the contemporary literature.

MSC:47H10, 54H25.

1 Introduction

Let E be a normed space. A subset K of E is called proximinal if for each $x\in E$ there exists $k\in K$ such that

$$\parallel x-k\parallel =inf\{\parallel x-y\parallel :y\in K\}=d(x,K).$$

It is well known that every closed convex subset of a uniformly convex Banach space is proximinal. For a nonempty set E, we shall denote the family of all nonempty proximinal subsets of E by $P(E)$, the family of all nonempty closed and bounded subsets of E by $\mathit{CB}(E)$, the family of all nonempty closed, convex, and bounded subsets of E by $\mathit{CVB}(E)$, the family of all nonempty closed subsets of E by $C(X)$, the family of all nonempty subsets of E by $2^E$, the identity on E by I, the weak topology of E by $\sigma (E,E^{\ast })$, and the norm (or strong) topology of E by $(E,\parallel \cdot \parallel )$.

Let H denote the Hausdorff metric induced by the metric d on E, that is, for every $A,B\in 2^E$,

$$H(A,B)=max\{\underset{a\in A}{sup}d(a,B),\underset{b\in B}{sup}d(b,A)\}.$$

If $A,B\in \mathit{CB}(E)$, then

$$H(A,B)=inf\{ϵ>0:A\subseteq N(ϵ,B)\text{ and }B\subseteq N(ϵ,A)\},$$

where $N(ϵ,C)={\bigcup }_{c\in C}\{x\in E:d(x,c)<ϵ\}$. Let E be a normed space. Let $T:D(T)\subseteq E\to 2^E$ be a multivalued mapping on E. A point $x\in D(T)$ is called a fixed point of T if $x\in Tx$. The set $F(T)=\{x\in D(T):x\in Tx\}$ is called the fixed point set of T. A point $x\in D(T)$ is called a strict fixed point of T if $Tx=\{x\}$. The set $F_s(T)=\{x\in D(T):Tx=\{x\}\}$ is called the strict fixed point set of T. A multivalued mapping $T:D(T)\subseteq E\to 2^E$ is called L-Lipschitzian if there exists $L\ge 0$ such that for all $x,y\in D(T)$

$$H(Tx,Ty)\le L\parallel x-y\parallel .$$

(1.1)

In (1.1) if $L\in [0,1)$, T is said to be a contraction, while T is nonexpansive if $L=1$. T is called quasi-nonexpansive if $F(T)=\{x\in D(T):x\in Tx\}\ne \mathrm{\varnothing }$ and for all $p\in F(T)$,

$$H(Tx,Tp)\le \parallel x-p\parallel .$$

(1.2)

Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive.

Several authors have studied various classes of multivalued mappings. In [1], Shahzad and Zegeye studied certain classes of multivalued nonself mappings in Banach spaces and constructed an appropriate net which converges strongly to a fixed point of the classes of the mappings. Recently, Isiogugu [2] introduced new classes of multivalued mappings as follows.

Definition 1.1 ([2])

Let X be a normed space. A multivalued mapping $T:D(T)\subseteq X\to 2^X$ is said to be k-strictly pseudocontractive-type in the sense of Browder and Petryshyn [3] if there exists $k\in [0,1)$ such that given any $x,y\in D(T)$ and $u\in Tx$, there exists $v\in Ty$ satisfying $\parallel u-v\parallel \le H(Tx,Ty)$ and

$$H^2(Tx,Ty)\le {\parallel x-y\parallel }^2+k{\parallel x-u-(y-v)\parallel }^2.$$

(1.3)

If $k=1$ in (1.3) T is said to be a pseudocontractive-type mapping. T is called nonexpansive-type if $k=0$. Clearly, every multivalued nonexpansive mapping is nonexpansive-type mapping.

From the definitions, it is clear that every multivalued nonexpansive-type mapping is k-strictly pseudocontractive-type and every k-strictly pseudocontractive-type mapping is pseudocontractive-type. Examples to show that the class of nonexpansive-type mappings is properly contained in the class of k-strictly pseudocontractive-type mappings and that the class of k-strictly pseudocontractive-type mappings is properly contained in the class of pseudocontractive-type mappings were given in [2]. The following theorems were also proved in [2].

Theorem 1.1 Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that $T:K\to P(K)$ is a k-strictly pseudocontractive-type mapping from K into the family of all proximinal subsets of K with $k\in (0,1)$ such that $F(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for all $p\in F(T)$. Suppose $(I-T)$ is weakly demiclosed at zero. Then the Mann-type sequence defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{y}_n$$

converges weakly to $q\in F(T)$, where $y_n\in T{x}_n$ with $\parallel x_n-y_n\parallel =d(x_n,T{x}_n)$ and ${\alpha }_n$ is a real sequence in $(0,1)$ satisfying: (i) ${\alpha }_n\to \alpha <1-k$; (ii) $\alpha >0$; (iii) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n(1-{\alpha }_n)=\mathrm{\infty }$.

Theorem 1.2 Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that $T:K\to P(K)$ is an L-Lipschitzian pseudocontractive-type mapping from K into the family of all proximinal subsets of K such that $F(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for all $p\in F(T)$. Suppose for any pair $x,y\in K$ and $u\in Tx$ with $\parallel x-u\parallel =d(x,Tx)$, there exists $v\in Ty$ with $\parallel y-v\parallel =d(y,Ty)$ satisfying the conditions of Definition  1.1. Suppose T satisfies condition (1) (i.e., if there exists a nondecreasing function $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $f(0)=0$ and $f(r)>0$ for all $r\in (0,\mathrm{\infty })$ such that $d(x,Tx)\ge f(d(x,F(T)))$, $\mathrm{\forall }x\in K$. Then the Ishikawa sequence defined by

$$\{\begin{array}{l}{y}_n=(1-{\beta }_n)x_n+{\beta }_n{u}_n,\\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{w}_n\end{array}$$

(1.4)

converges strongly to $p\in F(T)$, where $u_n\in T{x}_n$ with $\parallel x_n-u_n\parallel =d(x_n,T{x}_n)$, $w_n\in T{y}_n$ with $\parallel y_n-w_n\parallel =d(y_n,T{y}_n)$ satisfying the conditions in Definition  1.1 and $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are real sequences satisfying: (i) $0\le {\alpha }_n\le {\beta }_n<1$; (ii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n=\alpha >0$; (iii) ${sup}_{n\ge 1}{\beta }_n\le \beta \le \frac{1}{\sqrt{1+L^2}+1}$.

In [4], Chidume et al. also considered a class of multivalued k-strictly pseudocontractive mappings defined as follows.

Let H be a real Hilbert space. A multivalued mapping $T:D(T)\subseteq H\to \mathit{CB}(H)$ is said to be k-strictly pseudocontractive if there exists $k\in (0,1)$ such that for all $x,y\in D(T)$ one has

$$H^2(Tx,Ty)\le {\parallel x-y\parallel }^2+k{\parallel x-u-(y-v)\parallel }^2,\mathrm{\forall }u\in Tx,v\in Ty.$$

If $k=1$, T is said to be pseudocontractive mapping. They constructed a Mann-type iteration scheme which is an approximate fixed point sequence and obtain some strong convergence theorems for the class of k-strictly pseudocontractive mappings.

The following example shows that the class of multivalued pseudocontractive-type mappings considered by Isiogugu [2] is not a subclass of the multivalued pseudocontractive mappings considered by Chidume et al. [4].

Example 1.1 Let $X=\mathbb{R}$ (the reals with usual metric). Define $T:[0,\mathrm{\infty })\to \mathit{CB}(\mathbb{R})$ by

$$Tx=[-\frac{5x}{2},-2x].$$

(1.5)

It was shown in [2] that T is k-strictly pseudocontractive-type mapping hence pseudocontractive-type. However, for $x=3$, $y=2$ if we choose $u=-6\in Tx$ and $v=-5\in Ty$ then $H^2(Tx,Ty)=\frac{25}{4}$ and ${\parallel x-y\parallel }^2+{\parallel x-u-(y-v)\parallel }^2=5$. Consequently,

$$H^2(Tx,Ty)>{\parallel x-y\parallel }^2+{\parallel x-u-(y-v)\parallel }^2,$$

which implies that T is not pseudocontractive and hence not k-strictly pseudocontractive mapping in the sense of Chidume et al. [4].

It is our purpose in this work to introduce and study new classes of multivalued demicontractive-type and hemicontractive-type mappings which are more general than the class of multivalued quasi-nonexpansive mappings and are also related to the multivalued k-strictly pseudocontractive-type and pseudocontractive-type mappings of Isiogugu [2], single-valued mappings of Browder and Petryshyn [3], Hicks and Kubicek [5] and Naimpally and Singh [6]. We also prove weak and strong convergence theorems for approximation of fixed points of our classes of mappings.

2 Preliminaries

We shall need the following definitions and lemmas.

Definition 2.1 (see, e.g., [7])

Let E be a Banach space. Let $T:D(T)\subseteq E\to 2^E$ be a multivalued mapping. $I-T$ is said to be strongly demiclosed at zero if for any sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}\subseteq D(T)$ such that $x_n$ converges strongly to p and a sequence $\{y_n\}$ with $y_n\in T{x}_n$ for all $n\in \mathbb{N}$ such that $\{x_n-y_n\}$ converges strongly to zero, then $p\in Tp$ (i.e., $0\in (I-T)p$).

Observe that if T is a multivalued Lipschitzian mapping, then $I-T$ is strongly demiclosed.

Definition 2.2 (see, e.g., [7, 8])

Let E be a Banach space. Let $T:D(T)\subseteq E\to 2^E$ be a multivalued mapping. $I-T$ is said to be weakly demiclosed at zero if for any sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}\subseteq D(T)$ such that $\{x_n\}$ converges weakly to p and a sequence $\{y_n\}$ with $y_n\in T{x}_n$ for all $n\in \mathbb{N}$ such that $\{x_n-y_n\}$ converges strongly to zero. Then $p\in Tp$ (i.e., $0\in (I-T)p$).

Definition 2.3 (see, e.g., [7, 8])

Let E be a Banach space. Let $T:D(T)\subseteq E\to 2^E$ be a multivalued mapping. The graph of $I-T$ is said to be closed in $\sigma (E,E^{\ast })\times (E,\parallel \cdot \parallel )$ (i.e., $I-T$ is weakly demiclosed or demiclosed) if for any sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}\subseteq D(T)$ such that $\{x_n\}$ converges weakly to p and a sequence $\{y_n\}$ with $y_n\in T{x}_n$ for all $n\in \mathbb{N}$ such that $\{x_n-y_n\}$ converges strongly to y. Then $y\in (I-T)p$ (i.e., $y=p-v$ for some $v\in Tp$).

Definition 2.4 A Banach X is said to satisfy Opial’s condition if whenever a sequence $\{x_n\}$ converges weakly to $x\in X$ then it is the case that

$$lim\hspace{0.17em}inf\parallel x_n-x\parallel <lim\hspace{0.17em}inf\parallel x_n-y\parallel ,$$

for all $y\in X$, $y\ne x$.

Definition 2.5 ([9])

A multivalued mapping $T:K\to P(K)$ is said to satisfy condition (1) (see for example [9]) if there exists a nondecreasing function $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $f(0)=0$ and $f(r)>0$ for all $r\in (0,\mathrm{\infty })$ such that

$$d(x,Tx)\ge f\left(d(x,F(T))\right),\mathrm{\forall }x\in K.$$

Lemma 2.1 ([10])

Let $\{a_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be sequences of nonnegative real numbers satisfying the following relation:

$$a_{n+1}\le (1+{\beta }_n)a_n+{\gamma }_n,n\ge n_0,$$

where $n_0$ is a nonnegative integer. If $\sum {\beta }_n<\mathrm{\infty }$, $\sum {\gamma }_n<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

Lemma 2.2 ([11])

Let K be a normed space. Let $T:K\to P(K)$ be a multivalued mapping and $P_T(x)=\{y\in Tx:\parallel x-y\parallel =d(x,Tx)\}$. Then the following are equivalent:

• (1) $x\in Tx$;

• (2) $P_{T}x=\{x\}$;

• (3) $x\in F(P_T)$.

Moreover, $F(T)=F(P_T)$.

Lemma 2.3 ([12])

Let $A,B\in \mathit{CB}(X)$ and $a\in A$. If $\gamma >0$, then there exists $b\in B$ such that

$$d(a,b)\le H(A,B)+\gamma .$$

3 Main results

We now introduce the new classes of multivalued demicontractive-type and hemicontractive-type mappings and prove some convergence theorems for these classes of mappings.

Definition 3.1 Let X be a real normed space. A mapping $T:D(T)\subseteq X\to 2^X$ is said to be demicontractive in the terminology of Hicks and Kubicek [5] if $F(T)\ne \mathrm{\varnothing }$ and for all $p\in F(T)$, $x\in D(T)$ there exists $k\in [0,1)$ such that

$$H^2(Tx,Tp)\le {\parallel x-p\parallel }^2+k{d}^2(x,Tx),$$

(3.1)

where $H^2(Tx,Tp)={[H(Tx,Tp)]}^2$ and $d^2(x,p)={[d(x,p)]}^2$.

If $k=1$ in (3.1) then T is called a hemicontractive mapping.

The following are some examples of demicontractive mappings.

Example 3.1 Every multivalued quasi-nonexpansive mapping is demicontractive.

Example 3.2 Let X be a normed space. Suppose that T is a multivalued mapping such that $F(T)\ne \mathrm{\varnothing }$ and that $P_T$ is a k-strictly pseudocontractive-type mapping; then $P_T$ is demicontractive.

Example 3.3 Let X be a normed space. Let $T:D(T)\subseteq X\to P(X)$ be a multivalued k-strictly pseudocontractive-type with a nonempty fixed point set. Suppose $Tp=\{p\}$ for all $p\in F(T)$; then for any $x\in D(T)$, $p\in F(T)$ and $u\in Tx$ with $\parallel u-x\parallel =d(x,Tx)$ we have

$$H^2(Tx,Tp)\le {\parallel x-p\parallel }^2+k{\parallel x-u\parallel }^2={\parallel x-p\parallel }^2+k{d}^2(x,Tx);$$

therefore, T is demicontractive-type.

Example 3.4 Let $X=\mathbb{R}$ (the reals with usual metric). Define $T:\mathbb{R}\to 2^{\mathbb{R}}$ by

$$Tx=\{\begin{array}{ll}[-\frac{3x}{2},-2x],& x\in (-\mathrm{\infty },0],\\ [-2x,-\frac{3x}{2}],& x\in (0,\mathrm{\infty }).\end{array}$$

(3.2)

Then $F(T)=\{0\}$. For each $x\in (-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$,

$$\begin{array}{r}{H}^2(Tx,T0)={|-2x-0|}^2=4{|x-0|}^2={|x-0|}^2+3{|x|}^2,\\ d^2(x,Tx)=|x-(-\frac{3x}{2}){|}^2=|\frac{5x}{2}{|}^2=\frac{25}{4}{|x|}^2.\end{array}$$

Therefore,

$$H^2(Tx,T0)={|x-0|}^2+3{|x|}^2={|x-0|}^2+\frac{12}{25}{d}^2(x,Tx)\le {|x-0|}^2+k{d}^2(x,Tx).$$

Consequently, T is demicontractive-type with $k=\frac{12}{25}$. It then follows that T is hemicontractive. Observe that T is not quasi-nonexpansive so that the class of multivalued quasi-nonexpansive mappings is properly contained in the class of multivalued demicontractive-type mappings.

Next is an example of a multivalued mapping T with $F(T)\ne \mathrm{\varnothing }$, $Tp=\{p\}$ for all $p\in Tp$ for which $P_T$ is a demicontractive-type but not a k-strictly pseudocontractive-type mapping.

Example 3.5 Let $X=\mathbb{R}$ (the reals with usual metric). Define $T:[-1,1]\to 2^{[-1,1]}$ by

$$Tx=\{\begin{array}{ll}[-1,\frac{2}{3}xsin\frac{1}{x}],& x\in (0,1],\\ \{0\},& x=0,\\ [\frac{2}{3}xsin\frac{1}{x},1],& x\in [-1,0).\end{array}$$

(3.3)

Then $F(T)=\{0\}$. For each $x\in [-1,1]$,

$$P_{T}x=\{\begin{array}{ll}\{\frac{2}{3}xsin\frac{1}{x}\},& x\ne 0,\\ \{0\},& x=0,\end{array}$$

(3.4)

which is demicontractive-type but not k-strictly pseudocontractive-type (see for example [5]).

The following example shows that the class of demicontractive mapping is properly contained in the class of hemicontractive mappings.

Example 3.6 Let $X=\mathbb{R}$ (the reals with the usual metric). Define $T:\mathbb{R}\to 2^{\mathbb{R}}$ by

$$Tx=\{\begin{array}{ll}[-\sqrt{2}x,0],& x\in [0,\mathrm{\infty }),\\ [0,-\sqrt{2}x],& x\in (-\mathrm{\infty },0).\end{array}$$

(3.5)

Then $F(T)=\{0\}$. For each $x\in (-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$,

$$\begin{array}{c}{H}^2(Tx,T0)=|\sqrt{2}x-0{|}^2=2{|x-0|}^2={|x-0|}^2+{|x-0|}^2,\hfill \\ d^2(x,Tx)={|x-0|}^2={|x-0|}^2.\hfill \end{array}$$

Therefore,

$$H^2(Tx,T0)\le {|x-0|}^2+{|x-0|}^2={|x-0|}^2+d^2(x,Tx)>{|x-0|}^2+k{d}^2(x,Tx),$$

(3.6)

$\mathrm{\forall }x\in \mathbb{B}$ and $\mathrm{\forall }k\in [0,1)$. Therefore, T is hemicontractive but not demicontractive.

Other examples of hemicontractive mappings include the following.

Example 3.7 Let X be a normed space. Suppose T is a multivalued mapping such that $F(T)\ne \mathrm{\varnothing }$ and $P_T$ is pseudocontractive-type mapping; then $P_T$ is hemicontractive.

Example 3.8 Let X be a normed space. Let $T:D(T)\subseteq X\to P(X)$ be a multivalued pseudocontractive-type with a nonempty fixed point set. Suppose $Tp=\{p\}$ for all $p\in F(T)$; then for any $x\in D(T)$, $p\in F(T)$ and $u\in Tx$ with $\parallel u-x\parallel =d(x,Tx)$ we have

$$H^2(Tx,Tp)\le {\parallel x-p\parallel }^2+{\parallel x-u\parallel }^2={\parallel x-p\parallel }^2+d^2(x,Tx).$$

The following lemma shows that Lemma 2.3 is also valid for all $A,B\in P(E)$ and $\gamma =0$.

Lemma 3.1 Let E be a metric space. If $A,B\in P(E)$ and $a\in A$, then it is a simple consequence of the Hausdorff metric H that there exists $b\in B$ such that

$$d(a,b)\le H(A,B).$$

(3.7)

Proof Let E be a metric space and $P(E)$ be the family of all nonempty proximinal subsets of E. Let $A,B\in P(E)$ and $a\in A$. Since B is proximinal, there exists $b_a\in B$ such that

$$d(a,B)=d(a,b_a).$$

Observe that

$$\begin{array}{rcl}H(A,B)& =& max\{\underset{u\in A}{sup}d(u,B),\underset{v\in B}{sup}d(v,A)\}\\ \ge & \underset{u\in A}{sup}d(u,B)\ge d(a,B)=d(a,b_a).\end{array}$$

Hence the result follows. □

Remark 3.1 Lemma 3.1 holds if E is a reflexive real Banach space and $P(E)$ is replaced with $\mathit{CB}(K)$ with B weakly closed (see for example [4]).

We now prove the following theorems.

Theorem 3.1 Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that $T:K\to P(K)$ is a demicontractive mapping from K into the family of all proximinal subsets of K with $k\in (0,1)$ and $T(p)=\{p\}$ for all $p\in F(T)$. Suppose $(I-T)$ is weakly demiclosed at zero. Then the Mann type sequence defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{y}_n,$$

(3.8)

converges weakly to $q\in F(T)$, where $y_n\in T{x}_n$ and ${\alpha }_n$ is a real sequence in $(0,1)$ satisfying: (i) ${\alpha }_n\to \alpha <1-k$; (ii) $\alpha >0$.

Proof Using the well-known identity:

$${\parallel tx+(1-t)y\parallel }^2=t{\parallel x\parallel }^2+(1-t){\parallel y\parallel }^2-t(1-t){\parallel x-y\parallel }^2,$$

which holds for all $x,y\in H$ and for all $t\in [0,1]$, we obtain

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& =& {\parallel (1-{\alpha }_n)x_n+{\alpha }_n{y}_n-p\parallel }^2\\ =& {\parallel (1-{\alpha }_n)(x_n-p)+{\alpha }_n(y_n-p)\parallel }^2\\ =& (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{\parallel y_n-p\parallel }^2-{\alpha }_n(1-{\alpha }_n){\parallel x_n-y_n\parallel }^2\\ \le & (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{H}^2(T{x}_n,Tp)-{\alpha }_n(1-{\alpha }_n){\parallel x_n-y_n\parallel }^2\\ \le & (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n[{\parallel x_n-p\parallel }^2+k{d}^2(x_n,T{x}_n)]\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-y_n\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2+{\alpha }_{n}k{\parallel x_n-y_n\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-y_n\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-{\alpha }_n(1-({\alpha }_n+k)){\parallel x_n-y_n\parallel }^2.\end{array}$$

(3.9)

It then follows that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists; hence $\{x_n\}$ is bounded. Also,

$$\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n(1-({\alpha }_n+k)){\parallel x_n-y_n\parallel }^2\le {\parallel x_0-p\parallel }^2<\mathrm{\infty }.$$

Since $\alpha >0$ from (ii), we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$. Thus ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Also since K is closed and $\{x_n\}\subseteq K$ with $\{x_n\}$ bounded, there exist a subsequence $\{x_{n_t}\}\subseteq \{x_n\}$ such that $\{x_{n_t}\}$ converges weakly to some $q\in K$. Also ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ implies that ${lim}_{n\to \mathrm{\infty }}\parallel x_{n_t}-y_{n_t}\parallel =0$. Since $(I-T)$ is weakly demiclosed at zero we have $q\in Tq$. Since H satisfies Opial’s condition [13] we find that $\{x_n\}$ converges weakly to $q\in F(T)$. □

Corollary 3.1 Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that $T:K\to P(K)$ is k-strictly pseudocontractive-type mapping from K into the family of all proximinal subsets of K with $k\in (0,1)$ such that $F(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for all $p\in F(T)$. Suppose $(I-T)$ is weakly demiclosed at zero. Then the Mann sequence $\{x_n\}$ defined in Theorem  3.1 converges weakly to a point of $F(T)$.

Proof The proof follows easily from Example 3.3 and Theorem 3.1. □

Corollary 3.2 Let H be a real Hilbert space and K a nonempty closed and convex subset of H. Let $T:K\to P(K)$ be a multivalued mapping from K into the family of all proximinal subsets of K. Suppose $P_T$ is a demicontractive mapping with $k\in (0,1)$ and $(I-P_T)$ is weakly demiclosed at zero. Then the Mann sequence $\{x_n\}$ defined in Theorem  3.1 converges weakly to a point of $F(T)$.

Proof The proof follows easily from Lemma 2.2 and Theorem 3.1. □

Remark 3.2 Since the choice of $y_n\in T{x}_n$ in the Mann-type iteration scheme is independent of $d(x_n,T{x}_n)$, we can also replace $P(K)$ with $\mathit{CB}(K)$ in Theorem 3.1 and its corollaries. Furthermore, since ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$, one can impose standard conditions on T or K which guarantee strong convergence.

Theorem 3.2 Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that $T:K\to P(K)$ is an L-Lipschitzian hemicontractive mapping from K into the family of all proximinal subsets of K and $Tp=\{p\}$ for all $p\in F(T)$. Suppose T satisfies condition (1). Then the Ishikawa sequence defined by

$$\{\begin{array}{l}{y}_n=(1-{\beta }_n)x_n+{\beta }_n{u}_n,\\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{w}_n\end{array}$$

(3.10)

converges strongly to $p\in F(T)$, where $u_n\in T{x}_n$, $w_n\in T{y}_n$ satisfying the conditions of Lemma  3.1 and $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are real sequences satisfying: (i) $0\le {\alpha }_n\le {\beta }_n<1$; (ii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n=\alpha >0$; (iii) ${sup}_{n\ge 1}{\beta }_n\le \beta \le \frac{1}{\sqrt{1+L^2}+1}$.

Proof

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2={\parallel (1-{\alpha }_n)x_n+{\alpha }_n{w}_n-p\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2}={\parallel (1-{\alpha }_n)(x_n-p)+{\alpha }_n(w_n-p)\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2}=(1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{\parallel w_n-p\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2=}-{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2}\le (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{H}^2(T{y}_n,Tp)\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2=}-{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2}\le (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n[{\parallel y_n-p\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2=}+d^2(y_n,T{y}_n)]-{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2}=(1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{\parallel y_n-p\parallel }^2+{\alpha }_n{d}^2(y_n,T{y}_n)\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2=}-{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2,\hfill \end{array}$$

(3.11)

$$\begin{array}{c}{d}^2(y_n,T{y}_n)\le {\parallel y_n-w_n\parallel }^2\hfill \\ \phantom{d^2(y_n,T{y}_n)}={\parallel (1-{\beta }_n)x_n+{\beta }_n{u}_n-w_n\parallel }^2\hfill \\ \phantom{d^2(y_n,T{y}_n)}={\parallel (1-{\beta }_n)(x_n-w_n)+{\beta }_n(u_n-w_n)\parallel }^2\hfill \\ \phantom{d^2(y_n,T{y}_n)}=(1-{\beta }_n){\parallel x_n-w_n\parallel }^2+{\beta }_n{\parallel u_n-w_n\parallel }^2\hfill \\ \phantom{d^2(y_n,T{y}_n)\le }-{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2.\hfill \end{array}$$

(3.12)

Equations (3.11) and (3.12) imply that

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\le (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{\parallel y_n-p\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2\le }+{\alpha }_n[(1-{\beta }_n){\parallel x_n-w_n\parallel }^2+{\beta }_n{\parallel u_n-w_n\parallel }^2\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2\le }-{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2]\hfill \\ \phantom{{\parallel x_{n+1}-p\parallel }^2\le }-{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2,\hfill \end{array}$$

(3.13)

$$\begin{array}{c}{\parallel y_n-p\parallel }^2={\parallel (1-{\beta }_n)x_n+{\beta }_n{u}_n-p\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}={\parallel (1-{\beta }_n)(x_n-p)+{\beta }_n(u_n-p)\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}=(1-{\beta }_n){\parallel x_n-p\parallel }^2+{\beta }_n{\parallel u_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}\le (1-{\beta }_n){\parallel x_n-p\parallel }^2+{\beta }_n{H}^2(T{x}_n,Tp)-{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}\le (1-{\beta }_n){\parallel x_n-p\parallel }^2+{\beta }_n[{\parallel x_n-p\parallel }^2+d^2(x_n,T{x}_n)]\hfill \\ \phantom{{\parallel y_n-p\parallel }^2=}-{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}\le (1-{\beta }_n){\parallel x_n-p\parallel }^2+{\beta }_n{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel x_n-u_n\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2=}-{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}={\parallel x_n-p\parallel }^2+{\beta }_n^2{\parallel x_n-u_n\parallel }^2.\hfill \end{array}$$

(3.14)

Equations (3.13) and (3.14) imply that

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & (1-{\alpha }_n){\parallel x_n-p\parallel }^2\\ +{\alpha }_n[{\parallel x_n-p\parallel }^2+{\beta }_n^2{\parallel x_n-u_n\parallel }^2]\\ +{\alpha }_n[(1-{\beta }_n){\parallel x_n-w_n\parallel }^2+{\beta }_n{\parallel u_n-w_n\parallel }^2\\ -{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2]\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2\\ =& (1-{\alpha }_n){\parallel x_n-p\parallel }^2+{\alpha }_n{\parallel x_n-p\parallel }^2+{\alpha }_n{\beta }_n^2{\parallel x_n-u_n\parallel }^2\\ +{\alpha }_n(1-{\beta }_n){\parallel x_n-w_n\parallel }^2+{\alpha }_n{\beta }_n{\parallel u_n-w_n\parallel }^2\\ -{\alpha }_n{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2-{\alpha }_n(1-{\alpha }_n){\parallel x_n-w_n\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2+{\alpha }_n{\beta }_n^2{\parallel x_n-u_n\parallel }^2+{\alpha }_n{\beta }_n{H}^2(T{x}_n,T{y}_n)\\ -{\alpha }_n({\beta }_n-{\alpha }_n){\parallel x_n-w_n\parallel }^2\\ -{\alpha }_n{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2+{\alpha }_n{\beta }_n^2{\parallel x_n-u_n\parallel }^2+{\alpha }_n{\beta }_n^3{L}^2{\parallel x_n-u_n\parallel }^2\\ -{\alpha }_n{\beta }_n(1-{\beta }_n){\parallel x_n-u_n\parallel }^2\\ -{\alpha }_n({\beta }_n-{\alpha }_n){\parallel x_n-w_n\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-{\alpha }_n{\beta }_n[1-2{\beta }_n-L^2{\beta }_n^2]{\parallel x_n-u_n\parallel }^2\\ -{\alpha }_n({\beta }_n-{\alpha }_n){\parallel x_n-w_n\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-{\alpha }_n{\beta }_n[1-2{\beta }_n-L^2{\beta }_n^2]{\parallel x_n-u_n\parallel }^2.\end{array}$$

(3.15)

It then follows from Lemma 2.1 that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists. Hence $\{x_n\}$ is bounded so $\{u_n\}$ and $\{w_n\}$ also are. We then have from (3.15), (ii), and (iii)

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\alpha }^2[1-2\beta -L^2{\beta }^2]{\parallel x_n-u_n\parallel }^2& \le & \sum _{n=0}^{\mathrm{\infty }}{\alpha }_n{\beta }_n[1-2{\beta }_n-L^2{\beta }_n^2]{\parallel x_n-u_n\parallel }^2\\ \le & \sum _{n=0}^{\mathrm{\infty }}[{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2]\\ \le & {\parallel x_0-p\parallel }^2+D<\mathrm{\infty }.\end{array}$$

It then follows that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel =0$. Since $u_n\in T{x}_n$ we have $d(x_n,T{x}_n)\le \parallel x_n-u_n\parallel \to 0$ as $n\to \mathrm{\infty }$. Since T satisfies condition (1), ${lim}_{n\to \mathrm{\infty }}d(x_n,F(T))=0$. Thus there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $\parallel x_{n_k}-p_k\parallel \le \frac{1}{2^k}$ for some $\{p_k\}\subseteq F(T)$. From (3.10)

$$\parallel x_{n_{k+1}}-p_k\parallel \le \parallel x_{n_k}-p_k\parallel .$$

We now show that $\{p_k\}$ is a Cauchy sequence in $F(T)$. We have

$$\begin{array}{rcl}\parallel p_{k+1}-p_k\parallel & \le & \parallel p_{k+1}-x_{n_{k+1}}\parallel +\parallel x_{n_{k+1}}-p_k\parallel \\ \le & \frac{1}{2^{k+1}}+\frac{1}{2^k}\\ =& \frac{1}{2^{k-1}}.\end{array}$$

Therefore $\{p_k\}$ is a Cauchy sequence and converges to some $q\in K$ because K is closed. Now,

$$\parallel x_{n_k}-q\parallel \le \parallel x_n-p_k\parallel +\parallel p_k-q\parallel .$$

Hence $x_{n_k}\to q$ as $k\to \mathrm{\infty }$. We have

$$\begin{array}{rcl}d(q,Tq)& \le & \parallel q-p_k\parallel +\parallel p_k-x_{n_k}\parallel +d(x_{n_k},T{x}_{n_k})+H(T{x}_{n_k},Tq)\\ \le & \parallel q-p_k\parallel +\parallel p_k-x_{n_k}\parallel +d(x_{n_k},T{x}_{n_k})+L\parallel x_{n_k}-q\parallel .\end{array}$$

Hence, $q\in Tq$ and $\{x_{n_k}\}$ converges strongly to q. Since $lim\parallel x_n-q\parallel$ exists we see that $x_n$ converges strongly to $q\in F(T)$. □

Corollary 3.3 Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that $T:K\to P(K)$ is an L-Lipschitzian pseudocontractive-type mapping from K into the family of all proximinal subsets of K such that $F(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for all $p\in F(T)$. Suppose T satisfies condition (1). Then the Ishikawa sequence $\{x_n\}$ defined in (3.10) converges strongly to $p\in F(T)$.

Proof The proof follows easily from Example 3.8, Lemma 3.1, and Theorem 3.2. □

Corollary 3.4 Let H be a real Hilbert space and K a nonempty closed and convex subset of H. Let $T:K\to P(K)$ be a multivalued mapping from K into the family of all proximinal subsets of K such that $F(T)\ne \mathrm{\varnothing }$. Suppose $P_T$ is an L-Lipschitzian hemicontractive mapping. If T satisfies condition (1). Then the Ishikawa sequence $\{x_n\}$ defined in (3.10) converges strongly to $p\in F(T)$.

Proof The proof follows easily from Lemma 2.2 and Theorem 3.2. □

Remark 3.3 In Theorem 3.2 and its corollaries we can replace $P(K)$ with $\mathit{CB}(K)$ with additional condition that T is weakly closed for all $x\in D(T)=K$ in order to ensure that $u_n$ and $w_n$ satisfy Lemma 3.1 as indicated in Remark 3.1. Furthermore, since ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$, the additional requirement that $(I-T)$ is weakly demiclosed at zero in Theorem 3.2 yields weak convergence without condition (1).