We now obtain a demiclosedness property in the sense that if \(\{r_{n}\}_{n=1}^{\infty}\subseteq C\) is such that \(\{r_{n}\}\) weakly converges to *z* and a sequence \(\{g_{n}\}\) with \(g_{n}\in Sr_{n}\) and \(\|r_{n}-g_{n}\|=d(r_{n},Sr_{n})\) for all \(n\in\mathbb{N}\) is such that \(\{ r_{n}-g_{n}\}\) strongly converges to 0, then \(0\in(I-S)z\) (*i.e.*, \(z=v\) for some \(v\in Sz\)).

### Proposition 3.1

*Let*
*H*
*be a real Hilbert space*, *and*
*Ca nonempty weakly closed subset of H*. *Let*
\(S:C\subseteq H\rightarrow P(H)\)
*be a multivalued mapping from*
*C*
*into the family of all proximinal subsets of H*. *If*
*S*
*is a*
*k*-*strictly pseudocontractive mapping of type one*, *then*
\((I-S)\)
*is demiclosed at* zero (*i*.*e*., *the graph of*
\(I-S\)
*is closed at* zero *in*
\(\sigma(H,H^{*})\times(H,\|\cdot\|)\)
*or weakly demiclosed at* zero).

### Proof

We use a method similar to that of the proof of Proposition 3 in [5]. Let \(\{r_{n}\}_{n=1}^{\infty}\subseteq C\) be such that \(\{r_{n}\}\) weakly converges to *z*, and \(\{g_{n}\}\) be a sequence with \(g_{n}\in Sr_{n}\) and \(\|r_{n}-g_{n}\| =d(r_{n},Sr_{n})\) for all \(n\in\mathbb{N}\) such that \(\{ r_{n}-g_{n}\}\) strongly converges to 0. We prove that \(0\in(I-S)z\) (*i.e.*, \(z=v\) for some \(v\in Sz\)). Since \(\{r_{n}\}_{n=1}^{\infty}\) converges weakly, it is bounded. Let \(q\in Sz\) with \(\|z-q\|=d(z,Sz)\), From the definition of *k*-strictly pseudocontractive and type-one condition on *S*, for each \(n\in \mathbb{N}\), we have

$$ \|g_{n}-q\|\leq\Phi(Sr_{n},Sz) $$

(3.1)

and

$$ \Phi^{2}(Sr_{n},Sz)\leq\|r_{n}-z \|^{2}+k\bigl\| r_{n}-g_{n}-(z-q)\bigr\| ^{2}. $$

(3.2)

Therefore, for each \(r\in H\), define \(f:H\rightarrow[0,\infty)\) by

$$f(r):=\limsup_{n\to\infty}\|r_{n}-r\|^{2}. $$

Then from Lemma 2.1 we obtain

$$f(r)=\limsup_{n\to\infty}\|r_{n}-z\|^{2}+\|z-r \|^{2} ,\quad \forall r\in X. $$

Thus,

$$f(r)=f(z)+\|z-r\|^{2} , \quad \forall r\in X. $$

Therefore,

$$ f(q)=f(z)+\|z-q\|^{2}. $$

(3.3)

Observe also that

$$\begin{aligned} f(q) =&\limsup_{n\to\infty}\|r_{n}-q \|^{2} \\ =&\limsup_{n\to\infty}\bigl\| r_{n}-g_{n}+(g_{n}-q) \bigr\| ^{2} \\ =&\limsup_{n\to\infty}\|g_{n}-q\|^{2} \\ \leq& \limsup_{n\to\infty}\Phi^{2}(Sr_{n},Sz) \\ \leq&\limsup_{n\to\infty} \bigl[\|r_{n}-z \|^{2}+k\bigl\| r_{n}-g_{n}-(z-q)\bigr\| ^{2} \bigr] \\ =& \limsup_{n\to\infty}\|r_{n}-z\|^{2}+k\bigl\| (z-q) \bigr\| ^{2} \\ =&f(z)+k\|z-q\|^{2}. \end{aligned}$$

(3.4)

Hence, it follows from (3.3) and (3.4) that \((1-k)\|z-q\|^{2}=0\).

Therefore, \(z=q\in Sz\). □

We now obtain some strong and weak convergence results for the class of pseudocontractive mappings and *k*-strictly pseudocontractive mappings of Chidume *et al.* [1], respectively, in Hilbert spaces.

### Theorem 3.1

*Let*
*C*
*be a nonempty closed and convex subset of a real Hilbert space*
*X*. *If*
\(S:C\rightarrow P(C)\)
*is a type*-*one*
*L*-*Lipschitzian pseudocontractive mapping from*
*C*
*into the family of all proximinal subsets of*
*C*
*such that*
\(F(S)\neq\emptyset\). *Suppose that*
*S*
*satisfies condition *(1). *Then*, *the Ishikawa sequence defined by*

$$\left \{ \textstyle\begin{array}{@{}l} g_{n}=(1-\delta_{n})r_{n}+\delta_{n}u_{n},\\ r_{n+1}=(1-\mu_{n})r_{n}+\mu_{n}w_{n}, \end{array}\displaystyle \right . $$

*strongly converges to*
\(p\in F(S)\), *where*
\(u_{n}\in Sr_{n}\)
*with*
\(\|r_{n}-u_{n}\|=d(r_{n},Sr_{n})\), \(w_{n}\in Sg_{n}\)
*with*
\(\|g_{n}-w_{n}\|=d(g_{n},Sg_{n})\), *and*
\(\{\mu_{n}\}\)
*and*
\(\{\delta_{n}\}\)
*are real sequences satisfying* (i) \(\liminf_{n\to\infty}\mu_{n}=\mu>0\), (ii) \(0\leq\mu_{n}\leq\delta _{n}<1\), *and* (iii) \(\sup_{n\geq1}\delta_{n}\leq\delta\leq\frac{1}{\sqrt{1+L^{2}}+1}\).

### Proof

Using a method similar to that of the proof of Theorem 4 in [5], we have that

$$\begin{aligned} \|r_{n+1}-z\|^{2} =&\bigl\| (1-\mu_{n})r_{n}+ \mu_{n}w_{n}-z\bigr\| ^{2} \\ =&\bigl\| (1-\mu_{n}) (r_{n}-z)+\mu_{n}(w_{n}-z) \bigr\| ^{2} \\ =&(1-\mu_{n})\|r_{n}-z\|^{2}+\mu_{n} \|w_{n}-z\|^{2} -\mu_{n}(1-\mu_{n})\|r_{n}-w_{n} \|^{2} \\ \leq&(1-\mu_{n})\|r_{n}-z\|^{2}+ \mu_{n}\Phi^{2}(Sg_{n},Sz) \\ &{}-\mu_{n}(1-\mu_{n})\|r_{n}-w_{n} \|^{2} \\ \leq&(1-\mu_{n})\|r_{n}-z\|^{2}+ \mu_{n} \bigl[\|g_{n}-z\|^{2} +\|g_{n}-w_{n}\|^{2} \bigr] \\ &{}- \mu_{n}(1-\mu_{n})\|r_{n}-w_{n} \|^{2} \\ =&(1-\mu_{n})\|r_{n}-z\|^{2}+\mu_{n} \|g_{n}-z\|^{2}+\mu_{n}d^{2}(g_{n},Sg_{n}) \\ &{}-\mu_{n}(1-\mu_{n})\|r_{n}-w_{n} \|^{2}. \end{aligned}$$

(3.5)

Also,

$$\begin{aligned} \|g_{n}-w_{n}\|^{2} =&\bigl\| (1- \delta_{n})r_{n}+\delta_{n}u_{n}-w_{n} \bigr\| ^{2} \\ =&\bigl\| (1-\delta_{n}) (r_{n}-w_{n})+ \delta_{n}(u_{n}-w_{n})\bigr\| ^{2} \\ =&(1-\delta_{n})\|r_{n}-w_{n}\|^{2}+ \delta_{n}\|u_{n}-w_{n}\|^{2} \\ &{}-\delta_{n}(1-\delta_{n})\|r_{n}-u_{n} \|^{2}. \end{aligned}$$

(3.6)

From (3.5) and (3.6) it follows that

$$\begin{aligned}& \begin{aligned}[b] \|r_{n+1}-z\|^{2}\leq{}&(1- \mu_{n})\|r_{n}-z\|^{2}+\mu_{n} \|g_{n}-z\|^{2} \\ &{}+\mu_{n} \bigl[(1-\delta_{n})\|r_{n}-w_{n} \|^{2}+\delta_{n}\|u_{n}-w_{n} \|^{2} \\ &{}-\delta_{n}(1-\delta_{n})\|r_{n}-u_{n} \|^{2} \bigr] \\ &{}-\mu_{n}(1-\mu_{n})\|r_{n}-w_{n} \|^{2}, \end{aligned} \end{aligned}$$

(3.7)

$$\begin{aligned}& \begin{aligned}[b] \|g_{n}-z\|^{2}={}&\bigl\| (1- \delta_{n})r_{n}+\delta_{n}u_{n}-z \bigr\| ^{2} \\ ={}&\bigl\| (1-\delta_{n}) (r_{n}-z)+\delta_{n}(u_{n}-z) \bigr\| ^{2} \\ ={}&(1-\delta_{n})\|r_{n}-z\|^{2}+ \delta_{n}\|u_{n}-z\|^{2} -\delta_{n}(1-\delta_{n})\|r_{n}-u_{n} \|^{2} \\ \leq{}&(1-\delta_{n})\|r_{n}-z \|^{2}+\delta_{n}\Phi^{2}(Sr_{n},Sz) -\delta_{n}(1-\delta_{n})\|r_{n}-u_{n} \|^{2} \\ \leq{}&(1-\delta_{n}) \|r_{n}-z\|^{2}+\delta_{n} \bigl[ \|r_{n}-z\|^{2}+\|r_{n}-u_{n} \|^{2} \bigr] -\delta_{n}(1-\delta_{n})\|r_{n}-u_{n} \|^{2} \\ ={}&\|r_{n}-z\|^{2}+ \delta_{n}^{2}\|r_{n}-u_{n} \|^{2}. \end{aligned} \end{aligned}$$

(3.8)

From (3.7) and (3.8) it follows that

$$\begin{aligned} \|r_{n+1}-z\|^{2} \leq&(1-\mu_{n}) \|r_{n}-z\|^{2} \\ &{}+\mu_{n} \bigl[\|r_{n}-z\|^{2}+ \delta_{n}^{2}\|r_{n}-u_{n} \|^{2} \bigr] \\ &{}+\mu_{n} \bigl[(1-\delta_{n})\|r_{n}-w_{n} \|^{2}+\delta_{n}\|u_{n}-w_{n} \|^{2} -\delta_{n}(1-\delta_{n})\|r_{n}-u_{n} \|^{2} \bigr] \\ &{}-\mu_{n}(1-\mu_{n})\|r_{n}-w_{n} \|^{2} \\ =&(1-\mu_{n})\|r_{n}-z\|^{2}+\mu_{n} \|r_{n}-z\|^{2}+\mu_{n}\delta_{n}^{2} \|r_{n}-u_{n}\|^{2} \\ &{}+\mu_{n}(1-\delta_{n})\|r_{n}-w_{n} \|^{2}+\mu_{n}\delta_{n}\|u_{n}-w_{n} \|^{2} \\ &{}-\mu_{n}\delta_{n}(1-\delta_{n}) \|r_{n}-u_{n}\|^{2}-\mu_{n}(1- \mu_{n})\|r_{n}-w_{n}\|^{2} \\ \leq&\|r_{n}-z\|^{2}+\mu_{n} \delta_{n}^{2}\|r_{n}-u_{n} \|^{2}+\mu_{n}\delta_{n}\Phi ^{2}(Sr_{n},Sg_{n}) \\ &{}-\mu_{n}(\delta_{n}-\mu_{n}) \|r_{n}-w_{n}\|^{2} \\ &{}-\mu_{n}\delta_{n}(1-\delta_{n}) \|r_{n}-u_{n}\|^{2} \\ \leq&\|r_{n}-z\|^{2}+\mu_{n} \delta_{n}^{2}\|r_{n}-u_{n} \|^{2}+\mu_{n}\delta_{n}^{3}L^{2} \| r_{n}-u_{n}\|^{2} \\ &{}-\mu_{n}\delta_{n}(1-\delta_{n}) \|r_{n}-u_{n}\|^{2} \\ &{}-\mu_{n}(\delta_{n}-\mu_{n}) \|r_{n}-w_{n}\|^{2} \\ =&\|r_{n}-z\|^{2}-\mu_{n}\delta_{n} \bigl[1-2\delta_{n}-L^{2}\delta_{n}^{2} \bigr]\|r_{n}-u_{n}\|^{2} \\ &{}-\mu_{n}(\delta_{n}-\mu_{n}) \|r_{n}-w_{n}\|^{2} \\ =&\|r_{n}-z\|^{2}-\mu_{n}\delta_{n} \bigl[1-2\delta_{n}-L^{2}\delta_{n}^{2} \bigr]\|r_{n}-u_{n}\|^{2}. \end{aligned}$$

(3.9)

It then follows from Lemma 2.2 that \(\lim_{n\rightarrow\infty}\|r_{n}-z\|\) exists. Hence, \(\{r_{n}\}\) is bounded, so also are \(\{u_{n}\}\) and \(\{w_{n}\}\). We then have from (3.9)(i), (iii) that

$$\begin{aligned} \sum_{n=0}^{\infty}\mu^{2}\bigl[1-2 \delta-L^{2}\delta^{2}\bigr]\|r_{n}-u_{n} \|^{2} \leq& \sum_{n=0}^{\infty} \mu_{n}\delta_{n}\bigl[1-2\delta_{n}-L^{2} \delta _{n}^{2}\bigr]\|r_{n}-u_{n} \|^{2} \\ \leq&\sum_{n=0}^{\infty}\bigl[ \|r_{n}-z\|^{2}-\|r_{n+1}-z\|^{2}\bigr] \\ \leq&\|r_{0}-z\|^{2}+D< \infty. \end{aligned}$$

It then follows that \(\lim_{n\to\infty}\|r_{n}-u_{n}\|=0\). Since \(u_{n}\in Sr_{n}\), we have that \(d(r_{n},Sr_{n})\leq\|r_{n}-u_{n}\|\rightarrow 0\) as \(n\rightarrow\infty\). Since *S* satisfies condition (1), \(\lim_{n\rightarrow\infty}d(r_{n},F(S))=0\). Thus, there exists a subsequence \(\{r_{n_{k}}\}\) of \(\{r_{n}\}\) such that \(\|r_{n_{k}}-z_{k}\|\leq\frac{1}{2^{k}}\) for some \(\{z_{k}\}\subseteq F(S)\). From (3.9) we have

$$\|r_{n_{k+1}}-z_{k}\|\leq\|r_{n_{k}}-z_{k}\|. $$

We now show that \(\{z_{k}\}\) is a Cauchy sequence in \(F(S)\):

$$\begin{aligned} \|z_{k+1}-z_{k}\| \leq&\|z_{k+1}-r_{n_{k+1}}\|+ \|r_{n_{k+1}}-z_{k}\| \\ \leq&\frac{1}{2^{k+1}}+\frac{1}{2^{k}} \\ < &\frac{1}{2^{k-1}}. \end{aligned}$$

Therefore, \(\{z_{k}\}\) is a Cauchy sequence and converges to some \(q\in C\) because *C* is closed. Now, we have

$$\|r_{n_{k}}-q\|\leq\|r_{n}-z_{k}\|+ \|z_{k}-q\|. $$

Hence, \(r_{n_{k}}\rightarrow q\) as \(k\rightarrow\infty\). We have

$$\begin{aligned} d(q,Sq) \leq&\|q-z_{k}\|+\|z_{k}-r_{n_{k}} \|+d(r_{n_{k}},Sr_{n_{k}})+\Phi (Sr_{n_{k}},Sq) \\ \leq&\|q-z_{k}\|+\|z_{k}-r_{n_{k}} \|+d(r_{n_{k}},Sr_{n_{k}})+L\|r_{n_{k}}-q\|. \end{aligned}$$

Hence, \(q\in Sq\), and \(\{r_{n_{k}}\}\) strongly converges to *q*. Since \(\lim\|r_{n}-q\|\) exists, we have that \(r_{n}\) strongly converges to \(q\in F(S)\). □

### Theorem 3.2

*Let*
*C*
*be a nonempty closed and convex subset of a real Hilbert space*
*H*. *Suppose that*
\(S:C\rightarrow P(C)\)
*is*
*k*-*strictly pseudocontractive mapping from*
*C*
*into the family of all proximinal subsets of*
*C*
*with*
\(k\in(0,1)\)
*such that*
\(F(S)\neq\emptyset\). *If*
*S*
*is of type one*, *then the Mann*-*type sequence defined by*

$$r_{n+1}=(1-\mu_{n})r_{n}+\mu_{n}g_{n} $$

*weakly converges to*
\(q\in F(S)\), *where*
\(g_{n}\in Sr_{n}\)
*with*
\(\|r_{n}-g_{n}\|=d(r_{n},Sr_{n})\)
*and*
\(\mu_{n}\subseteq(0,1)\)
*satisfying* (i) \(\mu_{n}\rightarrow\mu<1-k\), (ii) \(\mu>0\), *and* (iii) \(\sum_{n=1}^{\infty}\mu_{n}(1-\mu_{n})=\infty\).

### Proof

Using the well-known identity and a method similar to that of the proof of Theorem 3 in [5], we have that

$$\bigl\| tr+(1-t)g\bigr\| ^{2}=t\|r\|^{2}+(1-t)\|g\|^{2}-t(1-t) \|r-g\|^{2}, $$

which holds for all \(r,g\in H\) and all \(t\in[0,1]\), from which we obtain

$$\begin{aligned} \|r_{n+1}-z\|^{2} =&\bigl\| (1-\mu_{n})r_{n}+ \mu_{n}g_{n}-z\bigr\| ^{2} \\ =&\bigl\| (1-\mu_{n}) (r_{n}-z)+\mu_{n}(g_{n}-z) \bigr\| ^{2} \\ =&(1-\mu_{n})\|r_{n}-z\|^{2}+\mu_{n} \|g_{n}-z\|^{2}-\mu_{n}(1-\mu_{n}) \|r_{n}-g_{n}\|^{2} \\ \leq&(1-\mu_{n})\|r_{n}-z\|^{2}+ \mu_{n}\Phi^{2}(Sr_{n},Sz)-\mu_{n}(1- \mu _{n})\|r_{n}-g_{n}\|^{2} \\ \leq&(1-\mu_{n})\|r_{n}-z\|^{2}+ \mu_{n} \bigl[\|r_{n}-z\|^{2}+k \|r_{n}-g_{n}\|^{2} \bigr] \\ &{}-\mu_{n}(1-\mu_{n})\|r_{n}-g_{n} \|^{2} \\ =&\|r_{n}-z \|^{2}-\mu_{n}\bigl(1-(\mu_{n}+k)\bigr) \|r_{n}-g_{n}\|^{2}. \end{aligned}$$

It then follows Lemma 2.2 that \(\lim_{n\rightarrow\infty} \| r_{n}-z\|\) exists, and hence \(\{r_{n}\}\) is bounded. Also,

$$\sum_{n=1}^{\infty}\mu_{n}\bigl(1-( \mu_{n}+k)\bigr)\|r_{n}-g_{n}\|^{2}\leq \|r_{0}-z\|^{2}\leq \infty. $$

Since \(\mu>0\) from (ii), we have that \(\lim_{n\to\infty}\|r_{n}-g_{n}\|=0\). Also, since *C* is closed and \(\{r_{n}\}\subseteq C\) with \(\{r_{n}\}\) bounded, there exists a subsequence \(\{r_{n_{t}}\}\subseteq\{r_{n}\}\) that weakly converges to some \(q\in C\). Also, \(\lim_{n\to\infty}\|r_{n}-g_{n}\|=0\) implies that \(\lim_{n\to\infty}\|r_{n_{t}}-g_{n_{t}}\|=0\). Since \((I-S)\) is weakly demiclosed at *zero*, we have that \(q\in Sq\). Since *H* satisfies Opial’s condition [15], we have that \(\{r_{n}\}\) weakly converges to \(q\in F(S)\). □

We now have the following corollaries with proofs easily following from the definition and Theorems 3.1 and 3.2, respectively.

### Corollary 3.1

([10])

*Let*
*C*
*be a nonempty closed and convex subset of a real Hilbert space H*. *Suppose that*
\(S:C\rightarrow P(C)\)
*is a type*-*one quasi*-*nonexpansive mapping from*
*C*
*into the family of all proximinal subsets of*
*C*. *If*
\((I-S)\)
*satisfies Proposition *
3.1, *then the Mann*-*type sequence defined by*

$$r_{n+1}=(1-\mu_{n})r_{n}+\mu_{n}g_{n} $$

*weakly converges to*
\(q\in F(S)\), *where*
\(g_{n}\in Sr_{n}\)
*with*
\(\|r_{n}-g_{n}\|=d(r_{n},Sr_{n})\), *and*
\(\mu_{n}\)
*is a real sequence in* (0,1) *satisfying*
\(\mu_{n}\rightarrow \mu\in(0,1)\).

### Corollary 3.2

([10])

*Let*
*C*
*be a nonempty closed and convex subset of a real Hilbert space H*. *Suppose that*
\(S:C\rightarrow P(C)\)
*is a type*-*one nonexpansive mapping from*
*C*
*into the family of all proximinal subsets of*
*C*
*such that*
\(F(S)\neq\emptyset\). *Then the Mann*-*type sequence defined by*

$$r_{n+1}=(1-\mu_{n})r_{n}+\mu_{n}g_{n} $$

*weakly converges to*
\(q\in F(S)\), *where*
\(g_{n}\in Sr_{n}\)
*with*
\(\|r_{n}-g_{n}\|=d(r_{n},Sr_{n})\)
*and*
\(\mu_{n}\subseteq(0,1)\)
*satisfying*
\(\mu _{n}\rightarrow \mu\in(0,1)\).

### Concluding Remark

It is not clear if Theorem 3.1 and Theorem 3.2 will still hold if the classes of pseudocontractive and *k*-strictly pseudocontractive mappings considered are replaced with the classes *k*-strictly pseudocontractive-type and pseudocontractive-type mappings, respectively, considered by Isiogugu [5].