1 Introduction

Let C be a nonempty subset of a real Hilbert space H. The set C is called proximinal if for each \(x\in H\) there exists \(u\in C\) such that

$$\begin{aligned} ||x-u||=\inf \{||x-y||:y\in C\}=d(x,C), \end{aligned}$$

where d is the metric on H generated by the inner product. It is well known that any nonempty closed and convex subset of a Hilbert space is proximinal. The family of nonempty proximinal bounded subsets of the set C is denoted by Prox(C).

Let \(A,B\in CB(H),\) where CB(H) is the set of nonempty, closed and bounded subsets of H. The Hausdorff distance between A and B,  denoted by D(AB), is defined as

$$\begin{aligned} D(A,B)=\max {\bigg \{\displaystyle \sup _{x\in B} d(x,A), \displaystyle \sup _{x\in A} d(x,B)}\bigg \}. \end{aligned}$$

A multi-valued mapping \(T:C\rightarrow 2^H\) is said to be L-Lipschitz if there exists \(L \ge 0\) such that

$$\begin{aligned} D(Tx,Ty)\le L||x-y||, \text{ for } \text{ all } x,y\in C. \end{aligned}$$

If \(L=1,\) then the mapping T is called nonexpansive mapping. It is immediate from the definition that every nonexpansive mapping is Lipschitz mapping.

A mapping \(T:C\rightarrow 2^H\) is said to be

  1. (a)

    k-strictly pseudocontractive if there exists \(k\in (0,1)\) such that for each \(x,y\in C\),

    $$\begin{aligned} D^2(Tx,Ty)\le ||x-y||^2+k||x-y-(u-v)||^2, \forall u\in Tx, v\in Ty. \end{aligned}$$
  2. (b)

    pseudocontractive if for each \(x,y\in C,\)

    $$\begin{aligned} D^2(Tx,Ty)\le ||x-y||^2+||x-y-(u-v)||^2, \forall u\in Tx, v\in Ty. \end{aligned}$$

We observe that the class of multi-valued pseudocontractive mappings includes the class of multi-valued k-strictly pseudocontractive mappings and hence the class of multi-valued nonexpansive mappings.

Given a multi-valued mapping \(T:C\rightarrow 2^H,\) a point \(x\in C\) is called a fixed point of T if \(x\in Tx.\) We denote the set of all fixed points of the mapping T by F(T).

If \(F(T)\not =\emptyset \) and \(D(Tx,Tp)\le ||x-p||, \forall x\in C, \forall p\in F(T),\) then T is said to be quasi-nonexpansive mapping. Clearly, every nonexpansive mapping T with \(F(T)\not =\emptyset \) is quasi-nonexpansive mapping. But the converse is not necessarily true (see, e.g., [23]).

Several physical problems in differential inclusions, economics, convex optimization, etc. can be transformed into finding fixed points of multi-valued mappings. As a result, researchers have studied the existence of fixed points and their approximations for different types of multi-valued mappings (see, e.g., [1, 3,4,5, 12, 13, 18, 19] and the references therein). For approximating fixed points of single-valued mappings, basically three iterative methods are in common use: Mann iteration method, Halpern iteration method and Ishikawa iteration method.

Mann iteration method, initially studied by Mann [17], is given by

$$\begin{aligned} x_{n+1} =\alpha _n x_n + (1 -\alpha _n) Tx_n, \end{aligned}$$
(1.1)

where the initial guess \(x_0\in C\) is arbitrary, T is single-valued self mapping on C and \(\{\alpha _n\}\subseteq [0,1]\) such that \(\displaystyle \lim _{n\rightarrow \infty }\alpha _n=0\) and \(\sum \alpha _{n}=\infty .\) This iteration method has been extensively investigated for nonexpansive mappings (see, e.g., [8, 20]). However, the Mann iteration scheme provides only weak convergence in an infinite-dimensional Hilbert space (see, e.g., [8]).

In 1967, Halpern [9] studied the following recursive formula:

$$\begin{aligned} x_{n+1} =\alpha _nu + (1-\alpha _n)Tx_n, n\ge 0, \end{aligned}$$
(1.2)

where T is single-valued self mapping on C and \(\alpha _n\) is a sequence of numbers in (0, 1) satisfying certain conditions. He proved strong convergence of \(\{x_n\}\) to a fixed point of T, provided that T is single-valued nonexpansive mapping. Halpern’s iterative method has been studied extensively by many authors (see, e.g., [14, 21, 26] and the references therein).

The Mann and Halpern methods were successful only for approximating fixed points of single-valued nonexpansive mappings. For approximating fixed points of single-valued Lipschitz pseudocontractive self-mapping T, in [10] Ishikawa introduced the following iterative method.

$$\begin{aligned} \, \, \, \, \, ~ \, \, \left\{ \begin{array}{lll} x_0 \in C, \\ y_n=\beta _n x_n+(1-\beta _n)Tx_n,\\ x_{n+1} =\alpha _n x_n + (1 -\alpha _n) Ty_n, \, n\ge 0, \end{array}\right. \end{aligned}$$
(1.3)

where \(\{\alpha _n\}, \{\beta _n\}\) are sequences of positive numbers satisfying the conditions:

(i) \(0 \le \alpha _n\le \beta _n\le 1\); (ii) \(\displaystyle \lim _{n\rightarrow \infty } \beta _n =0\); (iii) \( \sum \alpha _n\beta _n=\infty \). Then he showed that the sequence \(\{x_n\}\) converges strongly to a fixed point of T,  provided that C is compact convex subset of H. Several authors have extended the results of Ishikawa [10] to Banach spaces without compactness assumption on C (see, e.g., [15, 30]).

On the other hand, in 2005, Sastry and Babu [22] introduced Mann and Ishikawa-type iterative methods for multi-valued self mappings in a real Hilbert space H as follows.

  1. (i)

    Mann-type iterative method:

    $$\begin{aligned} x_0\in C, x_{n+1}=\alpha _n y_n + (1-\alpha _n)x_n, n\ge 0, \end{aligned}$$

    where \(y_n \in Tx_n\) such that \(||y_n-p||= d(p, Tx_n)\) and \(\alpha _n\in [0,1].\)

  2. (ii)

    Ishikawa-type iterative method:

    $$\begin{aligned} \,\,\,\left\{ \begin{array}{lll} x_0\in C,\\ y_n=\beta _nz_n+(1-\beta _n)x_n,\\ x_{n+1}=\alpha _n z'_n + (1-\alpha _n)x_n,n\ge 0,\\ \end{array}\right. \end{aligned}$$
    (1.4)

    where \(C\subset H, T:C\rightarrow Prox(C), ~p\in F(T), ~z_n \in Tx_n, ~ z'_n\in Ty_n\) such that \(||z_n-p||= d(p, Tx_n),||z'_n-p||= d(p, Ty_n)\) and \(\alpha _n,~ \beta _n\in [0,1].\)

Then they obtained strong convergence of the schemes to points in F(T) assuming that C is compact and convex subset of HT is nonexpansive mapping with \(F(T)\not =\emptyset \) and \(\alpha _n, \beta _n\in [0,1]\) satisfying certain conditions.

In [25], Song and Wang extended the result of Sastry and Babu [22] to uniformly convex Banach spaces assuming that \(F(T)\not =\emptyset \) and \(Tp=\{p\},\forall p\in F(T).\)

In [23], Shahzad and Zegeye extended the above results to multi-valued quasi-nonexpansive mappings and relaxed the compactness condition on C. In addition, they introduced the following new iterative scheme in an attempt to remove the end point condition, \(Tp=\{ p \}, \forall p\in F(T),\) in the result of Song and Wang [25].

Let C be a nonempty, closed and convex subset of a real Banach space E, \(T:C\rightarrow Prox(C)\) be a multi-valued mapping and \(P_Tx:=\{y\in Tx:||x-y||=d(x,Tx)\}.\) Let \(\{x_n\}\) be a sequence generated from \(x_0\in C\) as follows.

$$\begin{aligned} \, \, \, \, \, ~ \, \, \left\{ \begin{array}{lll} y_n=(1-\beta _n)x_n+\beta _n z_n,\\ x_{n+1} =(1-\alpha _n) x_n + \alpha _nz'_n, n\ge 0,\\ \end{array}\right. \end{aligned}$$
(1.5)

where \(z_n \in P_Tx_n,~ z'_n\in P_Ty_n\) and \(\{\alpha _n\},~ \{\beta _n\}\) are sequences in [0, 1]. Then they proved that \(\{x_n\}\) converges strongly to a fixed point of T under some mild conditions.

In 2016, Tufa and Zegeye [27] pointed out that the above results hold for approximating fixed points of self-mappings which are not always the cases in practical applications. Motivated by the result of Colao and Marino obtained in [6], Tufa and Zegeye introduced and studied Mann-type iterative scheme for multi-valued nonexpansive non-self mappings in a real Hilbert space. They obtained convergence results of the scheme to fixed points of the mappings.

Recently, Zegeye and Tufa [28] constructed a Halpern–Ishikawa type iterative scheme for single-valued Lipschitz pseudocontractive non-self mappings in Hilbert spaces and obtained strong convergence of the scheme to fixed points of the mappings under some mild conditions. Their result mainly extends the result of Colao et al. [7] from k-strictly pseudocontractive to pseudocontractive mapping.

Motivated by the above results, our purpose in this paper is to construct and study Halpern–Ishikawa type iterative schemes for multi-valued Lipschitz pseudocontractive non-self mappings in real Hilbert spaces. Strong convergence of the schemes to fixed points of the mappings are obtained under appropriate conditions. Our results extend and generalize many of the results in the literature.

2 Preliminaries

In this section, we collect some definitions and known results that we may use in the subsequent section.

Let C be a nonempty subset of a real Hilbert space H. A mapping \(T:C\rightarrow 2^H\) is said to be inward if for any \(x\in C,\) we have

$$\begin{aligned} Tx\subseteq I_C(x) := \{x + \lambda (w-x): \text{ for } \text{ some } w\in C \text{ and } \lambda \ge 1\}. \end{aligned}$$

The set \(I_C(x)\) is called inward set of C at x. A mapping \(I -T,\) where I is an identity mapping on C,  is called demiclosed at zero if for any sequence \(\{x_n\}\) in C such that \(x_n\rightharpoonup x\) and \(d(x_n, Tx_n)\rightarrow 0\) as \(n\rightarrow \infty \), then \(x\in Tx.\)

Lemma 2.1

For any \(x,y\in H,\) the following inequality holds:

$$\begin{aligned} ||x+y||^2\le ||x||^2+2\langle y,x+y\rangle . \end{aligned}$$

Lemma 2.2

[2] Let C be a convex subset of a real Hilbert space H and let \(x\in H.\) Then \(x_0=P_Cx\) if and only if

$$\begin{aligned} \langle z-x_0, x-x_0\rangle \le 0, \forall z\in C, \end{aligned}$$

where \(P_C\) is the metric projection of H onto C defined by

$$\begin{aligned} P_{C}x=\{y\in C:||x-y||=\inf ||x-z||, z\in C\}. \end{aligned}$$

Lemma 2.3

[32] Let H be a real Hilbert space. Then for all \(x,y\in H\) and \(\alpha \in [0,1]\) the following equality holds:

$$\begin{aligned} ||\alpha x +(1-\alpha )y||^2= \alpha ||x||^2+(1-\alpha )||y||^2-\alpha (1-\alpha )||x-y||^2. \end{aligned}$$

Lemma 2.4

[27] Let C be a nonempty, closed and convex subset of a real Hilbert space H and \(T:C\rightarrow CB(H)\) be a mapping and \(u\in Tx.\) Define \(h_u:C\rightarrow R\) by

$$\begin{aligned} h_u(x)=\inf \lbrace \lambda \ge 0: \lambda x + (1-\lambda ) u\in C \rbrace . \end{aligned}$$

Then for any \(x\in C\) the following hold:

  1. (1)

    \(h_u(x)\in [0,1]\) and \(h_u(x)=0\) if and only if \(u\in C;\)

  2. (2)

    if \(\beta \in [h_u(x), 1]\), then \(\beta x +(1-\beta ) u\in C;\)

  3. (3)

    if T is inward, then \(h_u(x)<1;\)

  4. (4)

    if \(u \not \in C,\) then \(h_u(x)x +(1-h(x))u\in \partial C.\)

Lemma 2.5

[19] Let E be a real Banach space. If \(A, B\in CB(E)\) and \(a\in A,\) then for every \(\gamma >0\) there exists \(b\in B\) such that \(||a-b||\le D(A,B)+\gamma .\)

Lemma 2.6

[11] Let E be a real Banach space. If \(A, B\in Prox(E)\) and \(a\in A,\) then there exists \(b\in B\) such that \(||a-b||\le D(A,B).\)

Lemma 2.7

[29] Let C be a closed convex nonempty subset of a real Hilbert space H and \( T : C \rightarrow CB(H) \) be a Lipschitz pseudocontractive mapping. Then F(T) is closed convex subset of C.

From the method of the proof of Lemma 1 of [24], we obtain the following lemma.

Lemma 2.8

Let C be a closed and convex subset of a real Hilbert space H and \(T:C\rightarrow Prox(H)\) be a multi-valued mapping. Define \(P_T:C\rightarrow Prox(H)\) by \(P_T(x)=\{y\in Tx:||x-y||=d(x,Tx)\}.\) Then the following are equivalent:

  1. (i)

    \(p\in F(T);\)

  2. (ii)

    \(P_T(p)=\{p\};\)

  3. (iii)

    \(p\in F(P_T).\)

Furthermore, \(F(T)= F(P_T).\)

Lemma 2.9

Let H be a real Hilbert space. Then the following equation holds: if \(\{x_n\}\) is a sequence in H such that \(x_n\rightharpoonup z\in H,\) then

$$\begin{aligned} \limsup _{n\rightarrow \infty }||x_n-y||^2=\limsup _{n\rightarrow \infty }||x_n-z||^2+||z-y||^2, \forall y\in H. \end{aligned}$$

Lemma 2.10

[31] Let \(\{a_{n}\}\) be a sequence of nonnegative real numbers satisfying the following relation:

$$\begin{aligned} a_{n+1} \le (1-\alpha _n)a_{n} + \alpha _n\delta _n , n\ge 0, \end{aligned}$$

where \(\{\alpha _n\} \subset (0,1)\) and \(\{\delta _n\}\subset IR\) satisfying the conditions: \(\sum _{n=0}^{\infty } \alpha _n=\infty \) and \(\limsup _{n\rightarrow \infty }\delta _n\le 0.\) Then \(\lim _{n\rightarrow \infty }a_{n}=0\).

Lemma 2.11

[16] Let \(\{a_{n}\}\) be sequences of real numbers such that there exists a subsequence \(\{n_i\}\) of \(\{n\}\) such that \(a_{n_i}<a_{{n_i}+1}\) for all \(i\in N\). Then there exists a nondecreasing sequence \(\{m_k\}\subset N\) such that \(m_k\rightarrow \infty \) and the following properties are satisfied by all (sufficiently large) numbers \(k\in N\):

$$\begin{aligned} a_{m_k}\le a_{{m_k}+1} \text{ and } a_k\le a_{{m_k}+1}. \end{aligned}$$

In fact, \(m_k=\max \{j\le k:a_j<a_{j+1}\}\).

3 Main results and discussion

Let C be a nonempty, closed and convex subset of a real Hilbert space H. In this section, we introduce a new iterative scheme for a multi-valued non-self mapping \(T:C\rightarrow CB(H)\) and prove strong convergence results of the scheme with end point condition, \(Tp=\{p\},\forall p\in F(T).\) We also construct an iterative sequence which strongly converges to a fixed point of a multi-valued mapping \(T:C\rightarrow Prox(H)\) without the end point condition.

3.1 Strong convergence results with end point condition

Let \(T:C\rightarrow CB(H)\) be a multi-valued inward Lipschitz mapping with Lipschitz constant L and \(\beta \in \bigg (1-\frac{1}{1+\sqrt{(L+1)^2+1}},1\bigg ).\) For a sequence \(\{\alpha _n\}\) in (0, 1),  we define Halpern–Ishikawa type iterative scheme as follows:

Given \(u,x_0\in C,\) let \(u_0\in Tx_0\) and

$$\begin{aligned} h_{u_0}(x_0):=\inf \lbrace \lambda \ge 0: \lambda x_0+ (1-\lambda )u_0\in C \rbrace . \end{aligned}$$

Now if we choose \(\lambda _0\in [\max \{\beta ,h_{u_0}(x_0)\},1),\) then it follows from Lemma 2.4 that

\(y_0:={\lambda _0} x_0+ (1-\lambda _0)u_0\in C.\)

By Lemma 2.5, we can choose \(v_0\in Ty_0\) such that

$$\begin{aligned} ||u_0-v_0||\le D(Tx_0,Ty_0)+||x_0-y_0||. \end{aligned}$$

Let \(g_{v_0}(y_0):=\inf \lbrace \theta \ge 0: \theta x_0 +(1-\theta )v_0\in C\rbrace .\) If we choose \(\theta _0\in [\max \{\lambda _0,g_{v_0}(y_0)\},1),\) then by Lemma 2.4, \(\theta _0 x_0+(1-\theta _0)v_0\in C.\) Thus, it follows that

$$\begin{aligned}x_{1}:=\alpha _0 u+ (1-\alpha _0)\big (\theta _0x_0+(1-\theta _0)v_0\big )\in C.\end{aligned}$$

Hence, by the principle of mathematical induction, we have

$$\begin{aligned} \quad \quad \quad \left\{ \begin{array}{lll} \lambda _{n}\in [\max \{\beta , h_{u_n}(x_{n})\}, 1); \\ y_n=\lambda _nx_n+(1-\lambda _n) u_n;\\ \theta _{n}\in [\max \{\lambda _n,g_{v_n}(y_{n})\},1);\\ x_{n+1}=\alpha _n u + (1-\alpha _n)\big (\theta _nx_n+(1-\theta _n) v_n\big ), \end{array}\right. \end{aligned}$$
(3.1)

where \(u_n\in Tx_n\) and \(v_n\in Ty_n\) such that \(||u_n-v_n||\le D(Tx_n, Ty_n)+||x_n-y_n||, \,h_{u_n}(x_n):=\inf \{\lambda \ge 0: \lambda x_n+ (1-\lambda ) u_n \in C\}\) and

 \(g_{v_n}(y_n):=\inf \{\theta \ge 0: \theta x_n+ (1-\theta ) v_n \in C\},\forall n\ge 0.\)

Now, we prove our main results.

Lemma 3.1

Let C be a nonempty, closed and convex subset of a real Hilbert space H\(T:C\rightarrow CB(H)\) be L-Lipschitz pseudocontractive inward mapping and let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences defined by (3.1) such that \(\displaystyle \lim _{n\rightarrow \infty } \alpha _n=0\) and \(\sum \alpha _n=\infty .\) Suppose that \(\displaystyle F(T)\not =\emptyset \) with \(Tp=\{p\}, \forall p\in F(T).\) Then \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded.

Proof

Let \(p\in F(T)\). Then from (3.1) and Lemma 2.3 and the fact that T is pseudocontractive, we have

$$\begin{aligned} ||x_{n+1}-p||^2= & {} ||\alpha _n u + (1-\alpha _n)(\theta _nx_n+ (1-\theta _n)v_n)-p||^2\nonumber \\\le & {} \alpha _n||u-p||^2+ (1-\alpha _n)|| \theta _n(x_n-p)+(1-\theta _n)(v_n-p) ||^2 \nonumber \\= & {} \alpha _n||u-p||^2+ (1-\alpha _n)\big [\theta _n||x_n-p||^2+(1-\theta _n ||v_n-p||^2\big ]\nonumber \\&-(1-\alpha _n)\theta _n (1-\theta _n) ||v_n-x_n||^2 \\\le & {} \alpha _n||u-p||^2+ (1-\alpha _n)\big [\theta _n||x_n-p||^2+(1-\theta _n) D^2(Ty_n, p)\big ]\nonumber \\&-(1-\alpha _n)\theta _n (1-\theta _n) ||v_n-x_n||^2 \\\le & {} \alpha _n||u-p||^2+(1-\alpha _n)\theta _n||x_n-p||^2 + (1-\alpha _n) (1-\theta _n)\nonumber \\&\times \big [||y_n-p||^2+||y_n-v_n||^2\big ]-(1-\alpha _n)\theta _n (1- \theta _n) ||v_n-x_n||^2\nonumber \\\le & {} \alpha _n||u-p||^2+ (1-\alpha _n) (1-\theta _n)\bigg (||y_n-p||^2+||y_n-v_n||^2\bigg )\nonumber \\&+(1-\alpha _n)\theta _n\bigg ( ||x_n-p||^2 -(1-\theta _n) ||v_n-x_n||^2\bigg ) \end{aligned}$$
(3.2)

and

$$\begin{aligned} ||y_{n}-p||^2= & {} ||\lambda _n (x_n-p)+(1-\lambda _n)(u_{n}-p)||^2\nonumber \\= & {} \lambda _n ||x_n-p||^2+(1-\lambda _n)||u_{n}-p||^2\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\\le & {} \lambda _n ||x_n-p||^2+(1-\lambda _n)D^2(Tx_n,p)^2\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\\le & {} \lambda _n||x_n-p||^2+(1-\lambda _n) \big [||x_{n}-p||^2+ ||x_n-u_n||^2\big ]\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\= & {} ||x_n-p||^2+(1-\lambda _n)^2||x_n-u_n||^2. \end{aligned}$$
(3.3)

On the other hand, since T is L-Lipschitz, it follows from (3.1) and Lemma 2.3 that

$$\begin{aligned} ||y_{n}-v_n||^2= & {} ||\lambda _n(x_n-v_n)+(1-\lambda _n) (u_{n}-v_n)||^2\nonumber \\= & {} \lambda _n ||x_n-v_n||^2+(1-\lambda _n) ||u_{n}-v_n||^2\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\\le & {} \lambda _n ||x_n-v_n||^2+(1-\lambda _n) \bigg (D(Tx_{n}, Ty_n)+||x_n-y_n||\bigg )^2\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\\le & {} \lambda _n ||x_n-v_n||^2+(1-\lambda _n) (L+1)^2||x_{n}-y_n||^2\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\= & {} \lambda _n ||x_n-v_n||^2+(1-\lambda _n)^2(L+1)^2||x_{n}-u_n||^2\nonumber \\&-\lambda _n (1-\lambda _n )||x_n-u_n||^2\nonumber \\= & {} \lambda _n||x_n-v_n||^2\nonumber \\&-(1-\lambda _n) \big (\lambda _n-(L+1)^2(1-\lambda _n)^2\big )||x_{n}-u_n||^2. \end{aligned}$$
(3.4)

Thus, from (3.2), (3.3) and (3.4), we obtain

$$\begin{aligned} ||x_{n+1}-p||^2\le & {} \alpha _n||u-p||^2+ (1-\alpha _n)(1-\theta _n) \bigg (||x_n-p||^2\nonumber \\&+(1-\lambda _n)^2||x_n-u_n||^2 \bigg ) + (1-\alpha _n)(1-\theta _n)\bigg (\lambda _n||x_n-v_n||^2 \nonumber \\&- (1-\lambda _n)(\lambda _n-(L+1)^2(1-\lambda _n)^2)||x_n-u_n||^2\bigg ) \nonumber \\&+(1-\alpha _n)\theta _n||x_n-p||^2 -(1-\alpha _n)\theta _n (1-\theta _n) ||v_n-x_n||^2\nonumber \\= & {} \alpha _n||u-p||^2+ (1-\alpha _n)||x_n-p||^2- (1-\alpha _n)(1-\theta _n)(1-\lambda _n)\nonumber \\&\times \bigg (1-(L+1)^2(1-\lambda _n)^2-2(1-\lambda _n)\bigg )||x_n-u_n||^2\nonumber \\&+(1-\alpha _n)(1-\theta _n)(\lambda _n-\theta _n)||v_n-x_n||^2. \end{aligned}$$
(3.5)

Since for each \( n\ge 0, ~~\theta _n\ge \lambda _n\) and

$$\begin{aligned} 1-2(1-\lambda _n)-(L+1)^2(1-\lambda _n)^2\ge 1-2(1-\beta )-(L+1)^2(1-\beta )^2>0, \end{aligned}$$
(3.6)

inequality (3.5) implies that

$$\begin{aligned} ||x_{n+1}-p||^2\le & {} \alpha _n||u-p||^2+ (1-\alpha _n)||x_n-p||^2. \end{aligned}$$
(3.7)

Hence, by induction,

$$\begin{aligned} ||x_{n+1}-p||^2\le & {} \max \{ ||u-p||^2, ||x_0-p||^2 \}, \forall n\ge 0. \end{aligned}$$

This implies that the sequence \(\{x_n\}\) is bounded which in turn implies that \(\{y_n\}\) is bounded. \(\square \)

Theorem 3.2

Let C be a nonempty, closed and convex subset of a real Hilbert space H\(T:C\rightarrow CB(H)\) be L-Lipschitz pseudocontractive inward mapping with \(\displaystyle F(T)\not =\emptyset .\) Let \(\{x_n\}\) be a sequence defined by (3.1) such that \(\displaystyle \lim _{n\rightarrow \infty } \alpha _n=0\) and \(\sum \alpha _n=\infty .\) Suppose that \(Tp=\{p\}, \forall p\in F(T)\) and \(I-T\) is demiclosed at zero. If there exists \(\epsilon >0\) such that \(\theta _n\le 1-\epsilon , \forall n\ge 0\), then \(\{x_{n}\}\) converges strongly to a fixed point \(x^*\) of T nearest to u in the sense that \(x^*=P_{F(T)}(u).\)

Proof

Let \(x^*=P_{F(T)}(u)\). Then by (3.1), Lemma 2.1, Lemma 2.3 and pseudocontractivity of T, we have

$$\begin{aligned} ||x_{n+1}-x^*||^2= & {} ||\alpha _n u+ (1-\alpha _n)\big (\theta _nx_n+(1-\theta _n)v_n\big ) -x^*||^2 \\= & {} ||\alpha _n (u-x^*) + (1-\alpha _n) \big [\theta _nx_n+ (1-\theta _n)v_n-x^*\big ]||^2\nonumber \\\le & {} (1-\alpha _n) || \theta _nx_n +(1-\theta _n)v_n -x^*||^2\\&+2\alpha _n\langle u- x^*, x_{n+1}-x^*\rangle \\= & {} (1-\alpha _n)\theta _n||x_n-x^*||^2+(1-\alpha _n)(1-\theta _n) ||v_n-x^*||^2 \\&-(1-\alpha _n)\theta _n(1-\theta _n)||v_n-x_n||^2 +2\alpha _n\langle u- x^*, x_{n+1}-x^*\rangle \\\le & {} (1-\alpha _n)\theta _n||x_n-x^*||^2+(1-\alpha _n)(1-\theta _n) D^2(Ty_n, x^*) \\&-(1-\alpha _n)\theta _n(1-\theta _n)||v_n-x_n||^2 +2\alpha _n\langle u- x^*, x_{n+1}-x^*\rangle \\\le & {} (1-\alpha _n)\theta _n ||x_n-x^*||^2 \\&+(1-\alpha _n)(1-\theta _n)\big [ ||y_n-x^*||^2+||y_n-v_n||^2\big ] \\&-(1-\alpha _n)\theta _n(1-\theta _n)||v_n-x_n||^2+2\alpha _n\langle u-x^*, x_{n+1}-x^*\rangle . \end{aligned}$$

Moreover, since \(x^*\in F(T),\) from (3.3) and (3.4) it follows that

$$\begin{aligned} ||y_{n}-x^*||^2\le & {} ||x_n-x^*||^2+(1-\lambda _n)^2||x_n-u_n||^2 \end{aligned}$$

and

$$\begin{aligned} ||y_{n}-v_n||^2\le & {} \lambda _n||x_n-v_n||^2-(1-\lambda _n) \bigg (\lambda _n-(L+1)^2(1-\lambda _n)^2\bigg )||x_{n}-u_n||^2. \end{aligned}$$

Hence, by substitution, we obtain

$$\begin{aligned} ||x_{n+1}-x^*||^2\le & {} (1-\alpha _n)\theta _n ||x_n-x^*||^2+ (1-\alpha _n)(1-\theta _n) \nonumber \\&\times \big [ ||x_n-x^*||^2+ (1-\lambda _n)^2||x_n-u_n||^2\big ]+(1-\alpha _n)(1-\theta _n)\nonumber \\&\times \big [ \lambda _n ||x_n-v_n||^2-(1-\lambda _n)(\lambda _n-(L+1)^2(1-\lambda _n)^2) ||x_n-u_n||^2\big ] \nonumber \\&-(1-\alpha _n)\theta _n(1-\theta _n)||v_n-x_n||^2+2\alpha _n\langle u-x^*, x_{n+1}-x^*\rangle \nonumber \\= & {} (1-\alpha _n) ||x_n-x^*||^2 -(1-\alpha _n)(1-\theta _n)(1-\lambda _n) \nonumber \\&\times [ 1-(L+1)^2(1-\lambda _n)^2 -2(1-\lambda _n)]||x_n-u_n||^2 \nonumber \\&+(1-\alpha _n) (1-\theta _n) ( \lambda _n-\theta _n)||x_n-v_n||^2\nonumber \\&+2\alpha _n\langle u-x^*, x_{n+1}-x^*\rangle \end{aligned}$$
(3.8)
$$\begin{aligned}\le & {} (1-\alpha _n) ||x_n-x^*||^2 +2\alpha _n\langle u-x^*, x_{n}-x^*\rangle \nonumber \\&+2\alpha _n||u-x^*||||x_{n+1}-x_n||. \end{aligned}$$
(3.9)

Next, we consider two possible cases.

Case 1. Suppose that there exists \(n_0\in {N}\) such that \(\{||x_n-x^*||\}\) is decreasing for all \(n\ge n_0\). Then it follows that \(\{||x_n-x^*||)\}\) is convergent. Thus, (3.8), (3.6) and the fact that \(\theta _n\ge \lambda _n\) and \(\displaystyle \lim _{n\rightarrow \infty } \alpha _n=0\) imply that

$$\begin{aligned} x_n-u_n\rightarrow 0 \text{ as } n\rightarrow \infty \text{. } \end{aligned}$$
(3.10)

Combining this with (3.1) yields

$$\begin{aligned}&||y_n-x_n||=(1-\lambda _n)||x_n-u_n||\rightarrow 0 \text{ as } n\rightarrow \infty \text{, } \end{aligned}$$
(3.11)

and so from Lipschitz continuity of T, we have

$$\begin{aligned} ||v_n-x_n||\le & {} ||v_n-u_n||+||u_n-x_n||\nonumber \\\le & {} D(Ty_n,Tx_n)+||x_n-y_n||+||u_n-x_n||\nonumber \\\le & {} (L+1)||y_n-x_n||+||u_n-x_n||\rightarrow 0 \text{ as } n\rightarrow \infty \text{. } \end{aligned}$$
(3.12)

Thus, from (3.1), it follows that

$$\begin{aligned} ||x_{n+1}-x_n||\le \alpha _n||u-x_n||+(1-\alpha _n)(1-\theta _n)||v_n-x_n||\rightarrow 0. \end{aligned}$$
(3.13)

On the other hand, since \(\{x_{n} \}\) is bounded and H is reflexive, we can choose a subsequence \(\{x_{n_i}\}\) of \(\{x_{n}\}\) such that

$$\begin{aligned}x_{n_i}\rightharpoonup w \text{ and } \displaystyle \limsup _{n\rightarrow \infty }\langle u-x^*, x_{n}-x^*\rangle =\lim _{i\rightarrow \infty }\langle u-x^*,x_{n_i}-x^*\rangle .\end{aligned}$$

Also from (3.1) and (3.10), we have \(d(x_n, Tx_n)\le ||x_n-u_n||\rightarrow 0.\) Then since \(I-T\) is demiclosed at 0, it follows that \(w\in F(T).\) Therefore, by Lemmas 2.7 and 2.2, we obtain

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle u-x^*,x_{n}-x^*\rangle= & {} \lim _{i\rightarrow \infty }\langle u-x^*, x_{n_i}-x^*\rangle \nonumber \\= & {} \langle u-x^*, w-x^*\rangle \le 0. \end{aligned}$$
(3.14)

Then it follows from (3.9), (3.14) and Lemma 2.10 that \(||x_n-x^*||\rightarrow 0\) as \(n\rightarrow \infty \). Consequently, \(x_n\rightarrow x^*=P_{F(T)}(u)\).

Case 2. Suppose that there exists a subsequence \(\{n_i\}\) of \(\{n\}\) such that

$$\begin{aligned} ||x_{n_i}-x^*|| <||x_{n_i+1}-x^*||, \forall i\in {N}. \end{aligned}$$

Then by Lemma 2.11, there exists a nondecreasing sequence \(\{m_k\}\subset {N}\) such that \(m_k\rightarrow \infty \) and

$$\begin{aligned} ||x_{m_k}-x^*||\le ||x_{m_k+1}-x^*|| \text{ and } ||x_{k}-x^*||\le ||x_{m_k+1}-x^*||, \forall k\in {N}. \end{aligned}$$
(3.15)

Thus, by (3.8) and (3.6), we have \(||x_{m_k}-u_{m_k}||]\rightarrow 0 \text{ as } k\rightarrow \infty ,\) which implies that

$$\begin{aligned} d(x_{m_k}, Tx_{m_k})\rightarrow 0 \text{ as } k\rightarrow \infty . \end{aligned}$$

Then using the methods we used in Case 1, we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }\langle u-x^*, x_{m_k}-x^*\rangle \le 0. \end{aligned}$$
(3.16)

Now, from (3.9), we have

$$\begin{aligned} ||x_{m_k+1}-x^*||^2\le & {} (1-\alpha _{m_k})||x_{m_k}-x^*||^2+ 2\alpha _{m_k}\langle u-x^*, x_{m_k}-x^*\rangle \nonumber \\&+ 2\alpha _{m_k}|| u-x^*|| || x_{m_k+1}-x_{m_k}||, \end{aligned}$$
(3.17)

and hence (3.15) and (3.17) imply that

$$\begin{aligned} \alpha _{m_k}||x_{m_k}-x^*||^2\le & {} ||x_{m_k}-x^*||^2-||x_{{m_k}+1}-x^*||^2 +2\alpha _{m_k}\langle u-x^*, x_{m_k}-x^*\rangle \\&+2\alpha _{m_k}|| u-x^*|| || x_{m_k+1}-x_{m_k}||\\\le & {} 2 \alpha _{m_k}\langle u-x^*, x_{m_k}-x^*\rangle + 2\alpha _{m_k}||u-x^*|| || x_{m_k+1}-x_{m_k}||. \end{aligned}$$

Then since \(\alpha _{m_k}>0\), we have

$$\begin{aligned} ||x_{m_k}-x^*||^2\le & {} 2 \langle u-x^*, x_{m_k}-x^*\rangle + 2||u-x^*|| || x_{m_k+1}-x_{m_k}||. \end{aligned}$$

Thus, using (3.13) and (3.16), we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }||x_{m_k}-x^*||^2\le 0 \text{ and } \text{ hence } || x_{m_k}-x^*||\rightarrow 0 \text{ as } k\rightarrow \infty . \end{aligned}$$

This together with (3.17) imply that \(|| x_{{m_k}+1}-x^*||\rightarrow 0\) as \(k\rightarrow \infty \). But, since \(|| x_{k}-x^*||\le || x_{{m_k}+1}-x^*||\), for all \(k\in {N}\), it follows that \(x_k\rightarrow x^*=P_{F(T)}(u).\) Therefore, the above two cases imply that \(\{x_n\}\) converges strongly to the fixed point of T nearest to u. \(\square \)

If T is assumed to be k-strictly pseudocontractive, then T is pseudocontractive and so, we have the following corollary.

Corollary 3.3

Let C be a nonempty, closed and convex subset of a real Hilbert space H and \(T:C\rightarrow CB(H)\) be L-Lipschitz k-strictly pseudocontractive inward mapping with \(\displaystyle F(T)\not =\emptyset .\) Let \(\{x_n\}\) be a sequence defined by (3.1) such that \(\displaystyle \lim _{n\rightarrow \infty } \alpha _n=0\) and \(\sum \alpha _n=\infty .\) Suppose that \(Tp=\{p\}, \forall p\in F(T)\) and \(I-T\) is demiclosed at zero. If there exists \(\epsilon >0\) such that \(\theta _n\le 1-\epsilon \, \forall n\ge 0\), then \(\{x_{n}\}\) converges strongly to a fixed point of T nearest to u.

Definition 3.4

A point \(x\in F(T)\) is said to be a minimum norm point of F(T) if \(||x||\le ||y||,\forall y\in F(T).\)

If C contains the zero element, then we have the following theorem for finding a point with minimum-norm in the set of fixed points of a Lipschitz pseudocontractive mapping.

Theorem 3.5

Let C be a nonempty, closed and convex subset of a real Hilbert space H containing 0,  \(T:C\rightarrow CB(H)\) be L-Lipschitz pseudocontractive inward mapping and let \(\{x_n\}\) be a sequence defined by (3.1) with \(u=0.\) Suppose that \(\displaystyle F(T)\not =\emptyset , Tp=\{p\}, \forall p\in F(T)\) and \(I-T\) is demiclosed at zero. If there exists \(\epsilon >0\) such that \(\theta _n\le 1-\epsilon \, \forall n\ge 0\), then \(\{x_{n}\}\) converges strongly to the minimum-norm point in F(T).

Proof

By Theorem 3.2, \(x_n\) converges to a fixed point \(x^*\) of T nearest to 0. Thus, \(||x^*||=||x^*-0||\le ||x-0||=||x||, \forall x\in C\) and hence the proof. \(\square \)

3.2 Strong convergence results without end point condition

Before introducing our algorithm, we prove the following lemmas.

Lemma 3.6

Let C be a nonempty, closed convex subset of a real Hilbert space H and \(T:C\rightarrow Prox(H)\) be a k-strictly pseudocontractive multi-valued mapping. Then T is Lipschitz mapping.

Proof

Let \(x, y\in C\) and \(u\in Tx.\) Then by Lemma 2.6, there is \(v\in Ty\) such that

$$\begin{aligned} ||u-v||\le D(Tx,Ty). \end{aligned}$$

Then since T is k-strictly pseudocontractive, we have

$$\begin{aligned} D^2(Tx,Ty)\le & {} ||x-y||^2+k||x-y-(u-v)||^2\\\le & {} \, \,\bigg (||x-y||+\sqrt{k}\big (||x-y||+||u-v||\big )\bigg )^2\\\,\le & {} \,\bigg (||x-y||+\sqrt{k}\big (||x-y||+D^2(Tx,Ty)\big )\bigg )^2 \end{aligned}$$

which implies that

$$\begin{aligned} D(Tx,Ty)\le & {} \frac{1+\sqrt{k}}{1-\sqrt{k}}||x-y||. \end{aligned}$$

Therefore, T is Lipschitzian with Lipschitz constant \(L=\frac{1+\sqrt{k}}{1-\sqrt{k}}.\) \(\square \)

Lemma 3.7

Let \(T: C\rightarrow Prox(H)\) be a multi-valued mapping such that \(P_T\) is k-strictly pseudocontractive. Then \(I-P_T\) is demiclosed at zero.

Proof

Let \(\{x_n\}\) be a sequence in C such that \(x_n\rightharpoonup p\) and \(d(x_n,P_Tx_n)\rightarrow 0.\) Let \(y\in P_Tp.\) By Lemma 2.6, for each \(n\in {N},\) there exists \(y_n\in P_Tx_n\) such that

$$\begin{aligned} ||y_n-y||\le D(P_Ty_n,P_Tp). \end{aligned}$$

Also, since \(y_n\in P_Tx_n,\) it follows that

$$\begin{aligned} ||x_n-y_n||=d(x_n,P_Tx_n)\rightarrow 0. \end{aligned}$$

Now, for each \(x\in H,\) define \(f: H\rightarrow [0,\infty ]\) by

$$\begin{aligned} f(x)=\limsup _{n\rightarrow \infty }||x_n-x||^2. \end{aligned}$$
(3.18)

Then from Lemma 2.9, we obtain

$$\begin{aligned} f(x)=\limsup _{n\rightarrow \infty }||x_n-p||^2+||p-x||^2, \forall x\in H, \end{aligned}$$

which implies that

$$\begin{aligned} f(x)=f(p)+||p-x||^2, \forall x\in H. \end{aligned}$$

Hence, we obtain that

$$\begin{aligned} f(y)=f(p)+||p-y||^2. \end{aligned}$$
(3.19)

In addition, by the definition of k-strictly pseudocontractive mapping, we have

$$\begin{aligned} f(y)= & {} \limsup _{n\rightarrow \infty }||x_n-y||^2\\\nonumber \\= & {} \limsup _{n\rightarrow \infty }||x_n-y_n+y_n-y||^2\\\nonumber \\= & {} \limsup _{n\rightarrow \infty }||y_n-y||^2\\\nonumber \\\le & {} \limsup _{n\rightarrow \infty }D^2(P_Tx_n, P_Tp)\\\nonumber \\\le & {} \limsup _{n\rightarrow \infty }\bigg (||x_n- p||^2+k||x_n-y_n-(p-y)||^2\bigg )\\\nonumber \\\le & {} \limsup _{n\rightarrow \infty }\bigg (||x_n- p||^2+k\big (||x_n-y_n||+||p-y||\big )^2\bigg )\\\nonumber \\= & {} \limsup _{n\rightarrow \infty }||x_n- p||^2+k||p-y||^2\\\nonumber \\= & {} f(p)+k||p-y||^2. \end{aligned}$$
(3.20)

Then it follows from (3.19) and (3.20) that \((1-k)||p-y||^2=0\) and hence, \(p = y \in P_Tp.\) Therefore, \(I-P_T\) is demiclosed at zero. \(\square \)

Now, we present our algorithm as follows. Let \(T:C\rightarrow Prox(H)\) be a multi-valued mapping such that \(P_T\) is inward Lipschitz mapping with Lipschitz constant L and \(\beta \in \bigg (1-\frac{1}{1+\sqrt{L^2+1}},1\bigg ).\) For a sequence \(\{\alpha _n\}\) in (0, 1),  we define Halpern–Ishikawa type iterative scheme as follows:

Given \(u,x_0\in C,\) let \(u_0\in P_Tx_0\) and

$$\begin{aligned} h_{u_0}(x_0):=\inf \lbrace \lambda \ge 0: \lambda x_0+ (1-\lambda )u_0\in C \rbrace . \end{aligned}$$

Now, if we choose \(\lambda _0\in [\max \{\beta ,h_{u_0}(x_0)\},1),\) then it follows from Lemma 2.4 that

$$\begin{aligned} y_0:=\lambda _0 x_0+ (1-\lambda _0)u_0\in C. \end{aligned}$$

By Lemma 2.6, we can choose \(v_0\in P_Ty_0\) such that

$$\begin{aligned} ||u_0-v_0||\le D(P_Tx_0,P_Ty_0). \end{aligned}$$

Let \(g_{v_0}(y_0):=\inf \lbrace \theta \ge 0: \theta x_0 +(1-\theta )v_0\in C\rbrace .\) If we choose \(\theta _0\in [\max \{\lambda _0,g_{v_0}(y_0)\},1),\) then by Lemma 2.4, \(\theta _0 x_0+(1-\theta _0)v_0\in C.\) Thus, it follows that

$$\begin{aligned} x_{1}:=\alpha _0 u+ (1-\alpha _0)\big (\theta _0x_0+(1-\theta _0)v_0\big )\in C.\end{aligned}$$

Inductively, \(\{x_n\}\) is defined as

$$\begin{aligned} \quad \quad \quad \left\{ \begin{array}{lll} \lambda _{n}\in [\max \{\beta , h_{u_n}(x_{n}\}, 1); \\ y_n=\lambda _nx_n+(1-\lambda _n) u_n;\\ \theta _{n}\in [\max \{\lambda _n,g_{v_n}(y_{n})\},1);\\ x_{n+1}=\alpha _n u + (1-\alpha _n)\big (\theta _nx_n+(1-\theta _n) v_n\big ),n\ge 0, \end{array}\right. \end{aligned}$$
(3.21)

where \(u_n\in P_Tx_n\) and \(y_n\in P_Ty_n\) such that \(||u_n-v_n||\le D(P_Tx_n, P_Ty_n), \)

  \(h_{u_n}(x_n):=\inf \{\lambda \ge 0: \lambda x_n+ (1-\lambda ) u_n \in C\}\) and

  \(g_{v_n}(y_n):=\inf \{\theta \ge 0: \theta x_n+ (1-\theta ) v_n \in C\}.\)

Theorem 3.8

Let C be a nonempty, closed and convex subset of a real Hilbert space H\(T:C\rightarrow Prox(H)\) be a multi-valued mapping such that \(P_T\) is k-strictly pseudocontractive inward mapping and \(\displaystyle F(T)\not =\emptyset .\) Let \(\{x_n\}\) be a sequence defined by (3.21) such that \(\displaystyle \lim _{n\rightarrow \infty } \alpha _n=0\) and \(\sum \alpha _n=\infty .\) If there exists \(\epsilon >0\) with \(\theta _n\le 1-\epsilon \, \forall n\ge 0\), then \(\{x_{n}\}\) converges strongly to a fixed point of T nearest to u.

Proof

By Lemma 3.6, \(P_T\) is Lipschitz with Lipschitz constant \(L=\frac{1+\sqrt{k}}{1-\sqrt{k}}\) and \(I-P_T\) is demiclosed at zero by Lemma 3.7. Moreover, by Lemma 2.8, \(F(T)=F(P_T)\) and \(P_Tp=\{p\}\) for all \(p\in F(T).\) The rest of the proof is very similar to the proof of Theorem 3.2. \(\square \)

In Theorem 3.8, if \(P_T\) is assumed to be nonexpansive mapping, then \(P_T\) is k-strictly pseudocontractive and hence we have the following corollary.

Corollary 3.9

Let C be a nonempty, closed and convex subset of a real Hilbert space H\(T:C\rightarrow Prox(H)\) be a multi-valued mapping such that \(P_T\) is nonexpansive inward mapping and \(\displaystyle F(T)\not =\emptyset .\) Let \(\{x_n\}\) be a sequence defined by (3.21) such that \(\displaystyle \lim _{n\rightarrow \infty } \alpha _n=0\) and \(\sum \alpha _n=\infty .\) If there exists \(\epsilon >0\) with \(\theta _n\le 1-\epsilon \, \forall n\ge 0\), then \(\{x_{n}\}\) converges strongly to a fixed point of T nearest to u.

The method of the proof of Theorem 3.2 also provides the following result.

Theorem 3.10

Let C be a nonempty, closed and convex subset of a real Hilbert space H and \(T:C\rightarrow \) Prox(H) be a multi-valued mapping such that \(P_T\) is an inward Lipschitz pseudocontractive mapping. Suppose that \(F(T)\not =\emptyset ,\) \(I-P_T\) is demiclosed at 0 and \(\{x_n\}\) be a sequence defined by (3.21). If there exists \(\epsilon >0\) such that \(\theta _n\le 1-\epsilon \, \forall n\ge 0\), then \(\{x_{n}\}\) converges strongly to a fixed point of T nearest to u.

Remark 3.11

Note that, in Algorithms (3.1) and (3.21), the coefficients \(\lambda _{n}\) and \(\theta _{n}\) can be chosen simply as follows: \(\lambda _n=\max \{\beta ,h_{u_n}(x_n)\}\) and \(\theta _{n}=\max \{\lambda _n,g_{v_n}(y_{n})\}.\)

4 Numerical example

Now, we give an example of a nonlinear mapping which satisfies the conditions of Theorem 3.2.

Example 4.1

Let \(H=\mathrm{IR}R\) with Euclidean norm. Let \(C=[-1,\frac{1}{2}]\) and \(T:C\rightarrow \mathrm{I\!R}\) be defined by

$$\begin{aligned} Tx= \, \left\{ \begin{array}{lll} \{-x,0\}, \, \, x\in [-1,0), \\ x,\,\,\,\,x\in [0, \frac{1}{2}]. \end{array}\right. \end{aligned}$$
(4.1)

Then we observe that T satisfies the inward condition and \(F(T)=[0,\frac{1}{2}].\) We first show that T is Lipschitz pseudocontractive mapping. We consider the following cases.

Case 1: Let \(x,y \in [-1,0).\) Then \(Tx=\{-x,0\}\) and \(Ty=\{-y,0\}.\) Thus, we have

$$\begin{aligned} D(Tx,Ty)= & {} \max \bigg \{\sup _{a\in Ty} d(a,Tx), \sup _{b\in Tx}d(b,Ty)\bigg \}\\\\= & {} \max \{\min \{|x-y|,|y|\}, \min \{|x-y|, |x|\}\}\\\\= & {} \, \left\{ \begin{array}{lll} \max \{\min \{|x-y|,|y|\}, |x-y|\}, \, \, \text{ if } x\le y, \\ \max \{|x-y|, \min \{|x-y|,|x|\}, \, \, \, \text{ if } y\le x, \end{array}\right. \\= & {} |x-y|. \end{aligned}$$

Case 2: Let \(x,y \in [0,\frac{1}{2}].\) Then \(Tx=\{x\}\) and \(Ty=\{y\}.\) Thus, we have

$$\begin{aligned} D(Tx,Ty)= & {} \max \bigg \{\sup _{a\in Ty} d(a,Tx), \sup _{b\in Tx}d(b,Ty)\bigg \}\\= & {} |x-y|. \end{aligned}$$

Case 3: Let \(x\in [-1,0)\) and \(y\in [0,\frac{1}{2}].\) Then \(Tx=\{-x,0\}\) and \(Ty=\{y\}.\) Thus, we have

$$\begin{aligned} D(Tx,Ty)= & {} \max \bigg \{\sup _{a\in Ty} d(a,Tx), \sup _{b\in Tx}d(b,Ty)\bigg \}\\= & {} \max \{\min \{|x+y|,y\}, \max \{|x+y|, y\}\}\\\le & {} |x-y|. \end{aligned}$$

From the above cases, it follows that T is L-Lipschitz pseudocontractive mapping with Lipschitz constant \(L=1.\) Then \(1-\frac{1}{1+\sqrt{(L+1)^2+1}}=0.691.\) Thus, we can choose \(\beta =\frac{5}{6}\) and \(\alpha _n=\frac{2}{n+5}.\) Now, let \(x_0=-1\) and \(u=0.5.\) Then \(Tx_0=\{0,1\}.\) Take \(u_0=0.\) Then we have

$$\begin{aligned} h_{u_0}(x_0)= & {} \inf \{\lambda \ge 0: \lambda x_0+(1-\lambda )u_0\in C\}\\= & {} \inf \big \{\lambda \ge 0: -\lambda \in C\big \}\\= & {} 0. \end{aligned}$$

Let \(\lambda _0=\max \{\beta ,h_{u_0}(x_0)\}=\frac{5}{6}.\) Then \(y_0=\lambda _0 x_0+(1-\lambda _0)u_0=-\frac{5}{6}\) and

\(Ty_0=\{0,\frac{5}{6}\}.\) If we take \(v_{0}=0,\) then we get

$$\begin{aligned} g_{v_0}(y_0)=\inf \{\theta \ge 0: \theta x_0+(1-\theta )v_0\in C\}=0. \end{aligned}$$

If we choose \(\theta _0=\max \{\lambda _0, g_{v_0}(y_0)\}=\frac{5}{6},\) then we have

$$\begin{aligned}x_1=\alpha _0 u+(1-\alpha _0)[\theta _0x_0+(1-\theta _0)v_0]=-\frac{3}{10}=-0.3. \end{aligned}$$

Then \(Tx_1=\{0,\frac{3}{10}\}.\) If we choose \(u_1=0,\) the we obtain \(h_{u_1}(x_1)=0.\) Now, we can choose \(\lambda _1=\frac{5}{6},\) which yields

$$\begin{aligned}y_1=\lambda _1x_1+(1-\lambda _1)u_1=-\frac{1}{4} \text{ and } Ty_1=\left\{ 0,\frac{1}{4}\right\} .\end{aligned}$$

Again, we can choose \(v_1=0\) and \(\theta _1=\frac{5}{6},\) which yields \(x_2=0.\) Then \(Tx_2=\{0\}.\) In this case \(u_2=Tx_2=0\) and hence \(h_{u_2}(x_2)=0.\) Thus, we can choose \(\lambda _2=\frac{5}{6}\) which yields \(y_2=0\) and \(x_3=0.14\) for \(\theta _2=\frac{5}{6}.\) In general, we observe that for \( x_0=-1,u=0.5\) and \(\alpha _{n}=\frac{2}{n+5}\), we can choose \(\lambda _n=\theta _{n}=\frac{5}{6}.\) Thus, all the conditions of Theorem 3.2 are satisfied and \(x_n\) converges to \(0.5=P_{F(T)}u\) (see Fig. 1).

Similarly, for \(x_0=0.5\) and \(u=0,\) the sequence \(\{x_n\}\) converges to \(0=P_{F(T)}u.\) Moreover, for \(x_0=-0.5\) and \(u=-1,\) \(x_n\) converges to \(0=P_{F(T)}u\) (see Fig. 1 which is obtained using MATLAB version 8.5.0.197613(R2015a)).

Fig. 1
figure 1

Convergence of \(x_n\) for different values of the initial point \(x_0\) and the constant u

5 Conclusion

In this paper, we have constructed Halpern–Ishikawa type iterative methods for approximating fixed points of multi-valued pseudocontractive non-self mappings in the setting of real Hilbert spaces. Strong convergence results of the scheme to a fixed points of multi-valued Lipschitz pseudocontractive mappings are obtained under appropriate conditions on the iterative parameter and an end point condition on the mappings under consideration. In addition, a Halpern–Ishikawa type iterative method for approximating fixed points of multi-valued k-strictly pseudocontractive mappings is introduced and strong convergence results of the scheme are obtained without the end point condition. Our results extend and generalize many of the results in the literature (see, e.g., [6, 7, 22, 23, 25, 27,28,29]). More particularly, Theorem 3.2 extends Theorem 3.2 of Zegeye and Tufa [28] from single-valued mapping to multi-valued mapping. Thus, if we assume that T is single-valued mapping in Theorem 3.2, then we get Theorem 3.2 of Zegeye and Tufa [28]. Theorem 3.8 extends Theorem 8 of Colao et al. [7] from single-valued mapping to multi-valued mapping.