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(A, B), 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 map** \(T:C\rightarrow 2^H\) 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 map** T is called nonexpansive map**. It is immediate from the definition that every nonexpansive map** is Lipschitz map**.

A map** \(T:C\rightarrow 2^H\) 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 map**s includes the class of multi-valued k-strictly pseudocontractive map**s and hence the class of multi-valued nonexpansive map**s.

Given a multi-valued map** \(T:C\rightarrow 2^H,\) 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 map** 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 map**. Clearly, every nonexpansive map** T with \(F(T)\not =\emptyset \) is quasi-nonexpansive map**. 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 map**s. As a result, researchers have studied the existence of fixed points and their approximations for different types of multi-valued map**s (see, e.g., [1, 3,4,5, 12, 13, 18, 19] and the references therein). For approximating fixed points of single-valued map**s, 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 map** 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 map**s (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 map** 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 map**. 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 map**s. For approximating fixed points of single-valued Lipschitz pseudocontractive self-map** 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 map**s 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 H,  T is nonexpansive map** 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 map**s 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 map** 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-map**s 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 map**s in a real Hilbert space. They obtained convergence results of the scheme to fixed points of the map**s.

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

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 map**s in real Hilbert spaces. Strong convergence of the schemes to fixed points of the map**s 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 map** \(T:C\rightarrow 2^H\) 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 map** \(I -T,\) where I is an identity map** 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 map** 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 map**. 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 map**. 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 map** \(T:C\rightarrow CB(H)\) 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 map** \(T:C\rightarrow Prox(H)\) 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 map** 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 map** 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 map** 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 map** 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 map**.

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 map** 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 map**. Then T is Lipschitz map**.

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 map** 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 map**, 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 map** such that \(P_T\) is inward Lipschitz map** 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 map** such that \(P_T\) is k-strictly pseudocontractive inward map** 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 map**, 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 map** such that \(P_T\) is nonexpansive inward map** 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 map** such that \(P_T\) is an inward Lipschitz pseudocontractive map**. 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 map** 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 map**. 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 map** 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

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5 Conclusion

In this paper, we have constructed Halpern–Ishikawa type iterative methods for approximating fixed points of multi-valued pseudocontractive non-self map**s in the setting of real Hilbert spaces. Strong convergence results of the scheme to a fixed points of multi-valued Lipschitz pseudocontractive map**s are obtained under appropriate conditions on the iterative parameter and an end point condition on the map**s under consideration. In addition, a Halpern–Ishikawa type iterative method for approximating fixed points of multi-valued k-strictly pseudocontractive map**s 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 map** to multi-valued map**. Thus, if we assume that T is single-valued map** 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 map** to multi-valued map**.