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On Noor-type iteration schemes for multivalued mappings in \(\operatorname{CAT}(0)\) spaces

Abstract

In this paper, we prove strong convergence theorems for Noor-type iteration schemes involving quasi-nonexpansive multivalued mappings in the framework of \(\operatorname{CAT}(0)\) spaces. The results we obtain are generalizations of Panyanak (Nonlinear Anal. 70:1547-1556, 2009), Sastry and Babu (Czechoslov. Math. J. 55:817-826, 2005), Shazhad and Zegeye (Nonlinear Anal. 71:838-844, 2009), Song and Wang (Comput. Math. Appl. 55:2999-3002, 2008; Nonlinear Anal. 70:1547-1556, 2009) and many others in the sense of Noor-type iteration process in the setting of \(\operatorname{CAT}(0)\) spaces.

1 Introduction

The study of metric spaces without linear structure has played a vital roll in various branches of pure and applied sciences. In particular, existence and approximation results in \(\operatorname{CAT}(0)\) spaces for classes of single-valued and multivalued mappings have been studied extensively by many researchers (see [1–8]).

Iteration schemes for numerical reckoning fixed points of diverse classes of nonlinear operators are available in the literature. The class of nonexpansive mappings via iteration methods has extensively been studied in this regard (see Tan and Xu [9]; Thakur et al. [10, 11]). The class of pseudocontractive mappings in their relation with iteration procedures has been studied by several researchers under suitable conditions (see Yao et al. [12, 13]; Thakur et al. [14, 15]; Dewangan et al. [16, 17]) and applications to variational inequalities are also considered [18, 19]. For nonexpansive multivalued mappings, Sastry and Babu [20] defined a Mann and Ishikawa iteration process in Hilbert spaces. Panyanak [21] and Song and Wang [22] (see also [23]) extended the result of Sastry and Babu [20] to uniformly convex Banach spaces. Recently, Shahzad and Zegeye [24] extended and improved results of [20–23].

In [25], Dhompongsa and Panyanak established Δ-convergence theorems for the Mann and Ishikawa iterations for nonexpansive single-valued mappings in \(\operatorname{CAT}(0)\) spaces. Inspired by Song and Wang [22], Laowang and Panyanak [2] extended the result of Dhompongsa and Panyanak [25] for multivalued nonexpansive mappings in a \(\operatorname{CAT}(0)\) space.

It is important to note here that several iteration processes having various number of steps have been employed for the purpose of the approximation of fixed points for various classes of nonlinear operators. The very famous Mann iteration process is a one-step process, while the Ishikawa process is a two-step process, among others.

In 2000, Noor [26] introduced a three-step iterative process and studied the approximate solution of variational inclusion in Hilbert spaces. This iteration process was further studied by many researchers to approximate fixed points for various classes of nonlinear operators (see e.g. [27–30]). It is observed that in many respects a three-step iterative process is better than a two- and a one-step iterative process in giving numerical results under certain conditions (see [31–33]). Thus we conclude that studying three-step iterative processes is very important in solving various numerical problems arising in pure and applied sciences.

Motivated by the above facts in this paper, we introduce a Noor-type iteration process for nonexpansive multivalued mappings and prove strong convergence theorems for the proposed iterative process in \(\operatorname{CAT}(0)\) spaces. The results we obtain are generalizations of Panyanak [21], Sastry and Babu [20], Shazhad and Zegeye [24] and Song and Wang [22] and many others in the sense of a Noor-type iteration process in the setting of \(\operatorname{CAT}(0)\) spaces.

2 Preliminaries

Let \((X, d)\) be a metric space. A geodesic path joining \(x \in X\) to \(y \in X\) is a map c from a closed interval \([0,l]\subset\mathbb{R}\) to X such that \(c(0) =x\), \(c(l) =y\), and \(d(c(t), c(t')) = \vert t - t'\vert\) for all \(t, t' \in[0, l]\).

In particular, c is an isometry and \(d(x, y) =l\). The image α of c is called a geodesic segment joining x and y; when it is unique this geodesic segment is denoted by \([x, y]\). The space \((X, d)\) is said to be a geodesic space if any two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each \(x, y \in X\). A subset \(Y \subseteq X\) is said to be convex if Y includes every geodesic segment joining any two points of itself.

A geodesic triangle \(\Delta(x_{1}, x_{2}, x_{3})\) in a geodesic metric space \((X, d)\) consists of three points \(x_{1}\), \(x_{2}\), \(x_{3} \) in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle \(\Delta(x_{1}, x_{2}, x_{3})\) in \((X, d)\) is a triangle \(\overline{\Delta}(x_{1}, x_{2}, x_{3}) := \Delta(\overline {x}_{1}, \overline{x}_{2}, \overline{x}_{3})\) in the Euclidean plane \(E^{2}\) such that \(d_{E^{2}}(\overline{x}_{i}, \overline{x}_{j})= d( x_{i}, x_{j})\) for \(i, j \in\{1, 2, 3\}\).

A geodesic metric space X is said to be a \(\operatorname{CAT}(0)\) space if all geodesic triangles of appropriate size satisfy the following comparison axiom.

\(\operatorname{CAT}(0)\): Let Δ be a geodesic triangle in X and let \(\overline {\Delta}\) be its comparison triangle for \(E^{2}\). Then Δ is said to satisfy the \(\operatorname{CAT}(0)\) inequality if for all \(x, y \in\Delta\) and all comparison points \(\overline{x}, \overline{y} \in\overline{\Delta}\),

$$d( x, y) \leq d_{E^{2}} ( \overline{x}, \overline{y}). $$

If x, \(y_{1}\), \(y_{2}\) are points in a \(\operatorname{CAT}(0)\) space and if \(y_{0}\) is the midpoint of the segment \([y_{1}, y_{2}]\), then the \(\operatorname{CAT}(0)\) inequality implies

$$ d( x, y_{0})^{2} \leq \frac{1}{2} d(x, y_{1})^{2} + \frac{1}{2} d( x, y_{2})^{2} - \frac{1}{4} d(y_{1}, y_{2})^{2}. $$
(CN)

This is the (CN) inequality of Bruhat and Tits [34]. In fact, a geodesic space is a \(\operatorname{CAT}(0)\) space if and only if it satisfies the (CN) inequality (cf. [35], p.163). We now collect some elementary facts about \(\operatorname{CAT}(0)\) spaces which will be used frequently in the proofs of our main results.

Lemma 2.1

(Lemma 2.1(iv), Lemma 2.4 and Lemma 2.5 in [25])

Let \((X,d)\) be a \(\operatorname{CAT}(0)\) space.

  1. (i)

    For \(x, y \in X\) and \(t \in[0, 1]\), there exists a unique point \(z \in[x, y]\) such that

    $$ d( x, z) = t d( x, y)\quad \textit{and}\quad d( y, z) = (1-t) d( x, y). $$
    (2.1)
  2. (ii)

    For \(x, y, z \in X\) and \(t \in[0, 1]\), we have

    $$d\bigl((1 - t)x \oplus ty, z\bigr) \leq(1 - t)d(x, z) + td(y, z). $$
  3. (iii)

    For \(x, y, z \in X\) and \(t \in[0, 1]\), we have

    $$d\bigl((1 - t)x \oplus ty, z\bigr)^{2} \leq(1 - t)d(x, z)^{2} + td(y, z)^{2} - t(1 - t)d(x, y)^{2}. $$

We will use the notation \((1-t) x \oplus ty \) for the unique point z satisfying (2.1). Now we define preliminaries for the construction of multivalued nonexpansive mapping.

Let K be the subset of \(\operatorname{CAT}(0)\) space X. Then:

  1. (i)

    The distance from \(x \in X \) to K is defined by

    $$\operatorname{dist} (x, K) = \inf\bigl\{ d(x, y): y \in K\bigr\} . $$
  2. (ii)

    The diameter of K is defined by

    $$\operatorname{diam} (K) = \sup\bigl\{ d(u, v): u, v \in K\bigr\} . $$

The set K is called proximinal if for each \(x \in X\), there exists an element \(y \in K\) such that \(d(x, y) = \operatorname{dist} (x,K)\). Let \(\mathit{CB}(K)\), \(C(K)\), and \(P(K)\) denote the family of nonempty closed bounded subsets, nonempty compact subsets and nonempty proximinal subsets of K, respectively. The Hausdorff metric H on \(\mathit{CB}(K)\) is defined by

$$H(A,B) = \max \Bigl\{ \sup_{x \in A} \operatorname{dist} (x, B), \sup_{y \in B} \operatorname{dist} (x, A) \Bigr\} $$

for \(A, B \in \mathit{CB}(K)\), where \(\operatorname{dist}(x, B)= \inf\{d( x, z), z \in B\}\).

Let \(T : X \to2^{X}\) be a multivalued mapping. An element \(x\in X\) is said to be a fixed point of T, if \(x \in Tx\). The set of fixed points will be denoted by \(\operatorname{Fix}(T)\).

Definition 2.2

A multivalued mapping \(T: K \to \mathit{CB}(K)\) is called:

  1. (i)

    nonexpansive, if \(H(T(x), T(y)) \leq d(x, y)\) for all \(x, y \in K\);

  2. (ii)

    quasi-nonexpansive, if \(\operatorname{Fix}(T) \neq\phi\), and \(H(x, T(p)) \leq d(x, p)\) for all \(x \in K\) and \(p \in\operatorname{Fix}(T)\).

The following example shows that every nonexpansive multivalued map T with \(\operatorname{Fix}(T)\neq\phi\) is quasi-nonexpansive. There exist quasi-nonexpansive mappings that are not nonexpansive.

Example 2.3

Let \(K= [0, \infty)\) with the usual metric and \(T : K \to \mathit{CB}(K)\) be defined by

$$Tx = \textstyle\begin{cases} \{0\}, &\mbox{if }x \leq1, \\ {[x-\frac{3}{4}, x -\frac{1}{3}]}, &\mbox{if } x >1. \end{cases} $$

Indeed, it is clear that \(\operatorname{Fix}(T)= \{0\}\) and for any x we have \(H(T(x), T(0)) \leq\vert x - 0 \vert\), hence, T is quasi-nonexpansive. However, if \(x =2\), \(y =1\) we get \(H(T(x), T(y)) > \vert x - y \vert=1\), and, hence, T is not nonexpansive.

3 Strong convergence theorems in \(\operatorname{CAT}(0)\) spaces

Now we introduce the notion of the proposed multivalued version of the Noor iteration process for a nonexpansive mapping T.

Let K be a nonempty convex subset of a complete \(\operatorname{CAT}(0)\) space X. The sequence of Noor iterates is defined by \(x_{0} \in K\),

$$\begin{aligned}& w_{n} = ( 1- \gamma_{n}) x_{n} \oplus\gamma_{n} z_{n}, \\ & y_{n} = ( 1- \beta_{n}) x_{n} \oplus \beta_{n} z_{n}', \\ & x_{n+1} = ( 1- \alpha_{n}) x_{n} \oplus \alpha_{n} z_{n}{''}, \end{aligned}$$
(3.1)

where \(z_{n} \in Tx_{n}\), \(z_{n}' \in Tw_{n}\), \(z_{n}'' \in Ty_{n}\), and \(\{\alpha _{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) are real sequences in \([a, b] \subset[0, 1]\).

Lemma 3.1

Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T : K \to \mathit{CB}(K)\) be a quasi-nonexpansive multivalued mapping with \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) be real sequences in \([a, b] \subset(0, 1)\). Then:

  1. (i)

    \(\lim_{n \to\infty} d(x_{n}, p)\) exists for each \(p \in \operatorname{Fix}(T)\).

  2. (ii)

    \(\lim_{ n \to\infty} \operatorname{dist}(Tx_{n} , x_{n}) =0\).

Proof

Let \(p \in\operatorname{Fix}(T) \). Then, using (3.1) and Lemma 2.1(ii), we have

$$\begin{aligned} d(w_{n}, p ) =& d\bigl(( 1- \gamma_{n}) x_{n} \oplus\gamma_{n}z_{n}, p\bigr) \\ \leq& ( 1- \gamma_{n})d( x_{n} , p) + \gamma_{n} d(z_{n}, p) \\ \leq& ( 1- \gamma_{n})d( x_{n} , p) + \gamma_{n} \operatorname{dist}\bigl(z_{n}, T(p)\bigr) \\ \leq& ( 1- \gamma_{n})d( x_{n} , p) + \gamma_{n} H\bigl(T(x_{n}) , T(p)\bigr) \\ \leq& ( 1- \gamma_{n})d( x_{n} , p) + \gamma_{n} d(x_{n}, p) \\ \leq& d(x_{n}, p). \end{aligned}$$
(3.2)

Also

$$\begin{aligned} d( y_{n}, p ) =& d\bigl(( 1- \beta_{n}) x_{n} \oplus\beta_{n} z_{n}', p \bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} d\bigl(z_{n}' , p\bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} \operatorname{dist}\bigl(z_{n}' , T(p)\bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} H\bigl(T(w_{n}) , T(p)\bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} d(w_{n} , p) \\ \leq& d(x_{n}, p). \end{aligned}$$
(3.3)

Again, using (3.1), (3.3), and Lemma 2.1(ii), we have

$$\begin{aligned} d( x_{n+1}, p ) =& d\bigl(( 1- \alpha_{n}) x_{n} \oplus\alpha_{n} z_{n}'', p\bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} d\bigl( z_{n}'', p \bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n}\operatorname{dist}\bigl( z_{n}'', T(p)\bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} H\bigl( T(y_{n}), T(p)\bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} d( y_{n}, p) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} d( x_{n}, p) \\ \leq& d(x_{n}, p). \end{aligned}$$
(3.4)

Hence, the sequence \(\{d(x_{n}, p)\}\) is decreasing and bounded below. It now follows that \(\lim_{n \to\infty} d(x_{n}, p)\) exists for any \(p \in \operatorname{Fix}(T)\). From Lemma 2.1(iii), we have

$$\begin{aligned} d^{2}( x_{n+1}, p ) =& d^{2}\bigl(( 1- \alpha_{n}) x_{n} \oplus\alpha_{n} z_{n}'', p\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} d^{2}\bigl( z_{n}'', p\bigr) - \alpha_{n} ( 1- \alpha_{n})d^{2} \bigl(x_{n}, z_{n}''\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} \operatorname{dist}^{2}\bigl( z_{n}'', T(p)\bigr) - \alpha_{n} ( 1- \alpha_{n})d^{2} \bigl(x_{n}, z_{n}''\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} H^{2}\bigl( T(y_{n}), T(p)\bigr) - \alpha_{n} ( 1- \alpha_{n})d^{2} \bigl(x_{n}, z_{n}''\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} d^{2}( y_{n}, p) - \alpha_{n} ( 1- \alpha_{n})d^{2} \bigl(x_{n}, z_{n}''\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} d^{2}( y_{n}, p). \end{aligned}$$
(3.5)

From Lemma 2.1(iii), we have

$$\begin{aligned} d^{2}( y_{n}, p ) =& d^{2} \bigl(( 1- \beta_{n}) x_{n} \oplus\beta_{n} z_{n}', p\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} d^{2}\bigl( z_{n}', p\bigr) - \beta_{n} ( 1- \beta_{n})d^{2} \bigl(x_{n}, z_{n}'\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} \operatorname{dist}^{2}\bigl( z_{n}', T(p)\bigr) - \beta_{n} ( 1- \beta_{n})d^{2} \bigl(x_{n}, z_{n}'\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} H^{2}\bigl( T(w_{n}), T(p)\bigr) - \beta_{n} ( 1- \beta_{n})d^{2} \bigl(x_{n}, z_{n}'\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} d^{2}( w_{n}, p) - \beta_{n} ( 1- \beta_{n})d^{2}\bigl(x_{n}, z_{n}'\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} d^{2}( w_{n}, p). \end{aligned}$$
(3.6)

Also

$$\begin{aligned} d^{2}( w_{n}, p ) =& d^{2} \bigl(( 1- \gamma_{n}) x_{n} \oplus\gamma_{n} z_{n}, p\bigr) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} d^{2}(z_{n}, p) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} \operatorname{dist}^{2}\bigl(z_{n}, T(p)\bigr) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} H^{2}\bigl(T(x_{n}), T(p)\bigr) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} d^{2}(x_{n}, p) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}). \end{aligned}$$
(3.7)

From (3.5), (3.6), and (3.7), we have

$$d^{2}( x_{n+1}, p ) \leq d^{2}( x_{n}, p) - \alpha_{n} \beta_{n} \gamma _{n} (1- \gamma_{n}) d^{2}( x_{n}, z_{n}). $$

This implies that

$$a^{3}(1 - b) d^{2}( x_{n}, z_{n}) \leq \alpha_{n} \beta_{n} \gamma_{n} (1- \gamma_{n}) d^{2}( x_{n}, z_{n}) \leq d^{2}( x_{n}, p ) - d^{2}( x_{n+1}, p) $$

and so

$$\sum_{n =1}^{\infty} a^{3} (1 - b) d^{2}(x_{n}, z_{n}) < \infty $$

and hence \(\lim_{ n \to\infty} d^{2}( x_{n} , z_{n}) =0\). Thus \(\lim_{n \to\infty} d( x_{n} , z_{n}) =0\). Hence, \(\operatorname{dist}(Tx_{n}, x_{n}) \leq d(x_{n}, z_{n}) \to0\) as \(n \to \infty\). □

Now we prove a strong convergence theorem for the Noor iteration process for multivalued mappings.

Theorem 3.2

Let K be nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to \mathit{CB}(K)\) be a quasi-nonexpansive multivalued mappings such that \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma _{n}\}\) be real sequences in \([a, b] \subset(0,1)\). Then \(\{x_{n}\}\) converges strongly to a fixed point of T if and only if \(\lim_{n \to \infty} \inf\operatorname{dist} (x_{n} , \operatorname{Fix}(T)) =0\).

Proof

Necessity is obvious. To prove the sufficiency, suppose that

$$\lim_{n \to\infty} \inf\operatorname{dist} \bigl(x_{n} , \operatorname{Fix}(T)\bigr) =0. $$

As in the proof of Lemma 3.1, we have

$$ d( x_{n+1} , p) \leq d( x_{n} , p) $$

for all \(p \in\operatorname{Fix}(T)\). This implies that

$$ \operatorname{dist} \bigl(x_{n} , \operatorname{Fix}(T)\bigr) \leq \operatorname{dist} \bigl(x_{n} , \operatorname{Fix}(T)\bigr) $$

so that \(\lim_{n \to\infty} \operatorname{dist} (x_{n} , \operatorname{Fix}(T))\) exists. Thus \(\lim_{n \to\infty} \operatorname{dist} (x_{n} , \operatorname{Fix}(T))=0\). Therefore, we can choose a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that

$$d(x_{n_{k}}, p_{k}) < \frac{1}{2^{k}} $$

for some \(\{p_{k}\} \subset\operatorname{Fix}(T) \) and for all k. By Lemma 3.1 we have

$$d(x_{n_{k+1}}, p_{k}) \leq d(x_{n_{k}}, p_{k}) < \frac{1}{2^{k}}. $$

Hence

$$ d( p_{k+1}, p_{k}) \leq d(x_{n_{k+1}}, p_{k+1} ) + d( x_{n_{k+1}}, p_{k}) < \frac{1}{2^{k+1}}+ \frac{1}{2^{k}}< \frac{1}{2^{k-1}}. $$

Consequently, \(\{p_{k}\}\) is a Cauchy sequence in K and hence converges to some \(q \in K\). Since

$$ \operatorname{dist} \bigl(p_{k}, T(q)\bigr) \leq H \bigl(T(p_{k}), T(q)\bigr) \leq d(q, p_{k}) $$

and \(p_{k}\to q\) as \(k \to\infty\), it follows that \(\operatorname{dist}(q, T(q)) = 0\) and so \(q \in\operatorname{Fix}(T)\) and thus \(\{x_{n_{k}}\}\) converges strongly to q. Since \(\lim_{n \to\infty}d(x_{n}, q) \) exists, it follows that \(\{ x_{n}\}\) converges strongly to q. This completes the proof. □

Theorem 3.3

Let K be nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to \mathit{CB}(K)\) be a quasi-nonexpansive multivalued mapping such that \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma _{n}\}\) be real sequences in \([a, b] \subset(0,1)\). Assume that T is hemicompact and continuous, then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Proof

By Lemma 3.1, we have \(\lim_{ n \to\infty} \operatorname{dist}(Tx_{n}, x_{n}) =0\). Since T is hemicompact, there exist a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) and \(q \in K\) such that \(\lim_{k \to\infty}x_{n_{k}} = q\). From continuity of T, we find that \(d(x_{n_{k}}, T(x_{n_{k}})) \to d(q, T(q))\). As a result, we have \(d(q, T(q)) = 0\) and so \(q \in\operatorname{Fix}(T)\). By Lemma 3.1, we find that \(\lim_{n \to\infty}d(x_{n}, p)\) exists for each \(p \in \operatorname{Fix}(T)\), hence \(\{x_{n}\}\) converges strongly to q. □

Theorem 3.4

Let K be nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to \mathit{CB}(K)\) be a quasi-nonexpansive multivalued mappings such that \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma _{n}\}\) be real sequences in \([a, b] \subset(0,1)\). Assume that there is a nondecreasing function \(f: [0, \infty) \to[0, \infty)\) with \(f(0) = 0\), \(f(r) > 0\) for \(r \in(0, \infty)\) such that

$$ \operatorname{dist}\bigl(x, T(x)\bigr) \geq f\bigl(\operatorname{dist}\bigl(x, F(T)\bigr)\bigr) \quad \textit{for all } x \in K. $$

Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Proof

By Lemma 3.1, we have \(\lim_{ n \to\infty} \operatorname{dist}(Tx_{n}, x_{n}) =0\). Hence, from the assumption we obtain \(\lim_{ n \to\infty} \operatorname{dist}(x_{n}, \operatorname{Fix}(T))=0\). The rest of the conclusion now follows from Theorem 3.2. □

The following corollaries are direct consequences of Theorems 3.2, 3.3, and 3.4.

Corollary 3.5

Let K be nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to \mathit{CB}(K)\) be a nonexpansive multivalued mappings such that \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) be real sequences in \([a, b] \subset(0,1)\). Then \(\{x_{n}\}\) converges strongly to a fixed point of T if and only if \(\lim_{n \to\infty} \operatorname{dist} (x_{n}, \operatorname{Fix}(T)) =0\).

Corollary 3.6

Let K be nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to \mathit{CB}(K)\) be a nonexpansive multivalued mappings such that \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) be real sequences in \([a, b] \subset(0,1)\). Assume that T is hemicompact and continuous, then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Corollary 3.7

Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to \mathit{CB}(K)\) be a nonexpansive multivalued mapping such that \(\operatorname{Fix}(T) \neq\phi\) and for which \(T(p) = \{p\}\) for each \(p \in\operatorname{Fix}(T)\). Let \(\{x_{n}\}\) be the Noor iterates defined by (3.1) and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) be sequences in \([a, b] \subset(0,1)\). Assume that there is a nondecreasing function \(f: [0, \infty) \to[0, \infty)\) with \(f(0) = 0\), \(f(r) > 0\) for \(r \in(0, \infty)\) such that

$$ \operatorname{dist}\bigl(x, T(x)\bigr) \geq f\bigl(\operatorname{dist}\bigl(x, F(T)\bigr)\bigr)\quad \textit{for all } x \in K. $$

Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

For a single-valued mapping, we obtain the following corollary.

Corollary 3.8

Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X. Let \(T: K \to K\) be a quasi-nonexpansive mappings such that \(\operatorname{Fix}(T) \neq\phi\). Let \(\{x_{n}\}\) be the Noor iterates defined by

$$\begin{aligned}& x_{n+1} = (1-\alpha_{n}) x_{n} \oplus \alpha_{n} Ty_{n} , \\& y_{n} = ( 1-\beta_{n}) x_{n} \oplus \beta_{n} Tz_{n} , \\& z_{n} = ( 1- \gamma_{n}) x_{n} \oplus \gamma_{n} Tx_{n}, \end{aligned}$$

where \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) are real sequences in \([a, b] \subset[0,1]\). Assume that there is a nondecreasing function \(f: [0, \infty) \to[0, \infty)\) with \(f(0) = 0\), \(f(r) > 0\) for \(r \in(0, \infty)\) such that

$$ d(x, Tx) \geq f\bigl(d\bigl(x,\operatorname{Fix}(T)\bigr)\bigr)\quad \textit{for all } x \in K. $$

Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Remark 3.9

Corollary 3.8 extends the results of Dhompongsa and Panyanak [25] and the results of Khan and Abbas [36] from the Ishikawa iteration process to the Noor iteration process.

In [24], Shahzad and Zegeye removed the restriction \(T(p) =\{p\} \) for each \(p \in\operatorname{Fix}(T) \) and defined a two-step iterative process. In view of this, we now define the following iteration process.

Let \(T:K \to P(K)\) and \(P_{T}(x) = \{y \in T(x): \Vert x - y\Vert =\operatorname{dist}(x, T(x))\}\). For \(x_{0} \in K\), the sequence \(\{x_{n}\}\) is defined iteratively in the following manner:

$$\begin{aligned} \begin{aligned} &w_{n} = ( 1- \gamma_{n}) x_{n} \oplus\gamma_{n} z_{n}, \\ &y_{n} = ( 1- \beta_{n}) x_{n} \oplus \beta_{n} z_{n}', \\ &x_{n+1} = ( 1- \alpha_{n}) x_{n} \oplus \alpha_{n} z_{n}{''}, \end{aligned} \end{aligned}$$
(3.8)

where \(z_{n} \in P_{T}(x_{n})\), \(z_{n}' \in P_{T}(w_{n})\), \(z_{n}'' \in P_{T}(y_{n})\), and \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) are real sequences in \([a, b] \subset[0,1]\).

Theorem 3.10

Let X be a complete \(\operatorname{CAT}(0)\) space, K a nonempty closed convex subset of X, and \(T : K \to P(K)\) a multivalued mapping with \(\operatorname{Fix}(T) \neq\phi\) such that \(P_{T}\) is nonexpansive. Let \(\{x_{n}\}\) be an iterative process defined by (3.8), where \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) are real sequences in \([a, b] \subset(0,1)\). Assume that there is a nondecreasing function \(f:[0, \infty) \to[0, \infty)\) with \(f(0) = 0\), \(f(r) > 0\) for \(r \in (0, \infty)\) such that

$$ \operatorname{dist}\bigl(x, T(x)\bigr) \geq f\bigl(\operatorname{dist}\bigl(x, \operatorname{Fix}(T)\bigr)\bigr)\quad \textit{for all } x \in K. $$

Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Proof

Let \(p \in P_{T}(p) = \{p\}\). Then, using (3.8) and Lemma 2.1(ii), we have

$$\begin{aligned} d(w_{n}, p ) =& d\bigl(( 1- \gamma_{n}) x_{n} \oplus\gamma_{n} z_{n}, p\bigr) \\ \leq& (1- \gamma_{n})d(x_{n} , p) + \gamma_{n} d(z_{n}, p) \\ \leq& (1- \gamma_{n})d( x_{n} , p) + \gamma_{n} \operatorname{dist}\bigl(z_{n}, P_{T}(p)\bigr) \\ \leq& (1- \gamma_{n})d( x_{n} , p) + \gamma_{n} H\bigl(P_{T}(x_{n}) , P_{T}(p)\bigr) \\ \leq& (1- \gamma_{n})d( x_{n} , p) + \gamma_{n} d(x_{n}, p) \\ \leq& d(x_{n}, p). \end{aligned}$$
(3.9)

Using (3.8), (3.9), and Lemma 2.1(ii), we have

$$\begin{aligned} d( y_{n}, p ) =& d\bigl(( 1- \beta_{n}) x_{n} \oplus\beta_{n} z_{n}', p \bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} d\bigl(z_{n}' , p\bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} \operatorname{dist}\bigl(z_{n}' , P_{T}(p)\bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} H\bigl(P_{T}(w_{n}) , P_{T}(p)\bigr) \\ \leq& ( 1- \beta_{n})d( x_{n} , p) + \beta_{n} d(w_{n} , p) \\ \leq& d(x_{n}, p). \end{aligned}$$
(3.10)

Using (3.8), (3.10), and Lemma 2.1(ii), we have

$$\begin{aligned} d( x_{n+1}, p ) =& d\bigl(( 1- \alpha_{n}) x_{n} \oplus\alpha_{n} z_{n}'', p\bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} d\bigl( z_{n}'', p \bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} \operatorname{dist}\bigl( z_{n}'', P_{T}(p)\bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} H\bigl( P_{T}(y_{n}), P_{T}(p)\bigr) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} d( y_{n}, p) \\ \leq& ( 1- \alpha_{n})d( x_{n} , p) + \alpha_{n} d( x_{n}, p) \\ \leq& d(x_{n}, p). \end{aligned}$$
(3.11)

Consequently, the sequence \(\{d(x_{n}, p)\}\) is decreasing and bounded below, and thus \(\lim_{n \to\infty} d(x_{n}, p)\) exists for any \(p \in \operatorname{Fix}(T)\). Applying Lemma 2.1(iii), we have

$$\begin{aligned} d^{2}( x_{n+1}, p ) =& d^{2} \bigl(( 1- \alpha_{n}) x_{n} \oplus\alpha_{n} z_{n}'', p\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} \operatorname{dist}^{2}\bigl( z_{n}'', p\bigr) - \alpha_{n} ( 1- \alpha_{n})d^{2} \bigl(x_{n}, z_{n}''\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} H^{2}\bigl( P_{T}(y_{n}), P_{T}(p)\bigr) - \alpha_{n} ( 1- \alpha_{n})d^{2} \bigl(x_{n}, z_{n}''\bigr) \\ \leq& ( 1- \alpha_{n})d^{2}( x_{n} , p) + \alpha_{n} d^{2}( y_{n}, p). \end{aligned}$$
(3.12)

From Lemma 2.1(iii), it follows that

$$\begin{aligned} d^{2}( y_{n}, p ) =& d^{2} \bigl(( 1- \beta_{n}) x_{n} \oplus\beta_{n} z_{n}', p\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} \operatorname{dist}^{2}\bigl( z_{n}', p\bigr) - \beta_{n} ( 1- \beta_{n})d^{2} \bigl(x_{n}, z_{n}'\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} H^{2}\bigl( P_{T}(w_{n}), P_{T}(p)\bigr) - \beta_{n} ( 1- \beta_{n})d^{2} \bigl(x_{n}, z_{n}'\bigr) \\ \leq& ( 1- \beta_{n})d^{2}( x_{n} , p) + \beta_{n} d^{2}( w_{n}, p). \end{aligned}$$
(3.13)

Also

$$\begin{aligned} d^{2}( w_{n}, p ) =& d^{2} \bigl(( 1- \gamma_{n}) x_{n} \oplus\gamma_{n} z_{n}, p\bigr) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} \operatorname{dist}^{2}(z_{n}, p) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} H^{2}\bigl(P_{T}(x_{n}), P_{T}(p)\bigr) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}) \\ \leq& ( 1- \gamma_{n})d^{2}( x_{n} , p) + \gamma_{n} d^{2}(x_{n}, p) - \gamma_{n}( 1- \gamma_{n})d^{2}( x_{n} , z_{n}). \end{aligned}$$
(3.14)

From (3.12), (3.13), and (3.14), we have

$$d^{2}( x_{n+1}, p ) \leq d^{2}( x_{n}, p) - \alpha_{n} \beta_{n} \gamma _{n} (1- \gamma_{n}) d^{2}( x_{n}, z_{n}). $$

This implies that

$$a^{3}(1 - b) d^{2}( x_{n}, z_{n}) \leq \alpha_{n} \beta_{n} \gamma_{n} (1- \gamma_{n}) d^{2}( x_{n}, z_{n}) \leq d^{2}( x_{n}, p ) - d^{2}( x_{n+1}, p) $$

and so

$$\sum_{n =1}^{\infty} a^{3} (1 - b) d^{2}(x_{n}, z_{n}) < \infty. $$

Thus \(\lim_{n \to\infty}d(x_{n}, z_{n}) = 0\). Also \(\operatorname{dist}(Tx_{n}, x_{n}) \leq d(x_{n}, z_{n}) \to0\) as \(n \to\infty\) and hence by assumption \(\lim_{ n \to\infty} \operatorname{dist}(Tx_{n} , \operatorname{Fix}(T)) =0\). Thus there is a subsequence \(\{x_{n_{k}}\}\) of \(\{ x_{n}\}\) such that \(d(x_{n_{k}}, p_{k}) < \frac{1}{2^{k}}\) for some \(\{p_{k}\} \subset F(T)\) and all k. As in the proof of Theorem 3.2, \(\{ p_{k}\}\) is a Cauchy sequence in K and thus converges to \(q \in K\). Since

$$\begin{aligned} \begin{aligned} d\bigl(p_{k}, T(q)\bigr) & \leq d\bigl(p_{k}, P_{T}(q)\bigr) \\ & \leq H\bigl(P_{T}(p_{k}), P_{T}(q)\bigr) \\ & \leq d(p_{k}, q), \end{aligned} \end{aligned}$$

and \(p_{k} \to q\) as \(k \to\infty\), it follows that \(\operatorname{dist}(q, T(q)) = 0\) and so \(q \in\operatorname{Fix}(T)\), and thus \(\{x_{n_{k}}\}\) converges strongly to q. Since \(\lim_{n \to\infty}d(x_{n}, q) = 0\) exists, it follows that \(\{ x_{n}\}\) converges strongly to q. This completes the proof. □

4 Conclusion

Remark 4.1

Theorems 3.2, 3.3, 3.4, and 3.10 improve and generalize the well-known results of Sastry and Babu (Theorem 5 in [20]), Panyanak (Theorem 3.1 and Theorem 3.8 in [21]), Song and Wang (Theorem 1 and Theorem 2 in [22]), Shahzad and Zegeye [24] and Laowang and Panyanak [2] and many others in the sense of Noor-type iteration process in the setting of \(\operatorname{CAT}(0)\) spaces.

Stability results established in metric spaces, normed linear spaces, and Banach spaces are available in the literature for single-valued mappings (see e.g., Haghi et al. [37], Olatinwo and Postolache [38] and references therein).

Open problem

It will be interesting to study the stability of iteration scheme (3.1).

References

  1. Dhompongsa, S, Kaewkhao, A, Panyanak, B: Lim’s theorems for multivalued mappings in \(\operatorname{CAT}(0)\) spaces. J. Math. Anal. Appl. 312, 478-487 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Laowang, W, Panyanak, B: Strong and Δ-convergence theorems for multivalued mappings in \(\operatorname{CAT}(0)\) spaces. J. Inequal. Appl. 2009, Article ID 730132 (2009)

    Article  MathSciNet  Google Scholar 

  3. Abbas, M, Khan, SH, Postolache, M: Existence and approximation results for SKC mappings in \(\operatorname{CAT}(0)\) spaces. J. Inequal. Appl. 2014, Article ID 212 (2014)

    Article  MathSciNet  Google Scholar 

  4. Saluja, GS, Postolache, M: Strong and Δ-convergence theorems for two asymptotically nonexpansive mappings in the intermediate sense in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2015, Article ID 12 (2015)

    Article  MathSciNet  Google Scholar 

  5. Shahzad, N: Invariant approximations in \(\operatorname{CAT}(0)\) spaces. Nonlinear Anal. 70, 4338-4340 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Shahzad, N, Markin, J: Invariant approximations for commuting mappings in hyperconvex and \(\operatorname{CAT}(0)\) spaces. J. Math. Anal. Appl. 337, 1457-1464 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Shahzad, N: Fixed point result for multimaps in \(\operatorname{CAT}(0)\) space. Topol. Appl. 156, 997-1001 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Abkar, A, Eslamian, M: Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in \(\operatorname{CAT}(0)\) spaces. Nonlinear Anal. 75, 1895-1903 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301-308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Thakur, BS, Thakur, D, Postolache, M: A new iteration scheme for approximating fixed points of nonexpansive mappings. Filomat (in press)

  11. Thakur, BS, Thakur, D, Postolache, M: New iteration scheme for numerical reckoning fixed points of nonexpansive mappings. J. Inequal. Appl. 2014, Article ID 328 (2014)

    Article  Google Scholar 

  12. Yao, Y, Postolache, M, Kang, SM: Strong convergence of approximated iterations for asymptotically pseudocontractive mappings. Fixed Point Theory Appl. 2014, Article ID 100 (2014)

    Article  MathSciNet  Google Scholar 

  13. Yao, Y, Postolache, M, Liou, YC: Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings. Fixed Point Theory Appl. 2013, Article ID 211 (2013)

    Article  Google Scholar 

  14. Thakur, BS, Dewangan, R, Postolache, M: Strong convergence of new iteration process for a strongly continuous semigroup of asymptotically pseudocontractive mappings. Numer. Funct. Anal. Optim. 34(12), 1418-1431 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thakur, BS, Dewangan, R, Postolache, M: General composite implicit iteration process for a finite family of asymptotically pseudocontractive mappings. Fixed Point Theory Appl. 2014, Article ID 90 (2014)

    Article  MathSciNet  Google Scholar 

  16. Dewangan, R, Thakur, BS, Postolache, M: A hybrid iteration for asymptotically strictly pseudocontractive mappings. J. Inequal. Appl. 2014, Article ID 374 (2014)

    Article  MathSciNet  Google Scholar 

  17. Dewangan, R, Thakur, BS, Postolache, M: Strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration. Appl. Math. Comput. 248, 160-168 (2014)

    Article  MathSciNet  Google Scholar 

  18. Thakur, BS, Postolache, M: Existence and approximation of solutions for generalized extended nonlinear variational inequalities. J. Inequal. Appl. 2013, Article ID 590 (2013)

    Article  MathSciNet  Google Scholar 

  19. Yao, Y, Postolache, M: Iterative methods for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory Appl. 155(1), 273-287 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sastry, KPR, Babu, GVR: Convergence of Ishikawa iterates for a multivalued mapping with a fixed point. Czechoslov. Math. J. 55, 817-826 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Panyanak, B: Mann and Ishikawa iterative processes for a multivalued mappings in Banach spaces. Nonlinear Anal. 70, 1547-1556 (2009)

    Article  MathSciNet  Google Scholar 

  22. Song, Y, Wang, H: Erratum to ‘Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces’. Comput. Math. Appl. 55, 2999-3002 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Song, Y, Wang, H: Convergence of iterative algorithms for multivalued mappings in Banach spaces. Nonlinear Anal. 70, 1547-1556 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shahzad, N, Zegeye, H: On Mann and Ishikawa iteration schemes for multivalued maps in Banach space. Nonlinear Anal. 71, 838-844 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dhompongsa, S, Panyanak, B: On Δ-convergence theorems in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 56, 2572-2579 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Noor, MA: New approximations scheme for general variational inequalities. J. Math. Anal. Appl. 251, 217-229 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cho, YJ, Zho, HY, Guo, G: Weak and strong convergence theorems for three step iteration with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, 707-717 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Khan, SH, Hussain, N: Convergence theorems for nonself asymptotically nonexpansive mappings. Comput. Math. Appl. 55, 2544-2553 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Suantai, S: Weak and strong convergence criteria of Noor iteration for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 311, 506-517 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, HK, Noor, MA: Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 267, 444-453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Abbas, M, Nazir, T: A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn. 106, 1-12 (2013)

    Article  Google Scholar 

  32. Glowinski, R, Le Tallec, P: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

  33. Haubruge, S, Nguyen, VH, Strodiot, JJ: Convergence analysis and applications of the Glowinski-Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 97(3), 645-673 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Bruhat, F, Tits, J: Groupes réductifs sur un corps local: I. Données radicielles valuées. Publ. Math. Inst. Hautes Études Sci. 41, 5-251 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bridson, M, Haefliger, A: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  36. Khan, SH, Abbas, M: Strong and Δ-convergence of some iterative schemes in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 61, 109-116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Haghi, RH, Postolache, M, Rezapour, S: On T-stability of the Picard iteration for generalized Ï•-contraction mappings. Abstr. Appl. Anal. 2012, Article ID 658971 (2012)

    Article  MathSciNet  Google Scholar 

  38. Olatinwo, MO, Postolache, M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 218(12), 6727-6732 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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The authors would like to thank the referee, who have made a number of valuable comments and suggestions which have improved the manuscript greatly.

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Pathak, R.P., Dashputre, S., Diwan, S.D. et al. On Noor-type iteration schemes for multivalued mappings in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl 2015, 133 (2015). https://doi.org/10.1186/s13663-015-0380-8

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