1 Introduction

The theory of fixed point for multi-valued mappings has become of great interest to many researchers, see, for instance, [3, 4, 8, 11, 13, 15] and references therein. This concept contributes significantly in convex approximation, fractals, optimal control, digital imaging, and economics.

The following statements are quoted from [26]: “Sastry and Babu [21] showed that under certain conditions, Mann and Ishikawa iterative algorithms for a multi-valued nonexpansive mapping with a fixed point x converge to a fixed point y of the mapping. Panyanak [18] extended their results to a uniformly convex Banach space. Song and Wang [23] improved the results of Panyanak [18]. Abbas et al. [1] proposed a one-step iterative scheme to find a common fixed point of two multi-valued nonexpansive mappings in a uniformly convex Banach space. Recently, Uddin et al. [26] discovered a few gaps in the scheme proposed in [1] and came up with a new one-step iterative scheme for fixed points of two multi-valued nonexpansive mappings in CAT(0) space and removed the gaps found in [1]”.

Motivated by [1, 26], we recommend a new scheme to approximate common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in CAT(0) spaces.

2 Preliminary lemmas

We refer to [2] for the following definitions. “Suppose (M, d) is a metric space and \(a,b\in M.\)” A geodesic path from a to b is a mapping \(s:[0,d(a,b)]\rightarrow X\), such that \(s(0)=a,s(d(a,b))=b\) and \( d(s(r),s(r^{\prime }))=|r-r^{\prime }|\) for all \(r,r^{\prime }\in [0,d(a,b)].\) A geodesic segment is defined as the image of the geodesic path. If every pair of points in M is joined by a unique geodesic segment, then the space is known as unique geodesic metric space. A geodesic triangle is denoted by \(\Delta (a,b,c)\) in a geodesic metric space (M, d) that consists of three points \(a,b,c\in M\) (the vertices of the geodesic triangle \(\Delta \) ) and three geodesic segments among these points. A comparison triangle for this geodesic triangle \(\Delta (a,b,c)\) in M is a triangle \(\overline{ \Delta }(\overline{a},\overline{b},\overline{c})\) in \( \mathbb {R}^{2}\), such that \(d_{\mathbb {R}^{2}} (\overline{i},\overline{j})=d(i,j)\) for \(i,j\in \{a,b,c\}.\)”

Definition 2.1

[2] A geodesic space M is a CAT(0) space if, for each pair \( \left( \Delta ,\overline{\Delta }\right) \in X\times \mathbb {R} ^{2},\) the following inequality:

$$\begin{aligned} d_{\mathbb {R}^{2}}(\overline{a},\overline{b})\ge d(a,b) \end{aligned}$$

holds for all \(\overline{a},\overline{b}\in \overline{\Delta }\) and \(a,b\in \Delta .\)

Definition 2.2

[2] The geodesic segment from a to b is denoted by [a, b],  that is:

$$\begin{aligned}{}[a,b]=\{\gamma a\oplus \left( 1-\gamma \right) b:\gamma \in [0,1]\}. \end{aligned}$$

For \(\gamma \in [0,1],\) we write \(\gamma a\oplus \left( 1-\gamma \right) b\) for the unique point c on [a, b], such that:

$$\begin{aligned} d(a,c)=\left( 1-\gamma \right) d(a,b)\text { and }d(b,c)=\gamma d(a,b). \end{aligned}$$

Let M be a CAT(0) space. A subset C of M is convex if \( [a,b]\subset C\) for all \(a,b\in C.\)

Definition 2.3

[13] For a fixed \(x\in M\) and a bounded sequence \(\{x_{n}\}\subset M,\) we define:

$$\begin{aligned} r(x,\{x_{n}\})=\limsup _{n\rightarrow \infty }d(x_{n},x). \end{aligned}$$

The asymptotic radius of \(\{x_{n}\}\) is defined by:

$$\begin{aligned} r(\{x_{n}\})=\inf _{x\in M}r(x,\{x_{n}\}); \end{aligned}$$

the asymptotic radius of \(\{x_{n}\}\) with respect to \(C\subset M\) is denoted by \(r_{C}(\{x_{n}\})\) and is defined by:

$$\begin{aligned} r_{C}(\{x_{n}\})=\inf _{x\in C}r(x,\{x_{n}\}). \end{aligned}$$

The asymptotic center of \(\{x_{n}\}\) is denoted by \(A(\{x_{n}\})\) and is given by the set:

$$\begin{aligned} A(\{x_{n}\})=\{\overline{x}\in X:\text { }r(\overline{x},\{x_{n}\})=r(\{x_{n} \})\}. \end{aligned}$$

The asymptotic center of \(\{x_{n}\}\) with respect to \(C\subset M\) is the set:

$$\begin{aligned} A_{C}(\{x_{n}\})=\{\overline{x}\in C:r(\overline{x},\{x_{n}\})=r(\{x_{n}\}) \}. \end{aligned}$$

Definition 2.4

[14] A sequence \(\{x_{n}\}\) in M is said to be \(\Delta \)-convergent to the point \(x\in M\) if x is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}.\) We call x the \(\Delta \)-limit of \(\{x_{n}\}\) and express this limit as \(\Delta -\lim _{n}x_{n}=x.\)

It has been observed in [12] that “if \(\{x_{n}\}\subset M\) and \( \{x_{n}\}\) is \(\Delta \)-convergent to \(x\in M\) and \(\overline{x}\left( \ne x\right) \in M,\) then:

$$\begin{aligned} \limsup _{n\rightarrow \infty }d(x_{n},x)<\limsup _{n\rightarrow \infty }d(x_{n},\overline{x}).\text {"} \end{aligned}$$

As mentioned in [12], “this concludes that \(CAT\left( 0\right) \) space satisfies a condition known as the Opial’s property in Banach spaces”.

Let C be a nonempty subset of a metric space (M, d). The set C is called proximinal if for each \(x\in M,\) there exists \(\overline{x}\in C\), such that \(d(x,\overline{x})=\mathrm{{dist}}(x,C),\) where \(\mathrm{{dist}}(x,C)=\inf _{z\in C}d(x,z).\) We denote the family of all bounded proximinal subsets of C by \(\mathcal {P}(C).\)

Let \(\mathcal {C}\mathcal {B}(C)\) and \(\mathcal {K}(C)\) be families of nonempty, closed, and bounded subsets of C and nonempty compact subsets of C, respectively. The Hausdorff metric H on \(\mathcal {C}\mathcal {B}(C)\) is defined as:

$$\begin{aligned} H(D,\overline{D})=\max \left\{ \sup _{x\in \overline{D}}\mathrm{{dist}}(x,D),\sup _{ \overline{x}\in D}\mathrm{{dist}}(\overline{x},\overline{D})\right\} \text {, where }D, \overline{D}\in \mathcal {C}\mathcal {B}(C). \end{aligned}$$

An element \(p\in C\) is a fixed point of \(S:C\rightarrow \mathcal {C}\mathcal {B }(C)\) if \(p\in Sp.\) The set of all fixed points of S is denoted by F(S).

Definition 2.5

[13] A mapping \(S:C\rightarrow \mathcal {C}\mathcal {B}(C)\) is a multi-valued nonexpansive if \(H(Sx,Sy)\le d(x,y)\) for all \(x,y\in C.\)

Definition 2.6

[22] A mapping \(S:C\rightarrow \mathcal {C}\mathcal {B}(C)\) is a multi-valued quasi-nonexpansive if \(H(Sx,Sp)\le d(x,p)\) for all \( x\in C,p\in F\left( S\right) .\)

Definition 2.7

[21] The mapping \(S:C\rightarrow \mathcal {C}\mathcal {B}(C)\) is called generalized multi-valued nonexpansive if

$$\begin{aligned} H(Sx,Sy)\le & {} a_{1}d(x,y)+a_{2}\left[ \mathrm{{dist}}(x,Sx)+\mathrm{{dist}}(y,Sy)\right] \nonumber \\&+a_{3}\left[ \mathrm{{dist}}(x,Sy)+\mathrm{{dist}}(y,Sx)\right] \end{aligned}$$
(2.1)

for all \(x,y\in C\) where \(a_{1}+2a_{2}+2a_{3}\le 1.\)

Example 2.8

Let \(G:\left[ 0,3\right] \rightarrow CB\left( \left[ 0,3 \right] \right) \) be defined as under:

$$\begin{aligned} G(x)=\left\{ \begin{array}{c} \left[ 0,\frac{x}{3}\right] \ \ \ \ \text { if }x\ne 3 \\ \left\{ 1.2\right\} \ \ \ \ \text { if }x=3. \end{array} \right. \end{aligned}$$

Then, G is a generalized nonexpansive mapping with \(F(G)=\left\{ 0\right\} . \)

Verification: We first show that G is nonexpansive \(\left( a_1=1,a_2=0=a_3\right) .\)

  1. Case 1.

    If \(x=y\ne 3,\) then \(H\left( Gx,Gy\right) =\left\| \frac{x }{3}-\frac{y}{3}\right\| \le \left\| x-y\right\| .\)

  2. Case 2.

    If \(x=3=y,\) then \(H\left( Gx,Gy\right) =0=\left\| x-y\right\| . \)

  3. Case 3.

    If \(x\ne 3\) and \(y=3,\) then \(H\left( Gx,Gy\right) =1.2-\frac{x}{3} \le 3-x=\left\| x-y\right\| .\) This relation holds if and only if \( x\le 2.7.\) Hence, G is a nonexpansive mapping.

    For generalized nonexpansivity of G,  we further investigate two cases.

  4. Case 4.

    If \(x\in \left[ 0,1.8\right) \) and \(y\in \left[ 0,3\right) ,\) then this case is similar to Case 1 as mentioned above.

  5. Case 5.

    If \(x\in \left[ 1.8,3\right) \) and \(y=3\) with \(a_1=\frac{1}{8}=a_2\) and \( a_3=\frac{1}{4},\) then we compute:

    $$\begin{aligned} H\left( Gx,Gy\right) =\left\| \frac{2x}{3}-1.2\right\| =\left\{ \begin{array}{l} 0 \qquad \quad \,\, \text { if }x=1.8 \\<0.8 \quad \text { if }x<3. \end{array} \right. \end{aligned}$$
    (2.2)

The other estimates are as follows:

  1. 1.

    \(\left\| x-y\right\| =\left\| x-3\right\| ;\)

  2. 2.

    \(\mathrm{{dist}}\left( x,Gx\right) =\left\| \frac{2x}{3}\right\| ;\)

  3. 3.

    \(\mathrm{{dist}}\left( y,Gy\right) =\left\| 3-1.2\right\| ;\)

  4. 4.

    \(\mathrm{{dist}}\left( x,Gy\right) =\left\| x-1.2\right\| ;\)

  5. 5.

    \(\mathrm{{dist}}\left( y,Gx\right) =\left\| 3-\left[ 0,1\right) \right\| .\)

Now, observe that:

$$\begin{aligned}&a_1\left\| x-y\right\| +a_2\left( \mathrm{{dist}}\left( x,Gx\right) +\mathrm{{dist}}\left( y,Gy\right) \right) +a_3\left( \mathrm{{dist}}\left( x,Gy\right) +\mathrm{{dist}}\left( y,Gx\right) \right) \nonumber \\&\quad =\frac{1}{8}\left\| x-3\right\| +\frac{1}{8}\left( \left\| \frac{ 2x}{3}\right\| +1.8\right) +\frac{1}{4}\left( \left\| x-1.2\right\| +\left\| 3-\left[ 0,1\right) \right\| \right) \nonumber \\&\quad \ge \frac{1}{8}\left\| 1.8-3\right\| +\frac{1}{8}\left( \left\| \frac{2\left( 1.8\right) }{3}\right\| +1.8\right) +\frac{1}{4}\left( \left\| 1.8-1.2\right\| +2\right) \nonumber \\&\quad =\frac{1.2}{8}+\frac{3}{8}+\frac{2.6}{4} \nonumber \\&\quad =1.175. \end{aligned}$$
(2.3)

It follows from (2.2) and (2.3) that:

$$\begin{aligned} H\left( Gx,Gy\right) \le&\, a_1\left\| x-y\right\| +a_2\left( \mathrm{{dist}}\left( x,Gx\right) +\mathrm{{dist}}\left( y,Gy\right) \right) \\&+a_3\left( \mathrm{{dist}}\left( x,Gy\right) +\mathrm{{dist}}\left( y,Gx\right) \right) . \end{aligned}$$

Hence, G is a generalized nonexpansive mapping.

If we choose \(a_{1}=1\) and \(a_{2}=0=a_{3}\) in (2.1), it becomes multi-valued nonexpansive, but the converse does not hold, in general. Furthermore, we can verify that every generalized multi-valued nonexpansive mapping having at least one fixed point is multi-valued quasi-nonexpansive. For \(x\in C,p\in F\left( S\right) ,\) we obtain:

$$\begin{aligned} H(Sx,Sp)\le & {} a_{1}d(x,p)+a_{2}\left[ \mathrm{{dist}}(x,Sx)+\mathrm{{dist}}(p,Sp)\right] \\&+a_{3}\left[ \mathrm{{dist}}(x,Sp)+\mathrm{{dist}}(p,Sx)\right] \\\le & {} a_{1}d(x,p)+a_{2}\left[ d(x,p)+\mathrm{{dist}}(p,Sx)\right] +a_{3}\left[ d(x,p)+\mathrm{{dist}}(p,Sx)\right] \\\le & {} (a_{1}+a_{2}+a_{3})d(x,p)+(a_{2}+a_{3})H(Sx,Sp). \end{aligned}$$

That is:

$$\begin{aligned} H(Sx,Sp)\le \dfrac{a_{1}+a_{2}+a_{3}}{1-(a_{2}+a_{3})}d(x,p)\le d(x,p). \end{aligned}$$

The following fixed point existence theorem has been proved by Lim [13].

Theorem 2.9

[13] Let X be a uniformly convex Banach space and C be a nonempty bounded, closed, and convex subset of X. Let \(T:C\rightarrow \mathcal {K}(C)\) be a multi-valued nonexpansive mapping. Then, T has a fixed point.

Due to the importance of fixed point problems, iterative schemes for approximating fixed points of multi-valued mappings remain to be a flourishing subject in fixed point theory. Unfortunately, the existence result of common fixed points of a family of mappings is not known in many situations. Therefore, it is natural to consider approximation results for these classes of mappings, see, for example, [10, 19, 20, 24, 25]. Many iterative schemes have been introduced for different classes of mappings with a nonempty set of common fixed points. To define new iterative scheme, the following lemma will be needed.

Lemma 2.10

[16] Let \(D,\overline{D}\in \mathcal {C}\mathcal {B}(C)\) and \(x\in D.\) For every \(\varepsilon >0,\) there exists \(y\in \overline{D}\), such that \(d(x,y)\le H(D,\overline{D})+\varepsilon .\)

Let \(\left( M,d\right) \) be a \(CAT\left( 0\right) \) space. For \(x_{i}\in M\) and \(\alpha _{i}\in \left( 0,1\right) ,\) \(i\in \{1,2,\ldots n\}\) and \( \sum _{i=1}^{n}\alpha _{i}=1,\) we can define, by induction, that:

$$\begin{aligned} \bigoplus \limits _{i=1}^{n}\alpha _{i}x_{i}=(1-\alpha _{n})\left( \dfrac{ \alpha _{1}}{1-\alpha _{n}}x_{1}\oplus \dfrac{\alpha _{2}}{1-\alpha _{n}} x_{2}\oplus \cdots \oplus \dfrac{\alpha _{n-1}}{1-\alpha _{n}}x_{n-1}\right) \oplus \alpha _{n}x_{n}. \end{aligned}$$

Now, we write up our new iterative scheme to approximate common fixed point of a finite family of multi-valued generalized nonexpansive mappings in a CAT(0) space as follows:

Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {C}\mathcal {B}(C)\) be multi-valued generalized nonexpansive mappings, such that \(\mathcal {F} =\bigcap _{i\in J}^{{}}F(S_{i})\ne \phi \), where \(J=\left\{ 1,2,3,...,m\right\} \) (we assume it throughout the paper).

In view of Lemma 2.10, we define the scheme \(\{x_{n}\}\) in C as follows:

$$\begin{aligned} x_{1}\in C,\text { }x_{n+1}=\bigoplus \limits _{i=0}^{m}\alpha _{n,i}y_{n,i}, \end{aligned}$$
(2.4)

where \(y_{n,0}=x_{n},\) \(y_{n,i}\in S_{i}x_{n}\), such that:

$$\begin{aligned} d(y_{n,i},p)\le H(S_{i}x_{n},S_{i}p)+\varepsilon _{n,i} \ \text { for }i\in J \ \text { and }p\in \mathcal {F} \end{aligned}$$

with \(\lim _{n\rightarrow \infty }\varepsilon _{n,i}=0\) and the sequences \( \{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are in (0, 1) with \(\sum _{i=0}^{m}\alpha _{n,i}=1.\)

We recall some needed propositions and lemmas.

Proposition 2.11

[7] If \(\{x_{n}\}\) is a bounded sequence in a complete CAT(0) space M and if C is nonempty closed convex subset of M,  then there exists a unique point \(u\in C\), such that:

$$\begin{aligned} r(u,\{x_{n}\})=\inf _{x\in C}r(x,\{x_{n}\}). \end{aligned}$$

The above fact immediately yields the following proposition.

Proposition 2.12

[17] Let \(\{x_{n}\},C\) and M be as in Proposition 2.11. Then, \(A(\{x_{n}\})\) and \(A_{C}(\{x_{n}\})\) are singleton.

Lemma 2.13

[6, 26] Let M be a CAT(0) space and \( \{x,x_{1},x_{2},\ldots ,x_{n}\}\subset M.\) If \(a_{i}\in \left( 0,1\right) \) for \(i=1,2,3,\ldots ,n\) with \(\sum _{i=1}^{n}a_{i}=1,\) then for each \(i,j\in \{1,2,\ldots ,n\}\), such that \(i<j,\) we have:

  1. (a)

    \(d\left( \bigoplus \nolimits _{i=1}^{n}a_{i}x_{i},x\right) \le \) \( \sum \nolimits _{i=1}^{n}a_{i}d(x_{i},x),\)

  2. (b)

    \(d\left( \bigoplus \nolimits _{i=1}^{n}a_{i}x_{i},x\right) ^{2}\le \sum \nolimits _{i=1}^{n}a_{i}d(x_{i},x)^{2}-a_{i}a_{j}d(x_{i},x_{j})^{2}.\)

Lemma 2.14

[12] Every bounded sequence in a complete CAT(0) space M admits a \(\Delta \)-convergent subsequence.

Lemma 2.15

[5] Let C be a closed and convex subset of a complete CAT(0) space M. If \(\{x_{n}\}\) is a bounded sequence in C,  then the asymptotic center of \(\{x_{n}\}\) lies in C.

In this paper, we prove some convergence theorems for a finite family of multi-valued generalized nonexpansive mappings using scheme (2.4). Our results generalize the corresponding results in [1, 26] and references cited therein.

3 Convergence theorems

We start with the following lemma.

Lemma 3.1

Let C be a nonempty closed and convex subset of a complete CAT(0) space M. Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {C} \mathcal {B}(C)\) be multi-valued generalized nonexpansive mappings, such that \( S_{i}p=\left\{ p\right\} \) for all \(p\in \mathcal {F}\). If \(\{x_{n}\}\) is a sequence in C given by (2.4), then \(\lim _{n\rightarrow \infty }d(x_{n},p)\) exists for each \(p\in \mathcal {F}.\)

Proof

Let \(p\in \mathcal {F}.\) Applying Lemma 2.13 (a) to the scheme (2.4), we have:

$$\begin{aligned} d(x_{n+1},p)= & {} d\left( \bigoplus \limits _{i=0}^{m}\alpha _{n,i}y_{n,i},p\right) \\\le & {} \sum \limits _{i=0}^{m}\alpha _{n,i}d(y_{n,i},p) \\= & {} \alpha _{n,0}d(x_{n},p)+\sum \limits _{i=1}^{m}\alpha _{n,i}\mathrm{{dist}}(y_{n,i},S_{i}p) \\\le & {} \alpha _{n,0}d(x_{n},p)+\sum \limits _{i=1}^{m}\alpha _{n,i}H(S_{i}x_{n},S_{i}p) \\\le & {} d(x_{n},p)\sum \limits _{i=0}^{m}\alpha _{n,i} \\= & {} d(x_{n},p). \end{aligned}$$

That is, \(\{d(x_{n},p)\}\) is decreasing and bounded below. Hence, \(\lim \nolimits _{n\rightarrow \infty }d(x_{n},p)\) exists.\(\square \)

Lemma 3.2

Let C be a nonempty closed convex subset of a complete CAT(0) space M. Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {C} \mathcal {B}(C)\) be multi-valued generalized nonexpansive mappings, such that \( S_{i}p=\left\{ p\right\} \) for all \(p\in \mathcal {F}\). If \(\{x_{n}\}\) is a sequence in C given by (2.4), where \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are sequences in \([\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \), such that \( \sum _{i=0}^{m}\alpha _{n,i}=1,\) then \(\ \lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},S_{j}x_{n})=0\) for each \(j\in J.\)

Proof

Let \(p\in \mathcal {F}.\) By the scheme (2.4) and Lemma 2.13 (b):

$$\begin{aligned} d(x_{n+1},p)^{2}= & {} d\left( \bigoplus \limits _{i=0}^{m}\alpha _{n,i}y_{n,i},p\right) ^{2} \\\le & {} \sum \limits _{i=0}^{m}\alpha _{n,i}d(y_{n,i},p)^{2}-\alpha _{n,1}\alpha _{n,j}d(x_{n},y_{n,j})^{2} \\\le & {} \alpha _{n,0}d(x_{n},p)^{2}+\sum \limits _{i=1}^{m}\alpha _{n,i}H(S_{i}x_{n},p)^{2}-\alpha _{n,1}\alpha _{n,j}d(x_{n},y_{n,j})^{2} \\\le & {} \sum \limits _{i=0}^{m}\alpha _{n,i}d(x_{n},p)^{2}-\alpha _{n,1}\alpha _{n,j}d(x_{n},y_{n,j})^{2} \\= & {} d(x_{n},p)^{2}-\alpha _{n,1}\alpha _{n,j}d(x_{n},y_{n,j})^{2}. \end{aligned}$$

That is:

$$\begin{aligned} \delta ^{2}d(x_{n},y_{n,j})^{2}\le d(x_{n},p)^{2}-d(x_{n+1},p)^{2}. \end{aligned}$$

For any positive integer \(N\ge 1,\)the above inequality provides that:

$$\begin{aligned} \delta ^{2}\sum \limits _{n=1}^{N}d(x_{n},y_{n,j})^{2}\le d(x_{1},p)^{2}-d(x_{N+1},p)^{2}\le d(x_{1},p)^{2}<\infty . \end{aligned}$$

If \(N\rightarrow \infty ,\) we have \(\delta ^{2}\sum _{n=1}^{\infty }d(x_{n},y_{n,j})^{2}<\infty .\)

Hence, for each \(j\in \{1,2,\ldots m\},\) we deduce that:

$$\begin{aligned} \lim _{n\rightarrow \infty }d(x_{n},y_{n,j})=0. \end{aligned}$$

Since \(\mathrm{{dist}}(x_{n},T_{j}x_{n})\le d(x_{n},y_{n,j}),\) it follows that:

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},T_{j}x_{n})=0. \end{aligned}$$

\(\square \)

Now, we prove the demiclosed principle for multi-valued generalized nonexpansive mappings.

Theorem 3.3

Let C be a nonempty closed and convex subset of a complete CAT(0) space M. Let \(S:C\rightarrow \mathcal {K}(C)\) be a multi-valued generalized nonexpansive mapping and \(\{x_{n}\}\subset C.\) If \(\Delta -\lim _{n}x_{n}=x\in C\) and \(\lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},Sx_{n})=0,\) then x is a fixed point of S.

Proof

For \(n\ge 1,\) choose \(y_{n}\in Sx\), such that \(d(x_{n},y_{n})=\mathrm{{dist}}(x_{n},Sx)\) as \(Sx\in \mathcal {K}(C).\) By the compactness of Sx,  there exists a subsequence \(\{y_{n_{i}}\}\) of \(\{y_{n}\}\), such that \(\lim _{i\rightarrow \infty }y_{n_{i}}=y\in Sx.\)

Now, we have:

$$\begin{aligned} d(x_{n_{i}},y_{n_{i}})= & {} \mathrm{{dist}}(x_{n_{i}},Sx) \\\le & {} \mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+H(Sx_{n_{i}},Sx) \\\le & {} \mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+a_{1}d(x,x_{n_{i}}) +a_{2}[\mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+\mathrm{{dist}}(x,Sx)]\\&+a_{3}[\mathrm{{dist}}(x_{n_{i}},Sx)+\mathrm{{dist}}(x,Sx_{n_{i}})] \\= & {} (1+a_{2})\mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+a_{1}d(x,x_{n_{i}})\\&+a_{2}[d(x_{n_{i}},x)+\mathrm{{dist}}(x_{n_{i}},Sx)]\\&+a_{3}[\mathrm{{dist}}(x_{n_{i}},Sx)+d(x_{n_{i}},x)+\mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})]. \end{aligned}$$

Thus:

$$\begin{aligned} d(x_{n_{i}},y_{n_{i}})= & {} \mathrm{{dist}}(x_{n_{i}},Sx) \\\le & {} \frac{1+a_{2}+a_{3}}{1-(a_{2}+a_{3})}\mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+\frac{ a_{1}+a_{2}+a_{3}}{1-(a_{2}+a_{3})}d(x_{n_{i}},x) \\\le & {} \frac{1+a_{2}+a_{3}}{1-(a_{2}+a_{3})} \mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+d(x_{n_{i}},x). \end{aligned}$$

Hence:

$$\begin{aligned} d(x_{n_{i}},y)\le & {} d(x_{n_{i}},y_{n_{i}})+d(y_{n_{i}},y) \\\le & {} \frac{1+a_{2}+a_{3}}{1-(a_{2}+a_{3})} \mathrm{{dist}}(x_{n_{i}},Sx_{n_{i}})+d(x_{n_{i}},x)+d(y_{n_{i}},y). \end{aligned}$$

Taking the superior limit on both sides of the above inequality, we obtain:

$$\begin{aligned} \limsup _{i\rightarrow \infty }d(x_{n_{i}},y)\le \limsup _{i\rightarrow \infty }d(x_{n_{i}},x)=r(x,\{x_{n_{i}}\}). \end{aligned}$$

By the uniqueness of the asymptotic center, we have \(x=y\in Sx.\) \(\square \)

Theorem 3.4

Let C be a nonempty closed convex subset of a complete CAT(0) space M. Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {K}(C)\) be multi-valued generalized nonexpansive mappings, such that \( S_{i}p=\left\{ p\right\} \) for all \(p\in \mathcal {F}\). If \(\{x_{n}\}\) is a sequence in C given by (2.4), where \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are sequences in \([\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \), such that \( \sum _{i=0}^{m}\alpha _{n,i}=1,\) then \(\Delta -\lim _{n}x_{n}=x\in \mathcal {F} . \)

Proof

It follows from Lemma 3.1 that \(\{x_{n}\}\) is bounded. By Proposition 2.11, \(\{x_{n}\}\) has a unique asymptotic center; that is, \( A_{C}(\{x_{n}\})=\{x\}\) and \(x\in C.\) By Lemmas 2.14 and 2.15, there must be a subsequence \(\{z_{n}\}\) of \(\{x_{n}\}\), such that \( \Delta -\lim _{n}z_{n}=z\in C.\) Lemma 3.2 implies that \( \lim _{n\rightarrow \infty }\mathrm{{dist}}(z_{n},S_{j}z_{n})=0\) for each \(j\in J.\) In view of Theorem 3.3, we have \(S_{j}z=z\) for each \(j\in J.\) That is, \(z\in \mathcal {F}.\)

If \(z\ne x,\) by the uniqueness of the asymptotic center, we have:

$$\begin{aligned} \limsup _{n\rightarrow \infty }d(z_{n},z)< & {} \limsup _{n\rightarrow \infty }d(z_{n},x)\le \limsup _{n\rightarrow \infty }d(x_{n},x) \\< & {} \limsup _{n\rightarrow \infty }d(x_{n},z)=\limsup _{n\rightarrow \infty }d(z_{n},z), \end{aligned}$$

which is a contradiction and, hence, \(z=x\in \mathcal {F}.\)

Let \(\{v_{n}\}\) be any subsequence of \(\{x_{n}\}\), such that \( A_{C}(\{v_{n}\})=\{v\}\) and \(v\in C.\) Suppose \(z\ne v;\) we have:

$$\begin{aligned} \limsup _{n\rightarrow \infty }d(z_{n},z)< & {} \limsup _{n\rightarrow \infty }d(z_{n},v) \\\le & {} \limsup _{n\rightarrow \infty }d(v_{n},v) \\< & {} \limsup _{n\rightarrow \infty }d(v_{n},z) \\= & {} \limsup _{n\rightarrow \infty }d(x_{n},z)=\limsup _{n\rightarrow \infty }d(z_{n},z), \end{aligned}$$

which is a contradiction. Hence, \(z=v=x\in \mathcal {F}.\) \(\square \)

Theorem 3.5

Let C be a nonempty closed convex subset of a complete CAT(0) space M. Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {C} \mathcal {B}(C)\) be multi-valued generalized nonexpansive mappings, such that \( S_{i}p=\left\{ p\right\} \) for all \(p\in \mathcal {F}\). If \(\{x_{n}\}\) is a sequence given by (2.4), where \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are sequences in \([\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \), such that \( \sum _{i=0}^{m}\alpha _{n,i}=1,\) then \(\{x_{n}\}\) converges to a point \(p\in \mathcal {F}\) if and only if:

$$\begin{aligned} \liminf _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0 \ \text { or } \ \limsup _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0. \end{aligned}$$
(3.1)

Proof

It is easy to see that if \(\{x_{n}\}\) converges to a point \(p\in \mathcal {F}\), then (3.1) holds.

Conversely, suppose that:

$$\begin{aligned} \liminf _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0\text { or } \limsup _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0. \end{aligned}$$

Lemma 3.1 implies that:

$$\begin{aligned} d(x_{n+1},p)\le d(x_{n},p) \ \ \ \text { for any }p\in \mathcal {F}, \end{aligned}$$

so that \(\mathrm{{dist}}(x_{n+1},\mathcal {F})\le \mathrm{{dist}}(x_{n},\mathcal {F}).\)Thus, \( \lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})\) exists. Since:

$$\begin{aligned} \liminf _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0 \ \text { or } \limsup _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0, \end{aligned}$$

we have \(\lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0.\) Let \( \varepsilon >0.\) We can find \(n_{0}\ge 1\), such that:

$$\begin{aligned} \mathrm{{dist}}(x_{n},\mathcal {F})<\frac{\varepsilon }{3},\text { for all }n\ge n_{0}. \end{aligned}$$

Therefore, there exists \(p\in \mathcal {F}\), such that:

$$\begin{aligned} d(x_{n_{0}},p)<\frac{\varepsilon }{2}. \end{aligned}$$

For \(m\ge n_{0},n\ge n_{0},\) we have that:

$$\begin{aligned} d(x_{m+n},x_{n})\le d(x_{m+n},p)+d(x_{n},p)<2d(x_{n_{0}},p)<\varepsilon . \end{aligned}$$

Thus, \(\{x_{n}\}\) is a Cauchy sequence in the closed subset C of X. This implies that \(\{x_{n}\}\) converges to \(x\in C.\) For each \(j\in \{1,2,\ldots m\}\):

$$\begin{aligned} \mathrm{{dist}}(x,S_{j}x)\le & {} d(x,x_{n})+\mathrm{{dist}}(x_{n},S_{j}x_{n})+H(S_{j}x_{n},S_{j}x)\\\le & {} d(x,x_{n})+\mathrm{{dist}}(x_{n},S_{j}x_{n})+a_{2}\left[ \mathrm{{dist}}(x_{n},S_{j}x_{n})+\mathrm{{dist}}(x,S_{j}x)\right] \\&+a_{1}d(x_{n},x)+a_{3}\left[ \mathrm{{dist}}(x_{n},S_{j}x)+\mathrm{{dist}}(x,S_{j}x_{n})\right] \\\le & {} \frac{a_{1}+2a_{3}+1}{1-(a_{2}+a_{3})}d(x,x_{n})+\frac{a_{2}+a_{3}+1}{ 1-(a_{2}+a_{3})}\mathrm{{dist}}(x_{n},S_{j}x_{n})\rightarrow 0, \end{aligned}$$

as \(n\rightarrow \infty .\) Hence, \(S_{j}x=x\) for each \(j\in J.\) That is, \(x\in \mathcal {F}\). \(\square \)

Definition 3.6

A finite family of multi-valued mappings \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {C}\mathcal {B}(C)\) satisfies condition (AV) if there exists a nondecreasing function \(f:[0,\infty )\rightarrow [0,\infty )\), such that \(f(0)=0,\) \(f(t)>0\) for \(t>0,\) and

$$\begin{aligned} \frac{1}{m}\sum _{i=1}^{m}\mathrm{{dist}}(x,S_{i}x)\ge f(\mathrm{{dist}}(x,\mathcal {F})), { \ \ \ }x\in C. \end{aligned}$$

The following theorem states the strong convergence of the algorithm.

Theorem 3.7

Let C be a nonempty closed and convex subset of a complete CAT(0) space M. Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {C} \mathcal {B}(C)\) be multi-valued generalized nonexpansive mappings satisfying condition(AV) and \(S_{i}p=\left\{ p\right\} \) for all \(p\in \mathcal {F}\). If \(\{x_{n}\}\) is a sequence given by (2.4) and \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are sequences in \( [\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \) such that \(\sum _{i=0}^{m}\alpha _{n,i}=1,\) then \(\{x_{n}\}\) converges to a point \( p\in \mathcal {F}.\)

Proof

As a consequence of Lemma 3.1, \(\lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n}, \mathcal {F})\) exists.

By condition (B):

$$\begin{aligned} \lim _{n\rightarrow \infty }f(\mathrm{{dist}}(x_{n},\mathcal {F}))\le \frac{1}{m} \sum _{i=1}^{m}\lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},S_{i}x)=0. \end{aligned}$$

That is, \(\lim _{n\rightarrow \infty }f(\mathrm{{dist}}(x_{n},\mathcal {F}))=0.\) Since f is a nondecreasing and \(f(0)=0,\) \(\lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n}, \mathcal {F})=0.\)The conclusion follows from Theorem 3.5. \(\square \)

4 Common fixed points on proximinal subsets

Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {P}(C)\) be multi-valued generalized nonexpansive mappings. For \(x_{1}\in C\) and \(i\in J.\) As \( S_{i}x_{1}\) is a proximinal subset of C,  there exists \(y_{1,i}\in S_{i}x_{1}\), such that \(d(q,y_{1,i})=\mathrm{{dist}}(q,S_{i}x_{1})\) whenever q is a fixed point of \(S_{i}.\) Hence, our scheme 2.4 reads as follows:

$$\begin{aligned} x_{n+1}=\bigoplus \limits _{i=0}^{m}\alpha _{n,i}y_{n,i}, \end{aligned}$$
(4.1)

where \(y_{n,0}=x_{n}.\) For \(i\in J,\) \(y_{n,i}\in S_{i}x_{n}\) with \( d(q,y_{n,i})=\mathrm{{dist}}(q,S_{i}x_{n})\) for \(p\in \mathcal {F}.\) Moreover, \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are sequences in (0, 1) , such that \(\sum \nolimits _{i=0}^{m}\alpha _{n,i}=1.\)

Using the scheme (4.1), we can prove the following results whose proofs carry on similar details to those in Sect. 2. In the following results, assume that C is a nonempty, convex, and closed subset of the space M.

Lemma 4.1

Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {P}(C)\) be multi-valued generalized nonexpansive mappings. If \(\{x_{n}\}\) is a sequence given by (4.1), then \(\lim _{n\rightarrow \infty }d(x_{n},q)\) exists for each \(q\in \mathcal {F}.\)

Lemma 4.2

Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {P}(C)\) be multi-valued generalized nonexpansive mappings. If \(\{x_{n}\}\) is a sequence given by (4.1), where \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots \) \(,\{\alpha _{n,m}\}\) are sequences in \([\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \), such that \( \sum _{i=1}^{m+1}\alpha _{n,i}=1,\) then \(\lim _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},S_{j}x_{n})=0\) for \(j\in J.\)

Theorem 4.3

Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal { P}(C)\) be multi-valued generalized nonexpansive mappings. If \(\{x_{n}\}\) is a sequence given by (4.1) where \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots \) \(,\{\alpha _{n,m}\}\) are sequences in \([\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \) with \(\sum _{i=1}^{m+1} \alpha _{n,i}=1,\) then \(\{x_{n}\}\) converges to a point \(q\in \mathcal {F}\) if and only if:

$$\begin{aligned} \liminf _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0 \ \ \text { or } \ \ \limsup _{n\rightarrow \infty }\mathrm{{dist}}(x_{n},\mathcal {F})=0. \end{aligned}$$

Theorem 4.4

Let \(S_{1},S_{2},\ldots S_{m}:C\rightarrow \mathcal {P}(C)\) be multi-valued generalized nonexpansive mappings satisfying condition (B). If \(\{x_{n}\}\) is a sequence given by (4.1) where \(\{\alpha _{n,0}\},\{\alpha _{n,1}\},\ldots ,\{\alpha _{n,m}\}\) are sequences in \([\delta ,1-\delta ]\) for some \(\delta \in \left( 0,\frac{1}{2}\right) \), such that \( \sum _{i=1}^{m+1}\alpha _{n,i}=1,\) then the sequence \(\{x_{n}\}\) converges to a point \(q\in \mathcal {F}\).

Remark 4.5

The above theorems are new in CAT(0) spaces and reform the results of Abbas et al. [1], Khan and Fukhar-ud-din [9] and some others. The reason is that we consider a nonlinear space, a family of generalized multi-valued nonexpansive mappings and a one-step iterative scheme (instead of multi-step).