Now we give two useful lemmas to prove our main results.

*Let D be a nonempty closed convex subset of a hyperbolic space X and* {*T*_{i} : *i* = 1,2,... ,*k*} *be a family of nonexpansive multivalued mappings such that ${P}_{{T}_{i}}$ is nonexpansive mapping and p ∈* *F*(*T*_{i})* for i =* 1, 2,..., *k. Suppose that $F={\bigcap}_{i=1}^{k}F\left({T}_{i}\right)\ne \mathrm{\varnothing},$ and the iterative sequence* {*x*_{n}}* is defined by* (3 *Then for p* ∈ *F, we get*

(1) $d({x}_{n},{P}_{{T}_{i}}{x}_{n})\le 2d({x}_{n},p)$* for all i =* 1,2*,... ,k*,

(2) $d({y}_{(i-1)n},{u}_{(i-1)n})\le 2d({y}_{(i-1)n},p)$* for all i =* 1,2*,... ,k*,

(3) $d({u}_{(i-1)n},p)\le d({y}_{(i-1)n},p)$* for all i =* 1,2*,... ,k*,

(4) $d({y}_{in},p)\le d({x}_{n},p)$* for all i =* 1,2*,... ,k -* 1,

(5) $d({x}_{n+1},p)\le d({x}_{n},p),$,

(6) ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)\phantom{\rule{thinmathspace}{0ex}}exists,$

Let *p ∈ F*.

(1). For *i* = 1, 2, 3,... *,k*, we have
$$\begin{array}{ll}d({x}_{n},{P}_{{T}_{i}}{x}_{n})& \le d({x}_{n},p)+d(p,{P}_{{T}_{i}}({x}_{n}))\\ & \le d({x}_{n},p)+H({P}_{{T}_{i}}(p),{P}_{{T}_{i}}({x}_{n}))\\ & \le d({x}_{n},p)+d(p,{x}_{n})\\ & =2d({x}_{n},p).\end{array}$$(4)

(2). In a similar way with part (1), we get
$$\begin{array}{ll}d({y}_{(i-1)n},{u}_{(i-1)n})& \le d({y}_{(i-1)n},p)+d(p,\phantom{\rule{thinmathspace}{0ex}}{u}_{(i-1)n})\\ & \le d({y}_{(i-1)n},p)+d(p,{P}_{{T}_{i}}({y}_{(i-1)n}))\\ & \le d({y}_{(i-1)n},p)+H({P}_{{T}_{i}}(p),{P}_{{T}_{i}}({y}_{(i-1)n}))\\ & \le d({y}_{(i-1)n},p)+d(p,{y}_{(i-1)n})\\ & =2d({y}_{(i-1)n},p).\end{array}$$(5)

(3). For *i* = 1, 2, 3,..., *k*, we have
$$\begin{array}{ll}d({u}_{(i-1)n},p)& \le d({u}_{(i-1)n},{P}_{{T}_{i}}(p))\\ & \le H({P}_{{T}_{i}}({y}_{(i-1)n}),{P}_{{T}_{i}}(p))\\ & \le d({y}_{(i-1)n},p).\end{array}$$(6)

(4). We prove this item in three parts. Firstly
$$\begin{array}{ll}d({y}_{1n},p)& =d(W({u}_{0n},{y}_{0n}{\alpha}_{1n}),p)\\ & \le (1-{\alpha}_{1n})d({u}_{0n},p)+{\alpha}_{1n}d({y}_{0n},p)\\ & \le (1-{\alpha}_{1n})d({u}_{0n},{P}_{{T}_{1}}(p))+{\alpha}_{1n}d({y}_{0n},p)\\ & \le (1-{\alpha}_{1n})H({P}_{{T}_{1}}({y}_{0n}),{P}_{{T}_{1}}(p))+{\alpha}_{1n}d({y}_{0n},p)\\ & \le (1-{\alpha}_{1n})d({y}_{0n},P)+{\alpha}_{1n}d({y}_{0n},p)\\ & =d({x}_{n},p)\end{array}$$(7)

Secondly, we assume that *d*(*y*_{jn}, p) ≤ *d*(*x*_{n}, p) holds for some 1 ≤ *j* ≤ *k — 2*. Then
$$\begin{array}{ll}d({y}_{(j+1)n},p)& =d(W({u}_{jn},{y}_{jn},{\alpha}_{(j+1)n}),p)\\ & \le (1-{\alpha}_{(j+1)n})d({u}_{jn},p)+{\alpha}_{(j+1)n}d({y}_{jn},p)\\ & \le (1-{\alpha}_{(j+1)n})H({P}_{{T}_{(j+1)n}}({y}_{jn}),{P}_{{T}_{(j+1)n}}(p))+{\alpha}_{(j+1)n}d({y}_{jn},p)\\ & \le (1-{\alpha}_{(j+1)n}d({y}_{jn},p)+{\alpha}_{(j+1)n}d({y}_{jn},p)\\ & \le d({y}_{jn},p)\\ & \le d({x}_{n},p).\end{array}$$(8)

Lastly,
$$\begin{array}{ll}d({y}_{(k-1)n},p)& =d(W({u}_{(k-2)n,}{y}_{y(k-2)n},{\alpha}_{(k-1)n}),p)\\ & \le (1-{\alpha}_{(k-1)n})d({u}_{(k-2)n},p)+{\alpha}_{(k-1)n}d({y}_{(k-2)n},p)\\ & \le (1-{\alpha}_{(k-1)n})d({u}_{(k-2)n},{P}_{{T}_{k-1}}(p))+{\alpha}_{(k-1)n}d({y}_{(k-2)n},p)\\ & \le (1-{\alpha}_{(k-1)n})H({P}_{{T}_{k-1}}({y}_{(k-2)n}),{P}_{{T}_{{}_{k-1}}}(p))+{\alpha}_{(k-1)n}d({y}_{(k-2)n},p)\\ & \le (1-{\alpha}_{(k-1)n})d({y}_{(k-2)n},p)+{\alpha}_{(k-1)n}d({y}_{(k-2)n},p)\\ & =d({y}_{(k-2)n},p).\end{array}$$(9)

So, by induction, we get
$$d({y}_{in},p)\le d({x}_{n},p)$$(10) for all *i* = 1, *2*,... *,k -* 1.

(5). By part (4), we have
$$\begin{array}{ll}d({{x}_{n}}_{+1},p)& =d(W({u}_{(k-1)n,}{y}_{(k-1)n},{\alpha}_{kn}),p)\\ & \le (1-{\alpha}_{kn})d({u}_{(k-1)n},p)+{\alpha}_{kn}d({y}_{(k-1)n},p)\\ & \le (1-{\alpha}_{kn})d({u}_{(k-1)n},{p}_{{T}_{k}}(p)+{\alpha}_{kn}d({y}_{(k-1)n}),p)\\ & \le (1-{\alpha}_{kn})H({p}_{{T}_{k}}({y}_{(k-1)n}),{p}_{{T}_{k}}(p))+{\alpha}_{kn}d({y}_{(k-1)n},p)\\ & =(1-{\alpha}_{kn})d({y}_{(k-1)n},p)+{\alpha}_{kn}d({y}_{(k-1)n},p)\\ & =d({y}_{(k-1)n},p)\\ & \le d({x}_{n},p).\end{array}$$(11)

(6). By part (5), we get
$$d({x}_{n+1},p)\le d({x}_{n},p)$$

for *i* = 1,2*,... ,k*. Thus, we obtain lim_{n}_{→∞} *d*(*x*_{n}, p)exists for each *p ∈ F*.

*Let D be a nonempty closed subset of a uniformly convex hyperbolic space X and T*_{i} : D → *Ρ* (*D*)* be a family of multivalued mappings such that ${P}_{{T}_{i}}$ is nonexpansive mapping for i =* 1,2,...,*k with a set of F ≠* ∅. *Then for the iterative process* {*x*_{n}}* defined in* (3), *we have*
$$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{P}_{{T}_{i}}{x}_{n})=0$$

*for i =* 1,2*,... ,k*.

By Lemma 3.1, lim_{n}_{→∞} *d*(*x*_{n}, p)exists for each *p ∈ F*. Therefore, for a *c ≥* 0, we have
$$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},p)=c.$$(12)

By taking lim sup on both sides of (10), we have
$$\underset{n\to \mathrm{\infty}}{lim\phantom{\rule{thinmathspace}{0ex}}sup}d\left({y}_{in},p\right)\le c$$(13)

for all *i* = 1,2,... *,k -* 1.

So, by (6) and (13), we get
$$\underset{n\to \mathrm{\infty}}{lim\phantom{\rule{thinmathspace}{0ex}}sup}d\left({u}_{\left(i-1\right)n},p\right)\le c$$(14)

forall *i* = 1,2,..., *k*.

Since lim_{n}_{→∞} *d*(*x*_{n+}_{1}*, p*)* = c*, we have $\underset{n\to \mathrm{\infty}}{lim}d(W\left({u}_{\left(k-1\right)n},{y}_{\left(k-1\right)n},{\alpha}_{kn}\right),p)=c.$. From Lemma 2.2, (13) and (14), we have
$$\underset{n\to \mathrm{\infty}}{lim}d\left({y}_{\left(k-1\right)n},{u}_{\left(k-1\right)n}\right)=0.$$

We claim that,
$$\underset{n\to \mathrm{\infty}}{lim}d\left({y}_{\left(j-1\right)n},{u}_{\left(j-1\right)n}\right)=0$$(15)

for all *j* = 2,3,..., *k* . By Lemma 3.1 we get
$$d\left({x}_{n+1},p\right)\le d\left({y}_{in},p\right)$$(16)

for all *i* = 1, 2 ,..., *k –* 1. Therefore, from (16), we obtain
$$c\le \underset{n\to \mathrm{\infty}}{liminf}d\left({y}_{\left(i-1\right)n},p\right)$$(17)

for *i* = 2,3,..., *k* it. By (3), (13) and (17), we obtain
$$\underset{n\to \mathrm{\infty}}{lim}d(W({u}_{\left(j-2\right)n},{y}_{\left(j-2\right)n},{\alpha}_{\left(j-1\right)n}),p)=\phantom{\rule{thickmathspace}{0ex}}\underset{n\to \mathrm{\infty}}{lim}d({y}_{(j-1)n},p)=c.$$

Using (13), (14) and Lemma 2.2, we get $\underset{n\to \mathrm{\infty}}{lim}d\left({u}_{\left(j-2\right)n},{y}_{\left(j-2\right)n}\right)=0.$ So, by induction
$$\underset{n\to \mathrm{\infty}}{lim}d({y}_{(i-1)n},{u}_{(i-1)n})=0$$(18)

for all *i* = 1,2*,... ,k*. From (3), we get
$$\begin{array}{ll}d({y}_{in},{y}_{(i-1)n})& =d(W({u}_{(i-1)n},{y}_{\left(i-1\right)n},{\alpha}_{in}),{y}_{\left(i-1\right)n})\\ & \le (1-{\alpha}_{in})d({u}_{(i-1)n},{y}_{\left(i-1\right)n})\phantom{\rule{thickmathspace}{0ex}}.\end{array}$$

By (18), we have
$$\underset{n\to \mathrm{\infty}}{lim}d({y}_{in},{y}_{\left(i-1\right)n})=0$$(19)

for *i* = 1,2*,... ,k*. Since
$$\begin{array}{ll}d({x}_{n},{y}_{1n})& =d({x}_{n},W({u}_{0n},{y}_{0n},{\alpha}_{1n})\\ & \le (1-{\alpha}_{1n})d({x}_{n},{u}_{0n})+{\alpha}_{1n}d({x}_{n},{y}_{0n})\\ & =(1-{\alpha}_{1n})d({x}_{n},{u}_{0n}).\end{array}$$

by taking *i* = 1 in (18), we get
$$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{y}_{1n})=0.$$(20)

By known triangle inequality,
$$d({x}_{n},{y}_{in})\le d({x}_{n},{y}_{1n})+d({y}_{1n},{y}_{2n})+\cdots +d({y}_{(i-1)n},{y}_{in})\phantom{\rule{thickmathspace}{0ex}}.$$

for all *i =* 1, 2,..., *k -* 1. It follows by (19) and (20) that
$$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{y}_{in})=0.$$(21)

For *i* = 1,2, 3,..., *k*
$$\begin{array}{ll}d({x}_{n},{P}_{{T}_{i}}{x}_{n})& \le d({x}_{n},{y}_{(i-1)n})+d({y}_{(i-1)n},{u}_{(i-1)n})+d({u}_{\left(i-1\right)n},{P}_{{T}_{i}}{x}_{n})\\ & \le d({x}_{n},{y}_{(i-1)n})+d({y}_{(i-1)n},{u}_{(i-1)n})+H({P}_{{T}_{i}}{y}_{(i-1)n},{P}_{{T}_{i}}{x}_{n})\\ & \le 2d({x}_{n},{y}_{(i-1)n})+d({y}_{(i-1)n},{u}_{(i-1)n}).\end{array}$$

From (18) and (21), we conclude
$$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{P}_{{T}_{i}}{x}_{n})=0.$$

*Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η. Let T*_{i}, ${P}_{{T}_{i}}$* and F be as in Lemma 3.2, Then the iterative process* {*x*_{n}}, Δ— *converges to p in F*.

*Proof. Let p ∈ F*. Then $p\in F\left({T}_{i}\right)=F\left({P}_{{T}_{i}}\right),$ for *i* = 1,2,... *,k*. By the Lemma 3.1, {*x*_{n}}is bounded and so lim_{n}_{→∞} *d*(*x*_{n}, p)exists. Thus {*x*_{n}}has a unique asymptotic center. In other words, we have *A*({*x*_{n}})* =* {*x*_{n}}. Let {*v*_{n}}be a subsequence of {*x*_{n}}such that *A*({*v*_{n}})* =* {*v*}. From Lemma 3.2, we get $\underset{n\to \mathrm{\infty}}{lim}\left({v}_{n},{P}_{{T}_{1}}\left({v}_{n}\right)\right)=0.$ We claim that υ is a fixed point of ${P}_{{T}_{1}}.$

To prove this, we take another sequence {*z*_{m}}in *P*_{T1}(*v*) Then,
$$\begin{array}{ll}r({z}_{m},\{{v}_{n}\})& =\underset{n\to \mathrm{\infty}}{limsup}d({z}_{m},{v}_{n})\\ & \le \underset{n\to \mathrm{\infty}}{lim}\{d({z}_{m},{P}_{{T}_{1}}({v}_{n})+d({P}_{{T}_{1}}({v}_{n}),{v}_{n})\}\\ & \le \underset{n\to \mathrm{\infty}}{lim}\{H({P}_{{T}_{1}}(v),{P}_{{T}_{1}}({v}_{n}))+d({P}_{{T}_{1}}({v}_{n}),{v}_{n}))\}\\ & \le \underset{n\to \mathrm{\infty}}{limsup}d(v,{v}_{n})\\ & =r(v,\{{v}_{n}\})\end{array}$$

this gives $|r({z}_{m},\{{v}_{n}\})-r(v,\{{v}_{n}\})|\to 0$ for *m →*. By Lemma 2.3, we get lim_{m→∞}_{→∞} *z*_{m} = v. Note that *T*_{1}(*υ*) ∈ *P*(*D*)being proximinal is closed, hence ${P}_{{T}_{1}}\left(v\right)$ is either closed or bounded. Consequently $\underset{m\to \mathrm{\infty}}{lim}{z}_{m}=v\in {P}_{{T}_{1}}\left(v\right).$ Similarly $v\in {P}_{{T}_{2}}\left(v\right),v\in {P}_{{T}_{3}}\left(v\right)\dots v\in {P}_{{T}_{k}}\left(v\right).$* So ν ∈ F*.

*Let D be a nonempty closed convex subset of a hyperbolic space E and Let T*_{i}, ${P}_{{T}_{i}}$* and F be as in Lemma 3.2 and let* {*x*_{n}}* be the iterative process defined in 3, then* {*x*_{n}}* converges to p in F if and only if* lim_{n}_{→∞} *d*(*x*_{n}, *F*)* =* 0.

If {*x*_{n}}converges to *p ∈ F*, then lim_{n}_{→∞} *d*(*x*_{n}, p)* =* 0. since 0 ≤ *d*(*x*_{n},F)* ≤* *d*(*x*_{n},* p*), we have lim_{n}_{→∞} *d*(*x*_{n}, *F*)* =* 0. Conversely, suppose that lim_{n}_{→∞} inf *d*(*x*_{n}, F)* =* 0. By Lemma 3.1, we have
$$d({x}_{n+1},p)\le d({x}_{n},p)$$

which implies
$$d({x}_{n+1},F)\le d({x}_{n},F).$$

This gives that lim_{n}_{→∞} *d*(*x*_{n}, *F*)exists. Therefore, by the hypothesis of our theorem, lim inf_{n}_{→∞} *d*(*x*_{n}, F)* =* 0. Thus we have lim_{n}_{→∞} *d*(*x*_{n}, *F*)* =* 0. Let us show that {*x*_{n}}is a Cauchy sequence in *D*. Let *m, n*, ∈ ℕ and assume *m > n*. Then it follows that *d*(*x*_{m}, p)* ≤* *d*(*x*_{n}, p)for all *p ∈ F*. Thus we get,
$$d({x}_{m},{x}_{n})\le d({x}_{m},p)+d(p,{x}_{n})\le 2d({x}_{n},p)\phantom{\rule{thickmathspace}{0ex}}.$$

Taking inf on the set *F*, we have *d*(*x*_{m},* x*_{n})≤ *d*(*x*_{n},* F*). We show that {*x*_{n}}is a Cauchy sequence in *D*. By taking as *m, n* →* ∞* in the inequality *d*(*x*_{m},* x*_{n})≤ *d*(*x*_{n},* F*). So, it converges to a *q ∈ D*. Now it is left to show that

*q ∈ F* (*Τ*_{1}). Indeed by $d({x}_{n},F({P}_{{T}_{1}}))={\displaystyle {inf}_{y\in F({P}_{{T}_{1}})}d({x}_{n},y).}$So for each ∊ > 0, there exists ${p}_{n}^{(\u03f5)}\in F({P}_{{T}_{1}})$ such that,
$$d({x}_{n},{p}_{n}^{(\u03f5)})<d({x}_{n},F({P}_{{T}_{1}}))+\frac{\u03f5}{2}.$$

This implies ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{p}_{n}^{(\u03f5)})\le \frac{\u03f5}{2}.\phantom{\rule{thinmathspace}{0ex}}\text{Since}\phantom{\rule{thinmathspace}{0ex}}d({p}_{n}{,}^{(\u03f5)}q)\le d({x}_{n},{p}_{0}^{(\u03f5)})+d({x}_{n},q)$ it follows that ${lim}_{n\to \mathrm{\infty}}d({p}_{n}^{(\u03f5)},q)\le \frac{\u03f5}{2}.$ Finally
$$\begin{array}{ll}d({P}_{{T}_{1}}(q),q)& \le d(q,{p}_{n}^{(\u03f5)})+d({p}_{n}^{(\u03f5)},{P}_{{T}_{1}}(q))\\ & \le d(q,{p}_{n}^{(\u03f5)})+H({P}_{{T}_{1}}({p}_{n}^{(\u03f5)}),{P}_{{T}_{1}}(q))\\ & \le 2d({p}_{n}^{(\u03f5)},q)\end{array}$$

which shows $d({P}_{{T}_{1}}(q),q)<\u03f5.\phantom{\rule{thinmathspace}{0ex}}\text{So},\phantom{\rule{thinmathspace}{0ex}}d({P}_{{T}_{1}}(q),q)=0.$ In a similar way, we get for any *i* = 1, 2..., *k* we obtain $d({P}_{{T}_{i}}(q),q)=0.$ Since *F* is closed, *q ∈ F*.

Now we give the definition of condition (B) of Senter and Dotson for a finite family of multivalued mappings to complete the proof of the following theorem.

([31]). *The multivalued nonexpansive mappings* *Τ*_{1}, *Τ*_{2},...,*T*_{k} :* D* → *CB*(*D*)* are said to satisfy condition* (*B*)*. If there exists a nondecreasing function* ƒ : [0, ∞) → [0, ∞) *with f*(*0*)* =* 0, *f*(*r*)* >* 0 *for all* r ∈ (0, ∞) *such that*
$$d({x}_{n},{T}_{i}{x}_{n})\ge f(d({x}_{n},F)),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}F\ne \mathrm{\varnothing}.$$

*A map Τ* : *D* → *Ρ*(*D*)* is called semi-compact if any bounded sequence* {*x*_{n}}* satisfying $d({x}_{n},{T}_{{x}_{n}})\to 0$* *as n* → ∞ *has a convergent subsequence*.

*Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η. Let T*_{i}, ${P}_{{T}_{i}}$ and F be as in Lemma 3.2. Suppose that each ${P}_{{T}_{i}}$ satisfies condition (*B*)*. Then the iterative process* {*x*_{n}}* defined in* (3) *converges strongly to p ∈* *F*.

*Proof*. By Lemma 3.1, lim_{n}_{→∞} *d*(*x*_{n}, p)exists for all *p ∈ F*. We call it *c* for some *c* ≥ 0.Then if *c* = 0,proof is completed. Assume *c* > 0. Now *d* (*x*_{n+}_{1}*, p*)* ≤* *d* (*x*_{n}, p)gives that
$$\underset{p\in F({T}_{\mathrm{i}})}{inf}d({x}_{n+1},p)\le \underset{p\in F({T}_{\mathrm{i}})}{inf}d({x}_{n},p)$$

which means that $d({x}_{n+1},F)\le d({x}_{n},F).\phantom{\rule{thinmathspace}{0ex}}\text{So},{\displaystyle {lim}_{n\to \mathrm{\infty}}d({x}_{n},F)}$ exists. By using the condition (*B*)and Lemma 3.2 we obtain,
$$\underset{n\to \mathrm{\infty}}{lim}f(d({x}_{n},F))\le \underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{P}_{{T}_{i}}({x}_{n})\to 0$$

and so lim_{n}_{→∞} (*d*(*x*_{n},* F*))* =* 0. By the properties of ƒ, we get lim_{n}_{→∞} *d*(*x*_{n}, F) *=* 0. Finally by applying Theorem 3.4, we obtain the result.

*Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η and let T*_{i}, ${P}_{{T}_{i}}$ F be as in Lemma 3.2. Suppose that ${P}_{{T}_{i}}$ is semi-compact then the iterative process {*x*_{n}}* defined in* (3) *converges strongly to p ∈* *F*.

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