Fixed point approximation for a class of generalized nonexpansive multi-valued mappings in Banach spaces

Abstract

In this paper, we propose a new iteration process, called multi-valued F-iteration process, for the approximation of fixed points. We introduce a new class of multi-valued generalized nonexpansive mappings satisfying a $B_{\gamma ,\mu }$ property. Moreover, we establish certain weak and strong convergence theorems in uniformly convex Banach spaces. We also discuss the stability of the modified F-iteration process. Furthermore, a numerical example is presented to illustrate the superiority of our results.

1 Introduction

Fixed point theory provides essential tools for solving various types of nonlinear problems. Fixed point theory for different types of single-valued and multi-valued mappings has attracted the attention of many researchers. Many types of iterative processes have been utilized to approximate the fixed points of multi-valued mappings in Banach spaces, (see, e.g., [1, 2, 6, 8, 16, 17]). Let U be a Banach space with norm $\Vert .\Vert$ and $\Re$ a nonempty subset of U. A mapping $\mathcal {S} : \Re \rightarrow \Re$ is contraction if and only if there is a real number $\alpha \in (0,1)$ such that

$$\begin{aligned} \Vert \mathcal {S} \ell -\mathcal {S}\eta \Vert \le \alpha \Vert \ell -\eta \Vert , \end{aligned}$$

(1)

for all $\ell , \eta \in \Re$. The mapping $\mathcal {S}$ is said to be nonexpansive if $\alpha = 1$ in (1). The set of all fixed points of $\mathcal {S}$ denote by $Fix (\mathcal {S}) := \{ \ell \in \Re : \mathcal {S} \ell = \ell \}$. It is well known that if $\Re$ is a closed, bounded, and convex subset of a uniformly convex Banach space $\Re$, then $Fix (\mathcal {S} )$ is nonempty for a nonexpansive mapping [5]. Many authors have years, several extensions and generalizations of nonexpansive mappings in recent years due to their diverse applications. Suzuki [15] introduced an interesting generalization of single-valued nonexpansive mappings and obtained some existence and convergence results. Such mappings are known as mappings satisfying condition (C). A mapping $\mathcal {S} : \Re \rightarrow \Re$ is said to be satisfy condition (C) if

$$\begin{aligned} \Vert \ell -\mathcal {S}\ell \Vert \le \Vert \ell -\eta \Vert \Rightarrow \Vert \mathcal {S}\ell -\mathcal {S}\eta \Vert \le \Vert \ell -\eta \Vert , \end{aligned}$$

for all $\ell , \eta \in \Re .$ Recently, in 2018, Patir et al. [11] suggested two parametric conditions, which they called Condition $B_{\gamma ,\mu }$. They proved that Condition $B_{\gamma ,\mu }$ is weaker than the corresponding condition (C). A self-mapping $\mathcal {S}$ of a subset $\Re$ of a metric space is said to satisfy Condition $B_{\gamma ,\mu }$ (or called Patir map) if there are some $\gamma \in [0, 1]$ and $\mu \in [0,\frac{1}{2}]$ with $2\mu \le \gamma$ such that for all $\ell , \eta \in \Re$,

$$\begin{aligned} \gamma \Vert \ell -\mathcal {S}\ell \Vert \le \Vert \ell -\eta \Vert + \mu \Vert \eta -\mathcal {S}\eta \Vert \Rightarrow \Vert \mathcal {S}\ell -\mathcal {S}\eta \Vert \le (1-\gamma )\Vert \ell -\eta \Vert +\mu (\Vert \ell -\mathcal {S}\eta \Vert +\Vert \eta -\mathcal {S}\ell \Vert ). \end{aligned}$$

In 2011, Abkar and Eslamian [1] extended the notion of condition (C) to the multi-valued mappings. To avoid the endpoint condition, Shahzad and Zegeye [14] introduced another Ishikawa iterative scheme using $\mathcal {P}_{\mathcal {S}}(\ell ) = \{\eta \in \mathcal {S}\ell :\Vert \ell -\eta \Vert = d(\ell ,\mathcal {S}\ell )\},$ where $\mathcal {S}$ is a given multi-valued mapping. Very recently, in 2020, Ali and Ali [3] introduced a new iteration process, called the F-iterative scheme for generalized contractions as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \ell _{1}\in \Re \\ \ell _{k+1}= \mathcal {S}\eta _{k}\\ \eta _{k}=\mathcal {S}\hbar _{k}\\ \hbar _{k}=\mathcal {S}((1-\delta _{k})\ell _{k}+\delta _{k}\mathcal {S}\ell _{k}), \end{array}\right. } \end{aligned}$$

(2)

where $\delta _{k}\in (0,1)$ and for all $k \in \mathbb {N}$. The authors showed that the sequence $\{\ell _{k}\}$ defined by iterative process (2) is stable and has a better rate of convergence when compared with the other iterations in the setting of generalized contractions.

Following are some basic definitions and results needed in the sequel.

Let $(U,\Vert .\Vert )$ be a Banach space and $\Re$ be a nonempty subset of U. The set $\Re$ is said to be a proximinal if there exists some $\eta$ in $\Re$ such that $d(\ell ,\eta )=d(\ell ,\Re )$, where $d(\ell ,\Re )=\inf \{d(\ell ,\eta ): \eta \in \Re \}$, for each $\ell \in U$. From now on, the notations $\mathcal {P}_{px}(\Re ), \mathcal {P}_{cb}(\Re )$ and $\mathcal {P}(\Re )$ denotes the families of nonempty proximinal subsets, closed bounded subsets and all possible subsets of $\Re$ respectively. A point $\ell \in \Re$ is called an endpoint of $\mathcal {S}$ if $\{\ell \} = \mathcal {S}(\ell )$. A multi-valued mapping $\mathcal {S}$ is said to satisfy the endpoint condition, if $\{\ell \} = \mathcal {S}(\ell )$ for all $\ell \in Fix(\mathcal {S})$. The Pompeiu–Hausdorff metric [4] on the set $\mathcal {P}_{cb}(\Re )$ is defined by

$$\begin{aligned} \mathcal {H}({\mathcal {M}},{\mathcal {N}})=\max \{\sup _{\ell \in {\mathcal {M}}}d(\ell ,{\mathcal {N}}),\sup _{\eta \in {\mathcal {N}}}d(\eta ,{\mathcal {M}})\}, \end{aligned}$$

for all ${\mathcal {M}},{\mathcal {N}}\in \mathcal {P}_{cb}(\Re )$.

Let $\Re$ be a subset of a Banach space and a multi-valued mapping $\mathcal {S}:\Re \rightarrow \mathcal {P}(\Re )$ is said to be:

(i) a contraction mapping if there exists an $\alpha \in [0, 1)$ such that

$$\begin{aligned} \mathcal {H}( \mathcal {S}\ell ,\mathcal {S}\eta ) \le \alpha \Vert \ell -\eta \Vert , \end{aligned}$$

for all $\ell , \eta \in \Re$.

(ii) a nonexpansive mapping if

$$\begin{aligned} \mathcal {H}(\mathcal {S}\ell ,\mathcal {S}\eta )\le \Vert \ell -\eta \Vert , \end{aligned}$$

for all $\ell , \eta \in \Re .$

(iii) a quasi-nonexpansive mapping if $Fix(\mathcal {S}) \ne \phi$ and

$$\begin{aligned} \mathcal {H}(\mathcal {S}\ell ,\mathcal {S}q) \le \Vert \ell - q\Vert , \end{aligned}$$

for every $q \in Fix(\mathcal {S})$.

A multi-valued mapping $\mathcal {S}: \Re \rightarrow \mathcal {P}_{cb}(\Re )$ is said to satisfy condition (C) if for all $\ell ,\eta \in \Re$ the following condition holds:

$$\begin{aligned} d(\ell , \mathcal {S}\ell ) \le \Vert \ell -\eta \Vert \Rightarrow \mathcal {H}(\mathcal {S}\ell ,\mathcal {S}\eta ) \le \Vert \ell -\eta \Vert . \end{aligned}$$

Every multi-valued nonexpansive mapping also satisfies condition (C).

Definition 1.1

A Banach space U is said to have Opial’s condition if and only if for each weakly convergent sequence $\{\ell _{k}\}\subset U$ with a weak limit $\ell$ in U, we have

$$\begin{aligned} \limsup _{k \rightarrow \infty }\Vert \ell _{k}-\ell \Vert < \limsup _{k \rightarrow \infty }\Vert \ell _{k}-\eta \Vert , \end{aligned}$$

for each $\eta$ in U and $\ell \ne \eta$.

Lemma 1.2

[14] Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{px}(\Re )$ and $\mathcal {P}_{\mathcal {S}}(\ell )=\{ \eta \in \mathcal {S}\ell : \Vert \ell -\eta \Vert = d(\ell , \mathcal {S}\ell )\}$. Then the following conditions are equivalent:

• (i) $q\in Fix(\mathcal {S})$.

• (ii) $\mathcal {P}_{\mathcal {S}}(q)=\{q\}.$

• (iii) $q \in Fix(\mathcal {P}_{\mathcal {S}}(q))$.

Moreover, $Fix(\mathcal {S}) = Fix(\mathcal {P}_{\mathcal {S}})$.

Definition 1.3

Let $\{\ell _{k}\}$ be a bounded sequence in U and $\Re$ be a subset of U. Then,

• (i) The asymptotic radius of $\{\ell _{k}\}$ at a point $\ell$ in U is defined as

  $$\begin{aligned} r(\ell ,\{\ell _{k}\}) = \limsup _{k \rightarrow \infty }\Vert \ell _{k}-\ell \Vert . \end{aligned}$$

• (ii) The asymptotic radius of $\{\ell _{k}\}$ with respect to $\Re$ is defined as

  $$\begin{aligned} r(\ell , \Re ) = \inf \{r(\ell , \{\ell _{k}\}): \ell \in \Re \}. \end{aligned}$$

• (iii) The asymptotic center of $\{\ell _{k}\}$ with respect to $\Re$ is defined as

  $$\begin{aligned} \mathcal {A}(\Re ,\{ \ell _{k}\}) = \{\ell \in \Re ; r(\ell ,\{\ell _{k}\}) = r(\Re ,\ell _{k})\}. \end{aligned}$$

Definition 1.4

Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{cb}(\Re )$. A sequence $\{\ell _{k}\}\in \Re$ is said to be an approximate fixed point sequence (or AFPS) for $\mathcal {S}$ provided that $d(\ell _{k}, \mathcal {S}\ell _{k}) \rightarrow 0$ as $k\rightarrow \infty$.

Definition 1.5

A multi-valued mapping $\mathcal {S}:\Re \rightarrow \mathcal {P}(\Re )$ is called demiclosed at $\eta \in \Re$ if for any sequence $\{\ell _{k}\}$ in $\Re$ weakly convergent to t in $\Re$ and $\eta _{k}\in \mathcal {S}$ $(\ell _{k})$ strongly convergent to $\eta$, we have $\eta \in \mathcal {S}(t)$.

In 1974, Senter and Dotson [13] provided the multi-valued version of condition (I).

Definition 1.6

[13] A multi-valued mapping $\mathcal {S}:\Re \rightarrow \mathcal {P}(\Re )$ is said to satisfy Condition (I) if there exists a continuous nondecreasing function $f:[0,\infty ) \rightarrow [0,\infty )$ with $f(0) = 0$, $f(s) > 0$ for all $s\in (0,\infty )$ such that $d(\ell , \mathcal {S}\ell )\ge f(d(\ell ,Fix(\mathcal {S}))$ for all $\ell \in \Re$.

Lemma 1.7

[12] Suppose that U is a uniformly convex Banach space. Assume that $\{\alpha _{k}\}$ is any sequence of real numbers such that $0< \beta \le \{\alpha _{k}\} \le \delta < 1$ for all $k \ge 1$. If $\{\ell _{k}\}$ and $\{\eta _{k}\}$ are any two sequences in U such that $\limsup _{k \rightarrow \infty } \Vert \ell _{k}\Vert \le j$, $\limsup _{k \rightarrow \infty } \Vert \eta _{k}\Vert \le j$ and $\lim _{k \rightarrow \infty } \Vert (1-\alpha _{k})\ell _{k}+\alpha _{k}\eta _{k}\Vert =j$ hold for some $j \ge 0$. Then $\lim _{k\rightarrow \infty } \Vert \ell _{k}-\eta _{k}\Vert =0$.

The purpose of this paper is to study a new class of multi-valued mappings generalized nonexpansive mappings satisfies Condition $B_{\gamma ,\mu }$ and present a fixed point result. We establish weak and strong convergence results for mapping which satisfies the Condition $B_{\gamma ,\mu }$, using the multi-valued version of F-iteration process in uniformly convex Banach spaces. Furthermore, we provide a stability of the modified iteration process and an interesting example to illustrate the results.

2 Main results

We define multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$ as follows:

Definition 2.1

Let $\Re$ be a nonempty subset of a Banach space U. A mapping $\mathcal {S}: \Re \rightarrow \mathcal {P}_{cb}(\Re )$ is called a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$ if there are some $\gamma \in [0, 1]$ and $\mu \in [0,\frac{1}{2}]$ with $2\mu \le \gamma$ such that for all $\ell , \eta \in \Re$,

$$\begin{aligned} \gamma d( \ell -\mathcal {S}\ell )\le \Vert \ell -\eta \Vert + \mu d(\eta -\mathcal {S}\eta ) \Rightarrow \mathcal {H}(\mathcal {S}\ell ,\mathcal {S}\eta )\le (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell )). \end{aligned}$$

Lemma 2.2

Let $\Re$ be a nonempty subset of a Banach space U and consider a multi-valued mapping $\mathcal {S} : \Re \rightarrow \mathcal {P}_{cb}(\Re )$. If $\mathcal {S}$ is mapping satisfying Condition $B_{\gamma ,\mu }$ with a fixed point $q\in Fix(\mathcal {S})$ and satisfies the endpoint condition, then $\mathcal {S}$ is quasi-nonexpansive.

Proof

Assume that $q \in Fix(\mathcal {S})$. Then

$$\begin{aligned} \gamma d(q,\mathcal {S}q)=0 \le \Vert q-\eta \Vert + \mu d(\eta ,\mathcal {S}\eta ), \end{aligned}$$

for any $\eta \in \Re$, so

$$\begin{aligned} \mathcal {H}(\mathcal {S}q,\mathcal {S}\eta )\le & {} (1-\gamma )\Vert q-\eta \Vert +\mu (d(q,\mathcal {S}\eta )+d(\eta ,\mathcal {S}q))\\\le & {} (1-\gamma )\Vert q-\eta \Vert +\mu (\mathcal {H}(\mathcal {S}q,\mathcal {S}\eta )+ \Vert \eta -q\Vert ). \end{aligned}$$

This yields

$$\begin{aligned} (1-\mu )\mathcal {H}(\mathcal {S}q,\mathcal {S}\eta )\le (1-\gamma +\mu )\Vert q-\eta \Vert . \end{aligned}$$

Since $2\mu \le \gamma$, we obtain the result. $\square$

Lemma 2.3

Let $\Re$ be a nonempty subset of a Banach space U and $\mathcal {S}:\Re \rightarrow \mathcal {P}_{cb}(\Re )$ be a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$. For any $\ell ,\eta \in \Re , \nu \in \mathcal {S}\ell$ and $h\in [0, 1]$. Then the following results hold:

• (i) $d(\nu , \mathcal {S}\nu ) \le \Vert \ell -\nu \Vert$.

• (ii) at least one of the following $((a)\;\; and\;\; (b))$ holds:

  • (a) $\frac{h}{2} d(\ell ,\mathcal {S}\ell )\le \Vert \ell -\eta \Vert$,

  • (b) $\frac{h}{2} d(\nu ,\mathcal {S}\nu )\le \Vert \nu -\eta \Vert$.

Proof

(i) Since $\gamma d(\ell ,\mathcal {S}\ell )\le \Vert \ell -\eta \Vert +\mu d(\eta ,\mathcal {S}\eta )$ for any $\nu \in \mathcal {S}\ell$, we obtain

$$\begin{aligned} \mathcal {H}(\mathcal {S}\ell ,\mathcal {S}\nu )\le & {} (1-\gamma )\Vert \ell -\nu \Vert +\mu (d(\ell ,\mathcal {S}\nu )+d(\nu ,\mathcal {S}\ell ))\\= & {} (1-\gamma )\Vert \ell -\nu \Vert +\mu d(\ell ,\mathcal {S}\nu )\\\le & {} (1-\gamma )\Vert \ell -\nu \Vert +\mu (\Vert \ell -\nu \Vert +d(\nu ,\mathcal {S}\nu ))\\= & {} (1-\gamma +\mu )\Vert \ell -\nu \Vert +\mu d(\nu ,\mathcal {S}\nu ). \end{aligned}$$

Since $\nu \in \mathcal {S}\ell$, $2\mu \le \gamma$, and $1-\mu >0$, we can write

$$\begin{aligned} d(\nu ,\mathcal {S}\nu )\le (1-\gamma +\mu )\Vert \ell -\nu \Vert +\mu d(\nu ,\mathcal {S}\nu ). \end{aligned}$$

which implies

$$\begin{aligned} d(\nu ,\mathcal {S}\nu )\le & {} \bigg (\frac{ 1-\gamma +\mu }{1-\mu }\bigg )\big \Vert \ell -\nu \big \Vert \\\le & {} \Vert \ell -\nu \Vert , \end{aligned}$$

for any $\nu \in \mathcal {S}\ell$.

(ii) In contrast, assume that for any $\ell ,\eta \in \Re$, $\nu \in \mathcal {S}\ell$, and $h\in [0,1]$, we have

$$\begin{aligned} \frac{h}{2}d(\ell ,\mathcal {S}\ell )> & {} \Vert \ell -\eta \Vert ,\nonumber \\ \frac{h}{2} d(\nu ,\mathcal {S}\nu )> & {} \Vert \nu -\eta \Vert . \end{aligned}$$

(3)

It follows from (3) and (i) that

$$\begin{aligned} \Vert \ell -\nu \Vert\le & {} \Vert \ell -\eta \Vert +\Vert \eta -\nu \Vert \\< & {} \frac{h}{2}d(\ell ,\mathcal {S}\ell )+\frac{h}{2}d(\nu ,\mathcal {S}\nu )\\\le & {} \frac{h}{2}\Vert \ell -\nu \Vert +\frac{h}{2}\Vert \ell -\nu \Vert , \end{aligned}$$

which is a contradiction. Hence, the result follows. $\square$

Lemma 2.4

Let $\Re$ be a nonempty subset of a Banach space U and $\mathcal {S} : \Re \rightarrow \mathcal {P}_{cb}(\Re )$ be a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$ .Then,

$$\begin{aligned} d(\ell ,\mathcal {S}\eta ) \le \left( \frac{3+\mu }{1-\mu }\right) d(\ell , \mathcal {S}\ell )+\Vert \ell -\eta \Vert , \end{aligned}$$

(4)

for $\ell ,\eta \in \Re$.

Proof

By Lemma 2.3, we have the following two cases: Case 1. If $\gamma d(\ell ,\mathcal {S}\ell )\le \Vert \ell -\eta \Vert$ (for $\gamma = \frac{h}{2},h\in [0,1])$, we obtain

$$\begin{aligned} d(\ell , \mathcal {S}\eta )\le & {} d(\ell ,\mathcal {S}\ell )+\mathcal {H}(\mathcal {S}\ell ,\mathcal {S}\eta )\\\le & {} d(\ell , \mathcal {S}\ell ) +(1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))\\\le & {} d(\ell ,\mathcal {S}\ell )+(1-\gamma )\Vert \ell -\eta \Vert + \mu (d(\ell , \mathcal {S}\eta ) + \Vert \ell -\eta \Vert + d(\ell ,\mathcal {S}\ell ))\\= & {} (1+\mu ) d(\ell ,\mathcal {S}\ell )+(1-\gamma +\mu )\Vert \ell -\eta \Vert + \mu d(\ell ,\mathcal {S}\eta ). \end{aligned}$$

From the previous inequalities, we obtain

$$\begin{aligned} (1-\mu )d(\ell ,\mathcal {S}\eta )\le (1+\mu )d(\ell ,\mathcal {S}\ell )+(1+\mu -\gamma )\Vert \ell -\eta \Vert , \end{aligned}$$

since $2\mu \le \gamma$, we have

$$\begin{aligned} d(\ell ,\mathcal {S}\eta )\le \left( \frac{1+\mu }{1-\mu }\right) d(\ell ,\mathcal {S}\ell )+\Vert \ell -\eta \Vert . \end{aligned}$$

The required result is proven.

Case 2. Let $\gamma d(\nu ,\mathcal {S}\nu )\le \Vert \ell - \nu \Vert$ (for $\gamma =\frac{h}{2}, h\in [0,1])$. Then, we have

$$\begin{aligned} d(\ell ,\mathcal {S}\eta )\le & {} \Vert \ell -\nu \Vert + d(\nu ,\mathcal {S}\nu )+\mathcal {H}(\mathcal {S}\nu ,\mathcal {S}\eta )\\\le & {} 2\Vert \ell -\nu \Vert +(1-\gamma )\Vert \nu -\eta \Vert +\mu (d(\nu ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\nu ))\\\le & {} 2\Vert \ell -\nu \Vert +(1-\gamma )\Vert \nu -\eta \Vert +\mu \Vert \nu -\ell \Vert +\mu (d(\ell ,\mathcal {S}\eta )+ \Vert \eta -\nu \Vert +d(\nu ,\mathcal {S}\nu )). \end{aligned}$$

Therefore, using Lemma 2.3, we have

$$\begin{aligned} (1-\mu )d(\ell ,\mathcal {S}\eta )\le & {} (2+2\mu )\Vert \ell -\nu \Vert +(1+\mu -\gamma )\Vert \nu -\eta \Vert \\\le & {} (2+2\mu )\Vert \ell -\nu \Vert +(1+\mu -\gamma )(\Vert \nu -\ell \Vert +\Vert \ell -\eta \Vert )\\= & {} (3+3\mu -\gamma )\Vert \ell -\nu \Vert +(1+\mu -\gamma )\Vert \ell -\eta \Vert , \end{aligned}$$

which implies

$$\begin{aligned} d(\ell ,\mathcal {S}\eta )\le \left( \frac{3+3\mu -\gamma }{1-\mu }\right) \Vert \ell -\nu \Vert +\frac{1+\mu -\gamma }{1-\mu }\Vert \ell -\eta \Vert , \end{aligned}$$

since $2\mu \le \gamma$, $\nu \in \mathcal {S}\ell$ and $\gamma d(\ell ,\mathcal {S}\ell )\le \Vert \ell -\eta \Vert$, we obtain

$$\begin{aligned} d(\ell ,\mathcal {S}\eta )\le \left( \frac{3+\mu }{1-\mu }\right) d(\ell ,\mathcal {S}\ell )+\Vert \ell -\eta \Vert . \end{aligned}$$

$\square$

Hence, in both cases the result is proven.

Lemma 2.5

Let U be a Banach space and $\Re$ be a nonempty closed convex and bounded subset of U. Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{cb}(\Re )$ be a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$. Let $\{\ell _{k}\}$ be a bounded approximate fixed point sequence for $\mathcal {S}$ in $\Re$ and $k \in \mathbb {N}$. Then, $\mathcal {A}(\Re ,\{\ell _{k}\})$ is $\mathcal {S}$-invariant.

Proof

Let $\ell \in \mathcal {A}(\Re ,\{\ell _{k}\})$. Since the mapping $\mathcal {S}$ satisfies (4), we have

$$\begin{aligned} d(\ell _{k},\mathcal {S}\ell )\le \left( \frac{3+\mu }{1-\mu }\right) d(\ell _{k},\mathcal {S}\ell _{k})+\Vert \ell _{k}-\ell \Vert . \end{aligned}$$

Then,

$$\begin{aligned} r(\mathcal {S}\ell ,\{\ell _{k}\})= & {} \limsup _{k \rightarrow \infty } d(\ell _{k},\mathcal {S}\ell )\\\le & {} \left( \frac{3+\mu }{1-\mu }\right) \limsup _{k\rightarrow \infty }d(\ell _{k},\mathcal {S}\ell _{k})+\limsup _{k \rightarrow \infty }\Vert \ell _{k}-\ell \Vert \\= & {} \limsup _{k \rightarrow \infty }\Vert \ell _{k}-\ell \Vert \\= & {} r(\ell ,\{\ell _{k}\}). \end{aligned}$$

We have, $\mathcal {S}\ell \in \mathcal {A}(\Re ,\{\ell _{k}\})$ by the definition of the asymptotic center. Hence, $\mathcal {A}(\Re ,\{\ell _{k}\})$ is $\mathcal {S}$-invariant. $\square$

Lemma 2.6

Let U be a Banach space and $\Re$ be a nonempty closed convex and bounded subset of U. Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{cb}(\Re )$ be a multi-valued mapping Satisfying Condition $B_{\gamma ,\mu }$. Suppose $\{\ell _{k}\}$ is an approximate fixed point sequence for $\mathcal {S}$. Then

$$\begin{aligned} \limsup _{k\rightarrow \infty }d(\ell _{k},\mathcal {S}\ell )\le \limsup _{k \rightarrow \infty }\Vert \ell _{k}-\ell \Vert , \end{aligned}$$

for each $\ell \in \Re$ and $k \in \mathbb {N}$.

Proof

Since $\mathcal {S}$ satisfies (4), for any $\ell \in \Re$, we obtain

$$\begin{aligned} d(\ell _{k},\mathcal {S}\ell )\le \left( \frac{3+\mu }{1-\mu }\right) d(\ell _{k},\mathcal {S}\ell _{k})+\Vert \ell _{k}-\ell \Vert . \end{aligned}$$

Since $\{\ell _{k}\}$ is an approximately fixed point sequence in $\Re$, we obtain

$$\begin{aligned} \limsup _{k \rightarrow \infty } d(\ell _{k},\mathcal {S}\ell )\le \left( \frac{3+\mu }{1-\mu }\right) \limsup _{k \rightarrow \infty }[d(\ell _{k},\mathcal {S}\ell _{k}) +\Vert \ell _{k}-\ell \Vert ], \end{aligned}$$

which implies

$$\begin{aligned} \limsup _{k \rightarrow \infty } d(\ell _{k},\mathcal {S}\ell )\le \limsup _{k \rightarrow \infty }\Vert \ell _{k}-\ell \Vert , \end{aligned}$$

$\square$

for $\ell \in \Re$.

We conclude next theorem with the property of demiclosedness.

Theorem 2.7

(Demiclosed principle) Let $\Re$ be a nonempty closed convex subset of a uniformly convex Banach space U with Opial’s property. $\mathcal {S}:\Re \rightarrow \mathcal {P}_{cb}(\Re )$ a multi-valued mapping satisfying the Condition $B_{\gamma ,\mu }$ and $\{\ell _{k}\}$ be a sequence in U. If $\{\ell _{k}\}$ converges weakly to some point $q \in \Re$ and $\limsup _{k\rightarrow \infty }d(\ell _{k}, \mathcal {S}\ell _{k})= 0$, then $q \in \mathcal {S}q$, i.e., $(I-\mathcal {S})$ is demiclosed at zero.

Proof

Since $q \in \Re$ and $\mathcal {S}q$ is closed and bounded, for each $k \in \mathbb {N}$ there exist $\ell _{k} \in \mathcal {S}q$ such that $\Vert q_{k}-\ell _{k}\Vert = d(q_{k}, \mathcal {S}q)$. Then by Lemma 2.4,

$$\begin{aligned} \Vert q_{k}-\ell _{k}\Vert= & {} d(q_{k}, \mathcal {S}q) \le d(q_{k}, \mathcal {S}q_{k}) + H(\mathcal {S}q_{k},\mathcal {S}q)\\\le & {} d(q_{k},\mathcal {S}q_{k})+ \Big (\frac{3+\mu }{1-\mu }\Big )d(q_{k}, \mathcal {S}q_{k})+\Vert q_{k}-q\Vert . \end{aligned}$$

Taking limsup on both sides and using $\limsup _{k\rightarrow \infty } d(q_{k}, \mathcal {S}q_{k}) = 0$, we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }\Vert q_{k}-\ell _{k}\Vert\le & {} \limsup _{k\rightarrow \infty }\Vert q_{k}-q\Vert ,\;\;\;\;\text { for all}\;\; k \in \mathbb {N}. \end{aligned}$$

(5)

As the sequence $\{q_{k}\}$ converges weakly to q and $\Re$ possesses Opail’s property, for any $K \in \mathbb {N}$ if $\ell _{k}\ne q$ then it follows that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\Vert q_{k}-q\Vert <\limsup _{k\rightarrow \infty }\Vert q_{k}-\ell _{k}\Vert , \end{aligned}$$

which contradicts (5), therefore we can infer $\ell _{k}= q$ for all $k \in \mathbb {N}$. As a consequence of $\ell _{k} \in \mathcal {S}q$ we have $q\in \mathcal {S}q$, i.e., $(I-\mathcal {S})$ is demiclosed at zero. $\square$

Now, we prove the existence of a fixed point of a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$.

Theorem 2.8

Let U be a Banach space and $\Re$ be a nonempty closed convex and bounded subset of U. Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{cb}(\Re )$ be a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$. Suppose $\{\ell _{k}\}$ is an approximate fixed point sequence $\{\ell _{k}\} \in \Re$ for $\mathcal {S}$, the asymptotic center $\mathcal {A}(\Re , \{\ell _{k}\})$ is nonempty and compact. Then, $\mathcal {S}$ has a fixed point.

Proof

Let $\{\ell _{k}\}$ be an approximate fixed point sequence in the asymptotic center $\mathcal {A}(\Re , \{\ell _{k}\})$. Since this center is compact, there exists a subsequence $\{\ell _{k_{j}}\}$ of $\{\ell _{k}\}$ such that

$$\begin{aligned} \{\ell _{k_{j}}\} \rightarrow q \in \mathcal {A}(\Re , \{\ell _{k}\}). \end{aligned}$$

As Lemma 2.5 the asymptotic center is $\mathcal {S}$-invariant, $\mathcal {S}q \in \mathcal {A}(\Re , \{\ell _{k}\})$. Additionally, by Lemma 2.6, we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }d(\ell _{k_{j}},\mathcal {S}q) \le \limsup _{k\rightarrow \infty }\Vert \ell _{k_{j}}-q\Vert , \end{aligned}$$

$\square$

which implies that $q\in \mathcal {S}q$.

3 Convergence results

We now define an F-iterative process as follow. Let $\Re$ be a nonempty closed and convex subset of a Banach space U and $\mathcal {S}:\Re \rightarrow \mathcal {P}(\Re )$ be a multi-valued mapping. Let $\{\ell _k\}$ be a sequence in $\Re$ defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \ell _{k+1}=p'''_{k},\\ \eta _{k}=p''_{k},\\ \hbar _{k}=p'_{k},\\ \xi _{k}=(1-\delta _{k})\ell _{k}+\delta _{k}p_{k}, \end{array}\right. } \end{aligned}$$

(6)

where $p_{k}\in \mathcal {P}_{\mathcal {S}}(\ell _{k}), p'_{k}\in \mathcal {P}_{\mathcal {S}}(\xi _{k}), p''_{k}\in \mathcal {P}_{\mathcal {S}}(\hbar _{k}), p'''_{k}\in \mathcal {P}_{\mathcal {S}}(\eta _{k})$, and $\delta _{k}\in (0,1)$ We start with the following lemmas:

Lemma 3.1

Let U be a uniformly convex Banach space and $\Re$ a nonempty closed convex subset of U. Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{px}(\Re )$ be a multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$ such that $Fix(\mathcal {S})\ne \phi$. Furthermore, assume that $\mathcal {P}_{\mathcal {S}}$ is a mapping satisfying Condition $B_{\gamma ,\mu }$. Let $\{\ell _{k}\}$ be the sequence defined by (6). Then, $\lim _{k \rightarrow \infty } \Vert \ell _{k}-q\Vert$ exists for all $q \in Fix(\mathcal {S})$ and $\lim _{k \rightarrow \infty } d(\ell _{k}, \mathcal {P}_{\mathcal {S}}(\ell _{k}))=0$.

Proof

Suppose we have the sequence $\{\ell _{k}\}$ generated by (6), and that $q\in Fix(\mathcal {S})$. Using Lemma 2.2 and (6), we have

$$\begin{aligned} \Vert \xi _{k}-q\Vert\le & {} (1-\delta _{k})\Vert \ell _{k}-q\Vert +\delta _{k}\Vert p_{k}-q\Vert \nonumber \\\le & {} (1-\delta _{n})\Vert \ell _{k}-q\Vert +\delta _{k}\mathcal {H}(\mathcal {P}_{\mathcal {S}}\ell _{k},\mathcal {P}_{\mathcal {S}}q) \nonumber \\\le & {} (1-\delta _{k})\Vert \ell _{k}-q\Vert +\delta _{k}\Vert \ell _{k}-q\Vert \nonumber \\\le & {} \Vert \ell _{k}-q\Vert , \end{aligned}$$

(7)

for all $k \in \mathbb {N}$, and

$$\begin{aligned} \Vert \hbar _{k}-q\Vert= & {} \Vert p'_{k}-q\Vert \le \mathcal {H}(\mathcal {P}_\mathcal {S}(\xi _{k}), \mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \Vert \xi _{k}-q\Vert . \end{aligned}$$

Furthermore

$$\begin{aligned} \Vert \eta _{k}-q\Vert= & {} \Vert p''_{k}-q\Vert \le \mathcal {H}(\mathcal {P}_\mathcal {S}(\hbar _{k}), \mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \Vert \hbar _{k}-q\Vert . \end{aligned}$$

These imply that

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert= & {} \Vert p'''_{k}-q\Vert \le \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\eta _{k}),\mathcal {P}_{\mathcal {S}}(q))\Vert \nonumber \\= & {} \Vert \eta _{k}-q\Vert \le \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\hbar _{k}),\mathcal {P}_{\mathcal {S}}(q))\Vert \nonumber \\= & {} \Vert \hbar _{k}-q\Vert \le \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\xi _{k}),\mathcal {P}_{\mathcal {S}}(q))\Vert \nonumber \\\le & {} \Vert \xi _{k}-q\Vert \nonumber \\\le & {} \Vert \ell _{k}-q\Vert . \end{aligned}$$

(8)

Thus, $\Vert \ell _{k}-q\Vert$ is non-increasing and bounded, implying that $\lim _{k\rightarrow \infty }\Vert \ell _{k}-q\Vert$ exists for all $q\in Fix(\mathcal {S})$. Then, we prove that

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert \ell _{k}-p_{k}\Vert =0. \end{aligned}$$

Assume that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \ell _{k}-q\Vert =c\;\;\;\text {where}\;\;\; q \in Fix(\mathcal {S}). \end{aligned}$$

If $c=0$, then the proof is trivial, we consider $c >0$, from  (7), we have

$$\begin{aligned}&\Vert \xi _{k}-q\Vert \le \Vert \ell _{k}-q\Vert \nonumber \\&\Rightarrow \limsup _{k \rightarrow \infty }\Vert \xi _{k}-q\Vert \le \limsup _{k \rightarrow \infty }\Vert \ell _{k}-q\Vert \le c. \end{aligned}$$

(9)

Since $q \in \mathcal {P}_{\mathcal {S}}(q)$ and $\Vert p_{k}-q\Vert =d(p_{k},\mathcal {P}_{\mathcal {S}}(q)),$ by Lemma 2.2, we have

$$\begin{aligned} \Vert p_{k}-q\Vert \le d(p_{k},\mathcal {P}_{\mathcal {S}}(q))\le \mathcal {H}(\mathcal {P}_\mathcal {S}(\ell _{k}),\mathcal {P}_\mathcal {S}(q))\le \Vert \ell _{k}-q\Vert . \end{aligned}$$

Taking limsup on both sides of the above inequality, we obtain

$$\begin{aligned} \limsup _{k \rightarrow \infty }\Vert p_{k}-q\Vert\le & {} \limsup _{k \rightarrow \infty }\Vert \ell _{k}-q\Vert = c. \end{aligned}$$

(10)

Again from  (8), we have

$$\begin{aligned}&\Vert \ell _{k+1}-q\Vert \le \Vert \xi _{k}-q\Vert \nonumber \\&\Rightarrow c = \liminf _{k \rightarrow \infty }\Vert \ell _{k+1}-q\Vert \le \liminf _{k\rightarrow \infty }\Vert \xi _{k}-q\Vert . \end{aligned}$$

(11)

Using (11) and (9)

$$\begin{aligned} c \le \liminf _{k \rightarrow \infty }\Vert \xi _{k}-q\Vert \le \limsup _{k\rightarrow \infty }\Vert \ell _{k}-q\Vert \le c. \end{aligned}$$

Thus

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert \xi _{k}-q\Vert =c \end{aligned}$$

which implies that

$$\begin{aligned} c=\lim _{k\rightarrow \infty }\Vert \xi _{k}-q\Vert =\lim _{k\rightarrow \infty }\Vert (1-\delta _{k})(\ell _{k}-q)+\delta _{k}(p_{k}-q)\Vert . \end{aligned}$$

By Lemma 1.7, we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \ell _{k}-p_{k}\Vert =0, \end{aligned}$$

which yields

$$\begin{aligned} \lim _{k \rightarrow \infty }d(\ell _{k}, \mathcal {P}_{\mathcal {S}}(\ell _{k})) = 0. \end{aligned}$$

$\square$

We now prove a strong convergence result for $\{\ell _{k}\}$ generated by (6) for multi-valued mapping satisfying Condition $B_{\gamma ,\mu }$

Theorem 3.2

Let U be a uniformly convex Banach space and $\Re$ be a nonempty compact convex subset of U. Let $\mathcal {S}: \Re \rightarrow \mathcal {P}_{px}(\Re )$ be such that $\mathcal {P}_{\mathcal {S}}$ is satisfying Condition $B_{\gamma ,\mu }$ and $Fix(\mathcal {S})\ne \phi$. Then $\{\ell _{k}\}$ generated by (6) converges strongly to a fixed point $\mathcal {S}$.

Proof

By Lemma 3.1, $\lim _{k\rightarrow \infty } d(\ell _{k},\mathcal {P}_{\mathcal {S}}(\ell _{k})) = 0$. Due to the compactness of $\Re$ we can find a subsequence $\{\ell _{k_{i}}\}$ of $\{\ell _{k}\}$ such that $\{\ell _{k_{i}}\}$ converges to some $q \in \Re$. In the view of Lemma 2.4, we have

$$\begin{aligned} d(q,\mathcal {P}_{\mathcal {S}}(q))\le & {} \Vert q-\ell _{k_{i}}\Vert + d(\ell _{k_{i}},\mathcal {P}_{\mathcal {S}}(q))\\\le & {} \Vert q-\ell _{k_{i}}\Vert +\left( \frac{3+\mu }{1-\mu }\right) d(\ell _{k_{i}}, \mathcal {P}_{\mathcal {S}}(\ell _{k_{i}}))+\Vert \ell _{k_{i}}-q\Vert \\= & {} 2\Vert \ell _{k_{i}}-q\Vert +\left( \frac{3+\mu }{1-\mu }\right) d(\ell _{k_{i}}, \mathcal {P}_{\mathcal {S}}(\ell _{k_{i}}))\rightarrow 0. \end{aligned}$$

Hence, $q \in \mathcal {P}_{\mathcal {S}}(q)$. By Lemma 1.2, $q \in Fix(\mathcal {P}_{\mathcal {S}})=Fix(\mathcal {S})$. By Lemma 3.1, $\lim _{k \rightarrow \infty }\Vert \ell _{k}-q\Vert$ exists. Hence, q is the strong limit of $\{\ell _{q}\}$. $\square$

The proof of the following result is elementary;

Theorem 3.3

Let U be a uniformly convex Banach space and $\Re$ be a nonempty closed convex subset of U. Let $\mathcal {S}: \Re \rightarrow \mathcal {P}_{px}(\Re )$ be such that $\mathcal {P}_{\mathcal {S}}$ satisfies Condition $B_{\gamma ,\mu }$. If $Fix(\mathcal {S})\ne \phi$. Let $\{\ell _{k}\}$ be the sequence defined by (6), and let $\liminf _{k \rightarrow \infty } d(\ell _{k}, Fix(\mathcal {S}))= 0$. Then $\{\ell _{k}\}$ converges strongly to a fixed point of $\mathcal {S}$.

We use condition (I) to prove another strong convergence theorem.

Theorem 3.4

Let U be a uniformly convex Banach space and $\Re$ be a nonempty closed convex subset of U. Let $\mathcal {S}: \Re \rightarrow \mathcal {P}_{px}(\Re )$ be a multi-valued mapping with $Fix(\mathcal {S})\ne \phi$. If $\mathcal {P}_{\mathcal {S}}$ satisfies condition $B_{\gamma ,\mu }$. Then $\{\ell _{k}\}$ generated by (6) converges strongly to a fixed point of $\mathcal {S}$ provided that $\mathcal {S}$ satisfies the condition (I).

Proof

By Lemma 3.1, $\lim _{k\rightarrow \infty } \Vert \ell _{k}-q\Vert$ exists for all $q \in Fix(\mathcal {S})$. Set $c= \lim _{k\rightarrow \infty } \Vert \ell _{k}-q\Vert$ for some $c \ge 0$. If $c = 0$ then the result is trivial. Moreover, suppose that $c > 0$. Then,

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert\le & {} \Vert \ell _{k}-q\Vert \\ \liminf _{k \rightarrow \infty }\Vert \ell _{k+1}-q\Vert\le & {} \liminf _{k \rightarrow \infty }\Vert \ell _{k}-q\Vert \\ d(\ell _{k+1},Fix(\mathcal {S}))\le & {} d(\ell _{k},Fix(\mathcal {S})). \end{aligned}$$

Hence $\lim _{k \rightarrow \infty } d(\ell _{k}, Fix(\mathcal {S}))$ exists. We show that $\lim _{k\rightarrow \infty }d(\ell _{k}, Fix(\mathcal {S})) = 0$. From Lemma 3.1, it follows that $\lim _{k \rightarrow \infty } d(\ell _{k},\mathcal {P}_{\mathcal {S}}(\ell _{k})) = 0$. Additionally, from Lemma 2.2, $Fix(\mathcal {S}) = Fix(\mathcal {P}_{\mathcal {S}})$. Using these facts and condition (I), we have

$$\begin{aligned} \lim _{k\rightarrow \infty }f(d(\ell _{k}, Fix(\mathcal {S})) = 0. \end{aligned}$$

Since f is nondecreasing and $f(0)=0$. We obtain

$$\begin{aligned} \lim _{k \rightarrow \infty }d(\ell _{k}, Fix(\mathcal {S})) = 0. \end{aligned}$$

$\square$

By Theorem 3.3, we obtain the required conclusions.

Finally, we prove a weak convergence of the sequence $\{\ell _{k}\}$.

Theorem 3.5

Let U be a uniformly convex Banach space satisfying Opial’s condition and $\Re$ be a nonempty closed convex subset of U. Assume $\mathcal {S}: \Re \rightarrow \mathcal {P}_{px}(\Re )$ is a multi-valued mapping with $Fix(\mathcal {S})\ne \phi$. If $\mathcal {P}_{\mathcal {S}}$ satisfies Condition $B_{\gamma ,\mu }$ and $I-\mathcal {P}_{\mathcal {S}}$ is demiclosed with respect to zero. Suppose $\{\ell _{k}\}$ is a sequence generated by (6). Then $\{\ell _{k}\}$ converges weakly to a fixed point of $\mathcal {S}$.

Proof

By the proof of Lemma 3.1$\{\ell _{k}\}$ is bounded. Since U is uniformly convex, so U is reflexive by Milman-Pettis’s Theorem. By Eberlin’s Theorem, every bounded sequence in U has a weakly convergent subsequence. Thus, we can find a weakly convergent subsequence $\{\ell _{k_{i}}\}$ of $\{\ell _{k}\}$ with weak limit say $q_{1}$ in $\Re$. By the demicloseness of $I-\mathcal {P}_{\mathcal {S}}$ at 0, $q_{1} \in Fix(\mathcal {P}_{\mathcal {S}})=Fix(\mathcal {S})$. We prove that $q_{1}$ is the unique weak limit of $\{\ell _{k}\}$. Let us find another weakly convergent subsequence $\{\ell _{k}\}$ of $\{\ell _{k}\}$ with a weak limit, say $q_{2} \in \Re$ and $q_{2}\ne q_{1}$. Again, $q_{2} \in Fix(\mathcal {P}_{\mathcal {S}}) = Fix(\mathcal {S})$. By Opial property and Lemma 3.1, we have

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert \ell _{k}-q_{1}\Vert= & {} \lim _{i \rightarrow \infty }\Vert \ell _{k_{i}}-q_{1}\Vert \\< & {} \lim _{i \rightarrow \infty }\Vert \ell _{k_{i}}-q_{2}\Vert \\= & {} \lim _{k \rightarrow \infty }\Vert \ell _{k}-q_{2}\Vert \\< & {} \lim _{j \rightarrow \infty }\Vert \ell _{k_{j}}-q_{2}\Vert \\< & {} \lim _{j \rightarrow \infty }\Vert \ell _{k_{j}}-q_{1}\Vert \\= & {} \lim _{k \rightarrow \infty }\Vert \ell _{k}-q_{1}\Vert . \end{aligned}$$

$\square$

This is a contradiction. Hence, $\{\ell _{k}\}$ converges weakly to $q_{1}$.

4 Stability analysis

This section concerns with the convergence and stability of the iteration process (6) for a multi-valued contraction mapping.

Theorem 4.1

Let $\Re$ be a nonempty, closed, and convex subset of a Banach space U. Let $\mathcal {S}:\Re \rightarrow \mathcal {P}_{px}(\Re )$ be a multi-valued mapping and $\mathcal {P}_{\mathcal {S}}$ a multi-valued contraction with $\vartheta \in [0, 1)$. If $\{\ell _{k}\}$ is a sequence defined in (6) with $\delta _{k}\in (0, 1)$ and $\sum _{k=0}^{\infty }\delta _{k}=\infty$, then $\{\ell _{k}\}$ converges to a fixed point of $\mathcal {S}$.

Proof

By Nadler contraction principle $\mathcal {P}_{\mathcal {S}}$ has a fixed point. Now, we will show that $\{\ell _{k}\}$ converges to a fixed point q. From Lemma 1.2, we have $q \in \mathrm{{Fix}}(\mathcal {P}_{\mathcal {S}})$. We use (6), to obtain

$$\begin{aligned} \Vert \xi _{k}-q\Vert\le & {} (1-\delta _{k})\Vert \ell _{k}-q\Vert + \delta _{k}\Vert p_{k}-q\Vert \nonumber \\\le & {} (1-\delta _{k})\Vert \ell _{k}-q\Vert +\delta _{k}d(p_{k},\mathcal {P}_\mathcal {S}(q))\nonumber \\\le & {} (1-\delta _{k})\Vert \ell _{k}-q\Vert +\delta _{k}\mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell _{k}),\mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} (1-\delta _{k})\Vert \ell _{k}-q\Vert +\delta _{k}\vartheta \Vert \ell _{k}-q\Vert \nonumber \\\le & {} (1-\delta _{k}(1-\vartheta ))\Vert \ell _{k}-q\Vert . \end{aligned}$$

(12)

Furthermore,

$$\begin{aligned} \Vert \hbar _{k}-q\Vert\le & {} \Vert p'-q\Vert \le d(p',\mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\xi _{k}), \mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \vartheta \Vert \xi _{k}-z\Vert . \end{aligned}$$

(13)

Similarly

$$\begin{aligned} \Vert \eta _{n}-q\Vert\le & {} \Vert p''-q\Vert \le d(p'',\mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\hbar _{k}), \mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \vartheta \Vert \hbar _{k}-q\Vert \nonumber \\\le & {} \vartheta ^{2}\Vert \xi _{k}-q\Vert . \end{aligned}$$

(14)

From (12),  (13),  (14), and using the fact that $(1-\delta _{k}(1-\vartheta ))<1$, for $\vartheta \in (0, 1)$ and $\{\alpha _{n}\} \in (0, 1)$, we obtain that

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert= & {} \Vert p'''-q\Vert \le \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\eta _{k}),\mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \vartheta \Vert \eta _{k}-q\Vert \le \vartheta \mathcal {H}(\mathcal {P}_\mathcal {S}(\hbar _{k}),\mathcal {P}_{\mathcal {S}}(q))\nonumber \\\le & {} \vartheta ^{2}\Vert \hbar _{k}-q\Vert \nonumber \\\le & {} \vartheta ^{3}\Vert \xi _{k}-q\Vert \nonumber \\\le & {} \vartheta ^{3}(1-\delta _{k}(1-\vartheta ))\Vert \ell _{k}-q\Vert . \end{aligned}$$

(15)

From (15), we have

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert\le & {} \vartheta ^{3}(1-\delta _{k}(1-\vartheta ))\Vert \ell _{k}-q\Vert \nonumber \\\le & {} \vartheta ^{3}(1-\delta _{k-1}(1-\vartheta ))\Vert \ell _{k-1}-q\Vert \nonumber \\\le & {} .\nonumber \\&.\nonumber \\&.\nonumber \\&.\nonumber \\\le & {} \vartheta ^{3}(1-\delta _{0}(1-\vartheta ))\Vert \ell _{0}-q\Vert . \end{aligned}$$

(16)

By (16), we obtain

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert \le \Vert \ell _{0}-q\Vert \left( \vartheta ^{3}\right) ^{k+1}\prod _{i=0}^{k}(1-\delta _{i}(1-\vartheta )). \end{aligned}$$

Since $\delta _{k}$ and $\vartheta \in (0,1)$ , we have $1-\delta _{i}(1-\vartheta )<1$, for all $k \in \mathbb {N}$. We know that $1-\ell \le e^{-\ell }$ for $0 \le u \le 1$. It follows that,

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert \le \Vert \ell _{0}-q\Vert \left( \vartheta ^{3}\right) ^{k+1}e^{-(1-\vartheta )\sum _{i=0}^{k}\delta _{i}}\rightarrow \infty \end{aligned}$$

(17)

If we take the limit in both sides of (17), we obtain $\lim \Vert \ell _{k}-q\Vert =0$, which implies that $\{\ell _{k}\}$ converges to q. Since $q \in Fix(\mathcal {P}_{\mathcal {S}} )$, from Lemma 1.2, we have $q \in Fix( \mathcal {S})$, and hence $\{\ell _{k}\}$ converges strongly to $q \in Fix( \mathcal {S})$. $\square$

Next, we give the definition of $\mathcal {S}$-stable iteration process.

Definition 4.2

[7] Let $\{x_k\}$ be any arbitrary sequence in U. Then, an iteration procedure $x_{k+1} = f(\mathcal {S}, x_{k})$, converging to fixed point q, is said to be $\mathcal {S}$-stable or stable with respect to $\mathcal {S}$, if for $\varepsilon _{k} =\Vert x_{k+1}- f(\mathcal {S}, x_{k})\Vert$, for all $k\in \mathbb {N}$, we have

$$\begin{aligned} \lim _{k \rightarrow \infty }\varepsilon _{k}=0 \Leftrightarrow \lim _{k \rightarrow \infty }x_{k}=q. \end{aligned}$$

Lemma 4.3

[18] Let $\{t_{k}\}$ and $\{\varepsilon _{k}\}$ be two nonnegative real sequences satisfying the following inequality:

$$\begin{aligned} t_{k+1} \le (1-\varpi _{k})t_{k} + \varepsilon _{k}, \end{aligned}$$

where $\varpi _{k} \in (0, 1)$ for all $k \in \mathbb {N}$, $\sum _{k=0}^{\infty }\varpi _{k}=\infty$ and $\lim _{k\rightarrow \infty }\frac{\varepsilon _{k}}{\varpi _{k}}=0$; then, $\lim _{k\rightarrow \infty }t_{k}=0.$

Theorem 4.4

[18] Let $\Re$ be a nonempty, closed and convex subset of a Banach space U, let $\mathcal {S}: \Re \rightarrow \mathcal {P}_{px}(\Re )$ and $\mathcal {P}_{\mathcal {S}}$ be a multi-valued contractions. If $\{\ell _{k}\}$ is a sequence given by (6) with $\delta _{k} \in (0,1)$ and $\sum _{k=0}^{\infty }\delta _{k}=\infty$; then, the iteration process (6) is $\mathcal {S}$-stable.

Proof

Let $\{\ell _{k}\}\subset \Re$ be any arbitrary sequence in U and suppose that the sequence generated by (6) is $\ell _{k+1} = f(\mathcal {S},\ell _{k})$ converging to a unique fixed point q and that $\varepsilon _{k}=\Vert \ell _{k+1}-f(\mathcal {S},\ell _{k})\Vert$. To establish that $\mathcal {S}$ is stable, we need to prove that $\lim _{k \rightarrow \infty }\varepsilon _{k}=0 \Longleftrightarrow \lim _{k\rightarrow \infty } \ell _{k}= q.$

Suppose that $\lim _{k \rightarrow \infty }\varepsilon _{k}=0$. Using triangular inequality and  (15), we have that

$$\begin{aligned} \Vert \ell _{k+1}-q\Vert\le & {} \Vert \ell _{k+1}-f(\mathcal {S},\ell _{k})\Vert + \Vert f(\mathcal {S},\ell _{k})-q\Vert \\\le & {} \varepsilon _{k} + \vartheta ^{3}(1-\delta _{k}(1-\vartheta ))\Vert \ell _{k}-q\Vert . \end{aligned}$$

If $d_{k} = \Vert \ell _{k}-q\Vert$, and $\varpi _{k}=\delta _{k}(1-\vartheta )$, then we have

$$\begin{aligned} d_{k+1} \le (1-\varpi _{k})d_{k}+ \varepsilon _{k}. \end{aligned}$$

As $\sum _{k=0}^{\infty } \varpi _{k}=\infty$ and $\lim _{k\rightarrow \infty } \varepsilon _{k} = 0$, $\lim _{k\rightarrow \infty }\frac{\varepsilon _{k}}{\varpi _{k}}=0$, by Lemma 4.3 we have that $\lim _{k\rightarrow \infty } \ell _{k}=q$.

Consequently, suppose that $\lim _{k \rightarrow \infty } \ell _{k}=q$. We have that

$$\begin{aligned} \varepsilon _{k}= & {} \Vert \ell _{k+1}- f(\mathcal {S},\ell _{k})\Vert \\\le & {} \Vert \ell _{k+1}-q\Vert + \Vert f(\mathcal {S},\ell _{k})-q\Vert \\\le & {} \Vert \ell _{k+1}-q\Vert +\vartheta ^{3}(1-\delta _{k}(1-\vartheta ))\Vert \ell _{k}-q\Vert \rightarrow 0, \;\;\; as\;\; k\rightarrow \infty . \end{aligned}$$

Using our hypothesis that $\lim _{k\rightarrow \infty } \ell _{k}=q$, we then have that $\lim _{k\rightarrow \infty } \varepsilon _{k}=0$. Hence, the iteration process (6) is stable with respect to $\mathcal {S}$. $\square$

5 Example

In this section we provide an example of multi-valued mapping for which best approximate operator $\mathcal {P}_{\mathcal {S}}$ is a generalized nonexpansive mapping satisfying Condition $B_{\gamma ,\mu }$.

Example 5.1

Let $\Re =[0,1]\subset \mathbb {R}$ be endowed with usual norm. Define $\mathcal {S}: \Re \rightarrow \mathcal {P}(\Re )$ by

$$\begin{aligned} \mathcal {S}\ell = \left\{ \begin{array}{ll} {0} &{} {\ell \in [0,\frac{1}{2})\setminus \{\frac{1}{4}\}}\\ {[}0,\frac{\ell }{4}] &{} {\ell \in [\frac{1}{2},1]}\\ {[}0,\frac{1}{12}]&{}{\ell =\{\frac{1}{4}\}}. \end{array}\right. \end{aligned}$$

If $\ell \in [0,\frac{1}{2}]\setminus \{\frac{1}{4}\}$, then $\mathcal {P}_{\mathcal {S}}(\ell )=\{0\}$. For $\ell \in [\frac{1}{2},1]$, then we have $\mathcal {P}_{\mathcal {S}}(\ell )=\{\frac{1}{4}\}$. If $\ell =\{\frac{1}{4}\}$, then $\mathcal {P}_{\mathcal {S}}(\ell )= \{\frac{1}{12}\}$. We show that $\mathcal {P}_{\mathcal {S}}$ is mapping satisfies Condition $B_{\gamma ,\mu }$. For this choose $\gamma =1$, and $\mu =\frac{1}{2}$. We shall consider the following three cases:

Case (1) For $\ell ,\eta \in [0,\frac{1}{2}]\setminus \{\frac{1}{4}\}$, we get

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))= & {} \frac{1}{2}\left( \left\| \ell \right\| +\left\| \eta \right\| \right) \\\ge & {} 0=\mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell ),\mathcal {P}_{\mathcal {S}}(\eta )). \end{aligned}$$

Case(2) For $\ell , \eta \in [\frac{1}{2},1]$, we have

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))= & {} \frac{1}{2}\left( \left\| \ell -\frac{\eta }{4}\right\| + \left\| \eta -\frac{\ell }{4}\right\| \right) \\= & {} \frac{1}{2}\left( \left\| \ell -\frac{\eta }{4}\right\| +\left\| \frac{\ell }{4}-\eta \right\| \right) \\\ge & {} \frac{1}{2}\left\| \frac{5\ell }{4}-\frac{5\eta }{4}\right\| \\= & {} \frac{5}{8}\left\| \ell -\eta \right\| > \frac{2}{8}\left\| \ell -\eta \right\| \\= & {} \frac{1}{4}\left\| \ell -\eta \right\| =\mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell ),\mathcal {P}_{\mathcal {S}}(\eta )). \end{aligned}$$

Case(3) For $\ell \in [0,\frac{1}{2}) \setminus \{\frac{1}{4}\}$ and $\eta =\{\frac{1}{4}\}$, we obtain

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))= & {} \frac{1}{2}\left( \Vert \ell -\frac{1}{12}\Vert +\Vert \frac{1}{4}\Vert \right) \\\ge & {} \frac{1}{2}\left\| \ell -\frac{1}{12}\right\| +\frac{1}{8} \\\ge & {} \frac{1}{12}= \mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell ),\mathcal {P}_{\mathcal {S}}(\eta )). \end{aligned}$$

Case(4) For $\ell \in [\frac{1}{2},1]$, $\eta =\{\frac{1}{4}\}$, we get

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))= & {} \frac{1}{2}\left( \Vert \ell -\frac{1}{12}\Vert +\Vert \frac{1}{4}-\frac{\ell }{4}\Vert \right) \\\ge & {} \frac{1}{2}\left\| \ell -\frac{1}{12}+\frac{1}{4}-\frac{\ell }{4}\right\| \\= & {} \frac{1}{2}\left\| \frac{3\ell }{4}+\frac{3}{12}\right\| =\frac{1}{4}\left\| \frac{3\ell }{2}+\frac{1}{2}\right\| \\\ge & {} \frac{1}{4}\left\| \frac{\ell }{2}-\frac{1}{12}\right\| =\mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell ),\mathcal {P}_{\mathcal {S}}(\eta )) \end{aligned}$$

Case(5) For $\ell \in [0,\frac{1}{2})\setminus \{\frac{1}{4}\}$, $\eta \in [\frac{1}{2},1]$, we obtain

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))= & {} \frac{1}{2}\left( \Vert \ell -\frac{\eta }{4}\Vert +\Vert \eta \Vert \right) . \end{aligned}$$

(18)

Then, we have two cases:

$$\begin{aligned} \Vert \ell -\frac{\eta }{4}\Vert ={\left\{ \begin{array}{ll}\ell -\frac{\eta }{4} &{} \text {if} \;\;\ell > \frac{\eta }{4}\\ \frac{\eta }{4}-\ell &{} \text {if}\,\;\;\ell \le \frac{\eta }{4}. \end{array}\right. } \end{aligned}$$

For the first case, $\ell >\frac{\eta }{4}$ and (18), imply

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))= & {} \frac{1}{2}\left( \ell -\frac{\eta }{4}+\eta \right) \\= & {} \frac{1}{2}\left( \ell +\frac{3\eta }{4}\right) \\\ge & {} \frac{3\eta }{8} \ge \frac{1}{4}\eta =\mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell ),\mathcal {P}_{\mathcal {S}}(\eta )). \end{aligned}$$

Then, from the second case, $\ell \le \frac{\eta }{4}$, and using (18) we obtain

$$\begin{aligned} (1-\gamma )\Vert \ell -\eta \Vert +\mu (d(\ell ,\mathcal {S}\eta )+d(\eta ,\mathcal {S}\ell ))\ge & {} \frac{1}{2}\eta \ge \frac{1}{4}\eta =\mathcal {H}(\mathcal {P}_{\mathcal {S}}(\ell ),\mathcal {P}_{\mathcal {S}}(\eta )). \end{aligned}$$

Thus, the mapping $\mathcal {P}_{S}$ satisfies Condition $B_{\gamma ,\mu }$, and $q=0$ is fixed point in $\Re$. Let $\delta _{k}$ be sequence such that $\delta _{k}=\frac{1}{2k}$ for all $k \in \mathbb {N}$, for any given $\ell _{1}\in [0,1]$, we can assume that $\ell _{1}=\frac{3}{4}$. Using the F-iteration defined by (6), we have $\mathcal {S}(\ell _{1})=\mathcal {S}(\frac{3}{4})=[0,\frac{3}{16}]$. Then

$$\begin{aligned} \xi _{1}= & {} (1-\delta _{1})\ell _{1}+\delta _{1}p_{1}\\= & {} \bigg (1-\frac{1}{2}\bigg )\bigg (\frac{3}{4}\bigg )+\bigg (\frac{1}{2}\bigg )\bigg (\frac{3}{16}\bigg )\\= & {} \frac{15}{32} \end{aligned}$$

implies $\hbar _{1}=0$, then we get $\eta _{1}=\ell _{2}=0$. Hence the sequence $\{ \ell _{n}\}$ converges to fixed point which is zero. If we take the initial point $\ell _{1} = \frac{1}{4}$, then $\mathcal {S}(\ell _{1})=\mathcal {S}(\frac{1}{4})=[0,\frac{1}{12}]$, again we have

$$\begin{aligned} \xi _{1}= & {} (1-\delta _{1})\ell _{1}+\delta _{1}p_{1}\\= & {} \bigg (1-\frac{1}{2}\bigg )\bigg (\frac{1}{4}\bigg )+\bigg (\frac{1}{2}\bigg )\bigg (\frac{1}{12}\bigg )\\= & {} \frac{1}{6} \end{aligned}$$

we have $\hbar _{1}=0$ and $\eta _{1}=\ell _{2}=0$. Continuing in this manner, $\ell _{n}=0$, for all $k\ge 2$ and hence the sequence $\{\ell _{n}\}$ generated by (6) converges to a fixed point.