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.
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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
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
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 \(\ell , \eta \in \Re \),
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:
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
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
for all \(\ell , \eta \in \Re \).
(ii) a nonexpansive mapping if
for all \(\ell , \eta \in \Re .\)
(iii) a quasi-nonexpansive mapping if \(Fix(\mathcal {S}) \ne \phi \) and
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:
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
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 \(\ell , \eta \in \Re \),
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
for any \(\eta \in \Re \), so
This yields
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)
\(\frac{h}{2} d(\nu ,\mathcal {S}\nu )\le \Vert \nu -\eta \Vert \).
-
(a)
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
Since \(\nu \in \mathcal {S}\ell \), \(2\mu \le \gamma \), and \(1-\mu >0\), we can write
which implies
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
It follows from (3) and (i) that
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,
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
From the previous inequalities, we obtain
since \(2\mu \le \gamma \), we have
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
Therefore, using Lemma 2.3, we have
which implies
since \(2\mu \le \gamma \), \(\nu \in \mathcal {S}\ell \) and \(\gamma d(\ell ,\mathcal {S}\ell )\le \Vert \ell -\eta \Vert \), we obtain
\(\square \)
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
Then,
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
for each \(\ell \in \Re \) and \(k \in \mathbb {N}\).
Proof
Since \(\mathcal {S}\) satisfies (4), for any \(\ell \in \Re \), we obtain
Since \(\{\ell _{k}\}\) is an approximately fixed point sequence in \(\Re \), we obtain
which implies
\(\square \)
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,
Taking limsup on both sides and using \(\limsup _{k\rightarrow \infty } d(q_{k}, \mathcal {S}q_{k}) = 0\), we obtain
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
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
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
\(\square \)
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
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
for all \(k \in \mathbb {N}\), and
Furthermore
These imply that
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
Assume that
If \(c=0\), then the proof is trivial, we consider \(c >0\), from (7), we have
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
Taking limsup on both sides of the above inequality, we obtain
Again from (8), we have
Thus
which implies that
By Lemma 1.7, we obtain
which yields
\(\square \)
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
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,
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
Since f is nondecreasing and \(f(0)=0\). We obtain
\(\square \)
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
\(\square \)
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
Furthermore,
Similarly
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
From (15), we have
By (16), we obtain
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,
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
Lemma 4.3
[18] Let \(\{t_{k}\}\) and \(\{\varepsilon _{k}\}\) be two nonnegative real sequences satisfying the following inequality:
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
If \(d_{k} = \Vert \ell _{k}-q\Vert \), and \(\varpi _{k}=\delta _{k}(1-\vartheta )\), then we have
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
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
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
Case(2) For \(\ell , \eta \in [\frac{1}{2},1]\), we have
Case(3) For \(\ell \in [0,\frac{1}{2}) \setminus \{\frac{1}{4}\}\) and \(\eta =\{\frac{1}{4}\}\), we obtain
Case(4) For \(\ell \in [\frac{1}{2},1]\), \(\eta =\{\frac{1}{4}\}\), we get
Case(5) For \(\ell \in [0,\frac{1}{2})\setminus \{\frac{1}{4}\}\), \(\eta \in [\frac{1}{2},1]\), we obtain
Then, we have two cases:
For the first case, \(\ell >\frac{\eta }{4}\) and (18), imply
Then, from the second case, \(\ell \le \frac{\eta }{4}\), and using (18) we obtain
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
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
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.
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Hussain, N., Alamri, H. & Alsulami, S. Fixed point approximation for a class of generalized nonexpansive multi-valued mappings in Banach spaces. Arab. J. Math. 12, 363–377 (2023). https://doi.org/10.1007/s40065-022-00403-y
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DOI: https://doi.org/10.1007/s40065-022-00403-y