Fixed Point Results for New Classes of k-Strictly Asymptotically Demicontractive and Hemicontractive Type Multivalued Mappings in Symmetric Spaces

Abstract

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature.

1. Introduction

The concept of fixed points has been extensively studied for single-valued mappings, leading to classical results such as Banach’s Fixed Point Theorem and Brouwer’s Fixed Point Theorem. However, real-world problems often involve scenarios where one state can lead to multiple possible subsequent states, necessitating the use of multivalued mappings. Examples include differential inclusions, control systems, and economic models where uncertainty or multiple choices are inherent. In recent years, various classes of multivalued mappings have been introduced to generalize and extend fixed point results. Notably, demicontractive and hemicontractive mappings have been studied due to their relevance in iterative methods and convergence analysis. However, the introduction of k-strictly asymptotically demicontractive and hemicontractive mappings presents new challenges and opportunities for further generalizations.

In a space $\mathcal{X}$, a distance function d is defined as symmetric if it fulfills the condition $d(u,v)=d(v,u)$ for every pair of points u and v within $\mathcal{X}$. A space that has such a symmetric distance function is known as a symmetric space. It is widely recognized that Hilbert spaces and metric spaces inherently have this symmetry. However, pseudo-metric spaces do not always exhibit this symmetric property. Several authors have proved fixed point results in symmetric spaces, for instance, see [1].

Let £ be a metric space. Fixed point theory is interested in the solution of the equation of the type

$$ᘐ\wp =\wp ,$$

(1)

where ᘐ is considered as a nonlinear mapping on £. Any point $\wp \in \pounds$ for which (1) holds is known as an invariant point of ᘐ, and a set consisting of all such points is represented with $F(ᘐ)$. Fixed point theory is no doubt an indispensable tool in the arsenal of modern analysis. Currently, there is still an intensive research in this direction, as it has several utilities in establishing the existence and uniqueness of solutions of diverse mathematically oriented models such as solutions of optimisation problems, ordinary differential equations, and variational analysis. Research has revealed that these models represent different phenomena arising in several fields such as neutron transport theory, steady state temperature distributions, economic theory, optimal control of systems, chemical equations, and the flow of fluids.

In recent times, fixed point theory for multivalued mappings has received unwavering attention from well-known mathematicians such as Nadler [2], Kakutani [3], Nash [4,5], Gaenakoplos [6], Downing and Kirk [7], etc.

Following the results of Nadler [2], who initiated fixed point for multivalued mappings, subsequent interesting works have been published by various researchers: for example, Markin [8], Hu et al. [9], Bunyawat and Suntain [10], Isogugu [11], etc.

In continuation, Agwu and Igbokwe [12] introduced a new class of type-one asymptotically demicontractive multivalued mappings, which guarantees strong convergence without the condition that the sum of the finite family of the iteration parameters is unity. Moreover, Agwu and Igbokwe [13] have obtained convergence results for finding common solutions for minimization problems and fixed point problems (FPPs) for asymptotically quasi-nonexpansive multivalued mappings (AQNMMs).

It is an established fact that the process involved in approximating fixed points of nonlinear single-valued mappings are relatively easier than that of multivalued mappings. Nevertheless, several researchers in this direction have deeply investigated fixed point results involving multivalued mappings due to their usefulness in the field of game theory and market economy (see, e.g., [4,5]), non-smooth differential equations (see, for instance, [14]), control theory, convex optimization, variational inequalities, and differential inclusions (see [2,8,15,16,17,18] for further study).

Let $\mathcal{X}$ be a normed space and $K\subseteq \mathcal{X}$. Then, K is known as proximinal if we can find a point $z\in K$ that satisfies the identity

$$\rho (\wp ,K)=inf\{\parallel \wp -\hslash \parallel :\hslash \in K\}=\rho (\wp ,z).$$

(2)

for each $\wp \in \mathcal{X}$. Let $\mathit{B}$ be a (uniformly convex) Banach space and $\mathit{C}\subset \mathit{B}$. If $\mathit{C}$ is either close and convex or weakly compact, then $\mathit{C}$ is proximinal.

Throughout this paper, we denote the family of nonempty bounded closed subsets of $\mathcal{X}$ by $CB\left(\mathcal{X}\right)$, the family of nonempty compact subsets of K by $C\left(K\right)$, the family of nonempty subsets of $\mathcal{X}$ by $2^{\mathcal{X}}$, and the family of nonempty bounded proximinal subsets of K by $\mathcal{P}\left(K\right)$. The Hausdorff distance induced by the metric $\rho$ on $\mathcal{X}$ for any $A,B\in CB\left(\mathcal{X}\right)$ is defined as follows:

$$\mathcal{D}(A,B)=max\left\{\underset{\wp \in A}{sup}\rho (\wp ,B),\underset{\hslash \in B}{sup}\rho (\hslash ,A)\right\},$$

(3)

where $\rho (\wp ,B)=inf\{\parallel \wp -\hslash \parallel :\hslash \in B\}$ is the distance of the point ℘ from the subset B. Throughout the remainder of this section of the paper, ${\left(\mathcal{D}(A,B)\right)}^2$ implies ${\mathcal{D}}^2(A,B)$.

Let $ᘐ:\overline{D}(ᘐ)\subseteq \mathcal{X}⟶2^{\mathcal{X}}$ be a multivalued mapping on $\mathcal{X}$, where $\overline{D}(ᘐ)$ denotes the domain of ᘐ (the range of ᘐ will be represented with $R(ᘐ)$). Then, $\vartheta \in \overline{D}(ᘐ)$ is said to be an invariant point of the multivalued mapping ᘐ if $\vartheta \in ᘐ\vartheta .$ The set of invariant points of ᘐ is represented with $F(ᘐ)=\{\wp \in \overline{D}(ᘐ):\wp \in ᘐ\wp \}$. A point $\wp \in \overline{D}(ᘐ)$ is known as a strict invariant point of ᘐ if $ᘐ\wp =\{\wp \}$.

Definition 1.

ᘐ is known according to the following:

(1)

uniformly ν-Lipschitzian if we can find $\nu \ge 0$ for which the inequality

$$\mathcal{D}({ᘐ}^n\wp ,{ᘐ}^n\hslash )\le \nu \parallel \wp -\hslash \parallel ,\forall x,y\in \overline{D}(ᘐ)$$

(4)

holds. If ν is replaced with ${\omega }_n$ in (4), where ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{\omega }_n=1$, then ᘐ becomes a multivalued asymptotically nonexpansive mapping $\left(ASM\right)$.

(2)

type-one [11] if, given $\wp ,\hslash \in \overline{D}(ᘐ)$ and $u\in P_{ᘐ}\wp$, we can find $v\in P_{ᘐ}\hslash$ for which the inequality

$$\parallel u-v\parallel \le \mathcal{D}(ᘐ\wp ,ᘐ\hslash )$$

(5)

holds, where $P_{ᘐ}\wp =\{u\in ᘐ\wp :\parallel \wp -u\parallel =\rho (\wp ,ᘐ\wp )\}.$

(3)

η-strictly asymptotically pseudocontractive (η-SAPC) if we can find a sequence ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{\omega }_n=1$ and a constant $\eta \in [0,1)$ such that, given $\wp ,\hslash \in \overline{D}(ᘐ)$ and $u\in {ᘐ}^n\wp ,$ we can find $v\in {ᘐ}^n\hslash$ for which the inequalities $\parallel u-v\parallel \le \mathcal{D}({ᘐ}^n\wp ,{ᘐ}^n\hslash )$ and

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\hslash )\le {\omega }_n\parallel \wp -\hslash \parallel +\eta {\parallel \wp -u-(\hslash -v)\parallel }^2$$

(6)

hold. If $\eta =1$ in (6), then ᘐ becomes asymptotically pseudocontractive, whereas ᘐ is 0-SAPC (or ASM) if $\eta =0$ in (6).

(4)

β-strictly asymptotically pseudononspreading (β-SAPN) if we can find a sequence ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{\omega }_n=1$ and a constant $\beta \in [0,1)$ such that given $\wp ,\hslash \in \overline{D}(ᘐ)$ and $u\in {ᘐ}^n\wp ,$ we can find $v\in {ᘐ}^n\hslash$ for which the inequalities $\parallel u-v\parallel \le \mathcal{D}({ᘐ}^n\wp ,{ᘐ}^n\hslash )$ and

$${\mathcal{D}}^2{({ᘐ}^n\wp ,{ᘐ}^n\hslash )}^2\le k_n\parallel \wp -\hslash \parallel +\beta \parallel \wp -u-(\hslash -v){\parallel }^2+2\langle \wp -u,\hslash -v\rangle$$

(7)

hold. Note that for $\beta =1$ in (7), ᘐ is called asymptotically pseudononspreading. ᘐ is called 0-SAPN if $\beta =0$ in (7).

(5)

asymptotically demicontrctive (ADM) if ᘐ admits a nonempty fixed point set and Definition 1 $\left(\right(3) and (4\left)\right)$ hold; that is, ᘐ is asymptotically demicontractive (see, [12]) if $F(ᘐ)\ne \varnothing$ and, $\forall (\wp ,\vartheta )\in \overline{D}(ᘐ)\times F(ᘐ)$ and $\mu \in [0,1)$, we can find a sequence ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{\omega }_n=1$ and $u\in {ᘐ}^n\wp$ for which the inequality

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\vartheta )\le k_n^2{\parallel \wp -\vartheta \parallel }^2+\mu {\parallel \wp -u\parallel }^2$$

(8)

holds.

Remark 1.

If $F(ᘐ)\ne \varnothing$ in Definition 1, then $\left(3\right)$ and $\left(4\right)$ reduce to $\left(5\right)$. Therefore, the class of multivalued ADMs includes the classes of multivalued η-SAPC mappings and multivalued β-SAPN mappings.

At this point, we must mention that assuming the fixed point of a particular operator during the approximation process does not guarantee sound judgment in application. In this regard, Lim [17] demonstrated a remarkable conclusion by establishing the existence of a fixed point for a multivalued nonexpansive mapping. Subsequently, several researchers have employed different fixed point algorithms to estimate the fixed point of multivalued nonexpansive mappings (MNN) using the Hausdorff metric (pick out [11,13,18,19] and the references therein for more clarification).

In the midst of this, Sastry and Babu [20] initiated the notion of a Mann and Ishikawa approximation sequence as follows:

Let $ᘐ:\mathcal{X}⟶\mathbb{P}\left(\mathcal{X}\right)$. The Mann iteration (MI) sequence developed from an arbitrary ${\wp }_0\in \mathcal{X}$ be given by

$${\wp }_{n+1}=(1-{\alpha }_n){\wp }_n+{\alpha }_n{\hslash }_n\forall n\ge 0,$$

(9)

where ${\hslash }_n\in ᘐ{\wp }_n$ is such that $\parallel {\hslash }_n-\vartheta \parallel =d(ᘐ{\wp }_n,\vartheta )$ and ${\left\{{\alpha }_n\right\}}_{n=0}^{\infty }$ is a real sequence in $(0,1)$ with ${\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n=\infty$.

The sequence of iterates due to Ishikawa (SII), developed from an arbitrary ${\wp }_0\in \mathcal{X}$, is defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\hslash }_n=(1-{\beta }_n){\wp }_n+{\beta }_n{\omega }_n\hfill & \\ {\wp }_{n+1}=(1-{\alpha }_n){\wp }_n+{\alpha }_n{u}_n\forall n\ge 0,\hfill \end{array}\right.\end{array}$$

(10)

where ${\omega }_n\in ᘐ{\wp }_n$, $u_n\in ᘐ{\hslash }_n$ are such that $\parallel {\omega }_n-\vartheta \parallel =d(ᘐ{\wp }_n,\vartheta ),\parallel u_n-\vartheta \parallel =d(ᘐ{\hslash }_n,\vartheta )$, and ${\left\{{\alpha }_n\right\}}_{n=0}^{\infty },{\left\{{\beta }_n\right\}}_{n=0}^{\infty }$ are real sequences in $(0,1)$ for which $\left(i\right)0\le {\alpha }_n,{\beta }_n<1,\left(ii\right){\displaystyle \underset{n\to \infty }{lim}}{\beta }_n=0,and\left(iii\right){\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n{\beta }_n=\infty$ are guaranteed. Using (9) and (10), the results obtained in [20] were generalized by Panyanak [21].

The following lemma, which was later used to modify the sequence of iterates defined by (9) and (10) (see [22]), was due to Nadler [2].

Lemma 1.

Let $A,B\in CB\left(\mathcal{X}\right)$ and $a\in A$. If $\delta >0$, then there exists $b\in B$ such that

$$d(A,B)\le \mathcal{D}(A,B)+\delta .$$

(11)

In addition to the modification of (9) and (10) using (11), Song and Wang [22] observed that generating a Mann and Ishikawa sequence of iterates in [20] is in some sense connected to the awareness of the invariant point. They presented their iteration scheme as follows:

Let $\varnothing \ne K\subset \mathcal{X},{\varrho }_n,{\phi }_n\in [0,1]$, and ${\delta }_n\in (0,\infty )$ with ${lim}_{n\to \infty }{\delta }_n=0$. Choose ${\wp }_1\in K,{\omega }_1\in ᘐ{\wp }_1$. Let

$${\hslash }_1=(1-{\phi }_1){\wp }_1+{\phi }_1{\omega }_1,$$

and choose $u_1\in ᘐ{\hslash }_1$ such that $\parallel {\omega }_1-u_1\parallel \le \mathcal{D}(ᘐ{\wp }_1,ᘐ{\hslash }_1)+{\delta }_1$ and

$${\wp }_2=(1-{\varrho }_1){\wp }_1+{\varrho }_1{u}_1.$$

Choose ${\omega }_2\in ᘐ{\wp }_2$ such that $\parallel {\omega }_2-u_1\parallel \le \mathcal{D}(ᘐ{\wp }_2,ᘐ{\hslash }_1)+{\delta }_1$ and

$${\hslash }_2=(1-{\phi }_2){\wp }_2+{\phi }_2{\omega }_2.$$

Choose $u_2\in ᘐ{\hslash }_2$ such that $\parallel {\omega }_2-u_2\parallel \le \mathcal{D}(ᘐ{\wp }_2,ᘐ{\hslash }_2)+{\delta }_2$ and

$${\wp }_3=(1-{\varrho }_2){\wp }_2+{\varrho }_2{u}_2.$$

Continuing in this manner, we obtain by induction that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\hslash }_n=(1-{\phi }_n){\wp }_n+{\phi }_n{\omega }_n\hfill & \\ {\wp }_{n+1}=(1-{\varrho }_n){\wp }_n+{\varrho }_n{u}_n\forall n\ge 0,\hfill \end{array}\right.\end{array}$$

(12)

where ${\omega }_n\in ᘐ{\wp }_n,u_n\in ᘐ{\hslash }_n$ are such that $\parallel {\omega }_n-u_n\parallel =\mathcal{D}(ᘐ{\wp }_n,ᘐ{\hslash }_n)+{\delta }_n,\parallel {\omega }_{n+1}-u_n\parallel =\mathcal{D}(ᘐ{\wp }_{n+1},ᘐ{\hslash }_n)$, and ${\left\{{\varrho }_n\right\}}_{n=0}^{\infty },{\left\{{\phi }_n\right\}}_{n=0}^{\infty }\in [0,1)$ and, satisfying ${\displaystyle \underset{n\to \infty }{lim}}{\phi }_n=0,{\displaystyle \sum _{n=1}^{\infty }}{\varrho }_n{\phi }_n=\infty$. Using (12), the authors in [22] proved the following theorem:

Theorem 1

([22]). Let $\mathit{B}$ be a Banach space and $\varnothing \ne K\subset \mathit{B}$ be compact and convex. Assume $ᘐ:K⟶CB\left(K\right)$ is an MNN such that $F(ᘐ)\ne \varnothing$ and $ᘐ\vartheta =\left\{\vartheta \right\}$ for all $\vartheta \in F(ᘐ).$ Then the iteration technique given by (12) guarantees a strong convergent point of $F(ᘐ)$.

In [23], the following observation was made: If $ᘐ:D(ᘐ)⟶\mathcal{P}\left(\mathcal{X}\right)$ is any multivalued mapping where $\mathcal{X}$ is a normed space, then $P_{ᘐ}:D(ᘐ)⟶\mathcal{P}\left(\mathcal{X}\right)$ given by

$$P_{ᘐ}\wp =\{\hslash \in ᘐ\wp :d(\wp ,ᘐ\wp )=\parallel \wp -\hslash \parallel \}$$

has the property that $P_{ᘐ}\left(\vartheta \right)=\left\{\vartheta \right\}$ for each ℘ and for all $\vartheta \in F(ᘐ)$. Using this idea, the restriction $ᘐ\left(\vartheta \right)=\left\{\vartheta \right\}$ for all $\vartheta \in F(ᘐ)$ imposed on $F(ᘐ)$ by Song and Wang [22] was removed.

In [15], Khan and Yildrim initiated the following scheme for multivalued nonexpansive mappings:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\wp }_1\in K\hfill & \\ {\hslash }_n=(1-\lambda ){\wp }_n+\lambda {\omega }_n\hfill & \\ {\wp }_{n+1}=(1-\eta ){\wp }_n+\eta u_n\forall n\ge 1,\hfill \end{array}\right.\end{array}$$

(13)

where ${\omega }_n\in P_{ᘐ}{\wp }_n,u_n\in P_{ᘐ}{\hslash }_n$, and $\lambda ,\eta \in [0,1)$. Further, by means of a lemma introduced in [24], the notion of removing the restriction imposed on $F(ᘐ)$ (i.e., $ᘐ\left(\vartheta \right)=\left\{\vartheta \right\}$) initiated in [23], and the method of direct formulation of a Cauchy sequence established in [25], they proved the following theorem:

Theorem 2

(Theorem 1, [25]). Let $\mathit{B}$, satisfying Opial condition and admitting $\varnothing \ne K\subset \mathit{B}$ is closed and convex, as described above. Let $ᘐ:D(ᘐ)⟶\mathcal{P}\left(\mathit{B}\right)$ be such that $ᘐ\left(\vartheta \right)=\left\{\vartheta \right\}$ and $P_{ᘐ}$ is nonexpansive. Let ${\left\{{\wp }_n\right\}}_{n\ge 1}$ be as described by (13). Let $(I-P_{ᘐ})$ be demiclosed at zero. Then, ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ converges weakly to an element of $F(ᘐ)$.

In [26], Isiogugu demonstrated a remarkable observation that gave birth to a new class of multivalued mappings known as nonexpansive-type (MNT), strictly pseudocontractive-type (MKSPT), and pseudocontractive-type (MPT). These classes of mappings stem from the fact that there are many multivalued mappings in which neither ᘐ nor $P_{ᘐ}$ are nonexpansive. Precisely, the following definitions were given:

Definition 2.

Let $\mathcal{X}$ be as described above. A mapping $ᘐ:D(ᘐ)\subseteq \mathcal{X}⟶2^{\mathcal{X}}$ is known as MKSPT in the sense of Browder and Petryshyn [27] if, for $\wp ,\hslash \in D(ᘐ)$ and $u\in ᘐ\wp$, there exist $k\in [0,1)$ and $v\in ᘐ\hslash$ satisfying $\parallel u-v\parallel \le \mathcal{D}(ᘐ\wp ,ᘐ\hslash )$ such that

$${\mathcal{D}}^2{(ᘐ\wp ,ᘐ\hslash )}^2\le {\parallel \wp -\hslash \parallel +k\parallel \wp -u-(\hslash -v)\parallel }^2.$$

(14)

Note that ᘐ becomes MPT if $k=1$ and MNT if $k=0$ in (14). It is not hard to see from Definition 2 that

$$\left(MNT\right)⊊\left(MKSPT\right)⊊\left(MPT\right).$$

(15)

The relationship illustrated by (15) was further buttressed with examples (see, for instance, Example 1 and Example 2) in [26].

Considering the invaluable nature of multivalued nonlinear mappings in a practical sense, it becomes pertinent to ask the following question:

Question. 

Is it possible to obtain the class of multivalued asymptotically k-strictly demicontractive mappings for which the associated $P_{ᘐ}$ fails to be asymptotically nonexpansive?

Motivated and inspired by the above results, in this study, we first present novel classes of multivalued asymptotically k-strictly demicontractive-type mappings (MASDTMs) and multivalued asymptotically hemicontractive-type mappings (MAHTMs), which includes the class of multivalued asymptotically nonexpansive-type mappings (MANTMs). Also, we show that the class of MASDT mappings and the class of mappings studied in [26] are independent (see our examples at the end of this paper).

In addition, if $\varnothing \ne K\subset \mathcal{H}$ is weakly closed ($\mathcal{H}$ denoting a Hilbert space) and $ᘐ:K\subseteq \mathcal{H}⟶\mathcal{P}\left(\mathcal{H}\right)$ is a MASDTM, then $(I-P_{ᘐ})$ is demiclosed at zero, where I represents the identity mapping on K. Finally, we present weak and strong convergence results for the aforementioned classes of mappings without any compactness assumption and without the application of Condition $\left(I\right)$ on the domain of the mappings via Mann and Ishikawa sequence of iterates. Our main results extend, improve, and generalize several results on single-valued and multivalued mappings in the current literature.

The organization of this paper is as follows: In Section 2, we provide some necessary preliminaries that will be helpful in proving our main results. In Section 3, strong and weak convergence of the new classes of mappings are obtained in the framework of $\mathcal{H}$. Some corollaries, as direct consequences of our main results, are provided.

2. Relevant Preliminaries

To prove our theorem, the following results are prerequisite:

• Let $\mathcal{H}$ be a real Hilbert space with the inner product $\langle .,.\rangle$ and the norm $\parallel .\parallel$ and let $\varnothing \ne K\subset \mathcal{H}$. In this paper, we shall denote by $\mathbb{N}$ and $\mathbb{R}$ the set of natural numbers and the set of real numbers, respectively. If ${\left\{{\wp }_n\right\}}_{n=1}^{\infty }\subset \mathcal{H}$, then the weak and strong convergence of ${\left\{{\wp }_n\right\}}_{n=1}^{\infty }$ shall be represented with ⇀ and $\to ,$ respectively.

Definition 3.

A mapping $ᘐ:\mathit{K}⟶\mathit{K}$ is said to be a k-strictly asymptotically pseudocontractive mapping (kSAPM) if we can find a sequence ${\left\{{\omega }_n\right\}}_{n=0}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{\omega }_n=1$ and a constant $k\in [0,1)$ that guarantee the inequality

$$\parallel {ᘐ}^n\wp -{ᘐ}^n{\hslash \parallel }^2\le {\omega }_n^2{\parallel \wp -\hslash \parallel }^2+k{\parallel (I-{ᘐ}^n)\wp -(I-{ᘐ}^n)\hslash \parallel }^2,\forall \wp ,\hslash \in \mathit{K},n\ge 1.$$

(16)

It is not difficult to see that, if $k=1,0$ in (16), then we obtain the class of asymptotically pseudocontractive mappings (APMs) and the class of asymptotically nonexpansive mappings (ANMs), respectively (where ᘐ is said to be an ANM if, for all $\wp ,\hslash \in K$, we can find a sequence ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{k}_n=1$ that guarantees $\parallel {ᘐ}^n\wp -{ᘐ}^n\hslash \parallel \le k_n\parallel \wp -\hslash \parallel ,\forall n\ge 1)$ initiated by Goebel and Kirk [28]. It is easy to see from (16) that

$$\left(ANM\right)⊊\left(kSAPM\right)⊊\left(APM\right).$$

(17)

Remark 2.

If $F(ᘐ)\ne \varnothing$ and (16) holds, then we obtain the class of mappings known as asymptotically demicontractive mappings (ADMs).

Example 1

([29]). Let $\mathbb{R}$ denote the reals with the usual norm. We define $ᘐ:\mathbb{R}⟶\mathbb{R}$ by

$$\begin{array}{c}\hfill ᘐ\wp =\left\{\begin{array}{c}-3\wp +1,\wp \in (-\infty ,0]\hfill \\ {\displaystyle \frac{1}{2}}(\wp +2)\wp \in (0,\infty ).\hfill \end{array}\right.\end{array}$$

Then, for all $\wp ,\hslash \in (-\infty ,0]$, we obtain ${|ᘐ\wp -ᘐ\hslash |}^2=9{|\wp -\hslash |}^2$ and ${|\wp -ᘐ\wp -(\hslash -ᘐ\hslash )|}^2$. As a consequence, we have

$$\begin{array}{ccc}\hfill {|ᘐ\wp -ᘐ\hslash |}^2& =& {9|\wp -\hslash |}^2={|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}+{|\wp -ᘐ\wp -(\hslash -ᘐ\hslash )|}^2.\hfill \end{array}$$

Observe also that, for all $\wp ,\hslash \in (0,\infty )$,

$$\begin{array}{ccc}\hfill {|ᘐ\wp -ᘐ\hslash |}^2& =& {\displaystyle \frac{1}{4}}{|\wp -\hslash |}^2\le {|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}+{|\wp -ᘐ\wp -(\hslash -ᘐ\hslash )|}^2.\hfill \end{array}$$

Moreover, for all $\wp \in (-\infty ,0]$ and $\hslash \in (0,\infty )$, we obtain

$$\begin{array}{c}\hfill {|ᘐ\wp -ᘐ\hslash |}^2=|-3\wp -{\displaystyle \frac{1}{2}}(\hslash +2){|}^2={\displaystyle \frac{1}{4}}{|-6\wp -y|}^2={\displaystyle \frac{1}{4}}(36{\wp }^2+12\wp \hslash +{\hslash }^2)=9{\wp }_2+3\wp \hslash +{\displaystyle \frac{1}{4}}{\hslash }^2\end{array}$$

and

$$\begin{array}{ccc}\hfill {|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}+{|\wp -ᘐ\wp -(\hslash -ᘐ\hslash )|}^2& =& {|\wp -\hslash |}^2+{\displaystyle \frac{1}{4}}+|4\wp -1-(\hslash -{\displaystyle \frac{1}{2}}(\hslash +2)\hslash ){|}^2\hfill \\ & =& {\wp }^2-2\wp \hslash +{\hslash }^2+{\displaystyle \frac{1}{8}}{|18\wp -\hslash |}^2\hfill \\ & =& {\wp }^2-2\wp \hslash +{\hslash }^2+{\displaystyle \frac{1}{8}}(64{\wp }^2-16\wp \hslash +{\hslash }^2)\hfill \\ & =& 9{\wp }^2-4\wp \hslash +{\hslash }^2+{\displaystyle \frac{1}{8}}{\hslash }^2\hfill \\ & =& 9{\wp }^2+3\wp \hslash +{\displaystyle \frac{1}{4}}{\hslash }^2-7\wp \hslash +{\displaystyle \frac{7}{8}}{\hslash }^2\hfill \\ & \ge & 9{\wp }^2+3\wp \hslash +{\displaystyle \frac{1}{4}}{\hslash }^2={|ᘐ\wp -ᘐ\hslash |}^2.\hfill \end{array}$$

(18)

Therefore,

$$\begin{array}{ccc}\hfill {|ᘐ\wp -ᘐ\hslash |}^2& =& \le {|\wp -\hslash |}^2{\displaystyle \frac{1}{2}}+{|\wp -ᘐ\wp -(\hslash -ᘐ\hslash )|}^2,\forall \wp ,\hslash \in \mathbb{R}.\hfill \end{array}$$

Observe that, for all integers $n\ge 2$, we have

$$\begin{array}{c}\hfill {ᘐ}^n\wp =\left\{\begin{array}{c}{\displaystyle \frac{1}{2^{n-1}}}(-3\wp +2^n-1),\wp \in (-\infty ,0]\hfill \\ {\displaystyle \frac{1}{2^n}}(\wp +2(2^n-1))\wp \in (0,\infty ).\hfill \end{array}\right.\end{array}$$

Now, for all $\wp ,\hslash \in (-\infty ,0]$, we have

$$\begin{array}{ccc}\hfill |{ᘐ}^n\wp -{ᘐ}^n{\hslash |}^2& =& {\displaystyle \frac{9}{2^{2^{(n-1)}}}}{|\wp -\hslash |}^2\hfill \end{array}$$

and

$$\begin{array}{c}\hfill |\wp -{ᘐ}^n\wp -(\hslash -{ᘐ}^n\hslash ){{|}^2={\left(1+{\displaystyle \frac{3}{2^{(n-1)}}}\right)}^2\wp -\hslash |}^2.\end{array}$$

Hence,

$$\begin{array}{ccc}\hfill {|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}{|\wp -{ᘐ}^n\wp -(\hslash -{ᘐ}^n\hslash )|}^2& =& \left[1+{\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)}^2\right]{|\wp -\hslash |}^2,\forall \wp ,\hslash \in \mathbb{R}.\hfill \\ & =& {\displaystyle \frac{3^2}{2^{2(n-1)}}}{|\wp -\hslash |}^2+[1+{\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)}^2\hfill \\ & & -{\displaystyle \frac{3^2}{2^{2(n-1)}}}]{|\wp -\hslash |}^2\hfill \\ & =& {\displaystyle \frac{3^2}{2^{2(n-1)}}}{|\wp -\hslash |}^2+[{\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{2^{n-1}}}+{\displaystyle \frac{3^2}{2^{2n-1}}}\hfill \\ & & -{\displaystyle \frac{3^2}{2^{2(n-1)}}}]{|\wp -\hslash |}^2\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{3^2}{2^{2(n-1)}}}{|\wp -\hslash |}^2+[{\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{2^{n-1}}}+{\displaystyle \frac{3^2}{2^{2n-1}}}\hfill \\ & & -{\displaystyle \frac{3^2\left(2\right)}{2^{2n-1}}}]{|\wp -\hslash |}^2\hfill \\ & =& {\displaystyle \frac{3^2}{2^{2(n-1)}}}{|\wp -\hslash |}^2+[{\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{2^{n-1}}}-{\displaystyle \frac{3^2}{2^{2n-1}}}{]|\wp -\hslash |}^2\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{3^2}{2^{2(n-1)}}}{|\wp -\hslash |}^2+\left[{\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{2^{n-1}}}\left(1-{\displaystyle \frac{3}{2^n}}\right)\right]{|\wp -\hslash |}^2\hfill \\ & \ge & {\displaystyle \frac{3^2}{2^{2(n-1)}}}{|\wp -\hslash |}^2={|ᘐ\wp -ᘐ\hslash |}^2.\hfill \end{array}$$

Next, for all $\wp ,\hslash \in (0,\infty )$, we obtain

$$\begin{array}{c}\hfill |{ᘐ}^n\wp -{ᘐ}^n{\hslash |}^2={\displaystyle \frac{1}{2^{2n}}}{|\wp -\hslash |}^2={|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}{|\wp -{ᘐ}^n\wp -(\hslash -{ᘐ}^n\hslash )|}^2.\end{array}$$

If, we now consider $\wp \in (-\infty ,0]$ and $\hslash \in (0,\infty )$, then we have

$$\begin{array}{c}\hfill |{ᘐ}^n\wp -{ᘐ}^n{\hslash |}^2={\displaystyle \frac{1}{2^{2(n-1)}}}\left[9{\wp }^2+3\wp \hslash +{\displaystyle \frac{{\hslash }^2}{4}}\right]\end{array}$$

(19)

and

$$\begin{array}{c}\hfill |\wp -{ᘐ}^n\wp -(\hslash -{ᘐ}^n\hslash ){|}^2={\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)}^2{\wp }^2+{\left(1-{\displaystyle \frac{1}{2^n}}\right)}^2{\hslash }^2+2\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)\left(1-{\displaystyle \frac{1}{2^n}}\right)\wp \hslash \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill {|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}{|\wp -{ᘐ}^n\wp -(\hslash -{ᘐ}^n\hslash )|}^2& =& \left[1+{\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)}^2\right]{\wp }^2+\left[1+{\displaystyle \frac{1}{2}}{\left(1-{\displaystyle \frac{1}{2^n}}\right)}^2\right]{\hslash }^2\hfill \\ & & -\left[2+2\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)\left(1-{\displaystyle \frac{1}{2^n}}\right)\right]\wp \hslash \hfill \\ & =& {\displaystyle \frac{1}{2^{2(n-1)}}}\left[9{\wp }^2+3\wp \hslash +{\displaystyle \frac{{\hslash }^2}{4}}\right]+[1+{\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)}^2\hfill \\ & & -{\displaystyle \frac{9}{2^{2(n-1)}}}]{\wp }^2+\left[1+{\displaystyle \frac{1}{2}}{\left(1-{\displaystyle \frac{1}{2^n}}\right)}^2-{\displaystyle \frac{1}{2^{2n}}}\right]{\hslash }^2\hfill \\ & & -\left[2+2\left(1+{\displaystyle \frac{3}{2^{n-1}}}\right)\left(1-{\displaystyle \frac{1}{2^n}}\right)+{\displaystyle \frac{3}{2^{2(n-1)}}}\right]\wp \hslash \hfill \\ \hfill & =& |{ᘐ}^n\wp -{ᘐ}^n{\hslash |}^2+\left[{\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{2^{n-1}}}\left(1-{\displaystyle \frac{3}{2^n}}\right)\right]{\wp }^2\hfill \\ & & +{\displaystyle \frac{3}{2}}\left(1-{\displaystyle \frac{1}{2^n}}\right){\hslash }^2-\left[4+{\displaystyle \frac{5}{2^{n-1}}}\right]\wp \hslash \hfill \\ & \ge & |{ᘐ}^n\wp -{ᘐ}^n{\hslash |}^2.\hfill \end{array}$$

(20)

Thus, for all $\wp ,\hslash \in \mathbb{R}$ and $n\in \mathbb{N}$, we obtain

$$\begin{array}{c}\hfill |{ᘐ}^n\wp -{ᘐ}^n{\hslash |}^2\le {|\wp -\hslash |}^2+{\displaystyle \frac{1}{2}}{|\wp -{ᘐ}^n\wp -(\hslash -{ᘐ}^n\hslash )|}^2.\end{array}$$

(21)

It, therefore, follows that ᘐ is k-strictly asymptotically pseudocontractive with $F(ᘐ)=\left\{2\right\}$, $k_n=1$, and $k={\displaystyle \frac{1}{2}}$ for all $n\ge 1$.

Definition 4.

Let $\mathcal{X}$ be a normed space and $\varnothing \ne K\subset \mathcal{X}$. A mapping $ᘐ:K⟶\mathcal{P}\left(K\right)$ is said to satisfy condition (I) (see [22]) if we can find a nondecreasing function $g:[0,\infty )⟶[0,\infty )$ such that

$$d(\wp ,ᘐ\wp )\ge g\left(d\right(\wp ,F(ᘐ)\left)\right),\forall \wp \in K.$$

Definition 5

([11]). Let $\mathcal{X}$ be as described in Section 1, and $S:D\left(S\right)\subseteq \mathcal{X}⟶2^{\mathcal{X}}$. $I-S$ is weakly demiclosed at zero if, for any sequence ${\left\{{\wp }_n\right\}}_{n=1}^{\infty }\subseteq D\left(S\right)$ that assures weak convergence to ϑ, we can find a sequence $\left\{{\hslash }_n\right\}$ with ${\hslash }_n\in S{\wp }_n$ for all $n\in \mathbb{N}$ that assures strong convergence to zero for which $\vartheta \in S\vartheta (i.e.,0\in (I-S\left)\vartheta \right).$

Lemma 2

(see [30]). Let ${\left\{{\zeta }_n\right\}}_{n=1}^{\infty },{\left\{{\kappa }_n\right\}}_{n=1}^{\infty },{\left\{{\varsigma }_n\right\}}_{n=1}^{\infty }\subseteq [0,\infty )$ be such that the following inequality holds:

$${\zeta }_{n+1}\le (1+{\kappa }_n){\zeta }_n+{\varsigma }_n,n\ge n_0,$$

where $n_0$ is a nonnegative integer. If ${\displaystyle \sum _{n=0}^{\infty }}{\kappa }_n<\infty ,{\displaystyle \sum _{n=0}^{\infty }}{\varsigma }_n<\infty$, then ${\displaystyle \underset{n\to \infty }{lim}}{\zeta }_n$ exists.

Lemma 3

(see [26]). Let $\mathcal{X}$ and $\mathit{K}$ be as described above. Let $ᘐ:K⟶\mathcal{P}\left(K\right)$ be a multivalued mapping and $P_{ᘐ}(\wp )=\{\hslash \in ᘐ\wp :\parallel \wp -\hslash \parallel =d(\wp ,ᘐ\wp )\}$. Then, the following statements are equivalent:

1. 

$\wp \in ᘐ\wp$;

2. 

$P_{ᘐ}(\wp )=\{\wp \}$;

3. 

$\wp \in F\left(P_{ᘐ}\right)$. Further, $F(ᘐ)=F\left(P_{ᘐ}\right)$.

Lemma 4

(see [26]). Let $\mathcal{H}$ retain its usual meaning. Then, the following identity holds: If ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }\subset \mathcal{H}$ such that $\wp \rightharpoonup \hslash \in \mathcal{H}$, then

$$\underset{n\to \infty }{lim\; sup}\parallel {\wp }_n{-\nu \parallel }^2=\underset{n\to \infty }{lim\; sup}\parallel {\wp }_n{-\hslash \parallel }^2+{\parallel \hslash -\nu \parallel }^2,\forall \nu \in \mathcal{H}.$$

3. Results

Definition 6.

Let £ be a normed space and $ᘐ:D(ᘐ)\subseteq \pounds ⟶2^{\pounds }$ be a multivalued mapping. Then, ᘐ is called a k-strictly asymptotically demicontractive-type mapping (kSADTM) in the sense of Isogugu et al. [31] if $F(ᘐ)\ne \varnothing$ and, for all $(\wp ,\hslash )\in \left(D\right(ᘐ)\times F(ᘐ\left)\right)$, we can find $k\in [0,1)$ and a sequence ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }\subseteq [1,\infty )$ with ${\displaystyle \underset{n\to \infty }{lim}}{\omega }_n=1$ such that

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\hslash )\le {\omega }_n^2{\parallel \wp -\hslash \parallel }^2+k{d}^2(\wp ,{ᘐ}^n\wp ).$$

(22)

If $k=1$, then ᘐ is called an asymptotically hemicontractive-type multivalued mapping (AHTM).

Remark 3.

From Definition 6, one can see that every multivalued asymptotically quasi-nonexpansive-type mapping (MAQNTM) is a kSADTM. The examples below show that the class of MAQNTMs is properly included into the class of kSADTMs and that the class of kSADTMs is properly included into the class of asymptotically hemicontrctive-type mappings (AHTMs).

Example 2.

Let $\pounds =\mathbb{R}$ be endowed with the usual metric. We define the mapping $ᘐ:[0,\infty )⟶2^{\mathbb{R}}$ by

$$ᘐ\wp =\left[-{\displaystyle \frac{\wp }{\sqrt{5}}},-{\displaystyle \frac{\wp }{\sqrt{3}}}\right].$$

Observe that $F(ᘐ)=\left\{0\right\}$. For n odd $(n\ge 2)$, we obtain

$${ᘐ}^n\wp =\left[-{\displaystyle \frac{\wp }{\sqrt{3^{2n-1}}}},-{\displaystyle \frac{\wp }{\sqrt{5^{2n-1}}}}\right].$$

Now,

$$\begin{array}{ccc}\hfill {\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^{n}0)& =& max\left\{|{\displaystyle \frac{1}{\sqrt{3^{2n-1}}}}(\wp -0){|}^2,|{\displaystyle \frac{1}{\sqrt{5^{2n-1}}}}(\wp -0){|}^2\right\}\hfill \\ & =& {\displaystyle \frac{1}{3^{2n-1}}}{|\wp -0|}^2\hfill \\ & <& {\displaystyle \frac{1}{3^{2n-3}}}{|\wp -0|}^2\hfill \\ & =& \left({\displaystyle \frac{9}{3^{2n}}}+{\displaystyle \frac{17}{3^{2n-1}}}\right){|\wp -0|}^2+{\displaystyle \frac{1}{3^{2n-1}}}{|\wp -0|}^2\hfill \\ & <& \left({\displaystyle \frac{9}{3^2}}+{\displaystyle \frac{17}{3^{2n-1}}}\right){|\wp -0|}^2+{\displaystyle \frac{1}{3^2}}{|\wp |}^2.\hfill \end{array}$$

(23)

Additionally, since

$$\begin{array}{c}\hfill d^2(\wp ,{ᘐ}^n\wp )=|\wp -\left(-{\displaystyle \frac{1}{\sqrt{5^{2n-1}}}}\wp \right){|}^2={\displaystyle \frac{5^{2n-1}+2\sqrt{5^{2n-1}}+1}{5^{2n-1}}}{|\wp |}^2.\end{array}$$

(24)

It follows from (23) and (24) that

$$\begin{array}{ccc}\hfill {\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^{n}0)& \le & \left({\displaystyle \frac{9}{3^2}}+{\displaystyle \frac{17}{3^{2n-1}}}\right){|\wp -0|}^2+{\displaystyle \frac{5^{2n-1}}{9(5^{2n-1}+2\sqrt{5^{2n-1}}+1)}}{d}^2(\wp ,{ᘐ}^n\wp )\hfill \\ & <& \left(1+{\displaystyle \frac{17}{3^{2n-1}}}\right){|\wp -0|}^2+{\displaystyle \frac{1}{9}}{d}^2(\wp ,{ᘐ}^n\wp ),\forall n\ge 1.\hfill \end{array}$$

(25)

Therefore, ᘐ is a kSADTM with $k={\displaystyle \frac{1}{9}}$ and $k_n=\left(1+{\displaystyle \frac{17}{3^{2n-1}}}\right)$. Observe that $k_n\to 1$ as $n\to \infty$. Consequently, ᘐ is an AHTM.

Example 3.

Let $\mathcal{H}=\mathbb{R}$ be endowed with the usual metric. We define the mapping $ᘐ:[-1.5,1]⟶2^{\mathbb{R}}$ by

$$\begin{array}{c}\hfill ᘐ\wp =\left\{\begin{array}{c}\left[-{\displaystyle \frac{2}{3}}\wp ,-{\displaystyle \frac{3}{2}}\wp \right],\wp \in \left[-{\displaystyle \frac{3}{2}},0\right]\hfill \\ \hfill \\ \left[-{\displaystyle \frac{3}{2}}\wp ,-{\displaystyle \frac{2}{3}}\wp \right],\wp \in [0,1].\hfill \end{array}\right.\end{array}$$

Then, for n odd $n\ge 2$, we obtain

$$\begin{array}{c}\hfill {ᘐ}^n\wp =\left\{\begin{array}{c}\left[-{\displaystyle \frac{2}{3}}\wp ,-{\displaystyle \frac{3}{2}}\wp \right],\wp \in \left[-{\displaystyle \frac{3}{2}},0\right]\hfill \\ \hfill \\ \left[-{\displaystyle \frac{3}{2}}\wp ,-{\displaystyle \frac{2}{3}}\wp \right],\wp \in [0,1].\hfill \end{array}\right.\end{array}$$

Then, $F(ᘐ)=\left\{0\right\}$ and $P_{{ᘐ}^n}\wp =\left\{-{\displaystyle \frac{3}{2}}\wp \right\}$ for all $\wp \in [-1.5,1]$. Additionally, observe that $F\left(P_{ᘐ}\right)=\left\{0\right\}$; hence, it is not asymptotically quasi-nonexpansive. Indeed,

$$\mathcal{D}({ᘐ}^n\wp ,{ᘐ}^n\hslash )= \parallel P_{{ᘐ}^n}\wp -P_{{ᘐ}^n}\hslash \parallel =|-{\displaystyle \frac{3}{2}}\wp -\left(-{\displaystyle \frac{3}{2}}\hslash \right)|={\displaystyle \frac{3}{2}}|\wp -\hslash |>|\wp -\hslash |$$

For $\wp \in \left[-{\displaystyle \frac{3}{2}},1\right]$, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^{n}0)& =& max\left\{|{\displaystyle \frac{2}{3}}(\wp -0){|}^2,|{\displaystyle \frac{3}{2}}(\wp -0){|}^2\right\}\hfill \\ & =& {\displaystyle \frac{9}{4}}{|\wp -0|}^2\hfill \\ & =& {|\wp -0|}^2+{\displaystyle \frac{5}{4}}{|\wp |}^2\hfill \end{array}$$

and

$$d^2(\wp ,{ᘐ}^n\wp )=|\wp -\left(-{\displaystyle \frac{3}{2}}\right)\wp {|}^2={\displaystyle \frac{25}{4}}{|\wp |}^2.$$

Therefore,

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^{n}0)={|\wp -0|}^2+{\displaystyle \frac{5}{4}}{|\wp |}^2={|\wp -0|}^2+{\displaystyle \frac{1}{5}}{d}^2(\wp ,{ᘐ}^n\wp ).$$

Therefore, ᘐ is a kSADTM with $k_n=1$ and $k={\displaystyle \frac{1}{5}}$. Note that ᘐ, not being an MAQNTM, demonstrates the conclusion that the class of MAQNTMs is included in the class of multivalued kSADTMs. The example that follows demonstrates the fact that the class of multivalued AHTMs possessed the class of multivalued kSADTMs.

Example 4.

Let $\mathcal{H}=\mathbb{R}$ be endowed with the usual metric and define the mapping $ᘐ:[0,\infty )⟶2^{\mathbb{R}}$ by

$$ᘐ\wp =\left[-{\displaystyle \frac{\wp }{\sqrt{2}}},-{\displaystyle \frac{\wp }{10}}\right].$$

Then, for n odd $(n\ge 2)$, we obtain

$${ᘐ}^n\wp =\left[-{\displaystyle \frac{\wp }{\sqrt{2^{2n-1}}}},-{\displaystyle \frac{\wp }{{10}^{2n-1}}}\right].$$

Observe that $F(ᘐ)=\left\{0\right\}$. Now, for any $\wp \in \mathcal{R}$, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^{n}0)& =& max\left\{|{\displaystyle \frac{1}{\sqrt{2^{2n-1}}}}(\wp -0){|}^2,|{\displaystyle \frac{1}{{10}^{2n-1}}}(\wp -0){|}^2\right\}\hfill \\ & =& {\displaystyle \frac{1}{2^{2n-1}}}{|\wp -0|}^2<{\displaystyle \frac{1}{2^{2n-4}}}{|\wp -0|}^2\hfill \\ & =& \left({\displaystyle \frac{8}{2^{2n}}}+{\displaystyle \frac{3}{2^{2n-1}}}\right){|\wp -0|}^2+{\displaystyle \frac{1}{2^{2n-1}}}{|\wp -0|}^2\hfill \\ & <& \left({\displaystyle \frac{8}{2^2}}+{\displaystyle \frac{3}{2^{2n-1}}}\right){|\wp -0|}^2+{\displaystyle \frac{1}{2^{2n-1}}}{|\wp -0|}^2\hfill \\ & <& \left(1+{\displaystyle \frac{3}{2^{2n-1}}}\right){|\wp -0|}^2+{\left(1+{\displaystyle \frac{1}{\sqrt{2^{2n-1}}}}\right)}^2{|\wp -0|}^2\hfill \end{array}$$

Since

$$d^2(\wp ,{ᘐ}^n\wp )=|\wp -\left(-{\displaystyle \frac{1}{\sqrt{2^{2n-1}}}}\right)\wp {|}^2={\left(1+{\displaystyle \frac{1}{\sqrt{2^{2n-1}}}}\right)}^2{|\wp |}^2,$$

it follows from the last inequality that

$$\begin{array}{ccc}\hfill {\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^{n}0)& \le & \left(1+{\displaystyle \frac{3}{2^{2n-1}}}\right){|\wp -0|}^2+{\left(1+{\displaystyle \frac{1}{\sqrt{2^{2n-1}}}}\right)}^2{|\wp -0|}^2\hfill \\ & =& \left(1+{\displaystyle \frac{3}{2^{2n-1}}}\right){|\wp -0|}^2+d^2(\wp ,{ᘐ}^n\wp )\hfill \\ & >& \left(1+{\displaystyle \frac{3}{2^{2n-1}}}\right){|\wp -0|}^2+k{d}^2(\wp ,{ᘐ}^n\wp ),\forall k\in (0,1).\hfill \end{array}$$

Therefore, ᘐ is an AHTM with $k_n=\left(1+{\displaystyle \frac{3}{2^{2n-1}}}\right)$ but not a kSADTM.

Proposition 1.

Let $\mathcal{H}$ be a real Hilbert space, $\varnothing \ne K\subset \mathcal{H}$ be compact and convex, and $ᘐ:D(ᘐ)\subseteq K⟶2^K$ be an asymptotically k-strictly demicontractive-type mapping such that $F(ᘐ),F_s(ᘐ)\ne \varnothing$. Then, $\left(a\right)F(ᘐ)$ is closed, and $\left(b\right)F_s(ᘐ)$ is closed and convex.

Proof. 

Let ${\left\{{\wp }_n\right\}}_{n=1}^{\infty }\subseteq F(ᘐ)$ such that ${\left\{{\wp }_n\right\}}_{n=1}^{\infty }$ converges to $\wp \in K$. It suffices to show that $\wp \in F(ᘐ)$.

Since

$$d(\wp ,ᘐ\wp )= \parallel \wp -{\wp }_n\parallel +d({\wp }_n,ᘐ{\wp }_n)+\mathcal{D}(ᘐ{\wp }_n,ᘐ\wp ),$$

it follows from the definition of the multivalued asymptotically demicontractiveness of ᘐ that

$$d(\wp ,ᘐ\wp )=(\sqrt{k_n}+\sqrt{k})\parallel \wp -{\wp }_n\parallel +\sqrt{k}d(\wp ,ᘐ\wp ).$$

Thus, taking limits as $n\to \infty$, we obtain that

$$d(\wp ,ᘐ\wp )=\sqrt{k}d(\wp ,ᘐ\wp ).$$

• Therefore, $\wp \in ᘐ\wp$.
• (a) The proof that $F_s(ᘐ)$ is closed follows an identical technique as the one described by (b) above. It is left for us to demonstrate the convexity of $F_s(ᘐ)$. Let ${\vartheta }_1,{\vartheta }_2\in F_s(ᘐ)$ and $w=\alpha {\vartheta }_1+(1-\alpha ){\vartheta }_2$. Then, $w-{\vartheta }_1=(1-\alpha )({\vartheta }_2-{\vartheta }_1)$ and $w-{\vartheta }_2=\alpha ({\vartheta }_1-{\vartheta }_2)$. Now,

$$\begin{array}{ccc}\hfill d^2(w,ᘐw)& \le & {\parallel w-u\parallel }^2,\forall u\in ᘐw\hfill \\ & =& \parallel \alpha {\vartheta }_1+(1-\alpha ){\vartheta }_2{-u\parallel }^2\hfill \\ & =& \alpha \parallel {\vartheta }_1{-u\parallel }^2+(1-\alpha )\parallel {\vartheta }_2{-u\parallel }^2-\alpha (1-\alpha ){\parallel {\vartheta }_2-{\vartheta }_1\parallel }^2.\hfill \end{array}$$

(26)

Since $ᘐ{\vartheta }_1=\left\{{\vartheta }_1\right\},ᘐ{\vartheta }_2=\left\{{\vartheta }_2\right\},\parallel u-{\vartheta }_1\parallel =d(u,ᘐ{\vartheta }_1)\le {sup}_{v\in ᘐw}d(v,ᘐ{\vartheta }_1)$ and $\parallel u-{\vartheta }_2\parallel =d(u,ᘐ{\vartheta }_2)\le {sup}_{v\in ᘐw}d(v,ᘐ{\vartheta }_2)$, it follows from the demicontractiveness of ᘐ and (26) that

$$\begin{array}{ccc}\hfill d^2(w,ᘐw)& \le & \alpha {\mathcal{D}}^2(ᘐw,ᘐ{\vartheta }_1)+(1-\alpha ){\mathcal{D}}^2(ᘐw,ᘐ{\vartheta }_2)-\alpha (1-\alpha ){\parallel {\vartheta }_2-{\vartheta }_1\parallel }^2\hfill \\ & \le & \alpha [k_n^2\parallel w-{\vartheta }_1{\parallel }^2+k{d}^2(w,ᘐw)]+(1-\alpha )[k_n^2\parallel w-{\vartheta }_2{\parallel }^2+k{d}^2(w,ᘐw)]\hfill \\ & & -\alpha (1-\alpha )\parallel {\vartheta }_2-{\vartheta }_1{\parallel }^2\hfill \\ & =& k_n^2\parallel \alpha {\vartheta }_1+(1-\alpha ){\vartheta }_2{-w\parallel }^2+(k_n^2-1)\alpha (1-\alpha ){\parallel {\vartheta }_2-{\vartheta }_1\parallel }^2+k{d}^2(w,ᘐw).\hfill \end{array}$$

Thus, taking limits as $n\to \infty$, we obtain that

$$d^2(w,ᘐw)\le {\parallel \alpha {\vartheta }_1+(1-\alpha ){\vartheta }_2-w\parallel }^2+k{d}^2(w,ᘐw)=k{d}^2(w,ᘐw).$$

Therefore, $w\in ᘐw$. Additionally, if $w\in ᘐw$, then we have

$${\parallel v-w\parallel }^2={\parallel v-[\alpha {\vartheta }_1+(1-\alpha ){\vartheta }_2]\parallel }^2=k{d}^2(w,ᘐw)=0.$$

Consequently, $ᘐw=\left\{w\right\}$. □

Next, we prove the following important property of asymptotically demicontractive-type mappings.

Proposition 2.

Let $\mathcal{H}$ be as in Proposition 1, $\varnothing \ne K\subset \mathcal{H}$ be compact and convex, and $ᘐ:D(ᘐ)\subseteq K⟶2^K$ be an asymptotically k-strictly demicontractive-type mapping. Then, ᘐ is uniformly $\mathcal{L}$-Lipschitz at a fixed point; i.e., for all $(\wp ,\vartheta )\in (K\times F(ᘐ\left)\right)$, there exists $L>0$ such that

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\vartheta )\le L\parallel \wp -\vartheta \parallel .$$

Proof. 

Since ᘐ is an asymptotically demidocontractive-type mapping, it follows that there exists $k\in [0,1)$ and a real sequence $\left\{k_n\right\}\subset [1,\infty )$ with ${\displaystyle \sum _{n=0}^{\infty }}(k_n^2-1)<\infty$ such that for any $(\wp ,\vartheta )\in (K\times F(ᘐ\left)\right)$, we have

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\vartheta )\le k_n^2{\parallel \wp -\vartheta \parallel }^2+k{d}^2(\wp ,{ᘐ}^n\wp )$$

From the last inequality, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\vartheta )& \le & k_n^2{\parallel \wp -\vartheta \parallel }^2+k{d}^2(\wp ,{ᘐ}^n\wp )\hfill \\ & \le & [k_n\parallel \wp -\vartheta \parallel +\sqrt{k}d(\wp ,{ᘐ}^n\wp ){]}^2\hfill \end{array}$$

It then follows that

$$\begin{array}{ccc}\hfill \mathcal{D}({ᘐ}^n\wp ,{ᘐ}^n\vartheta )& \le & k_n\parallel \wp -\vartheta \parallel +\sqrt{k}d(\wp ,{ᘐ}^n\wp )\vartheta \left)\right]\hfill \\ & \le & k_n\parallel \wp -\vartheta \parallel +\sqrt{k}[\parallel \wp -\vartheta \parallel +d(\vartheta ,{ᘐ}^n\vartheta )+\mathcal{D}({ᘐ}^n\wp ,{ᘐ}^n\vartheta )]\hfill \end{array}$$

Therefore,

$${\mathcal{D}}^2({ᘐ}^n\wp ,{ᘐ}^n\vartheta )\le {\displaystyle \frac{k_n+\sqrt{k}}{1-\sqrt{k}}}\parallel \wp -\vartheta \parallel .$$

□

Proposition 3.

Let $\mathcal{H}$ be as in Proposition 1 and $\varnothing \ne K\subset \mathcal{H}$ be weakly closed. Let $ᘐ:K\subseteq \mathcal{H}⟶P\left(\mathcal{H}\right)$ be a multivalued mapping. Suppose $P_{ᘐ}$ is a kSADM. Then $(I-P_{ᘐ})$ is demiclosed at zero.

Proof. 

Let ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }\subseteq K\ni \wp \rightharpoonup \vartheta$ and ${\left\{{\hslash }_n\right\}}_{n=0}^{\infty },$ with ${\hslash }_n\in P_{{ᘐ}^n}{\wp }_n$ $\forall n\in \mathbb{N}$, such that $\parallel {\wp }_n-{\hslash }_n\parallel \to 0$. We want to show that $0\in (I-P_{ᘐ})\vartheta$ (i.e., $\vartheta =\nu$ for a few $\nu \in P_{ᘐ}\vartheta$). Since ${\wp }_n\rightharpoonup \vartheta$, boundedness is assured. Let $q\in P_{{ᘐ}^n}\vartheta$ be arbitrary for all $n\in \mathbb{N}$. By Definition 6, for each n, we can find $u_{n\in P_{{ᘐ}^n}{\wp }_n}$, which guarantees the inequalities

$$\parallel u_n-q\parallel \le \mathcal{D}(P_{{ᘐ}^n}{\wp }_n,P_{{ᘐ}^n}\vartheta )$$

(27)

and

$${\mathcal{D}}^2(P_{{ᘐ}^n}{\wp }_n,P_{{ᘐ}^n}\vartheta )\le k_n^2{\parallel {\wp }_n-\vartheta \parallel }^2+k{d}^2(\vartheta ,ᘐ\vartheta )$$

(28)

for all $n\in \mathbb{N}$. Now, since $u_n,{\hslash }_n\in P_{{ᘐ}^n}{\wp }_n,\parallel {\wp }_n-u_n\parallel = \parallel {\wp }_n-{\hslash }_n\parallel$, it follows that

$$\underset{n\to \infty }{lim}\parallel {\wp }_n-u_n\parallel =\underset{n\to \infty }{lim}\parallel {\wp }_n-{\hslash }_n\parallel =0.$$

For each $\wp \in \mathcal{H}$, define $ð:\mathcal{H}⟶[0,\infty )$ by

$$ð(\wp )=\underset{n\to \infty }{lim\; sup}{\parallel {\wp }_n-\wp \parallel }^2.$$

Then, from Lemma 4, we obtain

$$ð(\wp )=\underset{n\to \infty }{lim\; sup}\parallel {\wp }_n{-\vartheta \parallel }^2+{\parallel \vartheta -\wp \parallel }^2,\forall \wp \in \mathcal{H}.$$

Thus, for any $\wp \in \mathcal{H}$,

$$ð(\wp )={ð\left(\vartheta \right)+\parallel \vartheta -\wp \parallel }^2,\forall \wp \in \mathcal{H}.$$

Consequently,

$$ð\left(q\right)={ð\left(\vartheta \right)+\parallel \vartheta -q\parallel }^2.$$

(29)

Further, observe that

$$\begin{array}{ccc}\hfill ð\left(q\right)& =& \underset{n\to \infty }{lim\; sup}{\parallel {\wp }_n-q\parallel }^2\hfill \\ & =& \underset{n\to \infty }{lim\; sup}{\parallel {\wp }_n-u_n+(u_n-q)\parallel }^2\hfill \\ & =& \underset{n\to \infty }{lim\; sup}{\parallel u_n-q\parallel }^2\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}{\mathcal{D}}^2(P_{{ᘐ}^n}{u}_n-P_{{ᘐ}^n}\vartheta )\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}[k_n^2\parallel {\wp }_n{-\vartheta \parallel }^2{+k\parallel \vartheta -q\parallel }^2]\hfill \\ & =& \underset{n\to \infty }{lim\; sup}{k}_n^2\parallel {\wp }_n{-\vartheta \parallel }^2+k{\parallel \vartheta -q\parallel }^2\hfill \\ & =& k_n^2ð\left(\vartheta \right)+k{\parallel \vartheta -q\parallel }^2.\hfill \end{array}$$

(30)

It then follows from (29) and (30) that

$${(1-k)\parallel \vartheta -q\parallel }^2\le (k_n^2-1)ð\left(\vartheta \right).$$

Hence,

$$\underset{n\to \infty }{lim}\parallel \vartheta -q\parallel =0$$

(31)

and $\vartheta =q\in P_{{ᘐ}^n}\vartheta$.

By Proposition 2, $P_{ᘐ}$ is uniformly L-Lipschitzian. Using this fact, we obtain the following estimates:

$$\begin{array}{ccc}\hfill d(\vartheta ,P_{ᘐ})& \le & \parallel \vartheta -q\parallel +\mathcal{D}(P_{{ᘐ}^n}\vartheta ,P_{ᘐ}\vartheta )\hfill \\ & =& \parallel \vartheta -q\parallel +\mathcal{D}(P_{ᘐ}\left(P_{{ᘐ}^{n-1}}\vartheta \right),P_{ᘐ}\vartheta )\hfill \\ & \le & \parallel \vartheta -q\parallel +Ld(P_{{ᘐ}^{n-1}}\vartheta ,\vartheta )\hfill \\ & \le & \parallel \vartheta -q\parallel +L[d(P_{{ᘐ}^{n-1}}\vartheta ,q)+\parallel \vartheta -q\parallel ]\hfill \\ & =& (1+L)\parallel \vartheta -q\parallel +Ld(P_{{ᘐ}^{n-1}}\vartheta ,q).\hfill \end{array}$$

Since $q\in P_{{ᘐ}^n}\vartheta$, it follows from (31) that

$$\underset{n\to \infty }{lim}d(\vartheta ,P_{ᘐ}\vartheta )=0,$$

and so $\vartheta \in P_{ᘐ}\vartheta$ as required. □

Theorem 3.

Let $\mathcal{H}$ be as in Proposition 1 and $\varnothing \ne K\subset \mathcal{H}$ be closed and convex. Let ᘐ be as described in Proposition 2 such that $\left\{k_n\right\}\subset [1,\infty )$ with ${\displaystyle \sum _{n=0}^{\infty }}(k_n^2-1)<\infty$, $k\in (0,1)$, and $ᘐ\vartheta =\left\{\vartheta \right\}$ for all $\vartheta \in F(ᘐ)$. Suppose $(I-ᘐ)$ is demiclosed at zero. Then, the Mann-type iteration sequence ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ generated from an arbitrary ${\wp }_0\in K$ by

$${\wp }_{n+1}=(1-{\varrho }_n){\wp }_n+{\varrho }_n{\hslash }_n$$

converges weakly to $\vartheta \in F(ᘐ)$, where ${\hslash }_n\in {ᘐ}^n{\wp }_n$ and ${\left\{{\varrho }_n\right\}}_{n=0}^{\infty }\subseteq (0,1)$, satisfying the conditions: $\left(i\right){\varrho }_n\to \varrho <1-k\left(ii\right)\varrho >0\left(iii\right){\displaystyle \sum _{n=0}^{\infty }}{\varrho }_n(1-{\varrho }_n)=\infty$.

Proof. 

From

$${\parallel t\wp +(1-t)\hslash \parallel }^2={t\parallel \wp \parallel }^2+{(1-t)\parallel \hslash \parallel }^2-t(1-t){\parallel \wp -\hslash \parallel }^2,$$

which is valid $\forall \wp ,\hslash \in \mathcal{H},\forall t\in [0,1]$, the following estimates ensued:

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+1}{-\vartheta \parallel }^2& =& \parallel (1-{\varrho }_n){\wp }_n+{\varrho }_n{\hslash }_n{-\vartheta \parallel }^2\hfill \\ & =& \parallel (1-{\varrho }_n)({\wp }_n-\vartheta )+{\varrho }_n({\hslash }_n-\vartheta ){\parallel }^2\hfill \\ & =& (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n\parallel {\hslash }_n{-\vartheta \parallel }^2-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-{\hslash }_n\parallel }^2\hfill \\ & \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{\mathcal{D}}^2({ᘐ}^n{\wp }_n,{ᘐ}^n\vartheta )-{\varrho }_n(1-{\alpha }_n){\parallel {\wp }_n-{\hslash }_n\parallel }^2\hfill \\ & \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n[k_n^2\parallel {\wp }_n{-\vartheta \parallel }^2+k{d}^2({\wp }_n,{ᘐ}^n{\wp }_n)]-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-{\hslash }_n\parallel }^2\hfill \\ & =& [1+(k_n^2-1){\varrho }_n]\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_{n}k\parallel {\wp }_n-{\hslash }_n{\parallel }^2-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-{\hslash }_n\parallel }^2\hfill \\ & =& [1+(k_n^2-1){\varrho }_n]\parallel {\wp }_n{-\vartheta \parallel }^2-{\varrho }_n[1-({\varrho }_n+k)]{\parallel {\wp }_n-{\hslash }_n\parallel }^2.\hfill \end{array}$$

(32)

It follows from (32) and Lemma 2 that ${lim}_{n\to \infty }\parallel {\wp }_n-\vartheta \parallel$ exists and hence ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ is bounded. Additionally, since

$$\sum _{n=1}^{\infty }{\varrho }_n[1-({\varrho }_n+k)]\parallel {\wp }_n-{\hslash }_n{\parallel }^2\le {\parallel {\wp }_0-\vartheta \parallel }^2+\sum _{n=1}^{\infty }{\varrho }_n(k_n^2-1)Q<\infty$$

and $\alpha >0$ from condition $\left(ii\right)$, it follows that

$$\underset{n\to \infty }{lim}\parallel {\wp }_n-{\hslash }_n\parallel =0.$$

Again, since K is closed and ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }\subseteq K$ with ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ bounded, we can find ${\left\{{\wp }_{n_j}\right\}}_{j=0}^{\infty }\subseteq {\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ such that ${\wp }_{n_j}\rightharpoonup \vartheta \in \mathit{K}$ as $j\to \infty$. In addition,

$$\underset{n\to \infty }{lim}\parallel {\wp }_n-{\hslash }_n\parallel =0\Rightarrow \underset{n\to \infty }{lim}\parallel {\wp }_{n_j}-{\hslash }_{n_j}\parallel =0.$$

Using the fact that $(I-ᘐ)$ is demiclosed at zero, we obtain $\vartheta \in ᘐ\vartheta$. Since $\mathcal{H}$ satisfies the Opial condition (see [32]), we obtain that ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ converges weakly to $\vartheta \in F(ᘐ)$. □

Corollary 1.

Let $\mathcal{H}$, K, and ᘐ be as described in Theorem 3. Suppose that $P_{ᘐ}$ is a multivalued kSADTM such that $\left\{k_n\right\}\subset [1,\infty )$ with ${\displaystyle \sum _{n=0}^{\infty }}(k_n^2-1)<\infty$, $k\in (0,1)$, and $ᘐ\vartheta =\left\{\vartheta \right\}$ for all $\vartheta \in F(ᘐ)$. Suppose $(I-P_{ᘐ})$ is weakly demiclosed at zero. Then, the Mann-type iteration sequence ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ generated from an arbitrary ${\wp }_0\in K$ by

$${\wp }_{n+1}=(1-{\varrho }_n){\wp }_n+{\varrho }_n{\hslash }_n$$

converges weakly to $\vartheta \in F(ᘐ)$, where ${\hslash }_n\in {ᘐ}^n{\wp }_n$ and ${\left\{{\varrho }_n\right\}}_{n=0}^{\infty }\subseteq (0,1)$, satisfying the conditions: $\left(i\right){\varrho }_n\to \varrho <1-k\left(ii\right)\varrho >0\left(iii\right){\displaystyle \sum _{n=1}^{\infty }}{\varrho }_n(1-{\varrho }_n)=\infty$.

Proof. 

The proof of the above corollary immediately follows from Lemma 3, Proposition 3, and Theorem 3. □

Remark 4.

Clearly, Examples 2 and 4 validate the requirement “given any pair $\wp ,\hslash \in D(ᘐ)$ and $u_n\in {ᘐ}^n\wp$ with $\parallel \wp -u_n\parallel =d(\wp ,{ᘐ}^n\wp )$, there exists $v_n\in {ᘐ}^n\hslash$ with $\parallel \hslash -v_n\parallel =d(\hslash ,{ᘐ}^n\hslash )$ validating the conditions of Definition 6”. In addition, if ᘐ is a multivalued mapping such that $P_{ᘐ}$ is an asymptotically hemicontractive-type mapping, then, given any pair $\wp ,\hslash \in D(ᘐ)$ and $u_n\in P_{{ᘐ}^n}\wp$ with the corresponding $v_n\in P_{{ᘐ}^n}\hslash$ validating the requirements of Definition 6, it is the case that $\parallel \wp -u_n\parallel =d(\wp ,P_{{ᘐ}^n}\wp )$ and $\parallel \hslash -v_n\parallel =d(\hslash ,P_{{ᘐ}^n}\hslash )$.

Dwelling on Remark 4, we present the following convergence results for the new class of multivalued asymptotically hemicontractive-type mappings (MAHTMs) based on these requirements: (i) given any pair $\wp ,\hslash \in D(ᘐ)$ and $u_n\in {ᘐ}^n\wp$ with $\parallel \wp -u_n\parallel =d(\wp ,{ᘐ}^n\wp )$, there exists $v_n\in {ᘐ}^n\hslash$ with $\parallel \hslash -v_n\parallel =d(\hslash ,{ᘐ}^n\hslash )$ validating the requirements of Definition 6; (ii) $ᘐ\left(\vartheta \right)=\left\{\vartheta \right\}$ for all $\vartheta \in F(ᘐ)$, which leads to the case for an arbitrary multivalued mapping ᘐ for which $P_{ᘐ}$ is an MAHTM devoid of the requirements of $P_{ᘐ}$ as a corollary.

Theorem 4.

Let $\mathcal{H}$ be as described in Theorem 3. Suppose that $ᘐ:K⟶P\left(K\right)$ is an L-Lipschitzian asymptotically hemicontractive-type multivalued mapping such that $\left\{k_n\right\}\subset [1,\infty )$ with ${\displaystyle \sum _{n=0}^{\infty }}(k_n^2-1)<\infty$, $k\in (0,1)$ and $ᘐ\vartheta =\left\{\vartheta \right\}$ for all $\vartheta \in F(ᘐ)$. Then, the Ishikawa-type iteration sequence ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ generated from an arbitrary ${\wp }_0\in K$ by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\hslash }_n=(1-{\phi }_n){\wp }_n+{\phi }_n{u}_n\hfill \\ {\wp }_{n+1}=(1-{\varrho }_n){\wp }_n+{\varrho }_n{v}_n\hfill \end{array}\right.\end{array}$$

(33)

converges strongly to $\vartheta \in F(ᘐ)$, where $u_n\in {ᘐ}^n{\wp }_n$ and $v_n\in {ᘐ}^n{\hslash }_n$, satisfying the condition of Lemma 3, and ${\left\{{\varrho }_n\right\}}_{n=0}^{\infty }$ and ${\left\{{\phi }_n\right\}}_{n=0}^{\infty }\subseteq 0,1)$, satisfying the following conditions: $\left(i\right)0\le {\varrho }_n\le {\phi }_n<1\left(ii\right){\displaystyle \underset{n\to \infty }{lim\; inf}}{\varrho }_n>0\left(iii\right){\displaystyle \underset{n\ge 1}{sup}}{\phi }_n\le \phi \le {\displaystyle \frac{2L}{(1+k^2)+\sqrt{{(1+k^2)}^2+4L^2}}},k=max{k}_1,k_2,\cdots ,k_n$.

Proof. 

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+1}{-\vartheta \parallel }^2& =& \parallel (1-{\varrho }_n){\wp }_n+{\varrho }_n{v}_n{-\vartheta \parallel }^2\hfill \\ & =& \parallel (1-{\varrho }_n)({\wp }_n-\vartheta )+{\varrho }_n(v_n-\vartheta ){\parallel }^2\hfill \\ & =& (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n\parallel v_n{-\vartheta \parallel }^2-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{\mathcal{D}}^2({ᘐ}^n{\hslash }_n,{ᘐ}^n\vartheta )-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n[k_n^2\parallel {\hslash }_n{-\vartheta \parallel }^2+d^2({\hslash }_n,{ᘐ}^n{\hslash }_n)]-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & =& (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{k}_n^2{\parallel {\hslash }_n-\vartheta \parallel }^2+{\varrho }_n{d}^2({\hslash }_n,{ᘐ}^n{\hslash }_n)\hfill \\ & & -{\varrho }_n(1-{\alpha }_n){\parallel {\wp }_n-v_n\parallel }^2.\hfill \end{array}$$

(34)

Moreover,

$$\begin{array}{ccc}\hfill d^2({\hslash }_n,{ᘐ}^n{\hslash }_n)& \le & \parallel {\hslash }_n-v_n{\parallel }^2\hfill \\ & =& \parallel (1-{\phi }_n)({\wp }_n-v_n)+{\phi }_n(u_n-v_n){\parallel }^2\hfill \\ & =& (1-{\phi }_n)\parallel {\wp }_n-v_n{\parallel }^2+{\phi }_n\parallel u_n-v_n{\parallel }^2-{\phi }_n(1-{\phi }_n){\parallel {\wp }_n-u_n\parallel }^2.\hfill \end{array}$$

(35)

From (34) and (35), we obtain

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+1}{-\vartheta \parallel }^2& \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{k}_n^2\parallel {\hslash }_n{-\vartheta \parallel }^2+{\varrho }_n[(1-{\phi }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & & +{\phi }_n\parallel u_n-v_n{\parallel }^2-{\varrho }_n(1-{\varrho }_n)\parallel {\wp }_n-v_n{\parallel }^2]\hfill \\ & & -{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2.\hfill \end{array}$$

(36)

Since

$$\begin{array}{ccc}\hfill \parallel {\hslash }_n{-\vartheta \parallel }^2& =& \parallel (1-{\phi }_n)({\wp }_n-\vartheta )+{\phi }_n(u_n-\vartheta ){\parallel }^2\hfill \\ & =& (1-{\phi }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n\parallel u_n{-\vartheta \parallel }^2-{\beta }_n(1-{\beta }_n){\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & \le & (1-{\phi }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n{\mathcal{D}}^2({ᘐ}^n{\hslash }_n,{ᘐ}^n\vartheta )-{\phi }_n(1-{\phi }_n){\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & \le & (1-{\phi }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n[k_n^2\parallel {\wp }_n-\vartheta {\parallel }^2+d^2({\wp }_n,{ᘐ}^n{\wp }_n)]\hfill \\ & & -{\phi }_n(1-{\beta }_n){\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & =& (1-{\phi }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n{k}_n^2\parallel {\wp }_n-\vartheta {\parallel }^2+{\phi }_n{d}^2{\wp }_n,{ᘐ}^n{\wp }_n)\hfill \\ & & -{\phi }_n(1-{\phi }_n){\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & =& [1+{\phi }_n(k_n^2-1)]\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n\parallel {\wp }_n-u_n{\parallel }^2-{\phi }_n(1-{\phi }_n){\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & =& [1+{\phi }_n(k_n^2-1)]\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n^2{\parallel {\wp }_n-u_n\parallel }^2,\hfill \end{array}$$

(37)

it follows from (36) that

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+1}{-\vartheta \parallel }^2& \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{k}_n^2\{[1+{\phi }_n(k_n^2-1)]{\parallel {\wp }_n-\vartheta \parallel }^2\hfill \\ & & +{\phi }_n^2\parallel {\wp }_n-u_n{\parallel }^2\}+{\varrho }_n[(1-{\phi }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & & +{\phi }_n\parallel u_n-v_n{\parallel }^2-{\phi }_n(1-{\phi }_n)\parallel {\wp }_n-u_n{\parallel }^2]-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & =& (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{k}_n^2\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n{\varrho }_n{k}_n^2(k_n^2-1){\parallel {\wp }_n-\vartheta \parallel }^2\hfill \\ & & +{\phi }_n^2{\varrho }_n{k}_n^2\parallel {\wp }_n-u_n{\parallel }^2+{\varrho }_n(1-{\phi }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & & +{\varrho }_n{\phi }_n\parallel u_n-v_n{\parallel }^2-{\varrho }_n{\phi }_n(1-{\phi }_n)\parallel {\wp }_n-u_n{\parallel }^2-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{k}_n^2\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n\varrho n{k}_n^2(k_n^2-1){\parallel {\wp }_n-\vartheta \parallel }^2\hfill \\ & & +{\phi }_n^2{\varrho }_n{k}_n^2\parallel {\wp }_n-u_n{\parallel }^2+{\varrho }_n(1-{\phi }_n){\parallel {\wp }_n-v_n\parallel }^2+{\varrho }_n{\phi }_n{\mathcal{D}}^2({ᘐ}^n{\wp }_n,{ᘐ}^n{\hslash }_n)\hfill \\ & & -{\varrho }_n{\phi }_n(1-{\phi }_n)\parallel {\wp }_n-u_n{\parallel }^2-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \\ & \le & (1-{\varrho }_n)\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{k}_n^2\parallel {\wp }_n{-\vartheta \parallel }^2+{\phi }_n{\varrho }_n{k}_n^2(k_n^2-1){\parallel {\wp }_n-\vartheta \parallel }^2\hfill \\ & & +{\phi }_n^2{\varrho }_n{k}_n^2\parallel {\wp }_n-u_n{\parallel }^2+{\varrho }_n(1-{\phi }_n)\parallel {\wp }_n-v_n{\parallel }^2+{\varrho }_n{\phi }_n{L}^2{\parallel {\wp }_n-{\hslash }_n\parallel }^2\hfill \\ & & -{\varrho }_n{\phi }_n(1-{\phi }_n)\parallel {\wp }_n-u_n{\parallel }^2-{\varrho }_n(1-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2\hfill \end{array}$$

$$\begin{array}{ccc}& =& \{1+[{\varrho }_n+{\varrho }_n{\phi }_n{k}_n^2](k_n^2-1)\}\parallel {\wp }_n{-\vartheta \parallel }^2+{\varrho }_n{\phi }_n^2{k}_n^2{\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & & +{\varrho }_n{\phi }_n^3{L}^2\parallel {\wp }_n-u_n{\parallel }^2-{\varrho }_n({\phi }_n-{\varrho }_n)\parallel {\wp }_n-v_n{\parallel }^2-{\varrho }_n{\phi }_n(1-{\phi }_n){\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & =& \{1+[{\varrho }_n+{\varrho }_n{\phi }_n{k}_n^2](k_n^2-1)\}\parallel {\wp }_n{-\vartheta \parallel }^2-{\varrho }_n{\phi }_n[1-{\phi }_n(k_n^2+1)-{\phi }_n^2{L}^2]{\parallel {\wp }_n-u_n\parallel }^2\hfill \\ & & -{\varrho }_n({\phi }_n-{\varrho }_n){\parallel {\wp }_n-v_n\parallel }^2.\hfill \end{array}$$

(38)

It then follows from the fact that ${\displaystyle \sum _{n=1}^{\infty }}(k_n^2-1)<\infty$ and Lemma 2 that ${lim}_{n\to \infty }\parallel {\wp }_n-\vartheta \parallel$ exists. Consequently, ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ is bounded, and so are the sequences ${\left\{u_n\right\}}_{n=0}^{\infty }$ and ${\left\{v_n\right\}}_{n=0}^{\infty }$. Next, we claim that the ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }\subset K$ is Cauchy. To prove our claim, observe in (38) that

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+1}-\vartheta \parallel & \le & \sqrt{\{1+[{\varrho }_n+{\varrho }_n{\phi }_n{k}_n^2](k_n^2-1)\}}\parallel {\wp }_n-\vartheta \parallel \hfill \\ & \le & \{1+[{\varrho }_n+{\varrho }_n{\phi }_n{k}_n^2](k_n^2-1)\}\parallel {\wp }_n-\vartheta \parallel .\hfill \end{array}$$

(39)

Now, setting ${\sigma }_n=[{\varrho }_n+{\varrho }_n{\phi }_n{k}_n^2](k_n^2-1)$ and using the fact that $e^{\gamma }\ge (1+\gamma )$, we obtain from (39) that

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+1}-\vartheta \parallel & \le & (1+{\sigma }_n)\parallel {\wp }_n-\vartheta \parallel \hfill \\ & \le & e^{{\sigma }_n}\parallel {\wp }_n-\vartheta \parallel ,\hfill \end{array}$$

which for $n\ge 1$ yields

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n+m}-\vartheta \parallel & \le & e^{{\sigma }_{n+m-1}}\parallel {\wp }_{n+m-1}-\vartheta \parallel \hfill \\ & \le & e^{{\sigma }_{n+m-1}+{\sigma }_{n+m-2}}\parallel {\wp }_{n+m-2}-\vartheta \parallel \hfill \\ & \le & e^{{\sigma }_{n+m-1}+{\sigma }_{n+m-2}+{\sigma }_{n+m-3}}\parallel {\wp }_{n+m-3}-\vartheta \parallel \hfill \\ & \le & \cdots \hfill \\ & \le & e^{{\sum }_{i=1}^{\infty }{\sigma }_i}\parallel {\wp }_0-\vartheta \parallel .\hfill \end{array}$$

(40)

Let $\mu ={\sum }_{i=1}^{\infty }{\sigma }_i$. Then, given any $ϵ>0$, it follows from the fact that ${\sum }_{i=1}^{\infty }{\sigma }_i<\infty$ that we can find a positive number $n_0$ and a point $\vartheta \in F$ such that

$$\parallel {\wp }_0-\vartheta \parallel <{\displaystyle \frac{ϵ}{2(1+\mu )}}.$$

Hence, for $m\ge 1$, we obtain that

$$\begin{array}{ccc}\hfill \parallel {\wp }_{n_0+m}-{\wp }_{n_0}\parallel & \le & \parallel {\wp }_{n_0+m}-\vartheta \parallel +\parallel {\wp }_{n_0}-\vartheta \parallel \hfill \\ & \le & 2\mu \parallel {\wp }_0-\vartheta \parallel \hfill \\ & <& ϵ.\hfill \end{array}$$

Thus, ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence in K and converges to some $\vartheta \in K$ because K is closed. Suppose ${lim}_{n\to \infty }{\wp }_n=\vartheta$. Then, it is left for us to show that $\vartheta \in F$. But, given any ${ϵ}_0>0$, we can find a positive integer $N_0\ge N$ for which

$$\parallel {\wp }_n-\vartheta \parallel =d({\wp }_0,\vartheta )\cap d({\wp }_n,F)<{\displaystyle \frac{{ϵ}_0}{2(1+3L)}}.$$

Similarly, we can find $\xi \in F$ for which

$$\parallel {\wp }_n-\xi \parallel =d({\wp }_0,\xi )\cap d({\wp }_n,F)<{\displaystyle \frac{{ϵ}_0}{2(1+L)}}.$$

From the above estimates, we get

$$\begin{array}{ccc}\hfill d(ᘐ\vartheta ,\vartheta )& \le & \mathcal{D}(ᘐ\vartheta ,\xi )+\mathcal{D}\left({ᘐ}^n{\wp }_{N_0},\xi )+\mathcal{D}\right(\xi ,{ᘐ}^n{\wp }_{N_0})+\parallel \xi -{\wp }_{n_0}|+\parallel {\wp }_{N_0}-\vartheta \parallel \hfill \\ & \le & L\parallel \vartheta -\xi \parallel +L\parallel {\wp }_{N_0}-\xi \parallel +L\parallel \xi -{\wp }_{N_0}+\parallel \xi -{\wp }_{n_0}+\parallel {\wp }_{N_0}-\vartheta \parallel \hfill \\ & \le & L[\parallel \vartheta -{\wp }_{N_0}\parallel +\parallel {\wp }_{N_0}-\xi \parallel ]+L\parallel {\wp }_{N_0}-\xi \parallel +L\parallel \xi -{\wp }_{N_0}+\parallel \xi -{\wp }_{n_0}\parallel +\parallel {\wp }_{N_0}-\vartheta \parallel \hfill \\ & =& (1+L)\parallel {\wp }_{N_0}-\xi \parallel +(1+3L)\parallel {\wp }_{N_0}-\vartheta \parallel \hfill \\ & <& {\displaystyle \frac{{ϵ}_0}{2(1+L)}}+{\displaystyle \frac{{ϵ}_0}{2(1+3L)}}<{ϵ}_0.\hfill \end{array}$$

Hence, $\vartheta \in ᘐ\vartheta$. Since ${lim}_{n\to \infty }\parallel {\wp }_n-\vartheta \parallel$ exists, we have that ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ converges strongly to $\vartheta \in F(ᘐ)$. □

Corollary 2.

Let $\mathcal{H}$ and K be as described in Theorem 3. Let $ᘐ:K⟶P\left(K\right)$ be a multivalued mapping such that $F(ᘐ)\ne \varnothing$. Suppose $P_{ᘐ}$ is an L-Lipschitzian and asymptotically hemicontractive-type mapping. Then, the Ishikawa-type iteration sequence ${\left\{{\wp }_n\right\}}_{n=0}^{\infty }$ generated by an arbitrary ${\wp }_0\in K$ defined by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\hslash }_n=(1-{\phi }_n){\wp }_n+{\phi }_n{u}_n\hfill \\ {\wp }_{n+1}=(1-{\varrho }_n){\wp }_n+{\varrho }_n{v}_n\hfill \end{array}\right.\end{array}$$

(41)

converges strongly to $\vartheta \in F(ᘐ)$.

Proof. 

By applying Lemma 3 and Theorem 4, the proof is completed. □

Remark 5.

We remark that the results obtained in the paper can be applied in the following way:

1. 

It is known that the class of asymptotically k-strictly demicontractive-type mappings contains the classes of asymptotically quasi- nonexpansive-type and quasi- nonexpansive-type mappings. Thus, the results established in the paper also hold for these classes of mappings so long as the indicated conditions are fulfilled.

2. 

Since every asymptotically k-strictly pseudocontractive-type multivalued mapping ᘐ with $F(ᘐ)\ne \varnothing$ and $ᘐ\left(\vartheta \right)=\left\{\vartheta \right\}$, $\forall \vartheta \in F(ᘐ)$, is an asymptotically k-strictly demicontractive-type multivalued mapping, our results can be applied for this class of mappings and hence for asymptotically nonexpansive multivalued mappings provided that the specified conditions are fulfilled.

3. 

Since every asymptotically pseudocontractive-type multivalued mapping ᘐ with $F(ᘐ)\ne \varnothing$ and $ᘐ\left(\vartheta \right)=\left\{\vartheta \right\}$, $\forall \vartheta \in F(ᘐ)$, is a asymptotically hemicontractive-type multivalued mapping, our results can be applied for this class of mappings provided that the specified conditions are fulfilled.

4. Conclusions

In this paper, we introduced and studied new classes of k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings. We also proved convergence theorems (without an imposition of compactness condition and condition (I) on the space or the mappings) and the demiclosedness property for these classes of mappings in the setup of a real Hilbert space. Additionally, we constructed some examples of the classes of mappings studied to demonstrate their existence. Finally, it was shown that the class of mappings studied in this paper is independent of the class of mappings studied in [26]. However, there are other spaces more general than the Hilbert space, and there are iterative methods involving an inertial term (or terms) together with some combination of other parameters for optimization problems. Consequently, our future research interest will focus on the aforementioned unexplored areas.