Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

Abstract

Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. Subsequently, we exhibit the role of additional properties of demicontractive operators, as well as the main features of the employed iterative algorithms to ensure weak convergence or strong convergence. We also include commentaries on the use of demicontractive mappings to solve some important nonlinear problems with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some brief statements on our view on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and some application challenges.

1. Introduction

Demicontractive mappings are ubiquitous objects in several important contexts of metric fixed-point theory with relevant applications in solving nonlinear problems, such as equilibrium problems, split problems, split common fixed-point problems, split feasibility problems with multiple output sets, split variational inequality problems, split common null point problems, etc. (see [1,2], among others). The main theoretical and applicative interest in studying this class of mappings lies in the fact that demicontractive mappings form one of the largest classes of discontinuous nonexpansive type mappings for which the fixed points can be approximated by means of iterative algorithms.

Throughout this paper, we shall consider H to be a real Hilbert space with its norm and inner product denoted as usual by $\parallel · \parallel$ and $\langle ·,·\rangle$, respectively. Let $C\subset H$ be a closed and convex set and let $T:C\to C$ be a self mapping. Denote by $Fix\left(T\right)=\{x\in C:Tx=x\}$ the set of fixed points of T.

In this survey, we shall refer mainly to the next classes of nonexpansive type mappings, which are defined as follows:

(1) T is called nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel , \mathrm{for} \mathrm{all} x,y\in C.$$

(1)

(2) T is called quasi-nonexpansive if $Fix\left(T\right)\ne Ø$ and

$$\parallel Tx-y\parallel \le \parallel x-y\parallel , \mathrm{for} \mathrm{all} x\in Candy\in Fix\left(T\right).$$

(2)

(3) T is called k-strictly pseudocontractive of the Browder-Petryshyn type ([3]) if there exists $k<1$ such that

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel x-y-Tx+Ty\parallel }^2,\forall x,y\in C.$$

(3)

(4) T is called demicontractive ([4,5]) or k-demicontractive if $Fix\left(T\right)\ne Ø$ and there exists a positive number $k<1$ such that

$${\parallel Tx-y\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel x-Tx\parallel }^2,$$

(4)

for all $x\in C$ and $y\in Fix\left(T\right)$.

It is easily seen that any nonexpansive mapping is k-strictly pseudocontractive, any nonexpansive mapping with $Fix\left(T\right)\ne Ø$ is quasi-nonexpansive, any quasi-nonexpansive mapping is demicontractive, and a k-strictly pseudocontractive mapping with $Fix\left(T\right)\ne Ø$ is demicontractive, but the reverses may not be true, as illustrated by the next example, which collects Examples 1.1 and 1.2 in [1].

Example 1

([1], Example 1.1; Example 1.2). Let H be a real line with the usual norm and $C=[0,1]$. Define T on C by $Tx={\displaystyle \frac{7}{8}}$ if $0\le x<1$ and $T1={\displaystyle \frac{1}{4}}$. Then, $Fix\left(T\right)=\left\{{\displaystyle \frac{7}{8}}\right\}$, and T is ${\displaystyle \frac{2}{3}}$-demicontractive, but T is neither nonexpansive nor quasi-nonexpansive or strictly pseudocontractive.

The natural ambient spaces for studying the iterative approximation of fixed points of demicontractive mappings were, for a long time, the linear settings, i.e., Hilbert and Banach spaces, but very recently, authors started to study this problem in nonlinear settings, as well.

Determining additional sufficient conditions for a demicontractive mapping and also designing a more suitable iterative algorithm for approximating its fixed points is a long-standing problem in the metric fixed-point theory of nonexpansive type mappings, which still presents many open questions and bustling, challenging research lines.

In this work, we provide a concise overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. We also exhibit the key role of the hosting settings, additional properties of demicontractive operators, and the main features of the employed iterative algorithms to ensure weak convergence or strong convergence towards their fixed points.

We also include commentaries on the use of demicontractive mappings to solve some nonlinear problems, with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some views on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and also with some application challenges.

Paper Organization

A brief account of the main historical events in the evolution of demicontractive mappings is presented in Section 2. Afterwards, the main developments arising from the problem of approximating fixed points of demicontractive mappings are discussed, as follows.

• In Section 3, the classical weak and strong convergence theorems attributed to Măruşter [5] and Hicks and Kubicek [4], respectively, are presented.
• In Section 4, some of the earliest subsequent fixed-point and common fixed-point results for single-valued demicontractive mappings in Hilbert spaces are surveyed.
• In Section 5, we expose some sample results on the use of demicontractive mappings in Hilbert spaces to solve relevant nonlinear problems (variational inequality problems, equilibrium problems etc.) over sets of fixed points or common fixed points of demicontractive mappings.
• In Section Section 6, we conclude with some comments and also indicate some future direc- tions of research for single-valued demicontractive mappings in Hilbert spaces.

2. Brief Historical Account

Demicontractive mappings did not emerge from the foam of the sea like Aphrodite. They were suggested by a concrete application, i.e., the solution of a complicated system of 8 nonlinear equations with 6 unknowns appearing in a problem of mechanics related to the production of cranes, in the case of the definition given by Măruşter (see [6] for more details), or were constructed by Hicks and Kubicek [4] by applying the technique of enriching quasi-nonexpansive mappings, similar to how Browder-Petryshyn [3] introduced the class of k-strictly pseudocontractive mappings by enriching nonexpansive mappings; see [7,8,9].

In the setting of a Hilbert space, demicontractive mappings were introduced independently in 1977 by Măruşter [5] and Hicks and Kubicek [4], respectively.

As the same notion was introduced in 1973 by Măruşter [10], in the particular case of ${\mathbb{R}}^n$, one can consider this year as the birth date of demicontractive mappings, thus having motivated the syntagm in the title of the current paper.

We now present Măruşter’s definition [5]. It is important to note that the term “demicontractive” was coined by Hicks and Kubicek [4], who introduced it by means of Inequality (4).

Definition 1

(Măruşter [5]). Let H be a real Hilbert space and let C be a closed convex subset of H. A mapping $T:C\to C$ such that $Fix\left(T\right)\ne Ø$ is said to satisfy Condition (A) if there exists $\lambda >0$ such that

$$\langle x-Tx,x-x^{\ast }\rangle \ge \lambda {\parallel Tx-x\parallel }^2,\forall x\in C,x^{\ast }\in Fix\left(T\right).$$

(5)

For a rather long time since their introduction, it was not apparent that the two Inequalities (4) and (5), which involve, in fact, different formulas, are equivalent in the setting of a Hilbert space. This fact was observed more than two decades later by Moore [11] and is based on the next identity, deduced from the properties that link the inner product and the norm in a Hilbert space:

$$\parallel x-x^{\ast }{\parallel }^2{+k\parallel x-Tx\parallel }^2-\parallel Tx-x^{\ast }{\parallel }^2=2\langle x-x^{\ast },x-Tx\rangle -(1-k){\parallel x-Tx\parallel }^2,$$

see [11] for more details.

An important moment in the subsequent development of demicontractive mappings was marked by considering them in a more general setting, i.e., in Banach spaces. To define demicontractive mappings in Banach spaces, it was necessary to use the notion of duality mapping, a fundamental concept in nonlinear functional analysis, which serves as the analogue in normed spaces of the inner product in Hilbert spaces.

At first sight, the definition of demicontractive mappings by (4), which involves only the norm $\parallel ·\parallel$, would have been more suitable to be transposed in Banach spaces, but this version did not offer real perspectives on further developments.

This was the main reason why Chidume in [12,13] (see also [14]) introduced demicontractive mappings in arbitrary real Banach spaces under the title Mappings Satisfying Property (A) by following Măruşter’s definition and terminology [5].

Let E be a real Banach space and let $J_q$, where $q>1$, denote the generalized duality mapping from E to $2^{E^{\ast }}$ given by

$$J_q\left(x\right)=\{f\in E^{\ast }:\langle x,f\rangle ={\parallel x\parallel }^q \mathrm{and} \parallel f\parallel ={\parallel x\parallel }^{q-1}\},$$

where $E^{\ast }$ denotes the dual space of E and $\langle ·,·\rangle$ denotes the generalized duality pairing.

It is well known that if $E^{\ast }$ is strictly convex, then $J_q$ is single-valued. In particular, $J_2$ is called the normalized duality mapping and is usually denoted by J, while the single-valued normalized duality map is denoted by j.

Definition 2

([12,13]). Let E be a real Banach space and let C be a nonempty closed convex subset of E. We say that the map $T:C\to C$ has Property (A) if $Fix\left(T\right)\ne Ø$ and T satisfies

$$\langle x-Tx,j(x-x^{\ast })\rangle \ge \lambda {\parallel x-Tx\parallel }^2,$$

(6)

for all $x\in C$, $x^{\ast }\in Fix\left(T\right)$ and for some $\lambda >0$, where $j(x-x^{\ast })\in J(x-x^{\ast })$.

It is well known (see, for example, [14]) that in a real Hilbert space, the normalized duality map is the identity map. This means that, in the particular case when E in Definition 2 is a real Hilbert space, by (6), we obtain exactly Inequality (5) appearing in the original definition of Măruşter’s [5] for demicontractive mappings.

If we look into the literature, we note that much more research has been conducted for demicontractive mappings in Hilbert spaces than in Banach spaces, as it is much easier to handle Inequality (5), or its equivalent version (4), than Inequality (6), which involves the normalized duality mapping assumed to possess some minimal properties, which are ensured by various geometric properties of Banach spaces; see [14].

Despite these difficulties, some relevant results were obtained in the later case by Chidume ([12,13,14]) and by his students and co-workers, but these will be reviewed in detail elsewhere.

Thus, Chidume [12] extended Măruşter’s results to real Banach spaces which are 2-uniformly smooth and which admit a weakly sequential continuous duality map, while in [13], he further extended the previous results to real Banach spaces that are $(m+1)$-uniformly smooth (m a positive integer) and admit a weakly sequential continuous duality map. We also mention in this context the results of Osilike [15,16], who extended Chidume’s results in [12,13] to all real q-uniformly ($q>1$) smooth Banach spaces which admit a weakly sequential continuous duality map.(Due to size limitations, the case of demicontractive mappings in Banach spaces will be the topic of a forthcoming review paper.)

3. Classical Weak and Strong Convergence Theorems in the Class of Demicontractive Mappings

In order to state the classical convergence results emerging from Măruşter [5] and Hicks and Kubicek ([4]), respectively, we recall the following notion.

Definition 3.

Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping $T:C\to C$ is said to be demiclosed at 0 on C if, for a sequence $\left\{x_n\right\}$ in C converging weakly to $u\in C$ and such that $\parallel T{x}_n\parallel \to 0$, as $n\to \infty$, one has $Tu=0$.

We state first the weak convergence theorem due to Măruşter.

Theorem 1

([5], Theorem 1). Let $T:C\to C$ be a nonlinear mapping, where C is a closed convex subset of a Hilbert space H. Suppose that T satisfies condition (A), $I-T$ is demiclosed at 0, and the sequence $\left\{x_n\right\}$ generated by

$$x_{n+1}=(1-t_n)x_n+t_{n}T{x}_n(x_0\in C)$$

(7)

with $0<a\le t_n\le b<2\lambda$ belongs to C.

Then, $\left\{x_n\right\}$ converges weakly to an element of $Fix\left(T\right)$.

Strong convergence may be achieved but with the price of an additional condition.

Theorem 2

([5], Theorem 2). Let T be as in Theorem 1. If, in addition, there is $h\in C$, where $h\ne 0$, such that

$$\langle x-Tx,h\rangle \le 0 \mathrm{for} \mathrm{all} x\in C,$$

(8)

then the sequence $\left\{x_n\right\}$ generated by (7) with $0<a\le t_n\le b<2\lambda$ and for a suitable $x_0\in C$ converges strongly to an element of $Fix\left(T\right)$.

Now we state the weak convergence result from Hicks and Kubicek [4] and note that, despite the fact Theorems 1 and 3 were discovered independently, their statements are quite similar. In our opinion, the explanation is that both results were obtained using an important convergence result for quasi-nonexpansive mappings;see [1] for more details.

Theorem 3

(Hicks and Kubicek [4]). Suppose C is a closed convex subset of a Hilbert space H. Suppose $T:C\to C$ such that:

$\left(a\right)$$Fix\left(T\right)\ne Ø$.

$\left(b\right)$ T is demicontractive with contraction coefficient k.

$\left(c\right)$ If any sequence $\left\{x_n\right\}$ converges weakly to x and $(I-T)x_n$ converges strongly to 0, then $(I-T)\left(x\right)=0$.

Then, for $v_1\in C$ and $d_n\to d$, $0<d<1-k$, the iteration process defined by

$$v_{n+1}=(1-d_n)v_n+d_{n}T{v}_n,$$

(9)

converges weakly to a fixed point of T.

We should also note that Theorem 1 is slightly more general than Theorem 3 due to the stronger assumptions on the parameter sequence $\left\{d_n\right\}$ involved in the Mann iterative scheme (9) in comparison to the assumptions on the parameter sequence $\left\{t_n\right\}$ involved in the Mann iterative scheme (7). Moreover, if we look at the relationship between the two demicontractivity constants $\lambda$ and k involved in Theorem 1 and Theorem 3, respectively, i.e.,

$$\lambda =\frac{1-k}{2},$$

one can see that the two parameter sequences $\left\{d_n\right\}$ and (9) satisfy exactly the same boundedness condition, which is due to the corresponding boundedness condition in the case of quasi-nonexpansive mappings; see Theorem 4.3 in [1].

4. Recent Fixed-Point and Common Fixed-Point Results for Demicontractive Mappings

In this section, we intend to expose some relevant recent fixed-point and common fixed-point results for single-valued demicontractive mappings. Within this category, we include the classical fixed-point and common fixed-point problems, as well as their generalizations: the split fixed-point problem, the split common fixed-point problem, the split equality fixed-point problem, the split equality common fixed-point problem, etc. As a remarkable exception, we include here (and not in the next section) the convex feasibility problem, which is an important application of the fixed-point problem, due to its fundamental role in the development of the all previous problems. In order to systematize a huge volume of literature, we have grouped the selected material on the following two main aspects:

• Considered problems that involve demicontractive mappings: the (split) fixed-point problems, the (split) common fixed-point problem, the convex feasibility problem, and the split common fixed-point problem, with some of its remarkable extensions and particular cases, etc.
• The algorithms used to solve the problems: the Krasnoselskij iteration, the Krasnoselskij–Mann iteration, the (CQ) algorithm, the projection type algorithm, cyclic projection algorithms, etc.

Subsequently, we shall also mention the type of convergence established by the surveyed results (weak convergence or strong convergence).

4.1. The Fixed-Point Problem

After the papers by Măruşter [5] and by Hicks and Kubicek [4], it took a rather long time until researchers understood the importance and generality of demicontractive mappings in the metric fixed-point theory.

Before that, some other related classes of nonexpansive type mappings, such as nonexpansive, quasi-nonexpansive, and strictly pseudocontractive mappings, which are all included in the class of demicontractive mappings, were intensively studied. Regardless, all these results (which will be surveyed separately in some forthcoming papers), especially those devoted to quasi-nonexpansive mappings, contributed to the coagulation of all the prerequisites necessary to the study of demicontractive mappings themselves.

We start with one of the first important contributions to the study of the fixed-point problem for demicontractive mappings, which came from Marino and Xu [17]. These authors used a more appropriate name to designate demicontractive mappings, i.e., k-quasi strict-pseudocontractions; see the recent paper devoted to the terminology used in fixed-point theory [18].

On the other hand, as we have seen in Section 3, the demicontractivity of the mapping T (or, similarly, its quasi-nonexpansivenes or strict pseudocontractivity) does not ensure the convergence of the Mann iteration (9), even in the case of finite dimensional spaces. Additional smoothness properties for T, such as demiclosedness, are necessary. Alternatively, a suitable modification of the algorithm could also ensure strong convergence.

Recall that, for a given closed and convex subset C of a real Hilbert space H, the nearest point projection$P_C$ from H onto C assigns to each $x\in H$ its nearest point in C, denoted by $P_{C}x$, which means that $P_{C}x$ is the unique point in C with the property

$$\parallel x-P_{C}x\parallel \le \parallel x-y\parallel , \mathrm{for} \mathrm{all} y\in C.$$

(10)

It is known (see Proposition 2.1 [17]) that if T is demicontractive, then the fixed-point set $Fix\left(T\right)$ of T is closed and convex, and hence, $P_{Fix\left(T\right)}$ is well defined.

Based on these facts, the next result provides a strong convergence theorem in the class of k-quasi strict-pseudocontractions (demicontractive mappings) obtained by considering the so-called (CQ) algorithm instead of the Mann iteration (7).

Theorem 4

( [17], Theorem 4.3). Let C be a closed convex subset of a Hilbert space H. Let $T:C\to C$ be a k-quasi strict pseudocontraction for some $0\le k<1$ and assume that the fixed-point set $Fix\left(T\right)$ of T is non-empty. Let $\left\{x_n\right\}$ be the sequence generated by the following (CQ) algorithm:

$$\left\{\begin{array}{c}{x}_0\in C \mathrm{chosen} \mathrm{arbitrarily},\hfill \\ y_n={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,\hfill \\ C_n=\{z\in C:\parallel y_n{-z\parallel }^2\le \parallel x_n{-z\parallel }^2+(1-{\alpha }_n)(k-{\alpha }_n)·\parallel x_n-T{x}_n{\parallel }^2\},\hfill \\ Q_n=\{z\in C:\langle x_n-z,x_0-x_n\rangle \ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap Q_n}{x}_0.\hfill \end{array}\right.$$

(11)

Assume that $I-T$ is demiclosed (at 0) and that the control sequence $\left\{{\alpha }_n\right\}$ is chosen so that ${\alpha }_n<1$ for all n.

Then, $\left\{x_n\right\}$ converges strongly to $P_{Fix\left(T\right)}{x}_0$.

It is easily seen that, in contrast to the assumptions of Theorem 2, the strong convergence in the above theorem was obtained only by means of the more complex (CQ) algorithm used without additionally asking the demiclosedness of the mapping $I-T$ and Property (8). However, we should also note that, in concrete applications, the computational complexity of the (CQ) algorithm may lead to very serious implementation difficulties, which are very often insurmountable, due to the need to compute at each step the involved orthogonal projections $P_{C_n\cap Q_n}$, while the Krasnoselskij–Mann iterative algorithm (7) is significantly easier to implement.

To our best knowledge, the first common fixed-point result for demicontractive mappings was established in 2008 by Maingé [19], Theorem 4.2. It extends Mărusţer’s convergence Theorem 1 from one mapping to a finite family of mappings in a Hilbert space H and uses the following inertial type algorithm:

$$\left\{\begin{array}{c}{x}_0\in H \mathrm{chosen} \mathrm{arbitrarily},\hfill \\ x_{n+1}=(1-w)v_n+w\sum _{i=0}^N{w}_i{T}_i{v}_n,n\ge 1,\hfill \\ v_n=(1-{\alpha }_n)x_n+{\theta }_n(x_n-x_{n-1}),n\ge 0,\hfill \end{array}\right.$$

(12)

where $w\in (0,1]$, $\left\{{\theta }_n\right\}$, $\left\{{\alpha }_n\right\}\subset [0,1]$, and ${\left\{w_i\right\}}_{i=0}^N\subset (0,1]$ are non-negative real numbers satisfying ${\sum }_{i=0}^N{w}_i=1$.

We note that in Berinde [1], it is explicitly proven that any convergence result for a Krasnoselkij-type fixed-point iterative algorithm in the class of demicontractive mappings in Hilbert spaces can be deduced from its counterpart in the class of quasi-nonexpansive mappings, a fact which was used implicitly long before by researchers working in this area. This fact is based on an auxiliary result (see Lemma 3.2 [1] or Lemma 2 [20]) that was previously implicitly or explicitly stated and/or used by several researchers dealing with the approximation of fixed points of demicontractive mappings in Hilbert spaces [1,5,20,21,22,23,24,25,26,27,28,29].

For other related developments on this topic, we refer the readers to Maingé [19], who obtained common fixed-point convergence results by extending Măruşter’s convergence theorem (Theorem 1) from one demicontractive mapping to a finite family of demicontractive mappings in a Hilbert space; Adamu and Adam [30]; Arfat et al. [31], where the authors investigate a fixed-point problem involving an infinite family of k-demicontractive operators in conjunction with the split common null point problems (SCNPP) in Hilbert spaces by using an accelerated variant of the hybrid shrinking projection algorithm; Maingé and Măruşter [32]; Anh et al. [33]; Charoensawan and Suparatulatorn [34]; Lin [35]; Mongkolkeha et al. [36]; Wang et al. [37]; Xiao et al. [38]; Zhu and Yao [39], etc.

4.2. The Convex Feasibility Problem

Let $C_i$, $i=1,...,m$, be a family of closed convex subsets of a Hilbert space H with a nonempty intersection, i.e., ${\bigcap }_{i=1}^m{C}_i\ne Ø$.

The convex feasibility problem (CFP), which is asking to find a point

$$x^{\ast }\in \bigcap _{i=1}^m{C}_i,$$

(13)

is an old problem; see for example Gurin et al. [40], which includes numerous important problems arising from various areas of pure and applied mathematics, physical sciences and engineering, such as approximation theory, signal processing, sensor network location problems, the inverse problem of intensity-modulated radiation therapy, dynamic emission tomographic image reconstruction, etc. This is the reason why the CFP has received much attention and has been studied extensively in the last decades.

The CFP is usually solved by employing projection-based algorithms, which use the projections onto the underlying sets $C_i$ (or, onto supersets of them) to generate a sequence of points convergent to a solution of the CFP. The geometric idea of the projection methods is to project the current iteration onto a certain set from the intersecting family and to take the next iteration on the straight line connecting the current iteration and this projection. A weight factor gives the exact position of the next iteration.

Different strategies concerning the selection of the set onto which the current iteration will be projected will give particular projection-type algorithms [26].

Due to the fact that the nearest point projections $P_{C_i}$ are of the nonexpansive type (non-expansive, firmly nonexpansive, and quasi-nonexpansive), such projection algorithms can exploit and use classical results to approximate the fixed points of nonexpansive-type mappings.

Around the year 2000, the World Mathematical Year, Prof. Măruşter—who worked mainly in computer science for almost three decades and ignored the impact of his mathematics papers—became aware of the importance of the class of demicontractive mappings he introduced about 25 years ago and started to study their potential for solving the CFP in a series of papers [24,25,26,27,41].

It appears that these are the first papers in the literature that exploit the great potential of demicontractive mappings for solving CFPs. The approach is based, as in many other instances, on the construction of a fixed-point problem that is equivalent to the original CFP in the following manner; see [25,27].

Let $P(x,i)$ denote the projection of a certain point $x\in H$ onto the set $C_i$ and also denote by $i_x$ the smallest index $i\in I=\{1,2,...,m\}$ such that

$$\parallel x-P(x,i_x)\parallel =\underset{i\in I}{max}\parallel x-P(x,i)\parallel .$$

Now, one defines a mapping $T:H\to H$ by

$$Tx=P(x,i_x),x\in H$$

and one proves, relatively easily, that $x^{\ast }\in {\bigcap }_{i=1}^m{C}_i$ if and only if $x^{\ast }\in Fix\left(T\right)$, which means that

$$\bigcap _{i=1}^m{C}_i=Fix\left(T\right).$$

Thus, the solution of the CFP is reduced to the solution of a fixed-point problem for the mapping T, for which there are many results available in the literature. We state now the main result in [25] (see also [24,26,27,41]) for other related aspects.

Theorem 5

([25], Theorem 2). Let $C_i$, $i=1,...,m$ be a family of closed and convex subsets of H such that $Int\left({\bigcap }_{i=1}^m{C}_i\right)$ is nonempty and bounded. Then, for $\lambda \in (0,2)$ and $x_0\in H$, the Krasnoselskij iteration

$$x_{n+1}=(1-\lambda )x_n+\lambda T{x}_n,n\ge 0,$$

converges strongly to a point $x^{\ast }\in {\bigcap }_{i=1}^m{C}_i$.

The proof of Theorem 5 is based on some important facts, of which we mention the following ones:

• The map T is shown to be demicontractive, which implies that the averaged map $T_{\lambda }$ defined by $T_{\lambda }x=(1-\lambda )x+\lambda Tx$ is quasi-nonexpansive;
• Contrary to the common situation in fixed-point approximation, where we always have $\lambda \in (0,1)$, for the CFP, it is necessary to work with $\lambda \in (0,2)$ due to the projections involved in the algorithm;
• The assumption $Int\left({\bigcap }_{i=1}^m{C}_i\right)\ne Ø$ is essential since, as shown by detailed examples in [27], ${\bigcap }_{i=1}^m{C}_i\ne Ø$ does not suffice;
• The convergence result in Theorem 5 is deduced by means of a convergence theorem in the class of quasi-nonexpansive mappings established in the same paper ([25], Theorem 1) and based on a regularity condition established in Lemma 1 [25], which reads as follows:

Lemma 1.

Let $C_i$, $i=1,...,m$ be a family of closed and convex subsets of H such that $Int\left({\bigcap }_{i=1}^m{C}_i\right)$ is nonempty and bounded and let $\left\{x_n\right\}$ be a sequence in H such that

$$d(x_n,C_i)\to 0asn\to \infty ,\forall i=1,\cdots ,m.$$

Then,

$$d(x_n,\bigcap _{i=1}^m{C}_i)\to 0asn\to \infty .$$

We note that the results reported in this subsection are very general and somehow ultimate in this line of research, as the demicontractive mappings appear to form one of the largest classes of discontinuous nonexpansive-type mappings for which one can approximate their fixed points by means of iterative algorithms. We are not aware of any direct extension or generalization of Theorem 5 given by other authors for the case of CFPs. However, the reference Tang et al. [20] could be considered in this context, as it is devoted to the study of the split common fixed-point problem, which is a generalization of the convex feasibility problem.

4.3. The Split Common Fixed-Point Problem for Demicontractive Mappings

The main results exposed in the previous subsection for CFPs were promptly extended by Moudafi [28] to the split common fixed-point problem (SCFP), which is a generalization of the convex feasibility problem (CFP) and of the split feasibility problem (SFP) (Moudafi’s work is a milestone of the subsequent interest of researchers in demicontractive mappings, a fact which is only in part reflected by its 172 citations in MathScinet, 297 citations in SCOPUS, 284 citations in Web of Science, and 317 citations in Google Scholar).

We first define the split feasibility problem (SFP) as follows. Let C be a closed convex subset of a Hilbert space $H_1$, Q a closed convex subset of a Hilbert space $H_2$, and $A:H_1\to H_2$ a bounded linear operator. The split feasibility problem (SFP) is asking to find a point

$$x\in CsuchthatAx\in Q.$$

(14)

Under the hypothesis that the SFP is consistent, i.e., (14) has a solution, it is possible to construct, as in the case of CFPs, a fixed-point problem equivalent to (14). Indeed, it was shown in [42] that $x\in C$ solves (14) if and only if it solves the fixed-point equation

$$x=P_C\left((I+\gamma A^{\ast }(P_Q-I)A\right)x,x\in C,$$

(15)

where $P_C$ and $P_Q$ are the orthogonal projections on the sets C and Q, respectively, I is the identity map, $\gamma$ is a positive constant, and $A^{\ast }$ denotes the adjoint of A.

Now, the Picard iteration corresponding to Problem (15) yields the so-called (CQ) algorithm, which provides the iterative sequence

$$x_{n+1}=P_C\left((I+\gamma A^{\ast }(P_Q-I)A\right)x_n,n\ge 0,$$

(16)

where $\gamma \in (0,\frac{2}{\lambda })$ and $\lambda$ is the spectral radius of the operator $A^{\ast }A$.

It is known that, for sufficiently small values of $\gamma >0$, the iteration operator appearing in the fixed-point problem (15), i.e., $P_C\left((I+\gamma A^{\ast }(P_Q-I)A\right)$, is nonexpansive.

It was the merit of Moudafi [28] to initiate the study of the split common fixed-point problem for the more general class of demicontractive mappings, which strictly includes nonexpansive mappings, direct mappings, and quasi-nonexpansive mappings.

Consider as before two Hilbert spaces $H_1$ and $H_2$; consider $A:H_1\to H_2$ a bounded linear operator and consider $U:H_1\to H_1$ and $T:H_2\to H_2$ two operators with $Fix\left(U\right)=C$ and $Fix\left(T\right)=Q$.

The two-sets SCFP is in fact a particular split fixed-point problem and consists of finding

$$x\in C \mathrm{such} \mathrm{that}Ax\in Q.$$

(17)

Denote by

$$\Gamma =\{y\in C:Ay\in Q\}$$

the solution set of the two-sets SCFP (17). We state now a strong convergence theorem for solving the SCFP.

Theorem 6

([28], Theorem 2.1). Given a bounded linear operator $A:H_1\to H_2$, let $U:H_1\to H_1$ and $T:H_2\to H_2$ be demicontractive (with constants β and γ, respectively) with nonempty fixed-point sets, $Fix\left(U\right)=C$ and $Fix\left(T\right)=Q$.

Assume that $U-I$ and $T-I$ are demiclosed at 0. If $\Gamma \ne Ø$, consider the sequence $\left\{x_n\right\}$ generated iteratively by the algorithm

$$\left\{\begin{array}{c}{x}_0\in H_{1}arbitrary,\hfill \\ u_n=x_n+\gamma A^{\ast }(P_Q-I)A{x}_n,n\ge 0,\hfill \\ x_{n+1}=(1-{\alpha }_n)u_n+{\alpha }_{n}U{u}_n,n\ge 0,\hfill \end{array}\right.$$

(18)

where $\gamma \in \left(0,\frac{1-\mu }{\lambda }\right)$, with λ being the spectral radius of the operator $A^{\ast }A$ and ${\alpha }_n\in (\delta ,1-\beta -\delta )$ for small enough $\delta >0$.

Then, $\left\{x_n\right\}$ converges strongly to a split common fixed point $x^{\ast }\in \Gamma$.

The general split common fixed-point problem (GSCFP) is formulated similarly to the case of two sets by considering two Hilbert spaces $H_1$ and $H_2$, a bounded linear operator $A:H_1\to H_2$, and two families of self mappings, ${\left\{U_i\right\}}_{i=1}^p$, ${\left\{T_j\right\}}_{j=1}^r$, $U_i:H_1\to H_1$, $T_j:H_2\to H_2$. It consists of finding

$$x^{\ast }\in \bigcap _{i=1}^{p}Fix\left(U_i\right)suchthatA{x}^{\ast }\in \bigcap _{j=1}^{r}Fix\left(T_j\right),$$

(19)

where $p,r$ are positive integers.

In the particular case $p=r=1$, the GSCFP problem (19) reduces to the two-sets SCFP problem (17) involved in Theorem 6.

Another particular case of the general split common fixed-point problem is the multiple-set split feasibility problem (MSSFP), which is formulated as

$$\mathrm{find} x^{\ast }\in \bigcap _{i=1}^p{C}_i\mathrm{such} \mathrm{that} A{x}^{\ast }\in \bigcap _{j=1}^r{Q}_j,$$

(20)

where $p,r$ are positive integers, ${\left\{C_i\right\}}_{i=1}^p$ are closed convex subsets of a Hilbert space $H_1$, ${\left\{Q_j\right\}}_{j=1}^r$ are closed convex subsets of a Hilbert space $H_2$, and $A:H_1\to H_2$ is a bounded linear operator. It is obvious that, in the particular case $p=r=1$, the MSSFP (20) reduces to the classical split feasibility problem (14).

Tang et al. [20] proposed a cyclic variant of Algorithm (18) and used it to solve the split common fixed-point Problem (17) in the class of demicontractive mappings. Their algorithm is defined in the following way:

$$\left\{\begin{array}{c}{x}_0\in H_1 \mathrm{arbitrary},\hfill \\ u_n=x_n+\gamma A^{\ast }(T_{j\left(n\right)}-I)A{x}_n,n\ge 0,\hfill \\ x_{n+1}=(1-{\alpha }_n)u_n+{\alpha }_n{U}_{i\left(n\right)}{u}_n,n\ge 0,\hfill \end{array}\right.$$

(21)

where $i\left(n\right)=n\left(modp\right)+1$, $j\left(n\right)=n\left(modr\right)+1$, and $\gamma \in \left(0,\frac{1-\mu }{\lambda }\right)$, with $\lambda$ being the spectral radius of the operator $A^{\ast }A$ and ${\alpha }_n\in (0,1)$.

The main result in [20] is based on a rather non-natural assumption in the class of demicontractive mappings involved in the algorithm, that is, on the continuity of ${\left\{U_i\right\}}_{i=1}^p$ and ${\left\{T_j\right\}}_{j=1}^r$, as it is well-known that a demicontractive mapping is, in general, not continuous.

We note that Wang and Cui [29] improved the convergence result in [20] for the cyclic Algorithm (21) by removing the continuity assumption.

For other important recent results regarding the study of the SCFP problem with its particular instances (FPP, SFP, CFP etc.), we refer the readers to Boikanyo [43], Chen [44], Chen et al. [45], Chidume et al. [46] (which deals with the split equality fixed-point problem, a generalization of the split feasibility problem), Cui [47], Cui et al. [48], Cui and Wang [49,50], Dang et al. [51], Eslamian [52,53], Eslamian et al. [54], Fan et al. [55], Gupta et al. [56], Hanjing and Suantai [57,58,59,60], He et al. [61,62,63,64], Jailoka and Suantai [65,66], Jirakitpuwapat et al. [67], Kitkuan et al. [68], Mouktonglang and Suparatulatorn [69], Padcharoen et al. [70], Qin and Wang [71], Shehu [72], Shehu and Cholamjiak [73], Shehu and Mewomo [74], Suparatulatorn et al. [75]-[76], Su and Zhao [77], Tang et al. [20,78], Wang [79], Wang and Zhao [80], Wang and Cui [29], Wang and Fang [81,82], Wang et al. [83,84,85,86,87,88], Xiao and Wang [89], Yao et al. [90,91,92,93], Ying et al. [94], Yu [95], Yu and Sheng [96], Zhao and Yao [97], Zheng et al. [98], Zhou et al. [99], Zong and Tang [100], etc.

5. Some Applications of Demicontractive Mappings for Solving Nonlinear Problems

From the diversity of nonlinear problems that were solved in connection with the class of demicontractive mappings, we shall expose in more details only two of them, namely, the variational inequality problems and equilibrium problems. For other problems, namely for minimization problems, we refer to Arfat et al. [31] and Chang et al. [101].

5.1. Variational Inequality Problems

The variational inequality theory has emerged as an important and useful branch of applied mathematics with a wide range of applications in economics, network analysis, optimization, engineering, pure and applied sciences, and so on.

The variational inequality formalism offers a unified framework for studying a large variety of problems with nonlinear analysis due to the fact that they can be recast as fixed-point problems and thus can use the available fixed-point iterative tools based on projection methods.

However, as we have seen in the previous section, the projection-type algorithms are difficult or impossible to apply in concrete problems as in several instances, it is hard to compute the projection operator or, in other instances, the projection operator cannot be computed explicitly. A feasible strategy to overcome such difficulties is to replace the projection operator by another mapping with a suitable fixed-point set and thus to solve, for instance, the variational inequality over the fixed-point set of a given operator.

We shall expose in the following some relevant results concerning the solution of variational inequalities over the set of fixed points of demicontractive mappings.

We consider, as usual in this paper, a real Hilbert space H with an inner product and induced norm denoted by $\langle ·,·\rangle$ and $\parallel · \parallel$, respectively. In this context, a mapping $F:H\to H$ is called monotone over $C\subset H$ if

$$\langle Fu-Fv,u-v\rangle \ge 0,\forall u,v\in C.$$

Given such a mapping F that is monotone over a nonempty closed convex subset C of H, the problem

$$\mathrm{find} u^{\ast }\in \mathrm{such} \mathrm{that} \langle u-u^{\ast },F{u}^{\ast }\rangle \ge 0,\forall u\in H$$

(22)

is called the variational inequality problem and is usually denoted by $VIP(C,F)$. It is well-known that (22) can be equivalently written as a fixed-point problem

$$u^{\ast }=P_C(u^{\ast }-\lambda F{u}^{\ast }),$$

where $\lambda$ is any positive real value and $P_C$ is the metric projection from H onto C.

To our best knowledge, Maingé [102] was the first one who studied the variational inequality Problem (22) in the case when $C=Fix\left(T\right)$ and T is a demicontractive mapping. Note that $VIP(C,F)$ is a well-posed problem, as it is known that if T is demicontractive, then $Fix\left(T\right)$ is a nonempty closed convex subset of H.

Consider C a closed convex subset of H, $F:C\to H$ and $T:C\to H$ a mapping with $Fix\left(T\right)\ne Ø$. In order to solve a special case of $VIP(C,F)$ (22), namely

$$\mathrm{find} u^{\ast }\in Fix\left(T\right)\mathrm{such} \mathrm{that} \langle u-u^{\ast },F{u}^{\ast }\rangle \ge 0,\forall u\in Fix\left(T\right),$$

(23)

Maingé [102] proposed the following hybrid steepest descent method:

$$x_{n+1}=P_C{T}_{w_n}{P}_C{x}_n-{\alpha }_{n}F\left(T_{w_n}{P}_C{x}_n\right),n\ge 0$$

(24)

where $\left\{w_n\right\},\left\{{\alpha }_n\right\}\subset (0,\infty )$ and $T_{w_n}=(1-w_n)I+w_{n}T$, with I being the identity map.

For this algorithm, he established a strong convergence theorem under very general assumptions on F and T.

Theorem 7

([102], Theorem 3.4). Let C be a closed convex subset of H and $T:C\to H$ a β-demicontractive mapping ($\beta <1$) with $Fix\left(T\right)\ne Ø$. Suppose that $F:C\to H$ verifies:

($i_1$) F is Hölder continuous, namely

$$\parallel Fu-Fv\parallel \le {\mu \parallel u-v\parallel }^{\gamma },\forall u,v\in C,$$

where $\mu \in (0,\infty )$ and $\gamma \in (0,1]$;

($i_2$) F is ϕ-strongly monotone, that is,

$$\langle Fu-Fv,u-v\rangle \ge \mathrm{\Phi }(\parallel u-v\parallel ),\forall u,v\in C,$$

where $\mathrm{\Phi }:[0,\infty )\to [0,\infty )$ has the following properties: Φ is continuous on $[0,\infty )$; $\mathrm{\Phi }\left(0\right)=0$, $\mathrm{\Phi }\left(\right(0,\infty \left)\right)\subset (0,\infty )$; and $\mathit{lim} {\mathit{inf}}_{t\to \infty }{t}^{-\frac{2}{(2-\gamma )}}\mathrm{\Phi }\left(t\right)>0$.

Suppose also that $\left\{{\alpha }_n\right\}$ and $\left\{w_n\right\}$ are sequences of positive real numbers such that

$\left({\alpha }_1\right)$${\alpha }_n\to 0$ as $n\to \infty$;

$\left({\alpha }_2\right)$${\sum }_{n\ge 0}{\alpha }_n=+\infty$,

and $w_n\in [w_a,w_b]$, where $w_b\in (0,1-\beta )$ and $w_a\in (0,w_b)$.

Then, the sequence generated by Algorithm (24) converges strongly to the unique solution $u^{\ast }$ of the variational inequality Problem (23).

Theorem 7 is very general and, in view of the following two properties of a demicontractive mapping $T:C\to H$ (see [2], Lemma 3.1),

$\left(i\right)$$Fix\left(T\right)=VIP(C,I-T)$; $\left(ii\right)$$Fix\left(T\right)=Fix(P_C(I-\lambda (I-T))$,

it includes in particular some important cases:

• The case of self-demicontractive mappings, when $T\left(D\right)\subset D$;
• The case when the variational operator F is strongly monotone ($\mathrm{\Phi }\left(t\right)=C{t}^2$);
• The case when the variational operator F is Lipschitizian ($\gamma =1$);
• The case when F is strongly monotone and Lipschitizian;
• The fixed-point problem for T for $F=I-T$.

For other important recent results regarding the study of VIP over the set of fixed points of demicontractive mappings and of finite families of demicontractive mappings, we refer to Maingé [22,103], Suantai and Phuengrattana [2], Thong and Hieu [104], Ogbuisi and Mewomo [105], Alakoya et al. [106], Meddahi et al. [107], Linh et al. [108], Okeke et al. [109,110], Rehman et al. [111], Sow [112,113,114], Tan et al. [115,116], Thong et al. [117,118], Thong and Hieu [104,119], Amarachi Uzor et al. [120], etc.

5.2. Equilibrium Problems

Let C be a nonempty closed convex subset of a Hilbert space H and let $F:C\times C\to R$ be a bifunction. The equilibrium problem denoted by $EP(C,F)$ consists of finding a point

$$u^{\ast }\in C \mathrm{such} \mathrm{that} F(u^{\ast },v)\ge 0,\forall v\in C.$$

(25)

The equilibrium Problem (25) is very general and includes, as special cases, various problems in nonlinear analysis, e.g., optimization problems, variational inequalities, minmax problems, Nash equilibrium problems in noncooperative games, and many others.

Due to the great generality of $EP(C,F)$, various methods for solving equilibrium problems have been studied extensively in recent years for diverse cases of bifunctions F.

Denote by $S_F$ the set of solutions of the $EP(C,F)$ (25) and assume $S_F\ne Ø$. For various applications, it is of interest to approximate a solution of the following mixed problem: find

$$u^{\ast }\in S_F\bigcap Fix\left(T\right),$$

(26)

where $T:H\to H$ is a mapping satisfying $S_F\bigcap Fix\left(T\right)\ne Ø$.

It is obvious that, in the particular case $Fix\left(T\right)\subset S_F$, the mixed Problem (26) reduces to the original equilibrium Problem (25).

In many instances, it is of interest to select a particular solution of (26), that is, to solve the problem $VIP(S_F\cap Fix\left(T\right),\mathcal{F})$:

$$\mathrm{find} x^{\ast }\in S_F\bigcap Fix\left(T\right)\mathrm{such} \mathrm{that} \langle v-x^{\ast },\mathcal{F}{x}^{\ast }\rangle \ge 0,\forall v\in S_F\bigcap Fix\left(T\right),$$

(27)

where $\mathcal{F}:C\to H$ satisfies the following two conditions:

$\left(LC\right)$$\mathcal{F}$ is Lipschitz-continuous, i.e., there exists $L>0$ such that

$$\parallel \mathcal{F}x-\mathcal{F}y\parallel \le L\parallel x-y\parallel ,\forall x,y\in C;$$

$\left(SM\right)$$\mathcal{F}$ is strongly monotone, i.e., there exists $\mu >0$ such that

$$\langle \mathcal{F}x-\mathcal{F}y,x-y\rangle \ge {\mu \parallel x-y\parallel }^2,\forall x,y\in C.$$

It is known that $VIP(SF\cap Fix(T),\mathcal{F})$ is a well-posed variational inequality problem as the existence and the uniqueness of the solution of (27) are ensured by the conditions $\left(LC\right)$ and $\left(SM\right)$ and by the fact that $S_F\bigcap Fix\left(T\right)$ is a nonempty closed and convex set.

In order to approximate the solution $x^{\ast }$ of (27), Maingé and Moudafi [23] considered the following algorithm:

$$\left\{\begin{array}{c}{x}_0\in H_1\mathrm{arbitrary},\hfill \\ \mathrm{compute}{u}_n\mathrm{such} \mathrm{that} F(u_n,y)+\frac{1}{r_n}\langle y-u_n,u_n-x_n\rangle \ge 0,\forall y\in C,\hfill \\ x_{n+1}=(1-w)v_n+wT{v}_n,n\ge 0,v_n:=u_n-{\alpha }_n\mathcal{F}{u}_n,\hfill \end{array}\right.$$

(28)

where $\{{\alpha }_n\subset [0,1)\}$, $\left\{r_n\right\}\subset (0,\infty )$ and $w\in (0,1)$.

For the needs of Algorithm (28), it is assumed that the bifunction F satisfies some usual conditions; see Lemma 2.1 [23]. Under these assumptions, $S_F$ is closed and convex, and $S_F=Fix\left(T_r\right)$, where $T_r:H\to C$, with $r>0$, is a single-valued firmly nonexpansive mapping defined by

$$T_{r}x=\{z\in C:F(z,y)+\frac{1}{r}\langle y-z,z-x_n\rangle \ge 0,\forall y\in C\}.$$

The next convergence theorem, dealing with Problem (26) in the case when T is demicontractive and demiclosed, was established by Maingé and Moudafi [23] and appears to be the first one of this type in the literature.

Theorem 8

([23], Theorem 3.5). Suppose $T:H\to H$ is β-demicontractive ($\beta <1$) and demiclosed with $S_F\cap Fix\left(T\right)\ne Ø$. Let $\mathcal{F}:H\to H$ satisfy $\left(LC\right)$ and $\left(SM\right)$. Assume the following conditions hold:

$\left(p_1\right)$$w\in \left(0,{\displaystyle \frac{1-\beta }{2}}\right)$;

$\left(p_2\right)$$\{{\alpha }_n\subset [0,1)\}$, ${\alpha }_n\to 0$ as $n\to \infty$ and ${\sum }_{n\ge 0}{\alpha }_n=\infty$;

$\left(p_3\right)$$\left\{r_n\right\}\subset [\delta ,\infty )$, where $\delta >0$.

Then, the sequences $\left\{x_n\right\}$ and $\left\{u_n\right\}$ generated by Algorithm (28) converge strongly to $x^{\ast }$, the unique solution of the variational Inequality (27).

We mention two important particular cases of Theorem 8:

(1)

If $F\equiv 0$, one has $u_n=P_C{x}_n$, and so, from the above theorem, we obtain a sequence $\left\{x_n\right\}$ that converges strongly to $x^{\ast }\in Fix\left(T\right)$, which solves the variational inequality

$$\langle v-x^{\ast },\mathcal{F}{x}^{\ast }\rangle \ge 0,\forall v\in Fix\left(T\right);$$

(2)

If $T=I$, from Theorem 8, we obtain that the sequences $\left\{x_n\right\}$ and $\left\{u_n\right\}$ generated by Algorithm (28) converge strongly to $x^{\ast }\in S_F$, which solves the variational inequality

$$\langle v-x^{\ast },\mathcal{F}{x}^{\ast }\rangle \ge 0,\forall v\in S_F;$$

For some other recent results on this topic, we refer to Abkar and Tavakkoli [121], Abkar and Shekarbaigi [122], Yao et al. [123], Hanjing and Suantai [60], Ogbuisi and Isiogugu [124], Hanjing et al. [125], Panyanak et al. [126], and Vuong et al. [127].

We end this section by mentioning another important application of the above results to convex minimization over the fixed-point set of demicontractive mappings (a topic which is not reviewed here); see Maingé [21], Mewomo [128], Okeke et al. [109], Chang et al. [101], Jailoka et al. [129], Arfat et al. [31], etc.

6. Conclusions and Future Directions of Development

1. Demicontractive mappings form one of the largest known classes of discontinuous nonexpansive-type mappings for which it is possible to approximate their fixed points by means of iterative algorithms and includes other important classes of mappings, e.g., nonexpansive mappings, quasi-nonexpansive mappings, and k-strictly pseudocontractive mappings, etc., which play a fundamental role in nonlinear analysis and were studied intensively in the literature. We mainly exposed the importance of the demicontractive mappings in solving various important nonlinear problems: fixed-point problems, common fixed-point problems, convex problems, split minimization problems, split common fixed-point problems, etc.

2. We provided a brief but substantial historical commentary on the introduction and first use and impact of demicontractive mappings in Hilbert spaces and also surveyed some important applications of the exposed theoretical results for solving convex feasibility problems, variational inequality problems, equilibrium problems, etc. With the aim of giving a better idea about the impressive literature developed mainly in the last fifteen years or so, we collected an almost exhaustive list of references strictly related to the study of various problems for single-valued demicontractive mappings in Hilbert spaces.

3. Due to the size limits of this paper, only a few algorithms used in connection to the study of single-valued demicontractive mappings in Hilbert spaces were exposed in detail, but, in order to help readers, for each surveyed topic, we also indicated a list of references related to that topic.

4. We noted that most of the obtained results on the study of single-valued demicontractive mappings in Hilbert spaces are theoretical in nature; most of the papers included as references do not accompany their results with examples that validate the set of assumptions (usually, quite numerous) in the established convergence theorems. Fortunately, in the last few years, most of the theoretical results were illustrated generally by non-trivial examples and by relevant numerical experiments. However, in the latter situation, only a few algorithms are usually considered in the numerical comparative studies.

5. One of the main challenges to incorporating single-valued demicontractive mappings in the study of various nonlinear problems has been the search for strong convergent algorithms (in infinite dimensional spaces, most of the iterative algorithms are known to converge only weakly). In order to achieve strong convergence, it was necessary to impose additional properties on the demicontractive mappings themselves (such as demiclosedness) or to design more complex algorithms. It is generally accepted that, in concrete applications, the computational complexity of such algorithms might lead to serious implementation difficulties, which are very often insurmountable.

6. It is therefore a very serious challenge to implement as many as possible computationally complex algorithms and to compare them numerically on a set of representative nonlinear problems involving demicontractive mappings. Our expectation is that the computational complexity does not always ensure fast convergence for an iterative algorithm; see for illustration the similar comparative study for enriched nonexpansive mappings (which coincide with k-strictly pseudocontractive mappings in a Hilbert space setting) in Berinde [130], where it has been shown that—at least in finite dimensional cases—the computationally simplest algorithm generally converges faster.

7. We would like to mention a few papers where the authors have included relevant examples in order to delineate the class of demicontractive mappings from other related classes (nonexpansive, quasi-nonexpansive, k-strictly pseudocontractive, etc.) and thus to illuminate the power of the proposed theoretical results. In this category are included the following works: Chidume and Măruşter [131], He and Du [132], Suantai and Phuengrattana [2], and Berinde [1].

8. There are some other classes of demicontractive-type mappings, such as the class of strongly demicontractive mappings (see Măruşter and Rus [133]) or that of asymptotically demicontractive mappings, which were also studied in the literature by various authors but which were not considered in this survey because of space limitations.

9. Having the privilege of following with a hawk’s eye the developments on demicontractive mappings over the last half a century, a clear conclusion has coagulated in this regard: there are some milestone papers that must be emphasized. In our opinion, the following researchers, works, and moments were crucial for the revival of the interest in studying demicontractive mappings at the beginning of the 21st century: Chidume (1984, [45]; 1994, [12]), Moore (1998, [11]), Măruşter (2003, [25]; 2005, [24]; 2005, [26]; 2008, [27]), Maingé (2008, [19,21,22,102,103]), Maingé and Moudafi (2008, [23]), and Moudafi (2008, [28]).

10. From the present work, it is evident that much more research has been conducted for single-valued demicontractive mappings in Hilbert spaces than in Banach spaces. Very recently, demicontractive mappings started to be studied in nonlinear settings (metric spaces, CAT(0) spaces, etc.) by appropriately transposing the concepts of linearity and convexity from linear settings (Hilbert spaces) to nonlinear settings (metric spaces); see Salisu et al. [134]. This direction of research appears to be extremely promising for the future study of demicontractive mappings in nonlinear settings; see Bantaojai et al. [135], Calderón et al. [136], and Kimura [137].

11. We can capture a better image of the developments on demicontractive mappings in the last 20 years if we search on MathScinet; by searching with the keyword “demicontractive”, one finds 203 papers indexed, with the following distribution per publication years: 2023 (10 papers); 2022 (20 papers); 2021 (21 papers); 2020 (27 papers); 2019 (23 papers); 2018 (19 papers); 2017 (20 papers); 2016 (5 papers); 2015 (7 papers); 2014 (4 papers); 2013 (3 papers); 2012 (3 papers); 2011 (3 papers); 2010 (4 papers); 2009 (1 paper); 2008 (9 papers); 2007 (3 papers); 2006 (4 papers); 2005 (3 papers); and 2004 (2 papers). Similar searches in SCOPUS and in Web of Science offer as a result 233 and 208 papers, respectively, with a similar distribution per years of publication, indicating the steady interest of researchers working in nonlinear analysis in the study of demicontractive mappings.