Developments in Modular Space Fixed Point Theory

Abstract

This survey article offers a snapshot view of the present state of fixed point theory within modular spaces, highlighting fundamental principles and their applications. The discussion primarily revolves around operators and their semigroups that satisfy pointwise asymptotic nonexpansive and contractive conditions in the modular sense, and the results can also be applied directly to Banach spaces. Utilizing the framework of regular and super-regular modular spaces, our research generalizes several established results concerning fixed points of nonlinear operators, applicable to both Banach spaces and modular function spaces. The study seeks to identify and discuss current challenges, knowledge gaps, and unresolved questions, providing insights into the potential of future research opportunities.

1. Introduction

To discuss the motivations for adopting the methods of modular spaces in fixed point theory, we shall begin with a concise overview of the origins of modular space theory. Early on, in their 1931 paper [1], Orlicz and Birnbaum noted that reliance solely on the methods of $L^p$-spaces led to various complications and, in some cases, hindered the resolution of certain non-power-type integral equations. Instead, they proposed considering spaces of measurable functions characterized by growth properties different from the power-type growth control offered by the $L^p$-norms. This resulted in the introduction of $L^{\phi }$ spaces, which later became known as Orlicz spaces, defined as follows:

$$L^{\phi }=\{x:\Omega \to \mathbb{R};\exists \lambda >0:{\int }_{\Omega }\phi \left(\lambda \right|x\left(\omega \right)\left|\right)d\mu \left(\omega \right)<\infty \},$$

(1)

where $(\Omega ,\mu )$ is a measurable space and $\phi :[0,\infty )\to [0,\infty )$ is a convex function that increases to infinity and satisfies $\phi \left(0\right)=0$, which means that it behaves somewhat similarly to power functions. However, $\phi$ may grow much faster than any power function, as illustrated by the following examples: ${\phi }_1\left(t\right)=e^t-t-1$, ${\phi }_2\left(t\right)=e^{t^2}-1$.

As a typical example of an application, consider the following Hammerstein nonlinear integral equation, which plays an important role in the elasticity theory, and is formulated as

$$x\left(t\right)={\int }_0^{1}k(t,s)\phi \left(x\left(s\right)\right)ds,$$

(2)

where $\phi \left(u\right)$ is a function which increases more rapidly than an arbitrary power function. Krasnosel’skii and Rutickii [2] provided an example of the Hammerstein operator, defined by the right-hand side of this integral equation, which does not operate within any of the $L^p$ spaces. Nevertheless, they identified an Orlicz space in which the Hammerstein operator is well-defined and possesses properties that enable the application of fixed point theorems for solving Equation (2).

A wide range of applications to differential and integral equations with non-power-type kernels contributed significantly to the development of Orlicz space theory, along with its applications and generalizations. The reader is directed to an extensive body of literature on the subject, such as [2,3,4,5,6]. The interest surrounding Orlicz spaces and their generalizations prompted Nakano to introduce the concept of modular spaces in his seminal book [7], which was further refined by Musielak and Orlicz in [8,9]. In the present paper, we will adopt their definition. Our focus will be limited to convex modulars, as they are most pertinent from the perspective of fixed point theory. For a fundamental exposition of modular space theory, the reader is directed to the monograph by Musielak [4].

Definition 1 

([4]). Let X be a real vector space. A functional $\rho :X\to [0,\infty ]$ is called a convex modular (or simply, a modular, when the context is clear) if

1. 

$\rho \left(x\right)=0$ if and only if $x=0$

2. 

$\rho (-x)=\rho \left(x\right)$

3. 

$\rho (\alpha x+\beta y)\le \alpha \rho \left(x\right)+\beta \rho \left(y\right)$ for any $x,y\in X$, and $\alpha ,\beta \ge 0$ with $\alpha +\beta =1$

The vector space $X_{\rho }=\{x\in X:\rho \left(\lambda x\right)\to 0,as\lambda \to 0\}$ is called a modular space (A more general definition of a modular is obtained when the convexity condition 3 of Definition 1 is replaced by the weaker condition $\rho (\alpha x+\beta y)\le \rho \left(x\right)+\rho \left(y\right)$ for any $x,y\in X$, and $\alpha ,\beta \ge 0$ with $\alpha +\beta =1$).

First, note that the function $\rho$, defined as

$$\rho \left(x\right)={\int }_{\Omega }\phi \left(\right|x\left(\omega \right)\left|\right)d\mu \left(\omega \right),$$

where $\phi$, $\Omega$, and $\mu$ are the same as those used in (1), satisfies the criteria of Definition 1. This confirms that Orlicz spaces are specific instances of modular spaces. The concept of modular spaces emerged as an innovative approach to systematically address Orlicz spaces, their generalizations, and other function and sequence spaces. Next, it is also important to recognize that every norm can be viewed as a modular in the sense of Definition 1. Nevertheless, modulars are fundamentally different from norms. Firstly, $\rho \left(x\right)$ can take on an infinite value for elements $x\in X_{\rho }$. Furthermore, modulars are not obliged to adhere to the triangle inequality. Such characteristics often arise in various important contexts, particularly within the theory of Orlicz spaces and their numerous generalizations. The unique attributes of modular spaces, coupled with their significant applications, have prompted the development of specialized methods and tools in modular-based fixed point theory and related approximation results over the past 35 years. Therefore, the findings of fixed point theory in modular spaces cannot be simply replaced with those from general metric space fixed point theory (see, for instance, [10,11]) or from generalized metric space theories, such as the theory of (E)-metric spaces (see, for example, [12,13]).

The concepts outlined in Definitions 2 below are well-established in the theory of modular spaces and their applications (see, for example, [6]). In particular, all properties defined in Definition 2 play a critical role in modular function space theory, which will be discussed later in this section.

Definition 2. 

Let ρ be a modular defined on a vector space X.

1. 

We say that $\left\{x_n\right\}$, a sequence of elements of $X_{\rho }$, is ρ-convergent to x, and write $x_n\stackrel{\rho }{\to }x$ if $\rho (x_n-x)\to 0$.

2. 

A sequence $\left\{x_n\right\}$ where $x_n\in X_{\rho }$ is called ρ-Cauchy if $\rho (x_n-x_m)\to 0$ as $n,m\to \infty$.

3. 

$X_{\rho }$ is called ρ-complete if every ρ-Cauchy is ρ-convergent to an $x\in X_{\rho }$.

4. 

A set $B\subset X_{\rho }$ is called ρ-closed (or simply closed) if for any sequence of $x_n\in B$, $x_n\stackrel{\rho }{\to }x$ implies that x belongs to B.

5. 

A set $B\subset X_{\rho }$ is called ρ-bounded (or simply bounded) if its ρ-diameter, defined as

$${\mathrm{diam}}_{\rho }\left(B\right)=sup\{\rho (x-y):x\in B,y\in B\}$$

is finite.

6. 

A set $B\subset X_{\rho }$ is called strongly ρ-bounded (or simply, strongly bounded) if there exists a $\beta >1$ such that $\beta B$ is bounded.

7. 

A set $K\subset X_{\rho }$ is called ρ-compact if for any sequence $\left\{x_n\right\}$ in K, there exists a subsequence $\left\{x_{n_k}\right\}$ and an $x\in K$ such that $\rho (x_{n_k}-x)\to 0$.

8. 

A ρ-ball $B_{\rho }(x,r)$ is defined by $B_{\rho }(x,r)=\{y\in X_{\rho }:\rho (x-y)\le r\}$.

The concept of $\rho$-convergence plays a significant role in fixed point theory and approximation theory within modular spaces. It is clear that if the $\rho$-limit of a sequence in a modular space exists, it is uniquely defined. Moreover, every subsequence of a $\rho$-convergent sequence also converges to the same limit. In addition, if $x_n\stackrel{\rho }{\to }x$, it follows that $x_n-y\stackrel{\rho }{\to }x-y$. Furthermore, if $x_n\stackrel{\rho }{\to }x$, $y_n\stackrel{\rho }{\to }y$, and $x_n-y_n\stackrel{\rho }{\to }0$, then it can be concluded that $x=y$ (see [14] [Proposition 2.1]). Similar to metric spaces, the $\rho$-compactness of a set $C\subset X$ ensures that C is $\rho$-closed.

It is evident that the properties outlined in Definition 2 closely resemble the analogous properties found in Banach spaces. In fact, when $\rho$ is a norm, they correspond exactly to those well-known properties. However, it is essential to proceed with caution, as—in general—some standard properties of convergence in topological vector spaces do not directly transfer to $\rho$-convergence. For instance, $x_n\stackrel{\rho }{\to }x$ does not generally imply that $\lambda x_n\stackrel{\rho }{\to }\lambda x$ for $\lambda >1$. Additionally, the $\rho$-compactness of a set does not necessarily guarantee its $\rho$-boundedness. Moreover, $\rho$-balls are not always $\rho$-closed, as illustrated in the following example.

Example 1. 

Let $X=\mathbb{R}$ and define $\rho \left(x\right)=\left|x\right|$ when $\left|x\right|<1$; otherwise, let $\rho \left(x\right)=+\infty$. Observe that ${\displaystyle x_n=1-\frac{1}{2n}\in B_{\rho }(0,1)}$, and that $x_n\stackrel{\rho }{\to }1$. However, $1\notin B_{\rho }(0,1)$ because $\rho \left(1\right)=+\infty$. Therefore, the ρ-ball $B_{\rho }(0,1)$ is not ρ-closed.

Nevertheless, in many noteworthy cases, $\rho$-balls are indeed $\rho$-closed. This highlights the significance of regular modular spaces (as discussed in Section 2), where $\rho$-balls are guaranteed to be $\rho$-closed.

It is well known, see [4], that every convex modular space can be equipped with a norm, referred to as the Luxemburg norm, which is defined as follows:

$${\parallel x\parallel }_{\rho }=inf\left\{\alpha >0:\rho \left({\displaystyle \frac{x}{\alpha }}\right)\le 1\right\}.$$

Convergence in the Luxemburg norm implies $\rho$-convergence; however, the converse is generally not true. Generally, without any additional requirements, we need to prove that $\rho \left(\lambda (x_n-x)\right)\to 0$ for every $\lambda >0$ to conclude that $\parallel x_n{-x\parallel }_{\rho }\to 0$. Consequently, these two types of convergence are typically not equivalent, with norm convergence being the stronger one. Also, the Luxemburg norm is not directly defined and can often be challenging to compute, whereas modulars are usually represented by explicit formulas, making calculations easier. Additionally, as was already demonstrated in 1990 in [15] [Example 2.15], there are mappings that are nonexpansive in the modular sense; that is, $\rho \left(T\right(x)-T(y\left)\right)\le \rho (x-y)$ (see Definition 15), but not with respect to the corresponding Luxemburg norm. For the sake of completeness, we reproduce this example in the version provided in [6] [Example 5.4]. Note that the modular defined in this example is a variable exponent Lebesgue modular (see, e.g., [16]), an important class of modular spaces that falls within the scope of the theory discussed in this paper.

Example 2. 

Let $X=(0,\infty )$ and Σ be the σ-algebra of all Lebesgue measurable subsets of X. Let $\mathcal{P}$ denote the δ-ring of subsets of finite measure. Define a function modular by

$$\rho \left(f\right)={\displaystyle \frac{1}{e^2}}{\int }_0^{\infty }{\left|f\left(x\right)\right|}^{x+1}dm\left(x\right).$$

Let B be the set of all measurable functions $f:(0,\infty )\to \mathbb{R}$ such that $0\le f\left(x\right)\le 1/2$. Consider the map

$$T\left(f\right)\left(x\right)=\left\{\begin{array}{cc}f(x-1),\hfill & for x\ge 1,\hfill \\ 0,\hfill & for x\in [0,1].\hfill \end{array}\right.$$

Clearly, we have $T\left(B\right)\subset B$. For every $f,g\in B$ and $\lambda \le 1$, we have

$$\rho \left(\lambda \left(T\left(f\right)-T\left(g\right)\right)\right)\le \lambda \rho \left(\lambda (f-g)\right),$$

which implies that T is ρ-nonexpansive. On the other hand, if we take $f=1_{[0,1]}$, where $1_{[0,1]}$ stands for the characteristic function of set $[0,1]$, then

$${\parallel T\left(f\right)\parallel }_{\rho }>e\ge {\parallel f\parallel }_{\rho },$$

which clearly implies that T is not ${\parallel .\parallel }_{\rho }$-nonexpansive. Note that T is linear.

As further demonstrated in [15] [Proposition 2.14], in the context of left-continuous convex modulars, it is only the significantly stronger condition

$$\rho \left(\lambda \right(T\left(x\right)-T\left(y\right)\left)\right)\le \rho \left(\lambda \right(x-y\left)\right)$$

for all $\lambda >0$, which generally guarantees the norm nonexpansiveness condition specified as

$${\parallel T\left(x\right)-T\left(y\right)\parallel }_{\rho }\le {\parallel x-y\parallel }_{\rho }.$$

Recall that a modular $\rho$ is called left-continuous if ${lim}_{\lambda \to 1^{-}}\rho \left(\lambda x\right)=\rho \left(x\right)$ for all $x\in X_{\rho }$; see [4] [Definition 1.7]. As we shall see, all modulars used in the context of this paper are left-continuous.

Given that nonexpansiveness and its generalizations are fundamental concepts in metric fixed point theory, these limitations of the norm ${\parallel ·\parallel }_{\rho }$ highlight the significance of adopting the modular approach.

Moreover, in numerous instances, an operator or a collection of operators can give rise to a modular space where these operators exhibit essential characteristics such as continuity, boundedness, or nonexpansiveness. For further details, see [6], as well as Example 7 in Section 5 of the current article.

From an application perspective, the growing interest in modulars stems from their similarity to norms, while they may lack properties such as the triangle inequality and homogeneity. Such ‘generalized’ norms are vital in empirical scientific pursuits that require objective differentiation among fundamentally different classes of objects—a common challenge in the empirical sciences (see, for example, [17,18]). They play a significant role in contemporary clustering techniques and artificial intelligence algorithms.

Consequently, due to these and similar considerations, standard practice in fixed point theory within modular spaces is to express all conditions placed on operators—such as various forms of nonexpansiveness or uniform convexity—exclusively in modular terms.

Fixed point theory in Banach spaces has been actively developed and applied since the 1920s, following the groundbreaking contributions of Banach [19], Browder [20], Kirk [21], Göhde [22], and many others. Numerous resources, including books [10,23,24,25,26,27,28], provide extensive insights into fixed point theory in Banach spaces. In contrast, the abstract theory of modular spaces, as defined in Definition 1, encounters notable limitations when compared to the rich geometry of Banach spaces; for instance, it lacks a corresponding notion of weak topology. Consequently, in its abstract form, it has been insufficient for developing a robust fixed point theory. This gap has led to the evolution of fixed point theory in modular function spaces as a dynamic research domain, initiated in [15]; see also [6] and numerous subsequent publications. As we will explore in detail below, elements of modular function spaces are measurable functions; thus, the structure of these spaces is much richer than that of abstract modular spaces. Consequently, modular function spaces serve as a natural generalization of spaces such as $L^p$, ${\ell }^p$, Orlicz spaces, Musielak–Orlicz spaces, Lorentz and Marcinkiewicz spaces, variable exponent Lebesgue spaces, and many others. These facts significantly contributed to the success of fixed point theory in modular function spaces.

The theory of modular function spaces began with the book [29] and a series of earlier papers referenced therein, later adapting to the requirements of fixed point theory as outlined in [6]. Below, we adopt this latter approach to provide an overview of the theory.

Let $\Omega$ be a nonempty set, and $\Sigma$ be a nontrivial $\sigma$-algebra of subsets of $\Omega$. Let $\mathcal{P}$ be a nontrivial $\delta$-ring of subsets of $\Omega$, which means that $\mathcal{P}$ is closed with respect to forming countable intersections, finite unions, and differences. Assume further that $E\cap A\in \mathcal{P}$ for any $E\in \mathcal{P}$ and $A\in \Sigma$. Let us assume that there exists an increasing sequence of sets $K_n\in \mathcal{P}$ such that $\Omega =\bigcup K_n$. By $\mathcal{E}$ we denote the linear space of all simple functions with supports from $\mathcal{P}$. By ${\mathcal{M}}_{\infty }$ we will denote the space of all extended measurable functions, that is, all functions $f:\Omega \to [-\infty ,\infty ]$ such that there exists a sequence $\left\{g_n\right\}\subset \mathcal{E}$, $|g_n|\le |f|$ and $g_n\left(\omega \right)\to f\left(\omega \right)$ for all $\omega \in \Omega$. By $1_A$, we denote the characteristic function of the set A.

Definition 3 

([6] Definition 3.1). Let $\rho :{\mathcal{M}}_{\infty }\to [0,\infty ]$ be a nontrivial, convex, and even function. We say that ρ is a regular convex function pseudomodular if:

(a) 

$\rho \left(0\right)=0$;

(b) 

ρ is monotone, that is, $\left|f\right(\omega \left)\right|\le \left|g\right(\omega \left)\right|$ for all $\omega \in \Omega$ implies $\rho \left(f\right)\le \rho \left(g\right)$, where $f,g\in {\mathcal{M}}_{\infty }$.

(c) 

ρ is orthogonally subadditive, that is, $\rho \left(f{1}_{A\cup B}\right)\le \rho \left(f{1}_A\right)+\rho \left(f{1}_B\right)$ for any $A,B\in \Sigma$ such that $A\cap B\ne \mathfrak{\varnothing }$, where $f\in {\mathcal{M}}_{\infty }$.

(d) 

ρ has the Fatou property: $|f_n\left(\omega \right)|\uparrow |f\left(\omega \right)|$ for all $\omega \in \Omega$ implies $\rho \left(f_n\right)\uparrow \rho \left(f\right)$, where $f\in {\mathcal{M}}_{\infty }$.

(e) 

ρ is order continuous in $\mathcal{E}$, that is, $g_n\in \mathcal{E}$ and $|g_n\left(\omega \right)|\downarrow 0$ implies $\rho \left(g_n\right)\downarrow 0$.

Similarly to the case of measure spaces, we say that a set $A\in \Sigma$ is $\rho$-null if $\rho \left(g{1}_A\right)=0$ for every $g\in \mathcal{E}$. We say that a property holds $\rho$-almost everywhere if the exceptional set is $\rho$-null. As usual, we identify any pair of measurable sets whose symmetric difference is $\rho$-null as well as any pair of measurable functions differing only on a $\rho$-null set. With this in mind, we define

$$\mathcal{M}=\{f\in {\mathcal{M}}_{\infty }:|f\left(\omega \right)|<\infty \rho -a.e\},$$

where each $f\in \mathcal{M}$ is actually an equivalence class of functions equal $\rho$-a.e. rather than an individual function.

Definition 4 

([6] Definition 3.2). Let ρ be a regular convex function pseudomodular. We say that ρ is a regular convex function modular if $\rho \left(f\right)=0$ implies that $f=0\rho -a.e.$

It is straightforward to verify that every regular convex function modular is a convex modular in the sense of Definition 1. Consequently, the modular space defined by such a function modular $\rho$ is referred to as a modular function space, traditionally denoted by $L_{\rho }$ rather than the more generic notation $X_{\rho }$ used throughout this paper. We consider both notations to be equivalent in the context of modular function spaces, but will reserve $X_{\rho }$ for discussions outside of this context.

In the next proposition, following [6], we collect some fundamental properties of modular function spaces. In the subsequent sections, we will see how these properties are reflected in abstract modular spaces within the framework introduced there, which will be discussed throughout the remainder of this paper.

Proposition 1. 

In modular function spaces, the following statements are true:

(a) 

$L_{\rho }$ is complete with respect to ρ-convergence.

(b) 

If $f_n\to f\rho -a.e.$ then $\rho \left(f\right)\le {lim\; inf}_{n\to \infty }\rho \left(f_n\right)$. This property is a direct consequence of the Fatou property.

(c) 

If $f_n\to f\rho -a.e.$ then there exists a subsequence $\left\{f_{n_k}\right\}$ of $\left\{f_n\right\}$ such that $f_{n_k}\to f$ρ-a.e.

It is fairly straightforward to recognize that many fixed point theorems established in the context of modular function spaces are analogs of classical results in Banach spaces. In this setting, function modulars serve the role of norms, while $\rho$-almost everywhere convergence can be interpreted as a substitute for convergence in the weak topology. Furthermore, property $\left(R\right)$ (as defined in Definition 33) becomes analogous to reflexivity, and so on. However, since not all Banach spaces qualify as modular function spaces, there is a need for a framework encompassing both normed and modular function spaces. This framework is provided by regular modular spaces, introduced in Section 2.1 as modular spaces $X_{\rho }$ that are $\rho$-complete, with a convex, $BC$-regular modular $\rho$. While many results in Banach spaces rely on weak topology, which parallels $\rho$-a.e. convergence in modular function spaces, we also employ the framework of super-regular spaces [30], introducing L-convergence aligned with the modular structure. These conditions are satisfied in Banach spaces, where norms are examples of super-regular modulars, and in modular function spaces with the Fatou property. It is important to recognize that super-regular modular spaces encompass spaces beyond Banach and modular function spaces, as illustrated by modular spaces defined by $\varphi$-variations, see Example 5.

To summarize the challenge addressed in this paper and its innovative contributions, we highlight the following key points:

• The paper provides a systematic survey of existing yet often fragmented fixed point results within the context of regular and super-regular modular spaces.
• The significance and novelty of the proposed framework lie in its ability to encompass both Banach and modular function spaces (and beyond), serving as a common denominator for theories that have previously been developed in isolation.
• Practically, this consolidation aims to eliminate unnecessary duplication and fragmentation within fixed point theory while maintaining the advantages of both norm and modular approaches.
• Notably, such unification also enhances our understanding of fixed point theory in Banach spaces and modular function spaces by introducing additional conceptual dimensions. For example, it clarifies the role of reflexivity by substituting it with the intuitive concept of property $\left(R\right)$, and it explains why weak convergence and $\rho$-almost everywhere convergence are so prominent.
• Lastly, the introduced framework demonstrates that $BC$-regularity (i.e., the $\rho$-closedness of $\rho$-balls) along with $\rho$-completeness constitutes the minimal requirement for a class of convex modular spaces to include both Banach and modular function spaces. It is important to note that, as shown in Proposition 2, $BC$-regularity is equivalent to the Fatou property.

This paper serves as a survey article, emphasizing the relationships among various properties and results rather than delving into their technical details. Our primary interest lies in identifying both similarities and differences, where applicable, among results from various fields. While the proofs of the theorems are omitted, relevant references are consistently provided. The aim is to offer a guide for readers interested in conducting research in this complex and often confusing field.

This paper is organized as follows: Section 2, ‘Modular Spaces’, offers the necessary background on regular modular spaces in Section 2.1, and then introduces super-regular modular spaces in Section 2.2. Section 3 deals with fixed point theorems for nonlinear operators acting in such spaces: Section 3.1 is devoted to contractive and nonexpansive mappings, while Section 3.2 is devoted to the foundations of the theory for asymptotic and pointwise asymptotic mappings. Section 4 surveys key results related to the existence of common fixed points for semigroups of nonlinear mappings operating in regular and super-regular modular spaces. Section 5 discusses fixed point construction processes in the context of super-regular modular spaces. Finally, Section 6, ‘Discussion’, summarizes the contents of this survey paper, pointing to the main notions and results discussed in the article. It also discusses gaps and open questions and identifies future research opportunities.

2. Regular and Super-Regular Modular Spaces

In this section, we present the framework of regular and super-regular spaces. The concepts discussed and their inter-relationships will lay the necessary foundations for the fixed point considerations in the following sections.

2.1. Regular Modular Spaces

In the Introduction, we highlighted the critical importance of $\rho$-closedness of $\rho$-balls. This concept is now formalized in the definition of $BC$-regularity presented below. Note that, based on this definition, every norm is a $BC$-regular modular.

Definition 5 

([31] Definition 2.3). A convex modular is called $BC$-regular if every ρ-ball $B_{\rho }(x,r)$, where $x\in X_{\rho }$, and $r>0$, is ρ-closed. In this context, we also refer to the modular space $X_{\rho }$ as possessing property $\left(BC\right)$.

The following result offers an important characterization of $BC$-regularity. It follows from Proposition 1 that every convex regular function modular satisfies condition 2 of Proposition 2 and, consequently, every function modular is $BC$-regular.

Proposition 2 

([14] Proposition 2.2). The following two conditions are equivalent:

1. 

ρ is $BC$-regular;

2. 

${\displaystyle \rho \left(x\right)\le \underset{n\to \infty }{lim\; inf}\rho \left(x_n\right)}$ provided $x_n\stackrel{\rho }{\to }x$.

Observe that, due to Proposition 2, given a $BC$-regular modular $\rho$ and a sequence of positive numbers $\left\{{\lambda }_n\right\}$ such that ${\lambda }_n\to 1^{-}$, we have

$$\rho \left({\lambda }_{n}x\right)\le {\lambda }_n\rho \left(x\right)\le \rho \left(x\right)\le \underset{k\to \infty }{lim\; inf}\rho \left({\lambda }_{k}x\right),$$

which implies that $\rho \left(x\right)={lim}_{\lambda \to 1^{-}}\rho \left(\lambda x\right)$ and, consequently, that $\rho$ is a left-continuous modular. Since all modulars discussed throughout the rest of this paper are assumed to be $BC$-regular, they will necessarily be left-continuous, as noted in the Introduction.

For the sake of brevity, the term ‘regular modular spaces’ is commonly used to refer to $\rho$-complete $BC$-regular modular spaces, as articulated in the following definition.

Definition 6 

([31] Definition 2.4). A modular space $X_{\rho }$ is called a regular modular space if ρ is a convex, $BC$-regular modular, and $X_{\rho }$ is ρ-complete.

In light of the above remarks, it is clear that the class of regular modular spaces includes all Banach spaces (where $\rho$ denotes a norm) and all modular function spaces, such as Lebesgue spaces, Orlicz spaces, Musielak–Orlicz spaces, and variable exponent Lebesgue spaces.

Uniform convexity plays a crucial role in Banach space theory, particularly in fixed point theory for mappings within Banach spaces. For example, it is a critical assumption in Browder’s fixed point theorem [20] [Theorem 1]. Thus, it is not surprising that we will need a modular equivalent of this essential property. The literature presents multiple, not always equivalent, definitions of modular uniform convexity (see, for example, [6,31,32]). In the context of regular modular spaces, the strongest version—often referred to as the $\left(UUC1\right)$ property—is commonly employed because, in the case when $\rho$ is a norm, it is equivalent to the standard definition of uniform convexity in normed spaces. In this paper, we adopt this perspective, as outlined in the following definition.

Definition 7. 

Let $X_{\rho }$ be a regular modular space. Let $r>0,\epsilon >0$. Set

$$D_1(r,\epsilon )=\{(x,y):x,y\in X_{\rho },\rho \left(x\right)\le r,\rho \left(y\right)\le r,\rho (x-y)\ge \epsilon r\}.$$

Let

$${\delta }_1(r,\epsilon )=inf\left\{1-{\displaystyle \frac{1}{r}}\rho \left({\displaystyle \frac{x+y}{2}}\right):(x,y)\in D_1(r,\epsilon )\right\},if{D}_1(r,\epsilon )\ne \mathfrak{\varnothing },$$

${\delta }_1(r,\epsilon )=1if{D}_1(r,\epsilon )=\mathfrak{\varnothing }$. We say that $X_{\rho }$ is uniformly convex if, for every $s\ge 0,\epsilon >0$, there exists

$${\eta }_1(s,\epsilon )>0,$$

depending only on s and ε such that

$${\delta }_1(r,\epsilon )>{\eta }_1(s,\epsilon )>0forr>s.$$

The significance of the above definition of modular uniform convexity is exemplified by the following discussion drawn from the theory of Orlicz spaces.

Example 3 

([33] Example 3). It is known that in Orlicz spaces, the Luxemburg norm is uniformly convex if and only if φ is uniformly convex and the ${\Delta }_2$ property holds. Furthermore, it is recognized that, under appropriate conditions, modular uniform convexity in Orlicz spaces is equivalent to the Orlicz function being very convex [5,34]. Note that a function φ is termed very convex if, for every $\epsilon >0$ and any $x_0>0$, there exists a $\delta >0$ such that

$$\phi \left({\displaystyle \frac{1}{2}}(x-y)\right)\ge {\displaystyle \frac{\epsilon }{2}}\left(\phi \left(x\right)+\phi \left(y\right)\right)\ge \epsilon \phi \left(x_0\right),$$

implies

$$\phi \left({\displaystyle \frac{1}{2}}(x+y)\right)\le {\displaystyle \frac{1}{2}}(1-\delta )\left(\phi \left(x\right)+\phi \left(y\right)\right).$$

Typical examples of Orlicz functions that do not satisfy the ${\Delta }_2$ condition but are very convex are: ${\phi }_1\left(t\right)=e^{\left|t\right|}-\left|t\right|-1$ and ${\phi }_2\left(t\right)=e^{t^2}-1$, [2,35]. Therefore, these are the examples of regular modular spaces that are not uniformly convex in the Luxemburg norm sense, and hence the classical Banach space fixed point theorems cannot be easily applied. However, these spaces are uniformly convex in the modular sense, and respective modular fixed point results can be applied.

We will also employ a weaker variant of uniform convexity, referred to as uniform convexity in every direction, which serves as a generalization of this concept within Banach spaces; see our Theorem 4 and the commentary around it.

Definition 8 

([15] Definition 3.9; [36] Definition 3.6). For any nonzero $u\in X_{\rho }$ and $r>0$, we define the r-modulus of uniform convexity of ρ in the direction of u as

$$\delta (r,u)=inf\left\{1-{\displaystyle \frac{1}{r}}\rho \left(y+{\displaystyle \frac{1}{2}}u\right)\right\},$$

where the infimum is taken over all $y\in X_{\rho }$ such that $\rho \left(y\right)\le r$ and $\rho (y+u)\le r$.

We say that $X_{\rho }$ is uniformly convex in every direction $\left(UCED\right)$ if $\delta (r,u)>0$ for every nonzero $u\in X_{\rho }$ and all $r>0$.

By a straightforward calculation, we find that $\delta (r,u)\ge {\delta }_1(r,\epsilon )$ for any $\epsilon >0$ such that $\rho \left(u\right)\ge r\epsilon$. Therefore, uniform convexity of a regular modular space implies that it is uniformly convex in every direction. As is known, even in the Banach space case, the converse is not true.

2.2. Super-Regular Modular Spaces

In regular modular spaces, there is no equivalent to the weak topology found in Banach spaces, nor is there a counterpart to $\rho$-almost everywhere convergence that plays a similar role in modular function spaces. Both concepts are fundamental to fixed point theory in their respective contexts, necessitating the development of an analogous framework for studying the existence of fixed points and the convergence of fixed point approximation processes in regular modular spaces. This brings us to the concept of super-regular modular spaces, introduced in [30]. These spaces are defined as modular spaces equipped with a sequential convergence structure, as proposed by the author in [37,38], and based on the framework of L-spaces developed by Kisyński in [39] (see also [40]), which builds upon earlier work by Fréchet [41] and Urysohn [42]. This setting allows for the utilization of convergence types that are not inherently tied to a topology, with convergence almost everywhere serving as a notable example.

Definition 9 

([37] Definition 2.4). Let X be any nonempty set. A relation ζ between sequences ${\left\{x_n\right\}}_{n=1}^{\infty }$ of elements of X and elements x of X, denoted by $x_n\stackrel{\zeta }{\to }x$, is called a sequential convergence on X if

1. 

if $x_n=x$ for all $n\in \mathbb{N}$ then $x_n\stackrel{\zeta }{\to }x$,

2. 

if $x_n\stackrel{\zeta }{\to }x$ and $\left\{x_{n_k}\right\}$ is a proper subsequence of $\left\{x_n\right\}$, then $x_{n_k}\stackrel{\zeta }{\to }x$.

The pair $(X,\zeta )$ (in short, X) is called a convergence space.

Given a sequential convergence $\zeta$ on X, we can introduce notions of closed and sequentially compact sets.

Definition 10 

([37] Definition 2.5). Let $(X,\zeta )$ be a convergence space. A set $K\subset X$ is called closed if whenever $x_n\in K$ all $n\in \mathbb{N}$ and $x_n\stackrel{\zeta }{\to }x$, then $x\in K$. Similarly, K is called sequentially compact if from every sequence $\left\{x_n\right\}$ of elements of K we can choose a subsequence $\left\{x_{n_k}\right\}$ such that $x_{n_k}\stackrel{\zeta }{\to }x$ for an $x\in K$.

Definition 11 

([37] Definition 2.6). A sequential convergence ζ is called an L-convergence on X if $x_n\stackrel{\zeta }{\to }x$ and $x_n\stackrel{\zeta }{\to }y$ imply that $x=y$. In this case, the pair $(X,\zeta )$ is called an L-space.

Let us define $LTI$-convergence, $LTI$-spaces, and modulated $LTI$-spaces.

Definition 12 

([37] Definition 2.8). Let X be a real vector space and let ζ be an L-convergence on X. We say that ζ is an $LTI$-convergence (translation invariant convergence) if $x_n\stackrel{\zeta }{\to }x$ implies that $x_n-y\stackrel{\zeta }{\to }x-y$ for any $y\in X$. In this case, the pair $(X,\zeta )$ is called an $LTI$-space.

Definition 13 

([37] Definition 2.9). Let ρ be a modular defined on X and let ζ be an L-convergence on $X_{\rho }$. The triplet $(X_{\rho },\rho ,\zeta )$ is called a modulated $LTI$-space if $(X_{\rho },\zeta )$ is an $LTI$-space and the following two conditions are satisfied:

1. 

${\displaystyle x_n\stackrel{\zeta }{\to }x\Rightarrow \rho \left(x\right)\le \underset{n\to \infty }{lim\; inf}\rho \left(x_n\right)}$,

2. 

if $x_n\stackrel{\rho }{\to }x$ then there exists a sub-sequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that $x_{n_k}\stackrel{\zeta }{\to }x$, where $x,x_n\in X$.

The statements in the following Proposition are straightforward consequences of the relevant definitions.

Proposition 3 

([30] Proposition 1). Let $(X_{\rho },\rho ,\zeta )$ be a modulated $LTI$-space. Then the following assertions are true.

1. 

Every ζ-closed set is also ρ-closed.

2. 

Every ρ-compact set is also sequentially ζ-compact.

3. 

Every ρ-ball $B_{\rho }(x,r)$ is ζ-closed (and hence also ρ-closed).

4. 

Every sequentially ζ-compact set is ζ-closed.

5. 

Every ζ-closed subset of a sequentially ζ-compact set is sequentially ζ-compact.

It follows from item 3 of Proposition 3 that every $\rho$-complete modulated $LTI$-space is regular. For the sake of simplicity, let us introduce the following definition:

Definition 14 

([30] Definition 10). By a super-regular modular space $(X_{\rho },\zeta )$ we will understand a ρ-complete modulated $LTI$-space $(X_{\rho },\rho ,\zeta )$.

Example 4. 

Typical examples of super-regular modular spaces include the following:

(a) 

Banach spaces where ρ is a norm and ζ represents convergence in the weak topology;

(b) 

Modular function spaces with the Fatou property and ζ being ρ-almost everywhere convergence;

(c) 

Lebesgue spaces, variable exponent Lebesgue spaces, Orlicz spaces, Musielak–Orlicz spaces with ζ corresponding to almost everywhere convergence with respect to a measure;

(d) 

Orlicz–Sobolev spaces with ζ corresponding to almost everywhere convergence with respect to a measure of all involved generalized derivatives.

The next example merits special attention as it demonstrates that the class of super-regular modular spaces encompasses certain classical modular spaces that are neither normed nor categorized as modular function spaces. It revisits the notion of $\phi$-variation, which was introduced by Musielak and Orlicz in [43] as a generalization of the classical quadratic variation defined by Wiener over a century ago [44]. The $\rho$-convergence illustrated here, referred to in the literature as convergence in $\phi$-variation, has been utilized in numerous applications.

Example 5 

([33] Example 2). Let $\phi :[0,\infty )\to [0,\infty )$ be a convex function such that $\phi \left(t\right)=0$ if and only if $t=0$. Let X be a space of all real-valued functions defined in the interval $[a,b]$ and vanishing at $t=a$. Musielak and Orlicz defined in [43] a φ-variation of a function $x\in X$ as follows:

$$\rho \left(x\right)=\underset{\Pi }{sup}\sum _{i=1}^{\infty }\phi \left(\right|x\left(t_i\right)-x\left(t_{t-1}\right)\left|\right),$$

where the supremum is taken over all partitions $\Pi :a=t_0<t_1<\dots <t_m=b$ of the interval $[a,b]$. It is easy to see that ρ is a convex modular on X and that the value of $\rho \left(x\right)$ may be infinite. Using results of [43] and [4] [Properties 10.7], it is straightforward to demonstrate that $(X_{\rho },\zeta )$ is a super-regular modular space, where ζ is the pointwise convergence over the interval $[a,b]$. The space $X_{\rho }$ is not a modular function space because φ-variation is not monotone.

3. Fixed Points of Nonlinear Operators in Modular Spaces

In this section, we present a comprehensive overview of the significant fixed point results for nonlinear operators operating within regular and super-regular modular spaces.

3.1. Contractions and Nonexpansive Mappings in Regular Modular Spaces

First, let us revisit the definitions of Lipschitzian mappings, contractions, and nonexpansive mappings within the context of regular modular spaces. It is important to note that these definitions are direct generalizations of the corresponding concepts from Banach spaces.

Definition 15 

([31] Definition 2.7). Let $X_{\rho }$ be a regular modular space and let $C\subset X_{\rho }$ be convex, nonempty, ρ-closed, and ρ-bounded. A mapping $T:C\to C$ is called

(i) 

Lipschitzian if there exists $\alpha >0$ such that

$$\rho \left(T\right(x)-T(y\left)\right)\le \alpha \rho (x-y)foranyx,y\in C.$$

(ii) 

a contraction if it is Lipschitzian with $\alpha <1$.

(iii) 

nonexpansive if it is Lipschitzian with $\alpha =1$.

An element $x\in C$ is called a fixed point of T whenever $T\left(x\right)=x$. The set of fixed points of T will be denoted by $F\left(T\right)$.

As a straightforward corollary to Definition 15, the following characterization of the sets of fixed points for Lipschitzian mappings in regular modular spaces is obtained.

Proposition 4 

([31] Proposition 2.8). $F\left(T\right)$ is ρ-closed for every Lipschitzian mapping T.

Variants of the renowned Banach Contraction Principle are regarded as vital tools for demonstrating the existence and uniqueness of solutions characterized as fixed points of self-mappings defined on metric, normed, and modular function spaces. Additionally, these theorems provide constructive approaches for locating such fixed points. Our next result provides a version of this esteemed theorem tailored to the context of regular modular spaces.

Theorem 1 

([33] Theorem 1). Let $X_{\rho }$ be a regular modular space, and let $C\subset X_{\rho }$ be nonempty, ρ-closed, and ρ-bounded. Let $T:C\to C$ be a contraction. Then, T has a unique fixed point $\overline{x}\in C$. Moreover, for any $x\in C$, $\rho \left(T^n\left(x\right)-\overline{x}\right)\to 0$ as $n\to \infty$, where $T^n$ is the n-th iterate of T.

Since Kirk’s seminal 1965 paper [21], the concept of normal structure has been crucial in the study of fixed point theory for nonexpansive operators in Banach spaces. Consequently, it is logical to adapt this technique for proving the existence of fixed points in super-regular modular spaces.

Definition 16 

([36] Definition 3.1). Let C be a ρ-bounded subset of $X_{\rho }$.

(1) 

The quantity $r_{\rho }(x,C)=sup\{\rho (x-y):y\in C\}$ will be called the $\rho$-Chebyshev radius of C with respect to x.

(2) 

The $\rho$-Chebyshev radius of C is defined by $R_{\rho }\left(C\right)=inf\{r_{\rho }(x,C):x\in C\}$.

Note that $R_{\rho }\left(C\right)\le r_{\rho }(x,C)\le {\mathrm{diam}}_{\rho }\left(C\right)$, for any $x\in C$ and any $\rho$-bounded nonempty subset C of $X_{\rho }$.

Let C be a $\rho$-bounded subset of $X_{\rho }$ such that ${\mathrm{diam}}_{\rho }\left(C\right)>0$. And let $\mathcal{A}$ be a class of subsets of C. With this in mind, let us introduce the following two definitions adapted from [36] [Definition 4.2].

Definition 17. 

A class $\mathcal{A}$ is said to be ρ-normal if for each $A\in \mathcal{A}$, not reduced to a single point, we have $R_{\rho }\left(A\right)<{\mathrm{diam}}_{\rho }\left(A\right)$.

Definition 18. 

We say that $\mathcal{A}$ is countably compact if any decreasing sequence ${\left\{A_n\right\}}_{n\ge 1}$ of nonempty elements of $\mathcal{A}$ has a nonempty intersection.

Our next definition extends the standard definition of normal structure to the class of super-regular modular spaces.

Definition 19 

([30] Definition 13). We say that $(X_{\rho },\zeta )$ possesses the ζ-normal structure if every nonempty, convex, ρ-bounded, ζ-sequentially compact set $C\subset X_{\rho }$ with ${\mathrm{diam}}_{\rho }\left(C\right)>0$ has a ρ-nondiametral point $x_0\in C$; that is,

$$r_{\rho }(x_0,C)=sup\{\rho (x_0-y):y\in C\}<{\mathrm{diam}}_{\rho }\left(C\right).$$

It is well known that weak normal structure—characterized by $\rho$ as a norm in a Banach space and $\zeta$ representing convergence in the weak topology—implies the weak fixed point property; however, the converse does not hold true. We will demonstrate that this situation extends to the broader context of super-regular modular spaces.

The proof of the generic existence theorem relies on a technical result that serves as a modular version of Kirk’s lemma [45] [Lemma 3]. This lemma, in turn, is an abstraction of a result originally established by Gillespie and Williams [46].

Lemma 1 

([47] Lemma 4.3). Let C be a ρ-bounded subset of $X_{\rho }$. Let $\mathcal{A}$ be a class of subsets of C that is stable under arbitrary intersections and contains all sets of the form $C\cap B_{\rho }(x,p)$, where $x\in C$ and $p>0$. Suppose $T:C\to C$ is ρ-nonexpansive. Then for each $\epsilon >0$ there exists $C_{\epsilon }\in \mathcal{A}$ such that $T\left(C_{\epsilon }\right)\subset C_{\epsilon }$ and for which

$${\mathrm{diam}}_{\rho }\left(C_{\epsilon }\right)\le R_{\rho }\left(C\right)+\epsilon {\mathrm{diam}}_{\rho }\left(C\right).$$

Using the above technical lemma, we can establish the following result, which generalizes a fixed point theorem for modular function spaces [6] [Theorem 5.9] and extends Kirk’s theorem [45] [Theorem I] to super-regular modular spaces. The proof of this result follows an identical path as demonstrated in [47].

Theorem 2 

([47] Theorem 4.4). Let C be a ρ-bounded subset of $X_{\rho }$. Let $\mathcal{A}$ be a class of subsets of C which is stable under arbitrary intersections and contains all sets of the form $C\cap B_{\rho }(x,p)$, where $x\in C$ and $p>0$. In addition, let us assume that $\mathcal{A}$ is normal and countably compact. If $T:C\to C$ is nonexpansive, then T has a fixed point in C.

Assume that C is a convex, $\rho$-bounded, and $\zeta$-closed subset of a super-regular modular space $(X_{\rho },\zeta )$. Denote by $\mathcal{A}$ the class of all nonempty, convex, $\rho$-bounded, and $\zeta$-closed subsets of $X_{\rho }$. Obviously, $\mathcal{A}$ is stable under arbitrary intersections and contains all sets of the form $C\cap B_{\rho }(x,p)$, where $x\in C$ and $p>0$, since $\rho$-balls are convex and $\zeta$-closed. Therefore, we can restate Theorem 2 in a more classical form.

Theorem 3. 

Assume that $(X_{\rho },\zeta )$ is a super-regular modular space with the ζ-normal structure property. Let C be a convex ρ-bounded, and ζ-sequentially compact subset of $X_{\rho }$. If $T:C\to C$ is nonexpansive, then T has a fixed point in C.

An alternative formulation of Theorem 3 is that the $\zeta$-normal structure property ensures the $\zeta$-fixed point property (denoted as $\zeta$-fpp).

As noted by Sims in [48], Banach spaces that are uniformly convex in every direction ($UCED$) possess weak normal structure, meaning that every weakly compact convex set exhibits normal structure. This crucial result stems from the work of Garkavi [49]. Consequently, by virtue of Kirk’s theorem [21], $UCED$ Banach spaces possess the weak fixed point property. As expected, the same is true for super-regular modular spaces.

Theorem 4. 

Let $(X_{\rho },\zeta )$ be a $UCED$ super-regular modular space. Let $C\subset X_{\rho }$ be convex, ρ-bounded, and ζ-sequentially compact. If $T:C\to C$ is nonexpansive, then T has a fixed point.

Theorem 4, being an extension of the Browder fixed point theorem [20] [Theorem 1] to super-regular modular spaces, directly follows from Theorem 3 and the following technical result.

Proposition 5 

([36] Proposition 3.7). Let a modular space $X_{\rho }$ be $UCED$, and let $C\subset X_{\rho }$ be convex, ρ-bounded, and not a singleton. Then C has a ρ-nondiametral point.

Note that, since every uniformly convex space is uniformly convex in every direction, Theorem 4 also applies to uniformly convex super-regular modular spaces.

Let us recall that a Banach space $(X,\parallel ·\parallel )$ is said to possess a weak Kadec–Klee property if every unit sphere sequence converges in norm whenever it converges weakly; see, for example, [50]. A stronger uniform variant is defined as follows: a Banach space $(X,\parallel ·\parallel )$ is said to possess a weak uniform Kadec–Klee property if for every $\epsilon >0$ there exists a $0<\eta <1$ such that for every sequence $\left\{x_n\right\}$ in the unit ball, weakly converging to x, we have $\parallel x\parallel \le 1-\eta$ provided that

$$\mathrm{sep}\left\{x_k\right\}:=inf\{\parallel x_n-x_m\parallel :n\ne m\}>\epsilon .$$

The weak uniform Kadec–Klee property is, in fact, a geometric feature of Banach spaces where weak convergence of sequences on the unit sphere implies norm convergence, in a uniform way. Beginning in the early 1980s, it became clear that the weak uniform Kadec–Klee property is connected to the fixed point property via the normal structure property [50,51,52,53]. Since the Kadec–Klee property is inherently sequential rather than topological, this naturally led to an expanded definition that includes various types of sequence convergence, beyond merely weak or weak*-topologies. Examples of such generalized, convergence-related, Kadec–Klee properties can be found in [6,54,55,56,57,58,59,60]. To explore the possibility of achieving analogous results in super-regular modular spaces, we will follow [30] and recall the following definition of the Kadec–Klee property for such spaces.

Definition 20 

([30] Definition 11). We say that a super-regular $(X_{\rho },\zeta )$ possesses the uniform ζ-KK1 property if for every $\epsilon >0$ and every $r>0$ there exists an ${\eta }_1>0$ such that for every sequence $\left\{x_n\right\}$ in $B_{\rho }(0,r)$, ζ-convergent to x, such that

$$\mathrm{sep}\left\{x_k\right\}:=inf\{\rho (x_n-x_m):n\ne m\}>r\epsilon ,$$

we have

$$\rho \left(x\right)\le r(1-{\eta }_1).$$

The following result demonstrates the connection between the $\zeta$-KK1 property and the $\zeta$-normal structure, thereby extending the previously mentioned Banach space findings to super-regular modular spaces.

Theorem 5 

([30] Theorem 1). Assume that $(X_{\rho },\zeta )$ possesses the uniform ζ-KK1 property. Then, $(X_{\rho },\zeta )$ possesses the ζ-normal structure.

Theorems 5 and 3 collectively lead to the following significant fixed point result.

Theorem 6 

([30] Theorem 3). Assume that $(X_{\rho },\zeta )$ possesses the uniform ζ-KK1 property. Then, $(X_{\rho },\zeta )$ possesses the ζ-fpp.

It is clear that the uniform $\zeta$-KK1 property is equivalent to the standard definition of the uniform Kadec–Klee property for a Banach space $(X,\parallel ·\parallel )$ with $\zeta$ standing for convergence in the weak topology of X. Consequently, Theorems 5 and 6 generalize well-known results that establish a link between the Kadec–Klee property, the normal structure property, and the weak fixed point property in Banach spaces. We will now discuss how these results relate to fixed point theory in modular function spaces. Before proceeding, we will need to revisit some important facts from this theory.

Definition 21. 

A function modular ρ is said to be orthogonally additive if $\rho \left(f{1}_{A\cup B}\right)=\rho \left(f{1}_A\right)+\rho \left(f{1}_B\right)$ whenever $A\cap B=\mathfrak{\varnothing }$.

Note that many classical function modulars, including Lebesgue, Orlicz, and Musielak–Orlicz modulars, are orthogonally additive.

Definition 22 

([6] Definition 3.7). We say that ρ satisfies the ${\Delta }_2$-type condition if there exists a finite constant $M_2$ such that $\rho \left(2f\right)\le M_2\rho \left(f\right)$ for every $f\in X_{\rho }$.

The class of modular spaces satisfying the ${\Delta }_2$-type condition includes spaces such as Lebesgue variable exponent spaces $L^p\left(t\right)$ for $1\le p\left(t\right)<\infty$ and Orlicz spaces $L^{\phi }$ for $\phi$ satisfying the ${\Delta }_2$-condition. The following important technical result is used in the proof of Theorem 8. However, it is worthwhile to keep this result in mind for its own merit, as it clearly demonstrates the strength of the assumption of orthogonal additivity.

Theorem 7 

([6] Theorem 4.7). Let ρ be an orthogonally additive function modular. Let $\left\{f_n\right\}$ be a strongly bounded, ρ-convergent-to-zero sequence of elements of $L_{\rho }$. For any $g\in E_{\rho }$, the following holds:

$$\underset{n\to \infty }{lim\; inf}\rho (f_n+g)=\underset{n\to \infty }{lim\; inf}\rho \left(f_n\right)+\rho \left(g\right).$$

Interestingly, many modular function spaces exhibit the uniform $\zeta$-KK1 property, where $\zeta$ denotes $\rho$-almost everywhere convergence (recall that in many examples, this is equivalent to almost everywhere convergence with respect to a measure). This fact leads to a significant result regarding fixed point existence, summarized in the following theorem.

Theorem 8 

([30] Theorem 5). Let $L_{\rho }$ be a modular function space, where ρ is orthogonally additive. Let ζ denote the ρ-a.e. convergence. If ρ satisfies the ${\Delta }_2$-type property, then the super-regular modular space $(L_{\rho },\zeta )$ possesses the uniform ζ-KK1 property. Consequently, via Theorem 6, $(L_{\rho },\zeta )$ possesses the ζ-fpp.

While there are numerous examples of modular function spaces that satisfy the conditions of Theorem 8—such as Lebesgue spaces $L^p$, Orlicz spaces $L^{\phi }$ meeting the ${\Delta }_2$ condition, and variable exponent Lebesgue spaces $L^{p(·)}$ with $1\le p\left(t\right)\le M<\infty$—it is essential to also consider cases where the function $\rho$ does not exhibit the ${\Delta }_2$-type property. With this task in mind, let us recall the concept of the strong $\zeta$-Opial property in super-regular modular spaces.

Definition 23 

([30] Definition 17). We say that a super-regular modular space $(X_{\rho },\zeta )$ has the strong ζ-Opial property if

$$\rho \left(x_0\right)+\underset{n\to \infty }{lim\; inf}\rho (x_n-x_0)\le \underset{n\to \infty }{lim\; inf}\rho \left(x_n\right),$$

(3)

provided $x_n\stackrel{\zeta }{\to }{x}_0$ and the sequence $\{x_n-x_0\}$ is strongly bounded.

Observe that, by the Fatou property of modular function spaces, it is always true that

$${\displaystyle \rho \left(x_0\right)\le \underset{n\to \infty }{lim\; inf}\rho \left(x_n\right),}$$

(4)

whenever $x_n\stackrel{\zeta }{\to }{x}_0$. In spaces exhibiting the strong $\zeta$-Opial property, inequality (3) provides a better estimate than that given in (4).

Remark 1. 

While the concept of the strong ζ-Opial property, as articulated in Definition 23 and related to the renowned weak Opial property [61], is relatively new, it has been known since Khamsi’s 1996 work [62] (see also [6] [Theorem 4.7]) that any ${\Delta }_2$ modular function space determined by a convex, orthogonally additive modular ρ, has the strong ζ-Opial property. Our Theorem 7 extends this result to spaces lacking the ${\Delta }_2$ property. These findings encompass a broad spectrum of function spaces possessing the strong ζ-Opial property, including $L^p$ spaces for $p\ge 1$, variable Lebesgue spaces $L^{p(·)}$ with $1\le p\left(t\right)<+\infty$, as well as Orlicz and Musielak–Orlicz spaces. It is also important to note, as pointed out in [61], that the weak Opial property does not hold in $L^p$ spaces for $1\le p\ne 2$, again demonstrating the advantages of the modular approach.

As is quite usual in modular space settings, a single concept used in normed spaces splits up into two distinct yet interrelated entities. Accordingly, let us introduce the following variant of the Kadec–Klee property.

Definition 24 

([30] Definition 12). We say that $(X_{\rho },\zeta )$ possesses the uniform ζ-KK2 property if for every $\epsilon >0$ and every $r>0$ there exists an ${\eta }_2>0$ such that for every strongly bounded sequence $\left\{x_n\right\}$ in $B_{\rho }(0,r)$, ζ-convergent to x, such that

$${\mathrm{sep}}_2\left\{x_k\right\}:=inf\left\{\rho \left({\displaystyle \frac{x_n-x_m}{2}}\right):n\ne m\right\}>r\epsilon$$

we have

$$\rho \left(x\right)\le r(1-{\eta }_2).$$

The above notions are utilized in the following result, which—together with Remark 1—ensures the existence of numerous examples of super-regular modular spaces exhibiting the uniform $\zeta$-KK2 property.

Theorem 9 

([30] Theorem 6). If $(X_{\rho },\zeta )$ has the strong ζ-Opial property, then $(X_{\rho },\zeta )$ possesses the uniform ζ-KK2 property.

Remark 2. 

Following [30], we raise the following open questions:

(Q1) 

What is the relationship between ζ-KK1 and ζ-KK2 properties? Recalling that both properties are equivalent in the Banach space context, it would greatly simplify and enhance future research on super-regular modular spaces to determine whether, and under what conditions, one of these properties implies the other. What are the minimal assumptions needed for their equivalence?

(Q2) 

Does ζ-KK2 imply some form of normal structure? This question pertains to Theorem 5: can we replace ζ-KK1 with ζ-KK2 in the assumptions of this theorem and still obtain ζ-normal structure? If not, perhaps another form of normal structure should be considered in this case. Any affirmative answer to this question should lead to a new fixed point theorem, as normal structure typically implies a relevant fixed point property (see Theorem 6 and the accompanying commentary).

(Q3) 

Does ζ-KK2 directly imply some fixed point property? Regardless of the answers to question (Q2), there may be other methods for proving the ζ-fpp under the assumption of ζ-KK2.

In this subsection, we discussed fixed point results for contractions and nonexpansive mappings in regular and super-regular modular spaces. In the next subsection, we will broaden our focus to encompass a more general framework involving asymptotic and pointwise asymptotic variants of nonlinear operators. While the techniques employed in this asymptotic context are inherently more complex, it is essential to recognize that all findings detailed in Section 3.2 also apply to contractions and nonexpansive mappings, as presented in Section 3.1. However, as is common in fixed point theory, we have treated the results for classical contractive and nonexpansive mappings separately in this subsection for three primary reasons: (1) the methods used for the classical, non-asymptotic cases are generally different and typically simpler; (2) many applications do not require the advanced asymptotic framework; and (3) some of the results discussed in this subsection have yet to be achieved outside the classical context (for example, all discussions related to the Kadec–Klee property).

3.2. Asymptotic Contractions and Asymptotically Nonexpansive Mappings in Regular Modular Spaces

Existence theorems for fixed points of contractions and nonexpansive mappings constitute the foundational elements of classical fixed point theory. In the context of normed, metric, and modular spaces, these results are still most valuable whenever an operator T can be easily associated with a suitable space X and its subset C that fulfills the criteria of these existence theorems. However, in practice, identifying such spaces and subsets can frequently prove to be quite challenging, if not outright impossible. This fact is a key reason why numerous scenarios with relaxed assumptions on operators have been explored over the past fifty years. We will discuss one of these scenarios, which employs the asymptotic approach. This approach was initiated by Goebel and Kirk in their 1972 paper [63]. In this paper, they defined an asymptotically nonexpansive mapping $T:C\to C$ as a mapping satisfying for each $x,y\in C$ the inequality $\parallel T^n\left(x\right)-T^n\left(y\right)\parallel \le a_n\parallel x-y\parallel$, where $\left\{a_n\right\}$ is a sequence of positive numbers such that ${lim}_{n\to \infty }{a}_n=1$. For such operators, they proved the following generalization of Browder’s theorem.

Theorem 10 

([63] Theorem 1). Let C be a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space X, and let $T:C\to C$ be asymptotically nonexpansive. Then T has a fixed point.

To illustrate the importance of the asymptotic approach, let us revisit the original example from this paper, along with commentary based on the discussion in the recent book [28] [Example 1.7]; see also the discussion around Example 1.6 in the same book.

Example 6. 

Let C denote the unit ball in the Hilbert space $l^2$ and let T be defined as follows:

$$T:(x_1,x_2,x_3,···)\mapsto (0,x_1^2,A_2{x}_2,A_3{x}_3,···),$$

where $\left\{A_i\right\}$ is a sequence of numbers such that $0<A_i<1$ and ${\prod }_{i=2}^{\infty }{A}_i={\displaystyle \frac{1}{2}}$. It is easy to see that T is not nonexpansive in C; however, the semigroup $\left\{T^k\right\}$ of iterations of T is asymptotic pointwise nonexpansive in C because for every $x,y\in C$,

$$\parallel T^k\left(x\right)-T^k\left(y\right)\parallel \le {\alpha }_k\parallel x-y\parallel ,$$

where ${\alpha }_k=2{\prod }_{i=2}^k{A}_i$ for $k=2,3,···$, and ${lim}_{k\to \infty }{\alpha }_k=1$. Thus, using Theorem 10 (or our modular result, Theorem 18, discussed later in this section), we are able to demonstrate the existence of a common fixed point for $\left\{T^k\right\}$, which will also serve as a fixed point of T itself. It is actually quite clear that $(0,0,0,···)$ is such a fixed point. The issue, which the authors of the original 1972 paper did not expand upon, is to understand why this fixed point was not identified using the classical Browder theorem. The choice of the unit ball as the set C is obviously a limiting factor. If, instead, we consider the mapping $T:D\to D$, where the set $D=\{x\in C:|x_1|\le {\displaystyle \frac{1}{2}}\}$, it becomes apparent that T is nonexpansive in D. Since D is bounded, closed, and convex, it follows from Browder’s theorem that T must have a fixed point in D. The critical point is that a priori, it was unclear which set should be selected. This straightforward example illustrates the power of the asymptotic approach, which can provide solutions in scenarios where classical methods may fall short.

An additional and significant advantage of the asymptotic approach is that the method of successive iterations $T^k$ of an operator T is commonly employed in fields such as science, information technology, robotics, and artificial intelligence. When a sequence of these iterations $\left\{T^k\left(x\right)\right\}$ converges to a fixed point $x_0$ of T, $x_0$ can be regarded as a stationary point of the system defined by this operator. From this perspective, we are primarily interested in the behavior of the system as k approaches infinity, rendering the properties of T and $T^k$ for small values of k less relevant. Let us also note that $\{T^k:k=0,1,\dots \}$ forms a semigroup of mappings, a topic explored in Section 4, where we will elaborate on this asymptotic behavior.

Based on the definition of asymptotically nonexpansive mappings in Banach spaces established by Goebel and Kirk in [63], and further generalized by Kirk and Xu in [64,65], the analogous (and, if $\rho$ is a norm, identical) concepts within the framework of regular modular spaces were introduced in [31]. We also refer to definitions in the narrower context of the modular function spaces, as discussed in [6,66,67].

Definition 25 

([31] Definition 2.7). Let $X_{\rho }$ be a regular modular space and let $C\subset X_{\rho }$ be convex, nonempty, ρ-closed, and ρ-bounded. A mapping $T:C\to C$ is called

(i) 

an asymptotic contraction if there exists a sequence of nonnegative numbers $\left\{{\alpha }_n\right\}$ with ${lim\; sup}_{n\to \infty }{\alpha }_n<1$ such that

$$\rho (T^n\left(x\right)-T^n\left(y\right))\le {\alpha }_n\rho (x-y)$$

for every $n\in \mathbb{N}$ and all $x,y\in C$.

(ii) 

asymptotically nonexpansive if there exists a sequence of nonnegative numbers $\left\{{\alpha }_n\right\}$ with ${lim\; sup}_{n\to \infty }{\alpha }_n=1$ such that

$$\rho (T^n\left(x\right)-T^n\left(y\right))\le {\alpha }_n\rho (x-y)$$

for every $n\in \mathbb{N}$ and all $x,y\in C$.

Building upon the foundational work of papers [65] for Banach spaces, [68] for metric spaces, and [66,67] for modular function spaces, extensive research has been conducted on the following pointwise variants.

Definition 26. 

Let $X_{\rho }$ be a regular modular space and let $C\subset X_{\rho }$ be convex, nonempty, ρ-closed, and ρ-bounded. A mapping $T:C\to C$ is called

(i) 

a pointwise contraction if for every $x\in C$ there exists an $\alpha \left(x\right)\in [0,1)$ such that

$$\rho \left(T\right(x)-T(y\left)\right)\le \alpha \left(x\right)\rho (x-y)$$

for every $x\in C$.

(i) 

an asymptotic pointwise contraction if for every $x\in C$ and every $n\in \mathbb{N}$ there exists an ${\alpha }_n\left(x\right)\in [0,1)$ such that ${lim\; sup}_{n\to \infty }{\alpha }_n\left(x\right)<1$ and

$$\rho (T^n\left(x\right)-T^n\left(y\right))\le {\alpha }_n\left(x\right)\rho (x-y).$$

(iii) 

an asymptotic pointwise nonexpansive mapping if for every $x\in C$ and every $n\in \mathbb{N}$ there exists an ${\alpha }_n\left(x\right)\in [0,1)$ such that ${lim\; sup}_{n\to \infty }{\alpha }_n\left(x\right)\le 1$ and

$$\rho (T^n\left(x\right)-T^n\left(y\right))\le {\alpha }_n\left(x\right)\rho (x-y).$$

Every pointwise contraction is inherently an asymptotic pointwise contraction, which further qualifies as an asymptotic pointwise nonexpansive mapping. Similarly, every contraction qualifies as a pointwise contraction, while every asymptotic contraction is an asymptotic pointwise contraction. Additionally, every nonexpansive mapping can be considered asymptotically nonexpansive, which also makes it an asymptotic pointwise nonexpansive mapping. An interest in pointwise variants stems partly from the following observation due to Kirk.

Proposition 6 

([69] Proposition 2.1). Let A be a bounded open convex subset of a Banach space X and let $T:A\to X$ be continuously Fréchet differentiable on A. Then, T is a pointwise contraction mapping on A if and only if $\parallel T_{x_0}^{\prime }\parallel <1$ for every $x_0\in A$.

Recall that, given a convex open subset A of X and $T:A\to X$, we say that T is continuously Fréchet differentiable on A if the mapping $x\mapsto T_x^{\prime }$ from A to the space of continuous linear operators on X is continuous. Here, $T_x^{\prime }$ denotes the Fréchet derivative of T at x, namely

$$T_x^{\prime }\left(y\right)=\underset{t\to 0}{lim}{\displaystyle \frac{T(x+ty)-T\left(x\right)}{t}},$$

provided the limit is uniform for all y with $\parallel y\parallel =1$.

Utilizing Proposition 6, one can easily establish analogous conditions for other types of pointwise contractions and pointwise nonexpansive mappings, including their asymptotic versions. This demonstrates that pointwise type conditions show up quite naturally in Banach spaces. Parallel results in the context of modular spaces are yet to be discovered.

The following result is known in the context of modular function spaces, as shown in [66] [Theorem 2.3]. An inspection of the proof of this theorem shows that both the theorem and the proof hold true in the case of regular modular spaces without any real modification.

Theorem 11. 

Let $X_{\rho }$ be a regular modular space, and let $C\subset X_{\rho }$ be nonempty, ρ-closed, and ρ-bounded. Assume that $T:C\to C$ is a pointwise contraction or asymptotic pointwise contraction. Then T has at most one fixed point in C. Moreover, if $x_0$ is a fixed point of T, then the orbit $\left\{T^n\left(x\right)\right\}$ ρ-converges to $x_0$ for any $x\in C$.

Later in this section, we will examine the question of the existence of fixed points for these types of mappings.

To deal with fixed points of asymptotic pointwise nonexpansive mappings, we define the following fixed point property.

Definition 27 

([31] Definition 2.9). We say that a regular modular space $X_{\rho }$ has the asymptotic fixed point property, or for short, the $\left(AFPP\right)$ property if every asymptotically nonexpansive mapping $T:C\to C$ defined on any convex, nonempty, ρ-closed, and ρ-bounded $C\subset X_{\rho }$ has a fixed point.

We extend this concept to the pointwise case.

Definition 28. 

We note that a regular modular space $X_{\rho }$ has the asymptotic pointwise fixed point property (in short, the $\left(APFPP\right)$ property) if every asymptotic pointwise nonexpansive mapping $T:C\to C$ defined on any convex, nonempty, ρ-closed, and ρ-bounded $C\subset X_{\rho }$ has a fixed point.

Our goal now is to demonstrate that, in the context of regular modular spaces, the uniform convexity $\left(UUC1\right)$ implies the $\left(APFPP\right)$ property, and consequently the $\left(AFPP\right)$ property as well, as formulated in Theorem 17 below. Fundamentally, this result can be viewed as a generalization of Browder’s fixed point theorem to the case of asymptotic pointwise nonexpansive mappings acting within a uniformly convex regular modular space, as stated in Theorem 18. Following the approach taken in this survey article, we will not provide a formal proof; instead, we will focus on defining some properties of regular modular spaces and explaining how these properties interact to guarantee the $\left(APFPP\right)$ property. This method aligns with the strategy previously devised for modular function spaces, as described in [6] [pp. 123–124]. Before we begin, let us introduce the following concepts.

Definition 29 

([31] Definition 3.1). Let $C\subset X_{\rho }$ be convex, nonempty, ρ-closed, and ρ-bounded.

1. 

A function $\tau :C\to [0,\infty ]$ is called a ρ-type (or, in short, a type) if there exists a sequence $\left\{y_m\right\}$ of elements of C such that for any $z\in C$:

$$\tau \left(z\right)=\underset{m\to \infty }{lim\; sup}\rho (y_m-z).$$

2. 

Let τ be a type. A sequence $\left\{g_n\right\}$ is called a minimizing sequence of τ if

$$\underset{n\to \infty }{lim}\tau \left(g_n\right)=inf\{\tau \left(f\right):f\in C\}.$$

Note that any type is convex provided $\rho$ is convex.

Definition 30 

([31] Definition 3.2). We say that a regular modular space $X_{\rho }$ has the minimizing sequence property, or for short, the $\left(MSP\right)$ property, if every minimizing sequence of any type is ρ-convergent, and its limit is independent of the minimizing sequence.

On a high level, our strategy can be described as a series of implications:

$$\left(UCC1\right)\Rightarrow \left(PP\right)+\left(R\right)\Rightarrow \left(MSP\right)\Rightarrow \left(AFPP\right),$$

where $\left(PP\right)$ stands for the parallelogram property (see Definition 32), a generalization of a known property of uniformly convex Banach spaces (and is reminiscent of the parallelogram law in Hilbert spaces), and $\left(R\right)$ denotes the modular equivalent of reflexivity (see Definition 33).

We will begin our discussion with the last of these implications.

Theorem 12 

([31] Theorem 3.3). $\left(MSP\right)$ property implies $\left(AFPP\right)$ property.

By examining the proof of Theorem 12 and considering related findings for modular function spaces, as detailed in proofs of [6] [Theorem 5.7] and [67] [Theorem 4.1], we can broaden this result to encompass asymptotic pointwise nonexpansive mappings.

Theorem 13. 

$\left(MSP\right)$ property implies $\left(APFPP\right)$ property.

Theorem 13 provides a comprehensive framework for establishing fixed point results in regular modular spaces. To apply this theorem effectively, we must identify the conditions under which these spaces possess the $\left(MSP\right)$ property. To achieve this, we will first review the following two properties of regular modular spaces.

Definition 31 

([31] Definition 4.1). The Parallelogram Function Ψ associated with a regular modular space $X_{\rho }$ is defined as follows:

$$\Psi (r,s,\epsilon )=inf\left\{{\displaystyle \frac{1}{2}}{\rho }^2\left(x\right)+{\displaystyle \frac{1}{2}}{\rho }^2\left(y\right)-{\rho }^2\left({\displaystyle \frac{x+y}{2}}\right)\right\},$$

where $0<s<r$ and $\epsilon \ge 0$, and the infimum is taken over all $x,y\in X_{\rho }$ such that $\rho \left(x\right)\le r$, $\rho \left(y\right)\le r$, $max\left\{\rho \right(x),\rho (y\left)\right\}\ge s$, and $\rho (x-y)\ge r\epsilon$.

Remark 3 

([31] Remark 4.2). It easily follows from the above definition that

1. 

$\Psi (r,s,\epsilon )\ge 0$,

2. 

$\Psi (r,s,0)=0$,

3. 

$\Psi (r,s,\epsilon )$ is a nondecreasing function of ε with $0<s<r$ fixed,

4. 

if ${\displaystyle \underset{n\to \infty }{lim}\Psi (r,s,t_n)=0}$, then ${\displaystyle \underset{n\to \infty }{lim}{t}_n=0}$.

Due to the evident similarity with the parallelogram law in Hilbert spaces, we define the following property for regular modular spaces.

Definition 32 

([31] Definition 4.3). We say that a regular modular space $X_{\rho }$ has the parallelogram property (that is, $\left(PP\right)$ property) if

$$\Psi (r,s,\epsilon )>0$$

whenever $\epsilon >0$.

Following [6] [Definition 3.9], let us recall the definition of the property $\left(R\right)$ in modular spaces.

Definition 33. 

We say that $X_{\rho }$ has property $\left(R\right)$ if and only if every non-increasing sequence $\left\{C_n\right\}$ of nonempty, ρ-bounded, ρ-closed, convex subsets of $X_{\rho }$ has a nonempty intersection.

Note that in the case $\rho$ is a norm, it follows from the Eberlein–Šmulian theorem that the property $\left(R\right)$ is equivalent to reflexivity.

We are now prepared to present the following fundamental result, which extends Theorem 4.6 in [31] to the case of asymptotic pointwise nonexpansive mappings.

Theorem 14. 

Let a regular modular space $X_{\rho }$ possess the properties $\left(PP\right)$ and $\left(R\right)$. Then, $X_{\rho }$ also has the $\left(MSP\right)$ property, which, according to Theorem 13, implies the $\left(APFPP\right)$ property.

The proof of Theorem 14 follows the path of the proof of [6] [Lemma 4.3]. Note also that the $\left(MSP\right)$ property in abstract modular spaces with the Fatou property was considered in [32]. Browder’s original result from 1965, as presented in [20] [Theorem 1], along with many subsequent generalizations, is articulated in terms of uniform convexity. Utilizing Theorems 12, 13, and 14, we will now show that this holds true for regular modular spaces as well. To accomplish this, let us first revisit the following two results.

Theorem 15 

([31] Theorem 5.5). Every uniformly convex regular modular space has property $\left(PP\right)$.

Theorem 16 

([31] Theorem 5.6). Every uniformly convex regular modular space has property $\left(R\right)$.

Let $X_{\rho }$ be a uniformly convex regular modular space. According to Theorem 16, $X_{\rho }$ possesses property $\left(R\right)$, and by Theorem 15, it also exhibits property $\left(PP\right)$. With the application of Theorem 14, we can conclude that $X_{\rho }$ has property $\left(APFPP\right)$, which is articulated in the following fundamental fixed point result.

Theorem 17. 

Every uniformly convex regular modular space has the $\left(APFPP\right)$ property.

Following the argument presented in the proof of [6] [Theorem 5.7], we can demonstrate that the set of common fixed points of an asymptotic pointwise nonexpansive mapping is both $\rho$-closed and convex. Considering this, we can express our findings in a more traditional manner.

Theorem 18. 

Let C be a nonempty, ρ-closed, convex, and ρ-bounded subset of a uniformly convex regular modular space $X_{\rho }$, and let $T:C\to C$ be an asymptotic pointwise nonexpansive mapping. Then, T has a fixed point. Furthermore, the set of all fixed points, $F\left(T\right)$, is ρ-closed and convex.

Since every asymptotic pointwise contraction is an asymptotic pointwise nonexpansive mapping, Theorem 18 addresses the earlier question regarding the existence of fixed points for asymptotic pointwise contractions. Specifically, when combined with Theorem 11, it yields the following fixed point result.

Theorem 19. 

Let C be a nonempty, ρ-closed, convex, and ρ-bounded subset of a uniformly convex regular modular space $X_{\rho }$, and let $T:C\to C$ be an asymptotic pointwise contraction. Then, T has a unique fixed point $\overline{x}\in C$. Moreover, for any $x\in C$, $\rho \left(T^n\left(x\right)-\overline{x}\right)\to 0$ as $n\to \infty$, where $T^n$, is the n-th iterate of T.

While many regular modular spaces that are important from an applications perspective are uniformly convex, this requirement is still quite restrictive. Our next result offers a more general and less restrictive condition.

Theorem 20. 

Let C be a nonempty ρ-bounded, ρ-closed subset of a regular modular space $X_{\rho }$. Assume $T:C\to C$ to be an asymptotic pointwise contraction. Moreover, assume that there exists an $x\in C$ such that the ρ-type Φ defined for $u\in C$ by $\Phi \left(u\right)={lim\; sup}_{k\to \infty }\rho (T^k\left(x\right)-u)$ attains its minimum in C. Then, there exists a unique fixed point $z\in F\left(T\right)$. Moreover, $\rho (T^k\left(u\right)-z)\to 0$ for every $u\in C$.

Proof. 

By Theorem 11, we need only to prove the existence of a fixed point. By our assumption, there exists $z\in C$ such that $\Phi \left(z\right)=inf\{\Phi (y):y\in C\}$. We will prove now that $\Phi \left(z\right)=0$. First, note that for any $m,k\in \mathbb{N}$

$$\rho (T^{m+k}\left(x\right)-T^m\left(z\right))\le a_m\left(z\right)\rho (T^k\left(x\right)-z),$$

and that by letting $k\to +\infty$ we get $\Phi \left(T^m\left(z\right)\right)\le a_m\left(z\right)\Phi \left(z\right)$, which, after passing with m to infinity, gives

$$\Phi \left(z\right)\le \underset{m\to +\infty }{lim\; sup}\Phi \left(T^m\left(z\right)\right)\le \underset{m\to +\infty }{lim\; sup}{a}_m\left(z\right)\Phi \left(z\right).$$

Since ${lim\; sup}_{m\to +\infty }{a}_m\left(z\right)<1$, we conclude that $\Phi \left(z\right)=0$. Consequently, using the definition of $\Phi$, we have

$$\begin{array}{ccc}\hfill 0\le \underset{k\to +\infty }{lim\; sup}\rho (T^k\left(x\right)-T\left(z\right))& =& \underset{k\to +\infty }{lim\; sup}\rho (T^{k+1}\left(x\right)-T\left(z\right))\hfill \\ & \le & \underset{k\to +\infty }{lim\; sup}\rho (T^k\left(x\right)-z)=\Phi \left(z\right)=0.\hfill \end{array}$$

Hence, $T^k\left(x\right)\stackrel{\rho }{\to }z$ and $T^k\left(x\right)\stackrel{\rho }{\to }T\left(z\right)$, which, by the uniqueness of the $\rho$-limit, implies that $T\left(z\right)=z$, as claimed. □

To effectively apply Theorem 20, we need practical methods for determining when $\rho$-types reach their minimum. In the context of Banach spaces, this is satisfied when C is nonempty, convex, bounded, and weakly compact, as established in sources such as as [24,70]. However, this conclusion relies on specific Banach space characteristics, such as the norm’s triangle property, and the principle that a closed convex subset of a weakly compact set remains weakly compact. Since these attributes are generally absent in modulated topological vector spaces, we will introduce a powerful method involving uniformly continuous modulars.

Definition 34 

([6] Definition 5.4). A modular ρ is called uniformly continuous if for every $\epsilon >0$ and every $0<L<\infty$ there exists $\delta >0$ such that

$$\left|\rho \right(x+y)-\rho (x\left)\right|<\epsilon ,$$

whenever $x\in X_{\rho }$, $y\in X_{\rho }$, $\rho \left(y\right)<\delta$, and $\rho \left(x\right)\le L$.

The following two lemmas, originally proven for function modulars, can be easily extended to regular modular spaces:

Lemma 2 

([6] Lemma 5.1). Let C be a nonempty ρ-bounded, ρ-closed, convex subset of a regular modular space $X_{\rho }$, where ρ is uniformly continuous. Then, any ρ-type defined in C is ρ-lower semicontinuous on C.

Lemma 3 

([6] Lemma 3.3). Let C be a nonempty ρ-bounded, ρ-closed, convex subset of a regular modular space $X_{\rho }$, where ρ possesses property $\left(R\right)$. If φ is a ρ-lower semicontinuous ρ-type defined in C then φ attains its minimum in C.

Our next fixed point theorem for asymptotic pointwise contractions follows immediately from Lemma 2, Lemma 3, and Theorem 20.

Theorem 21. 

Let C be a nonempty ρ-bounded, ρ-closed, convex subset of a regular modular space $X_{\rho }$. Assume that ρ is uniformly continuous and has property $\left(R\right)$. Let $T:C\to C$ be an asymptotic pointwise contraction. Then, there exists a unique common fixed point $z\in F\left(T\right)$. Moreover, $\rho (T^k\left(u\right)-z)\to 0$ for every $u\in C$.

Every norm is inherently a uniformly continuous modular, which means that in the context of Banach spaces, the only additional requirement is the possession of property $\left(R\right)$. In this case, this property equates to reflexivity. This aligns with our earlier observation, as reflexivity is equivalent to the weak compactness of the unit ball in a Banach space.

Generally, for convex modulars, the uniform continuity of $\rho$ is equivalent to $\rho$ satisfying the ${\Delta }_2$-type condition; see, for example, [71], hence Theorem 21 can be applied to a wide class of spaces including Lebesgue spaces $L^p$, Orlicz spaces $L^{\phi }$ with $\phi$ satisfying a relevant ${\Delta }_2$ condition, and variable exponent Lebesgue spaces $L^{p(·)}$ with $1\le p\left(t\right)\le M<\infty$.

4. Semigroups of Nonlinear Operators in Modular Spaces

In this section, we will outline fixed-point considerations related to semigroups of nonlinear operators acting within modular spaces. Let us start with the following basic definitions.

Definition 35 

([14] Definition 2.10). Let $X_{\rho }$ be a regular modular space and let $C\subset X_{\rho }$ be convex, nonempty, ρ-closed, and ρ-bounded. Let J be a parameter semigroup of nonnegative numbers, that is, a subsemigroup of $[0,+\infty )$ with the usual addition such that $0\in J$ and there exists $0<t\in J$. A one-parameter family $\mathcal{T}=\{T_t:t\in J\}$ of mappings from C into itself is called a pointwise Lipschitzian semigroup on C if $\mathcal{T}$ satisfies the following conditions:

1. 

$T_0\left(x\right)=x$ for $x\in C$;

2. 

$T_{t+s}\left(x\right)=T_t\left(T_s\left(x\right)\right)$ for $x\in C$ and $t,s\in J$;

3. 

For each $t\in J$, $T_t$ is a pointwise Lipschitzian mapping, i.e., for every $t\in J$ and every $x\in C$ there exists an ${\alpha }_t\left(x\right)>0$ such that for every $y\in C,$

$$\rho (T_t\left(x\right)-T_t\left(y\right))\le {\alpha }_t\left(x\right)\rho (x-y),$$

4. 

For each $x\in C$, the mapping $t\to T_t\left(x\right)$ is ρ-continuous at every $t\in J$, i.e,

$$\rho (T_{t_n}\left(x\right)-T_t\left(x\right))\to 0,$$

whenever $t_n\in J$ for every $n\in \mathbb{N}$, and $t_n\to t$.

For each $t\in J$, let $F\left(T_t\right)$ denote the set of all fixed points of $T_t$. Define the set of all common fixed points set for mappings from $\mathcal{T}$ as the intersection

$$F\left(\mathcal{T}\right)=\bigcap _{t\in J}F\left(T_t\right).$$

Definition 36. 

Using assumptions and notations from Definition 35, we say that a pointwise Lipschitzian semigroup $\mathcal{T}$ is asymptotic pointwise nonexpansive if, in addition,

$$\underset{t\to \infty }{lim\; sup}{\alpha }_t\left(x\right)\le 1.$$

(5)

Similarly, $\mathcal{T}$ is called asymptotic pointwise contractive if

$$\underset{t\to \infty }{lim\; sup}{\alpha }_t\left(x\right)<1.$$

(6)

Remark 4. 

Let us note the following:

1. 

Assumptions on J immediately imply that $+\infty$ is a cluster point of J in the sense of the natural topology inherited by J from $[0,+\infty )$.

2. 

Typical examples of the parameter semigroups J satisfying conditions from Definitions 35 and 36 are: $J=[0,+\infty )$ and ideals of the form $J=\{n\alpha :n=0,1,2,3,\dots \}$ for a given $\alpha >0$.

3. 

In (5) and (6), as in the whole of this section, we use a convention that the notation $t\to \infty$ means that t tends to infinity over J.

4. 

Without loss of generality we may assume ${\alpha }_t\left(x\right)\ge 1$ for any $t\in J$ and any $x\in C$, as well as that ${lim\; sup}_{t\to \infty }{\alpha }_t\left(x\right)={lim}_{t\to \infty }{\alpha }_t\left(x\right)=1$.

In mathematics and its applications, scenarios involving continuous semigroups of operators are prevalent. For instance, in dynamical systems theory, the space X represents the state space, while the mapping $(t,x)\to T_t\left(x\right)$ models the evolution function of a dynamical system. The parameter t can signify either continuous or discrete time (recall the case of the semigroup of iterations $\{T^k:k=0,1,2,\dots \}$), contingent on the nature of the parameter set J. Common fixed points of the semigroup can be interpreted as the system’s stationary points that remain unchanged under the transformation $T_t$ for all $t\in J$. The exploration of semigroups of nonlinear operators as resolving semigroups has been a significant area of research in recent decades. Since the state space X can be infinite-dimensional, these findings are relevant to both deterministic and stochastic dynamical systems. In this framework, creating algorithms to identify common fixed points of such semigroups is intrinsically connected to the challenge of solving stochastic evolution equations.

The fixed point theory for semigroups of nonlinear operators operating within normed and metric spaces has been thoroughly developed and documented. For detailed references, see works such as [6,10,23,24,26]. Ref. [28] offers an extensive discussion of the fixed point theory of semigroups of pointwise Lipschitzian operators acting within Banach spaces. This book also provides a comprehensive justification for employing the pointwise asymptotic approach (as presented in this section), supported by a variety of examples. Let us note that in applications, we are often interested not only in stationary points (where the system behaves stably) but also in situations where orbits ${T_t\left(x\right)}_{t\in J}$ do not converge over large sets. Such systems tend to exhibit chaotic behavior in the sense of Devaney [72]. For an illustrative example, we direct the reader to [73] (see also [74]). In this paper, the authors, working in the realm of Orlicz spaces, establish chaos and stability criteria for the operator semigroup induced by the von Foerster–Lasota partial differential equation. In population biology, this equation is used to describe the dynamics of an age-dependent population.

In the realm of modular function spaces, the semigroup theory has been developed across numerous works. To highlight a few, notable references include [6,15,32,75,76,77,78,79]. Some of these results can be adapted to regular modular spaces, while others are contingent upon the specifics of modular function space theory.

Foundations of the semigroup theory for regular modular spaces can be found in [31,80]. The existence of a common fixed point was established in the author’s recent paper [14], as summarized in the following theorem:

Theorem 22 

([14] Theorem 3.12). Let $X_{\rho }$ be a uniformly convex regular modular space. Let $\mathcal{T}$ be an asymptotic pointwise nonexpansive semigroup on C, where $C\subset X_{\rho }$ is convex, nonempty, ρ-closed, and ρ-bounded. Then, there exists a common fixed point for $\mathcal{T}$, which means that the set $F\left(\mathcal{T}\right)$ is nonempty. Moreover, $F\left(\mathcal{T}\right)$ is convex and ρ-closed.

This theorem naturally leads to a parallel result for asymptotic pointwise contractive semigroups.

Theorem 23. 

Let $X_{\rho }$ be a uniformly convex regular modular space. Let $\mathcal{T}$ be an asymptotic pointwise contractive semigroup on C, where $C\subset X_{\rho }$ is convex, nonempty, ρ-closed, and ρ-bounded. Then, there exists a unique common fixed point $z\in F\left(\mathcal{T}\right)$. Moreover, for every $u\in C$, $\rho (T_t\left(u\right)-z)\to 0$ as $t\to \infty$.

For contractive semigroups, a result from [80] can be adapted from the context of modular topological vector spaces, where $\zeta$-convergence is defined as sequential convergence with respect to a linear Hausdorff topology.

Theorem 24 

([80] Theorem 3.5). Let C be a nonempty ρ-bounded subset of a regular modular space $X_{\rho }$ and let $\mathcal{T}=\{T_t:t\ge 0\}$ be a contractive semigroup on C. Assume that there exists an $x\in C$ such that the ρ-type Φ defined for $u\in C$ by $\Phi \left(u\right)={lim\; sup}_{t\to \infty }\rho (T_t\left(x\right)-u)$ attains its minimum in C. Then there exists a unique common fixed point $z\in F\left(\mathcal{T}\right)$. Moreover, $\rho (T_t\left(u\right)-z)\to 0$ for every $u\in C$.

To effectively utilize Theorem 24, it is crucial to identify practical criteria for determining the $\rho$-types that achieve their minimum for specific sets C. In the context of Banach spaces, it is well-established that this holds when C is a nonempty, convex, bounded, and weakly compact set, as noted in sources such as [24,70]. Again, this conclusion relies on certain characteristics unique to Banach spaces, such as the triangle inequality for norms and the property that closed convex subsets of weakly compact sets remain weakly compact. Given that these properties may be absent in super-regular modular spaces, we will employ the robust techniques of $\zeta$-Opial sets (see Definition 37) modified after the analogue concept used in the theory of modulated topological vector spaces. This approach will enable us to establish our next common fixed point theorem for contractive semigroups and provide a comprehensive list of examples and applications.

Definition 37 

([80] Definition 3.7). Let C be a nonempty subset of a super-regular modular space $(X_{\rho },\zeta )$. We say that C is a ζ-Opial set if for every $y\in C$ and every sequence $\left\{x_n\right\}$ of elements of C with $x_n\stackrel{\zeta }{\to }x$ for an $x\in C,$

$$\underset{n\to \infty }{lim\; inf}\rho (x_n-y)=\underset{n\to \infty }{lim\; inf}\rho (x_n-x)+\rho (x-y).$$

(7)

We will start by revisiting a standard result expressed in the context of super-regular modular spaces. Here, we denote $\zeta$-l.s.c as lower semi-continuity with respect to the $\zeta$ convergence. Specifically, this means that for a given function $\Psi :X\to [0,+\infty )$, the inequality $\Psi \left(y\right)\le {lim\; inf}_{k\to \infty }\Psi \left(y_k\right)$ holds whenever the sequence $y_k$ of elements of C converges to y with respect to $\zeta$ (i.e., $y_k\stackrel{\zeta }{\to }y$).

Lemma 4 

([80] Lemma 3.6). Let C be a nonempty, sequentially ζ-compact subset of a super-regular modular space $(X_{\rho },\zeta )$. Let $\Psi :C\to [0,+\infty )$. If Ψ is ζ-l.s.c. then Ψ attains its minimum in C.

Using this lemma and following the flow of the proof of Theorem 3.8 in [80], one can prove the following important result.

Theorem 25. 

Let C be a nonempty, sequentially ζ-compact, ρ-bounded subset of a super-regular modular space $(X_{\rho },\zeta )$. If C is a ζ-Opial set, then every ρ-type Φ on C is sequentially ζ-l.s.c. and attains its minimum in C. Moreover, if $\left\{y_n\right\}$ is a sequence of elements of C such that $y_n\stackrel{\zeta }{\to }y$ then

$$\Phi \left(y\right)+\underset{n\to \infty }{lim\; inf}\rho (y_n-y)\le \underset{n\to \infty }{lim\; inf}\Phi \left(y_n\right).$$

(8)

Combining Theorems 24 and 25, we immediately obtain the following fixed point result (compare with the proof of Theorem 3.9 in [80]).

Theorem 26 

([80] Theorem 3.9). Let C be a ρ-bounded and sequentially ζ-compact subset of a super-regular modular space $(X_{\rho },\zeta )$. Let $\mathcal{T}=\{T_t:t\ge 0\}$ be a contractive semigroup on C. If C is a ζ-Opial set, then there exists a unique common fixed point $z\in F\left(\mathcal{T}\right)$. Moreover, $\rho (T_t\left(u\right)-z)\to 0$ for every $u\in C$.

Remark 5. 

Observe that if a super-regular modular space $(X_{\rho },\zeta )$ has a strong ζ-Opial property, then every strongly bounded closed convex set is a ζ-Opial set. By Remark 1, there are many important examples of scenarios where Theorem 26 can be applied, including spaces like $L^p$ for $p\ge 1$, variable Lebesgue spaces $L^{p(·)}$ where $1\le p\left(t\right)<+\infty$, Orlicz spaces, and Musielak–Orlicz spaces.

5. Fixed Point Construction Processes

It is well known that Picard’s technique of iterates and orbits, effective in establishing fixed points for contraction mappings (as demonstrated by our Theorems 1, 20, 21, 23, 24, and 26) does not typically lead to a convergent process in the case of nonexpansive mappings. However, as demonstrated in [81], this convergence holds for typical elements (in the sense of Baire’s category) of a large class of nonexpansive mappings in Banach spaces.It is not yet known whether similar results can be obtained for nonexpansive mappings in modular spaces. To address this issue for mappings operating in Hilbert, Banach, metric, and modular function spaces, a vast array of effective algorithms has been developed over the past few decades. Due to space constraints in this brief survey, we cannot do full justice to this extensive body of work; therefore, we will highlight just a few resources that may be beneficial. See, for example, [6,28,61,79,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97].

It is important to note that many iterative processes, such as the generalized Krasnosel’skii–Mann and Ishikawa processes, which have been thoroughly studied in the context of modular function spaces, can be cautiously adapted to the framework of super-regular modular spaces. This is one of the promising future research areas that holds great potential for offering deeper insights and applications of the fixed point theory discussed in this paper.

In the 2024 paper [33], the author began an investigation into implicit iterative processes in regular modular spaces. Below, we summarize the key findings of that work.

Definition 38 

([33] Definition 8). Let C be a convex, nonempty, ρ-closed, and ρ-bounded subset of a regular modular space $X_{\rho }$. Given a ρ-nonexpansive semigroup $\mathcal{T}=\{T_t:t\in [0,\infty )\}$ on C, the implicit iteration process $P(C,\mathcal{T},x_0,\left\{c_k\right\},\left\{t_k\right\})$ is defined by the following formula:

$$\left\{\begin{array}{c}{x}_0\in C\hfill \\ x_{k+1}=c_k{T}_{t_{k+1}}\left(x_{k+1}\right)+(1-c_k)x_k,fork\in {\mathbb{N}}_0,\hfill \end{array}\right.$$

(9)

where ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, the sequence ${\left\{c_k\right\}}_{k\in {\mathbb{N}}_0}$ of real numbers from $(0,1)$ is bounded away from 0 and 1, and ${\left\{t_k\right\}}_{k\in \mathbb{N}}$ is a sequence of positive real numbers. We will also say that the sequence ${\left\{x_k\right\}}_{k\in {\mathbb{N}}_0}$ is generated by the process $P(C,\mathcal{T},x_0,\left\{c_k\right\},\left\{t_k\right\})$ and write

$${\left\{x_k\right\}}_{k\in {\mathbb{N}}^0}=P(C,\mathcal{T},x_0,\left\{c_k\right\},\left\{t_k\right\}).$$

For $k\in {\mathbb{N}}_0$, $u\in C$, $w\in C$, let us introduce the following notation:

$$P_{k,w}\left(u\right)=c_k{T}_{t_{k+1}}\left(u\right)+(1-c_k)w.$$

Since each $P_{k,w}\left(u\right):C\to C$ is a ρ-contraction, it follows from the Banach Contraction Principle (Theorem 1) that each $x_{k+1}$ in (9) is uniquely defined.

Theorem 27 

([33] Theorem 3). Let $X_{\rho }$ be a uniformly convex regular modular space, and let $C\subset X_{\rho }$ be convex, nonempty, ρ-compact, and ρ-bounded. Assume that $\mathcal{T}$ is a ρ-nonexpansive semigroup on C. Let ${\left\{x_k\right\}}_{k\in {\mathbb{N}}^0}=P(C,\mathcal{T},x_0,\left\{c_k\right\},\left\{t_k\right\})$ be an implicit iteration process, where

(i) 

$t_n>0$ for every $n\in \mathbb{N}$

(ii) 

${\displaystyle \underset{n\to \infty }{lim\; inf}{t}_n=0}$

(iii) 

${\displaystyle \underset{n\to \infty }{lim\; sup}{t}_n>0}$

(iv) 

${\displaystyle \underset{n\to \infty }{lim}(t_{n+1}-t_n)=0.}$

There then exists a common fixed point $x\in F\left(\mathcal{T}\right)$, such that $\rho (x_k-x)\to 0$ when $k\to \infty$.

Building on [33], we would like to offer a few remarks regarding the applicability of Theorem 27. It is straightforward to construct a sequence $\left\{t_k\right\}$ that satisfies conditions $\left(i\right)$ through $\left(iv\right)$. Furthermore, we have previously emphasized that the class of regular modular spaces is quite broad, encompassing all Banach spaces and modular function spaces. Consequently, it includes a variety of significant spaces, such as $L^p$, $l^p$, variable exponent versions of these spaces, Orlicz spaces, Musielak–Orlicz spaces, and many other important function and sequence spaces. The existence of natural examples of $\rho$-nonexpansive semigroups within these modular spaces has been established since the work of Khamsi [75], as well as in [98] and the associated literature.

The following example shows how the modular space fixed point results can be employed for the construction of a stationary point of a system defined by the Urysohn integral operator. It is also worthwhile noting that the modular itself is constructed from the components of the operator in question, emphasizing the flexibility advantage of the modular techniques over a traditional norm approach, which would necessitate additional constraints on the operator.

Example 7 

([33] Example 4). Consider the Urysohn operator:

$$T\left(f\right)\left(x\right)={\int }_0^{1}k(x,y,|f\left(y\right)\left|\right)dy+f_0\left(x\right),$$

(10)

where $f_0$ is a fixed function and $f:[0,1]\to \mathbb{R}$ is Lebesgue measurable. For kernel k, we assume that

(a) 

$k:[0,1]\times [0,1]\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is Lebesgue measurable;

(b) 

$k(x,y,0)=0$;

(c) 

$k(x,y,.)$ is continuous, convex, and increasing to $+\infty$;

(d) 

${\displaystyle {\int }_0^{1}k(x,y,t)dx>0}$ for $t>0$ and $y\in (0,1)$.

In addition, assume that for almost all $t\in [0,1]$, and any two measurable functions f and g, the following inequality holds:

$${\int }_0^1\left\{{\int }_0^{1}k(t,u,|k(u,v,|f\left(v\right)\left|\right)-k(u,v,|g\left(v\right)\left|\right)\left|\right)dv\right\}du\le {\int }_0^{1}k(t,u,|f\left(u\right)-g\left(u\right)\left|\right)du.$$

Define

$${\displaystyle \rho \left(f\right)={\int }_0^1\left\{{\int }_0^{1}k(x,y,|f\left(y\right)\left|\right)dy\right\}dx.}$$

It is not difficult to show that ρ is a regular modular, and that $\rho \left(T\right(f)-T(g\left)\right)\le \rho (f-g)$, that is, T is ρ-nonexpansive. To ensure that the Urysohn operator given by $\left(10\right)$ generates a dynamical system, it is essential to verify that, for every $f\in C$, where C is a ρ-ball centered at f, the following initial value problem,

$$\left\{\begin{array}{c}u\left(0\right)=f\hfill \\ u^{\prime }\left(t\right)+(I-T)u\left(t\right)=0,\hfill \end{array}\right.$$

(11)

has a solution $u_f:[0,+\infty ]\to C$. It can be seen that—under certain technical assumptions—Equation $\left(11\right)$ has a solution $u_f$ for every initial value f. Denote $S_t\left(f\right)=u_f\left(t\right)$. Following the pattern given in [75], we can demonstrate that ${\left\{S_t\right\}}_{t\ge 0}$ constitutes a ρ-nonexpansive semigroup of nonlinear operators on C. Therefore, the results of our theory enable us to construct a common fixed point for the semigroup ${\left\{S_t\right\}}_{t\ge 0}$ by applying the implicit iterative process defined in $\left(9\right)$.

As previously noted, this topic remains underexplored and necessitates further theoretical foundations, as well as more compelling application examples. For additional insights, please refer to items (D) and (E) in the future research directions discussed in the following section.

6. Discussion

This article establishes the framework of regular and super-regular modular spaces, offering a comprehensive overview of significant advances in fixed point theory for nonlinear operators within these spaces. Our goal was to develop an open framework applicable not only to Banach spaces and modular function spaces, but also to a broader setting. The discussions presented, along with the underlying and referenced research, demonstrate that the framework of regular and super-regular modular spaces provides a sufficiently rich yet minimalist setting for developing a comprehensive fixed point theory that combines the advantages of both norm and modular methods. The advantage of regular and super-regular modular spaces over established theories, such as generalized Orlicz spaces and modular function spaces, lies in the fact that regular and super-regular modular spaces do not impose any specific assumptions on the nature of their elements. This generality permits the inclusion of all Banach spaces within this framework, resulting in a significant unification of previously isolated and frequently fragmented research areas. At the same time, by imposing the natural restriction of $\rho$-closedness on $\rho$-balls and introducing $\zeta$-convergence as a generalization of weak or almost everywhere convergence, one obtains a more realistic and applicable fixed point theory than is possible in the basic ‘naked’ modular spaces. The documented openness of this framework, along with the established foundations of applicable fixed point theory, creates promising prospects for the further development of both theoretical results and practical applications. Several future research directions are outlined at the end of this section.

The core principles of the framework are outlined in Section 2. This section began with an examination of regular modular spaces in Section 2.1, which starts with Definition 5. Next, the discussion advanced to the crucial concepts of super-regular modular spaces in Section 2.2, introduced with Definition 14. To revisit the fundamental concept of super-regular modular spaces, it centers on the introduction of convergence that enables results analogous to those derived from weak convergence in Banach spaces and $\rho$-almost everywhere convergence in modular function spaces. This advancement is made possible through the framework of modular $LTI$-spaces (as defined in Definition 12), which were introduced in [37] and are based on the principles of L-convergence (defined in Definition 11), originally presented in [39]. Examples 4, 5, and 7 illustrate that the class of super-regular modular spaces is extremely vast, underscoring the significance of advancing this area for both theoretical exploration and practical applications.

The rich structure of super-regular modular spaces facilitates the application of the $\zeta$-normal structure concept, enabling the generalization and unification of Kirk’s Fixed Point Theorem (Theorem 3) and Browder’s Fixed Point Theorems (Theorem 4) within the context of super-regular modular spaces. It also gives rise to the concepts of Kadec–Klee properties for these spaces, as outlined in Definitions 20 and 24. In a manner analogous to established findings in Banach spaces, we observe that the uniform $\zeta -KK1$ property entails the $\zeta$-normal structure (as stated in Theorem 5). This, in turn, leads to the conclusion—through Theorem 3—that the analogous $\zeta$-fixed point property also holds, as demonstrated in Theorem 6. In addition to its applicability to Banach spaces, this fixed point result extends to a broad category of orthogonally additive modular function spaces with the ${\Delta }_2$-type property, as illustrated in Theorem 8 and the accompanying discussion. Despite its broad applicability, we must also take into account scenarios where the function $\rho$ lacks the ${\Delta }_2$-type property. By employing the innovative concept of the strong $\zeta$-Opial property within super-regular modular spaces (see Definition 23 and Remark 1), we established in Theorem 9 that the strong $\zeta$-Opial property guarantees the uniform $\zeta -KK2$ property. This finding raises pertinent open questions (Remark 2).

Expanding beyond the scope of contractions and nonexpansive mappings in super-regular modular spaces, we drew inspiration from works such as [6,31,63,64,65,66,67] to introduce the concepts of asymptotic contractions and asymptotic nonexpansive mappings, along with their pointwise equivalents, as defined in Definitions 25 and 26. The key results in this area generalize the classical fixed point existence results to this new context, as illustrated in Theorem 18 for asymptotic pointwise nonexpansive mappings and in Theorem 19 for the case of asymptotic pointwise contractions. These theorems are established through a series of non-trivial results pertaining to the geometry of super-regular modular spaces, such as the modular uniform convexity property, the minimizing sequence property, and the parallelogram property. For asymptotic pointwise contractions, we reproduced a less restrictive result, where the only assumption was the existence of a special type of function that attains its minimum (Theorem 20). This general result can be applied in situations where the modular $\rho$ is uniformly continuous and has property $\left(R\right)$, which is a modular equivalent of reflexivity, as demonstrated in Theorem 21. We observe that the uniform continuity of convex modulars is equivalent to the ${\Delta }_2$-type property. This equivalence, along with its clear relevance to Banach spaces, shows that Theorem 21 is applicable to a broad range of spaces. These include Lebesgue spaces $L^p$, Orlicz spaces $L^{\phi }$ where the function $\phi$ meets an appropriate ${\Delta }_2$-condition, and variable exponent Lebesgue spaces $L^{p(·)}$ satisfying $1\le p\left(t\right)\le M<\infty$.

Due to the significance of semigroups of nonlinear operators in various fields of pure and applied mathematics, including dynamical systems, differential equations, probability, and stochastic evolution equations, research on this topic within the framework of modular spaces has been ongoing since the 1990s. The common fixed points of such a semigroup ${\left\{T_t\right\}}_{t\in J}$ can be understood as the stationary points of the corresponding dynamical system, remaining invariant under each transformation $T_t$. The theory of super-regular modular spaces provides an opportunity to unify disparate results into a cohesive and comprehensive framework. Theorems 22 and 23, referenced following the recent paper [14], provide foundational results for the existence of common fixed points in the context of asymptotic pointwise nonexpansive and asymptotic pointwise contractive semigroups in uniformly convex regular modular spaces.

Analogous to the situation involving a single operator addressed in Theorem 20, we refer to the common fixed point existence result (Theorem 24), as noted after [80], which does not necessitate uniform convexity. By utilizing the innovative concept of $\zeta$-Opial sets, as defined in Definition 37, we arrive at an additional existence result, outlined in Theorem 26. Furthermore, Remark 5 highlights the broad applicability of these findings to a wide range of significant spaces.

In Section 5, we briefly addressed the issue of fixed point construction processes for operators and semigroups of operators operating within regular and super-regular modular spaces. In contrast to well-established results in Banach, metric, and modular function spaces, similar research in the context of regular modular spaces has only recently begun, with a notable example being the implicit iteration process outlined in Theorem 27.

As mentioned at the beginning of this section, this survey article also aims to identify and discuss current challenges, knowledge gaps, and unresolved questions, offering insights into potential future research opportunities. At this stage of theory development, the following topics represent important and promising areas for future research:

(A) As indicated in Remark 2, which highlights specific open questions, the entire field of research concerning Kadec–Klee, Opial, normal structure, and fixed point properties stands to gain significantly from further investigation. This research should particularly focus on exploring the relationships among these concepts from diverse perspectives.

(B) As noted in the commentary following Proposition 6, it is very important to identify modular conditions for pointwise features that resemble those outlined by the Fréchet derivatives in this Proposition. This would underscore the importance of further developing the theory of pointwise and asymptotic pointwise contractions, as well as pointwise and asymptotic pointwise nonexpansive mappings.

(C) An entire field of research related to monotone nonexpansive operators and operators in modular spaces with graphs is not covered in this survey. Given its wide range of applications, this area, particularly within the context of regular and super-regular modular spaces, is of significant interest. For additional background information, the reader is referred to [99,100,101,102,103,104,105,106] and the literature referenced therein.

(D) The area of fixed point construction processes in regular and super-regular modular spaces is ripe for systematic exploration. This includes a focus on both “strong” $\rho$-convergence and “weak” $\zeta$-convergence to fixed points. In the case of semigroups, it also includes the study of the relevant notions of convergence to common fixed points. Existing methods like the Krasnosel’skii–Mann and Ishikawa processes for modular function spaces can provide a reliable starting point; see, e.g., [6,79,107].

(E) Moreover, the questions surrounding the stability of such processes in the context of regular modular spaces remain largely unresolved. Recent works such as [90,91,92,93,94,95,96,108,109] offer valuable, albeit in different contexts, background information on these topics.

(F) The entire field of fixed point theory for multivalued mappings in regular and super-regular modular spaces remains open for exploration. The following works can serve as starting points [110,111,112,113,114].

(G) Last but certainly not least, the topics concerning the applications of fixed point theory in regular and super-regular modular spaces, frequently mentioned in this study and the cited literature, represent a fascinating area for future research development. In addition to the topics discussed earlier in the paper, areas such as the modeling of electrorheological fluids [115,116], image processing [16,117], and image reconstruction [118,119,120] could also be explored. However, this is by no means an exhaustive list of possibilities.