1. Introduction
The initial breakthrough in addressing the fixed-circle problem was achieved in a study conducted in [
1], which proposed a solution specifically for metric spaces. Since then, subsequent research has focused on exploring new solutions for both metric spaces and generalized metric spaces. For example, Özgür et al. [
1] introduced fixed-circle results using the Caristi-type contraction on metric spaces. This approach was further developed by researchers in [
2,
3], who proved new fixed-circle theorems by employing Wardowski’s technique and classical contractive conditions. The fixed-circle problem was also investigated in the context of S-metric spaces in studies conducted in [
4,
5], where a modified Khan-type contractive condition was utilized to establish a novel fixed-circle theorem in another study [
6]. Additionally, generalized fixed-circle results were obtained for Sb-metric spaces and parametric
-metric spaces, incorporating a geometric perspective. Furthermore, Mlaiki et al. [
7] proposed investigating fixed-circle theorems on extended
-metric spaces. In 2020, Taş extended the concept of fixed circles to include Jaggi-type, Dass–Gupta-type-I, and Dass–Gupta-type-II bilateral contractions. For a more comprehensive understanding of this research direction, we recommend referring to the studies conducted in [
1,
2,
3,
4,
5,
6,
8,
9,
10] and the references therein.
Activation functions play a crucial role in neural networks as they significantly influence decision-making processes. Therefore, selecting the most suitable activation function is essential for effective network analysis. Several studies, such as [
11,
12], have provided comprehensive analyses of different activation functions and their real-world applications. It is worth noting that commonly used activation functions, including the Ramp function, ReLU function, and LeakyReLU function, have fixed-point sets consisting of fixed discs and fixed circles [
10,
13,
14,
15,
16]. Fixed-circle theorems have been established and extended to various aspects, including discontinuous activation functions, as well as rectified linear unit activation functions employed in neural networks. Theoretical studies on neural networks often utilize well-known fixed-point theorems like the Banach fixed-point theorem and Brouwer’s fixed-point theorem. For instance, Li et al. [
17] demonstrated the existence of a fixed point for every recurrent neural network using a geometric approach, with Brouwer’s fixed-point theorem ensuring the existence of a fixed point. This study highlights the significance of adopting a geometric viewpoint and leveraging theoretical fixed-point results in practical applications.
Operator enrichment techniques have emerged as a new avenue in fixed-point theory, inspired by Krasnoselskii’s fixed-point theorem [
18] for nonexpansive operators. These techniques have motivated the exploration of various enriched classes of operators, such as enriched contractions and enriched
-contractions [
19], enriched Kannan contractions [
20], enriched Chatterjea operators [
21], enriched nonexpansive operators in Hilbert spaces [
22], enriched multivalued contractions [
23], enriched Ćirić–Reich–Rus contractions [
24], enriched cyclic contractions [
25], enriched modified Kannan pairs [
26], and enriched quasi contractions [
27]. Notably, Abbas et al. [
23] established fixed-point results by imposing the condition that the orbital subset is a complete subset of a normed space (Theorem 3 of [
23]), while Gronicki and Bisht [
28] considered enriched Ćirić–Reich–Rus contraction operators and proved a fixed-point theorem by imposing the condition that the average operator is an asymptotically regular operator (Theorem 3.1 of [
28]).
Motivated by the research mentioned above, this paper presents innovative solutions to the fixed-circle problem by utilizing Caristi-type, general Jaggi-type bilateral, general Dass–Gupta-type-I, and Dass–Gupta-type-II bilateral contractions enriched with operators. In
Section 2, a brief survey related to the fixed-circle problem is provided.
Section 3 then modifies the known contractive conditions, specifically the Jaggi-type bilateral contraction and the Dass–Gupta-type bilateral contraction, in order to derive new fixed-circle (fixed-disc, common fixed-circle, common fixed-disc) results. Additionally,
Section 4 demonstrates the practical effectiveness of our theoretical findings by applying them to rectified linear units activation functions, thus highlighting their reliance on fixed circles.
2. Definition of the Problem
Consider a metric space and a self-operator A point that satisfies is known as a fixed point of We can denote the set of all fixed points of as .
Self-operators can have either a unique fixed point or multiple fixed points. For example, let us consider the metric space with the function defined as , where . Now, consider the self-operators and defined as and , respectively, for all . In this case, has a unique fixed point , while has two fixed points and . When a self-operator has multiple fixed points, it raises the question of the geometric properties of these fixed points.
This question is known as the “fixed-circle problem” and has been extensively studied from a geometric perspective. The problem was initially discussed in [
1] and has gained significant importance in both theoretical mathematical studies and various practical applications.
Now, let us define the notion of a fixed circle. Consider the metric space , and let be an operator. A circle is defined as the set of all points such that , where is the center and is the radius of the circle .
Now, we can formally state the fixed-circle problem ([
1]) in the context of a metric space. Let
be an operator and
be a circle. If
for every
, then the circle
is referred to as the fixed circle of
.
3. A Survey of Recent Solutions
In 2017, Özgür and Taş initiated the study of fixed circles and proved the results for Caristi-type contractions on a metric space in [
1].
The main results in [
1] are stated as follows:
Theorem 1 ([
1])
. Consider a metric space and let be any circle on . Let us define the operator as If there exists a self-operator on satisfying the following conditions for all - (C1)
;
- (C2)
then the circle is a fixed circle for .
Theorem 2 ([
1])
. Consider a metric space and let be any circle on . Let us define the operator as If there exists a self-operator on satisfying the following conditions for all - (C1)*
;
- (C2)*
then the circle is a fixed circle for .
Theorem 3 ([
1])
. Consider a metric space and let be any circle on . Let us define the operator as If there exists a self-operator on satisfying the following conditions for all and some - (C1)**
;
- (C2)**
then the circle is a fixed circle for .
For more results in this direction, we refer the reader to [
2,
3,
4,
5,
8] and the references mentioned therein. These fixed circle results made a significant contribution in fixed-point theory.
In 2020, Taş [
10] extended the concept of fixed circles to Jaggi-type, Dass–Gupta-type-I, and Dass–Gupta-type-II bilateral contractions in [
10].
The main definitions and results of [
10] are the following:
Definition 1 ([
10])
. If there exist functions and such that for all wherethen is called a Jaggi-type bilateral -contractive operator. Theorem 4 ([
10])
. Let be a Jaggi-type bilateral -contractive operator with and r defined asIf then fixes the circle
Definition 2 ([
10])
. If there exists a function and such that for all where,then is called a Dass–Gupta-type-II bilateral -contractive operator. Theorem 5 ([
10])
. Let be a Dass–Gupta-type-II bilateral -contractive operator with and r defined as in Equation (2). If then fixes the circle 4. New Fixed-Circle Theorems
Throughout this paper, we denote
as the normed space over the field
(the set of all real numbers). Let
be a given operator and
An operator
given by
is called an averaged operator of
. Note that
4.1. Fixed Circle Results for Caristi-Type Contractions
Throughout this section, we denote
as an operator defined by
We start with the following result.
Theorem 6. Let be a normed space and Assume that there exists such that following conditions are satisfied for each ;
;
Then, is the fixed circle of
Proof. Let us denote
Clearly,
for any
and condition
becomes
which can be written equivalently as
Based on the same procedure, for any
, the condition
becomes
which can also be written as
By using (
7) and (
6), we get
Thus, we get for all Hence, is a fixed circle for □
Example 1. Let be a finite measure space. The classical Lebesgue space is defined as the set of all Borel measurable functions such that . It is known that , equipped with the norm , is a Banach space. Let α be a constant, and consider the Borel measurable function for all , satisfying Define the operator as follows: Taking , we have . It follows from the proof of Theorem 6 that the contractive conditions and are equivalent to (7) and (6), respectively. On the other hand, for , the averaged operator becomes It can be easily seen that conditions (7) and (6) are satisfied. Clearly, is a fixed circle of . Theorem 7. Let be a normed space and Assume that there exists such that following conditions are satisfied for each ;
;
Then, is the fixed circle of
Proof. We omit the proof since it follows the same pattern as the proof of Theorem 6. □
Theorem 8. Let be a normed space and Assume that there exist and such that following conditions are satisfied for each ;
Then, is the fixed circle of
Proof. Let us denote
Based on the same procedure as in proof of Theorem 6, the contractive condition
becomes
Similarly, for the value of
the condition
can also be written equivalently in the form of
Using the conditions (
8) and (
9), we obtain
which is contradiction to our assumption since
Therefore, we get
and
is the fixed circle of
□
Remark 1. If we take in Theorems 6–8, we obtain Theorems 1–3, respectively, of [1] in the setting of normed spaces. 4.2. Fixed Circle/Disc Results for Bilateral-Type Contractions
We introduce the following idea.
Definition 3. An operator is called a general Jaggi-type bilateral symmetric contraction if there is an operator and there exist and such that for all we havewhereprovided that for all we have In order to denote the involvement of parameters in (10), we shall also call a -general Jaggi-type bilateral symmetric contraction. Theorem 9. Let be a -general Jaggi-type bilateral symmetric contraction and r be defined as Then, fixes the circle provided that
Proof. Let us denote
Clearly,
The contractive condition (
10) becomes
which can be equivalently written as
where
Moreover, also notice that for the value of
we get
We divide the proof into two following cases.
- Case (i):
Assume that . Then, in this case, we have Clearly, fixes
- Case (ii):
Assume that Clearly, we have Indeed,
By using (
14) and
we obtain
and therefore, we have
Note that
Therefore, condition (
15) becomes
which is a contradiction, so our supposition is wrong and we have
Hence,
for all
; that is,
fixes the circle
□
Corollary 1. Assume satisfies all the assumptions of Theorem 9. If then fixes the disc
Proof. The proof is obvious and hence omitted. □
Definition 4. An operator is called a general Dass–Gupta-type-I bilateral symmetric contraction if there is an operator and there exist and such that for all we havewhereprovided that for all we have In order to denote the involvement of parameters in (17), we shall also call a -Dass–Gupta-type-I bilateral symmetric contraction. We state the following theorem for the class of -Dass–Gupta-type-I bilateral symmetric contractions.
Theorem 10. Let be a -Dass–Gupta-type-I bilateral symmetric contraction and r be defined as (11). Then, fixes the circle provided that Proof. Let us denote
Clearly,
The contractive condition (
17) becomes
where
We divide the proof into the following two cases.
- Case (i):
Assume that Then, in this case, we have Clearly, fixes
- Case (ii):
Assume that Clearly, we have Indeed,
By using (
18) and
we obtain
Since
we have
or
which is a contradiction, so our supposition is wrong and we have
Hence,
that is,
fixes the circle
□
Remark 2. If we take in Theorems 9 and 10, we obtain Theorems 3.2 and 3.10 in [10], respectively, in the setting of normed spaces. Corollary 2. Assume satisfies all the assumptions of Theorem 10. If then fixes the disc
Remark 3. If we take in Corollaries 1 and 2, we obtain Corollaries 3.3 and 3.11 in [10], respectively, in the setting of normed spaces. Definition 5. An operator is called a general -Dass–Gupta-type-II bilateral symmetric contraction if there is an operator and there exist and such that for all and we havewhereprovided that for all we have In order to denote the involvement of parameters in (20), we shall also call a -Dass–Gupta-type-II bilateral symmetric contractions. We state the following theorem for the class of -Dass–Gupta-type-II bilateral symmetric contraction.
Theorem 11. Let be a -Dass–Gupta-type-II bilateral symmetric contraction and r be defined as (11). Then, fixes the circle , provided that Proof. We omit the proof because it follows from similar arguments as used in the proof of Theorem 10. □
Corollary 3. Assume satisfies all the assumptions of Theorem 11. If then fixes the disk
Now, we present an example which supports our result.
Example 2. Let be the usual metric space and be defined by If we take then and thus we have . However, and hence for , we have Thus, does not satisfy the condition of Definition 1.
If we take then In this case, we obtain Moreover, the contractive condition (10) reduces towhereprovided that for all we have Clearly, is a -general Jaggi-type bilateral symmetric contraction. Indeed, for , we have and Hence, satisfies the conditions of Theorem 9 and Corollary 1. Consequently, fixes and
Example 3. Let be the usual metric space and be defined by If we take then and thus we have . However, and hence for we have Thus, does not satisfy the condition of Definition 2.
If we take then In this case, we obtain Moreover, the contractive condition (20) reduces towhereprovided that for all we have Clearly, is a -general Dass–Gupta-type-II bilateral symmetric contraction. Indeed, for , we have and Hence, satisfies the conditions of Theorem 11 and Corollary 3. Consequently, fixes and disk
5. Application
Fixed-circle theorems are relevant to activation functions in neural networks because they offer valuable insights into the behavior of iterative processes and their convergence. Activation functions have a vital role in determining the output of a neuron based on its input within a neural network.
By examining fixed-circle theorems, we can analyze the characteristics of activation functions in terms of fixed points or stable cycles. These theorems establish conditions under which iterative processes, such as forward propagation in a neural network, converge to a fixed point or cycle.
Comprehending the properties of fixed-circle theorems can aid in the design and selection of appropriate activation functions for neural networks. It enables us to evaluate the convergence properties of the network and ensure that the learning process attains the desired solution.
In conclusion, the connection between fixed-circle theorems and activation functions lies in their ability to provide insights into the convergence behavior of neural networks, enabling us to choose suitable activation functions for effective and efficient learning.
Activation functions are pivotal in neural networks as they facilitate learning and interpretation. Their primary function is to transform the input signal of a node in the neural network into an output signal. There are numerous examples of activation functions being utilized in neural networks, with “rectified linear units (ReLUs)” [
10,
29] being one of the most widely used activation functions.
Now, we consider the following activation function
where
The average operator satisfies Theorem 9 on the usual metric space with
and the function
as
for all
. Indeed, for
, we get
and
Hence,
is a Jaggi-type bilateral
-contractive operator. Furthermore, we obtain
Hence, the average operator satisfies the conditions of Theorem 9, so the operator , which is the activation function , fixes the circle and the disc .
6. Conclusions and Future Directions
In this study, we have introduced a comprehensive class of contractive operators called Caristi-type contractions. We have also defined and explored general Jaggi-type bilateral-type contractions, Dass–Gupta-type-I bilateral contractions, and Dass–Gupta-type-II bilateral contractions. These newly defined classes of contractions provide a broader framework for analyzing and studying various types of contractive operators.
We have derived several fixed-circle theorems for different types of contractions. Theorems 6–11 present these results for general Jaggi-type bilateral contractions, Caristi-type contractions, general Dass–Gupta-type-I bilateral contractions, and Dass–Gupta-type-II bilateral contractions, respectively. These theorems demonstrate the versatility and applicability of our approach across different types of symmetric contraction operators.
To support our results, we have provided Examples 2 and 3, which illustrate the effectiveness and applicability of our theorems. These examples highlight the fact that our newly defined contractions are genuine generalizations of other contractions studied in the literature. The obtained results are generalizations of corresponding results in the literature and can be applied to other research areas.
Furthermore, we have showcased the practical application of our main result, Theorem 9, in the context of a novel activation function for neural networks. By demonstrating the effectiveness of this new approach, we contribute to the advancement of neural network research and its potential for improving the performance of various applications.
In terms of future research, it would be interesting to explore the applicability of the contractions defined by Definitions 4 and 5 in the context of partial metric spaces and m-metric spaces. Investigating fixed-circle theorems in these more general settings would further expand our understanding of the geometric properties of fixed points.
Overall, our study provides valuable insights into the theory of contractive operators and opens up new avenues for research and application in various mathematical and applied fields.
Author Contributions
Conceptualization, R.A., M.A., S.R. and M.Z.; formal analysis, S.R., M.D. and M.Z.; investigation, R.A. and M.Z.; writing—original draft preparation, H.S. and M.Z.; writing—review and editing, M.D., S.R. and M.Z.; funding acquisition, M.Z. All authors have read and approved the final manuscript.
Funding
Mi Zhou was partially supported by Key Research and Development Project of Hainan Province (Grant No. ZDYF2023GXJS007); the High-Level Project of Hainan Provincial Natural Science Foundation (Grant No. 621RC602); Sanya City Science and Technology Innovation Special Project (Grant No. 2022KJCX22); and the Key Special Project of University of Sanya (Grant No. USY22XK-04).
Data Availability Statement
Data are contained in the article.
Acknowledgments
The authors are thankful to the reviewers for their useful comments and constructive remarks, which helped to improve the presentation of the paper.
Conflicts of Interest
The authors declare no conflicts of interest.
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