1. Introduction
The Banach contraction theorem, introduced in [
1], is a fundamental result in metric fixed point theory with widespread implications across various mathematical and applied disciplines. It serves as a powerful tool in solving both linear and nonlinear ordinary and integral equations and has meaningful uses in fields such as image processing, fractal geometry, engineering, physics, and biology. The theorem asserts that if a self-mapping
T on a complete metric space
fulfills a contractive condition of the form
for all
, where
, then
T has a unique fixed point in
X. Over time, extensive research has led to various generalizations and refinements of this theorem, including extensions to broader domains [
2] and the introduction of more generalized contractive conditions [
3,
4], further expanding its applicability in both theoretical and practical contexts. Fixed point theory is very important in the study of economics, particularly in the analysis of models of economic growth and production–consumption equilibrium. It provides a mathematical framework within the production–consumption equilibrium framework to guarantee the existence of the equilibrium solutions or locations in markets where supply and demand are equal. Our findings allow economists to study how market forces adjust on their own to reach equilibrium without outside intervention. This technique, which is essential to the study of general equilibrium theory, examines producer–consumer interactions in multiple markets at the same time. Recent studies demonstrate that the Caputo fractional derivative provides a flexible framework for modeling dynamic systems with memory and hereditary properties, capturing long-term dependencies and delayed effects in economic growth more effectively than classical models. This approach has been successfully applied in various fields, such as fractional-order delayed zooplankton–phytoplankton dynamics [
5], glycerol hydrogenolysis via heterogeneous catalysis [
6], the (SEIrIsR) model with fractional infection stages [
7], the seventh-order Caputo fractional KdV equation (including Sawada–Kotera–Ito and Lax applications) [
8], the fractal fractional nonlinear coupled Burgers equation using a hybrid transform approach [
9], and the dynamical analysis of a novel 4D hyperchaotic system with both integer and fractional-order methods [
10].
Recently, Berinde and Pǎcurar [
11] initiated an extended class of contractive operators, known as enriched contractions, which extend classical Banach contraction mappings and incorporate various non-expansive operators found in the literature.
A mapping
on a linear normed space
is called an enriched contraction if there exist
and
such that, for all
, the contractive condition involving
and
holds. Notice that non-expansive mappings do not always guarantee fixed point existence; in contrast, enriched contractions consistently ensure the presence of a unique fixed point. For further insights, we refer to Example 1(2) in [
11]. Let
K be a convex subset of the normed space
X, and consider
with a mapping
. The operator
, defined by
, is the averaged mapping of
T. Notably, the set of fixed points of
coincides with that of
T. Extensive research has been conducted in this domain, with contributions from various scholars; see, for instance, [
12,
13,
14] and the references therein.
Recently, Zhou et al. in [
15] discussed the equivalence and convergence of several higher-order iteration schemes, including the Krasnoselskij iteration, under more general conditions. Their work extends classical fixed point results by considering generalized enriched contraction mappings in closed convex Banach spaces, which broadens the applicability of the Krasnoselskij and other iteration schemes. This contribution serves as a key reference for understanding the theoretical foundation of these iterative methods; additionally, see the related developments in [
16,
17,
18].
Alam and Imdad [
19] established fixed point results by introducing a weak contractive condition rather than the classical Banach contraction condition. This technique requires the contractive condition to be met only for elements belonging to a specific binary relation, rather than the whole space. Because of this study, fixed point theory has been extensively studied in the context of binary relations. Notably, Din, along with collaborators in [
4,
20,
21], investigated Perov’s classical fixed point theorems under this setting. Their work included illustrative examples and applications within vector-valued spaces, highlighting the adaptability of this approach. Furthermore, Alam et al. [
22] expanded the results of [
19] by introducing even more instances of weaker conditions, thereby establishing the existence and uniqueness of fixed points for self-mappings governed by binary relations; see [
23].
2. Preliminaries
We begin by revisiting some fundamentals of order-theoretic metric fixed point theory.
Definition 1 ([
24]).
Consider , and let . For all , either , that is, ς is related to η under , commonly denoted as or , which means that ς is not related to η in .By definition, , ∅, and are well-known examples of binary relations on X, called, respectively, the universal relation, empty relation, and diagonal relation.
Throughout the study, denotes a non-empty binary relation. For simplicity, we will call it “binary relation” instead of explicitly stating “non-empty binary relation”. In the following sections, we introduce essential definitions and propositions that serve as the foundation for metric order-theoretic fixed point theorems. These fundamental notions form the necessary groundwork for further theoretical developments.
Definition 2 ([
24]).
Suppose and ; then, ς and η are said to be -comparative if or . We represent this relation as . Proposition 1 ([
19]).
Consider a metric space with , a self-mapping T on X, and . The following contractive conditions for all are equivalent:- (I)
;
- (II)
.
Definition 3 ([
24]).
Let and . The inverse relation of is given by . The reflexive closure is denoted as , and the symmetric closure is represented as . Proposition 2 ([
19,
24,
25]).
If and , then Definition 4 ([
19]).
A sequence is called -preserving if Definition 5 ([
19]).
Consider a metric space with . The relation is said to be d-self-closed if for any -preserving sequence with under the metric d, there exists a subsequence such that for all . Definition 6 ([
19]).
Let T be a self-mapping on X and . The relation is said to be closed under T if, for all , whenever , then Proposition 3 ([
26]).
If T is -closed, then for all is also -closed. Proposition 4 ([
19]).
Let X, T, and be defined as per Definition 6. If is closed under T, then its symmetric closure also retains the property of T-closure. Before proceeding, we discuss the concepts of -self-closedness and -completeness in the context of normed spaces as follows:
Definition 7 ([
12]).
A normed space is said to be -complete if every -preserving Cauchy sequence converges in norm. Definition 8 ([
12]).
A mapping T is termed -continuous at ς if for any -preserving sequence with implies . If this condition is met for all , then T is said to be -continuous. Every complete metric space is naturally -complete, irrespective of the chosen binary relation . In particular, when corresponds to the universal relation, -completeness corresponds to the classical notion of completeness in metric spaces. Additionally, any continuous mapping is inherently -continuous for all , and if is universal, -continuity is equivalent to standard continuity.
Definition 9 ([
12]).
In a normed space X with norm , the relation is said to be -self-closed if, for any -preserving sequence converging to ς in norm, there exists a subsequence such that for all . Definition 10 ([
27]).
Let with . A subset is known as -directed if, for every , ∃, which implies and Definition 11 ([
28]).
Let with . For , a path of length k (where k is a natural number) in from ς to η is a finite sequence satisfying , , and for each i with . Not long ago, Jleli and Samet [
3] presented a well-known fixed point theorem in the complete metric space
. It uses the following generalized contraction condition:
where
, and
is a mapping satisfying specific properties, which are given below, and
. The contraction defined above is referred to as a
-contraction. The set of functions
satisfy the following conditions:
is non-decreasing;
For any sequence , ;
There exist and such that .
This is denoted by
in [
3]. Examples of functions belonging to
include
- (i)
, for all ;
- (ii)
, for and .
It was shown in [
3] that any
-contraction over a Banach space possesses a unique fixed point. Afterwards, numerous generalizations and extensions of this result have been developed, as seen in [
29,
30,
31].
Building upon the work of Rizwan et al. [
14] and the contributions of Alam and Imdad with their collaborators [
19,
22], this study introduces enriched
-contractions and generalized enriched
-contractions, which extend the concepts of enriched contractions,
-contractions, enriched
-contractions, Kannan contractions, and many more. To illustrate the applicability of our findings, we provide examples that validate the theoretical framework. The stability of the production–consumption equilibrium and the economic growth model are also examined in this study.
Finally, we revisit certain fundamentals and useful properties, which are necessary for our next discussions. Throughout this paper, we employ the following notations:
: The collection of all fixed points of T;
;
: The set of all paths in connecting and ;
For , we define .
3. Main Results
In this section, we introduce the notions of enriched -contractions and generalized enriched -contractions and present fixed point results for these newly defined contractions, accompanied by illustrative examples.
Definition 12. Let be a normed space, and let be a binary relation on , where . A mapping is called an enriched -contraction if there exist , , and such that for all with , and such that To highlight the role of binary relation , constant , and control function in Definition 12, the mapping T is often referred to as an enriched -contraction.
Definition 13. Let be a normed space, and let be a binary relation on , where . A mapping is called a generalized enriched -contraction if there exist , , and such that for all with , and such that To highlight the role of binary relation , constant , and control function in Definition 13, the mapping T is often referred to as a generalized enriched -contraction.
Remark 1. It is important to note that conditions (2) and (3) are not required to hold universally across the entire space X; rather, they apply only to points that are related under a specified binary relation . Consequently, this formulation provides a weaker variant of the enriched -contractions introduced by Rizwan et al. [14], offering a more generalized framework for fixed point analysis within relational metric structures. However, if one considers , then both definitions become equivalent, as the condition (2) would hold universally for all pairs of points in X. By the symmetry property of the norm , the following result follows immediately. Consequently, we leave the details of the proof to the interested readers.
Proposition 5. A mapping T is an -contraction if and only if T is also an -contraction.
We now present the first main result involving the enriched -contractions.
Theorem 1. Let be a normed space, and let be a binary relation on , where . Further suppose that
- 1.
is -complete;
- 2.
T is an enriched -contraction;
- 3.
is -closed, where ;
- 4.
is -self-closed, or is -continuous;
- 5.
.
Then, T has a fixed point in , and the sequence , given by for all , converges to some fixed point of T, depending on the choice of initial point .
Proof. It follows from (
2) that for all
, with
and
, we obtain
This demonstrates that is an enriched -contraction with .
In lieu of assumption (5), there exists an
such that
. Define a Krasnoselskij iterative scheme
starting from the initial guess
by
As
, using the
-closedness of
and Proposition 3; then, we obtain
which further implies that
This shows that
is an
-preserving sequence. Next, we suppose that
for all
. Otherwise, if
for some
, it follows that
, and the proof is complete. Therefore, for
for all
, Equation (
4) implies that
Continuing this process, we obtain
Thus, we have
which, together with
, implies
Moreover, from
, there exist
and
for which
Let
, and define
. Then, according to the definition of the limit and (
10), there exists
such that for all
,
so that we obtain
Equivalently, for all
, we have
Next, assume
, and let
. Therefore, there exists
whereby for all
,
Then, for all
, we obtain
Using (
8), (
11), and (
12), we derive
Taking the limit as
in (
13), we obtain
Thus, from (
13), there exists
such that for all
,
Next, to show
is a Cauchy sequence, let
and
such that
. Then, from (
14), we obtain
Since the series converges, it follows that is a Cauchy sequence. Therefore, according to the -completeness of X, there exists such that converges to . Finally, from assumption (4), we will show that is a fixed point of T. Since is -continuous and is an -preserving sequence with in norm, it follows that in norm. By the uniqueness of the limit, we conclude that , meaning is a fixed point of , and thus, also a fixed point of T.
On the other hand, taking into account that
is
-self-closed, and since
is
-preserving with
in norm, there exists a subsequence
such that
Using assumption (2) with property
and Proposition 5, along with the convergence
, we obtain that
Hence, Thus, which shows that is a fixed point of . Consequently, stands as a fixed point for both and T. □
Next, we focus on the uniqueness of the fixed point for our enriched -contractions. To achieve uniqueness, we impose additional conditions on the contraction mapping and the underlying relation. These extra constraints ensure that no two distinct fixed points can exist.
Theorem 2. Under all the assumptions of Theorem 1, and additionally supposing thatthe self-mapping T admits a unique fixed point. Proof. The existence part follows directly from Theorem 1. We now proceed to demonstrate the uniqueness part. In contrast, assume that
T has two fixed points:
and
. Then, there exists a path (denoted as
) of some finite length
m in
from
to
such that
So, by letting
and using the triangle inequality together with property
and (
14), we obtain
Thus, . □
If is complete or is -directed, the following result is noteworthy.
Corollary 1. If T fulfills the assumptions of Theorem 2, then T possesses a unique fixed point, provided that at least one of the following holds: Proof. Suppose condition
holds. Then, for all
, the pair
belongs to
and, hence, forms a path of length 1 in
. This guarantees that
and the conclusion follows directly from Theorem 2.
Now, suppose that condition
holds. Then, for any
, there exists
such that both
, and
belong to
. These pairs form a path of length 2 in
, ensuring that
Consequently, Theorem 2 implies that T has a unique fixed point. □
By setting
in Theorem 1, we obtain a fixed point result of theoretic order, which is the main result of Din et al. [
21] in normed spaces.
Corollary 2. Let be a normed space, and let be a binary relation on , where . Further, suppose that
- 1.
is -complete;
- 2.
T is a theoretic-order χ-contraction;
- 3.
is T-closed;
- 4.
is -self-closed, or T is -continuous;
- 5.
.
Then, T has a fixed point in X, and the iteration scheme , given by for all , converges to some fixed point of T, depending on the choice of initial point .
By setting the binary relation
in Theorem 1, our corollary recovers the main result of Rizwan et al. [
14].
Corollary 3. Let be a normed space where
- 1.
is complete;
- 2.
T is an enriched -contraction.
Then, T has a unique fixed point in X, and the scheme , given by for all , converges to a unique fixed point of T for any choice of initial point , where .
Setting
in Theorem 1 yields fixed point results for JS-contractions (i.e.,
-contractions), as introduced by Jleli and Samet in [
3] in normed spaces.
Corollary 4. Let be a normed space. Suppose the following conditions hold:
- 1.
is complete;
- 2.
T is a χ-contraction.
Then, T has a unique fixed point in X, and the Picard iteration scheme , given by for all , converges to a unique fixed point of T for any choice of .
Similarly, by imposing different conditions on , , and a subset of X or the control function , many fixed point results available in the literature can be recovered as special cases of our framework. In what follows, we present an illustrative example that demonstrates our definition is a proper generalization of the existing contraction mappings.
Example 1. Let be the usual normed space, and set . Consider the mapping defined by
If we set , then the parameter λ becomes , yielding the averaged operator
By routine calculation, we can verify that T is a non-self-enriched -contraction, where and with . Moreover, note that , being a closed subset of X, is complete. Next, whenever , we have and . Then, for such ς and η, we obtain
;
when , and when .
Thus, in all cases, . Hence, is -closed, and (since ). Finally, it is evident that is not continuous. Let be a -preserving sequence in X such thatwith for all . Since , it follows that for every j. As the interval is closed in X, we have for all j. Therefore, is -self-closed. It is also easy to prove that Consequently, all the assumptions of Theorem 2 are met; therefore, T has a unique fixed point, namely . The convergence behavior of the Krasnoselskij iteration is clearly illustrated in Table 1, showcasing the numerical results for selected initial points, and this is further visualized in Figure 1, demonstrating the iterations. These representations highlight the method’s efficiency and stability. Remark 2. It should be noted that Example 1 plays a crucial role. In this example, the mapping T is neither a χ-contraction in the sense of Jleli and Samet, (since T is not continuous) nor an enriched contraction, as proposed by Berinde and Pecurar, as it is neither a self-mapping nor continuous. It is also clear that by taking and with , we observe thatindicating the absence of any , satisfying (1). Moreover, it does not satisfy the conditions of an enriched -contraction, as defined by Rizwan et al. [14]. However, within our framework, T qualifies as an enriched -contraction. Hence, our results provide a proper generalization of the existing work. Example 2. Let and with the usual norm . Let be a mapping defined byand construct a binary relation . Since T is not continuous, it is not a Banach contraction, an enriched contraction, nor even a JS-contraction, as discussed in [1,3,11]. It can be also seen that for and , we have and so we cannot find any for any such that it satisfies the following inequality with : Therefore, it is not an enriched -contraction, introduced by Rizwan et al. in [14]. However, taking implies , and hence the operator T coincides with its averaged operator . Notice that there exists such that ; moreover, whenever , then , showing that the binary relation is T-closed. Finally, for any , we obtainand hence the inequality in Definition 12 holds for all , that is, T is an enriched -contraction. As all assumptions of Theorem 1 are satisfied, T has a fixed point. Next, we present another fixed point result, which is the generalization of Theorem 1, and it is stated as follows:
Theorem 3. Let be a normed space, and let be a binary relation on , where . Suppose the following conditions hold:
- 1.
is -complete;
- 2.
T is a generalized enriched -contraction;
- 3.
is -closed, where ;
- 4.
is -continuous;
- 5.
.
Then, T has a fixed point in and the Krasnoselskij iteration scheme , given by for all , converges to some fixed point of T, depending on the choice of initial point .
Proof. According to the definition of the generalized enriched
-contraction, it follows that for all
, with
and
that
This demonstrates that is a generalized enriched -contraction with .
In lieu of assumption (5), there exists an
such that
. Define a Krasnoselskij iterative scheme
starting from the initial guess
by
As
, by using the
-closedness of
and Proposition 3, we obtain
which further implies that
This shows that
is a
-preserving sequence. Next, we suppose that
for all
. Otherwise, if
for some
, it follows that
, and the proof is complete. Therefore, for
for all
, Equation (
15) implies that
If for some
,
. Then, using (
18), we obtain a contradiction. Indeed,
Hence, for all
, we have
Therefore, (
18) implies for all
that
Continuing this process, we obtain
Thus, we have
which, together with
, implies
Next, by using the same steps as in the proof of Theorem 1, finally, for all
, we obtain
To show
is a Cauchy sequence, take two natural numbers:
m and
such that
. Then, from the triangular inequality and (
21), we obtain
Since the series converges, it follows that is a Cauchy sequence. Therefore, according to the -completeness of X, there exists such that converges to . Using assumption (4), we show that is a fixed point of T. Since is -continuous and is an -preserving sequence with in norm, it follows that in norm. According to the uniqueness of the limit, we conclude that , meaning is a fixed point of , and thus, also a fixed point of T. □
By setting
in Theorem 3, we obtain a generalized fixed point result of theoretic order, which is an extension of the main result of Din et al. [
21] in normed spaces.
Corollary 5. Let be a normed space, and let be a binary relation on , where . Suppose the following conditions hold:
- 1.
is -complete;
- 2.
T is a generalized theoretic-order -contraction;
- 3.
is T-closed;
- 4.
T is -continuous;
- 5.
.
Then, T has a fixed point in X, and the iteration scheme , given by for all , converges to some fixed point of T, depending on the choice of initial point .
By setting the binary relation in Theorem 3, we obtain the following corollary:
Corollary 6. Let be a normed space, where is complete and T is a generalized enriched -contraction. Then, T has a unique fixed point in X, and the Krasnoselskii iteration scheme , given by for all , converges to a unique fixed point of T for any choice of initial point , where .
Setting in Theorem 3 yields extended fixed point results for JS-contractions (i.e., generalized -contractions) in normed spaces.
Corollary 7. Let be a Banach space, and let T be a generalized χ-contraction. Then, T has a unique fixed point in X, and the Picard iteration scheme , given by for all , converges to a unique fixed point of T for any choice of .
Similarly, by imposing different conditions on , , and a subset of X or the control function , many generalized fixed point results can be recovered as special cases from our study.
5. Conclusions and Future Directions
In conclusion, we have introduced two new classes: enriched -contractions and generalized enriched -contractions over normed spaces connected with a binary relation. Our results expand on prior ideas and offer a more comprehensive contractive structure, which advances the field of fixed point theory. These classes generalize several well-known contraction types, including -contractions, Picard–Banach contractions, enriched contractions, Kannan contractions, Zamfirescu contractions, non-expansive mappings, and -enriched contractions. We establish fixed point results in normed spaces equipped with binary relations, preserving key symmetric properties, ensuring consistency, and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched -contractions, employing a suitable binary relation, or control function , our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, we apply our findings to guarantee the stability of the production-consumption market equilibrium and the economic growth model. This study provides a new direction in this field and expands the scope of applicable solutions.
It would be interesting in future work to explore numerical simulations, as they could help support and illustrate the theoretical results we’ve established in this study. We hope to include such simulations in future work to better understand how the proposed contraction types perform in practice. Moreover, it would be interesting to generalize these results further by combining enriched and generalized enriched contractions with other fixed point techniques, for example, by replacing the Krasnoselskij iteration with Mann or Ishikawa iterations, adopting Suzuki-type contractions, or introducing suitable control functions. These extensions could provide a broader framework and deeper insights into the behavior of iterative schemes in normed spaces.