In this study, we characterize a novel contraction mapping referred to as αji-(DC(PˆE))-contraction in light of DC-contraction mappings associated with the Geraghty-type contraction and E-type contraction. Besides, a novel common fixed-point theorem providing such mappings is demonstrated in the context of partial ♭-metric spaces. It is stated that the main theorem is a generalization of the existing literature, and its comparisons with the results are expressed. Additionally, the efficiency of the result of this study is demonstrated through some examples and an application to homotopy theory.
Citation: Leyla Sağ Dönmez, Abdurrahman Büyükkaya, Mahpeyker Öztürk. Fixed-point results via αji-(DC(PˆE))-contractions in partial ♭-metric spaces[J]. AIMS Mathematics, 2023, 8(10): 23674-23706. doi: 10.3934/math.20231204
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In this study, we characterize a novel contraction mapping referred to as αji-(DC(PˆE))-contraction in light of DC-contraction mappings associated with the Geraghty-type contraction and E-type contraction. Besides, a novel common fixed-point theorem providing such mappings is demonstrated in the context of partial ♭-metric spaces. It is stated that the main theorem is a generalization of the existing literature, and its comparisons with the results are expressed. Additionally, the efficiency of the result of this study is demonstrated through some examples and an application to homotopy theory.
Metric fixed-point theory is a prominent and essential topic of functional analysis. As expected, this field has had a flood of scientific activity. Because of its elegance, simplicity, and ease of application in various mathematics disciplines, the Banach fixed-point theorem [1], also known as the Banach contraction principle, is unquestionably crucial for the metric fixed-point theory. The existence and uniqueness of a fixed-point of contraction mappings on a complete metric space are guaranteed by this theorem, which also gives a constructive approach to finding the fixed-point.
One beneficial aspect of studying the metric fixed-point theory is generalizing the metric space structure under consideration. Furthermore, generalizing the metric function has emerged as one of the most intriguing and profitable research topics. New structures have arisen through the modification of certain aspects of the distance function or the addition of some new features to this function, and many new topological structures have been added to the literature. Additionally, researchers successfully apply these novel metric functions by studying summability theory, sequence spaces, Banach space geometry, fuzzy theory, and so on. The ♭-metric function, which has a constant in the triangle inequality, has been the most visible extension of the metric function in the previous 40 years. Bakhtin [2] and Czerwik [3] expressed the ♭-metric function as an expanded form of metric functions. The triangle inequality of the known metric function has been replaced by a more general inequality by using a constant ς≥1. It corresponds to the standard metric function with ς=1. Some researchers have studied fixed-point theorems in ♭-metric spaces, as detailed in [4,5,6,7,8,9,10,11,12,13].
In 1992, Matthews [14] proposed a partial metric function, which is an intriguing expansion of the metric function. Partial metric spaces are metric space extensions in which any of the points has a non-zero self-distance. This topic has a wide range of applications in various branches of mathematics, computer science and semantics and has also become a thriving area in metric fixed-point theory. Currently, a number of researchers depend on the partial metric as a crucial idea for investigating the presence and uniqueness of a fixed-point for mappings that satisfy various contractive conditions; for details, please see [15,16,17,18,19,20,21,22,23].
Many new metric functions have appeared in the literature as a result of combining various distance functions. These structures have significance in the study of metric fixed-point theory, summability theory, fuzzy theory and other related topics. The study conducted by Brzdek et al. [24] in 2018 is recognized as an example of one of the most significant studies in which metric frameworks are employed simultaneously.
The partial ♭-metric, which is a new concept combining the above-mentioned partial metric and ♭-metric structures, was developed in 2013 by Mustafa et al. [25], and its modification was introduced by Shukla [26] in 2014. The properties of this newly defined space were examined, and generalizations of the metric fixed-point theory were obtained, see [27,28].
On the other hand, many authors have attempted to generalize the Banach contraction principle by applying auxiliary functions in various abstract spaces to gain more constructive results in fixed-point theory. This research is still garnering attention today.
The Geraghty-type contraction, one of the most significant variations of the Banach contraction, was highlighted in 1973 by Geraghty [29]. Like the Banach contraction principle, the Geraghty contraction principle appealed to the researchers. It has found a place in the literature in several investigations, notably, the ones mentioned in [30,31,32].
Samet et al. [33] proposed the ideas of α-admissibility and α-ψ-contraction mappings and some fixed-point insights for these contractions were put forward. Soon after, various researchers' evaluations appeared in the literature [34,35,36]. Cho et al. founded diverse fixed-point theorems by combining α-admissibility and Geraghty contractions in [37]. Afshari et al. [38] focused on these results by implementing the idea of generalized α-ψ-Geraghty contraction mappings and investigating the existence and uniqueness of a fixed-point for such mappings in the context of ♭-metric spaces.
Meanwhile, Fulga and Proca [39] proposed the premise of an E-contraction in 2017. In the following year, Fulga and Proca [40] setup a fixed-point theorem for the PE-Geraghty contraction, and some findings were brought about by implementing this concept; see [41,42,43]. In 2018, Alqahtani et al. [44] confirmed a common fixed-point theorem on complete metric spaces by applying the Geraghty contraction of type ES,O. The following year, Aydi et al. [45] presented the α-PE-Geraghty contraction on a ♭-metric space, and some fixed-point findings were achieved. Lang and Guan [46] recently introduced αji-PES,O-Geraghty contraction mappings and established common fixed-point findings for generalized αji-PES,O-Geraghty contraction mappings in ♭-metric spaces.
Liu et al. [47] identified the DC-class, which encompasses the F-contractions given by Wardowski [48]. See [49,50] for additional knowledge on DC-contractions.
This study aims to expound upon the generalized αji-(DC(PˆE))-contraction by using the aforementioned concepts and to prove some fixed-point and common fixed-point theorems for such a contraction in partial ♭-metric spaces. It is seen that the obtained results generalize and improve many of the already existing results in the literature. At the same time, the presented examples support the accuracy of the results obtained. Finally, the application of the homotopy theory also illustrates the conclusion that the proposed study is multifaceted.
The fundamental principles that are essential to this study are set forth below.
Definition 2.0.1. ([14]) A function ℘:U×U→R+, where U is a nonempty set, is called a partial metric if the following assertions are provided:
℘1. ℘(ȷ,ȷ)=℘(ȷ,ρ)=℘(ρ,ρ)⇔ȷ=ρ;
℘2. ℘(ȷ,ȷ)≤℘(ȷ,ρ);
℘3. ℘(ȷ,ρ)=℘(ρ,ȷ); ℘4. ℘(ȷ,ρ)≤℘(ȷ,r)+℘(r,ρ)−℘(r,r)
for all ȷ,ρ,r∈U. The pair (U,℘) denotes a partial metric space.
It is evident from (℘1) and (℘2) that ȷ=ρ provided that ℘(ȷ,ρ)=0. However, ℘(ȷ,ρ)=0 might not hold for each ȷ∈U. This means that every metric space is a partial metric space, but the converse is not necessarily accurate.
Remark 2.0.2. ([17]) In (U,℘), for all ȷ∈U and ε>0, the ensuing sets
B℘(ȷ,ε)={ρ∈U:℘(ȷ,ρ)<℘(ȷ,ȷ)+ε} |
and
B℘[ȷ,ε]={ρ∈U:℘(ȷ,ρ)≤℘(ȷ,ȷ)+ε} |
denote ℘-open balls and ℘-closed balls, respectively. A T0 topology arises over (U,℘) via a base family of ℘-open balls
{B℘(ȷ,ε):ȷ∈U,ε>0}. |
Example 2.0.3. ([20]) Let U=[0,1] and ℘(ȷ,ρ)=max{ȷ,ρ} for all ȷ,ρ∈U. Then, (U,℘) denotes a partial metric space. However, this is not a metric space.
Example 2.0.4. ([14]) Let U=[0,1]∪[2,3] and define ℘:U×U→[0,∞) by
℘(ȷ,ρ)={max{ȷ,ρ},{ȷ,ρ}∩[2,3],|ȷ−ρ|,{ȷ,ρ}⊂[0,1]. |
Then, ℘ fulfills all partial metric conditions. As a result, (U,℘) is a partial metric space.
Definition 2.0.5. ([3]) Assume that U is a nonempty set. A ♭-metric has been identified as the function ♭:U×U→[0,∞), which possesses the subsequent attributes for any ȷ,ρ,r∈U:
♭1. ♭(ȷ,ρ)=0⇔ȷ=ρ;
♭2. ♭(ȷ,ρ)=♭(ρ,ȷ);
♭3. there is a real constant ς≥1 that implements ♭(ȷ,ρ)≤ς[♭(ȷ,r)+♭(r,ρ)].
(U,♭) stands for a ♭-metric space with the coefficient ς.
If ς=1, the function ♭ is an ordinary metric. In this circumstance, each metric is a ♭-metric. Nevertheless, the reverse is not generally accurate.
Example 2.0.6. ([10]) Let (U,d) be a metric space. The function ♭:U×U→[0,∞), which is defined as ♭(ȷ,ρ)=[d(ȷ,ρ)]μ, is a ♭-metric for all ȷ,ρ,r∈U and μ>1. So, (U,♭) is a ♭-metric space with ς=2μ−1.
In 2013, Mustafa et al. [25] introduced the idea of a partial ♭-metric, which is considered an improvement of the partial metric and ♭-metric, which was subsequently refined by Shukla [26] in 2014.
Definition 2.0.7. ([25]) Assume that U is a nonempty set. If the succeeding characteristics are met for all ȷ,ρ,r∈U, ℘♭:U×U→R+ is referred to as a partial ♭-metric:
(℘♭1) ℘♭(ȷ,ρ)=℘♭(ȷ,ȷ)=℘♭(ρ,ρ)⇔ȷ=ρ;
(℘♭2) ℘♭(ȷ,ȷ)≤℘♭(ȷ,ρ);
(℘♭3) ℘♭(ȷ,ρ)=℘♭(ρ,ȷ);
(℘♭4) a real number ς≥1 exists such that
℘♭(ȷ,ρ)≤ς[℘♭(ȷ,r)+℘♭(r,ρ)−℘♭(r,r)]+1−ς2(℘♭(ȷ,ȷ)+℘♭(ρ,ρ)). | (2.1) |
Thus, the pair (U,℘♭) specifies a partial ♭-metric space via the coefficient ς.
Shukla [26] revised the partial ♭-metric approach by using the property (℘♭4′) instead of (℘♭4), which is given below:
(℘♭4′) for all ȷ,ρ,r∈U
℘♭(ȷ,ρ)≤ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). | (2.2) |
Throughout the paper, we use the definition of a partial ♭-metric in the sense of Shukla [26]. The notations P♭M and P♭MS, will be used throughout this work to designate the ideas of the partial ♭-metric and partial ♭-metric space, respectively.
Remark 2.0.8. ([25,26,27,28]) With the same coefficient ς≥1, every ♭-metric space is a P♭MS; taking the coefficient ς=1, every partial metric space is a P♭MS. Moreover, a P♭M on U is not a partial metric, nor a ♭-metric, in general. As far as we comprehend, a P♭MS includes the set of a ♭-metric space and partial metric space.
Example 2.0.9. ([26])
i. Let U=[0,∞) and ω>1 be a fixed element and define ℘♭:U×U→R+ by ℘♭(ȷ,ρ)={max{ȷ,ρ}}ω+|ȷ−ρ|ω for all ȷ,ρ∈U. Then, (U,℘♭) is a P♭MS with the coefficient ς=2ω>1. It is not a ♭-metric or a partial metric space.
ii. Assume that a>0 is a fixed element and U=[0,∞). Define ℘♭:U×U→R+ in order to become ℘♭(ȷ,ρ)=max{ȷ,ρ}+a for all ȷ,ρ∈U. Then, (U,℘♭) is a P♭MS with the coefficient ς≥1.
iii. Let q≥1 and (U,℘) be a partial metric space. ℘♭:U×U→R+ is a P♭M with the coefficient ς=2q−1 if this mapping is defined by ℘♭(ȷ,ρ)=[℘(ȷ,ρ)]q.
iv. Let U={1,2,3,4} and ℘♭:U×U→R+ be a P♭M with the coefficient ς=4, where P♭M is given by
℘♭(ȷ,ρ)={|ȷ−ρ|2+max{ȷ,ρ},ȷ≠ρ;ȷ,ȷ=ρ≠1;0,ȷ=ρ=1. |
It is clear that ℘♭(2,2)=2≠0, so it is not a ♭-metric. Also,
℘♭(3,1)=7>5=℘♭(3,2)+℘♭(2,1)−℘♭(2,2) |
is obtained. As seen here, ℘♭ is not a partial metric.
v. With ς≥1, assume that ℘ is a partial metric and ♭ is a ♭-metric on a nonempty set U. ℘♭ is a P♭M with the coefficient ς≥1 on U, where ℘♭ is characterized with ℘♭(ȷ,ρ)=℘(ȷ,ρ)+♭(ȷ,ρ).
We confer two novel instances to diversify the partial ♭-metric idea.
Example 2.0.10. i. Let U=(0,1]⋃{2,3,4,…}, and for q≥1, ℘♭:U×U→R+is a P♭M with the coefficient ς=2q−1, where ℘♭ is specified by the subsequent expression:
℘♭(ȷ,ρ)={|1ȷ−1ρ|q+1,ȷ,ρ∈(0,1],ȷ≠ρ;emax{ȷ,ρ}−min{ȷ,ρ},ȷ,ρ∈{2,3,4,…},ȷ≠ρ;1,ȷ=ρ. | (2.3) |
ii. Let U=N and ℘♭:U×U→R+ be a P♭M with the coefficient ς=4, where ℘♭ is defined as follows:
℘♭(ȷ,ρ)={emax{|ȷ−ρ|,ȷ+ρ2},ȷ≤ρ;e|ȷ−ρ|2+ȷ+ρ2,ȷ>ρ. | (2.4) |
Proof. i. The axioms (℘♭1)–(℘♭3) are apparent. In order to show the validity of (℘♭4), the following six cases will be examined.
Case 1: For ȷ,ρ,r∈(0,1] and ȷ≠ρ≠r, since q≥1, 2q−1≥1; we have
℘♭(ȷ,ρ)=1+|1ȷ−1ρ|q=1+|1ȷ−1r+1r−1ρ|q≤(2q−1)+2q−1[|1ȷ−1r|q+|1r−1ρ|q]≤2q−1[|1ȷ−1r|q+|1r−1ρ|q+2]−1≤2q−1[|1ȷ−1r|q+1+|1r−1ρ|q+1]−1−|1r−1r|q=2q−1[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r)=ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 2: For ȷ,ρ∈(0,1], ȷ≠ρ and ȷ=r, we conclude that
℘♭(ȷ,ρ)=1+|1ȷ−1ρ|q=1+|1r−1ρ|q+1−1−|1r−1r|q≤2q−1[1+|1r−1ρ|q+1]−[1+|1r−1r|q]=2q−1[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r)=ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 3: A similar consequence is true for ȷ,r∈(0,1], ȷ≠r and ȷ=ρ.
Case 4: For ȷ,ρ,r∈{2,3,4,...} and ȷ≠ρ≠r, owing to fact that
max{ȷ,ρ}−min{ȷ,ρ}=max{ȷ+r−r,ρ+r−r}−min{ȷ+r−r,ρ+r−r}≤max{ȷ,r}+max{r,ρ}−max{r,r}−min{ȷ,r}−min{r,ρ}+min{r,r}=max{ȷ,r}−min{ȷ,r}+max{r,ρ}−min{r,ρ}−[max{r,r}−min{r,r}], |
we derive
emax{ȷ,ρ}−min{ȷ,ρ}≤emax{ȷ,r}−min{r,ρ}+emax{r,ρ}−min{r,ρ}−emax{r,r}−min{r,r}, |
which is the desired inequality.
Case 5: For ȷ,ρ∈{2,3,4,...}, ȷ≠ρ and ȷ=r, since
max{ȷ,ρ}−min{ȷ,ρ}=max{r,ρ}−min{r,ρ}=0+max{r,ρ}−min{r,ρ}−0, |
we have
emax{ȷ,ρ}−min{ȷ,ρ}=e0+emax{r,ρ}−min{r,ρ}−e0℘♭(ȷ,ρ)=1+℘♭(r,ρ)−1≤ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 6: This case is provided for ȷ,r∈{2,3,4,...}, ȷ≠r and ȷ=ρ, as in Case 5.
Consequently, since (℘♭1)–(℘♭4) are provided, the mapping indicated in (2.3) is a P♭M with ς=2q−1.
However, for ȷ=ρ, by definition, since ℘♭(ȷ,ρ)=1, ℘♭(ȷ,ρ)=0 does not occur. The first axiom of the ♭-metric does not hold. So, a partial ♭-metric need not be a ♭-metric.
ii. The axioms of (℘♭1)–(℘♭3) are evident. To demonstrate (℘♭4), we need to examine the six cases mentioned below.
Case 1: Let ȷ≤r≤ρ. Then,
max{|ȷ−ρ|,ȷ+ρ2}=max{|ȷ−r+r−ρ|,ȷ+r+r+ρ2−r}≤max{|ȷ−r|+|r−ρ|,ȷ+r2+r+ρ2−r}=max{|ȷ−r|,ȷ+r2}+max{|r−ρ|,r+ρ2}−max{|r−r|,r+r2}; |
we have
emax{|ȷ−ρ|,ȷ+ρ2}≤emax{|ȷ−r|,ȷ+r2}+emax{|r−ρ|,r+ρ2}−emax{|r−r|,r+r2}. |
Thereby, the following expression is ensured for all ς≥1:
℘♭(ȷ,ρ)≤℘♭(ȷ,r)+℘♭(r,ρ)−℘♭(r,r)≤ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 2: Let ȷ≤ρ≤r. Because
max{|ȷ−ρ|,ȷ+ρ2}=max{|ȷ−r+r−ρ|,ȷ+r+r+ρ2−r}≤max{|ȷ−r|+|r−ρ|,ȷ+r2+r+ρ2−r}=max{|ȷ−r|,ȷ+r2}+max{|r−ρ|,r+ρ2}−max{|r−r|,r+r2}≤max{|ȷ−r|,ȷ+r2}+|r−ρ|2+r+ρ2−max{|r−r|,r+r2}, |
we have
emax{|ȷ−ρ|,ȷ+ρ2}≤emax{|ȷ−r|,ȷ+r2}+e|r−ρ|2+r+ρ2−emax{|r−r|,r+r2}. |
Then, for all ς≥1, we procure
℘♭(ȷ,ρ)≤℘♭(ȷ,r)+℘♭(r,ρ)−℘♭(r,r)≤ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 3: Let r≤ȷ≤ρ. Then, for all ς≥1, we achieve
max{|ȷ−ρ|,ȷ+ρ2}=max{|ȷ−r+r−ρ|,ȷ+r+r+ρ2−r}≤max{|ȷ−r|+|r−ρ|,ȷ+r2+r+ρ2−r}=max{|ȷ−r|,ȷ+r2}+max{|r−ρ|,r+ρ2}−max{|r−r|,r+r2}≤|ȷ−r|2+ȷ+r2+max{|r−ρ|,r+ρ2}−max{|r−r|,r+r2}, |
and
emax{|ȷ−ρ|,ȷ+ρ2}≤e|ȷ−r|2+ȷ+r2+emax{|r−ρ|,r+ρ2}−emax{|r−r|,r+r2}. |
Case 4: Let ȷ>r>ρ. Because
|ȷ−ρ|2+ȷ+ρ2=|ȷ−r+r−ρ|2+ȷ+r−r+ρ2−r≤22[|ȷ−r|22+|r−ρ|22]+ȷ+r2+r+ρ2−r≤22[|ȷ−r|2+ȷ+r2+|r−ρ|2+r+ρ2]−max{|r−r|,r+r2}, |
for ς=4, we gain
e|ȷ−ρ|2+ȷ+ρ2≤22[e|ȷ−r|2+ȷ+r2+e|r−ρ|2+r+ρ2]−emax{|r−r|,r+r2},℘♭(ȷ,ρ)≤4[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 5: Let ȷ>ρ>r. Then, for all ς≥1, we attain
|ȷ−ρ|2+ȷ+ȷ2=|ȷ−r|2+ȷ+r−r+ȷ2−r=|ȷ−r|2+ȷ+r2+r+ȷ2−r≤|ȷ−r|2+ȷ+r2+max{|r−ȷ|,r+ȷ2}−max{|r−r|,r+r2}, |
and
e|ȷ−ρ|2+ȷ+ρ2≤e|ȷ−r|2+ȷ+r2+emax{|r−ρ|,r+ρ2}−emax{|r−r|,r+r2}℘♭(ȷ,ρ)≤℘♭(ȷ,r)+℘♭(r,ρ)−℘♭(r,r)≤ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Case 6: Let r>ȷ>ρ. Since
|ȷ−ρ|2+ȷ+ρ2≤|r−ρ|2+r+ρ2+2max{|ȷ−r|,ȷ+r2}−max{|r−r|,r+r2}≤2[max{|ȷ−r|,ȷ+r2}+|r−ρ|2+r+ρ2]−max{|r−r|,r+r2}, |
and
e|ȷ−ρ|2+ȷ+ρ2≤2[emax{|ȷ−r|,ȷ+r2}+e|r−ρ|2+r+ρ2]−emax{|r−r|,r+r2}, |
for ς=2, we derive
℘♭(ȷ,ρ)≤2[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r)≤ς[℘♭(ȷ,r)+℘♭(r,ρ)]−℘♭(r,r). |
Contemplating the above six cases, we conclude that the mapping given in (2.4) is a P♭M with ς=4.
Proposition 2.0.11. ([25]) Every partial ♭-metric ℘♭ defines a ♭-metric d℘♭, where
d℘♭(ȷ,ρ)=2℘♭(ȷ,ρ)−℘♭(ȷ,ȷ)−℘♭(ρ,ρ) |
for all ȷ,ρ∈U.
Remark 2.0.12. Let ε>0 {B℘♭(ȷ,ε):ȷ∈U,ε>0} be the family of ℘♭-open balls, where B℘♭(ȷ,ε)={ρ∈U:℘♭(ȷ,ρ)<℘♭(ȷ,ȷ)+ε} for all ȷ∈U. Each P♭M brings on a T0 topology T℘♭ on U; however, it need not be T1. To explain this, in Example 2.0.10 (i), for 1=ȷ≠ρ=3 and q=2, we currently possess
B℘♭(12,1)={ρ∈U:℘♭(12,ρ)<℘♭(12,12)+1}={ρ∈U:|2−1ρ|2+1<1+1}={ρ∈U:|2−1ρ|2<1}={ρ∈U:12≤ρ<1}=[12,1) |
and
B℘♭(14,4)={ρ∈U:℘♭(14,ρ)<℘♭(14,14)+4}={ρ∈U:|4−1ρ|2+1<1+4}={ρ∈U:|4−1ρ|2<4}={ρ∈U:0≤4−1ρ<2}=[14,12). |
12∈B℘♭(12,1), but 14∉B℘♭(12,1), and 14∈B℘♭(14,4), but 12∉B℘♭(14,4). Thus, it is deduced that a P♭M on a set U need not to be T1.
Definition 2.0.13. ([25]) Let (U,℘♭) be a P♭MS with a coefficient ς and {ȷn}n∈N be a sequence in (U,℘♭).
i. Provided that the equality limn→∞℘♭(ȷn,ȷ)=℘♭(ȷ,ȷ) holds, {ȷn}n∈N is called a ℘♭-convergent sequence in U and ȷ is termed the ℘♭-limit of {ȷn}n∈N.
ii. {ȷn}n∈N is a ℘♭-Cauchy sequence if limn,m→∞℘♭(ȷn,ȷm) exists and is finite.
iii. If, for each ℘♭-Cauchy sequence {ȷn}n∈N, a point ȷ exists in ∈U such that
limn,m→∞℘♭(ȷn,ȷm)=lim℘♭n→∞(ȷn,ȷ)=℘♭(ȷ,ȷ), |
then (U,℘♭) is called a ℘♭-complete space.
Remark 2.0.14. In a P♭MS, a convergent sequence does not require a unique limit. In Example 2.0.10 (ii), consider
(ȷn)n∈N=(1n2+1)n∈N∈U. |
Then,
limn→∞℘♭(1n2+1,0)=limn→∞e|1n2+1−0|2+1n2+1+02=1=℘♭(0,0)=emax{|0−0|,0+02},limn→∞℘♭(1n2+1,1)=limn→∞emax{|1n2+1−1|,1n2+1+12}=e1=℘♭(1,1)=emax{|1−1|,1+12}. |
As can be seen, the sequence 1n2+1 has more than one limit.
Example 2.0.15. ([20]) Let U=R+ and ℘♭:U×U→R+ denote a mapping determined by ℘♭(ȷ,ρ)=max{ȷ,ρ} for all ȷ,ρ∈U. In such a case, (U,℘♭) has become a P♭MS with the coefficient ς≥1. We obtain the following for the sequence {1n+1}n∈N in (U,℘♭):
limn→∞℘♭(1,11+n)=1=℘♭(1,1),lim℘♭n→∞(2,11+n)=2=℘♭(2,2). |
The sequence {1n+1}n∈N owns two limits in U.
Lemma 2.0.16. ([25]) Presume that (U,℘♭) and (U,d℘♭) are a P♭MS and ♭-metric space, respectively.
1) {ȷn}n∈N is a ℘♭-Cauchy sequence in (U,℘♭) if and only if it is also a ♭-Cauchy sequence in (U,d℘♭).
2) (U,℘♭) is ℘♭-complete if and only if (U,d℘♭) is ♭-complete. As well, limn→∞d℘♭(ȷ,ȷn)=0 if and only if
limm→∞℘♭(ȷ,ȷm)=limn→∞℘♭(ȷ,ȷn)=℘♭(ȷ,ȷ). |
Remark 2.0.17. ([27]) The mapping ℘♭ is not continuous in general.
Presume that U=N∪{∞} and ℘♭: U×U→R+ is a mapping defined as follows:
℘♭(ȷ,ρ)={0,ȷ=ρ;|1ȷ−1ρ|,ifoneofȷandρoddandtheotherisoddorȷρ=∞;5,ifoneofȷandρevenandtheotherisevenȷ≠ρor∞;2,otherwise. |
(U,℘♭) is a P♭MS with ς=3. For each n∈N, consider that ȷn=2n+3. Then,
℘♭(2n+3,∞)=12n+3→0 |
as n→∞; so, ȷn→∞; however, limn→∞℘♭(ȷn,4)=2≠5=℘♭(∞,4), which means that ℘♭ does not have the continuity property.
The subsequent lemma is essential in regarding the ℘♭-convergent sequences as the confirmation of our findings because a P♭M is not continuous in general.
Lemma 2.0.18. ([27]) Ensure that (U,℘♭) is a P♭MS with the coefficient ς>1 and the sequences {ȷn}n∈N and {ρn}n∈N are ℘♭-convergent to ȷ and ρ, respectively. Afterward, we obtain
1ς2℘♭(ȷ,ρ)−1ς℘♭(ȷ,ȷ)−℘♭(ρ,ρ)≤limn→∞inf℘♭(ȷn,ρn)≤limn→∞sup℘♭(ȷn,ρn)≤ς℘♭(ȷ,ȷ)+ς2℘♭(ρ,ρ)+ς2℘♭(ȷ,ρ). |
Wardowski [48] proposed a novel notion associated with the F-contraction in 2012. As a result, several investigations have been conducted to obtain more extended contractive mappings on metric spaces and other generalized metric spaces.
Definition 2.0.19. ([48]) Let (U,d) be a metric space. The mapping O:U→U is an F-contraction provided that F∈F and κ>0 exist such that, for all ȷ,ρ∈U,
d(Oȷ,Oρ)>0⇒κ+F(d(Oȷ,Oρ))≤F(d(ȷ,ρ)), |
where F is the set of functions F:(0,∞)→R fulfilling the subsequent statements:
(F1) F is strictly increasing, i.e., for all a,b∈(0,∞), such that a<b, F(a)<F(b);
(F2) for each {an}n∈N of positive numbers, limn→∞an=0 ⇔ limn→∞F(an)=−∞;
(F3) a constant c∈(0,1) exists such that lima→0+acF(a)=0.
Subsequently, Wardowski attested that in [48], any F-contraction enjoys a unique fixed-point in a complete metric space (U,d).
As an extension of the family F, Piri and Kumam [51] put forth a new set of functions F∈F∗ by substituting the term (F3) with (F3′) in Definition 2.0.19, as noted below:
(F3′) F is continuous.
Briefly, set F∗={F:(0,∞)→(−∞,+∞):Fholds(F1),(F2)and(F3′)}.
In 2014, Jleli and Samet [52] introduced the concept of a θ-contraction, and the class Θ={θ:(0,∞)→(1,∞)} represents all such functions provided to fulfill the below circumstances:
(θ1) θ is non-decreasing;
(θ2) for each {az}⊂(0,∞), limz→∞θ(az)=1 ⇔ limz→∞az=0+;
(θ3) a constant r∈(0,1) and s∈(0,∞] exist such that lima→0+θ(a)ar=s.
Theorem 2.0.20. Let O:U→U be a mapping on a complete metric space (U,d). Provided that a function θ∈Θ and a constant τ∈(0,1) exist such that
d(Oȷ,Oρ)≠0⇒θ(d(Oȷ,Oρ))≤[θ(d(ȷ,ρ))]τ |
for all ȷ,ρ∈U, then O owns a unique fixed-point.
Furthermore, Liu et al. [47] identified the set ˜Θ={θ:(0,∞)→(1,∞):θholds(θ1′)and(θ2′)}, where
(θ1′)θ is non-decreasing and continuous;
(θ2′)infa∈(0,∞)θ(a)=1.
Theorem 2.0.21. [47] Let O:U→U be a self-mapping on a complete metric space (U,d). Thereby, the following statements are equivalent:
i. the mapping O is a θ−contraction with θ∈˜Θ;
ii. the mapping O is an F−contraction with F∈F∗.
The concept of a DC-contraction was proposed by Liu et al. [47] as follows.
Presume that D:(0,∞)→(0,∞) is a function and fulfills the terms (D1)−(D3).
(D1) D is non-decreasing;
(D2) limn→∞D(an)=0⇔limn→∞an=0;
(D3) D is continuous.
Set Δ={D:(0,∞)→(0,∞):D satisfies (D1)−(D3)}.
C:(0,∞)→(0,∞) is a comparison function that has the features (C1) and (C2).
(C1) C is monotonically increasing, that is, a<b⇒C(a)<C(b).
(C2) limn→∞Cn(a)=0 for all a>0, where Cn denotes the nth-iteration of C.
If C is a comparison function, then C(a)<a for all a>0. The mappings
● Cx(a)=χa, 0<χ<1, a>0,
● Cy(a)=a1+a
can be given as examples of comparison functions.
Definition 2.0.22. ([47]) Let (U,d) be a metric space and O be a self-mapping on this space. Let ℑ={(ȷ,ρ)∈U2:d(Oȷ,Oρ)>0}. O is named a DC-contraction if it ensures the following expression:
D(d(Oȷ,Oρ))≤C(D(d(ȷ,ρ))) | (2.5) |
for all ȷ,ρ∈ℑ.
In 2021, Nazam et al. [50] presented a new definition of the DC-contraction, including two self-mappings with a binary relation in a P♭MS, referring to Liu's definition of a DC-contraction.
Definition 2.0.23. ([50]) Let O and S be two self-mappings on a P♭MS and ℜ be a binary relation on U. Define the set ℑ={(ȷ,ρ)∈ℜ:℘♭(Oȷ,Sρ)>0}. The mappings O and S form a DC-contraction if there exist D∈Δ and a continuous comparison function C such that
D(ς2℘♭(O(ȷ),S(ρ)))≤C(D(℘♭(ȷ,ρ))) | (2.6) |
for all ȷ,ρ∈ℑ.
Definition 2.0.24. ([46]) Let U be a nonempty set and O:U→U and α:U2→R be given mappings. O is α-orbital-admissible provided that the expression given below is true:
α(ȷ,Oȷ)≥1⇒α(Oȷ,O2ȷ)≥1 |
for all ȷ∈U.
Definition 2.0.25. ([35]) Let U be a nonempty set and α:U2→R be a given function.
i. O is an α-orbital-admissible mapping;
ii. α(ȷ,ρ)≥1 and α(ρ,Oρ)≥1⇒α(ȷ,Oρ)≥1, ȷ,ρ∈U.
A mapping O:U→U that satisfies the above features is named a triangular α-orbital-admissible mapping.
Definition 2.0.26. ([46]) Let (U,d) be a complete metric space. The mapping O:U→U is a Geraghty-type contraction if a function P:[0,∞)→[0,1) exists, which ensures the following term:
limn→∞P(tn)=1⇒limn→∞tn=0 | (2.7) |
such that
d(Oȷ,Oρ)≤P(d(ȷ,ρ))d(ȷ,ρ) |
for all ȷ,ρ∈U. The family of P:[0,∞)→[0,1) satisfying (2.7) is represented as B.
Definition 2.0.27. ([46]) In a metric space (U,d), if the mappings O,S:U→U satisfy the below statements for all ȷ,ρ∈U:
d(Oȷ,Sρ)≤P(ES,O(ȷ,ρ))ES,O(ȷ,ρ), |
where
ES,O(ȷ,ρ)=d(ȷ,ρ)+|d(ȷ,Oȷ)−d(ρ,Sρ)| |
whenever a function P∈B exists, then O,S are called Geraghty contractions of type ES,O.
Definition 2.0.28. ([46]) The set of P:[0,∞)→[0,1ς) functions fulfilling the constraint limn→∞P(tn)=1ς for ς≥1 implies that limn→∞tn=0 and it is stated as Bς.
Definition 2.0.29. ([46]) Let α:U2→R be a function in (U,d). Provided that the subsequent statement
α(ȷ,ρ)≥1⇒d(Oȷ,Oρ)≤P(EO(ȷ,ρ))EO(ȷ,ρ), |
where
EO(ȷ,ρ)=d(ȷ,ρ)+|d(ȷ,Oȷ)−d(ρ,Oρ)| |
is satisfied for all ȷ,ρ∈U whenever P∈Bς, then O:U→U is called an α-PE-Geraghty contraction.
Unless otherwise stated, i and j will be treated as arbitrary positive integers throughout this study.
Definition 2.0.30. ([46]) Let αji:U2→R be a function in a ♭-metric space (U,♭) with the coefficient ς≥1. O,S:U→U are two self-mappings which are called αji-PˆES,O-Geraghty contractions if a function P∈Bς exists, which ensures the following expressions:
αji(ȷ,ρ)≥ςf⇒αji(ȷ,ρ)♭(Oiȷ,Sjρ)≤P(ES,O(ȷ,ρ))ES,O(ȷ,ρ), |
where
ˆES,O(ȷ,ρ)=♭(ȷ,ρ)+|♭(ȷ,Oiȷ)−♭(ρ,Sjρ)| |
for all ȷ,ρ∈U, and f≥2 is a constant.
Remark 2.0.31. ([46])
i. If ς=1, αji(ȷ,ρ)=1 and i=j=1, we obtain the Geraghty contraction of type ES,O.
ii. If S=O and i=j=1, we obtain the α-PE-Geraghty contraction.
Definition 2.0.32. ([46]) Let αji:U2→[0,∞) be a function in a ♭-metric space (U,♭) with the coefficient ς≥1. The mappings O,S:U→U are called αji-orbital-admissible if the following circumstances hold:
αji(ȷ,Oiȷ)≥ςf⇒αji(Oiȷ,SjOiȷ)≥ςf,αji(ȷ,Sjȷ)≥ςf⇒αji(Sjȷ,OiSjȷ)≥ςf |
for all ȷ∈U, where f≥2 is a constant.
Definition 2.0.33. ([46]) Let αji:U2→[0,∞) be a function in a complete ♭-metric space (U,♭) with the coefficient ς≥1. When O,S:U→U are two self-mappings, the pair (O,S) is a triangular αji-orbital-admissible pair if
i. O,S are αji-orbital-admissible,
ii. αji(ȷ,ρ)≥ςf,αji(ρ,Oiρ)≥ςf and αji(ρ,Sjρ)≥ςf imply that αji(ȷ,Oiρ)≥ςf and αji(ȷ,Sjρ)≥ςf, where f≥2 is a constant.
Lemma 2.0.34. ([46]) Consider that O,S:U→U are two mappings in a complete ♭-metric space (U,♭) with the coefficient ς≥1. Assume that (O,S) is a triangular αji-orbital-admissible pair and an element ȷ0 in U exists with the property αji(ȷ0,Oiȷ0)≥ςf. Setup a sequence {ȷn}n∈N in (U,b) as follows:
ȷ2n=Sjȷ2n−1,ȷ2n+1=Oiȷ2n, |
where n=0,1,2,...; afterward, for n,m∈N∪{0} with m>n, αji(ȷn,ȷm)≥ςf exists.
This is the leading part of the study, and it contains a novel definition and a common fixed-point theorem. Furthermore, some relevant results will be presented. Illustrative examples are provided to indicate the accuracy and validity of the findings.
In the course of the study, the set of fixed-points of O and the set of common fixed-points of O and S will be denoted with the notations Fix(O) and CFix(O,S), respectively.
Definition 3.0.1. Let (U,℘♭) be a P♭MS with the coefficient ς≥1 and αji:U2→[0,∞) be a function. The mappings O,S:U→U are called generalized αji-(DC(PˆE))-contractions if D∈Δ and a continuous comparison function C and P∈Bς exist such that, for all ȷ,ρ∈U, αji(ȷ,ρ)≥ςf and ℘♭(Oiȷ,Sjρ)>0,
D(αji(ȷ,ρ)℘♭(Oiȷ,Sjρ))≤C(D(P(ˆEO,S(ȷ,ρ))ˆEO,S(ȷ,ρ))), | (3.1) |
where
ˆES,O(ȷ,ρ)=℘♭(ȷ,ρ)+|℘♭(ȷ,Oiȷ)−℘♭(ρ,Sjρ)| |
and f≥2 is a constant.
Theorem 3.0.2. Ensure that (U,℘♭) is a ℘♭-complete P♭MS, O,S:U→U are two mappings and αji:U2→[0,∞) is a function. Assume that the subsequent statements are satisfied:
i. O,S are generalized αji-(DC(PˆE))-contractions mappings;
ii. the pair (O,S) is triangular αji-orbital-admissible;
iii. ȷ0∈U exists, satisfying αji(ȷ0,Oiȷ0)≥ςf;
iv. one of the below terms is provided:
iva. Oi and Sj are ℘♭-continuous,
or
ivb. if {ȷn}n∈N is a sequence in U such that αji(ȷn,ȷn+1)≥ςf for each n∈N and ȷn→r∈U as n→∞, then a subsequence {ȷnk} of {ȷn} exists such that αji(ȷnk,r)≥ςf for each k∈N;
v. for all r∈Fix(Oi) or w∈Fix(Sj) we have αji(r,w)≥ςf.
Then, the set of CFix(O,S) consists of a unique element belonging to (U,℘♭).
Proof. An initial point ȷ0∈U with the property αji(ȷ0,Oiȷ0)≥ςf exists from (ⅲ). Consider {ȷn}n∈N in U, which is constructed as
ȷ2n+2=Oiȷ2n+1andȷ2n+1=Sjȷ2n |
for each n∈N. Hypothetically, a natural number n0 exists such that ȷn0=ȷn0+1. Let ȷ2n0=ȷ2n0+1; then, ȷ2n0=ȷ2n0+1=Sjȷ2n0. Hence, Fix(Sj)={ȷ2n0}. Suppose that ℘♭(ȷ2n0+2,ȷ2n0+1)>0 and Oiȷ2n0+1≠Sjȷ2n0. According to Lemma 2.0.34, we have
αji(ȷ2n0,ȷ2n0+1)=αji(ȷ2n0+1,ȷ2n0)≥ςf. |
In (3.1), writing ȷ=o2n0+1 and ρ=o2n0, we attain
D(℘♭(Oiȷ2n0+1,Sjȷ2n0))≤D(αi,j(ȷ2n0+1,ȷ2n0)℘♭(Oiȷ2n0+1,Sjȷ2n0))≤C(D(P(ˆES,O(ȷ2n0+1,ȷ2n0))ˆES,O(ȷ2n0+1,ȷ2n0)))<C(D(1ςˆES,O(ȷ2n0+1,ȷ2n0))), |
where
ˆES,O(ȷ2n0+1,ȷ2n0)=℘♭(ȷ2n0+1,ȷ2n0)+|℘♭(ȷ2n0+1,Oiȷ2n0+1)−℘♭(ȷ2n0,Sjȷ2n0)|=℘♭(ȷ2n0+1,ȷ2n0+1)+|℘♭(ȷ2n0+1,ȷ2n0+2)−℘♭(ȷ2n0+1,ȷ2n0+1)|=℘♭(ȷ2n0+1,ȷ2n0+1)+℘♭(ȷ2n0+1,ȷ2n0+2)−℘♭(ȷ2n0+1,ȷ2n0+1)=℘♭(ȷ2n0+1,ȷ2n0+2). |
Hence, employing the property of C(a)<a, we obtain the following contradictory expression:
D(℘♭(ȷ2n0+2,ȷ2n0+1))=D(℘♭(Oiȷ2n0+1,Sjȷ2n0))<C(D(1ς℘♭(ȷ2n0+1,ȷ2n0+2)))<D(1ς℘♭(ȷ2n0+1,ȷ2n0+2)). |
In this case, we obtain ℘♭(ȷ2n0+2,ȷ2n0+1)=0, with the property (D1) of the function D. By (℘♭2), we have that ℘♭(ȷ2n0+2,ȷ2n0+2)=℘♭(ȷ2n0+1,ȷ2n0+1)=0, and it yields that
℘♭(ȷ2n0+2,ȷ2n0+2)=℘♭(ȷ2n0+1,ȷ2n0+1)=℘♭(ȷ2n0+2,ȷ2n0+1). |
By (℘♭1), ȷ2n0+2=ȷ2n0+1 and Oiȷ2n0+1=ȷ2n0+1. Therefore, ȷ2n0+1 is a fixed-point of Oi, and then ȷ2n0=ȷ2n0+1 belongs to CFix(Oi,Sj). For some natural numbers n0, we achieve a similar result for ȷ2n0=ȷ2n0−1. Hereafter, we presume that ȷn≠ȷn+1 for all n∈N. It is needed to investigate the subsequent two cases:
Case 1: Presume that ȷ2n≠ȷ2n−1 for all n∈N. ℘♭(Oiȷ2n−1,Sjȷ2n)>0 and, moreover, αji(ȷ2n−1,ȷ2n)≥ςf. Therefore, employing (3.1), we gain
D(℘♭(ȷ2n,ȷ2n+1))=D(℘♭(Oiȷ2n−1,Sjȷ2n))≤D(αji(ȷ2n−1,ȷ2n)℘♭(Oiȷ2n−1,Sjȷ2n))≤C(D((ˆES,O(ȷ2n−1,ȷ2n))ˆES,O(ȷ2n−1,ȷ2n)))<C(D(1ςˆES,O(ȷ2n−1,ȷ2n))), |
where
ˆES,O(ȷ2n−1,ȷ2n)=℘♭(ȷ2n−1,ȷ2n)+|℘♭(ȷ2n−1,Oiȷ2n−1)−℘♭(ȷ2n,Sjȷ2n)|=℘♭(ȷ2n−1,ȷ2n)+|℘♭(ȷ2n−1,ȷ2n)−℘♭(ȷ2n,ȷ2n+1)|. |
If ℘♭(ȷ2n,ȷ2n+1)≥℘♭(ȷ2n−1,ȷ2n), the following expression is obtained:
ˆES,O(ȷ2n−1,ȷ2n)=℘♭(ȷ2n−1,ȷ2n)−℘♭(ȷ2n−1,ȷ2n)+℘♭(ȷ2n,ȷ2n+1). |
Thereby, the inequality given below is obtained:
D(℘♭(ȷ2n,ȷ2n+1))<C(D(1ς℘♭(ȷ2n,ȷ2n+1)))<D(1ς℘♭(ȷ2n,ȷ2n+1)). |
However, this produces a discrepancy. Accordingly, the following statement must be true:
℘♭(ȷ2n,ȷ2n+1)<℘♭(ȷ2n−1,ȷ2n). |
Case 2: Presume that ȷ2n≠ȷ2n+1 for all n∈N. For all n∈N, ℘♭(Oiȷ2n,Sjȷ2n+1)>0 and αji(ȷ2n,ȷ2n+1)≥ςf, we procure
D(℘♭(ȷ2n+1,ȷ2n+2))=D(℘♭(Oiȷ2n,Sjȷ2n+1))≤D(αji(ȷ2n,ȷ2n+1)℘♭(Oiȷ2n,Sjȷ2n+1))≤C(D(P(ˆES,O(ȷ2n,ȷ2n+1))ˆES,O(ȷ2n,ȷ2n+1)))<C(D(1ςˆES,O(ȷ2n,ȷ2n+1))), |
where
ˆES,O(ȷ2n,ȷ2n+1)=℘♭(ȷ2n,ȷ2n+1)+|℘♭(ȷ2n,Oiȷ2n)−℘♭(ȷ2n+1,Sjȷ2n+1)|=℘♭(ȷ2n,ȷ2n)+|℘♭(ȷ2n,ȷ2n+1)−℘♭(ȷ2n+1,ȷ2n+2)|. |
If ℘♭(ȷ2n+1,ȷ2n+2)≥℘♭(ȷ2n,ȷ2n+1), the following equation is obtained:
ˆES,O(ȷ2n,ȷ2n+1)=℘♭(ȷ2n,ȷ2n+1)−℘♭(ȷ2n,ȷ2n+1)+℘♭(ȷ2n+1,ȷ2n+2). |
Thus, we conclude that
D(℘♭(ȷ2n+1,ȷ2n+2))<C(D(1ς℘♭(ȷ2n+1,ȷ2n+2)))<D(1ς℘♭(ȷ2n+1,ȷ2n+2)). |
This indicates a contradiction. Thus, the subsequent situation is provided:
℘♭(ȷ2n+1,ȷ2n+2)<℘♭(ȷ2n,ȷ2n+1). |
As can be seen from the above two cases, {℘♭(ȷn,ȷn+1)}n∈N is a non-increasing sequence. Therefore, L≥0 exists such that limn→∞℘♭(ȷn,ȷn+1)=L. Suppose that L>0. If the limit is taken on both sides of the subsequent inequality owing to the fact that D and C are continuous functions, we gain
D(limn→∞℘♭(ȷ2n,ȷ2n+1))<C(D(1ςlimn→∞ˆES,O(ȷ2n−1,ȷ2n))), |
where
ˆES,O(ȷ2n−1,ȷ2n)=℘♭(ȷ2n−1,ȷ2n)+|℘♭(ȷ2n−1,Oiȷ2n−1)−℘♭(ȷ2n,Sjȷ2n)|=℘♭(ȷ2n−1,ȷ2n)+|℘♭(ȷ2n−1,ȷ2n)−℘♭(ȷ2n,ȷ2n+1)|=2℘♭(ȷ2n−1,ȷ2n)−℘♭(ȷ2n,ȷ2n+1) |
and
limn→∞infˆES,O(ȷ2n−1,ȷ2n)=liminfn→∞[2℘♭(ȷ2n−1,ȷ2n)−℘♭(ȷ2n,ȷ2n+1)]≤limsupn→∞[2℘♭(ȷ2n−1,ȷ2n)−℘♭(ȷ2n,ȷ2n+1)]=L. |
Hence, we attain
D(L)=D(limn→∞℘♭(ȷ2n,ȷ2n+1))<C(D(1ςlimn→∞supˆES,O(ȷ2n−1,ȷ2n)))<C(D(1ςL))<D(Lς), |
which is a contradiction. The same argument is true for ℘♭(ȷ2n,ȷ2n−1). Therefore,
limn→∞℘♭(ȷn,ȷn+1)=0. |
Because of (℘♭2), we have that ℘♭(ȷn,ȷn)≤℘♭(ȷn,ȷn+1) and ℘♭(ȷn+1,ȷn+1)≤℘♭(ȷn,ȷn+1); then,
limn→∞℘♭(ȷn,ȷn)=limn→∞℘♭(ȷn+1,ȷn+1)=0. |
Using Proposition 2.0.11, we write
d℘♭(ȷn,ȷn+1)=2.℘♭(ȷn,ȷn+1)−℘♭(ȷn,ȷn)−℘♭(ȷn+1,ȷn+1). |
If we take the limit in the above equation as n tends to infinity, we get
0≤limn→∞d℘♭(ȷn,ȷn+1)=limn→∞2[℘♭(ȷn,ȷn+1)−℘♭(ȷn,ȷn)−℘♭(ȷn+1,ȷn+1)]≤limn→∞sup2[℘♭(ȷn,ȷn+1)−℘♭(ȷn,ȷn)−℘♭(ȷn+1,ȷn+1)]=0. |
Thus, limn→∞d℘♭(ȷ2n,ȷ2n+1)=0 for all n∈N. Furthermore, we currently possess the subsequent expression for all n,m≥1:
limn,m→∞d℘♭(ȷ2m,ȷ2n)=2limn,m→∞sup℘♭(ȷ2m,ȷ2n). |
It is necessary to indicate that the sequence {ȷn}n∈N is a ℘♭-Cauchy sequence in (U,℘♭). Instead, it is needed to verify that {ȷ2n}n∈N is a ℘♭-Cauchy sequence. According to Lemma 2.0.16 (1), it is required to explicitly state that {ȷ2n}n∈N is a Cauchy sequence in (U,d℘♭). Assume that {ȷ2n}n∈N is not a Cauchy sequence in (U,d℘♭). In this instance, two subsequences of positive numbers {ȷ2mk} and {ȷ2nk} exist such that n(k)>m(k)>k and the number ε>0, which yield that
d℘♭(ȷ2mk,ȷ2nk)≥ε, | (3.2) |
and, for k∈N,
d℘♭(ȷ2mk,ȷ2nk−2)<ε. | (3.3) |
Applying ♭3, we deduce that
ε≤d℘♭(ȷ2mk,ȷ2nk)≤ςd℘♭(ȷ2mk,ȷ2mk+1)+ςd℘♭(ȷ2mk+1,ȷ2nk). |
By taking the limit in the above as k→∞, we write
ες≤limk→∞infd℘♭(ȷ2mk+1,ȷ2nk)≤limk→∞supd℘♭(ȷ2mk+1,ȷ2nk). | (3.4) |
Also, employing ♭3, we gain
d℘♭(ȷ2mk,ȷ2nk−1)≤ςd℘♭(ȷ2mk,ȷ2nk−2)+ςd℘♭(ȷ2nk−2,ȷ2nk−1); |
again, taking the limit in the above as k→∞, we get
limk→∞supd℘♭(ȷ2mk,ȷ2nk−1)≤ςε. | (3.5) |
Moreover, the following statement is derived:
d℘♭(ȷ2mk,ȷ2nk)≤ςd℘♭(ȷ2mk,ȷ2nk−2)+ςd℘♭(ȷ2nk−2,ȷ2nk)≤ςd℘♭(ȷ2mk,ȷ2nk−2)+ς2d℘♭(ȷ2nk−2,ȷ2nk−1)+ς2d℘♭(ȷ2nk−1,ȷ2nk). |
Similarly, if the limit is taken for k→∞, we obtain
limk→∞supd℘♭(ȷ2mk,ȷ2nk)≤ςε. | (3.6) |
Again, from ♭3, it is achievable to produce
d℘♭(ȷ2mk+1,ȷ2nk−1)≤ςd℘♭(ȷ2mk+1,ȷ2mk)+ςd℘♭(ȷ2mk,ȷ2nk−1). |
Likewise, taking the limit as k→∞, it is concluded that
limk→∞supd℘♭(ȷ2mk+1,ȷ2nk−1)≤ς2ε. | (3.7) |
From the inequalities (3.4)–(3.7), the subsequent expressions are attained:
ε2ς≤limk→∞inf℘♭(ȷ2mk+1,ȷ2nk)≤limk→∞sup℘♭(ȷ2mk+1,ȷ2nk), | (3.8) |
limk→∞sup℘♭(ȷ2mk,ȷ2nk−1)≤ςε2, | (3.9) |
limk→∞sup℘♭(ȷ2mk,ȷ2nk)≤ςε2, | (3.10) |
limk→∞sup℘♭(ȷ2mk+1,ȷ2nk−1)≤ς2ε2. | (3.11) |
Due to Lemma 2.0.18, it is known that αji(ȷ2mk,ȷ2nk−1)≥ςf and
D(℘♭(ȷ2mk+1,ȷ2nk))≤D(αji(ȷ2mk,ȷ2nk−1)℘♭(Oiȷ2mk,Sjȷ2nk−1))≤C(D(P(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1))), |
where
ˆES,O(ȷ2mk,ȷ2nk−1)=℘♭(ȷ2mk,ȷ2nk−1)+|℘♭(ȷ2mk,Oiȷ2mk)−℘♭(ȷ2nk−1,Sjȷ2nk−1)|=℘♭(ȷ2mk,ȷ2nk−1)+|℘♭(ȷ2mk,ȷ2mk+1)−℘♭(ȷ2nk−1,ȷ2nk)|, |
and, taking the limit in the above as k→∞, we procure
limk→∞infˆES,O(ȷ2mk,ȷ2nk−1)=limk→∞inf℘♭(ȷ2mk,ȷ2nk−1)≤limk→∞sup℘♭(ȷ2mk,ȷ2nk−1)=limk→∞supˆES,O(ȷ2mk,ȷ2nk−1)≤ςε2. |
In light of the results obtained above, using (3.1), we write
D(ε2)=D(ςε2ς)≤D(ςlimk→∞inf℘♭(ȷ2mk+1,ȷ2nk))≤D(ςplimk→∞inf℘♭(ȷ2mk+1,ȷ2nk))≤D(limk→∞infαji(ȷ2mk,ȷ2nk−1)℘♭(Oiȷ2mk,Sjȷ2nk−1))≤C(D(limk→∞infP(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)))≤D(limk→∞inf1ςςε2)=D(ε2). | (3.12) |
Then,
D(ε2)≤D(αi,j(ȷ2mk,ȷ2nk−1)limk→∞sup℘♭(Oiȷ2mk,Sjȷ2nk−1))≤limk→∞supD(αji(ȷ2mk,ȷ2nk−1)℘♭(Oiȷ2mk,Sjȷ2nk−1))≤limk→∞supC(D(P(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)))=C(D(limk→∞supP(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)))<D(limk→∞sup1ςςε2)=D(ε2). | (3.13) |
Therefore, from (3.12) and (3.13), we obtain
limk→∞P(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)=ε2. | (3.14) |
Similarly,
D(ςε2)=D(ς2ε2ς)≤D(ς2limk→∞inf℘♭(ȷ2mk+1,ȷ2nk))≤D(ςplimk→∞inf℘♭(ȷ2mk+1,ȷ2nk))≤D(limk→∞infαji(ȷ2mk,ȷ2nk−1)℘♭(Oiȷ2mk,Sjȷ2nk−1))≤C(D(limk→∞infP(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)))<D(limk→∞infˆES,O(ȷ2mk,ȷ2nk−1))≤D(ςε2), | (3.15) |
and
D(ςε2)≤D(αji(ȷ2mk,ȷ2nk−1)limk→∞sup℘♭(Oiȷ2mk,Sjȷ2nk−1))≤limk→∞supD(αji(ȷ2mk,ȷ2nk−1)℘♭(Oiȷ2mk,Sjȷ2nk−1))≤limk→∞supC(D(P(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)))=C(D(limk→∞supP(ˆES,O(ȷ2mk,ȷ2nk−1))ˆES,O(ȷ2mk,ȷ2nk−1)))<D(limk→∞supˆES,O(ȷ2mk,ȷ2nk−1))≤D(ςε2). | (3.16) |
From (3.15) and (3.16), it is obtained that
limk→∞ˆES,O(ȷ2mk,ȷ2nk−1)=ςε2. | (3.17) |
Because of (3.14) and (3.17), we conclude that
limk→∞P(ˆES,O(ȷ2mk,ȷ2nk−1))=1ς, | (3.18) |
and, in (3.18), it is deduced by using the property of P that
limk→∞ˆES,O(ȷ2mk,ȷ2nk−1)=0. |
This contradiction brings about the sequence {ȷn}n∈N as a Cauchy sequence in (U,d℘♭). The completeness of (U,d℘♭) provides that the sequence {ȷn}n∈N converges to r∈U. Thus,
limn→∞d℘♭(ȷn,r)=0. |
By Lemma 2.0.16, we derive
℘♭(r,r)=limn→∞℘♭(ȷn,r)=limn,m→∞℘♭(ȷn,ȷm). |
Then, {ȷn}n∈N converges to r∈(U,℘♭). Additionally, by ♭3, the following expression is evident:
d℘♭(ȷn,ȷm)≤ς[d℘♭(ȷn,r)+d℘♭(r,ȷm)]. |
If the limit for n,m→∞ is taken in the above equation, we have
0≤limn,m→∞supd℘♭(ȷn,ȷm)≤limn,m→∞supς[d℘♭(ȷn,r)+d℘♭(r,ȷm)] |
and limn,m→∞d℘♭(ȷn,ȷm)=0. If we consider the fact that limn→∞℘♭(ȷn,ȷn)=0, and, taking the limit as n,m→∞ in the following equations, we procure
d℘♭(ȷn,ȷm)=2℘♭(ȷn,ȷm)−℘♭(ȷn,ȷn)−℘♭(ȷm,ȷm) | (3.19) |
and
limn,m→∞℘♭(ȷn,ȷm)=0. |
So,
℘♭(r,r)=limn→∞℘♭(ȷn,r)=limn,m→∞℘♭(ȷn,ȷm)=0. | (3.20) |
Because of the hypothesis (iv), a subsequence {ȷ2nk} of {ȷn}n∈N exists such that αji(ȷ2nk,r)≥ςf and
D(℘♭(ȷ2nk+1,Sjr))≤D(αji(ȷ2nk,r)℘♭(Oiȷ2nk,Sjr))≤C(D(P(ˆES,O(ȷ2nk,r))ˆES,O(ȷ2nk,r)))<D(P(ˆES,O(ȷ2nk,r))ˆES,O(ȷ2nk,r))<D(1ςˆES,O(ȷ2nk,r)), |
where
ˆES,O(ȷ2nk,r)=℘♭(ȷ2nk,r)+|℘♭(ȷ2nk,Oiȷ2nk)−℘♭(r,Sjr)|=℘♭(ȷ2nk,r)+|℘♭(ȷ2nk,ȷ2nk+1)−℘♭(r,Sjr)|. |
In the last equality, by taking the limit as k→∞, we have
limk→∞supˆES,O(ȷ2nk,r)=℘♭(r,Sjr), | (3.21) |
and, using this, we get the following expression:
D(limk→∞sup℘♭(ȷ2nk+1,Sjr))<D(1ς℘♭(r,Sjr)). | (3.22) |
Further, since 1ς℘♭(r,Sjr)≤limk→∞sup℘♭(ȷ2nk+1,Sjr), we attain
D(1ς℘♭(r,Sjr))≤D(limk→∞sup℘♭(ȷ2nk+1,Sjr)). | (3.23) |
Therefore, using the expressions (3.22) and (3.23), we obtain
D(1ς℘♭(r,Sjr))≤D(limk→∞sup℘♭(ȷ2nk+1,Sjr))≤D(limk→∞P(ˆES,O(ȷ2nk,r))ˆES,O(ȷ2nk,r))≤D(1ς℘♭(r,Sjr)). |
From here, one can deduce that
limk→∞P(ˆES,O(ȷ2nk,r))ˆES,O(ȷ2nk,r)=1ς℘♭(r,Sjr). | (3.24) |
Because of (3.21) and (3.24), it is concluded that
limk→∞P(ˆES,O(ȷ2nk,r))=1ς. | (3.25) |
In (3.25), due to the property of P, we obtain that limk→∞ˆES,O(ȷ2nk,r)=0. Thus, ℘♭(r,Sjr)=0. From the axiom (℘♭2), ℘♭(r,r)≤℘♭(r,Sjr) and ℘♭(Sjr,Sjr)≤℘♭(r,Sjr). Obviously,
℘♭(r,r)=℘♭(r,Sjr)=0. |
Therefore, we get that ℘♭(r,r)=℘♭(r,Sjr)=℘♭(Sjr,Sjr). Because of (℘♭1), r=Sjr. This also yields that r is the fixed-point of Sj. In the same way, one can see that ℘♭(r,Oir)=0. In this instance, r is the fixed-point of Oi. So, r is the common fixed-point of Sj and Oi. Now, to demonstrate the uniqueness of the fixed-point, let Sj have another fixed-point, provided that w∈U satisfies Sjw=w≠r. Because of the fifth hypothesis of the theorem, we have that αji(r,w)≥ςf. By (3.1), we obtain
D(℘♭(r,w))≤D(αji(r,w)℘♭(r,w))=D(αi,j(r,w)℘♭(Oir,Sjw))≤C(D(P(ˆES,O(r,w))ˆES,O(r,w)))≤C(D(1ςˆES,O(r,w)))<D(1ςˆES,O(r,w)), |
where
ˆES,O(r,w)=℘♭(r,w)+|℘♭(r,Oir)−℘♭(w,Sjw)|=℘♭(r,w)+|℘♭(r,r)−℘♭(w,w)|=℘♭(r,w). |
Hence, this indicates that D(℘♭(r,w))<D(1ς℘♭(r,w)). However, this causes a contradiction. So, assuming that Sj has a different fixed-point is inaccurate. Thus, Fix(Sj)={r}. With the same method, it turns out that Fix(Oi)={r}. Accordingly, CFix(Sj,Oi)={r}. Due to the fact that
Sr=SSjr=SjSr |
and
Or=OOjr=OjOr, |
it is obvious to verify that S and O own a unique common fixed-point owing to the uniqueness of the common fixed-point of Sj and Oi.
Example 3.0.3. Assume that U=[0,1] and ℘♭:U×U→R+ is determined as ℘♭(ȷ,ρ)=(ȷ−ρ)2. Then, (U,℘♭) is a complete P♭MS with the coefficient ς=2. αji:U2→[0,∞) is defined as follows:
αji(ȷ,ρ)={ςf,ȷ,ρ∈[0,1],0,otherwise, |
with f≥2. Let O,S:U→U be defined by O(ȷ)=ȷ2, S(ȷ)=ȷ4 and take P(t)=132,t>0,P∈Bς. The pair (O,S) is triangular αji-orbital-admissible. Define D:(0,∞)→(0,∞) by D(θ)=θeθ for each θ>0; then, D∈Δ. C:(0,∞)→(0,∞), which is defined by C(r)=r2 for all r∈(0,∞), is a continuous comparison function. It shall be determined that O and S constitute αji-(DC(PˆE))-contraction mappings. If f=2, i=4 and j=2, then we have that αji(ȷ,ρ)=4. The above choices ensure that
℘♭(Oiȷ,Sjρ)=℘♭(ȷ16,ρ16)=1162(ȷ−ρ)2,ˆES,O(ȷ,ρ)=℘♭(ȷ,ρ)+|℘♭(ȷ,Oiȷ)−℘♭(ρ,Sjρ)|=(ȷ−ρ)2+|(15ȷ16)2−(15ρ16)2|, |
and
D(αi,j(ȷ,ρ)℘♭(Oiȷ,Sjρ))=D(41162(ȷ−ρ)2)=D(164(ȷ−ρ)2)=164(ȷ−ρ)2e164(ȷ−ρ)2. |
Then,
C(D(P(ˆES,O(ȷ,ρ))ˆES,O(ȷ,ρ)))=C(D(132[(ȷ−ρ)2+|(15ȷ16)2−(15ρ16)2|]))=12132[(ȷ−ρ)2+|(15ȷ16)2−(15ρ16)2|]e132[(ȷ−ρ)2+|(15ȷ16)2−(15ρ16)2|]. |
Therefore, (3.1) is achieved; so, CFix(O,S)={0}, where 0 is the unique common fixed-point of O and S.
Example 3.0.4. Assume that U=[0,1] and ℘♭:U×U→R+ is determined as ℘♭(ȷ,ρ)=(max{ȷ,ρ})2. Then, (U,℘♭) is a complete P♭MS with the coefficient ς=2. αji:U2→[0,∞) is defined as follows:
αji(ȷ,ρ)={ςf,ȷ,ρ∈[0,1],0,otherwise, |
with f≥2. Let O,S:U→U be respectively defined by O(ȷ)=ȷ16, S(ȷ)=ȷ4, and take P(t)=1256, t>0,P∈Bς. The pair (O,S) is triangular αji-orbital-admissible. Define D:(0,∞)→(0,∞) by D(ϑ)=ϑ for each ϑ>0; then, D∈Δ. C:(0,∞)→(0,∞), which is defined by C(ȷ)=ȷ16 for all ȷ∈(0,∞), is a continuous comparison function. Now, it is shown that O,S are αji-(DC(PˆE))-contraction mappings. If f=4,i=2 and j=4, then we currently have that αji(ȷ,ρ)=16. Based on these, we acquire
℘♭(Oiȷ,Sjρ)=℘♭(ȷ162,ρ162)=1164(max{ȷ,ρ})2, |
ˆES,O(ȷ,ρ)=℘♭(ȷ,ρ)+|℘♭(ȷ,Oiȷ)−℘♭(ρ,Sjρ)|=(max{ȷ,ρ})2+|ȷ−ρ|, |
and
D(αi,j(ȷ,ρ)℘♭(Oiȷ,Sjρ))=D(161164(max{ȷ,ρ})2)=D(1163(max{ȷ,ρ})2)=1163(max{ȷ,ρ})2. |
Thus, we derive that
C(D(P(ˆES,O(ȷ,ρ))ˆES,O(ȷ,ρ)))=C(D(1256[(max{ȷ,ρ})2+|ȷ−ρ|]))=1161256[(max{ȷ,ρ})2+|ȷ−ρ|]. |
As a result, (3.1) becomes apparent; then, CFix(O,S)={0}.
In this section, contemplating our main theorem, we list some conclusions involving an E-contraction endowed with various auxiliary functions.
Initially, the subsequent corollary generalizes Theorem 5 in [44].
Corollary 4.0.1. Consider that (U,℘♭) indicates a ℘♭-complete P♭MS and O,S represent two self-mappings on this space. Presume that C:(0,∞)→(0,∞) is a continuous comparison function, D:(0,∞)→(0,∞), D∈Δ, and P:[0,∞)→[0,1ς), P∈Bς. If the pair (O,S) provides the following statement:
℘♭(Oȷ,Sρ)>0,D(℘♭(Oȷ,Sρ))≤C(D(P(ES,O(ȷ,ρ))ES,O(ȷ,ρ))) |
for all ȷ,ρ∈U, then the set of CFix(O,S) has a unique element.
Proof. Consider that αji(ȷ,ρ)=ςf=1 and i=j=1 in Theorem 3.0.2; then, the result is obvious.
Moreover, by taking S=O in Corollary 4.0.1, we gain the below consequence.
Corollary 4.0.2. Consider that (U,℘♭) indicates a ℘♭-complete P♭MS and O represents a self-mapping on this space. Suppose that C:(0,∞)→(0,∞) is a continuous comparison function, D:(0,∞)→(0,∞), D∈Δ, and P:[0,∞)→[0,1ς), P∈Bς. If O provides the following statement:
℘♭(Oȷ,Oρ)>0,D(℘♭(Oȷ,Oρ))≤C(D(P(EO(ȷ,ρ))EO(ȷ,ρ))) |
for all ȷ,ρ∈U, then Fix(O) consists of a unique element which belongs to (U,d).
The following consequence is an analysis of Theorem 2.1 presented by Aydi et al. in [45].
Corollary 4.0.3. Consider that (U,℘♭) indicates a ℘♭-complete P♭MS, O:U→U represents a mapping and α:U×U→[0,∞) is a function. Let C:(0,∞)→(0,∞) be a continuous comparison function, D:(0,∞)→(0,∞), D∈Δ, and P:[0,∞)→[0,1ς), P∈Bς. Presume that O fulfills the following conditions for ℘♭(Oȷ,Oρ)>0:
i. α(ȷ,ρ)≥1⇒ D(℘♭(Oȷ,Oρ))≤C(D(P(EO(ȷ,ρ))EO(ȷ,ρ))) for all ȷ,ρ∈U.
ii. O is a triangular α-orbital-admissible mapping.
iii. ȷ0 exists in U satisfying α(ȷ0,Oȷ0)≥1.
iv. O is ℘♭-continuous, or, if {ȷn}n∈N is a sequence in U such that α(ȷn,ȷn+1)≥1 for all n∈N and ȷn→ȷ as n→∞, then a subsequence {ȷnk} of {ȷn} exists such that α(ȷnk,ȷ)≥1 for all k∈N.
Then, Fix(O) consists of a unique element which belongs to (U,d).
Proof. By selecting αji(ȷ,ρ)≥ςf≥1 with i=j=1 and S=O in Theorem 3.0.2, the proof is completed.
The subsequent conclusion is an enhancement of the Banach fixed-point theorem [1] by taking the E-contraction into account.
Corollary 4.0.4. Consider that (U,℘♭) indicates a ℘♭-complete P♭MS and O represents a self-mapping. Suppose that, for all ȷ,ρ∈U and ℏ∈[0,1), the following statement holds:
℘♭(Oȷ,Oρ)≤ℏ(EO(ȷ,ρ)). | (4.1) |
Then, Fix(O) consists of a unique element which belongs to (U,d).
Proof. By taking C(t)=ℏt and D∈Δ with D(t)=t, as well as α(ȷ,ρ)=1, and keeping P:[0,∞)→[0,1ς) in mind, in Corollary 4.0.3, we achieve the desired conclusion.
Now, we state a new concept.
Definition 4.0.5. Consider that (U,℘♭) indicates a P♭MS and O:U→U represents a mapping. Presume that P∈Bς, θ∈˜Θ and τ∈(0,1) exist such that
℘♭(Oȷ,Oρ)≠0⇒θ(℘♭(Oȷ,Oρ))≤[θ(P(EO(ȷ,ρ))EO(ȷ,ρ))]τ | (4.2) |
for all ȷ,ρ∈U. Thus, O:U→U is termed as a Geraghty θE-contraction.
Theorem 4.0.6. Consider that (U,℘♭) indicates a ℘♭-complete P♭MS and O:U→U represents a Geraghty θE-contraction mapping. Thus, Fix(O) consists of a unique element which belongs to (U,d).
Proof. It is enough to take in Corollary 4.0.2 C(t)=(lnk)t and D∈Δ with D(t)=lnθ:(0,∞)→(0,∞); then, the proof is evident.
On the other hand, we characterize a new notation, which is an extension of [39], introduced by Fulga and Proca in 2017.
Definition 4.0.7. Consider that (U,℘♭) indicates a P♭MS and O:U→U represents a mapping. If P∈Bς, F∈F and κ>0 exist satisfying the inequality
κ+F(℘♭(Oȷ,Oρ))≤F(P(EO(ȷ,ρ))EO(ȷ,ρ)) | (4.3) |
for all ȷ,ρ∈U, then O is a Geraghty FE-contraction.
Theorem 4.0.8. Let (U,℘♭) be a ℘♭-complete P♭MS and O:U→U be a Geraghty FE-contraction. Thus, O has a unique fixed point.
Proof. By selecting C(t)=e−κt and D∈Δ with D(t)=eF:(0,∞)→(0,∞) in Corollary 4.0.2, we attain the claim.
We verify the following theorem, which offers an application of Corollary 4.0.2 to homotopy theory.
Theorem 5.0.1. Let (U,℘♭) be a ℘♭-complete P♭MS and Υ,Λ be open and closed subsets of U, respectively. Presume that R:Λ×[0,1]→U is an operator ensuring the subsequent statements.
i. ȷ≠R(ȷ,ι) for every ȷ∈Λ∖Υ and ι∈[0,1).
ii. For all ȷ,ρ∈Λ and ι,ℏ∈[0,1), we have
D(℘♭(R(ȷ,ι),R(ρ,ι)))≤C(D(P(EO(ȷ,ρ))EO(ȷ,ρ))), |
where
EO(ȷ,ρ)=℘♭(ȷ,ρ)+|℘♭(ȷ,R(ȷ,ι))−℘♭(ρ,R(ρ,ι))|. |
iii. A function ψ:[0,1]→R, which has the continuity property, exists such that
ς℘♭(R(ȷ,ι),R(ȷ,ι∗))≤|ψ(ι)−ψ(ι∗)| |
for all ι,ι∗∈[0,1) and ∀o∈Λ.
Then, R(.,0) holds a fixed-point ⇔ R(.,1) holds a fixed-point.
Proof. Define the subsequent set
X={ι∈[0,1]:ȷ=R(ȷ,ι)forsomeo∈Υ}. |
(⇒:) Presume that R(.,0) has a fixed-point. Then, X is nonempty, which signifies that 0∈X. Our claim is that X is both open and closed in [0,1]. As a result of utilizing the connectedness aspect, we derive X=[0,1]. In this case, R(.,1) allows a fixed-point in Υ.
Initially, the closedness of X in [0,1] is verified. Assume that {ιn}∞n=1⊆X with ιn→ι∈[0,1] as n→∞. It is essential to point out that ι∈X. For this reason, ιn∈X for n=1,2,3,…; ȷn∈Υ exists with ȷn=R(ȷn,ιn). Also, for n,m∈{1,2,3,…}, we have
℘♭(ȷn,ȷm)=℘♭(R(ȷn,ιn),R(ȷm,ιm))≤ς℘♭(R(ȷn,ιn),R(ȷn,ιm))+ς℘♭(R(ȷn,ιm),R(ȷm,ιm)). | (5.1) |
Also, by considering the function Cx with ℏ∈(0,1), from (b), we get
D(℘♭(R(ȷn,ιm),R(ȷm,ιm)))≤C(D(P(EO(ȷn,ȷm))EO(ȷn,ȷm)))=ℏD(P(EO(ȷn,ȷm))EO(ȷn,ȷm))≤ℏD(1ς[℘♭(ȷn,ȷm)+|℘♭(ȷn,R(ȷn,ιm))−℘♭(ȷm,R(ȷm,ιm))|])=ℏD(1ς[℘♭(ȷn,ȷm)+℘♭(R(ȷn,ιn),R(ȷn,ιm))]), |
which, by (D1), implies that
ς℘♭(R(ȷn,ιm),R(ȷm,ιm))<ℏ[℘♭(ȷn,ȷm)+℘♭(R(ȷn,ιn),R(ȷn,ιm))]. |
So, by using the above inequality and (iii), (5.1) becomes
℘♭(ȷn,ȷm)≤|ψ(ιn)−ψ(ιm)|+ℏ[℘♭(ȷn,ȷm)+1ς|ψ(ιn)−ψ(ιm)|] |
such that
℘♭(ȷn,ȷm)≤(1+ςς(1−ℏ))|ψ(ιn)−ψ(ιm)|. |
Then, employing the convergence of {ιn}n∈N with n,m→∞, we procure
limn,m→∞℘♭(ȷn,ȷm)=0. |
This confirms that {ȷn} is a ℘♭-Cauchy sequence in U. ℘♭-completeness of (U,℘♭) ensures that ȷ∗∈Λ exists such that
℘♭(ȷ∗,ȷ∗)=limn→∞℘♭(ȷ∗,ȷn)=limn,m→∞℘♭(ȷn,ȷm)=0. |
Moreover,
℘♭(ȷn,R(ȷ∗,ι))=℘♭(R(ȷn,ιn),R(ȷ∗,ι))≤ς℘♭(R(ȷn,ιn),R(ȷn,ι))+ς℘♭(R(ȷn,ι),R(ȷ∗,ι)). | (5.2) |
Likewise, we have
D(℘♭(R(ȷn,ι),R(ȷ∗,ι)))≤C(D(P(EO(ȷn,ȷ∗))EO(ȷn,ȷ∗)))≤ℏD(1ςEO(ȷn,ȷ∗))=ℏD(1ς[℘♭(ȷn,ȷ∗)+|℘♭(ȷn,R(ȷn,ι))−℘♭(ȷ∗,R(ȷ∗,ι))|]) |
such that
ς℘♭(R(ȷn,ι),R(ȷ∗,ι))<ℏ[℘♭(ȷn,ȷ∗)+|℘♭(ȷn,R(ȷn,ι))−℘♭(ȷ∗,R(ȷ∗,ι))|]. |
Then, by (5.2), we obtain
℘♭(ȷn,R(ȷ∗,ι))≤|ψ(ιn)−ψ(ι)|+ℏ℘♭(ȷn,ȷ∗). |
By taking the limit as n→∞ in the above equation, we have that limn→∞℘♭(ȷn,R(ȷ∗,ι))=0; hence,
℘♭(ȷ∗,R(ȷ∗,ι))=limn→∞℘♭(ȷn,R(ȷn,ι))=0. |
This means that ȷ∗=R(ȷ∗,ι). As (ⅰ) is provided, we obtain ȷ∗∈Υ.
Thus, ι∈X and X is closed in [0,1]. Second, the openness of X in [0,1] will be verified. Let ι0∈X. Then, ȷ0∈Υ exists with ȷ0=R(ȷ0,ι0). Because Υ is open, a non-negative δ exists such that B℘♭(ȷ0,δ)⊆Υ in U. Considering that ε=ς(1−ℏ)ς+ℏ(℘♭(ȷ0,ȷ0)+δ)>0 with ℏ∈[0,1) and ς≥1, there exists ϑ(ε)>0 such that |ψ(ι)−ψ(ι0)|<ε for all ι∈(ι0−ϑ(ε),ι0+ϑ(ε)) owing to fact of the continuity of ψ on ι0.
Let ι∈(ι0−ϑ(ε),ι0+ϑ(ε)); for
p∈¯B℘♭(ȷ0,δ)={ȷ∈U:℘♭(ȷ,ȷ0)≤℘♭(ȷ0,ȷ0)+δ}, |
we obtain
℘♭(R(ȷ,ι),ȷ0)=℘♭(R(ȷ,ι),R(ȷ0,ι0))≤ς℘♭(R(ȷ,ι),R(ȷ,ι0))+ς℘♭(R(ȷ,ι0),R(ȷ0,ι0)). | (5.3) |
Furthermore,
D(℘♭(R(ȷ,ι0),R(ȷ0,ι0)))≤C(D(P(EO(ȷ,ȷ0))EO(ȷ,ȷ0)))≤ℏD(1ςEO(ȷ,ȷ0))=ℏD(1ς[℘♭(ȷ,ȷ0)+|℘♭(ȷ,R(ȷ,ι0))−℘♭(ȷ0,R(ȷ0,ι0))|]), |
which implies that
ς℘♭(R(ȷ,ι0),R(ȷ0,ι0))<ℏ[℘♭(ȷ,ȷ0)+℘♭(R(ȷ,ι),R(ȷ,ι0))]. |
Finally, taking the above inequalities into account, from (5.3), we gain
℘♭(R(ȷ,ι),ȷ0)≤|ψ(ι)−ψ(ι0)|+ℏ[℘♭(ȷ,ȷ0)+1ς|ψ(ι)−ψ(ι0)|]≤(1+ℏς)|ψ(ι)−ψ(ι0)|+ℏ(℘♭(ȷ0,ȷ0)+δ)≤(1+ℏς)ε+ℏ(℘♭(ȷ0,ȷ0)+δ)≤℘♭(ȷ0,ȷ0)+δ, |
and R(ȷ,ι)∈¯B℘♭(ȷ0,δ). Therefore,
R(.,ι):¯B℘♭(ȷ0,δ)→¯B℘♭(ȷ0,δ) |
holds for every fixed ι∈(ι0−ϑ(ε),ι0+ϑ(ε)). We can now apply Corollary 4.0.2, contemplating the function C as Cx; then, R(.,ι) has a fixed-point in Λ. However, this point belongs to Υ, as (ⅰ) is true. Therefore, (ι0−ϑ(ε),ι0+ϑ(ε))⊆X, and we induce that X is open in [0,1].
The authors appreciate the anonymous reviewers' recommendations for improving the study.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare that there is no conflict of interest regarding the publication of this paper.
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