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Research article

Fixed-point results via αji-(DC(PˆE))-contractions in partial -metric spaces

  • Received: 03 May 2023 Revised: 05 July 2023 Accepted: 11 July 2023 Published: 01 August 2023
  • MSC : 47H10, 54H25

  • 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 ς1. 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×UR+, 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,rU. 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 ȷU. 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 ȷU and ε>0, the ensuing sets

    B(ȷ,ε)={ρU:(ȷ,ρ)<(ȷ,ȷ)+ε}

    and

    B[ȷ,ε]={ρU:(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):ȷU,ε>0}.

    Example 2.0.3. ([20]) Let U=[0,1] and (ȷ,ρ)=max{ȷ,ρ} for all ȷ,ρU. 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],|ȷρ|,{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,rU:

    1. (ȷ,ρ)=0ȷ=ρ;

    2. (ȷ,ρ)=(ρ,ȷ);

    3. there is a real constant ς1 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,rU and μ>1. So, (U,) is a -metric space with ς=2μ1.

    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,rU, :U×UR+ 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 ς1 exists such that

    (ȷ,ρ)ς[(ȷ,r)+(r,ρ)(r,r)]+1ς2((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,rU

    (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 PM and PMS, 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 ς1, every -metric space is a PMS; taking the coefficient ς=1, every partial metric space is a PMS. Moreover, a PM on U is not a partial metric, nor a -metric, in general. As far as we comprehend, a PMS 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×UR+ by (ȷ,ρ)={max{ȷ,ρ}}ω+|ȷρ|ω for all ȷ,ρU. Then, (U,) is a PMS 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×UR+ in order to become (ȷ,ρ)=max{ȷ,ρ}+a for all ȷ,ρU. Then, (U,) is a PMS with the coefficient ς1.

    iii. Let q1 and (U,) be a partial metric space. :U×UR+ is a PM with the coefficient ς=2q1 if this mapping is defined by (ȷ,ρ)=[(ȷ,ρ)]q.

    iv. Let U={1,2,3,4} and :U×UR+ be a PM with the coefficient ς=4, where PM is given by

    (ȷ,ρ)={|ȷρ|2+max{ȷ,ρ},ȷρ;ȷ,ȷ=ρ1;0,ȷ=ρ=1.

    It is clear that (2,2)=20, 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 ς1, assume that is a partial metric and is a -metric on a nonempty set U. is a PM with the coefficient ς1 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 q1, :U×UR+is a PM with the coefficient ς=2q1, where is specified by the subsequent expression:

    (ȷ,ρ)={|1ȷ1ρ|q+1,ȷ,ρ(0,1],ȷρ;emax{ȷ,ρ}min{ȷ,ρ},ȷ,ρ{2,3,4,},ȷρ;1,ȷ=ρ. (2.3)

    ii. Let U=N and :U×UR+ be a PM with the coefficient ς=4, where is defined as follows:

    (ȷ,ρ)={emax{|ȷρ|,ȷ+ρ2},ȷρ;e|ȷρ|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 ȷρr, since q1, 2q11; we have

    (ȷ,ρ)=1+|1ȷ1ρ|q=1+|1ȷ1r+1r1ρ|q(2q1)+2q1[|1ȷ1r|q+|1r1ρ|q]2q1[|1ȷ1r|q+|1r1ρ|q+2]12q1[|1ȷ1r|q+1+|1r1ρ|q+1]1|1r1r|q=2q1[(ȷ,r)+(r,ρ)](r,r)=ς[(ȷ,r)+(r,ρ)](r,r).

    Case 2: For ȷ,ρ(0,1], ȷρ and ȷ=r, we conclude that

    (ȷ,ρ)=1+|1ȷ1ρ|q=1+|1r1ρ|q+11|1r1r|q2q1[1+|1r1ρ|q+1][1+|1r1r|q]=2q1[(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], ȷr and ȷ=ρ.

    Case 4: For ȷ,ρ,r{2,3,4,...} and ȷρr, owing to fact that

    max{ȷ,ρ}min{ȷ,ρ}=max{ȷ+rr,ρ+rr}min{ȷ+rr,ρ+rr}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,...}, ȷρ 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)1ς[(ȷ,r)+(r,ρ)](r,r).

    Case 6: This case is provided for ȷ,r{2,3,4,...}, ȷr and ȷ=ρ, as in Case 5.

    Consequently, since (1)(4) are provided, the mapping indicated in (2.3) is a PM with ς=2q1.

    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 ȷrρ. Then,

    max{|ȷρ|,ȷ+ρ2}=max{|ȷr+rρ|,ȷ+r+r+ρ2r}max{|ȷr|+|rρ|,ȷ+r2+r+ρ2r}=max{|ȷr|,ȷ+r2}+max{|rρ|,r+ρ2}max{|rr|,r+r2};

    we have

    emax{|ȷρ|,ȷ+ρ2}emax{|ȷr|,ȷ+r2}+emax{|rρ|,r+ρ2}emax{|rr|,r+r2}.

    Thereby, the following expression is ensured for all ς1:

    (ȷ,ρ)(ȷ,r)+(r,ρ)(r,r)ς[(ȷ,r)+(r,ρ)](r,r).

    Case 2: Let ȷρr. Because

    max{|ȷρ|,ȷ+ρ2}=max{|ȷr+rρ|,ȷ+r+r+ρ2r}max{|ȷr|+|rρ|,ȷ+r2+r+ρ2r}=max{|ȷr|,ȷ+r2}+max{|rρ|,r+ρ2}max{|rr|,r+r2}max{|ȷr|,ȷ+r2}+|rρ|2+r+ρ2max{|rr|,r+r2},

    we have

    emax{|ȷρ|,ȷ+ρ2}emax{|ȷr|,ȷ+r2}+e|rρ|2+r+ρ2emax{|rr|,r+r2}.

    Then, for all ς1, we procure

    (ȷ,ρ)(ȷ,r)+(r,ρ)(r,r)ς[(ȷ,r)+(r,ρ)](r,r).

    Case 3: Let rȷρ. Then, for all ς1, we achieve

    max{|ȷρ|,ȷ+ρ2}=max{|ȷr+rρ|,ȷ+r+r+ρ2r}max{|ȷr|+|rρ|,ȷ+r2+r+ρ2r}=max{|ȷr|,ȷ+r2}+max{|rρ|,r+ρ2}max{|rr|,r+r2}|ȷr|2+ȷ+r2+max{|rρ|,r+ρ2}max{|rr|,r+r2},

    and

    emax{|ȷρ|,ȷ+ρ2}e|ȷr|2+ȷ+r2+emax{|rρ|,r+ρ2}emax{|rr|,r+r2}.

    Case 4: Let ȷ>r>ρ. Because

    |ȷρ|2+ȷ+ρ2=|ȷr+rρ|2+ȷ+rr+ρ2r22[|ȷr|22+|rρ|22]+ȷ+r2+r+ρ2r22[|ȷr|2+ȷ+r2+|rρ|2+r+ρ2]max{|rr|,r+r2},

    for ς=4, we gain

    e|ȷρ|2+ȷ+ρ222[e|ȷr|2+ȷ+r2+e|rρ|2+r+ρ2]emax{|rr|,r+r2},(ȷ,ρ)4[(ȷ,r)+(r,ρ)](r,r).

    Case 5: Let ȷ>ρ>r. Then, for all ς1, we attain

    |ȷρ|2+ȷ+ȷ2=|ȷr|2+ȷ+rr+ȷ2r=|ȷr|2+ȷ+r2+r+ȷ2r|ȷr|2+ȷ+r2+max{|rȷ|,r+ȷ2}max{|rr|,r+r2},

    and

    e|ȷρ|2+ȷ+ρ2e|ȷr|2+ȷ+r2+emax{|rρ|,r+ρ2}emax{|rr|,r+r2}(ȷ,ρ)(ȷ,r)+(r,ρ)(r,r)ς[(ȷ,r)+(r,ρ)](r,r).

    Case 6: Let r>ȷ>ρ. Since

    |ȷρ|2+ȷ+ρ2|rρ|2+r+ρ2+2max{|ȷr|,ȷ+r2}max{|rr|,r+r2}2[max{|ȷr|,ȷ+r2}+|rρ|2+r+ρ2]max{|rr|,r+r2},

    and

    e|ȷρ|2+ȷ+ρ22[emax{|ȷr|,ȷ+r2}+e|rρ|2+r+ρ2]emax{|rr|,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 PM 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,ρU.

    Remark 2.0.12. Let ε>0 {B(ȷ,ε):ȷU,ε>0} be the family of -open balls, where B(ȷ,ε)={ρU:(ȷ,ρ)<(ȷ,ȷ)+ε} for all ȷU. Each PM 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=ȷρ=3 and q=2, we currently possess

    B(12,1)={ρU:(12,ρ)<(12,12)+1}={ρU:|21ρ|2+1<1+1}={ρU:|21ρ|2<1}={ρU:12ρ<1}=[12,1)

    and

    B(14,4)={ρU:(14,ρ)<(14,14)+4}={ρU:|41ρ|2+1<1+4}={ρU:|41ρ|2<4}={ρU:041ρ<2}=[14,12).

    12B(12,1), but 14B(12,1), and 14B(14,4), but 12B(14,4). Thus, it is deduced that a PM on a set U need not to be T1.

    Definition 2.0.13. ([25]) Let (U,) be a PMS with a coefficient ς and {ȷn}nN be a sequence in (U,).

    i. Provided that the equality limn(ȷn,ȷ)=(ȷ,ȷ) holds, {ȷn}nN is called a -convergent sequence in U and ȷ is termed the -limit of {ȷn}nN.

    ii. {ȷn}nN is a -Cauchy sequence if limn,m(ȷn,ȷm) exists and is finite.

    iii. If, for each -Cauchy sequence {ȷn}nN, a point ȷ exists in U such that

    limn,m(ȷn,ȷm)=limn(ȷn,ȷ)=(ȷ,ȷ),

    then (U,) is called a -complete space.

    Remark 2.0.14. In a PMS, a convergent sequence does not require a unique limit. In Example 2.0.10 (ii), consider

    (ȷn)nN=(1n2+1)nNU.

    Then,

    limn(1n2+1,0)=limne|1n2+10|2+1n2+1+02=1=(0,0)=emax{|00|,0+02},limn(1n2+1,1)=limnemax{|1n2+11|,1n2+1+12}=e1=(1,1)=emax{|11|,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×UR+ denote a mapping determined by (ȷ,ρ)=max{ȷ,ρ} for all ȷ,ρU. In such a case, (U,) has become a PMS with the coefficient ς1. We obtain the following for the sequence {1n+1}nN in (U,):

    limn(1,11+n)=1=(1,1),limn(2,11+n)=2=(2,2).

    The sequence {1n+1}nN owns two limits in U.

    Lemma 2.0.16. ([25]) Presume that (U,) and (U,d) are a PMS and -metric space, respectively.

    1) {ȷn}nN 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, limnd(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×UR+ is a mapping defined as follows:

    (ȷ,ρ)={0,ȷ=ρ;|1ȷ1ρ|,ifoneofȷandρoddandtheotherisoddorȷρ=;5,ifoneofȷandρevenandtheotherisevenȷρor;2,otherwise.

    (U,) is a PMS with ς=3. For each nN, consider that ȷn=2n+3. Then,

    (2n+3,)=12n+30

    as n; so, ȷn; however, limn(ȷn,4)=25=(,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 PM is not continuous in general.

    Lemma 2.0.18. ([27]) Ensure that (U,) is a PMS with the coefficient ς>1 and the sequences {ȷn}nN and {ρn}nN are -convergent to ȷ and ρ, respectively. Afterward, we obtain

    1ς2(ȷ,ρ)1ς(ȷ,ȷ)(ρ,ρ)limninf(ȷn,ρn)limnsup(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:UU is an F-contraction provided that FF and κ>0 exist such that, for all ȷ,ρU,

    d(Oȷ,Oρ)>0κ+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}nN of positive numbers, limnan=0 limnF(an)=;

    (F3) a constant c(0,1) exists such that lima0+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 FF 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,), limzθ(az)=1 limzaz=0+;

    (θ3) a constant r(0,1) and s(0,] exist such that lima0+θ(a)ar=s.

    Theorem 2.0.20. Let O:UU be a mapping on a complete metric space (U,d). Provided that a function θΘ and a constant τ(0,1) exist such that

    d(Oȷ,Oρ)0θ(d(Oȷ,Oρ))[θ(d(ȷ,ρ))]τ

    for all ȷ,ρU, 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:UU be a self-mapping on a complete metric space (U,d). Thereby, the following statements are equivalent:

    i. the mapping O is a θcontraction with θ˜Θ;

    ii. the mapping O is an Fcontraction with FF.

    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) limnD(an)=0limnan=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<bC(a)<C(b).

    (C2) limnCn(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 ={(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.

    In 2021, Nazam et al. [50] presented a new definition of the DC-contraction, including two self-mappings with a binary relation in a PMS, referring to Liu's definition of a DC-contraction.

    Definition 2.0.23. ([50]) Let O and S be two self-mappings on a PMS and be a binary relation on U. Define the set ={(ȷ,ρ):(Oȷ,Sρ)>0}. The mappings O and S form a DC-contraction if there exist DΔ and a continuous comparison function C such that

    D(ς2(O(ȷ),S(ρ)))C(D((ȷ,ρ))) (2.6)

    for all ȷ,ρ.

    Definition 2.0.24. ([46]) Let U be a nonempty set and O:UU and α:U2R be given mappings. O is α-orbital-admissible provided that the expression given below is true:

    α(ȷ,Oȷ)1α(Oȷ,O2ȷ)1

    for all ȷU.

    Definition 2.0.25. ([35]) Let U be a nonempty set and α:U2R be a given function.

    i. O is an α-orbital-admissible mapping;

    ii. α(ȷ,ρ)1 and α(ρ,Oρ)1α(ȷ,Oρ)1, ȷ,ρU.

    A mapping O:UU 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:UU is a Geraghty-type contraction if a function P:[0,)[0,1) exists, which ensures the following term:

    limnP(tn)=1limntn=0 (2.7)

    such that

    d(Oȷ,Oρ)P(d(ȷ,ρ))d(ȷ,ρ)

    for all ȷ,ρU. 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:UU satisfy the below statements for all ȷ,ρU:

    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 PB 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 limnP(tn)=1ς for ς1 implies that limntn=0 and it is stated as Bς.

    Definition 2.0.29. ([46]) Let α:U2R be a function in (U,d). Provided that the subsequent statement

    α(ȷ,ρ)1d(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,ρU whenever PBς, then O:UU 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:U2R be a function in a -metric space (U,) with the coefficient ς1. O,S:UU are two self-mappings which are called αji-PˆES,O-Geraghty contractions if a function PBς exists, which ensures the following expressions:

    αji(ȷ,ρ)ςfαji(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,ρU, and f2 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 ς1. The mappings O,S:UU are called αji-orbital-admissible if the following circumstances hold:

    αji(ȷ,Oiȷ)ςfαji(Oiȷ,SjOiȷ)ςf,αji(ȷ,Sjȷ)ςfαji(Sjȷ,OiSjȷ)ςf

    for all ȷU, where f2 is a constant.

    Definition 2.0.33. ([46]) Let αji:U2[0,) be a function in a complete -metric space (U,) with the coefficient ς1. When O,S:UU 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 f2 is a constant.

    Lemma 2.0.34. ([46]) Consider that O,S:UU are two mappings in a complete -metric space (U,) with the coefficient ς1. 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}nN in (U,b) as follows:

    ȷ2n=Sjȷ2n1,ȷ2n+1=Oiȷ2n,

    where n=0,1,2,...; afterward, for n,mN{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 PMS with the coefficient ς1 and αji:U2[0,) be a function. The mappings O,S:UU are called generalized αji-(DC(PˆE))-contractions if DΔ and a continuous comparison function C and PBς exist such that, for all ȷ,ρU, 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 f2 is a constant.

    Theorem 3.0.2. Ensure that (U,) is a -complete PMS, O,S:UU 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. ȷ0U exists, satisfying αji(ȷ0,Oiȷ0)ςf;

    iv. one of the below terms is provided:

    iva. Oi and Sj are -continuous,

    or

    ivb. if {ȷn}nN is a sequence in U such that αji(ȷn,ȷn+1)ςf for each nN and ȷnrU as n, then a subsequence {ȷnk} of {ȷn} exists such that αji(ȷnk,r)ςf for each kN;

    v. for all rFix(Oi) or wFix(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 ȷ0U with the property αji(ȷ0,Oiȷ0)ςf exists from (ⅲ). Consider {ȷn}nN in U, which is constructed as

    ȷ2n+2=Oiȷ2n+1andȷ2n+1=Sjȷ2n

    for each nN. 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+1Sjȷ2n0. 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=ȷ2n01. Hereafter, we presume that ȷnȷn+1 for all nN. It is needed to investigate the subsequent two cases:

    Case 1: Presume that ȷ2nȷ2n1 for all nN. (Oiȷ2n1,Sjȷ2n)>0 and, moreover, αji(ȷ2n1,ȷ2n)ςf. Therefore, employing (3.1), we gain

    D((ȷ2n,ȷ2n+1))=D((Oiȷ2n1,Sjȷ2n))D(αji(ȷ2n1,ȷ2n)(Oiȷ2n1,Sjȷ2n))C(D((ˆES,O(ȷ2n1,ȷ2n))ˆES,O(ȷ2n1,ȷ2n)))<C(D(1ςˆES,O(ȷ2n1,ȷ2n))),

    where

    ˆES,O(ȷ2n1,ȷ2n)=(ȷ2n1,ȷ2n)+|(ȷ2n1,Oiȷ2n1)(ȷ2n,Sjȷ2n)|=(ȷ2n1,ȷ2n)+|(ȷ2n1,ȷ2n)(ȷ2n,ȷ2n+1)|.

    If (ȷ2n,ȷ2n+1)(ȷ2n1,ȷ2n), the following expression is obtained:

    ˆES,O(ȷ2n1,ȷ2n)=(ȷ2n1,ȷ2n)(ȷ2n1,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)<(ȷ2n1,ȷ2n).

    Case 2: Presume that ȷ2nȷ2n+1 for all nN. For all nN, (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)}nN is a non-increasing sequence. Therefore, L0 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(1ςlimnˆES,O(ȷ2n1,ȷ2n))),

    where

    ˆES,O(ȷ2n1,ȷ2n)=(ȷ2n1,ȷ2n)+|(ȷ2n1,Oiȷ2n1)(ȷ2n,Sjȷ2n)|=(ȷ2n1,ȷ2n)+|(ȷ2n1,ȷ2n)(ȷ2n,ȷ2n+1)|=2(ȷ2n1,ȷ2n)(ȷ2n,ȷ2n+1)

    and

    limninfˆES,O(ȷ2n1,ȷ2n)=liminfn[2(ȷ2n1,ȷ2n)(ȷ2n,ȷ2n+1)]limsupn[2(ȷ2n1,ȷ2n)(ȷ2n,ȷ2n+1)]=L.

    Hence, we attain

    D(L)=D(limn(ȷ2n,ȷ2n+1))<C(D(1ςlimnsupˆES,O(ȷ2n1,ȷ2n)))<C(D(1ςL))<D(Lς),

    which is a contradiction. The same argument is true for (ȷ2n,ȷ2n1). 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

    0limnd(ȷn,ȷn+1)=limn2[(ȷn,ȷn+1)(ȷn,ȷn)(ȷn+1,ȷn+1)]limnsup2[(ȷn,ȷn+1)(ȷn,ȷn)(ȷn+1,ȷn+1)]=0.

    Thus, limnd(ȷ2n,ȷ2n+1)=0 for all nN. Furthermore, we currently possess the subsequent expression for all n,m1:

    limn,md(ȷ2m,ȷ2n)=2limn,msup(ȷ2m,ȷ2n).

    It is necessary to indicate that the sequence {ȷn}nN is a -Cauchy sequence in (U,). Instead, it is needed to verify that {ȷ2n}nN is a -Cauchy sequence. According to Lemma 2.0.16 (1), it is required to explicitly state that {ȷ2n}nN is a Cauchy sequence in (U,d). Assume that {ȷ2n}nN 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 kN,

    d(ȷ2mk,ȷ2nk2)<ε. (3.3)

    Applying 3, we deduce that

    εd(ȷ2mk,ȷ2nk)ςd(ȷ2mk,ȷ2mk+1)+ςd(ȷ2mk+1,ȷ2nk).

    By taking the limit in the above as k, we write

    εςlimkinfd(ȷ2mk+1,ȷ2nk)limksupd(ȷ2mk+1,ȷ2nk). (3.4)

    Also, employing 3, we gain

    d(ȷ2mk,ȷ2nk1)ςd(ȷ2mk,ȷ2nk2)+ςd(ȷ2nk2,ȷ2nk1);

    again, taking the limit in the above as k, we get

    limksupd(ȷ2mk,ȷ2nk1)ςε. (3.5)

    Moreover, the following statement is derived:

    d(ȷ2mk,ȷ2nk)ςd(ȷ2mk,ȷ2nk2)+ςd(ȷ2nk2,ȷ2nk)ςd(ȷ2mk,ȷ2nk2)+ς2d(ȷ2nk2,ȷ2nk1)+ς2d(ȷ2nk1,ȷ2nk).

    Similarly, if the limit is taken for k, we obtain

    limksupd(ȷ2mk,ȷ2nk)ςε. (3.6)

    Again, from 3, it is achievable to produce

    d(ȷ2mk+1,ȷ2nk1)ςd(ȷ2mk+1,ȷ2mk)+ςd(ȷ2mk,ȷ2nk1).

    Likewise, taking the limit as k, it is concluded that

    limksupd(ȷ2mk+1,ȷ2nk1)ς2ε. (3.7)

    From the inequalities (3.4)–(3.7), the subsequent expressions are attained:

    ε2ςlimkinf(ȷ2mk+1,ȷ2nk)limksup(ȷ2mk+1,ȷ2nk), (3.8)
    limksup(ȷ2mk,ȷ2nk1)ςε2, (3.9)
    limksup(ȷ2mk,ȷ2nk)ςε2, (3.10)
    limksup(ȷ2mk+1,ȷ2nk1)ς2ε2. (3.11)

    Due to Lemma 2.0.18, it is known that αji(ȷ2mk,ȷ2nk1)ςf and

    D((ȷ2mk+1,ȷ2nk))D(αji(ȷ2mk,ȷ2nk1)(Oiȷ2mk,Sjȷ2nk1))C(D(P(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1))),

    where

    ˆES,O(ȷ2mk,ȷ2nk1)=(ȷ2mk,ȷ2nk1)+|(ȷ2mk,Oiȷ2mk)(ȷ2nk1,Sjȷ2nk1)|=(ȷ2mk,ȷ2nk1)+|(ȷ2mk,ȷ2mk+1)(ȷ2nk1,ȷ2nk)|,

    and, taking the limit in the above as k, we procure

    limkinfˆES,O(ȷ2mk,ȷ2nk1)=limkinf(ȷ2mk,ȷ2nk1)limksup(ȷ2mk,ȷ2nk1)=limksupˆES,O(ȷ2mk,ȷ2nk1)ςε2.

    In light of the results obtained above, using (3.1), we write

    D(ε2)=D(ςε2ς)D(ςlimkinf(ȷ2mk+1,ȷ2nk))D(ςplimkinf(ȷ2mk+1,ȷ2nk))D(limkinfαji(ȷ2mk,ȷ2nk1)(Oiȷ2mk,Sjȷ2nk1))C(D(limkinfP(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)))D(limkinf1ςςε2)=D(ε2). (3.12)

    Then,

    D(ε2)D(αi,j(ȷ2mk,ȷ2nk1)limksup(Oiȷ2mk,Sjȷ2nk1))limksupD(αji(ȷ2mk,ȷ2nk1)(Oiȷ2mk,Sjȷ2nk1))limksupC(D(P(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)))=C(D(limksupP(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)))<D(limksup1ςςε2)=D(ε2). (3.13)

    Therefore, from (3.12) and (3.13), we obtain

    limkP(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)=ε2. (3.14)

    Similarly,

    D(ςε2)=D(ς2ε2ς)D(ς2limkinf(ȷ2mk+1,ȷ2nk))D(ςplimkinf(ȷ2mk+1,ȷ2nk))D(limkinfαji(ȷ2mk,ȷ2nk1)(Oiȷ2mk,Sjȷ2nk1))C(D(limkinfP(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)))<D(limkinfˆES,O(ȷ2mk,ȷ2nk1))D(ςε2), (3.15)

    and

    D(ςε2)D(αji(ȷ2mk,ȷ2nk1)limksup(Oiȷ2mk,Sjȷ2nk1))limksupD(αji(ȷ2mk,ȷ2nk1)(Oiȷ2mk,Sjȷ2nk1))limksupC(D(P(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)))=C(D(limksupP(ˆES,O(ȷ2mk,ȷ2nk1))ˆES,O(ȷ2mk,ȷ2nk1)))<D(limksupˆES,O(ȷ2mk,ȷ2nk1))D(ςε2). (3.16)

    From (3.15) and (3.16), it is obtained that

    limkˆES,O(ȷ2mk,ȷ2nk1)=ςε2. (3.17)

    Because of (3.14) and (3.17), we conclude that

    limkP(ˆES,O(ȷ2mk,ȷ2nk1))=1ς, (3.18)

    and, in (3.18), it is deduced by using the property of P that

    limkˆES,O(ȷ2mk,ȷ2nk1)=0.

    This contradiction brings about the sequence {ȷn}nN as a Cauchy sequence in (U,d). The completeness of (U,d) provides that the sequence {ȷn}nN converges to rU. Thus,

    limnd(ȷn,r)=0.

    By Lemma 2.0.16, we derive

    (r,r)=limn(ȷn,r)=limn,m(ȷn,ȷm).

    Then, {ȷn}nN 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

    0limn,msupd(ȷn,ȷm)limn,msupς[d(ȷn,r)+d(r,ȷm)]

    and limn,md(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}nN 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

    limksupˆES,O(ȷ2nk,r)=(r,Sjr), (3.21)

    and, using this, we get the following expression:

    D(limksup(ȷ2nk+1,Sjr))<D(1ς(r,Sjr)). (3.22)

    Further, since 1ς(r,Sjr)limksup(ȷ2nk+1,Sjr), we attain

    D(1ς(r,Sjr))D(limksup(ȷ2nk+1,Sjr)). (3.23)

    Therefore, using the expressions (3.22) and (3.23), we obtain

    D(1ς(r,Sjr))D(limksup(ȷ2nk+1,Sjr))D(limkP(ˆES,O(ȷ2nk,r))ˆES,O(ȷ2nk,r))D(1ς(r,Sjr)).

    From here, one can deduce that

    limkP(ˆES,O(ȷ2nk,r))ˆES,O(ȷ2nk,r)=1ς(r,Sjr). (3.24)

    Because of (3.21) and (3.24), it is concluded that

    limkP(ˆES,O(ȷ2nk,r))=1ς. (3.25)

    In (3.25), due to the property of P, we obtain that limkˆES,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 wU satisfies Sjw=wr. 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×UR+ is determined as (ȷ,ρ)=(ȷρ)2. Then, (U,) is a complete PMS with the coefficient ς=2. αji:U2[0,) is defined as follows:

    αji(ȷ,ρ)={ςf,ȷ,ρ[0,1],0,otherwise,

    with f2. Let O,S:UU be defined by O(ȷ)=ȷ2, S(ȷ)=ȷ4 and take P(t)=132,t>0,PBς. The pair (O,S) is triangular αji-orbital-admissible. Define D:(0,)(0,) by D(θ)=θeθ for each θ>0; then, DΔ. 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(ȷρ)2,ˆES,O(ȷ,ρ)=(ȷ,ρ)+|(ȷ,Oiȷ)(ρ,Sjρ)|=(ȷρ)2+|(15ȷ16)2(15ρ16)2|,

    and

    D(αi,j(ȷ,ρ)(Oiȷ,Sjρ))=D(41162(ȷρ)2)=D(164(ȷρ)2)=164(ȷρ)2e164(ȷρ)2.

    Then,

    C(D(P(ˆES,O(ȷ,ρ))ˆES,O(ȷ,ρ)))=C(D(132[(ȷρ)2+|(15ȷ16)2(15ρ16)2|]))=12132[(ȷρ)2+|(15ȷ16)2(15ρ16)2|]e132[(ȷρ)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×UR+ is determined as (ȷ,ρ)=(max{ȷ,ρ})2. Then, (U,) is a complete PMS with the coefficient ς=2. αji:U2[0,) is defined as follows:

    αji(ȷ,ρ)={ςf,ȷ,ρ[0,1],0,otherwise,

    with f2. Let O,S:UU be respectively defined by O(ȷ)=ȷ16, S(ȷ)=ȷ4, and take P(t)=1256, t>0,PBς. The pair (O,S) is triangular αji-orbital-admissible. Define D:(0,)(0,) by D(ϑ)=ϑ for each ϑ>0; then, DΔ. 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+|ȷρ|,

    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+|ȷρ|]))=1161256[(max{ȷ,ρ})2+|ȷρ|].

    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 PMS and O,S represent two self-mappings on this space. Presume that C:(0,)(0,) is a continuous comparison function, D:(0,)(0,), DΔ, and P:[0,)[0,1ς), PBς. 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,ρU, 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 PMS and O represents a self-mapping on this space. Suppose that C:(0,)(0,) is a continuous comparison function, D:(0,)(0,), DΔ, and P:[0,)[0,1ς), PBς. If O provides the following statement:

    (Oȷ,Oρ)>0,D((Oȷ,Oρ))C(D(P(EO(ȷ,ρ))EO(ȷ,ρ)))

    for all ȷ,ρU, 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 PMS, O:UU represents a mapping and α:U×U[0,) is a function. Let C:(0,)(0,) be a continuous comparison function, D:(0,)(0,), DΔ, and P:[0,)[0,1ς), PBς. 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,ρU.

    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}nN is a sequence in U such that α(ȷn,ȷn+1)1 for all nN and ȷnȷ as n, then a subsequence {ȷnk} of {ȷn} exists such that α(ȷnk,ȷ)1 for all kN.

    Then, Fix(O) consists of a unique element which belongs to (U,d).

    Proof. By selecting αji(ȷ,ρ)ςf1 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 PMS and O represents a self-mapping. Suppose that, for all ȷ,ρU and [0,1), the following statement holds:

    (Oȷ,Oρ)(EO(ȷ,ρ)). (4.1)

    Then, Fix(O) consists of a unique element which belongs to (U,d).

    Proof. By taking C(t)=t and DΔ 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 PMS and O:UU represents a mapping. Presume that PBς, θ˜Θ and τ(0,1) exist such that

    (Oȷ,Oρ)0θ((Oȷ,Oρ))[θ(P(EO(ȷ,ρ))EO(ȷ,ρ))]τ (4.2)

    for all ȷ,ρU. Thus, O:UU is termed as a Geraghty θE-contraction.

    Theorem 4.0.6. Consider that (U,) indicates a -complete PMS and O:UU 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 DΔ 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 PMS and O:UU represents a mapping. If PBς, FF and κ>0 exist satisfying the inequality

    κ+F((Oȷ,Oρ))F(P(EO(ȷ,ρ))EO(ȷ,ρ)) (4.3)

    for all ȷ,ρU, then O is a Geraghty FE-contraction.

    Theorem 4.0.8. Let (U,) be a -complete PMS and O:UU be a Geraghty FE-contraction. Thus, O has a unique fixed point.

    Proof. By selecting C(t)=eκt and DΔ 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 PMS 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. ȷR(ȷ,ι) for every ȷΛΥ and ι[0,1).

    ii. For all ȷ,ρΛ and ι,[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 oΛ.

    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)forsomeoΥ}.

    (⇒:) Presume that R(.,0) has a fixed-point. Then, X is nonempty, which signifies that 0X. 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=1X with ιnι[0,1] as n. It is essential to point out that ιX. For this reason, ιnX for n=1,2,3,; ȷnΥ 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 (0,1), from (b), we get

    D((R(ȷn,ιm),R(ȷm,ιm)))C(D(P(EO(ȷn,ȷm))EO(ȷn,ȷm)))=D(P(EO(ȷn,ȷm))EO(ȷn,ȷm))D(1ς[(ȷn,ȷm)+|(ȷn,R(ȷn,ιm))(ȷm,R(ȷm,ιm))|])=D(1ς[(ȷn,ȷm)+(R(ȷn,ιn),R(ȷn,ιm))]),

    which, by (D1), implies that

    ς(R(ȷn,ιm),R(ȷm,ιm))<[(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)|+[(ȷn,ȷm)+1ς|ψ(ιn)ψ(ιm)|]

    such that

    (ȷn,ȷm)(1+ςς(1))|ψ(ιn)ψ(ιm)|.

    Then, employing the convergence of {ιn}nN 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 ȷΛ 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)))D(1ςEO(ȷn,ȷ))=D(1ς[(ȷn,ȷ)+|(ȷn,R(ȷn,ι))(ȷ,R(ȷ,ι))|])

    such that

    ς(R(ȷn,ι),R(ȷ,ι))<[(ȷn,ȷ)+|(ȷn,R(ȷn,ι))(ȷ,R(ȷ,ι))|].

    Then, by (5.2), we obtain

    (ȷn,R(ȷ,ι))|ψ(ιn)ψ(ι)|+(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 ȷΥ.

    Thus, ιX and X is closed in [0,1]. Second, the openness of X in [0,1] will be verified. Let ι0X. Then, ȷ0Υ 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)ς+((ȷ0,ȷ0)+δ)>0 with [0,1) and ς1, there exists ϑ(ε)>0 such that |ψ(ι)ψ(ι0)|<ε for all ι(ι0ϑ(ε),ι0+ϑ(ε)) owing to fact of the continuity of ψ on ι0.

    Let ι(ι0ϑ(ε),ι0+ϑ(ε)); for

    p¯B(ȷ0,δ)={ȷU:(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)))D(1ςEO(ȷ,ȷ0))=D(1ς[(ȷ,ȷ0)+|(ȷ,R(ȷ,ι0))(ȷ0,R(ȷ0,ι0))|]),

    which implies that

    ς(R(ȷ,ι0),R(ȷ0,ι0))<[(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)|+[(ȷ,ȷ0)+1ς|ψ(ι)ψ(ι0)|](1+ς)|ψ(ι)ψ(ι0)|+((ȷ0,ȷ0)+δ)(1+ς)ε+((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(ι0ϑ(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, (ι0ϑ(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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