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Research article Special Issues

Optimal pair of fixed points for a new class of noncyclic mappings under a (φ,Rt)-enriched contraction condition

  • In the present study, we commenced by presenting a new class of maps, termed noncyclic (φ,Rt)-enriched quasi-contractions within metric spaces equipped with a transitive relation Rt. Subsequently, we identified the conditions for the existence of an optimal pair of fixed points pertaining to these mappings, thereby extending and refining a selection of contemporary findings documented in some articles. Specifically, our analysis will encompass the outcomes pertinent to reflexive and strictly convex Banach spaces.

    Citation: A. Safari-Hafshejani, M. Gabeleh, M. De la Sen. Optimal pair of fixed points for a new class of noncyclic mappings under a (φ,Rt)-enriched contraction condition[J]. Electronic Research Archive, 2024, 32(4): 2251-2266. doi: 10.3934/era.2024102

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  • In the present study, we commenced by presenting a new class of maps, termed noncyclic (φ,Rt)-enriched quasi-contractions within metric spaces equipped with a transitive relation Rt. Subsequently, we identified the conditions for the existence of an optimal pair of fixed points pertaining to these mappings, thereby extending and refining a selection of contemporary findings documented in some articles. Specifically, our analysis will encompass the outcomes pertinent to reflexive and strictly convex Banach spaces.



    Let F and G be subsets of a metric space (X,d). A self-mapping Γ on FG is said to be noncyclic whenever Γ(F)F and Γ(G)G. In this situation, a point (p,q)F×G is called an optimal pair of fixed points of Γ provided that

    (Γp,Γq)=(p,q)andd(p,q)=Dist(F,G),

    where Dist(F,G)=inf{d(p,q):pF,qG}. We denote the set of all optimal pairs of fixed points of Γ in F×G by Fix(ΓF×G).

    Γ:FGFG is called a noncyclic contraction if there exists λ[0,1) such that

    d(Γp,Γq)λd(p,q)+(1λ)Dist(F,G), (1.1)

    for all (p,q)F×G.

    In 2013, Espínola and Gabeleh proved that if F and G are nonempty, weakly compact, and convex subsets of a strictly convex Banach space X, then Fix(ΓF×G) for every noncyclic contraction Γ defined on FG (see Theorem 3.10 of [1]).

    After that, Gabeleh used the projection operators and proved both existence and convergence of an optimal pair of fixed points for noncyclic contractions in the setting of uniformly convex Banach spaces (see Theorem 3.2 of [2]).

    We refer to [3,4,5,6,7,8,9,10] to study the problem of the existence of an optimal pair of fixed points for various classes of noncyclic mappings.

    Recently, the authors of [10] introduced a new class of noncyclic mappings called noncyclic Fisher quasi-contractions, which contains the class of noncyclic contractions as a subclass, and they surveyed the existence and convergence of an optimal pair of fixed points in metric spaces by using a geometric notion of property WUC (Definition 2.2) on a nonempty pair of subsets of a metric space.

    In this article, we extend the main conclusion of the paper [10] by considering an appropriate control function and equipping the metric space (X,d) with a transitive relation Rt. Indeed, we introduce a new class of noncyclic mappings called noncyclic (φ,Rt)-enriched quasi-contractions, which is a kind of contraction at a point defined first in [11] and generalized later on in [12,13]. We then study the existence, uniqueness, and convergence of an optimal pair of fixed points for such mappings in metric spaces equipped with a transitive relation Rt. This idea to consider a contractive condition only for points in some transitive relations was first introduced in [14] in order to generalize the ideas of coupled fixed points in partially ordered spaces, and further developed in a sequence of articles [15,16,17]. We will also examine some other existence conclusions of an optimal pair of fixed points in the framework of reflexive and strictly convex Banach spaces.

    In this section, we point out some definitions and notations, which will be used in our coming arguments.

    In what follows, BX and SX denote the unit closed ball and the unit sphere in a Banach space X.

    Definition 2.1. ([18]) A Banach space X is said to be

    (i) uniformly convex provided that for every ε(0,2], one can find a corresponding δ=δ(ε) with the property that, whenever p,qBX with pqε, it follows that

    p+q2<1δ;

    (ii) strictly convex if for any two distinct elements p,qSX, we have

    p+q2<1.

    It is evident that every uniformly convex Banach space X is strictly convex. However, the reverse does not universally hold. For instance, the Banach space 1, which is equipped with its standard norm

    u=u21+u22, ul1,

    where .1 and .2 are the norms on l1 and l2, respectively, is strictly convex, which is not uniformly convex (see [19] for more details). Also, Hilbert spaces and lp spaces (1<p<) are well-known examples of uniformly convex Banach spaces. It is worth noticing that by the Milman-Pettis theorem, every uniformly convex Banach space is reflexive, too.

    Definition 2.2. ([20,21]) Let F and G be subsets of a metric space (X,d), then (F,G) is said to satisfy

    (i) property UC, if for all sequences {pn},{pn}F in F and {qn}G, we have

    limnd(pn,qn)=Dist(F,G),limnd(pn,qn)=Dist(F,G),}limnd(pn,pn)=0;

    (ii) property WUC, if for any sequence {pn}F such that

    ϵ>0, qG ; d(pn,q)Dist(F,G)+ϵ, for nn0,

    {pn} is Cauchy.

    In [22], it was disclosed that each nonempty, closed, and convex pair in a uniformly convex Banach space X possesses the property UC. Additionally, if F and G are subsets in a metric space (X,d), with F being complete and the pair (F,G) exhibiting the property UC, then the pair (F,G) is also endowed with the property WUC (see [20]). For more information and properties of the geometric notions of UC, we refer to [23] and the most recent results in [24], where the authors have found a connection between the properties UC and uniform convexity and have introduced some generalizations of these properties.

    Here, we sate the main result of [10].

    Theorem 2.3. ([10]) Given nonempty and complete subsets F and G of a metric space (X,d), suppose that the pairs (F,G) and (G,F) have the property WUC. Let noncyclic continuous self-mapping Γ on FG, be a noncyclic Fisher quasi-contraction, that is, for some α,βN, there exists λ[0,1) such that

    d(Γαx,Γβq)λΔ[Cpα,Cqβ]+(1λ)Dist(F,G)pF, qG, (2.1)

    where Cun:={u,Γu,Γ2u,,Γnu} for uX, nN, and

    Δ[Cpα,Cqβ]:=sup{d(p,q): (p,q)Cpα×Cqβ}.

    There exists (p,q)F×G such that Fix(ΓF×G)={(p,q)}, (Γnp0,Γnq0)(p,q) as n for every (p0,q0)F×G.

    Throughout this section, we assume that I is an identity function defined on [0,+) and φ[ϕ], such that

    [ϕ]:={φ:[0,+)[0,+):φ is a strictly increasing function and Iφ is increasing}.

    For instance, if we define φ1(t)=λt for some λ[0,1) and φ2(t)=(t+2)ln(t+2) and φ3(t)=tt+1+3, then φj[ϕ] for j=1,2,3.

    It is worth noticing that if φ[ϕ], then for all t>0, we have

    φ(t)>φ(t2)0. (3.1)

    So, (Iφ)(t)<t for all t>0. Since Iφ is increasing, it can be easily proven that φ is continuous.

    Also, for given nonempty subsets F and G of a metric space (X,d), we set

    d(p,q):=d(p,q)Dist(F,G),(p,q)F×G,Δ[F,G]:=sup{d(p,q): (p,q)F×G}.

    Definition 3.1. Let F and G be subsets of a metric space (X,d) and "Rt" be a transitive relation on F. Let Γ be a noncyclic mapping on FG, then

    (i) we say that Γ is Rt-continuous at pF if for every sequence {pn} in F with pnp and pnRtpn+1, for all nN, we have ΓpnΓp;

    (ii) we say that Γ preserves "Rt" on F whenever TuRtTv for every u,vF with uRtv;

    (iii) we say that "Rt" has a property () on F, if for any sequence {pn} in F with pnpF and pnRtpn+1 for all nN, we have pnRtp for all nN.

    Now, with these prerequisites and inspired by the main existence results of [10], we introduce the following new family of noncyclic mappings. Henceforth, we denote a metric space (X,d) equipped with a transitive relation "Rt" by Xd,t.

    Definition 3.2. Let F,GXd,t. A mapping Γ:FGFG is said to be a noncyclic (φ,Rt)-enriched quasi-contraction if Γ is noncyclic. For some α,βN,

    d(Γαp,Γβq)(Iφ)(Δ[Cpα,Cqβ]), (3.2)

    for all (p,q)F×G that are comparable with respect to "Rt".

    Example 3.3. Let F and G be subsets of a metric space (X,d) and let Γ:FGFG be a noncyclic Fisher quasi-contraction in the sense of Theorem 2.3, then Γ is a noncyclic (φ,Rt)-enriched quasi-contraction with Rt:=X×X and φ(t):=(1λ)t for t0 and λ[0,1).

    Remark 3.4. Let F,GXd,t and Γ:FGFG be a noncyclic mapping. Set D:=Dist(F,G). If for any (p,q)F×G, we have

    d(Γp,Γq)(Iφ)(max{d(p,q),d(p,Γq),d(q,Γp)})+φ(D),

    then

    d(Γp,Γq)max{(Iφ)(d(p,q)),(Iφ)(d(p,Γq)),(Iφ)(d(q,Γp))}(Iφ)(D)+D=max{(Iφ)(d(p,q)+D)(Iφ)(D),(Iφ)(d(p,Γq)+D)(Iφ)(D),(Iφ)(d(q,Γp)+D)(Iφ)(D)}+D. (3.3)

    Now, define φ:[0,+)[0,+) with φ(t):=φ(t+D)φ(D) for all t0. In view of the fact that (Iφ)(t)=(Iφ)(t+D)(Iφ)(D), we can see that φ is strictly increasing and Iφ is increasing. So from (3.3), we get

    d(Γp,Γq)max{(Iφ)(d(p,q)),(Iφ)(d(p,Γq)),(Iφ)(d(q,Γp))}+D(Iφ)(max{d(p,q),d(p,Γq),d(q,Γp)}).

    Example 3.5. Given complete subsets F and G of a metric space (X,d), let Γ:FGFG be a noncyclic φ-contraction ([8]), that is, Γ is noncyclic on FG and

     φ[ϕ];d(Γx,Γy)d(p,q)φ(d(p,q))+φ(Dist(F,G)),(p,q)F×G.

    From Remark 3.4, Γ is a noncyclic (φ,Rt)-enriched quasi-contraction with Rt:=X×X.

    The following lemmas play essential roles in proving our main result in this section.

    Lemma 3.6. Let F,GXd,t be complete. Let Γ be a noncyclic (φ,Rt)-enriched quasi-contraction mapping on FG, and Γ preserves "Rt". Let p0F and q0G be such that p0Rtq0RtΓp0. Define pn+1:=Γpn and qn+1:=Γqn for each n0, then for any m,nN, we have

    Δ[Cp0n,Cq0m]=d(Γkp0,Γlq0),wherek<αorl<β. (3.4)

    Proof. Since Γ preserves "Rt" on FG and p0Rtq0Rtp1, we get

    p0Rtq0Rtp1Rtq1Rtp2Rtq2Rtp3Rt. (3.5)

    So, from transitivity of Rt, for all i,jN, we have

    piandqjare comparable w.r.t."Rt". (3.6)

    Suppose that Δ[Cp0n,Cq0m]=d(Γip0,Γjq0), where αin and βjm. From (3.2) and (3.6), we have

    d(Γip0,Γjq0)=d(Γαpiα,Γβqjβ)(Iφ)(Δ[Cpiαα,Cqjββ])(Iφ)(Δ[Cp0n,Cq0m]). (3.7)

    Thus, we must have φ(Δ[Cp0n,Cq0m])0. Strictly increasing of the function φ causes Δ[Cp0n,Cq0m]=0 and Δ[Cp0n,Cq0m]=d(p0,q0), which ensures that (3.4) holds.

    Lemma 3.7. Under the assumptions and notations of Lemma 3.6, for every m,nN, we have

    Δ[Cp0n,Cq0m]Mp0,q0, (3.8)

    where

    Mp0,q0=max0i,jmax{α,β}{d(Γip0,Γjq0),φ1(d(Γip0,Γαp0))φ1(d(Γiq0,Γβq0))}.

    Proof. From Lemma 3.6, we have Δ[Cp0n,Cq0m]=d(Γip0,Γjq0), for some i,j0 where i<α or j<β. In the case that i<α and j<β, (3.8) clearly holds. Therefore, without loss of generality, it can be assumed that 0i<α and βjm. Using (3.7), we obtain

    Δ[Cp0n,Cq0m]=d(Γip0,Γjq0)d(Γip0,Γαp0)+d(Γαp0,Γjq0)d(Γip0,Γαp0)+(Iφ)(Δ[Cp0n,Cq0m]),

    which deduces that

    φ(Δ[Cp0n,Cq0m])d(Γip0,Γαp0).

    Since φ[ϕ], φ1 exists. Therefore,

    Δ[Cp0n,Cq0m]φ1(d(Γip0,Γαp0)),

    and so (3.8) holds.

    Lemma 3.8. Under the assumptions and notations of Lemma 3.6, for each m,n,r,s0 with m,nmax{α,β}, we have

    Δ[Cpnr,Cqms](Iφ)(Δ[Cpnαr+α,Cqmβs+β]). (3.9)

    Proof. It follows from the relation (3.7) that for some 0rr,0ss,

    Δ[Cpnr,Cqms]=d(Γrpn,Γsqm)=d(Γp+rpnα,Γq+sqmβ)(Iφ)(Δ[Cpnαr+α,Cqmβs+β]).

    Hence, (3.9) holds.

    Lemma 3.9. Under the assumptions and notations of Lemma 3.6,

    ϵ>0,mN ;d(pn,qm)Dist(F,G)+ϵ,fornm.

    Proof. From Lemma 3.8, for n,mmax{2α,2β}, we have

    d(pn,qm)=Δ[Cpn0,Cqm0](Iφ)(Δ[Cpnαα,Cqmββ])(Iφ)((Iφ)(Δ[Cpn2α2α,Cqm2β2β]))=(Iφ)2(Δ[Cpn2α2α,Cβm2β2q]).

    Continuing this process and using Lemma 3.7, we get

    0d(pn,qm)(Iφ)kn,m(Δ[Cpnkn,mαkn,mα,Cqmkn,mβkn,mβ])(Iφ)kn,m(Δ[Cp0n,Cq0m])(Iφ)kn,m(Mp0,q0), (3.10)

    where On the other hand, for the purposes of this discussion, it is permissible to presume that . Since is increasing and for all , we obtain

    (3.11)

    Additionally, from (3.10), for every there exist such that , and so (3.11) implies that

    Thus,

    which deduces that the sequence is decreasing. Since is bounded below, we assume that

    for some . If for some , then . Otherwise, if for each , from continuity of , we get

    hence, , and from (3.1), we get . Therefore, from (3.10), we conclude that

    and, in addition, the lemma.

    The next result is a direct consequence of Lemma 3.9.

    Corollary 3.10. Under the assumptions and notations of Lemma 3.6, if has the property , then the sequence is Cauchy.

    We have now reached a level of preparedness that allows us to demonstrate the main existential finding of this segment, an expanded variant of Theorem 2.3.

    Theorem 3.11. Under the assumptions and notations of Lemma 3.6, the following statements hold:

    If the pair satisfies the property , the set is complete, and is -continuous on , then there exists such that ;

    If the pair satisfies the property , the set is complete, and is -continuous on , then there exists such that ;

    If, in addition to and , every pair of elements are comparable w.r.t. "", then .

    Proof. (ⅰ) Let for each . From Corollary 3.10 and completeness of , the sequence converges to some . Also from (3.5), we have for each . Since is -continuous, it follows that .

    (ⅱ) By using a similar argument (ⅰ), the result is obtained.

    (ⅲ) If and are the fixed points of , then from Lemma 3.9 we have

    that is, . Now, assume that each elements and are comparable with respect to "". Suppose is another fixed point of in and let . From Lemma 3.9, we have

    Since satisfies the property , we get . In a similar fashion, it becomes apparent that is a unique fixed point of in .

    Example 3.12. Consider with the usual metric and let

    For and , define a noncyclic mapping with

    If for , then and . Let be comparable w.r.t. "", then we must have for some , which implies that

    that is, is a noncyclic -enriched quasi-contraction map, which is not a noncyclic -contraction. It is not difficult to see that all conditions of the part of Theorem 3.11 are satisfied, and is a fixed point of in . Note that since every pair of elements are not comparable w.r.t. "", the fixed point of in is not unique.

    Example 3.13. Again, consider with the usual metric and let . For and , define a noncyclic mapping by

    If for , then . A similar argument of the previous example shows that for all . Hence, is a noncyclic -enriched quasi-contraction map. It now follows from Theorem 3.11 that is a unique fixed point of in .

    The next theorem shows that if (resp., ) in Definition 3.2, then we can drop the continuity of (resp., ) in Theorem 3.11. In this way, we obtain a real generalization of Theorem 3 in [6] as well as Theorem 2.7 in [10].

    Theorem 3.14. Let be such that is complete and satisfies the property . Let "" be a transitive relation on with the property on , and is a noncyclic -enriched quasi-contraction mapping on with , for which preserves "" on . Let and be such that , then there exists such that . If every pair of elements and are comparable with respect to "", then has a unique fixed point in .

    Proof. From the proof of Theorem 3.11, the sequence is convergent to some . By Lemma 3.9, for each . By using property , we get for each . Now, from the relation (3.6), we obtain for each . Thus, for each , and by the fact that is a noncyclic -enriched quasi-contraction from (3.2), we have

    Therefore,

    By Lemma 3.9, we get

    Hence,

    So, from (3.1), we obtain

    (3.12)

    Since , from (3.12) and by taking into account that has the property , we conclude that . The uniqueness of a fixed point of in follows from an equivalent discussion of Theorem 3.11.

    Corollary 3.15. Let and be complete subsets of a metric space such that and satisfy the property . Let "" be a transitive relation on with the property on . Assume that is a noncyclic mapping on satisfying

    for each that are comparable with respect to "". Let be such that and preserves "" on , then there exists . If every pair of elements and are comparable with respect to "", then .

    Building upon the foundations laid by the preceding theorem, we arrive at a subsequent finding that serves as a generalization of Corollary 2.8 of [10].

    Corollary 3.16. Let and be complete subsets of a metric space such that and satisfy the property . Assume that is a noncyclic mapping on satisfying

    for each and . There exists such that , and for every and , the sequences and converge to and , respectively.

    The following common fixed point results are obtained from Theorem 3.11 and Corollary 3.15, immediately. These results are extensions of Corollaries 2.10 and 2.11 of [10].

    Corollary 3.17. Let and be two continuous self-mappings on a complete metric space such that for some ,

    for all , then and have a unique common fixed point such that for every .

    Corollary 3.18. Let and be two self-mappings on a complete metric space satisfying

    for all , then and have a unique common fixed point in .

    In the latest section of this article, motivated by the results of [25,26], we present some other existence, convergence, and uniqueness of an optimal pair of fixed points of noncyclic -quasi-contractions in the setting of reflexive and strictly convex Banach spaces. We also refer to [27,28,29] for different approaches to the same problems for cyclic mappings and some interesting applications in game theory.

    Throughout this section, we assume that . Also, by , we mean the weak convergence in a Banach space .

    Theorem 4.1. Suppose that and are weakly closed subsets of a reflexive Banach space and let be a noncyclic -quasi-contraction map, that is,

    for all . There exists such that .

    Proof. In the case that , the result follows from Theorem 3.14. Otherwise, if , for an arbitrary element , define

    From Lemma 3.9, the sequence is bounded in . Since is weakly closed in a reflexive Banach space , there exists a subsequence of with . As is a bounded sequence in a weakly closed set , without loss of generality, one may assume that as . Since as , one can find a bounded linear functional with the property that

    It follows from Lemma 3.9 that

    So, .

    Definition 4.2. Suppose that and are subsets of a normed linear space and is a noncyclic self-mapping on . We say that satisfies the -property on if is a sequence in and is a sequence in , such that

    then .

    Note that if or has the property , then the conditions of the above definition require that

    Therefore, in these cases, the -property of on is equal to demiclosedness property of at .

    Theorem 4.3. Suppose that and are weakly closed subsets of a reflexive and strictly convex Banach space and let be a noncyclic -quasi-contraction map. Assume that one of the following conditions is satisfied:

    (a) is convex and is weakly continuous on ;

    (b) satisfies the -property on .

    Thus has a fixed point in .

    Proof. In the case that , there is nothing to prove by Theorem 3.14, so assume that . Let be an arbitrary element and define

    From Theorem 4.1, there exists a point and subsequences and such that , , and as .

    (a) Since is weakly continuous on and , we have as . Since as , one can find a bounded linear functional with the property that

    It follows from Lemma 3.9 that

    So, . We assume the contrary, , and it follows from the strict convexity of that

    (4.1)

    Since is convex, , so (4.1) is a contradiction.

    (b) It follows from Lemma 3.9 that

    and by the -property of on , we get .

    Theorem 4.4. Suppose that and are weakly closed and convex subsets of a reflexive and strictly convex Banach space , and let be a noncyclic -quasi-contraction map. Let one of the following conditions be satisfied:

    (a) is weakly continuous on ;

    (b) satisfies the -property on .

    Thus, . Also, if , then for some .

    Proof. According to Theorems 4.1 and 4.3, it is enough to prove the uniqueness of an optimal pair of fixed points . Suppose that there exists another point for which . As , we obtain that (since , we have or . Hence, or ), so . From the strict convexity of , we have

    (4.2)

    which is a contradiction.

    The next result guarantees the uniqueness of an optimal pair of fixed points in Theorem 3.5 of [5].

    Theorem 4.5. Suppose that and are closed and convex subsets of a reflexive and strictly convex Banach space and let be a noncyclic -contraction map, that is,

    (4.3)

    for all . If , then there exists such that .

    Proof. In the case that , the result concludes from Theorem 3.14 directly. Otherwise, if , since is closed and convex, it is weakly closed. It follows from Theorem 4.1 that there exists such that . The proof of uniqueness of with is concluded from a similar discussion of Theorem 4.4. It follows from (4.3) that

    which ensures that . Thus, and , and we are finished.

    In this paper, we defined a new class of noncyclic mappings and investigated the existence, uniqueness, and convergence of an optimal pair fixed point for such maps in the framework of metric spaces equipped with a transitive relation. We also presented the counterpart results under some other sufficient conditions in strictly convex and reflexive Banach spaces. In this way, we obtained some real extensions of previous results that appeared in [2,10,22,25].

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Manuel De La Sen is thankful for the support of Basque Government (Grant No. IT1555-22).

    The authors declare there are no conflicts of interest.



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