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 F∪G 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):p∈F,q∈G}. We denote the set of all optimal pairs of fixed points of Γ in F×G by Fix(Γ∣F×G).
Γ:F∪G→F∪G is called a noncyclic contraction if there exists λ∈[0,1) such that
d(Γp,Γq)≤λd(p,q)+(1−λ)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(Γ∣F×G)≠∅ for every noncyclic contraction Γ defined on F∪G (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,q∈BX with ‖p−q‖≥ε, it follows that
‖p+q2‖<1−δ; |
(ii) strictly convex if for any two distinct elements p,q∈SX, 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 ℓ1, which is equipped with its standard norm
‖u‖=√‖u‖21+‖u‖22, ∀u∈l1, |
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},{p′n}⊆F in F and {qn}⊆G, we have
limn→∞d(pn,qn)=Dist(F,G),limn→∞d(p′n,qn)=Dist(F,G),}⇒limn→∞d(pn,p′n)=0; |
(ii) property WUC, if for any sequence {pn}⊆F such that
∀ϵ>0, ∃q∈G ; d(pn,q)≤Dist(F,G)+ϵ, for n≥n0, |
{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 F∪G, be a noncyclic Fisher quasi-contraction, that is, for some α,β∈N, there exists λ∈[0,1) such that
d(Γαx,Γβq)≤λΔ[Cpα,Cqβ]+(1−λ)Dist(F,G)∀p∈F, q∈G, | (2.1) |
where Cun:={u,Γu,Γ2u,⋯,Γnu} for u∈X, n∈N, and
Δ[Cpα,Cqβ]:=sup{d(p′,q′): (p′,q′)∈Cpα×Cqβ}. |
There exists (p∗,q∗)∈F×G such that Fix(Γ∣F×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 I−φ 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)=t−√t+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, (I−φ)(t)<t for all t>0. Since I−φ 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 F∪G, then
(i) we say that Γ is Rt-continuous at p∈F if for every sequence {pn} in F with pn→p and pnRtpn+1, for all n∈N, we have Γpn→Γp;
(ii) we say that Γ preserves "Rt" on F whenever TuRtTv for every u,v∈F with uRtv;
(iii) we say that "Rt" has a property (∗) on F, if for any sequence {pn} in F with pn→p∈F and pnRtpn+1 for all n∈N, we have pnRtp for all n∈N.
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,G⊆Xd,t. A mapping Γ:F∪G→F∪G is said to be a noncyclic (φ,Rt)-enriched quasi-contraction if Γ is noncyclic. For some α,β∈N,
d∗(Γαp,Γβq)≤(I−φ)(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:F∪G→F∪G 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):=(1−λ)t for t≥0 and λ∈[0,1).
Remark 3.4. Let ∅≠F,G⊆Xd,t and Γ:F∪G→F∪G be a noncyclic mapping. Set D:=Dist(F,G). If for any (p,q)∈F×G, we have
d(Γp,Γq)≤(I−φ)(max{d(p,q),d(p,Γq),d(q,Γp)})+φ(D), |
then
d(Γp,Γq)≤max{(I−φ)(d(p,q)),(I−φ)(d(p,Γq)),(I−φ)(d(q,Γp))}−(I−φ)(D)+D=max{(I−φ)(d∗(p,q)+D)−(I−φ)(D),(I−φ)(d∗(p,Γq)+D)−(I−φ)(D),(I−φ)(d∗(q,Γp)+D)−(I−φ)(D)}+D. | (3.3) |
Now, define φ∗:[0,+∞)→[0,+∞) with φ∗(t):=φ(t+D)−φ(D) for all t≥0. In view of the fact that (I−φ∗)(t)=(I−φ)(t+D)−(I−φ)(D), we can see that φ∗ is strictly increasing and I−φ∗ is increasing. So from (3.3), we get
d(Γp,Γq)≤max{(I−φ∗)(d∗(p,q)),(I−φ∗)(d∗(p,Γq)),(I−φ∗)(d∗(q,Γp))}+D≤(I−φ∗)(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:F∪G→F∪G be a noncyclic φ-contraction ([8]), that is, Γ is noncyclic on F∪G 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,G⊆Xd,t be complete. Let Γ be a noncyclic (φ,Rt)-enriched quasi-contraction mapping on F∪G, and Γ preserves "Rt". Let p0∈F and q0∈G be such that p0Rtq0RtΓp0. Define pn+1:=Γpn and qn+1:=Γqn for each n≥0, then for any m,n∈N, we have
Δ∗[Cp0n,Cq0m]=d∗(Γkp0,Γlq0),wherek<αorl<β. | (3.4) |
Proof. Since Γ preserves "Rt" on F∪G and p0Rtq0Rtp1, we get
p0Rtq0Rtp1Rtq1Rtp2Rtq2Rtp3Rt⋯. | (3.5) |
So, from transitivity of Rt, for all i,j∈N, we have
piandqjare comparable w.r.t."Rt". | (3.6) |
Suppose that Δ∗[Cp0n,Cq0m]=d∗(Γip0,Γjq0), where α≤i≤n and β≤j≤m. From (3.2) and (3.6), we have
d∗(Γip0,Γjq0)=d∗(Γαpi−α,Γβqj−β)≤(I−φ)(Δ∗[Cpi−αα,Cqj−ββ])≤(I−φ)(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,n∈N, we have
Δ∗[Cp0n,Cq0m]≤Mp0,q0, | (3.8) |
where
Mp0,q0=max0≤i,j≤max{α,β}{d∗(Γip0,Γjq0),φ−1(d(Γip0,Γαp0))φ−1(d(Γiq0,Γβq0))}. |
Proof. From Lemma 3.6, we have Δ∗[Cp0n,Cq0m]=d∗(Γip0,Γjq0), for some i,j≥0 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 0≤i<α and β≤j≤m. Using (3.7), we obtain
Δ∗[Cp0n,Cq0m]=d∗(Γip0,Γjq0)≤d(Γip0,Γαp0)+d∗(Γαp0,Γjq0)≤d(Γip0,Γαp0)+(I−φ)(Δ∗[Cp0n,Cq0m]), |
which deduces that
φ(Δ∗[Cp0n,Cq0m])≤d(Γip0,Γαp0). |
Since φ∈[ϕ], φ−1 exists. Therefore,
Δ∗[Cp0n,Cq0m]≤φ−1(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,s≥0 with m,n≥max{α,β}, we have
Δ∗[Cpnr,Cqms]≤(I−φ)(Δ∗[Cpn−αr+α,Cqm−βs+β]). | (3.9) |
Proof. It follows from the relation (3.7) that for some 0≤r′≤r,0≤s′≤s,
Δ∗[Cpnr,Cqms]=d∗(Γr′pn,Γs′qm)=d(Γp+r′pn−α,Γq+s′qm−β)≤(I−φ)(Δ∗[Cpn−αr+α,Cqm−βs+β]). |
Hence, (3.9) holds.
Lemma 3.9. Under the assumptions and notations of Lemma 3.6,
∀ϵ>0,∃m∈N ;d(pn,qm)≤Dist(F,G)+ϵ,forn≥m. |
Proof. From Lemma 3.8, for n,m≥max{2α,2β}, we have
d∗(pn,qm)=Δ∗[Cpn0,Cqm0]≤(I−φ)(Δ∗[Cpn−αα,Cqm−ββ])≤(I−φ)((I−φ)(Δ∗[Cpn−2α2α,Cqm−2β2β]))=(I−φ)2(Δ∗[Cpn−2α2α,Cβm−2β2q]). |
Continuing this process and using Lemma 3.7, we get
0≤d∗(pn,qm)≤(I−φ)kn,m(Δ∗[Cpn−kn,mαkn,mα,Cqm−kn,mβkn,mβ])≤(I−φ)kn,m(Δ∗[Cp0n,Cq0m])≤(I−φ)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.
[1] |
R. Espínola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numer. Funct. Anal. Optim., 34 (2013), 845–860. https://doi.org/10.1080/01630563.2013.763824 doi: 10.1080/01630563.2013.763824
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[2] |
M. Gabeleh, Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indagationes Math., 30 (2019), 227–239. https://doi.org/10.1016/j.indag.2018.11.001 doi: 10.1016/j.indag.2018.11.001
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