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
[1] | Shuguan Ji, Yanshuo Li . Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion. Electronic Research Archive, 2023, 31(12): 7182-7194. doi: 10.3934/era.2023363 |
[2] | Mingliang Song, Dan Liu . Common fixed and coincidence point theorems for nonlinear self-mappings in cone $ b $-metric spaces using $ \varphi $-mapping. Electronic Research Archive, 2023, 31(8): 4788-4806. doi: 10.3934/era.2023245 |
[3] | Fabian Ziltener . Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, 2021, 29(4): 2553-2560. doi: 10.3934/era.2021001 |
[4] | Zhen Zhang, Shance Wang . Relative cluster tilting subcategories in an extriangulated category. Electronic Research Archive, 2023, 31(3): 1613-1624. doi: 10.3934/era.2023083 |
[5] | Mohammed Shehu Shagari, Faryad Ali, Trad Alotaibi, Akbar Azam . Fixed point of Hardy-Rogers-type contractions on metric spaces with graph. Electronic Research Archive, 2023, 31(2): 675-690. doi: 10.3934/era.2023033 |
[6] | Duraisamy Balraj, Muthaiah Marudai, Zoran D. Mitrovic, Ozgur Ege, Veeraraghavan Piramanantham . Existence of best proximity points satisfying two constraint inequalities. Electronic Research Archive, 2020, 28(1): 549-557. doi: 10.3934/era.2020028 |
[7] | Huali Wang, Ping Li . Fractional integral associated with the Schrödinger operators on variable exponent space. Electronic Research Archive, 2023, 31(11): 6833-6843. doi: 10.3934/era.2023345 |
[8] | Xinguang Zhang, Yongsheng Jiang, Lishuang Li, Yonghong Wu, Benchawan Wiwatanapataphee . Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative. Electronic Research Archive, 2024, 32(3): 1998-2015. doi: 10.3934/era.2024091 |
[9] | Yajun Ma, Haiyu Liu, Yuxian Geng . A new method to construct model structures from left Frobenius pairs in extriangulated categories. Electronic Research Archive, 2022, 30(8): 2774-2787. doi: 10.3934/era.2022142 |
[10] | Haiyu Liu, Rongmin Zhu, Yuxian Geng . Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28(4): 1563-1571. doi: 10.3934/era.2020082 |
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 kn,m=min{⌊nα⌋,⌊mβ⌋}. On the other hand, for the purposes of this discussion, it is permissible to presume that Mp0,q0>0. Since I−φ is increasing and (I−φ)(t)<t for all t>0, we obtain
Mp0,q0≥(I−φ)(Mp0,q0)≥(I−φ)2(Mp0,q0)≥⋯. | (3.11) |
Additionally, from (3.10), for every i∈N there exist ni,mi∈N such that kni,mi≥i, and so (3.11) implies that
(I−φ)i(Mp0,q0)≥(I−φ)kni,mi(Mp0,q0)≥0. |
Thus,
Mp0,q0≥(I−φ)(Mp0,q0)≥(I−φ)2(Mp0,q0)≥⋯≥0, |
which deduces that the sequence {(I−φ)k(Mp0,q0)} is decreasing. Since {(I−φ)k(Mp0,q0)} is bounded below, we assume that
limk→∞(I−φ)k(Mp0,q0)=s, |
for some s≥0. If (I−φ)k0(Mp0,q0)=0 for some k0≥1, then s=0. Otherwise, if (I−φ)k(Mp0,q0)>0 for each k∈N, from continuity of I−φ, we get
(I−φ)(s)=s, |
hence, φ(s)=0, and from (3.1), we get s=0. Therefore, from (3.10), we conclude that
∀ϵ>0,∃m∈N:d∗(pn,qm)≤ϵ,forn≥m, |
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 (F,G) has the property WUC, then the sequence {pn} 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:
(i) If the pair (F,G) satisfies the property WUC, the set F is complete, and Γ∣F:F→F is Rt-continuous on F, then there exists p∗∈F such that Γp∗=p∗;
(ii) If the pair (G,F) satisfies the property WUC, the set G is complete, and Γ∣G:G→G is Rt-continuous on G, then there exists q∗∈G such that Γq∗=q∗;
(iii) If, in addition to (i) and (ii), every pair of elements (p,q)∈F×G are comparable w.r.t. "Rt", then Fix(Γ∣F×G)={(p∗,q∗)}.
Proof. (ⅰ) Let pn+1:=Γpn for each n≥0. From Corollary 3.10 and completeness of F, the sequence {pn} converges to some p∗∈F. Also from (3.5), we have pnRtpn+1 for each n≥0. Since Γ∣F is Rt-continuous, it follows that Γp∗=p∗.
(ⅱ) By using a similar argument (ⅰ), the result is obtained.
(ⅲ) If p∗∈F and q∗∈G are the fixed points of T, then from Lemma 3.9 we have
d(p∗,q∗)=limn→∞d(Γnp0,Γnq0)=Dist(F,G), |
that is, (p∗,q∗)∈Fix(Γ∣F×G). Now, assume that each elements p∈F and q∈G are comparable with respect to "Rt". Suppose ¯p is another fixed point of Γ in F and let q0∈G. From Lemma 3.9, we have
limn→∞d(p∗,Γnq0)=limn→∞d(Γnp∗,Γnq0)=Dist(F,G)=limn→∞d(Γn¯p,Γnq0)=limn→∞d(¯p,Γnq0). |
Since (F,G) satisfies the property WUC, we get p∗=¯p. In a similar fashion, it becomes apparent that q∗ is a unique fixed point of Γ in G.
Example 3.12. Consider X:=R with the usual metric and let
Rt:={(±1n+1,±1m+1)∈X×X:n,m∈N}. |
For F=[0,1] and G=[−1,0], define a noncyclic mapping Γ:F∪G→F∪G with
Γ(p)={p1+2pif p∈{1n+1:n∈N},1if p∈F∖{0,1n+1:n∈N},0if p=0. |
Γ(q)={q1−2qif q∈{−1m+1:m∈N},−1if p∈G∖{0,−1m+1:m∈N},0if q=0. |
If φ(t)=t21+2t for t≥0, then (I−φ)(t)=t+t21+2t and φ∈[ϕ]. Let (p,q)∈F×G be comparable w.r.t. "Rt", then we must have (p,q)=(1n+1,−1m+1) for some n,m∈N, which implies that
d∗(Γp,Γq)=|1n+11+2n+1+1m+11+2m+1|=1n+1+1m+1+4(n+1)(m+1)1+2(1n+1+1m+1)+4(n+1)(m+1)≤1n+1+1m+1+4(n+1)(m+1)1+2(1n+1+1m+1)≤(1n+1+1m+1)+(1n+1+1m+1)21+2(1n+1+1m+1)=(I−φ)(1n+1+1m+1)=(I−φ)(d∗(p,q)), |
that is, Γ is a noncyclic (φ,Rt)-enriched quasi-contraction map, which is not a noncyclic φ-contraction. It is not difficult to see that all conditions of the part (i) of Theorem 3.11 are satisfied, and p∗=0 is a fixed point of Γ in F. Note that since every pair of elements (p,q)∈F×G are not comparable w.r.t. "Rt", the fixed point of Γ in F is not unique.
Example 3.13. Again, consider X:=R with the usual metric and let Rt:=X×X. For F=[0,1] and G=[−1,0], define a noncyclic mapping Γ:F∪G→F∪G by
Γ(p)={p1+2pif p∈F,q1−2qif q∈G. |
If φ(t)=t21+2t for t≥0, then φ∈[ϕ]. A similar argument of the previous example shows that d∗(Γp,Γq)≤(I−φ)(d∗(p,q)) for all (p,q)∈F×G. Hence, Γ is a noncyclic (φ,Rt)-enriched quasi-contraction map. It now follows from Theorem 3.11 that p∗=0 is a unique fixed point of Γ in F.
The next theorem shows that if α=1 (resp., β=1) in Definition 3.2, then we can drop the continuity of T|F (resp., T|G) 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 ∅≠F,G⊆Xd,t be such that F is complete and (F,G) satisfies the property WUC. Let "Rt" be a transitive relation on F∪G with the property (∗) on F, and Γ is a noncyclic (φ,Rt)-enriched quasi-contraction mapping on F∪G with α=1, for which Γ preserves "Rt" on F∪G. Let p0∈F and q0∈G be such that p0Rtq0RtΓp0, then there exists p∗∈F such that Γp∗=p∗. If every pair of elements p∈F and q∈G are comparable with respect to "Rt", then Γ has a unique fixed point in F.
Proof. From the proof of Theorem 3.11, the sequence {Γnp0} is convergent to some p∗∈F. By Lemma 3.9, pnRtpn+1 for each n≥0. By using property (∗), we get pnRtp∗ for each n≥0. Now, from the relation (3.6), we obtain qnRtpn+1Rtp∗ for each n≥0. Thus, qnRtp∗ for each n≥0, and by the fact that Γ is a noncyclic (φ,Rt)-enriched quasi-contraction from (3.2), we have
d∗(Γp∗,Γnq0)=d∗(Γp∗,Γβqn−β)≤(I−φ)(Δ∗[Cp∗1,Cqn−ββ]). |
Therefore,
lim supn→∞d∗(Γp∗,Γnq0)≤(I−φ)(max{lim supm→∞d∗(p∗,Γmq0),lim supm→∞d∗(Γp∗,Γmq0)}). |
By Lemma 3.9, we get
lim supn→∞d∗(Γp∗,Γnq0)≤(I−φ)(max{0,lim supn→∞d∗(Γp∗,Γnq0)}). |
Hence,
φ(lim supn→∞d∗(Γp∗,Γnq0))=0. |
So, from (3.1), we obtain
limn→∞d(Γp∗,Γnq0)=Dist(F,G). | (3.12) |
Since limn→∞d(p∗,Γnq0)=Dist(F,G), from (3.12) and by taking into account that (F,G) has the property WUC, we conclude that Γp∗=p∗. The uniqueness of a fixed point of Γ in F follows from an equivalent discussion of Theorem 3.11.
Corollary 3.15. Let F≠∅ and G≠∅ be complete subsets of a metric space (X,d) such that (F,G) and (G,F) satisfy the property WUC. Let "Rt" be a transitive relation on F∪G with the property (∗) on F∪G. Assume that Γ is a noncyclic mapping on F∪G satisfying
d∗(Γp,Γq)≤(I−φ)(max{d∗(p,q),d∗(p,Γq),d∗(q,Γp)}), |
for each (p,q)∈F×G that are comparable with respect to "Rt". Let (p0,q0)∈F×G be such that p0Rtq0RtΓp0 and Γ preserves "Rt" on F∪G, then there exists (p∗,q∗)∈Fix(Γ∣F×G). If every pair of elements p∈F and q∈G are comparable with respect to "Rt", then Fix(Γ∣F×G)={(p∗,q∗)}.
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 F≠∅ and G≠∅ be complete subsets of a metric space (X,d) such that (F,G) and (G,F) satisfy the property WUC. Assume that Γ is a noncyclic mapping on F∪G satisfying
d∗(Γp,Γq)≤(I−φ)(max{d∗(p,q),d∗(p,Γq),d∗(q,Γp)}), |
for each p∈F and q∈G. There exists (p∗,q∗)∈F×G such that Fix(Γ∣F×G)={(p∗,q∗)}, and for every p0∈F and q0∈G, the sequences {Γnp0} and {Γnq0} converge to p∗ and q∗, 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 Γ and Λ be two continuous self-mappings on a complete metric space (X,d) such that for some α,β∈N,
d(Γαp,Λβq)≤(I−φ)(max{d(Γip,Λjq):0≤i≤α,0≤j≤β}), |
for all p,q∈X, then Λ and Γ have a unique common fixed point p∗∈X such that limn→∞Γnp0=limn→∞Λnp0=p∗ for every p0∈X.
Corollary 3.18. Let Γ and Λ be two self-mappings on a complete metric space (X,d) satisfying
d(Γp,Λq)≤(I−φ)(max{d(p,q),d(p,Λq),d(q,Γp)}), |
for all p,q∈X, then Λ and Γ have a unique common fixed point in X.
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 φ-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 φ∈[ϕ]. Also, by "w→", we mean the weak convergence in a Banach space X.
Theorem 4.1. Suppose that F≠∅ and G≠∅ are weakly closed subsets of a reflexive Banach space X and let Γ:F∪G→F∪G be a noncyclic φ-quasi-contraction map, that is,
‖Γp−Γq‖≤(I−φ)(max{‖x−y‖,‖x−Γq‖,‖Γp−y‖})+φ(Dist(F,G)), |
for all (p,q)∈F×G. There exists (p∗,q∗)∈F×G such that ‖p∗−q∗‖=Dist(F,G).
Proof. In the case that Dist(F,G)=0, the result follows from Theorem 3.14. Otherwise, if Dist(F,G)>0, for an arbitrary element (p0,q0)∈F×G, define
(pn+1,qn+1):=(Γpn,Γqn),∀n≥0. |
From Lemma 3.9, the sequence {(pn,qn)} is bounded in F×G. Since F is weakly closed in a reflexive Banach space X, there exists a subsequence {pnk} of {pn} with pnkw→p∗∈F. As {qnk} is a bounded sequence in a weakly closed set G, without loss of generality, one may assume that qnkw→q∗∈G as k→∞. Since pnk−qnkw→p∗−q∗≠0 as k→∞, one can find a bounded linear functional f:X→[0,+∞) with the property that
‖f‖=1andf(p∗−q∗)=‖p∗−q∗‖. |
It follows from Lemma 3.9 that
‖p∗−q∗‖=|f(p∗−q∗)|=limk→∞|f(pnk−qnk)|≤limk→∞‖f‖‖pnk−qnk‖=limk→∞‖pnk−qnk‖=Dist(F,G). |
So, ‖p∗−q∗‖=Dist(F,G).
Definition 4.2. Suppose that F and G are subsets of a normed linear space X and Γ is a noncyclic self-mapping on F∪G. We say that Γ satisfies the D-property on F if {pn} is a sequence in F and {qn} is a sequence in G, such that
pnw→p∗∈F,‖pn−qn‖→Dist(F,G), and ‖Γpn−qn‖→Dist(F,G), |
then Γp∗=p∗.
Note that if Dist(F,G)=0 or (F,G) has the property UC, then the conditions of the above definition require that
pnw→p∗∈F,and‖Γpn−pn‖→0. |
Therefore, in these cases, the D-property of Γ on F is equal to demiclosedness property of I−Γ∣F at 0.
Theorem 4.3. Suppose that F≠∅ and G≠∅ are weakly closed subsets of a reflexive and strictly convex Banach space X and let Γ:F∪G→F∪G be a noncyclic φ-quasi-contraction map. Assume that one of the following conditions is satisfied:
(a) F is convex and Γ is weakly continuous on F;
(b) Γ satisfies the D-property on F.
Thus Γ has a fixed point in F.
Proof. In the case that Dist(F,G)=0, there is nothing to prove by Theorem 3.14, so assume that Dist(F,G)>0. Let (p0,q0)∈F×G be an arbitrary element and define
(pn+1,qn+1):=(Γpn,Γqn),∀n≥0. |
From Theorem 4.1, there exists a point (p∗,q∗)∈F×G and subsequences {pnk} and {qnk} such that ‖p∗−q∗‖=Dist(F,G), pnkw→p∗∈F, and qnkw→q∗∈G as k→∞.
(a) Since Γ is weakly continuous on F and Γ(F)⊆F, we have pnk+1w→Γp∗∈F as k→∞. Since pnk+1−qnkw→Γp∗−q∗≠0 as k→∞, one can find a bounded linear functional f:X→[0,+∞) with the property that
‖f‖=1,andf(Γp∗−q∗)=‖Γp∗−q∗‖. |
It follows from Lemma 3.9 that
‖Γp∗−q∗‖=|f(Γp∗−q∗)|=limk→∞|f(pnk+1−qnk)|≤limk→∞‖f‖‖pnk+1−qnk‖=limk→∞‖pnk+1−qnk‖=Dist(F,G). |
So, ‖Γp∗−q∗‖=Dist(F,G). We assume the contrary, Γp∗≠p∗, and it follows from the strict convexity of X that
‖p∗+Γp∗2−q∗‖<Dist(F,G). | (4.1) |
Since F is convex, p∗+Γp∗2∈F, so (4.1) is a contradiction.
(b) It follows from Lemma 3.9 that
limk→∞‖pnk−qnk‖=limk→∞‖Γpnk−qnk‖=Dist(F,G), |
and by the D-property of Γ on F, we get Γp∗=p∗.
Theorem 4.4. Suppose that F≠∅ and G≠∅ are weakly closed and convex subsets of a reflexive and strictly convex Banach space X, and let Γ:F∪G→F∪G be a noncyclic φ-quasi-contraction map. Let one of the following conditions be satisfied:
(a) Γ is weakly continuous on F∪G;
(b) Γ satisfies the D-property on F∪G.
Thus, Fix(Γ∣F×G)≠∅. Also, if (F−F)∩(G−G)={0}, then Fix(Γ∣F×G)={(p∗,q∗)} for some (p∗,q∗)∈F×G.
Proof. According to Theorems 4.1 and 4.3, it is enough to prove the uniqueness of an optimal pair of fixed points (p∗,q∗)∈F×G. Suppose that there exists another point (p′,q′)∈F×G for which ‖p′−q′‖=Dist(F,G). As (F−F)∩(G−G)={0}, we obtain that p′−p∗≠q′−q∗ (since (p′,q′)≠(p∗,q∗), we have p′≠p∗ or q′≠q∗. Hence, p′−p∗≠0 or q′−q∗≠0), so p∗−q∗≠p′−q′. From the strict convexity of X, we have
‖p′+p∗2−q′+q∗2‖<Dist(F,G). | (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 F≠∅ and G≠∅ are closed and convex subsets of a reflexive and strictly convex Banach space X and let Γ:F∪G→F∪G be a noncyclic φ-contraction map, that is,
‖Γp−Γq‖≤‖p−q‖−φ(‖p−q‖)+φ(Dist(F,G)), | (4.3) |
for all (p,q)∈F×G. If (F−F)∩(G−G)={0}, then there exists (p∗,q∗)∈F×G such that Fix(Γ∣F×G)={(p∗,q∗)}.
Proof. In the case that Dist(F,G)=0, the result concludes from Theorem 3.14 directly. Otherwise, if Dist(F,G)>0, since F is closed and convex, it is weakly closed. It follows from Theorem 4.1 that there exists (p∗,q∗)∈F×G such that ‖p∗−q∗‖=Dist(F,G). The proof of uniqueness of (p∗,q∗)∈F×G with ‖p∗−q∗‖=Dist(F,G) is concluded from a similar discussion of Theorem 4.4. It follows from (4.3) that
‖Γp∗−Γq∗‖=‖p∗−q∗‖=Dist(F,G), |
which ensures that (Γp∗,Γq∗)=(p∗,q∗). Thus, Γp∗=p∗ and Γq∗=q∗, 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
![]() |
[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
![]() |
[3] |
A. Abkar, M. Gabeleh, Global optimal solutions of noncyclic mappings in metric space, J. Optim. Theory Appl., 153 (2012), 298–305. https://doi.org/10.1007/s10957-011-9966-4 doi: 10.1007/s10957-011-9966-4
![]() |
[4] |
L. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer.Math. Soc., 45 (1974), 267–273. https://doi.org/10.1090/s0002-9939-1974-0356011-2 doi: 10.1090/s0002-9939-1974-0356011-2
![]() |
[5] | A. Fernández-León, M. Gabeleh, Best proximity pair theorems for noncyclic mappings in Banach and metric spaces, Fixed Point Theory, 17 (2016), 63–84. |
[6] |
B. Fisher, Quasicontractions on metric spaces, Proc. Amer. Math. Soc., 75 (1979), 321–325. https://doi.org/10.1090/s0002-9939-1979-0532159-9 doi: 10.1090/s0002-9939-1979-0532159-9
![]() |
[7] |
M. Gabeleh, C. Vetro, A new extension of Darbo's fixed point theorem using relatively Meir-Keeler condensing operators, Bull. Aust. Math. Soc., 98 (2018), 286–297. https://doi.org/10.1017/s000497271800045x doi: 10.1017/s000497271800045x
![]() |
[8] | A. Safari-Hafshejani, Existence and convergence of fixed point results for noncyclic φ-contractions, AUT J. Math. Comput., Amirkabir University of Technology, in press. https://doi.org/10.22060/AJMC.2023.21992.1127 |
[9] |
A. Safari-Hafshejani, The existence of best proximi points for generalized cyclic quasi-contractions in metric spaces with the UC and ultrametric properties, Fixed Point Theory, 23 (2022), 507–518. https://doi.org/10.24193/fpt-ro.2022.2.06 doi: 10.24193/fpt-ro.2022.2.06
![]() |
[10] |
A. Safari-Hafshejani, A. Amini-Harandi, M. Fakhar, Best proximity points and fixed points results for noncyclic and cyclic Fisher quasi-contractions, Numer. Funct. Anal. Optim., 40 (2019), 603–619. https://doi.org/10.1080/01630563.2019.1566246 doi: 10.1080/01630563.2019.1566246
![]() |
[11] |
V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc., 23 (1969), 631–634. https://doi.org/10.1090/S0002-9939-1969-0250292-X doi: 10.1090/S0002-9939-1969-0250292-X
![]() |
[12] |
S. Karaibyamov, B. Zlatanov, Fixed points for mappings with a contractive iterate at each point, Math. Slovaca, 64 (2014), 455–468. https://doi.org/10.2478/s12175-014-0218-6 doi: 10.2478/s12175-014-0218-6
![]() |
[13] |
L. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 26 (1970), 615–618. https://doi.org/10.1090/S0002-9939-1970-0266010-3 doi: 10.1090/S0002-9939-1970-0266010-3
![]() |
[14] |
B. Samet, C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct. Anal., 1 (2010), 46–56. https://doi.org/10.15352/afa/1399900586 doi: 10.15352/afa/1399900586
![]() |
[15] |
A. Petrusel, Fixed points vs. coupled fixed points, J. Fixed Point Theory Appl., 20 (2018), 150. https://doi.org/10.1007/s11784-018-0630-6 doi: 10.1007/s11784-018-0630-6
![]() |
[16] | A. Petrusel, G. Petrusel, B. Samet, J. C. Yao, Coupled fixed point theorems for symmetric contractions in b-metric spaces with applications to a system of integral equations and a periodic boundary value problem, Fixed Point Theory, 17 (2016), 459–478. |
[17] | A. Petrusel, G. Petrusel, Y. B. Xiao, J. C. Yao, Fixed point theorems for generalized contractions with applications to coupled fixed point theory, J. Nonlinear Convex Anal., 19 (2018), 71–88. |
[18] | K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511526152 |
[19] | V. Zizler, On Some Rotundity and Smoothness Properties of Banach Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1971. |
[20] | R. Espínola, A. Fernández-León, On best proximity points in metric and Banach space, preprint, arXiv: 0911.5263. |
[21] |
T. Suzuki, M. Kikawa, C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal. Theory Methods Appl., 71 (2009), 2918–2926. https://doi.org/10.1016/j.na.2009.01.173 doi: 10.1016/j.na.2009.01.173
![]() |
[22] |
A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006. https://doi.org/10.1016/j.jmaa.2005.10.081 doi: 10.1016/j.jmaa.2005.10.081
![]() |
[23] |
W. Sintunavarat, P. Kumam, Coupled best proximity point theorem in metric spaces, Fixed Point Theory Appl., 2012 (2012), 93. https://doi.org/10.1186/1687-1812-2012-93 doi: 10.1186/1687-1812-2012-93
![]() |
[24] | V. Zhelinski, B. Zlatanov, On the UC and UC∗ properties and the existence of best proximity points in metric spaces, preprint, arXiv: 2303.05850. |
[25] |
M. A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. Theory Methods Appl., 70 (2009), 3665–3671. https://doi.org/10.1016/j.na.2008.07.022 doi: 10.1016/j.na.2008.07.022
![]() |
[26] |
M. Petric, B. Zlatanov, Best proximity points for p-cyclic summing iterated contractions, Filomat, 32 (2018), 3275–3287. https://doi.org/10.2298/fil1809275p doi: 10.2298/fil1809275p
![]() |
[27] |
L. Ajeti, A. Ilchev, B. Zlatanov, On coupled best proximity points in reflexive Banach spaces, Mathematics, 10 (2022), 1304. https://doi.org/10.3390/math10081304 doi: 10.3390/math10081304
![]() |
[28] |
S. Kabaivanov, V. Zhelinski, B. Zlatanov, Coupled fixed points for Hardy-Rogers type of maps and their applications in the investigations of market equilibrium in duopoly markets for non-differentiable, Symmetry, 14 (2022), 605. https://doi.org/10.3390/sym14030605 doi: 10.3390/sym14030605
![]() |
[29] |
Y. Dzhabarova, S. Kabaivanov, M. Ruseva, B. Zlatanov, Existence, uniqueness and stability of market equilibrium in oligopoly markets, Adm. Sci., 10 (2020), 70. https://doi.org/10.3390/admsci10030070 doi: 10.3390/admsci10030070
![]() |