New results on contractive type in cone 2-metric space
ABDALLAH M.M. BADR1,2,∗, BASEL HARDAN3, AHMED A. HAMOUD4, BADR SALEH AL-ABDI5,
FAISAL A.M. ALI6, JAYASHREE PATIL7, ALAA A. ABDALLAH8
1Department of Administration, College of Business, King Khalid University, SAUDI ARABIA.
2Department of Statistics, College of Commerce, Al-Azhar University, EGYPT.
3Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, INDIA.
4Department of Mathematics, Taiz University, Taiz P.O. Box 6803, YEMEN.
5Department of Administration, College of Business, King Khalid University, SAUDI ARABIA.
6Department of Data Sciences and Technology, College of Administrative Sciences, Taiz University, YEMEN.
7Department of Mathematics, Vasantrao Naik Mahavidyalaya, Cidco, Aurangabad, INDIA.
8Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, INDIA.
Abstract: - The common fixed point for self-contractive mappings in cone 2-metric spaces over Banach algebra
is established in this study. The acquired results enhance and generalise the corresponding conclusions from the
literature. A numerical example and a counterexample were then provided at the end.
Key-Words: - Metric spaces; contraction principle; fixed point; contractive mapping.
Received: January 8, 2023. Revised: June 14, 2023. Accepted: July 8, 2023. Published: August 2, 2023.
1 Introduction
The principle of 2-metric space (2-MS) was estab-
lished in [1] and [2], using generalizing the metric
space(MS) and showed numerous fixed point theo-
rems (FPTs) in such space. Many papers have in-
vestigated the necessary factors for the existence /
uniqueness of FPT for contraction mappings in 2-MS,
[3], [4], [5], [6], [7], [8]. On another hand, [9], in-
troduced sundry FPTs in cone 2-MS. The authors in
[9], [10], [11], [12], [13], established various FPTs in
new MSs for an ordered Banach space (BS) in the co-
domain. Over Banach algebras, [14] and [15], worked
on cone MS. [16] presented cone 2-MS generaliz-
ing both 2-MS and cone MS and proved some FPTs
for self-mappings satisfying certain contractive con-
ditions, [16], [17], [18]. The analysis of the existence
/ uniqueness of coincide /common points of diverse
operators in the context of MS is also one of the most
alluring research topics in FPTs, [19], [20], [21], [22].
Banach contraction principle to prove the exist a FP
for a given space was introduced by Banach [23]. The
method of FP development is either developing a type
of used space or a type of contractive mapping. The
development of space depends on decrease or chang-
ing the metric conditions. Consider that abusing or
debilitating a portion of the metric conditions rise to
the loss of some topological advantages, thus getting
hard in proving some FPTs. Hardy-Rogers’ theory
(H-R) [24], is one of the most main findings that de-
veloped the Banach contraction principle by contrac-
tive type, many researchers have developed various
FPTs on this important finding, [25], [26], [27], [28].
For this reason, we have seen generalize some FPTs
in a cone 2-MS by using H-R’ mappings, which opens
the entrance to a similar study on cone n-MS.
2 Preliminaries
Definition 2.1. [29] Suppose Gbe a Banach algebra
(BG), then ∀u1,u2,u3∈G, α ∈ R:
(i) (u1u2)u3=u1(u2u3);
(ii) u1(u2+u3) = u1u2+u1u3and (u1+u2)u3=
u1u3+u2u3;
(iii) α(u1u2) = (αu1)u2=u1(αu2);
(iv) ∥u1u2∥ ≤ ∥u1∥∥u2∥.
In this work, a BG has a unit e:eu1=u1e=u1
∀u1∈G, where u1if there is an inverse element, then
is said to be invertible. u2∈G,u1u2=u2u1=e.
u1’s inverse is represented by u−1
1. see [31] for
further information. The set {u1,u2,· · · ,un} ⊂ Gis
commute if uiuj=ujui∀i, j ∈ {1,2,· · · , n}.
Definition 2.2. [30] Suppose that Ube a non-
empty set and the mapping δ:U × U × U → G
satisfies
(i) δ(u1,u2,u3)= 0 for every pair u1=u2∈ U, and
u3∈ U,
(ii) δ(u1,u2,u3)≥0for all u1,u2,u3∈ U and
δ(u1,u2,u3)=0if and only if at least two of
u1,u2,u3are equal,
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DOI: 10.37394/23206.2023.22.66
Abdallah M. M. Badr, Basel Hardan,
Ahmed A. Hamoud, Badr Saleh Al-Abdi,
Faisal A. M. Ali, Jayashree Patil, Alaa A. Abdallah
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(iii) δ(u1,u2,u3) = δ(p(u1,u2,u3)) for all
u1,u2,u3∈ U and for all permutations
p(u1,u2,u3)of (u1,u2,u3),
(iv) δ(u1,u2,u3)≼δ(u1,u2,u4) + δ(u1,u4,u3) +
δ(u4,u2,u3), for all u1,u2,u3,u4∈ U.
Then δis called a cone 2-M on Uand (U, δ)is called
a cone 2-MS.
Definition 2.3. [30] Suppose (U, δ)be a cone 2-
MS. Let u∈ U and {un}be a sequence in U. Then
(i) {un}is convergence sequence if un→uwhen-
ever for every c∈Gwith 0≪c, there is a natu-
ral number Nsuch that
δ(un,u,u3)≪c, for all u3∈ U and n≥ N .
(ii) {un}is a Cauchy sequence if for every c∈G
with 0≪c, there is a natural number Nsuch that
δ(un,uk,u3)≪c, for all u3∈ U and n, k ≥ N .
(iii) (U, δ)is a complete cone 2-MS if every Cauchy
sequence is convergent in U.
Proposition 2.4. [31] Let Gbe a BG with a unite
eand u∈G. If the spectral radius r(u)<1, which
implies that
r(u) = lim
n→∞ ∥un∥1
n=inf
n→∞ ∥un∥1
n<1.
Then (e−u)is invertible. Actually, (e−u)−1=
+∞
i=0 ui.
Remark 2.5.
(i) r(u)≤ ∥u∥for any u∈G, refer [31].
(ii) In Proposition 2.4, if r(u)<1is replaced by
∥u∥<1then the conclusion remains true.
Lemma 2.6. [33] If Gis a real BS with a solid
cone Pand if ∥un∥ → 0as n → ∞, then for any
0≪c, there exists n1∈ N such that for all n>n1,
we have un≪c.
3 Main Results
In this section, we will prove the uniqueness of the
common FP in con 2-MS using H-R contractive self
mappings of BG.
Theorem 3.1. Let (U, δ)be a complete cone 2-
MS on a BG Gand Pthe underlying cone. Assume
that, T, F are self-mappings of Usatisfying the con-
dition
δ(Tki
iu1, T kj
ju2,u3)
≼α1δ(Fki
iu1, F kj
ju2,u3) + α2δ(Fki
iu1, T ki
iu1,u3)
+α3δ(Fkj
ju2, T kj
ju2,u3) + α4δ(Fki
iu1, T kj
ju2,u3)
+α5δ(Fkj
ju2, T ki
iu1,u3),(1)
where α1, α2, α3, α4, α5∈ P, for all u1,u2,u3∈ U.
If α1, α2, α3, α4, α5are commute and r(α1)+r(α2)+
r(α3) + r(α4) + r(α5)<1. Then {Tki
i}∞
i=1 and
{Fki
i}∞
i=1 have a unique common FP.
Proof. Consider Tki
i=hiand Fki
i=fi, for all i∈
N. Inequality (1) it will become
δ(hiu1,hju2,u3)
≼α1δ(fiu1,fju2,u3) + α2δ(fiu1,hiu1,u3)
+α3δ(fju2,hju2,u3) + α4δ(fiu1,hju2,u3)
+α5δ(fju2,hiu1,u3).(2)
Let u0∈ U be arbitrary and define the sequence un
as un=hn(un−1) = fnun, for all n∈ N . Now we
prove that {un}is a Cauchy sequence in U. Take
δ(un+1,un,u3)
=δ(hn+1un,hnun−1,u3)
≼α1δ(fnun,fnun−1,u3) + α2δ(fnun,hnun,u3)
+α3δ(fnun−1,hnun−1,u3) + α4δ(fnun,hnun−1,u3)
+α5δ(fnun−1,hnun,u3)
≼α1δ(un,un−1,u3) + α2δ(un,un+1,u3) + α3δ(un−1,un,u3)
+α4δ(un,un,u3) + α5δ(un−1,un+1,u3)
≼α1δ(un,un−1,u3) + α2δ(un,un+1,u3) + α3δ(un−1,un,u3)
+α5[δ(un−1,un,u3) + δ(un,un+1,u3)]
≼α1δ(un,un−1,u3) + α2δ(un,un+1,u3) + α3δ(un−1,un,u3)
+α5δ(un−1,un,u3) + α5δ(un,un+1,u3)
≼(e−α2−α5)−1(α1+α3+α5)δ(un−1,un,u3)
≼η1δ(un−1,un,u3),
where
η1= (e−α2−α5)−1(α1+α3+α5)∈ P.
By symmetrical probability of cone 2-MS, we have
δ(un+1,un,u3)
=δ(un,un+1,u3)
=δ(hnun−1,hn+1un,u3)
≼α1δ(fnun−1,fnun,u3) + α2δ(fnun−1,hnun−1,u3)
+α3δ(fnun,hnun,u3) + α4δ(fnun−1,hnun,u3)
+α5δ(fnun,hnun−1,u3)
≼α1δ(un−1,un,u3) + α2δ(un−1,un,u3) + α3δ(un,un+1,u3)
+α4δ(un−1,un+1,u3) + α5δ(un,un,u3)
≼α1δ(un−1,un,u3) + α2δ(un−1,un,u3) + α3δ(un,un+1,u3)
+α4[δ(un−1,un,u3) + δ(un,un+1,u3)]
≼α1δ(un−1,un,u3) + α2δ(un−1,un,u3) + α3δ(un,un+1,u3)
+α4δ(un−1,un,u3) + α4δ(un,un+1,u3)
≼(e−α3−α4)−1(α1+α2+α4)δ(un−1,un,u3)
≼η2δ(un−1,un,u3),
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where
η2= (e−α3−α4)−1(α1+α2+α4)∈ P.
We pretend that, either r(η1)<1or r(η2)<1. If
r(η1)>1, we obtain
r(α1) + r(α3) + r(α5)1−r(α2)−r(α5)−1
≥r(η1)>1.
Which leads to,
r(α1) + r(α2) + r(α3)+2r(α5)>1.(3)
If r(η2)>1, we obtain
r(α1) + r(α2) + r(α4)1−r(α3)−r(α4)−1
≥r(η2)>1.
Which implies
r(α1) + r(α2) + r(α3)+2r(α4)>1.(4)
By adding (3) and (4), we have r(α1)+r(α2)+r(α3)+
r(α4) + r(α5)>1. Which is a contradiction. Hence
our pretension is correct.
δ(un+1,un,u3)≼αδ(un,un−1,u3),(5)
for all n≥1and r(α)<1. Jointly with Proposition
2.4 we have
δ(un+1,un,u3)≼αδ(un,un−1,u3)
≼α2δ(un−1,un−2,u3)
.
.
.
≼αnδ(u1,u0,u3),
where
α=η1, when r(η1)<1,
η2, when r(η2)<1,
η1or η2when r(η1)<1and r(η2)<1.
Then α∈ P and r(α)<1. For all τ < n we have
δ(un,un−1,uτ)≼αδ(un−1,un−2,uτ)
≼α2δ(un−2,un−3,uτ)
.
.
.
≼αn−τ−1δ(uτ+1,uτ,uτ).
Therefore, for all τ < n, we obtain δ(un,un−1,uτ) =
0. Now, for such n > s we have
δ(un, us,u3)
≼δ(un,us,un−1) + δ(un,un−1,u3) + δ(un−1,us,u3)
≼αn−1δ(u1,u0,u3) + δ(un−1,us,un−2)
+δ(un−1,un−2,u3) + δ(un−2,us,u3)
≼(αn−1+αn−1)δ(u1,u0,u3) + δ(un−2,us,u3)
≼(αn−1+αn−1+· · · +αs+1)δ(u1,u0,u3)
+δ(us+1,us,u3)
≼(αn−1+αn−1+· · · +αs+1 +αs)δ(u1,u0,u3)
=(e+α+· · · +αn−s+1)αsδ(u1,u0,u3)
≼
∞
i=1
αiαsδ(u1,u0,u3)
=αs(e−α)−1δ(u1,u0,u3).
From Lemma 2.6 and the actuality
∥αs(e−α)−1δ(u1,u0,u3)∥= 0, as n → ∞.
We get for any β∈Gwith 0≪β, there exist n∈ N .
Such that for all n, s > N, we get
δ(un, us,u3)≼αs(e−α)−1δ(u1,u0,u3)≪β.
Which proves that, {un}is a Cauchy sequence in U.
There exists u∈ U such that un→uas n→ ∞ since
Uis complete. We pretend that uis common FP of
{Tki
i}∞
i=1 and {Fki
i}∞
i=1 for all i∈ N . Inequality (2)
become,
δ(hnun,hmum,u)
≼α1δ(fnun,fmum,u) + α2δ(fnun,hnun,u)
+α3δ(fmum,hmum,u)
+α4δ(fnun,hmum,u) + α5δ(fmum,hnun,u)
≼α1δ(un,um,u) + α2δ(un,un+1,u) + α3δ(um,um+1,u)
+α4δ(un,um+1,u) + α5δ(um,un+1,u).
Hence, δ(hnun,hmum,u)≼0. By (ii) in Defini-
tion 2.1 we have: hnun=hnu=uwhen n→ ∞
or hmun=hmu=uwhen m→ ∞. Which mean u
is common FP of hn={Tki
i}∞
i=1. Also, we can see
that, fnun=fu =u. From this we conclude that uis
a common FP of {Tki
i}∞
i=1 and {Fki
i}∞
i=1 on U. As-
sume That, wis any another common FP of hnand fn
on U, such that w=ufor all n∈ N . Then by (2) we
obtain,
δ(w, u,u3)
=δ(hnun,hmum,u3)
≼α1δ(fnun,fmum,u3) + α2δ(fnun,hnun,u3)
+α3δ(fmum,hmum,u3)
+α4δ(fnun,hmum,u3) + α5δ(fmum,hnun,u3)
≼0.
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This implies that w=uwhich proven the uniqueness
of a common FP uof {Tki
i}∞
i=1 and {Fki
i}∞
i=1 on U.
The next conclusions can be gained from our
main result.
Corollary 1. Let (U, δ)be a complete cone 2-MS
in a BG Gand Pthe underlying cone. Assume that
T, F are self-mappings of Usatisfying the condition
δ(Tki
iu1, T kj
ju2,u3)
≼α1δ(Fki
iu1, F kj
ju2,u3) + α2δ(Fki
iu1, T ki
iu1,u3)
+α3δ(Fkj
ju2, T kj
ju2,u3),(6)
where α1, α2, α3∈ P, for all u1,u2,u3∈ U. If
α1, α2, α3are commute and r(α1) + r(α2) + r(α3)<
1, then {Tki
i}∞
i=1 and {Fki
i}∞
i=1 have a unique com-
mon FP.
The above FPT is development and generalization
for Wang’s result in [30].
Corollary 2. Let (U, δ)be a complete cone 2-MS
in a BG Gand Pthe underlying cone. Assume that,
T, F are self-mappings of Usatisfying the condition
δ(Tki
iu1, T kj
ju2,u3)(7)
≼αδ(Fki
iu1, T ki
iu1,u3) + δ(Fkj
ju2, T kj
ju2,u3),
where α∈ P, for all u1,u2,u3∈ U. If r(α)<1/2,
then {Tki
i}∞
i=1 and {Fki
i}∞
i=1 have a unique common
FP.
Corollary 3. Let (U, δ)be a complete cone 2-MS
in a BG Gand Pthe underlying cone. Assume that
T, F are self-mappings of Usatisfying the condition
δ(Tki
iu1, T kj
ju2,u3)≼
αδ(Fki
iu1, T kj
ju2,u3) + δ(Fkj
ju2, T ki
iu1,u3),
where α∈ P, for all u1,u2,u3∈ U. If r(α)<1/2,
then {Tki
i}∞
i=1 and {Fki
i}∞
i=1 have a unique common
FP.
The result of Mlaiki of FP [32] can be developed and
generalized as follows.
Corollary 4. Let (U, δ)be a complete cone 2-MS
in a BGa Gand Pthe underlying cone. Assume that
T, F are self-mappings of Usatisfying the condition
δ(Tki
iu1, T kj
ju2,u3)≼
αmax δ(Fki
iu1, T kj
ju1,u3), δ(Fkj
ju2, T ki
iu2,u3),
where, α∈ P, for all u1,u2,u3∈ U. If r(α)∈(0,1),
then {Tki
i}∞
i=1 and {Fki
i}∞
i=1 have a unique common
FP.
Example 1. Suppose that G=R2and (u1,u2)∈
Gsuch that ∥u1,u2∥=|u1|+|u2|. Consider the mul-
tiplication as
uw= (u1.u2)(w1, w2) = κ1w1+κ2w2
Then Gis a BG with unite e= (1,0). Assume that
P={(u1,u2)∈ R2|u1,u2,≥0}. Then Pis a cone
on G. Let, U={(α, 0) ∈R2|0≤α < 1}∪{(0, α)∈
R2|0≤α < 1}.
Define the metric as
δ(u1,u2,u3) = δ(µ1, µ2),
where, u1,u2,u3∈ U and µ1, µ2∈(u1,u2,u3). Such
that
∥µ1−µ2∥=min{∥u1−u2∥,∥u2−u3∥,∥u3−u1∥},
and,
δ1(α, 0),(u,0)=|α−u|,5
4|α−u|,
δ1(0, α),(0,u)=3
4|α−u|,|α−u|,
δ1(α, 0),(0,u)=δ1(0,u),(α, 0)
=α+3
4u,5
4α+u.
Thus, (U, δ)is a complete cone 2-MS on the BG G.
Define Ti, Fi:U → U where i∈ N as
Ti(Fiα, 0)=0,3i
2
−1
i−
1
221
2
−
i
2
i−
1
2α,
and
Ti(0, Fiα)=3i
2
−1
i−
1
221
2
−
i
2
i−
1
2α, 0.
Therefore, T2i−1
i(F2i−1
iα, 0) = (0,1
12 α)and
T2i−1
i(0, F 2i−1
iα) = ( 1
18 α, 0). Thus, it con-
cludes that Ti, Fiachieves the contractive condition
(1), where Ki = 2i−1, α1= ( 1
4,0), α2=
α3=α4=α5= (1
6,0). Furthermore,
r(α1) = 1
4,r(α2) = r(α3) = r(α4) = r(α5) = 1
6.
Hence, by Theorem 3.1, we get (0,0) is a unique FP
for Ti, Fion Ufor all i≥1.
We will show that the normality condition of cone
2- metric is necessary to ensure the existence of a
common FP in our result.
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Example 2. Suppose that G=C1
R[0,1],
and (u1,u2)∈Gwith ∥(u, v)∥=∥(u1,u2)∥∞+
∥(u1,u2)′∥∞, then Gis a BG with unite e= (1,0).
Let, P={u1,u2∈G:u(t)≥0,u2(t)≥
0, t ∈[0,1]}be a non-normal solid cone. Consider,
Tu1n(t) = tn
nand Fu1n(t) = 1
n, then Fu1n≥
Tu1n≥0and lim
n→∞ Fu1n= 0, but ∥u1n∥=
maxt∈[0,1] |tn
n|+maxt∈[0,1] ||tn−1|=1
n+ 1 >1.
Thus, undoes not converges to 0. Hence, Tand F
does not have common FP.
4 Conclusion
Thus, in this work, we have obtained a unique
common fixed point result in cone 2-metric space on
Banach algebras. Also, we have generalized some
fixed point theorems in the literature. Using the idea
of n-inner product spaces on n-normed metric spaces
we will make an analog study concerning the unique
common fixed point of Hardy-Rogers contraction
type in cone n-metric spaces over a Banach Algebra.
Acknowledgment: Their authors extend their ap-
preciation to the Deanship of Scientific Research
at King Khalid University for funding this work
through Larg Groups. (Project under grant number
(1/371/43).
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A.B., B.H. and A.H.; methodology, B.H. and A.H.;
validation, B.H., A.H. and B.A.; formal analysis,
B.H.; resources, F.A. and A.A.; data curation,
A.H.; writing-original draft preparation, B.H.;
writing-review and editing, A.H., F.A. and A.A.;
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.66
Abdallah M. M. Badr, Basel Hardan,
Ahmed A. Hamoud, Badr Saleh Al-Abdi,
Faisal A. M. Ali, Jayashree Patil, Alaa A. Abdallah
E-ISSN: 2224-2880
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Volume 22, 2023
