@
Appl. Gen. Topol. 23, no. 1 (2022), 145-156

doi:10.4995/agt.2022.15571

© AGT, UPV, 2022

Common new fixed point results on b-cone

Banach spaces over Banach algebras

Hojjat Afshari a , Hadi Shojaat b and Andreea Fulga c

a Department of Mathematics, Faculty of Sciences, University of Bonab, Bonab, Iran

(hojat.afshari@yahoo.com)
b Department of Mathematics, Farhangian University, Qazvin, Iran. (hadishojaat@yahoo.com)
c Department of Mathematics and Computer Sciences, Transilvania University of Brasov, Brasov,

Romania (afulga@unitbv.ro)

Communicated by E. Karapinar

Abstract

Recently Zhu and Zhai studied the concepts of cone b-norm and cone b-
Banach space as generalizations of cone b-metric spaces and they gave
a definition of φ-operator and obtained some new fixed point theorems
in cone b-Banach spaces over Banach algebras by using φ-operator. In
this paper we propose a notion of quasi-cone over Banach algebras,
then by utilizing some new conditions and following their work with
introducing two mappings T and S we improve the fixed point theorems
to the common fixed point theorems. An example is given to illustrate
the usability of the obtained results.

2020 MSC: 47H10; 54H25; 55M20.

Keywords: common fixed point; φ-operator; cone b-norm; cone b-Banach
space.

1. Introduction

The notion of b-metric was proposed by Czerwik [12, 13] to generalize the
concept of distance. The analog of the famous Banach fixed point theorem was
proved by Czerwik in the frame of complete b-metric spaces, see also [9, 10, 11].
In [19] E. Karapinar generalized some conclusions on the cone Banach space,

Received 06 May 2021 – Accepted 27 September 2021

http://dx.doi.org/10.4995/agt.2022.15571
https://orcid.org/0000-0003-1149-4336
https://orcid.org/0000-0002-6838-072X
https://orcid.org/0000-0002-6689-0355


H. Afshari, H. Shojaat and A. Fulga

in the literature [2] and obtained the existence results of fixed points for self-
mappings. Also, the cone metric space over Banach algebra, proposed by Liu
and Xu (see [23]) and they considered some fixed point results on such new
space.

In 2001 Hussain and Shah [17] introduced the notation of cone b-metric
space. Many researchers continued the work of Hussain and Shah, and proved
some fixed point theorems and common fixed point theorems for multiple op-
erators on these new spaces, and also used them to investigate the existence of
the solutions of fractional integral equations (see [3, 5, 4, 6, 8, 14, 15, 16, 18,
24, 25, 27, 28, 26, 20, 21]).

Recently Zhu and Zhai [30] studied the concepts of cone b-norm and cone b-
Banach space as generalizations of cone b-metric spaces. Also they introduced
the operator φ and obtained some new fixed point theorems in cone b-Banach
spaces over Banach space utilizing the φ-operator.
In this paper by introducing a notion of quasi-cone over Banach space and also
with applying different conditions we examine the existence of some common
fixed points of two self-mappings S and T that has led to the development of
similar results in the literature.

2. Preliminaries

Let (E,‖ · ‖) be a real Banach space, P ⊂ E a cone and θ be the zero of E,
also there is a partial ordering ≤ such that ξ ≤ ζ iff ζ − ξ ∈ P. Write ξ � ζ
for ζ − ξ ∈ intP, where intP is the interior set of P. We say that P is normal
if there exists N > 0 such that θ ≤ ξ ≤ ζ implies ‖ ξ ‖≤ N ‖ ζ ‖, for ξ,ζ ∈ E.
P is called to be solid if intP 6= ∅.
Definition 2.1 (see [17]). Let X 6= ∅ and s ≥ 1, a mapping % : X × X → E is
a cone b-metric if;
(i ) θ < %(ξ,ζ) with ξ 6= ζ and %(ξ,ζ) = θ iff ξ = ζ;
(ii ) %(ξ,ζ) = %(ζ,ξ);
(iii ) %(ξ,ζ) ≤ s[%(ξ,η) + %(η,ζ)],
for all ξ,ζ,η ∈ X. The pair (X,%) is said a cone b-metric space, in short,
CBMS.

Lemma 2.2 (see [17]). If (X,%) is a CBMS. Then;
(p1) If ξ � ζ and ζ � η, then ξ � η.
(p2) If ξ � ζ and ζ � η, then ξ � η.
(p3) If θ ≤ ξ � c for c ∈ intP, then ξ = θ.
(p4) If c ∈ intP, θ ≤ ξn and ξn → θ, then there exists n0 such that ξn � c for
n > n0.
(p5) Suppose θ � c, if θ ≤ %(ξn,ξ) ≤ ζn and ζn → θ, then eventually %(ξn,ξ) �
c, where ξ ∈ X and {ξn}n≥1 is a sequence in X.
(p6) If θ ≤ ξn ≤ ζn and ξn → ξ, ζn → ζ, then ξ ≤ ζ, for each cone P.
(p7) If ξ ≤ λξ where ξ ∈ P and 0 ≤ λ < 1, then ξ = θ.
Definition 2.3 (see [19]). Let X be a vector space over R. For a cone P ⊂ E
and a mapping ‖ · ‖E: X → E if we have;

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 146



Common new fixed point results on b-cone Banach spaces over Banach algebras

(i ) ‖ ξ ‖E≥ θ for ξ ∈ X and ‖ ξ ‖E= θ iff ξ = θ;
(ii ) ‖ ξ + ζ ‖E≤‖ ξ ‖E + ‖ ζ ‖E for ξ,ζ ∈ X;
(iii )‖ kξ ‖E=| k |‖ ξ ‖E for k ∈ R.
Then ‖ · ‖E is said a cone norm on X, and (X,‖ · ‖E) is said a cone normed
space (CNS). If we set %(ξ,ζ) =‖ ξ − ζ ‖E, then every CNS is a CMS.
Definition 2.4 (see [19]). Let 1 ≤ s ≤ 2, X be a vector space over R, cone
P ⊂ E. If ‖ · ‖P: X → E satisfies;
(i ) ‖ ξ ‖P≥ θ for ξ ∈ X and ‖ ξ ‖P= θ iff ξ = θ;
(ii ) ‖ ξ − ζ ‖P=‖ ζ − ξ ‖P for ξ,ζ ∈ X;
(iii ) ‖ ξ + ζ ‖P≤ s[‖ ξ ‖P + ‖ ζ ‖P] for ξ,ζ ∈ X;
(i v)‖ kξ ‖P=| k |s‖ ξ ‖P for k ∈ R.
Then we call ‖ · ‖P a cone-norm on X, and (X,‖ · ‖P), we call it a cone-normed
space (CNS). Obviously, each CNS is a CMS. In fact, we only need to set
%(ξ,ζ) =‖ ξ − ζ ‖P.
Definition 2.5 (see [30]). Suppose that (X,‖ · ‖P) is a cone b-normed space,
P ⊂ E is a solid cone, ξ ∈ X and {ξn}n≥1 is a sequence in X. Then;
(i ) we say that {ξn}n≥1 converges to ξ if for c ∈ E with θ � c, there is a natural
number N satisfying ‖ ξn − ξ ‖P� c for n ≥ N. We denote lim

n→∞
ξn = ξ or

ξn → ξ;
(ii ) we say that {ξn}n≥1 is a Cauchy if for c ∈ E with θ � c, there exists a
natural number N satisfying ‖ ξn − ξm ‖P� c for all n,m ≥ N;
(iii ) we say that (X,‖ · ‖P) is complete if every Cauchy is convergent.
Lemma 2.6 (see [30]). Suppose (X,‖ · ‖P) is a cone b-normed space, P is a
solid cone, ξ ∈ X and {ξn}n≥1 is a sequence in X. Then the following conclu-
sions hold:
(i ) ‖ ξn − ξ ‖P→ θ(n →∞) iff {ξn} converges to ξ.
(ii )‖ ξn − ξm ‖P→ θ(n,m →∞) iff {ξn} is a Cauchy.
Lemma 2.7 ([29]). Suppose (E,‖ · ‖) is a real Banach space and P is a
normal cone in E, then there is an equivalent norm ‖ · ‖1, which satisfies
θ ≤ ξ ≤ ζ =⇒‖ ξ ‖1≤‖ ζ ‖1, for ξ,ζ ∈ E, that is, norm ‖ · ‖1 is monotonous.
Remark 2.8. Suppose E is a linear space, ‖ · ‖1 and ‖ · ‖2 are two given norms
in E, we say that ‖ · ‖2 is stronger than ‖ · ‖1 if ‖ ξn ‖2→ 0 =⇒‖ ξn ‖1→
0 (n → ∞). If ‖ · ‖2 is stronger than ‖ · ‖1, and ‖ · ‖1 is stronger than ‖ · ‖2,
then ‖ · ‖1 is equivalent ‖ · ‖2.
Definition 2.9 ([29, 7]). Let E be a real Banach algebra, that is, for ξ,ζ,η ∈ E,
a ∈ R,
(i ) ξ(ζη) = (ξζ)η;
(ii ) ξ(ζ + η) = ξζ + ξη, (ξ + ζ)η = ξη + ζη;
(iii ) a(ξζ) = (aξ)ζ = (aζ)ξ;
(i v) ‖ ξζ ‖≤‖ ξ ‖‖ ζ ‖.
If Banach algebra E with unit element e, such that ξe = eξ = ξ for all ξ ∈ E,
then ‖ e ‖= 1. If every non-zero element of E has an inverse element in E, then
E is called a divisible Banach algebra.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 147



H. Afshari, H. Shojaat and A. Fulga

Definition 2.10 ([7]). Let E be a Banach algebra with unit element e and
P ⊆ E be a cone. P is called algebra cone if e ∈ P and for each ξ,ζ ∈ P, ξζ ∈ P.

In our following discussions, ξ = (X,‖ · ‖P) is a cone b-Banach space, P is
a solid cone and S is a operator defined on D of X. Let E := (E,‖ · ‖) be a
divisible Banach algebra with unit element e. Let PE be a normal algebra cone
in E with a normal constant N.

Definition 2.11. Let (E,‖ · ‖) be a divisible Banach algebra. PE is a normal
algebra cone in E. We call the mapping φ : PE → PE is a φ-operator if it
satisfies
(i ) φ is an increasing operator;
(ii ) φ is a continuous bijection and has an inverse mapping φ−1 which is also
continuous and increasing;
(iii ) φ(ξ + ζ) ≤ φ(ξ) + φ(ζ) for all ξ,ζ ∈ PE;
(i v) φ(ξζ) = φ(ξ)φ(ζ) for all ξ,ζ ∈ PE.

Remark 2.12. By Definition 2.11, the part of (iii ), we can get φ−1(ξ)+φ−1(ζ) ≤
φ−1(ξ + ζ) for all ξ,ζ ∈ PE. In fact, note that φ(ξ + ζ) ≤ φ(ξ) + φ(ζ) for all
ξ,ζ ∈ PE and φ−1 is also a continuous and increasing operator, then

φ−1(φ(ξ + ζ)) ≤ φ−1(φ(ξ) + φ(ζ)),
which yields that

ξ + ζ ≤ φ−1(φ(ξ) + φ(ζ)).
Hence,

φ−1(φ(ξ)) + φ−1(φ(ζ)) ≤ φ−1(φ(ξ) + φ(ζ)).
Since φ : PE → PE is a continuous bijection, thus φ−1(ξ)+φ−1(ζ) ≤ φ−1(ξ+ζ),
for all ξ,ζ ∈ PE.

Remark 2.13. By Definition 2.11, the part of (i v), we can obtain φ−1(ξζ) =
φ−1(ξ)φ−1(ζ), for all ξ,ζ ∈ PE.
Indeed, from φ(ξζ) = φ(ξ)φ(ζ) for all ξ,ζ ∈ PE and φ−1 : PE → PE is also a
continuous, we get

φ−1(φ(ξζ)) = φ−1(φ(ξ)φ(ζ)),

which yields that

ξζ = φ−1(φ(ξ)φ(ζ)).

Then

φ−1(φ(ξ))φ−1(φ(ζ)) = φ−1(φ(ξ)φ(ζ)).

Thanks to that φ : PE → PE is a continuous bijection, φ−1(ξζ) = φ−1(ξ)φ−1(ζ),
for all ξ,ζ ∈ PE.

Remark 2.14. For example, let E = R, a divisible Banach algebra, PE = {ξ ∈
E | ξ ≥ 0} be a normal cone in E, suppose φ : PE → PE, defined by φ(ξ) = ξ

1
5

and then φ−1(ξ) = ξ5, for ξ ∈ PE. We can prove it also satisfies the above
conditions.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 148



Common new fixed point results on b-cone Banach spaces over Banach algebras

3. Main results

Theorem 3.1. Let X be a cone-b-Banach space with the coefficient 1 ≤ s ≤ 2,
E1 and E2 be divisible Banach space with identity elements e1 and e2, also PE1
and PE2 be normal algebra cones in E1 and E2 (respectively). If D and D

′ ⊂ X
with D ∩ D′ 6= ∅ are closed and convex, also φ : PE1 ∪ PE2 → PE1 ∪ PE2 is
φ-operator and T : D → D′, S : D′ → D satisfying the followings

φ(%(η,Sξ′)) + φ(%(η′,T ξ)) ≤ k1φ(%(η,ξ)),(3.1)
φ(%(ξ′,T η)) + φ(%(ξ,Sη′)) ≤ k2φ(%(ξ′,η′)),

for all ξ,η ∈ D, ξ′,η′ ∈ D′, where φ(2se1) ≤ k1 < φ(2s+1e1) , φ(2se2) ≤ k2 <
φ(2s+1e2) in PE1 and PE2 (respectively). Then S and T have a common fixed
point in D ∩ D′.

Proof. Let ξ1 ∈ D, η1 ∈ D′ be arbitrary. We introduce two sequences {ξn},{ηn}∈
D ∪ D′, defined by

ξ2 =
η1+T ξ1

2
∈ D′,

η2 =
ξ1+Sη1

2
∈ D,

ξ3 =
η2+Sξ2

2
∈ D′,

η3 =
ξ2+Tη2

2
∈ D,

...

ξ2n =
η2n−1+T ξ2n−1

2
, n = 1, 2, · · · ,

η2n =
ξ2n−1+Sη2n−1

2
, n = 1, 2, · · · ,

ξ2n +1 =
η2n +Sξ2n

2
, n = 1, 2, · · · ,

η2n +1 =
ξ2n +Tη2n

2
, n = 1, 2, · · · .

We get

η2n −Sξ2n = 2(η2n − (
η2n + Sξ2n

2
)) = 2(η2n − ξ2n +1),

ξ2n −T η2n = 2(ξ2n − (
ξ2n + T η2n

2
)) = 2(ξ2n −η2n +1),

η2n +1 −T ξ2n +1 = 2(η2n +1 − (
η2n +1 + T ξ2n +1

2
)) = 2(η2n +1 − ξ2n +2),

ξ2n +1 −Sη2n +1 = 2(ξ2n +1 − (
ξ2n +1 + Sη2n +1

2
)) = 2(ξ2n +1 −η2n +2),

which is equivalent to

%(η2n,Sξ2n ) = ‖ η2n −Sξ2n ‖PE1
= ‖ 2(η2n − ξ2n +1) ‖PE1
= 2s ‖ η2n − ξ2n +1 ‖PE1
= 2s%(η2n,ξ2n +1),

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 149



H. Afshari, H. Shojaat and A. Fulga

%(ξ2n,T η2n ) = ‖ ξ2n −T η2n ‖PE2
= ‖ 2(ξ2n −η2n +1) ‖PE2
= 2s ‖ ξ2n −η2n +1 ‖PE2
= 2s%(ξ2n,η2n +1)

and

%(η2n +1,T ξ2n +1) = ‖ η2n +1 −T ξ2n +1 ‖PE2(3.2)
= ‖ 2(η2n +1 − ξ2n +2) ‖PE2
= 2s ‖ η2n +1 − ξ2n +2 ‖PE2
= 2s%(η2n +1,ξ2n +2),

%(ξ2n +1,Sη2n +1) = ‖ ξ2n +1 −Sη2n +1 ‖PE1(3.3)
= ‖ 2(ξ2n +1 −η2n +2) ‖PE1
= 2s ‖ ξ2n +1 −η2n +2 ‖PE1
= 2s%(ξ2n +1,η2n +2).

Substituting ξ′ = ξ2n,ξ = ξ2n +1 and η = η2n,η
′ = η2n +1 in (3.1), we can obtain

φ(%(η2n,Sξ2n )) + φ(%(η2n +1,T ξ2n +1)) ≤ k1φ(%(η2n,ξ2n +1)).

We get

φ(2s%(η2n,ξ2n +1)) + φ(2
s%(η2n +1,ξ2n +2)) ≤ k1φ(%(η2n,ξ2n +1)).

According to the condition (iii ) of φ-operator,

φ(2s(%(η2n,ξ2n +1) + %(η2n +1,ξ2n +2))) ≤ k1φ(%(η2n,ξ2n +1)).

Remark 2.13 and the property of φ−1 operator, we can get

2s(%(η2n,ξ2n +1) + %(η2n +1,ξ2n +2)) ≤ φ−1(k1)%(η2n,ξ2n +1),

by simplifying, we get

%(η2n +1,ξ2n +2) ≤ (
φ−1(k1)

2s
−e1)%(η2n,ξ2n +1).

Substituting ξ′ = ξ2n,ξ = ξ2n +1 and η = η2n,η
′ = η2n +1 in (3.1). Then one can

obtain

φ(%(ξ2n,T η2n )) + φ(%(ξ2n +1,Sη2n +1)) ≤ k2φ(%(ξ2n,η2n +1)).

By (3.2) and (3.3) we get

φ(2s%(ξ2n,η2n +1)) + φ(2
s%(ξ2n +1,η2n +2)) ≤ k2φ(%(ξ2n,η2n +1)).

According to the condition (iii ) of φ-operator,

φ(2s(%(ξ2n,η2n +1) + %(ξ2n +1,η2n +2))) ≤ k2φ(%(ξ2n,η2n +1)).

By Remark 2.13 and the property of φ−1 operator, we can get

2s(%(ξ2n,η2n +1) + %(ξ2n +1,η2n +2)) ≤ φ−1(k2)%(ξ2n,η2n +1),

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 150



Common new fixed point results on b-cone Banach spaces over Banach algebras

by simplifying, we get

%(ξ2n +1,η2n +2) ≤ (
φ−1(k2)

2s
−e2)%(ξ2n,η2n +1).

Thus, %(η2n +1,ξ2n +2) ≤ k′1%(η2n,ξ2n +1) and %(ξ2n +1,η2n +2) ≤ k′2%(ξ2n,η2n +1),
where k′1 =

φ−1(k1)
2s

−e1 and k′2 =
φ−1(k2)

2s
−e2. Repeating this relations, we get

%(η2n +1,ξ2n +2) ≤ k′2
n
k′1
n
%(η1,ξ2),(3.4)

%(ξ2n +1,η2n +2) ≤ k′2
n
k′1
n
%(ξ1,η2).

For any m ≥ 1,p ≥ 1, we have one of the following two cases:
(i ) m + p = 2r − 1, r ≥ 1,r ∈ N, then we get

%(ηm+p,ξm) ≤ s[%(ηm+p,ξm+p−1) + %(ξm+p−1,ξm)]
≤ s%(ηm+p,ξm+p−1) + s2[%(ξm+p−1,ηm+p−2) + %(ηm+p−2,ξm)]
≤ s%(ηm+p,ξm+p−1) + s2%(ξm+p−1,ηm+p−2) + s3%(ηm+p−2,ξm+p−3) + · · ·

+sp−1%(ξm+2,ηm+1) + s
p−1%(ηm+1,ξm)

≤ sk′2
r−2

k′1
r−1

%(η2,ξ1) + s
2k′2

r−3
k′1
r−2

%(ξ2,η1) + s
3k′2

r−4
k′1
r−3

%(η2,ξ1) + · · ·
+sp−1k′2

2r−p−1
k′1

2r−p
%(ξ2,η1) + s

p−1k′2
2r−p−2

k′1
2r−p−1

%(η2,ξ1)

= (sk′2
r−2

k′1
r−1

+ s3k′2
r−4

k′1
r−3

+ · · · + sp−1k′2
2r−p−2

k′1
2r−p−1

)%(η2,ξ1)

+(s2k′2
r−3

k′1
r−2

+ · · · + sp−1k′2
2r−p−1

k′1
2r−p

)%(ξ2,η1)

=
sk′2

r−2
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(η2,ξ1)

+
s2k′2

r−3
k′1
r−2

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1).

(ii ) m + p = 2r, r ≥ 1,r ∈ N, then we get

%(ξm+p,ηm) ≤ s[%(ξm+p,ηm+p−1) + %(ηm+p−1,ηm)]
≤ s%(ξm+p,ηm+p−1) + s2[%(ηm+p−1,ξm+p−2) + %(ξm+p−2,ηm)]
≤ s%(ξm+p,ηm+p−1) + s2%(ηm+p−1,ξm+p−2) + s3%(ξm+p−2,ηm+p−3) + · · ·

+sp−1%(ηm+2,ξm+1) + s
p−1%(ξm+1,ηm)

≤ sk′2
r−1

k′1
r−1

%(ξ2,η1) + s
2k′2

r−2
k′1
r−2

%(η2,ξ1) + s
3k′2

r−3
k′1
r−3

%(ξ2,η1) + · · ·
+sp−1k′2

2r−p
k′1

2r−p
%(η2,ξ1) + s

p−1k′2
2r−p−1

k′1
2r−p−1

%(ξ2,η1)

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 151



H. Afshari, H. Shojaat and A. Fulga

= (sk′2
r−1

k′1
r−1

+ s3k′2
r−3

k′1
r−3

+ · · · + sp−1k′2
2r−p−1

k′1
2r−p−1

)%(ξ2,η1)

+(s2k′2
r−2

k′1
r−2

+ · · · + sp−1k′2
2r−p

k′1
2r−p

)%(η2,ξ1)

=
sk′2

r−1
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1)

+
s2k′2

r−2
k′1
r−2

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e1 − s
2

(k′2k
′
1)

2

%(η2,ξ1).

Since φ(2se1) ≤ k′2 < φ(2s+1e1) in PE1 and φ(2se2) ≤ k′1 < φ(2s+1e2) in PE2
with 1 ≤ s ≤ 2, we know θ2 ≤ k′2 < e1, θ1 ≤ k′1 < e2, thus θ2 < se2 −k′2 ≤ se2,
θ1 < se1 −k′1 ≤ se1. Further,

‖ k′2
r−p −θ2 ‖=‖ k′2

r−p ‖≤‖ k′2 ‖
r−p,

‖ k′1
r−p −θ1 ‖=‖ k′1

r−p ‖≤‖ k′1 ‖
r−p,(3.5)

since θ2 ≤ k′2 < e2, θ1 ≤ k′1 < e1 and PE is a normal cone in E, by Lemma 2.7
we know there is an equivalent norm ‖ · ‖1 and thus

0 ≤‖ k′2 ‖1<‖ e2 ‖1= 1,(3.6)
0 ≤‖ k′1 ‖1<‖ e1 ‖1= 1.

By (3.5) and (3.6), we get

‖ k′2
r−p −θ2 ‖1≤‖ k′2 ‖

r−p
1 → 0((r − p) →∞),

‖ k′1
r−p −θ1 ‖1≤‖ k′1 ‖

r−p
1 → 0((r − p) →∞).(3.7)

From Remark (2.8) and (3.7),

‖ k′2
r−p −θ2 ‖≤‖ k′2 ‖

r−p→ 0((r − p) →∞),

‖ k′1
r−p −θ1 ‖≤‖ k′1 ‖

r−p→ 0((r − p) →∞).
Thus,

lim
(r−p)→∞

k′2
(r−p) → θ2,

lim
(r−p)→∞

k′1
(r−p) → θ1. (3.8)

Let θ1,θ2 � c be given. By (3.8),

sk′2
r−1

k′1
r
(e1e2 − s

2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(η2,ξ1)+
s2k′2

r−2
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1) → θ1,

sk′2
r
k′1
r
(e1e2 − s

2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1)+
s2k′2

r−1
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e1 − s
2

(k′2k
′
1)

2

%(η2,ξ1) → θ2,

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 152



Common new fixed point results on b-cone Banach spaces over Banach algebras

as (r − p) → ∞. Making full use of Lemma 2.2 (p4), we find m0 ∈ N, such
that

sk′2
r−1

k′1
r
(e1e2 − s

2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(η2,ξ1)+
s2k′2

r−2
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1) � c,

sk′2
r
k′1
r
(e1e2 − s

2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1)+
s2k′2

r−1
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e1 − s
2

(k′2k
′
1)

2

%(η2,ξ1) � c,

for each m > m0. Thus

sk′2
r−1

k′1
r
(e1e2 − s

2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(η2,ξ1)+
s2k′2

r−2
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1) � c,

sk′2
r
k′1
r
(e1e2 − s

2

(k′2k
′
1)

2 )
p+1
2

e1e2 − s
2

(k′2k
′
1)

2

%(ξ2,η1)+
s2k′2

r−1
k′1
r−1

(e1e2 − s
2

(k′2k
′
1)

2 )
p−1
2

e1e1 − s
2

(k′2k
′
1)

2

%(η2,ξ1) � c,

for m > m0 and each p. Considering the upper relations we can get;

%(ξm+p,ξm) ≤ s(%(ξm+p,ηm) + %(ηm + ξm)),
%(ηm+p,ηm) ≤ s(%(ηm+p,ξm) + %(ξm + ηm)).

Now by Lemma ?? part (p1), we can claim that {ξn} and {ηn} are Cauchy
sequences in D. Note that D and D′ are closed and convex and {ξ2n}, {η2n}
converges to some ζ,ζ′, that is, ξ2n,η2n → ζ,ζ′ ∈ D ∪ D′. Regarding the
inequality

%(ζ,Sη2n +1) ≤ s[%(ζ,ξ2n +1) + %(ξ2n +1,Sη2n +1)],
%(ζ′,T ξ2n +1) ≤ s[%(ζ′,η2n +1) + %(η2n +1,T ξ2n +1)],

and from (3.3), we obtain

%(ζ,Sη2n +1) ≤ s[%(ζ,ξ2n +1) + 2s%(ξ2n +1,η2n +2)],(3.8)

%(ζ′,T ξ2n +1) ≤ s[%(ζ′,η2n +1) + 2s%(η2n +1,ξ2n +2)],
let n → ∞, then Sη2n +1 → ζ, T ξ2n +1 → ζ. Finally, replacing η2n +1 = ζ in
(3.8). Then one can obtain

φ(%(ζ,Sζ′)) ≤ s[%(ζ,ξ2n +1) + 2s%(ξ2n +1,η2n +2)],

and if ζ = ξ2n +1

φ(%(ζ′,T ζ)) ≤ s[%(ζ′,η2n +1) + 2s%(η2n +1,ξ2n +2)],

and by making use of the property iv of φ-operator, we obtain, φ(e1) =
e1,φ(e2) = e2. So we get

φ(%(ζ,Sζ′) = e1,
φ(%(ζ′,T ζ) = e2.

Therefore as n → ∞, we can obtain Sζ′ = ζ,T ζ = ζ′. Hence considering
ζ = ζ′, we conclude ζ = Tζ = Sζ. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 153



H. Afshari, H. Shojaat and A. Fulga

Corollary 3.2. Let X be a cone-b-Banach space with the coefficient 1 ≤ s ≤ 2,
E be a divisible Banach algebra with identity element e, and also PE be a normal
algebra cone in E. If D ⊂ X is closed and convex, φ : PE → PE is an φ-operator
and S,T : D → D are mappings satisfying the conditions

φ(%(η,Sξ)) + φ(%(η,T ξ)) ≤ kφ(%(η,ξ)),(3.9)
φ(%(ξ,T η)) + φ(%(ξ,Sη)) ≤ kφ(%(ξ,η)),

for all ξ,η ∈ D, where φ(2se) ≤ k < φ(2s+1e) in PE. Then S and T have a
common fixed point in D.

Proof. If in Theorem 3.1 we set, T = S and D = D′, considering the condition
of (3.10) and by the proof similar to the proof of Theorem 3.1 we deduce the
result. �

Example 3.3. Let X = R2 and E = R2 endowed with partial ordered ξ =
(ξ1,ξ2) ≤ ζ = (ζ1,ζ2) iff ξ1 ≤ ζ1,ξ2 ≤ ζ2. If P = {(ξ1,ξ2) ∈ E : ξ1 ≥ 0,ξ2 ≥ 0},
we define ‖ (ξ1,ξ2) ‖P= (| ξ1 |2, | ξ2 |2). Then (X,‖ · ‖P) is a cone b-Banach
space with s = 2.
For ξ = (ξ1,ξ2) and ζ = (ζ1,ζ2) we define; ξ.ζ = (ξ1ξ2,ζ1ζ2). By the the
mentioned definition P is a Banach algebra and E := (E,‖ · ‖) is a divisible
Banach algebra with unit element e = (1, 1), because ξe = eξ = ξ,‖e‖ = 1 and
hence e is a multiplicative identity. If we put φ : P → P with φ(ξ = (ξ1,ξ2)) =
(
√
ξ1,
√
ξ2), then φ satisfies the conditions (i )-(i v) of Definition 3.4. Also we set;

%(ξ,ζ) =‖ ξ − ζ ‖P= (| (ξ1 − ζ1 |2, | ξ2 − ζ2 |2),%(ξ,A) = inf{%(ξ,ζ) : ζ ∈ A}
and Sξ = ξ

2
,T ξ = ξ

2

4
. Now we define the region D as the following;

D = {(ξn,ηn ) : |ηn−
ξn
2
|+|ηn−

ξn
2

4
| ≤ 2.8|ηn−ξn|, |ξn−

ηn
2
|+|ξn−

ηn
2

4
| ≤ 2.8|ξn−ηn|,n = 1, 2}.

Obviously D is closed and convex.

φ(%(η,Sξ)) + φ(%(η,T ξ)) ≤ φ(%(η, ξ
2
)) + φ(%(η, ξ

2

4
))

≤ φ(|η1 − ξ12 |
2, |η2 − ξ22 |

2) + φ(|η1 − ξ1
2

4
|2, |η2 − ξ2

2

4
|2)

= (|η1 − ξ12 |, |η2 −
ξ2
2
|) + (|η1 − ξ1

2

4
|, |η2 − ξ2

2

4
|)

= (|η1 − ξ12 | + |η1 −
ξ1

2

4
|, |η2 − ξ22 | + |η2 −

ξ2
2

4
|)

≤ (|η1 − ξ12 | + |η1 −
ξ1

2

4
|, |η2 − ξ22 | + |η2 −

ξ2
2

4
|),

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 154



Common new fixed point results on b-cone Banach spaces over Banach algebras

also

φ(%(ξ,T η)) + φ(%(ξ,Sη)) ≤ φ(%(ξ, η
2

4
)) + φ(%(ξ, η

2
))

≤ φ(|ξ1 − η1
2

4
|2, |ξ2 − η2

2

4
|2) + φ(|ξ1 − η12 |

2, |ξ2 − η22 |
2)

= (|ξ1 − η1
2

4
|, |ξ2 − η2

2

4
|) + (|ξ1 − η12 |, |ξ2 −

η2
2
|)

= (|ξ1 − η12 | + |ξ1 −
η1

2

4
|, |ξ2 − η22 | + |ξ2 −

η2
2

4
|)

≤ (|ξ1 − η12 | + |ξ1 −
η1

2

4
|, |ξ2 − η22 | + |ξ2 −

η2
2

4
|).

Considering

φ(2se) = φ(4, 4) = (2, 2) ≤ k < φ(2s+1e) = φ(8, 8) = (2
√

2, 2
√

2),

we should have

(|η1 − ξ12 | + |η1 −
ξ1

2

4
|, |η2 − ξ22 | + |η2 −

ξ2
2

4
|)

≤ (2.8, 2.8)(|η1 − ξ1|, |η2 − ξ2|) = (2.8|η1 − ξ1|, 2.8|η2 − ξ2|),

(|ξ1 − η12 | + |ξ1 −
η1

2

4
|, |ξ2 − η22 | + |ξ2 −

η2
2

4
|)

≤ (2.8, 2.8)(|ξ1 −η1|, |ξ2 −η2|) = (2.8|ξ1 −η1|, 2.8|ξ2 −η2|).
So according to the definition of region D the conditions of Corollary 3.2 are
satisfied. Hence S and T have a common fixed point.

Corollary 3.4. Let X be a cone-b-Banach space with the coefficient 1 ≤ s ≤ 2,
E be a divisible Banach algebra with identity element e, and also PE be a
normal algebra cone in E . If D ⊂ X is closed and convex, φ : PE → PE is an
φ-operator and T : D → D is a mapping satisfying the condition

φ(%(η,T η)) + φ(%(ξ,T ξ)) ≤ kφ(%(η,ξ)),
for all ξ,η ∈ D, where φ(2se) ≤ k < φ(2s+1e) in PE. Then T has a fixed point
in D.

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