Some Fixed-Point Results of α-Admissible Mappings in Partial Cone
Metric Spaces over Banach Algebra
Jerolina Fernandez
Department of Mathematics and Statistics, The Bhopal School of Social Sciences, Bhopal, India
Keywords: PCMS over BA,
𝛼 -Admissible Mappings, Fixed-Point.
Abstract: The article presents α-admissible mappings in Partial Cone Metric Spaces (PCMS) over Banach Algebra
(BA), exploring fixed-point results in this context. It builds upon the foundational work of Liu and Xu [2013],
which introduced CMS over BA, marking a significant step in fixed-point theory. Fernandez et al. [2016]
extended this to PCMS over BA, examining fixed-point results for generalized Lipschitz mappings. This study
defines and explores α-admissible mappings in the newly established space, offering results that extend
previous findings. The main theorem establishes conditions for the existence of fixed-points for α-admissible,
generalized Lipschitz self-maps in θ-complete PCMS over BA, contributing to this evolving area of
mathematics.
1 INTRODUCTION
Liu and Xu (2013) introduced the notion of CMS over
BA by replacing the Banach space by Banach algebra
which became a milestone in the study of fixed-point
theory. Moreover, they gave some examples to
elucidate that fixed-point results in CMS over BA are
not equivalent to metric spaces (in usual sense).
Recently, Fernandez et al., (2016) introduced the
concept of PCMS over BA and studied some fixed-
point results for generalized Lipschitz mappings.
Inspired by the previous notion, in this paper we
establish some fixed-point results of 𝛼-admissible
mappings in the newly defined space. Our results
generalize and extend the recent result of Malhotra et
al. (2015).
2 PRELIMINARIES
First, we define PCMS over BA.
Definition 2.1.[1] A partial cone metric on a
nonempty set M is a function 𝑝:M×M A such that
for all 𝛽,𝛾,𝛿 M:
(𝑝

) 𝛽 = 𝛾 ⇔ 𝑝(𝛽, 𝛽) = 𝑝 (𝛽, 𝛾) = 𝑝(𝛾, 𝛾),
(𝑝

) ≼ 𝑝(𝛽, 𝛽) ≼ 𝑝(𝛽, 𝛾),
(𝑝

) 𝑝(𝛽, 𝛾) =𝑝(𝛾, 𝛽),
(𝑝

)𝑝(𝛽, 𝛾) ≼ 𝑝 (𝛽, 𝛿)+ p(𝛿, 𝛾) - 𝑝(𝛿, 𝛿).
The pair (M, p) is called a PCMS over BA.
Lemma 2.2. ([5]). Let A be a Banach algebra with
a unit e, k A, then 𝑙𝑖𝑚
→
‖𝑘

‖


exists and the
spectral radius (k) satisfies
(k) =𝑙𝑖𝑚
→
‖𝑘

‖


= inf ‖𝑘

‖


.
If (k) <
||
, then ( e- k) is invertible in A,
moreover,
𝑒 –𝑘

=
∑




where is a complex constant.
Lemma 2.3. ([2]). If E is a real Banach space with
a solid cone P and {u
n
} P be a sequence with ‖𝑢

‖
0 (n ∞), then {u
n
} is a c-sequence.
Lemma 2.4. ([2]). Let A be a Banach algebra with
a unit e and kA. If is a complex constant and
(k) <
||
,then
(𝑒  𝑘



||
 
.
192
Fernandez, J.
Some Fixed-Point Results of A-Admissible Mappings in Partial Cone Metric Spaces over Banach Algebra.
DOI: 10.5220/0012610300003739
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 1st International Conference on Artificial Intelligence for Internet of Things: Accelerating Innovation in Industry and Consumer Electronics (AI4IoT 2023), pages 192-194
ISBN: 978-989-758-661-3
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.

3 DISCUSSION AND MAIN
RESULTS
In this section, we introduce the concept of 𝛼-
admissible mappings in PCMS over BA.
Definition 3.1. Let M be a nonempty set and 𝛼:
M M →[0;∞) be a function. We say that T is 𝛼-
admissible if (𝛽, 𝛾) ∈M, 𝛼(𝛽, 𝛾) 1 ⇒ 𝛼(T 𝛽, T
𝛾)1.
Example 3.2. Let M = [0, ) and A be the set of
all real valued function on M which also have
continuous derivatives on M with the norm ‖𝛽‖=
‖𝛽‖ + ‖𝛽

‖ . Define multiplication in the usual
way. Let P = { 𝛽 A: 𝛽 (t) 0, t M}. It is clear
that P is a nonnormal cone and A is a Banach algebra
with a unit e = 1. Define a mapping 𝑝: MM A by
𝑝 (𝛽, 𝛾) = 𝛽𝑒

, 𝛽  𝛾 𝛽  𝑦𝑒

,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Define a self-map T on M as follows
T 𝛽 = 


, 


 



,  ∈,
and 𝛼𝛽,𝛾=
, 
 ,, ∈,
Since ln(1 + t) t for each t [0, 1), for all 𝛽, 𝛾
X, we have
when 𝛽 𝛾
𝑝(T 𝛽, T 𝛾)(t) =

𝑙𝑛 𝑙𝑛

1



 𝑙𝑛𝑙𝑛

1


 
𝑒

≼







𝑒

=


𝛽  𝛾𝑒

≼


𝑝(𝛽, 𝛾) 𝑡
and when 𝛽 = 𝛾,
𝑝(T 𝛽, T 𝛽)(t) =

𝑙𝑛 𝑙𝑛

1


 
𝑒

≼




𝑒

=


𝛽𝑒

≼


𝑝(𝛽, 𝛽)(t).
Therefore, 𝑝(T 𝛽, T 𝛾)(t) ≼


𝑝(𝛽, 𝛾)(t). Thus, T is
a Generalized Lipschitz map in M where 𝜌(k) =


< 1
and α(𝛽,𝛾) ≥ 1.
Theorem 3.3. Let (M, 𝑝) be a 𝜃-complete PCMS
over BA Suppose T be a generalized Lipschitz self-
map with Lipschitz vector k satisfying:
(i) T is 𝛼-admissible;
(ii) there exists 𝛽

∈ X such that 𝛼(𝛽

, T𝛽

)  1;
(iii) T is continuous.
Then T has a fixed-point 𝛽
∗
∈ M.
Proof. Let 𝛽

∈ M such that 𝛼(𝛽

, T𝛽

)  1.
Define a sequence {𝛽

} in M such that 𝛽

= T𝛽

∀
n ∈ N. If 𝛽

= 𝛽

∀ n ∈ N, then 𝛽
∗
𝛽

is a fixed-
point for T. Suppose 𝛽

= 𝛽

for all n ∈ N. Since T
is 𝛼-admissible we deduce
𝛼(𝛽

, 𝛽

) = 𝛼(𝛽

, T𝛽

)  1 ⇒ 𝛼(T𝛽

,𝑇

𝛽

) =
𝛼(𝛽

, 𝛽

)  1:
Continuing, we get
𝛼(𝛽

,𝛽

)  1 for all n ∈ N.
Now,
𝑝 (𝛽

, 𝛽
 
) = 𝑝 (𝑇𝛽

, 𝑇𝛽

)
≼ k 𝑝(𝛽

, 𝛽

)
≼ 𝑘

𝑝(𝛽

, 𝛽

).
For n < m we have
𝑝 (𝛽

, 𝛽

)≼ 𝑝 (𝛽

, 𝛽
  
) + 𝑝 (𝛽

, 𝛽

) - 𝑝
(𝛽

, 𝛽
  
)
≼ 𝑝 (𝛽

, 𝛽
  
) + 𝑝 (𝛽

, 𝛽

)
≼ 𝑝(𝛽

,𝛽
  
) + 𝑝 (𝛽

,𝛽
  
) +
𝑝 (𝛽

,𝛽

)-𝑝 (𝛽

, 𝛽
  
)
….
≼ 𝑝 (𝛽

, 𝛽
  
) + 𝑝 (𝛽

, 𝛽
  
)
+ 𝑝 (𝛽

, 𝛽

)
≼ 𝑝(𝛽

, 𝛽
  
) + 𝑝(𝛽

, 𝛽
  
)
+𝑝(𝛽

, 𝛽
  
)+ ……+ 𝑝(𝛽

, 𝛽
  
) + 𝑝(𝛽

, 𝛽

)
≼ 𝑘

𝑝(𝛽

, 𝛽

) +𝑘

𝑝(𝛽

, 𝛽

)
+ 𝑘

𝑝(𝛽

, 𝛽

)+…....+ 𝑘

𝑝(𝛽

, 𝛽

)
= 𝑘

 e + k + 𝑘

+ ………+ 𝑘

]𝑝(𝛽

, 𝛽

)
≼ 𝑘

𝑒  𝑘

𝑝(𝛽

, 𝛽

).
Some Fixed-Point Results of A-Admissible Mappings in Partial Cone Metric Spaces over Banach Algebra
193

Then ‖𝑘

𝑝𝛽

,𝛽

‖‖𝑘

‖‖𝑝𝛽

,𝛽

‖ 0 (n
), by Lemma 2.3, 𝑘

𝑝𝛽

,𝛽

} is a c-
sequence. By Lemma 2.2 and Lemma 2.4, {𝛽

} is a
𝜃-Cauchy sequence. Since M is complete, ∃ 𝛽
∗
∈ M
such that 𝛽

→𝛽
∗
as n →∞. Therefore,
𝑝 (𝛽

, 𝛽
∗
 = 𝑝 (𝛽

, 𝛽

 = 𝑝(𝛽
∗
, 𝛽
∗
 = .
Since T is continuous, we have 𝛽

= T𝛽

→
T𝛽
∗
as n →∞. By uniqueness, 𝛽
∗
= T𝛽
∗
, that is 𝛽
∗
is
a fixed-point of T.
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Fernandez J. (2016), Partial cone metric spaces over
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st
M.P.
Young Scientist Congress.
Huang H, and Radenovic S. (2015), Common fixed-point
theorems of Generalized Lipschitz mappings in cone
metric spaces over Banach algebras, Appl. Math. Inf.
Sci. 9, No. 6, 2983-2990.
Liu, H, Xu, S. (2013), Cone metric spaces with Banach
algebras and fixed-point theorems of generalized
Lipschitz mappings, Fixed-point Theory Appl., 320.
Malhotra S.K., Sharma J.B. and Shukla S. (2015), Fixed-
points of 𝛼-admissible mappings in Cone metric spaces
with Banach algebra, International Journal of Analysis
and Applications, Volume 9, Number 1, 9-18.
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