Research Article
Cyclic b-Multiplicative (A, B)-Hardy–Rogers-Type Local
Contraction and Related Results in b-Multiplicative and
b-Metric Spaces
Abdullah Eqal Al-Mazrooei ,
1
Abdullah Shoaib ,
2
and Jamshaid Ahmad
1
1
Department of Mathematics, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
2
Department of Mathematics & Statistics, Riphah International University, Islamabad, Pakistan
Correspondence should be addressed to Jamshaid Ahmad; jamshaid_jasim@yahoo.com
Received 2 June 2020; Revised 28 July 2020; Accepted 10 August 2020; Published 12 October 2020
Academic Editor: Nan-Jing Huang
Copyright © 2020 Abdullah Eqal Al-Mazrooei et al. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
e aim of this paper is to define cyclic b-multiplicative Hardy–Rogers-type local contraction in the context of generalized spaces
named as b-multiplicative spaces to extend various results of the literature including the main results of Yamaod et al. In this way,
we apply a new generalized contractive condition only on a closed set instead of a whole set and by using b-multiplicative space
instead of multiplicative metric space. We apply our results to obtain new results in b-metric spaces. Examples are given to show
the usability of our results, when others cannot.
1. Introduction and Preliminaries
Bakhtin [1] was the first who gave the idea of b-metric. After
that, Czerwik [2] gave an axiom and formally defined a
b-metric space. For further results on the b-metric space, see
[3, 4]. Ozaksar and Cevical [5] investigated the multiplicative
metric space and proved its topological properties. Mon-
gkolkeha and Sintunavarat [6] described the concept of
multiplicative proximal contraction mapping and proved
some results for such mappings. Recently, Abbas et al. [7]
proved some common fixed point results of quasi-weak
commutative mappings on a closed ball in the setting of
multiplicative metric spaces. For further results on the
multiplicative metric space, see [8–12]. In 2017, Ali et al. [13]
introduced the notion of the b-multiplicative space and
proved some fixed point results. As an application, they
established an existence theorem for the solution of a system
of Fredholm multiplicative integral equations. For further
results on the b-multiplicative space, see [14]. Shoaib et al.
[4] discussed some results for the mappings satisfying
contraction condition on a closed ball in a b-metric space.
For further results on a closed ball, see [15–25]. In this paper,
we generalized the results in [12] by using cyclic b-multi-
plicative (A, B)-Hardy–Rogers-type local contraction on a
closed ball in a b-multiplicative space. Moreover, we show
that our results can be applied on those mappings where the
other results cannot be applied. e following definitions
and results will be used to understand this paper.
Definition 1 (see [13]). Let W be a nonempty set, and let
s ≥ 1 be a given real number. A mapping
m
b
: W × W ⟶ [1, ∞) is called b-multiplicative with co-
efficient s, if the following conditions hold:
(i) m
b
(w, μ) > 1 for all w, μ ∈ W with w ≠ μ and
m
b
(w, μ)� 1 if and only if w � μ
(ii) m
b
(w, μ)� m
b
(μ,w) for all w, μ ∈ W
(iii) m
b
(w, z) ≤ [m
b
(w, μ).m
b
(μ,z)]
s
for all w, μ,z ∈ W
e pair (W, m
b
) is called a b-multiplicative space. If
r > 1, u ∈ W, then B
m
b
(u, r)� v: m
b
(u, v) ≤ r and
B
m
b
(u, r)� v: m
b
(u, v) < r are called the closed ball and
the open ball in (W, m
b
), respectively. Note that if r ≤ 1, then
we obtain empty open balls because m
b
(u, v) ≥ 1.
Hindawi
Journal of Mathematics
Volume 2020, Article ID 2460702, 9 pages
https://doi.org/10.1155/2020/2460702