Cyclic 
                     b
                  -Multiplicative 
                     
                        
                           A
                           ,
                           B
                        
                     
                  -Hardy–Rogers-Type Local Contraction and Related Results in 
                     b
                  -Multiplicative and 
                     b
                  -Metric Spaces

The 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.


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. Mongkolkeha 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][9][10][11][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][16][17][18][19][20][21][22][23][24][25]. In this paper, we generalized the results in [12] by using cyclic b-multiplicative (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 coefficient s, if the following conditions hold: where a > 1 is any fixed real number. en, for each a, m a is b-multiplicative on W with s � 2. Note that m a is not a multiplicative metric on W. Considering a � 2, r � 2 16 , and Definition 2 (see [13]). Let (W, m b ) be a b-multiplicative space.
Definition 3 (see [3]). Let W ≠ ϕ and s ≥ 1 be a real number. A mapping d: { } is said to be b-metric with coefficient "s," if for all w, μ, z ∈ W, the following assertions hold: Remark 1 (see [13]).
is implies that By using (6), we have Journal of Mathematics Now, using inequality (19), we get By using triangle inequality and inequality (20), we get By using inequality (12), we have is implies that þ j+1 ∈ B m b (þ°, r). By induction on n, we conclude that þ n ∈ B m b (þ°, r) for all n ∈ N. By a similar method, for all n ∈ N, we get is implies that Now, inequality (20) implies that Now, we prove that þ n is a b-multiplicative Cauchy sequence in K. Let m > n, so m � n + p; p ∈ N. By using the triangle inequality, we have By using inequality (25), we get Taking limit as m, n ⟶ ∞, we get m b (þ n , þ m ) ⟶ 1. Hence, the sequence þ n is a b-multiplicative Cauchy sequence. By the completeness of (K, m b ), it follows that þ n ⟶ þ * ∈ B m b (þ°, r). Suppose that g is continuous. us, we get þ * � lim n⟶∞ þ n+1 � lim n⟶∞ gþ n � g(lim n⟶∞ þ n ) � gþ * . Now, we assume that condition (a) of Definition 7 holds. As B(þ n ) ≥ 1 and þ n ⟶ þ * ∈ B m b (þ°, r), so B(þ * ) ≥ 1. en, we have Letting n ⟶ ∞, we get Hence, m b (qþ * , þ * ) � 1, that is, gþ * � þ * . is proves that þ * is a fixed point of g. Eventually we prove that þ * is the unique fixed point of g. Suppose that μ is another fixed point of g. By the hypothesis, we find that A(þ * ) ≥ 1 and B(μ) ≥ 1. us, 4 Journal of Mathematics is proves that m b (þ * , μ) � 1 and then þ * � μ. us, þ * is the unique fixed point of g. □ Example 6. In Example 3, we have proved that g is a cyclic b-multiplicative (A, B)-Hardy-Rogers-type local contraction on B m b (þ°, r). It has been proved in Example 5 that the mapping g in Example 3 is cyclic regular on a closed ball Hence, all the conditions of eorem 1 are satisfied, and zero is the unique fixed point of the mapping g. Note that the results in [12] cannot ensure the existence of a fixed point of mapping g because g cannot satisfy the contractive condition of any theorem in [12]. e following results for various other contractions on bmultiplicative spaces can be proved by following the proof of eorem 1.  B m b (þ°, r). Moreover, if B(z) ≥ 1 and A(z) ≥ 1, for all z in the set of fixed points of g, then the fixed point of g will be unique.

Journal of Mathematics
where h � (λ + θ + ]/1 − θ − ]). en, there exists a multiplicative convergent sequence in B b (þ°, r). Also, if one of the following conditions holds: (a) If B m (þ°, r) contains a sequence þ n such that B(þ n ) ≥ 1 for all n ∈ N and þ n ⟶ þ * ∈ B m (þ°, r) is continuous on B m (þ°, r) en, there exists a fixed point of g in B m (þ°, r). Moreover, if B(z) ≥ 1 and A(z) ≥ 1, for all z in the set of fixed points of g, then the fixed point of g will be unique.

Theorem 8. Let
where Q: [a, b] × [a, b] × R + ⟶ R + is an integrable function. Assume that the following conditions hold: (1) A( g°( u)) ≥ 1 and B( g°( u)) ≥ 1; (2) S is a cyclic (A, B)-admissible mapping on E g 0 ,r ; Also, if one of the following conditions holds: (5) S is continuous on E g 0 ,r or (6) If g n (u) is a sequence in E g 0 ,r such that g n (u) ⟶ g * (u) ∈ E g 0 ,r as n ⟶ ∞ and B(g n (u)) ≥ 1 for all n ∈ N, then B(g * (u)) ≥ 1. en, the integral equation (38) has a solution. Moreover, if A(g) ≥ 1 and B(g) ≥ 1 for all g in the set of fixed points of S, then equation (38) has a unique solution.