International Journal of Analysis and Applications Volume 18, Number 3 (2020), 439-447 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-439 SOME FIXED POINT RESULTS FOR MULTIVALUED MAPPINGS IN b−MULTIPLICATIVE AND b−METRIC SPACE MAZHAR MEHMOOD∗, ABDULLAH SHOAIB AND HAMZA KHALID Department of Mathematics and Statistics,Riphah International University, Islamabad - 44000, Pakistan ∗Corresponding author: mazharm53@gmail.com Abstract. The main outcome of this paper is to introduce the notion of Hausdorff b-multiplicative metric space and to present some fixed point results for multivalued mappings in this space. Moreover, we obtain some fixed point results satisfying rational type contractive condition on closed ball for multivalued mappings in b-metric space.The proven results are original in nature. 1. Introduction Bourbaki and Bakhtin [6], were the first ones who gave the idea of b-metric. After that, Czerwik [7] gave an axiom and formally defined a b-metric space. For further results on b-metric space, see [11–13]. Ozaksar and Cevical [10] investigated multiplicative metric space and proved its topological properties. Mongkolkeha et al. [9] described the concept of multiplicative proximal contraction mapping and proved best proximity point theorems for such mappings. Recently, Abbas et al. [1] proved some common fixed points results of quasi weak commutative mappings on a closed ball in the setting of multiplicative metric spaces. They also describe the main conditions for the existence of common solution of multiplicative boundary value problem. For further results on multiplicative metric space, see [2, 3, 8]. In 2017, Ali et al. [4] introduced the notion of b-multiplicative and proved some fixed point result. As an application, they established an existence theorem for the solution of a system of Fredholm multiplicative integral equations. Shoaib et Received January 7th, 2020; accepted January 27th, 2020; published May 1st, 2020. 2010 Mathematics Subject Classification. 47H09, 47H10. Key words and phrases. fixed point; closed ball; b-multiplicative metric space; b-metric space; multivalued mapping; con- tractive condition. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 439 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-439 Int. J. Anal. Appl. 18 (3) (2020) 440 al. [13], discussed the result for fuzzy mappings on a closed ball in a b-metric space. For further results on closed ball, see [5, 12, 14, 15]. In this paper, we proved some fixed point results for multivalued mappings in b-multiplicative and b-metric space. 2. Preliminaries and Basic Definitions In this section we include some basic definitions and theorems which are useful to understand the results presented in this paper. First we give definition of b-metric space and its relevant results. Definition 2.1. [11] Let W be a non-empty set and s ≥ 1 be a real number. A mapping d : W × W → <+ ∪{0} is said to be b-metric with coefficient ”s”, if for all w,y,z ∈ W , the following conditions hold i. d(w,y) = 0 if and only if w = y. ii. d(w,y) = d(y,w). iii. d(w,z) ≤ s[d(w,y) + d(y,z)]. The pair (W,d) is called b-metric space. Example 2.1. [11] Let (W,d) be a metric space. Then for a real number k > 1, we define a function d1(a,b) = (d(a,b)) k, then d1 is a b-metric with b = 2 k−1. Definition 2.2. [11] Let (W,b) be a b-metric space. i. A sequence {wn} in (W,b) is called convergent if and only if there exists w ∈ W such that b(wn,w) → 0, as n → +∞. ii. A sequence {wn} in (W,b) is a Cauchy sequence, if and only if b(wn,wm) → 0, as n,m → +∞. iii. A b-metric space (W,b) is said to be complete if every Cauchy sequence in W converge to a point of W. Definition 2.3. Let (W,d) be a b-metric space. A be a non empty subset of W with some w0 ∈ W . An element a ∈ A is called a best approximation in A if d(wo,A) = d(wo,a), where d(w0,A) = infw∈A d(wo,w), if each wo ∈ W has at least one best approximation in A, then A is called a proximinal set. We denote P(W), the set of all closed proximinal subsets of W . Definition 2.4. Let (W,d) be b-metric space. The function H : P(W) ×P(W) →<+,defined by H(A,B) = max{supw∈A d(w,B), supy∈B d(A,y)}, is called Hausdorff b-metric on P(W). Lemma 2.1. Let (W,d) be b-metric space. Let H be a Hausdorff b-metric on P(W). Then for all A,B ∈ P(W) and for each w ∈ A there exist y ∈ B satisfying d(w,B) = d(w,y) then H(A,B) ≥ d(w,y). Now, we include the definition of b-multiplicative metric space and its relevant results. Int. J. Anal. Appl. 18 (3) (2020) 441 Definition 2.5. [4] Let W be a non-empty set and let s ≥ 1 be a given real number. A mapping m : W ×W → [1,∞) is called a b-multiplicative metric with coefficient s, if the following conditions hold: i. m(w,y) > 1 for all w,y ∈ W with w 6= y and m(w,y) = 1 if and only if w = y. ii. m(w,y) = m(y,w) for all w,y ∈ W . iii. m(w,z) ≤ [m(w,y).m(y,z)]s for all w,y,z ∈ W . The triplet (W,m,s) is called b-multiplicative metric space. Example 2.2. [4] Let W = [ 0,∞) . Define a mapping ma : W ×W → [ 1,∞) , ma(w,y) = a(w−y) 2 , where a > 1 is any fixed real number. Then for each a, ma is b-multiplicative metric on W with s = 2. Note that ma is not a multiplicative metric on W . Definition 2.6. [4] Let (W,m) be a b-multiplicative metric space. i. A sequence {wn} is convergent iff there exists w ∈ W , such that m(wn,w) → 1, as n → +∞. ii. A sequence {wn} is called b-multiplicative Cauchy, iff m(wm,wn) → 1, as m,n → +∞. iii. A b-multiplicative metric space (W,m) is said to be complete if every multiplicative Cauchy sequence in W is convergent to some y ∈ W . Definition 2.7. Let (W,d) be a b-multiplicative metric space. A be a non empty subset of W with some wo ∈ W . An element a ∈ A is called a best approximation in A if d(wo,A) = d(wo,a), where d(wo,A) = infw∈A d(wo,w), if each wo ∈ W has at least one best approximation in A, then A is called a proximinal set. We denote P(W), the set of all closed proximinal subsets of W . Definition 2.8. Let (W,d) be a b-multiplicative metric space. The function H : P(W) × P(W) → <+, defined by H(A,B) = max{supw∈A d(w,B), supy∈B d(A,y)}, is called Hausdorff b-multiplicative metric on P(W). Lemma 2.2. Let (W,d) be a b-multiplicative metric. Let H be a Hausdorff b-multiplicative metric on P(W). Then for all A,B ∈ P(W) and for each w ∈ A there exist y ∈ B satisfying d(w,B) = d(w,y) then H(A,B) ≥ d(w,y). Definition 2.9. Let (W,d) be a complete b-multiplicative metric and w0 ∈ W and S : W → P(W) be the multivalued mapping on W , then there exist w1 ∈ Sw0 be an element such that d(w0,Sw0) = d(w0,w1). Let w2 ∈ Sw1 be such that d(w1,Sw1) = d(w1,w2). Let w3 ∈ Sw2 be such that d(w2,Sw2) = d(w2,w3). Continuing this process, we construct a sequence {wn} of points in W such that wn+1 ∈ Swn, d(wn,Swn) = d(wn,wn+1). We denote this iterative sequence by {WS(wn)}. We say that {WS(wn)} is a sequence in W generated by w0. Int. J. Anal. Appl. 18 (3) (2020) 442 3. Results for b−Multiplicative Metric Space Definition 3.1. Let (W,d) be a b-multiplicative metric space and S : W → P(W) be the multivalued mapping. Let w0 ∈ W and {WS(wn)} be a sequence in W generated by w0. We define the family M(S) of all functions a : W ×W → [ 0, 1) which satisfy the following property a( wn,wn+1) ≤ a( w0,w1) , for all n ∈ N ∪{0}. Also, if {WS(wn)}→ h, then a(wn,h) ≤ a(w0,h). Theorem 3.1. Let (W,d) be a complete b-multiplicative metric space with coefficient s, S : W → P(W) be a multivalued mapping on W and β,σ,ψ ∈ M(S). If the following relations hold: H(Swn,Swn+1) ≤ [d(wn,wn+1)]β(wn,wn+1).[d(wn,Swn+1).d(wn+1,Swn)]σ(wn,wn+1) .[d(wn,Swn).d(wn+1,Swn+1)] ψ(wn,wn+1), (3.1) for all wn,wn+1 ∈{WS(wn)}, n ∈ N ∪{0}, a,b ≥ 0, and sβ(w0,w1) + (s 2 + s)σ(w0,w1) + (s + 1)ψ(w0,w1) < 1 for w0,w1 ∈{WS(wn)}, (3.2) then {WS(wn)}→ w∗ ∈ W . Also, if inequalities 3.1 and 3.2 hold for h, then S has a fixed point w∗. Proof. Considering a sequence {WS(wn)} in W generated by w0, then we have wn+1 ∈ Swn, where n = 0, 1, 2, · · · now by using Lemma 2.2, we can write d(wn,wn+1) ≤ H(Swn−1,Swn) ≤ [d(wn−1,wn)]β(wn−1,wn).[d(wn−1,Swn).d(wn,Swn−1)]σ(wn−1,wn) .[d(wn−1,Swn−1).d(wn,Swn)] ψ(wn−1,wn) ≤ [d(wn−1,wn)]β(wn−1,wn).[d(wn−1,wn+1).d(wn,wn)]σ(wn−1,wn) .[d(wn−1,wn).d(wn,wn+1)] ψ(wn−1,wn) by using the Definition 3.1 and triangle inequality, we can write d(wn,wn+1) ≤ [d(wn−1,wn)]β(w0,w1).[d(wn−1,wn).d(wn,wn+1)]sσ(w0,w1) .[d(wn−1,wn).d(wn,wn+1)] ψ(w0,w1).[d(wn,wn+1)] 1−sσ(w0,w1)−ψ(w0,w1) ≤ [d(wn−1,wn)]β(w0,w1)+sσ(w0,w1)+ψ(w0,w1) d(wn,wn+1) ≤ [d(wn−1,wn)] β(w0,w1) + sσ(w0,w1) + ψ(w0,w1) 1 −sσ(w0,w1) −ψ(w0,w1) = [d(wn−1,wn)]K. (3.3) Now, d(wn−1,wn) ≤ H(Swn−2,Swn−1) ≤ [d(wn−2,wn−1)]β(wn−2,wn−1).[d(wn−2,Swn−1).d(wn−1,Swn−2)]σ(wn−2,wn−1) .[d(wn−2,Swn−2).d(wn−1,Swn−1)] ψ(wn−2,wn−1) ≤ [d(wn−2,wn−1)]β(w0,w1).[d(wn−2,wn−1).d(wn−1,wn)]sσ(w0,w1).[d(wn−2,wn−1).d(wn−1,wn)]ψ(w0,w1) Int. J. Anal. Appl. 18 (3) (2020) 443 [d(wn−1,wn)] 1−sσ(w0,w1)−ψ(w0,w1) ≤ [d(wn−2,wn−1)]β(w0,w1)+sσ(w0,w1)+ψ(w0,w1) d(wn−1,wn) ≤ [d(wn−2,wn−1)] β(w0,w1) + sσ(w0,w1) + ψ(w0,w1) 1 −sσ(w0,w1) −ψ(w0,w1) = [d(wn−2,wn−1)]K. (3.4) From 3.3 and 3.4, we can write d(wn,wn+1) ≤ [d(wn−1,wn)]K ≤ [d(wn−2,wn−1)K]K = [d(wn−2,wn−1)]K 2 ≤ [d(wn−3,wn−2)]K 3 ≤ ···≤ [d(w0,w1)]K n (3.5) Now, for m,n ∈ N, with m > n, we have d(wn,wm) ≤ d(wn,wn+1)s.d(wn+1,wn+2)s 2 · · · .d(wm−2,wm−1)s m−n−1 .d(wm−1,wm) sm−n by using the inequality 3.5, we have d(wn,wm) ≤ d(wn,wn+1)s.d(wn+1,wn+2)s 2 · · · .d(wm−2,wm−1)s m−n−1 .d(wm−1,wm) sm−n ≤ [d(w0,w1)]sK n .[d(w0,w1)] s2Kn+1 · · · .[d(w0,w1)]s m−n−1Km−2.[d(w0,w1)] sm−nKm−1 ≤ [d(w0,w1)]sK n(1+sK+(sK)2+···+sm−n−2Km−n−2+sm−n−1Km−n−1) ≤ [d(w0,w1)]sK n(1+sK+(sK)2+···+(sK)m−n−2+(sK)m−n−1) < [d(w0,w1)] sKn(1+sK+(sK)2+···) = [d(w0,w1)] sKn( 1 1−sK ). Taking limm,n→∞, we get d(wn,wm) → 1. Hence {WS(wn)} is a b-multiplicative Cauchy sequence. By completeness of (W,d), we have wn → w∗ ∈ W . Also lim n→∞ d(wn,w ∗) = 1. (3.6) Now, d(w∗,Sw∗) ≤ [d(w∗,wn+1).d(wn+1,Sw∗)]s ≤ d(w∗,wn+1)s.H(Swn,Sw∗)s ≤ d(w∗,wn+1)s.[d(wn,w∗)]sβ(wn,w ∗).[d(wn,Sw ∗).d(w∗,Swn)] sσ(wn,w ∗).[d(wn,Swn).d(w ∗,Sw∗)]sψ(wn,w ∗) ≤ d(w∗,wn+1)s.[d(wn,w∗)]sβ(wn,w ∗).[d(wn,w ∗).d(w∗,Sw∗)]s 2σ(wn,w ∗) .d(w∗,wn+1) sσ(wn,w ∗).[d(wn,wn+1).d(w ∗,Sw∗)]sψ(wn,w ∗). By using Definition 2.8, we can write d(w∗,Sw∗) ≤ d(w∗,wn+1)s.[d(wn,w∗)]sβ(w0,w ∗).[d(wn,w ∗).d(w∗,Sw∗)]s 2σ(w0,w ∗).d(w∗,wn+1) sσ(w0,w ∗) .[d(wn,wn+1).d(w ∗,Sw∗)]sψ(w0,w ∗). On taking limn→∞ and by using inequality 3.6, we get d(w∗,Sw∗) ≤ [d(w∗,Sw∗)]s 2σ(w0,w ∗).[d(w∗,Sw∗)]sψ(w0,w ∗) [d(w∗,Sw∗)]1−s 2σ(w0,w ∗)−sψ(w0,w∗) ≤ 1 d(w∗,Sw∗) ≤ (1) 1 1 −s2σ(w0,w∗) −sψ(w0,w∗) ≤ 1. This implies that d(w∗,Sw∗) = 1 and hence w∗ is a fixed point of mapping S. � Int. J. Anal. Appl. 18 (3) (2020) 444 Theorem 3.2. Let (W,d) be a complete b-multiplicative metric space with coefficient s, S : W → P(W) be the multivalued mapping on W and β ∈ M(S). If the following relations hold: H(Swn,Swn+1) ≤ [d(wn,wn+1)]β(wn,wn+1), (3.7) for all wn,wn+1 ∈{WS(wn)}, n ∈ N ∪{0},a,b ≥ 0 and sβ(w0,w1) < 1 for w0,w1 ∈{WS(wn)}, then {WS(wn)}→ w∗ ∈ W. (3.8) Also if inequalities 3.7 and 3.8 hold for h, then S has a fixed point w∗. Theorem 3.3. Let (W,d) be a complete b-multiplicative metric space with coefficient s, S : W → P(W) be the multivalued mapping on W and σ ∈ M(S). If the following relations hold: H(Swn,Swn+1) ≤ [d(wn,Swn+1).d(wn+1,Swn)]σ(wn,wn+1), (3.9) for all wn,wn+1 ∈{WS(wn)}, n ∈ N ∪{0}, a,b ≥ 0 and (s2 + s)σ(w0,w1) < 1 for w0,w1 ∈{WS(wn)}, then {WS(wn)}→ w∗ ∈ W. (3.10) Also if inequalities 3.9 and 3.10 hold for h, then S has a fixed point w∗. Theorem 3.4. Let (W,d) be a complete b-multiplicative metric space with coefficient s, S : W → P(W) be the multivalued mapping on W and ψ ∈ M(S). If the following relations hold: H(Swn,Swn+1) ≤ [d(wn,Swn).d(wn+1,Swn+1)]ψ(wn,wn+1), (3.11) for all wn,wn+1 ∈{WS(wn)}, n ∈ N ∪{0}, a,b ≥ 0 (s + 1)ψ(w0,w1) < 1 for w0,w1 ∈{WS(wn)}, then {WS(wn)}→ w∗ ∈ W. (3.12) Also if inequalities 3.11 and 3.12 hold for h, then S has a fixed point w∗ 4. Results for b−Metric Space Theorem 4.1. Let (W,d) be a complete b-metric space with coefficient s and w0 be any point in W . Let the mapping S : W → P(W) satisfy the following relations: H(Swn,Swn+1) ≤ a1d(wn,wn+1) + a2[ a + d(wn,Swn) b + d(wn,wn+1) ]d(wn,Swn) + a3[ c + d(wn,wn+1) d ′ + d(wn,Swn) ]d(wn+1,Swn+1) + a4[d(wn,Swn+1) + d(wn+1,Swn)], (4.1) for all wn,wn+1 ∈ B(w0; r) ∩{WS(w0)} and a,b,c,d ′ ,a1,a2,a3,a4 > 0 with a ≤ b, c ≤ d ′ . Also d(w0,Sw0) ≤ β(1 −sβ)r, where β = a1 + a2 + sa4 1 −a3 −sa4 , r > 0 and sβ < 1. (4.2) Int. J. Anal. Appl. 18 (3) (2020) 445 Then {WS(wn)} is a sequence in B(w0; r) and {WS(wn)}→ h ∈ B(w0; r). Also if inequality 4.1 holds for h, then S has a fixed point h in B(w0; r). Proof. Considering a sequence {WS(wn)} in W generated by w0, then, we have wn+1 ∈ Swn, where n=0,1,2,... From 4.2, we have d(w0,w1) ≤ β(1 − sβ)r ≤ r. This implies w1 ∈ B(w0; r). Now by using Lemma 2.1 and inequality 4.1, we can write d(w1,w2) = d(w1,Sw1) ≤ H(Sw0,Sw1) ≤ a1d(w0,w1) + a2[ a + d(w0,Sw0) b + d(w0,w1) ]d(w0,Sw0) + a3[ c+d(w0,w1) d ′ +d(w0,Sw0) ]d(w1,Sw1) +a4[d(w0,Sw1) + d(w1,Sw0)] ≤ a1d(w0,w1) + a2[ a + d(w0,w1) b + d(w0,w1) ]d(w0,w1) + a3[ c+d(w0,w1) d ′ +d(w0,w1) ]d(w1,w2) + sa4d(w0,w1) +sa4d(w1,w2). As a ≤ b and c ≤ d ′ , we have [1−a3−sa4]d(w1,w2) ≤ [a1+a2+sa4]d(w0,w1)d(w1,w2) ≤ [a1+a2+sa41−a3−sa4 ]d(w0,w1) ≤ βd(w0,w1) ≤ β 2(1−sβ)r. Now, d(w0,w2) ≤ s[d(w0,w1)+d(w1,w2)] ≤ s[β(1−sβ)r+β2(1−sβ)r] ≤ βs(1−sβ)(1+β)r ≤ βs(1−sβ)(1+sβ)r ≤ βs[1 − (sβ)2]r ≤ r. This implies w2 ∈ B(w0; r). Considering w3,w4,w5, · · · ,wj ∈ B(w0; r). Now, d(wj,wj+1) ≤ H(Swj−1,Swj) ≤ a1d(wj−1,wj) + a2[ a + d(wj−1,Swj−1) b + d(wj−1,wj) ]d(wj−1,Swj−1) + + a3[ c + d(wj−1,wj) d ′ + d(wj−1,Swj−1) ]d(wj,Swj) + a4[d(wj−1,Swj) + d(wj,Swj−1)] ≤ a1d(wj−1,wj) + a2[ a + d(wj−1,wj) b + d(wj−1,wj) ]d(wj−1,wj) + a3[ c + d(wj−1,wj) d ′ + d(wj−1,wj) ]d(wj,wj+1) + sa4[d(wj−1,wj) + d(wj,Swj+1)]. As a ≤ b and c ≤ d ′ , we have (1 −a3 −sa4)d(wj,wj+1) ≤ (a1 + a2 + sa4)d(wj−1,wj)d(wj,wj+1) ≤ [ a1 + a2 + sa4 1 −a3 −sa4 ]d(wj−1,wj) = βd(wj−1,wj) ≤ β[βd(wj−2,wj−1)] ≤ β2[βd(wj−3,wj−2)]. (1 −a3 −sa4)d(wj,wj+1) ≤ βjd(w0,w1) ∀ j ∈ N (4.3) Now, d(w0,wj+1) ≤ sd(w0,w1) + s2d(w1,w2) + · · · + sjd(wj,wj+1) ≤ sd(w0,w1) + s2βd(w′,w1) + · · · + sj+1βjd(w0,w1) ≤ s(1 + sβ + (sβ)2 + · · · + (sβ)jd(w0,w1)) ≤ s[ 1 − (sβ)j 1 −sβ ]β(1 −sβ)r ≤ sβ[1 − (sβ)j]r ≤ r. Thus wj+1 ∈ B(w0; r). Hence wn ∈ B(w0; r) for all n ∈ N∪{0}, therefore {WS(wn)} is a sequence in B(w0; r). Now, the inequality 4.3 can be written as d(wn,wn+1) ≤ βnd(w0,w1) for all n ∈ N. (4.4) Hence for any m > n d(wn,wm) ≤ sd(wn,wn+1) + s2d(wn+1,wn+2) + · · · + sm−1d(wm−1,wm) ≤ [sβn + s2βn+1 + · · · + Int. J. Anal. Appl. 18 (3) (2020) 446 sm−nβm−1]d(w0,w1) by using 4.4 ≤ sβn[1 + sβ + (sβ)2 + · · · + sm−n−1βm−n−1]d(w0,w1) ≤ sβn[1 + sβ + (sβ)2 + · · · ]d(w0,w1) ≤ [ (sβ)n 1−sβ ]d(w0,w1) → 0 as m, n → ∞. Thus, we prove that {WS(wn)} is a Cauchy sequence in B(w0; r). As every closed ball in complete b-metric space is complete, so there exists h ∈ B(w0; r) such that {WS(wn)} → h or limn→∞d(wn,h) = 0. If h ∈ Sh, the desired result is obvious and straight forward. If h /∈ Sh then d(h,Sh) = z > 0, that is d(h,Sh) ≤ s[d(h,wn+1)+d(wn+1,Sh)] ≤ s[d(h,wn+1)+H(Swn,Sh)] ≤ sd(h,wn+1)+sa1d(wn,h)+a2[ a+d(wn,Swn) b+d(wn,h) ]d(wn,Swn)+a3[ c+d(wn,h) d ′ +d(wn,Swn) ]d(h,Sh)+a4[d(wn,Sh)+d(h,Swn)] ≤ sd(h,wn+1) + sa1d(wn,h) + a2[ a+d(wn,wn+1) b+d(wn,h) ]d(wn,wn+1) + a3[ c+d(wn,h) d ′ +d(wn,wn+1) ]d(h,Sh) + sa4[d(wn,h) + d(h,Sh)]. Taking n →∞, it follows that d(h,Sh) ≤ a3d(h,Sh) + sa4d(h,Sh) (1 −a3 −sa4)d(h,Sh) ≤ 0. This implies d(h,Sh) = z ≤ 0, a contradiction, so h ∈ Sh. Hence proved � Corollary 4.1. Let (W,d) be a complete metric space and w0 be any point in W . Let S : W → W be a mapping and wn = Swn−1 be a Picard sequence. If d(Swn,Swn+1) ≤ a1d(wn,wn+1) + a2[ a + d(wn,Swn) b + d(wn,wn+1) ]d(wn,Swn) + a3[ c + d(wn,wn+1) d ′ + d(wn,Swn) ]d(wn+1,Swn+1) + a4[d(wn,Swn+1) + d(wn+1,Swn)], (4.5) for all wn,wn+1 ∈ B(w0; r) ∪{wn} and a,b,c,d ′ ,a1,a2,a3,a4 > 0 with a ≤ b, c ≤ d ′ . Also d(w0,Sw0) ≤ β(1 −β)r, where β = a1+a2+a4 1−a3−a4 , β < 1. Then {wn} is a sequence in B(w0; r) and wn → h ∈ B(w0; r). Also if inequality 4.5 holds for h, then S has a fixed point h in B(w0; r). Corollary 4.2. Let (W,d) be a complete b-metric space with coefficient s and w0 be any point in W . Let the mapping S : W → P(W) satisfy the following: H(Swn,Swn+1) ≤ a1d(wn,wn+1) + a2[( a + d(wn,Swn) b + d(wn,wn+1) ]d(wn,Swn) + a3[ c + d(wn,wn+1) d ′ + d(wn,Swn) ]d(wn+1,Swn+1) + a4[d(wn,Swn+1) + d(wn+1,Swn)], (4.6) for all wn,wn+1 ∈{WS(wn)} and a,b,c,d ′ ,a1,a2,a3,a4 > 0 with sa1 + sa2 + a3 + (s 2 + s)a4 < 1, and a ≤ b, c ≤ d ′ . Then {WS(wn)}→ h ∈ W . Also if inequality 4.6 holds for h, then S has a fixed point h in W . Authors Contribution: All authors contributed equally and signifcantly in writing this paper. All authors read and approved the origenal manuscript. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. Int. J. Anal. Appl. 18 (3) (2020) 447 References [1] M. Abbas, B. Ali, and YI. Suleiman, Common Fixed Points of Locally Contractive Mappings in Multiplicative Metric Spaces with Application, Int. J. Math. Math. Sci. 2015 (2015), Article ID 218683. [2] M. Abbas, M. D. Sen, and T. 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