International Journal of Analysis and Applications Volume 16, Number 6 (2018), 904-920 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-904 FIXED POINT RESULTS OF RATIONAL TYPE CONTRACTIONS IN b-METRIC SPACES MIAN BAHADUR ZADA1, MUHAMMAD SARWAR1,∗ AND POOM KUMAM2,∗ 1Department of Mathematics University of Malakand, Khyber Pakhtunkhwa, Chakdara Dir(L), Pakistan 2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand ∗Corresponding authors: sarwar@uom.edu.pk, poom.kum@kmutt.ac.th Abstract. The aim of this manuscript is to establish fixed point results satisfying contractive conditions of rational type in the setting of b-metric spaces. The results proved herein are the generalization and extension of some well known results in the existing literature. Example is also given in order to illustrate the validity of the presented results. 1. Introduction and Preliminaries The Banach contraction principle [2] is considered to be the pioneering result of the fixed point theory, and plays an important role for solving existence problems in many branches of nonlinear analysis. This principle asserts if (X,d) is a complete metric space and K : X → X satisfies d(Kx,Ky) ≤ λd(x,y), (1.1) for all x,y ∈ X with λ ∈ [0, 1), then K has a unique fixed point. This principle have been improved and extended by several mathematicians in different directions some of them are as follows: Let K be a mapping on a metric space (X,d) and x,y ∈ X, then K is said to be Received 2018-06-11; accepted 2018-08-13; published 2018-11-02. 2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25. Key words and phrases. b–metric spaces; common fixed points; self-maps; Cauchy sequence; contractive conditions. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 904 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-904 Int. J. Anal. Appl. 16 (6) (2018) 905 (1) Kannan type contraction [10], if there exists a number λ ∈ [ 0, 1 2 ) such that d(Kx,Ky) ≤ λ[d(x,Kx) + d(y,Ky)]. (1.2) (2) Chatterjee type contraction [4], if there exists a number λ ∈ [ 0, 1 2 ) such that d(Kx,Ky) ≤ λ[d(x,Ky) + d(y,Kx)]. (1.3) (3) Reich type contraction [12], if there exists a number λ,µ,ν ∈ [0, 1) with λ + µ + ν < 1 such that d(Kx,Ky) ≤ λd(x,y) + µd(x,Kx) + νd(y,Ky). (1.4) (4) Das and Gupta [7] rational type contraction, if there exists a number λ,µ ∈ [0, 1) with λ + µ < 1 such that d(Kx,Ky) ≤ λd(x,y) + µ d(y,Ky)[1 + d(x,Kx)] 1 + d(x,y) . (1.5) The contractive conditions on underlying functions play an important role for finding solutions of metric fixed point problems. Inspired from the impact of this natural idea, the above contractions have been extended and generalized by several researchers in various spaces such as quasi-metric spaces, cone metric spaces, G-metric spaces, partial metric spaces and vector valued metric spaces etc. Following this trend, Bakhtin [1] and Czerwik [5] generalized metric space with non Hausdorff topology called b–metric space to overcome the problem of measurable functions with respect to measure and their convergence. They presented the generalization of the Banach contraction principle in b–metric spaces. Since then, several papers has been studied by many authors dealing with the existence of fixed point in b–metric spaces (see, [3, 6, 8, 9, 11, 13, 14] and the references therein). The aim of this contribution is to investigate some fixed point results using the concept of the contractive conditions of rational type in the context of b–metric spaces. Moreover, an example is given here to illustrate the validity of the obtained results. Actually the derived results generalizes the results of [2, 4, 7, 10, 12]. Now, we recall some essential notations, definitions and primary results known in the literature. Through- out this manuscript, R = set of real numbers, R+ = [0,∞) and N = set of positive integers. Definition 1.1. [1, 5] Let X be a nonempty set. A function d : X × X → R+ is called a b–metric with coefficient s ≥ 1 if: (1) d(x,y) = 0 ⇔ x = y; (2) d(x,y) = d(y,x) ∀ x,y ∈ X; (3) d(x,y) ≤ s[d(x,y) + d(z,y)] ∀ x,y,z ∈ X. Then the pair (X,d) is called a b-metric space. Int. J. Anal. Appl. 16 (6) (2018) 906 Remark 1.1. Every metric space is b–metric space with s = 1, but in general, a b–metric space need not necessarily be a metric space, as in below example 1.1, (X,d) is b–metric space but not a metric space (see also, examples in [6, 13]). Thus, the class of b–metric spaces is larger than the class of metric spaces. Example 1.1. Let X = R and let the mapping d : X × X → R+ be defined by d(x,y) = |x − y|2 for all x,y ∈ X. Then (X,d) is a b–metric space with coefficient s = 2. Sintunavarat [14] generalized Example 1.1 as: Example 1.2. Let (X,ρ) be a metric space and p ≥ 1 be a given real number. Then d(x,y) = [ρ(x,y)]p is a b–metric on X with parameter s ≤ 2p−1. The following example 1.3 shows that b-metric is not continuous in general (see also, examples in [9, 11]). Example 1.3. [8] Let X = N∪{∞} and d : X ×X → R be defined by d(m,n) =   0, if m = n,∣∣ 1 m + 1 n ∣∣ , if one of m,n is even and the other is even or ∞, 5, if one of m,n is odd and the other is odd (and m = n) or ∞, 2, otherwise. Then, considering all possible cases, it can be checked that (X,d) is a b–metric space with s = 5 2 . However, let xn = 2n for each n ∈ N. Then lim n→∞ d(2n,∞) = lim n→∞ 1 2n = 0, that is, xn →∞, but d(xn, 1) = 2 9 5 = d(∞, 1) as n →∞. Definition 1.2. [3] Let {xn} be a sequence in b–metric space (X,d) and x ∈ X. Then (1) {xn} converges to x if and only if for every ε > 0, there exists n(ε) ∈ N, such that d(xn,x) < ε for all n > n(ε) and we write lim n→∞ d(xn,x) = 0 or lim n→∞ xn = x. (2) {xn} is a Cauchy sequence if for every ε > 0, there exists n(ε) ∈ N, such that d(xn,xm) < ε for all m,n > n(ε). Proposition 1.1. [3] In a b–metric space (X,d), the following assertions hold: • a convergent sequence has a unique limit; • each convergent sequence is Cauchy; • a metric space (X,d) is complete if every Cauchy sequence is convergent in X. 2. Fixed point Results in b-metric spaces To present the main results, we need the following lemma. Int. J. Anal. Appl. 16 (6) (2018) 907 Lemma 2.1. Let (X,d) be a complete b-metric space and L : X → X. Let x0 ∈ X and define the sequence {xn} by Lxn = xn+1 ∀ n = 0, 1, 2, ... Let there exists a mapping λ : X ×X → [0, 1) satisfying λ(Lx,y) ≤ λ(x,y) and λ(x,Ly) ≤ λ(x,y), for all x,y ∈ X. Then λ(xn,y) ≤ λ(x0,y) and λ(x,xn+1) ≤ λ(x,x1) for all x,y ∈ X and n = 0, 1, 2, .... Proof. Let x,y ∈ X and n = 0, 1, 2, ..., then λ(xn,y) = λ(Lxn−1,y) ≤ λ(xn−1,y) = λ(Lxn−2,y) ≤ λ(xn−2,y) = ... ≤ λ(x0,y). Similarly, λ(x,xn+1) = λ(x,Lxn) ≤ λ(x,xn) = λ(x,Lxn−1)) = ... ≤ λ(x,x0). � Now, we prove the main result. Theorem 2.1. Let (X,d) be a complete b-metric space and λi : X×X → [0, 1), i = 1, 2, ..., 6. If L : X → X be a self-map such that for all x,y ∈ X the following conditions are satisfied: (i) λi(Lx,y) ≤ λi(x,y) and λi(x,Ly) ≤ λi(x,y); (ii) d(Lx,Ly) ≤λ1(x,y)d(x,y) + λ2(x,y) [d(x,Ly) + d(y,Lx)] s + λ3(x,y)[d(x,Lx) + d(y,Ly)] + λ4(x,y) d(y,Ly)[1 + d(x,Lx)] 1 + d(x,y) + λ5(x,y) d(x,Ly)d(x,Lx) s[1 + d(x,y)] + λ6(x,y) d(x,Ly)d(y,Lx) s[1 + d(x,y)d(y,Lx)] , where λ2(x,y) + λ3(x,y) + λ5(x,y) + s 6∑ i=1 λi(x,y) < 1, with 0 ≤ 6∑ i=1 λi(x,y) < 1. Then the mapping L has a unique fixed point in X. Proof. Let x0 ∈ X and construct a sequence {xn} by the rule Lxn = xn+1, ∀ n = 0, 1, 2, ... (2.1) First, we show that {xn} is a Cauchy sequence in X. For this, consider d(xn+1,xn+2) = d(Lxn,Lxn+1), Int. J. Anal. Appl. 16 (6) (2018) 908 by using condition (ii) of Theorem 2.1 with x = xn and y = xn+1, we have d(Lxn,Lxn+1) ≤λ1(xn,xn+1)d(xn,xn+1) + λ2(xn,xn+1) [d(xn,Lxn+1) + d(xn+1,Lxn)] s + λ3(xn,xn+1)[d(xn,Lxn) + d(xn+1,Lxn+1)] + λ4(xn,xn+1) d(xn+1,Lxn+1)[1 + d(xn,Lxn)] 1 + d(xn,xn+1) + λ5(xn,xn+1) d(xn,Lxn+1)d(xn,Lxn) s[1 + d(xn,xn+1)] + λ6(xn,xn+1) d(xn,Lxn+1)d(xn+1,Lxn) s[1 + d(xn,xn+1)d(xn+1,Lxn)] , using (2.1), we get d(xn+1,xn+2) ≤λ1(xn,xn+1)d(xn,xn+1) + λ2(xn,xn+1) [d(xn,xn+2) + d(xn+1,xn+1)] s + λ3(xn,xn+1)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(xn,xn+1) d(xn+1,xn+2)[1 + d(xn,xn+1)] 1 + d(xn,xn+1) + λ5(xn,xn+1) d(xn,xn+2)d(xn,xn+1) s[1 + d(xn,xn+1)] + λ6(xn,xn+1) d(xn,xn+2)d(xn+1,xn+1) s[1 + d(xn,xn+1)d(xn+1,xn+1)] , with the help of condition (i) of Theorem 2.1, we get d(xn+1,xn+2) ≤λ1(x0,x0)d(xn,xn+1) + λ2(x0,x0) [d(xn,xn+2) + d(xn+1,xn+1)] s + λ3(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(x0,x0) d(xn+1,xn+2)[1 + d(xn,xn+1)] 1 + d(xn,xn+1) + λ5(x0,x0) d(xn,xn+2)d(xn,xn+1) s[1 + d(xn,xn+1)] ≤λ1(x0,x0)d(xn,xn+1) + λ2(x0,x0) d(xn,xn+2) s + λ3(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(x0,x0)d(xn+1,xn+2) + λ5(x0,x0) d(xn,xn+2) s . Using triangular inequality, we get d(xn+1,xn+2) ≤λ1(x0,x0)d(xn,xn+1) + λ2(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ3(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(x0,x0)d(xn+1,xn+2) + λ5(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)], Int. J. Anal. Appl. 16 (6) (2018) 909 which implies that d(xn+1,xn+2) ≤ 3∑ i=1 λi(x0,x0) + λ5(x0,x0) 1 − 5∑ i=2 λi(x0,x0) d(xn,xn+1). Let h = 3∑ i=1 λi(x0,x0)+λ5(x0,x0) 1− 5∑ i=2 λi(x0,x0) < 1 s · Then d(xn+1,xn+2) ≤ hd(xn,xn+1). Similarly, d(xn,xn+1) ≤ hd(xn−1,xn). Consequently, d(xn+2,xn+1) ≤ hd(xn+1,xn) ≤ h2d(xn,xn−1) ≤ ... ≤ hn+1d(x1,x0). Now, for m > n and sh < 1, we have d(xn,xm) ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + ... + sm−nd(xm−1,xm), ≤ shnd(x1,x0) + s2hn+1d(x1,x0) + ... + sm−nhm−1d(x1,x0) ≤ [shn + s2hn+1 + ... + sm−nhm−1]d(x1,x0) ≤ shn [ 1 + (sh)1 + (sh)2 + ... + (sh)m−n−1 ] d(x1,x0) ≤ shn 1 −sh d(x1,x0). Therefore lim n→∞ d(xn,zm) = 0. Hence, {xn} is a cauchy sequence. But X is complete, so there exists t ∈ X such that xn → t as n →∞. Next, we show that t is a fixed point of L. For this, assume that Lt 6= t, then d(t,Lt) 6= 0. Now d(t,Lt) ≤ d(t,Lxn) + d(Lxn,Lt). (2.2) By applying condition (ii) of Theorem 2.1, equation (2.2) become d(t,Lt) ≤d(t,Lxn) + λ1(xn, t)d(xn, t) + λ2(xn, t) [d(xn,Lt) + d(t,Lxn)] s + λ3(xn, t)[d(xn,Lxn) + d(t,Lt)] + λ4(xn, t) d(t,Lt)[1 + d(xn,Lxn)] 1 + d(xn, t) + λ5(xn, t) d(xn,Lt)d(xn,Lxn) s[1 + d(xn, t)] + λ6(xn, t) d(xn,Lt)d(t,Lxn) s[1 + d(xn, t)d(t,Lxn)] , Int. J. Anal. Appl. 16 (6) (2018) 910 with the help of equation (2.1) and condition (i) of Theorem 2.1, we can write d(t,Lt) ≤d(t,xn+1) + λ1(x0, t)d(xn, t) + λ2(x0, t) [d(xn,Lt) + d(t,xn+1)] s + λ3(x0, t)[d(xn,xn+1) + d(t,Lt)] + λ4(x0, t) d(t,Lt)[1 + d(xn,xn+1)] 1 + d(xn, t) + λ5(x0, t) d(xn,Lt)d(xn,xn+1) s[1 + d(xn, t)] + λ6(x0, t) d(xn,Lt)d(t,xn+1) s[1 + d(xn, t)d(t,xn+1)] . Taking limit as n →∞, we get d(t,Lt) ≤λ2(x0, t) d(t,Lt) s + λ3(x0, t)d(t,Lt) + λ4(x0, t)d(t,Lt). d(t,Lt) ≤ [λ2(z0, t) + sλ3(z0, t) + sλ4(z0, t)] d(t,Lt) s . (2.3) But λ2(z0, t) + sλ3(z0, t) + sλ4(z0, t) < 1, so the above inequality (2.3) contradict the fact that d(t,Lt) 6= 0. Thus Lt = t. Hence t is a fixed point of L. Finally, we have to show that t is a unique fixed point of L. For this, let t∗ 6= t be another fixed point of L. Then on putting x = t and y = t∗ in condition (ii) of Theorem 2.1, we get d(t,t∗) =d(Lt,Lt∗) ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) [d(t,Lt∗) + d(t∗,Lt)] s + λ3(t,t ∗)[d(t,Lt) + d(t∗,Lt∗)] + λ4(t,t ∗) d(t∗,Lt∗)[1 + d(t,Lt)] 1 + d(t,t∗) + λ5(t,t ∗) d(t,Lt∗)d(t,Lt) s[1 + d(t,t∗)] + λ6(t,t ∗) d(t,Lt∗)d(t∗,Lt) s[1 + d(t,t∗)d(t∗,Lt)] ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) [d(t,t∗) + d(t∗, t)] s + λ6(t,t ∗) d(t,t∗)d(t∗, t) s[1 + d(t,t∗)d(t∗, t)] , implies that d(t,t∗) ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) 2d(t,t∗) s + λ6(t,t ∗) d(t,t∗)d(t∗, t) s[1 + d(t,t∗)d(t∗, t)] ≤[sλ1(t,t∗) + 2λ2(t,t∗) + λ6(t,t∗)] d(t,t∗) s , which is contradiction because sλ1(t,t ∗) + 2λ2(t,t ∗) + λ6(t,t ∗) < 1. Hence t is a unique fixed point of L. � From Theorem 2.1, we can easily derive the following corollaries and the proofs of which are simple, so we omit it. Corollary 2.1. Let (X,d) be a complete b-metric space and λi : X ×X → [0, 1), i = 1, 3. If L : X → X be a self-map such that for all x,y ∈ X the following conditions are satisfied: (i) λi(Lx,y) ≤ λi(x,y) and λi(x,Ly) ≤ λi(x,y); Int. J. Anal. Appl. 16 (6) (2018) 911 (ii) d(Lx,Ly) ≤ λ3(x,y)[d(x,Lx) + d(y,Ly)], where 0 ≤ λ3(x,y) < 1s+1 . Then the mapping L has a unique fixed point in X. Corollary 2.2. Let (X,d) be a complete b-metric space and λi : X×X → [0, 1), i = 1, 2, ..., 8. If L : X → X be a self-map such that for all x,y ∈ X the following conditions are satisfied: (i) λi(Lx,y) ≤ λi(x,y) and λi(x,Ly) ≤ λi(x,y); (ii) d(Lx,Ly) ≤ λ2(x,y) [d(x,Ly) + d(y,Lx)] s , where 0 ≤ λ2(x,y) < 1s+1 . Then the mapping L has a unique fixed point in X. Corollary 2.3. Let (X,d) be a complete b-metric space and λi : X ×X → [0, 1), i = 1, 4. If L : X → X be a self-map such that for all x,y ∈ X the following conditions are satisfied: (i) λi(Lx,y) ≤ λi(x,y) and λi(x,Ly) ≤ λi(x,y); (ii) d(Lx,Ly) ≤λ1(x,y)d(x,y) + λ4(x,y) d(y,Ly)[1 + d(x,Lx)] 1 + d(x,y) , where 0 ≤ sλ1(x,y) + λ4(x,y) < 1. Then the mapping L has a unique fixed point in X. Corollary 2.4. Let (X,d) be a complete b-metric space and λi : X ×X → [0, 1), i = 1, 2, 3. If L : X → X be a self-map such that for all x,y ∈ X the following conditions are satisfied: (i) λi(Lx,y) ≤ λi(x,y) and λi(x,Ly) ≤ λi(x,y); (ii) d(Lx,Ly) ≤ λ1(x,y)d(x,y) + λ2(x,y)d(x,Lx) + λ3(x,y)d(y,Ly), where 0 ≤ s[λ1(x,y) + λ2(x,y)] + λ3(x,y) < 1. Then the mapping L has a unique fixed point in X. Corollary 2.5. Let λ6 = 0 and all other conditions of Theorem 2.1 are satisfied, then L has a unique fixed point in X. Corollary 2.6. Let λ5 = λ6 = 0 and all other conditions of Theorem 2.1 are satisfied, then L has a unique fixed point in X. Corollary 2.7. Let λ2 = λ3 = 0 and all other conditions of Theorem 2.1 are satisfied, then L has a unique fixed point in X. Int. J. Anal. Appl. 16 (6) (2018) 912 Corollary 2.8. Let λ4 = λ5 = λ6 = 0 and all other conditions of Theorem 2.1 are satisfied, then L has a unique fixed point in X. Remark 2.1. (1) In Theorem 2.1, if s = 1,λi = 0, for i = 2, 3, 4, 5, 6 and λ1(·) = λ1, we get the Banach Theorem [2]. (2) In Corollary 2.1, if λi = 0, for i = 1, 2, 4, 5, 6, λ3(·) = λ and s = 1, we get the Kannan Theorem [10]. (3) In Corollary 2.2, if λi = 0, for i = 3, 4, 5, 6, λ2(·) = λ and s = 1, we get the Chatterjee Theorem [4]. (4) In Corollary 2.3, if λi = 0, for i = 2, 3, 5, 6, λj(·) = λj for j = 1, 4 and s = 1, we get the result of Dass and Gupta [7]. (5) In Corollary 2.4, if s = 1 and λi(·) = λi for i = 1, 2, 3, we get Theorem 3 of [12]. 3. Common Fixed point Results in b-metric spaces For the proof of our next result we use the following Lemma. Lemma 3.1. Let (X,d) be a complete b-metric space and K,L : X → X. Let x0 ∈ X and define the sequence {xn} by Kx2n = x2n+1 and Lx2n+1 = x2n+2 ∀ n = 0, 1, 2, ... Let there exists a mapping λ : X ×X → [0, 1) satisfying λ(LKx,y) ≤ λ(x,y) and λ(x,KLy) ≤ λ(x,y), for all x,y ∈ X. Then λ(x2n,y) ≤ λ(x0,y) and λ(x,x2n+1) ≤ λ(x,x1) for all x,y ∈ X. Proof. The proof easily follows from lemma 2.1 . � Our next result is proved for a pair of self-maps. Theorem 3.1. Let (X,d) be a complete b-metric space with s ≥ 1 and λi : X ×X → [0, 1), i = 1, 2, ..., 5. If K,L : X → X be two self-mappings such that for all x,y ∈ X the following conditions are satisfied: (i) λi(LKx,y) ≤ λi(x,y) and λi(x,KLy) ≤ λi(x,y); (ii) d(Kx,Ly) ≤λ1(x,y)d(x,y) + λ2(x,y) d(x,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ3(x,y) d(y,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ4(x,y) d(y,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] + λ5(x,y) d(x,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] ; Int. J. Anal. Appl. 16 (6) (2018) 913 where 5∑ i=2 λi(x,y) + s 5∑ i=1 λi(x,y) + 1 s [λ2(x,y) + λ4(x,y)] < 1, with 0 ≤ 5∑ i=1 λi(x,y) < 1. Then K and L have a unique common fixed point in X. Proof. Let x0 ∈ X and construct a sequence {xn} by the rule Kx2n = x2n+1 and Lx2n+1 = x2n+2, ∀ n = 0, 1, 2, ... (3.1) First we to show that {xn} is a Cauchy sequence in X. For this, consider d(x2k+1,x2k) = d(KLx2k−1,Lx2k−1). By using condition (ii) of Theorem 3.1 with x = Lx2k−1 and y = x2k−1, we have d(KLx2k−1,Lx2k−1) ≤λ1(Lx2k−1,x2k−1)d(Lx2k−1,x2k−1) +λ2(Lx2k−1,x2k−1) d(Lx2k−1,KLx2k−1)[d(Lx2k−1,Lx2k−1) + d(x2k−1,Lx2k−1)] s[1 + d(Lx2k−1,x2k−1)] +λ3(Lx2k−1,x2k−1) d(x2k−1,KLx2k−1)[d(Lx2k−1,Lx2k−1) + d(x2k−1,Lx2k−1)] s[1 + d(Lx2k−1,x2k−1)] +λ4(Lx2k−1,x2k−1) d(x2k−1,Lx2k−1)[d(Lx2k−1,KLx2k−1) + d(x2k−1,KLx2k−1)] s[1 + d(Lx2k−1,x2k−1)] +λ5(Lx2k−1,x2k−1) d(Lx2k−1,Lx2k−1)[d(Lx2k−1,KLx2k−1) + d(x2k−1,KLx2k−1)] s[1 + d(Lx2k−1,x2k−1)] ; By equation (3.1), we get d(x2k+1,x2k) ≤λ1(x2k,x2k−1)d(x2k,x2k−1) + λ2(x2k,x2k−1) d(x2k,x2k+1)[d(x2k,x2k) + d(x2k−1,x2k)] s[1 + d(x2k,x2k−1)] + λ3(x2k,x2k−1) d(x2k−1,x2k+1)[d(x2k,x2k) + d(x2k−1,x2k)] s[1 + d(x2k,x2k−1)] + λ4(x2k,x2k−1) d(x2k−1,x2k)[d(x2k,x2k+1) + d(x2k−1,x2k+1)] s[1 + d(x2k,x2k−1)] + λ5(x2k,x2k−1) d(x2k,x2k)[d(x2k,x2k+1) + d(x2k−1,x2k+1)] s[1 + d(x2k,x2k−1)] ≤λ1(x2k,x2k−1)d(x2k,x2k−1) + λ2(x2k,x2k−1) d(x2k,x2k+1)d(x2k−1,x2k) s[1 + d(x2k,x2k−1)] + λ3(x2k,x2k−1) d(x2k−1,x2k+1)d(x2k−1,x2k) s[1 + d(x2k,x2k−1)] + λ4(x2k,x2k−1) d(x2k,x2k+1) + d(x2k−1,x2k+1) s Int. J. Anal. Appl. 16 (6) (2018) 914 ≤λ1(x2k,x2k−1)d(x2k,x2k−1) + λ2(x2k,x2k−1) d(x2k,x2k+1) s + λ3(x2k,x2k−1) d(x2k−1,x2k+1) s + λ4(x2k,x2k−1) d(x2k,x2k+1) + d(x2k−1,x2k+1) s . From Lemma 3.1 and triangular inequality, we can write d(x2k+1,x2k) ≤λ1(x0,x1)d(x2k,x2k−1) + λ2(x0,x1) d(x2k,x2k+1) s + λ3(x0,x1)[d(x2k−1,x2k) + d(x2k,x2k+1)] + λ4(x0,x1) d(x2k,x2k+1) s + λ4(x0,x1)[d(x2k−1,x2k) + d(x2k,x2k+1)]. Finally one can get d(x2k+1,x2k) ≤ λ1(x0,x1) + λ3(x0,x1) + λ4(x0,x1) 1 − ( 1 s λ2(x0,x1) + λ3(x0,x1) + (1+s) s λ4(x0,x1) )d(x2k,x2k−1). Let h = 5∑ i=1 λi(x0,x1) 1−( (1+s)s λ2(x0,x1)+λ3(x0,x1)+ (1+s) s λ4(x0,x1)+λ5(x0,x1)) < 1 s · Then d(x2k+1,x2k) ≤ hd(x2k,x2k−1). (3.2) Similarly, consider d(x2k−1,x2k) = d(Kx2k−2,LKx2k−2). (3.3) By applying condition (ii) of Theorem 3.1 with x = x2k−2 and y = Kx2k−2 to equation (3.3) , we get d(Kx2k−2,LKx2k−2) ≤λ1(x2k−2,Kx2k−2)d(x2k−2,Kx2k−2) + λ2(x2k−2,Kx2k−2) d(x2k−2,Kx2k−2)[d(x2k−2,LKx2k−2) + d(Kx2k−2,LKx2k−2)] s[1 + d(x2k−2,Kx2k−2)] + λ3(x2k−2,Kx2k−2) d(Kx2k−2,Kx2k−2)[d(x2k−2,LKx2k−2) + d(Kx2k−2,LKx2k−2)] s[1 + d(x2k−2,Kx2k−2)] + λ4(x2k−2,Kx2k−2) d(Kx2k−2,LKx2k−2)[d(x2k−2,Kx2k−2) + d(Kx2k−2,Kx2k−2)] s[1 + d(x2k−2,Kx2k−2)] + λ5(x2k−2,Kx2k−2) d(x2k−2,LKx2k−2)[d(x2k−2,Kx2k−2) + d(Kx2k−2,Kx2k−2)] s[1 + d(x2k−2,Kx2k−2)] , Int. J. Anal. Appl. 16 (6) (2018) 915 with the help of (3.1), we get d(x2k−1,x2k) ≤λ1(x2k−2,x2k−1)d(x2k−2,x2k−1) + λ2(x2k−2,x2k−1) d(x2k−2,x2k−1)[d(x2k−2,x2k) + d(x2k−1,x2k)] s[1 + d(x2k−2,x2k−1)] + λ3(x2k−2,x2k−1) d(x2k−1,x2k−1)[d(x2k−2,x2k) + d(x2k−1,x2k)] s[1 + d(x2k−2,x2k−1)] + λ4(x2k−2,x2k−1) d(x2k−1,x2k)[d(x2k−2,x2k−1) + d(x2k−1,x2k−1)] s[1 + d(x2k−2,x2k−1)] + λ5(x2k−2,x2k−1) d(x2k−2,x2k)[d(x2k−2,x2k−1) + d(x2k−1,x2k−1)] s[1 + d(x2k−2,x2k−1)] ≤λ1(x2k−2,x2k−1)d(x2k−2,x2k−1) + λ2(x2k−2,x2k−1) d(x2k−2,x2k) + d(x2k−1,x2k) s + λ4(x2k−2,x2k−1) d(x2k−1,x2k) s + λ5(x2k−2,x2k−1) d(x2k−2,x2k) s . Using Lemma 3.1, one can get d(x2k−1,x2k) ≤λ1(x0,x1)d(x2k−2,x2k−1) + λ2(x0,x1)[d(x2k−2,x2k−1) + d(x2k−1,x2k)] + λ2(x0,x1) d(x2k−1,x2k) s + λ4(x0,x1) d(x2k−1,x2k) s + λ5(x0,x1)[d(x2k−2,x2k−1) + d(x2k−1,x2k)]. Finally, d(x2k−1,x2k) ≤ λ1(x0,x1) + λ2(x0,x1) + λ5(x0,x1) 1 − ( (1+s) s λ2(x0,x1) + 1 s λ4(x0,x1) + λ5(x0,x1) )d(x2k−2,x2k−1). Implies that d(x2k−1,x2k) ≤ hd(x2k−2,x2k−1). (3.4) Now, from equations (3.2) and (3.4), we have d(x2k+1,x2k) ≤ hd(x2k,x2k−1) ≤ h2d(x2k−1,x2k−2). Consequently, we can write d(xn+1,xn) ≤ hd(xn,xn−1) ≤ h2d(xn−1,xn−2) ≤ ... ≤ hnd(x1,x0). Now, for m > n and sh < 1, we have Int. J. Anal. Appl. 16 (6) (2018) 916 d(xn,xm) ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + ... + sm−nd(xm−1,xm), ≤ shnd(x1,x0) + s2hn+1d(x1,x0) + ... + sm−nhm−1d(x1,x0) ≤ [shn + s2hn+1 + ... + sm−nhm−1]d(x1,x0) ≤ shn [ 1 + (sh)1 + (sh)2 + ... + (sh)m−n−1 ] d(x1,x0) ≤ shn 1 −sh d(x1,x0). Therefore lim n→∞ d(xn,zm) = 0. Hence, {xn} is a Cauchy sequence. But X is complete, so there exists t ∈ X such that xn → t as n →∞. Next, to show that t is a fixed point of K. For this, consider d(t,Kt) ≤ d(t,Lx2n+1) + d(Lx2n+1,Kt). Using condition (ii) of Theorem 3.1 with x = t and y = x2n+1, we have d(t,Kt) ≤d(t,Lx2n+1) + λ1(t,x2n+1)d(t,x2n+1) + λ2(t,x2n+1) d(t,Kt)[d(t,Lx2n+1) + d(x2n+1,Lx2n+1)] s[1 + d(t,x2n+1)] + λ3(t,x2n+1) d(x2n+1,Kt)[d(t,Lx2n+1) + d(x2n+1,Lx2n+1)] s[1 + d(t,x2n+1)] + λ4(t,x2n+1) d(x2n+1,Lx2n+1)[d(t,Kt) + d(x2n+1,Kt)] s[1 + d(t,x2n+1)] + λ5(t,x2n+1) d(t,Lx2n+1)[d(t,Kt) + d(x2n+1,Kt)] s[1 + d(t,x2n+1)] . Using equation (3.1) and Proposition 3.1, we get d(t,Kt) ≤d(t,x2n+2) + λ1(t,x1)d(t,x2n+1) + λ2(t,x1) d(t,Kt)[d(t,x2n+2) + d(x2n+1,x2n+2)] s[1 + d(t,x2n+1)] + λ3(t,x1) d(x2n+1,Kt)[d(t,x2n+2) + d(x2n+1,x2n+2)] s[1 + d(t,x2n+1)] + λ4(t,x1) d(x2n+1,x2n+2)[d(t,Kt) + d(x2n+1,Kt)] s[1 + d(t,x2n+1)] + λ5(t,x1) d(t,x2n+2)[d(t,Kt) + d(x2n+1,Kt)] s[1 + d(t,x2n+1)] . Taking limit as n →∞, we get d(Kt,t) ≤ 0. Thus d(Kt,t) = 0 implies that Kt = t. Hence t is a fixed point of K. Analogously, using condition (ii) of Theorem 3.1 with x = x2n and y = t one can show that t is a fixed point of L. Therefore Kt = Lt = t, that is t is a common fixed point of K and L. Int. J. Anal. Appl. 16 (6) (2018) 917 Finally, we prove that t is a unique common fixed point of K and L. For this, suppose that t∗ 6= t be another fixed point of K and L. Then putting x = t and y = t∗ in condition (ii) of Theorem 3.1, we have d(Kt,Lt∗) ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) d(t,Kt)[d(t,Lt∗) + d(t∗,Lt∗)] s[1 + d(t,t∗)] + λ3(t,t ∗) d(t∗,Kt)[d(t,Lt∗) + d(t∗,Lt∗)] s[1 + d(t,t∗)] + λ4(t,t ∗) d(t∗,Lt∗)[d(t,Kt) + d(t∗,Kt)] s[1 + d(t,t∗)] + λ5(t,t ∗) d(t,Lt∗)[d(t,Kt) + d(t∗,Kt)] s[1 + d(t,t∗)] , which implies that d(t,t∗) ≤λ1(t,t∗)d(t,t∗) + λ3(t,t∗) d(t∗, t)d(t,t∗) s[1 + d(t,t∗)] + λ5(t,t ∗) d(t,t∗)d(t∗, t) s[1 + d(t,t∗)] ≤λ1(t,t∗)d(t,t∗) + λ3(t,t∗) d(t∗, t) s + λ5(t,t ∗) d(t,t∗) s ≤ [sλ1(t,t∗) + λ3(t,t∗) + λ5(t,t∗)] d(t,t∗) s . Which is contradiction because sλ1(t,t ∗) + λ3(t,t ∗) + λ5(t,t ∗) < 1, thus d(t∗, t) = 0 and hence t∗ = t. Therefore t is a unique common fixed point of K and L. � From Theorem 3.1, we can derive the following corollaries and the proof of which is simple, so we omit it. Corollary 3.1. If K = L and all other conditions of Theorem 3.1 are satisfied, then L has a unique fixed point in X. Corollary 3.2. Let (X,d) be a complete b-metric space with s ≥ 1 and λi : X × X → [0, 1), i = 1, 2. If K,L : X → X be two self-mappings such that for all x,y ∈ X the following conditions are satisfied: (i) λi(LKx,y) ≤ λi(x,y) and λi(x,KLy) ≤ λi(x,y); (ii) d(Kx,Ly) ≤ λ1(x,y)d(x,y) + λ2(x,y) d(x,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] ; where 0 ≤ sλ1(x,y) + ( s2+s+1 S ) λ2(x,y) < 1. Then K and L have a unique common fixed point in X. Corollary 3.3. Let (X,d) be a complete b-metric space with s ≥ 1 and λi : X × X → [0, 1), i = 1, 4. If K,L : X → X be two self-mappings such that for all x,y ∈ X the following conditions are satisfied: (i) λi(LKx,y) ≤ λi(x,y) and λi(x,KLy) ≤ λi(x,y); (ii) d(Kx,Ly) ≤ λ1(x,y)d(x,y) + λ4(x,y) d(y,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] ; Int. J. Anal. Appl. 16 (6) (2018) 918 where 0 ≤ sλ1(x,y) + ( s2+s+1 S ) λ4(x,y) < 1. Then K and L have a unique common fixed point in X. Corollary 3.4. Let (X,d) be a complete b-metric space with s ≥ 1 and λi : X ×X → [0, 1), i = 1, 2, 4. If K,L : X → X be two self-mappings such that for all x,y ∈ X the following conditions are satisfied: (i) λi(LKx,y) ≤ λi(x,y) and λi(x,KLy) ≤ λi(x,y); (ii) d(Kx,Ly) ≤λ1(x,y)d(x,y) + λ2(x,y) d(x,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ4(x,y) d(y,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] ; where 0 ≤ sλ1(x,y) + ( s2+1 S ) λ2(x,y) + ( s2+s+1 S ) λ4(x,y) < 1. Then K and L have a unique common fixed point in X. For the validity of Theorem 3.1, we construct the following example. Example 3.1. Let X = [0, 1] and d : X × X → R+ defined by d(x,y) = (α|x−y|)2 = α2|x − y|2 with α ≥ 8,s = 2. Define K,L : X → X by Kx = x 4 and Lx = x 5 . Let λi : X × X → [0, 1), i = 1, 2, ..., 5 are defined as: λ1(x,y) = x 19 + y 21 , λ2(x,y) = x 17 + y 23 , λ3(x,y) = x2 29 + y2 37 , λ4(x,y) = x3 + y3 41 , λ5(x,y) = xy 43 . To check condition (i), we have, since LKx = x 20 and KLy = y 20 . Then by routine calculation, one can easily obtained that λi(LKx,y) ≤ λi(x,y) and λi(x,KLy) ≤ λi(x,y) for all i = 1, 2, ..., 5; To check condition (ii), we have, d(Kx,Ly) =α2|Kx−Ly|2 = α2 ∣∣∣x 4 − y 5 ∣∣∣2 ≤ ( x 19 + y 21 ) α2|x−y|2 + ( x 17 + y 23 ) α4 ∣∣x− x 4 ∣∣2 [∣∣x− y 5 ∣∣2 + ∣∣y − y 5 ∣∣2] 2 [1 + α2|x−y|2] + ( x 29 + y 37 ) α4 ∣∣y − x 4 ∣∣2 [∣∣x− y 5 ∣∣2 + ∣∣y − y 5 ∣∣2] 2 [1 + α2|x−y|2] + ( x3 + y3 41 ) α4 ∣∣y − y 5 ∣∣2 [∣∣x− x 4 ∣∣2 + ∣∣y − x 4 ∣∣2] 2 [1 + α2|x−y|2] + (xy 43 ) α4 ∣∣x− y 5 ∣∣2 [∣∣x− x 4 ∣∣2 + ∣∣y − x 4 ∣∣2] 2 [1 + α2|x−y|2] Int. J. Anal. Appl. 16 (6) (2018) 919 d(Kx,Ly) =λ1(x,y)d(x,y) + λ2(x,y) d(x,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ3(x,y) d(y,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ4(x,y) d(y,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] + λ5(x,y) d(x,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] ; where 5∑ i=2 λi(x,y) + 2 5∑ i=1 λi(x,y) + 1 2 [λ2(x,y) + λ4(x,y)] < 1. Thus all the conditions of Theorem 3.1 are satisfied, hence K and L has a unique fixed point 0 ∈ X. To state the next result, we need the following Lemma the proof of which can be easily obtained from Lemma 2.1. Lemma 3.2. Let (X,d) be a complete b-metric space with s ≥ 1 and K,L : X → X. Let x0 ∈ X and define the sequence {xn} by Kx2n = x2n+1 and Lx2n+1 = x2n+2 ∀ n = 0, 1, 2, ... Assume that there exists a mapping λ : X → [0, 1) such that λ(LKx) ≤ λ(x), for all x ∈ X. Then λ(x2n) ≤ λ(x0) and λ(x2n+1) ≤ λ(x1) for all x ∈ X and n = 0, 1, 2, .... Theorem 3.2. Let (X,d) be a complete b-metric space with s ≥ 1 and λi : X → [0, 1), i = 1, 2, ..., 9. If K,L : X → X be two self-mappings such that for all x,y ∈ X the following conditions are satisfied: (i) λi(LKx) ≤ λi(x); (ii) d(Kx,Ly) ≤λ1(x)d(x) + λ2(x) d(x,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ3(x) d(y,Kx)[d(x,Ly) + d(y,Ly)] s[1 + d(x,y)] + λ4(x) d(y,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] + λ5(x) d(x,Ly)[d(x,Kx) + d(y,Kx)] s[1 + d(x,y)] ; where 5∑ i=2 λi(x) + s 5∑ i=1 λi(x) + 1 s [λ2(x) + λ4(x)] < 1, with 0 ≤ 5∑ i=1 λi(x) < 1. Then K and L have a unique common fixed point in X. Proof. By using Lemma 2.1 and following the same steps as in Theorem 3.1 one can easily prove the Theorem. � One can deduce corollaries from Theorem 3.2 in the same way as derived from Theorem 3.1. Int. J. Anal. Appl. 16 (6) (2018) 920 Acknowledgements The authors wish to thank the editor and anonymous referees for their comments and suggestions, which helped to improve this paper. References [1] I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., Unianowsk Gos. Ped. 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