International Journal of Analysis and Applications Volume 18, Number 1 (2020), 33-49 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-33 AN EXTENDED S-ITERATION SCHEME FOR G-CONTRACTIVE TYPE MAPPINGS IN b-METRIC SPACES WITH GRAPH NILAKSHI GOSWAMI1, NEHJAMANG HAOKIP1, VISHNU NARAYAN MISHRA2,∗ 1Department of Mathematics, Gauhati University, Guwahati - 781014, India 2Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur 484 887, India ∗Corresponding author: vishnunarayanmishra@gmail.com Abstract. In this paper, we introduce an extended S-iteration scheme for G-contractive type mappings and prove ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph. We also give a numerical example in support of our result and compare the convergence rate between the studied iteration and the modified S-iteration. 1. Introduction In 1922, Banach gave the proof of a fixed point result, which later on came to be known as the celebrated Banach contraction principle. He showed that a contraction mapping T on a complete metric space (X,d) has a unique fixed point. Moreover, for an arbitrary point x0 in X, the sequence of Picard iterates given by the relation xn = Txn−1 n = 1, 2, 3, . . . (1.1) converges to the unique fixed point. In the last few decades, many authors have extended this result by considering a more generalized space, altering the condition of the contraction or by considering different Received 2019-05-01; accepted 2019-09-30; published 2020-01-02. 2010 Mathematics Subject Classification. Primary 47H10, Secondary 54E50. Key words and phrases. b-metric space; extended S-iteration scheme; G-contractive type mapping; ∆-convergence; strong convergence. c©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 33 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-33 Int. J. Anal. Appl. 18 (1) (2020) 34 iteration processes (one may refer to [6]– [8], [9], [10], [13], [14], [19], [21]– [24], [26], [27], [28], [34], [29]– [33] and the references therein). The increasing interest in the study of iteration schemes is accelerated by the advancement in computa- tional mathematics aided by computer programming. We list some of the prominent iteration schemes which are generalizations of (1.1). For x0 ∈ X, the iteration scheme given by xn+1 = (1 −αn)xn + αnTxn, n = 0, 1, 2, . . . where {αn}⊂ [0, 1] is called the Mann iteration scheme (refer to [20]). For x0 ∈ X, the iteration scheme given by xn+1 = (1 −αn)xn + αnTyn yn = (1 −βn)xn + βnTxn, n = 0, 1, 2, . . . where {αn} and {βn} are sequences in [0, 1] is called the Ishikawa iteration scheme (refer to [15]). In 1976, Jungck [16] proved a common fixed point theorem for a pair of mappings S and T satisfying d(Tx,Ty) ≤ αd(Sx,Sy) with α ∈ (0, 1) and T(X) ⊂ S(X) in a complete metric space. For x0 in a linear space X, the sequence {Sxn} defined by Sxn+1 = αnSxn + (1 −αn)Txn, n = 0, 1, 2, . . . where {αn} is a sequence in [0, 1] is called the Jungck-Mann iteration scheme (refer to [39]). If αn = 0, we get the Jungck iteration scheme. For x0 ∈ X, the sequence {Sxn} defined by Sxn+1 = (1 −αn)Sxn + αnTyn Syn = (1 −βn)Sxn + βnTxn, n = 0, 1, 2, . . . where {αn} and {βn} are sequences in [0, 1] is called the Jungck-Ishikawa iteration scheme (refer to [25]). In 2007, Agarwal et al. [2] introduced the S-iteration scheme and studied its convergence. For a convex subset K of a linear space X and a self mapping T on K, the iterative sequence {xn} of the S-iteration scheme is generated from x1 ∈ K, and is defined by xn+1 = (1 −λn)Txn + λnTyn, yn = (1 −µn)xn + µnTxn, n ∈ N where {λn} and {µn} are real sequences in (0, 1), satisfying ∞∑ n=1 λnµn(1 −µn) = ∞. Int. J. Anal. Appl. 18 (1) (2020) 35 Recently, Suparatulatorn et al. [40] introduced the modified S-iteration scheme for G-nonexpansive map- pings S1 and S2 in Banach spaces with graphs. Here, for x0 ∈ K, n ≥ 0  xn+1 = (1 −λn)S1xn + λnS2yn, yn = (1 −µn)xn + µnS1xn (1.2) where {λn} and {µn} are sequences in (0, 1). Motivated by [40], in this paper, we consider a convex b-metric space (X,d) with graph and define an extended S-iteration scheme for a triplet of three G-contractive type self mappings on a nonempty closed convex subset K of X. The convergence of this iteration scheme in comparison to the existing modified S-iteration scheme is also discussed with a numerical example. In the following we reproduce the concepts of some of the terms used in this paper. Definition 1.1. [5], Let X be a non empty set and s ≥ 1 be a given real number. A function d : X×X −→ [0,∞) is called a b-metric if it satisfies the following properties. (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x); and (3) d(x,z) ≤ s[d(x,y) + d(y,z)], for all x,y,z ∈ X. The pair (X,d) is called a b-metric space with coefficient s. In 1970, Takahashi [41] introduced the following concept of convex structure in a metric space. Definition 1.2. [41] Let (X,d) be a metric space. A mapping W : X2 × [0, 1] −→ X satisfying d (z,W(x,y,t)) ≤ td(z,x) + (1 − t)d(z,y) for all x,y,z ∈ X and t ∈ [0, 1] is called a convex structure on X. The above notion of convex structure can as well be extended naturally to b-metric spaces with the condition sd (z,W(x,y,t)) ≤ td(z,x) + (1 − t)d(z,y). (1.3) Kirk & Ray [17], in 1977, defined a metric space (X,d) to be metrically convex or simply convex if for every distinct elements x and y in X, there exists z in X, distinct from x and y such that d(x,y) = d(x,z) + d(z,y). A natural extension of this notion to b-metric spaces is by the equation d(x,y) = s [d(x,z) + d(z,y)] . Int. J. Anal. Appl. 18 (1) (2020) 36 The mapping W satisfies d(x,y) = s { d (x,W(x,y,t)) + d (x,W(x,y,t)) } for all x,y ∈ X and t ∈ [0, 1] (refer to [41]). This can be seen in proving the following assertion as is done in [1] for metric spaces. If (X,d) is a b-metric space on which a convex structure W is defined, then for all x,y ∈ X and t ∈ [0, 1], sd (x,W(x,y,t)) = td(x,y) and sd (y,W(x,y,t)) = (1 − t)d(x,y). Let α = d (x,W(x,y,t)), β = d (y,W(x,y,t)) and γ = d(x,y). Then, from (1.3), we get sα ≤ tγ and sβ ≤ (1 − t)γ. Now, by the triangle inequality of a b-metric, we have γ ≤ s(α + β) ≤ tγ + (1 − t)γ = γ, that is, γ = s(α + β). Now, if sα < tγ, then γ ≤ sα + sβ < γ, which is a contradiction. Hence sα = tγ, and consequently, sβ = (1 − t)γ. Definition 1.3. [1] A b-metric space (X,d) on which a convex structure W is defined is called a convex b-metric space, denoted by (X,W,d). A subset K of X is called convex if W(x,y,λ) ∈ K whenever x,y ∈ K and λ ∈ [0, 1]. Definition 1.4. [38] A convex metric space (X,W,d) is called uniformly convex if for every δ > 0, c > 0 and x,y,z ∈ X, there exists µ > 0 such that d(z,x) ≤ c, d(z,y) ≤ c and d(x,y) ≥ cδ implies d ( z,W ( x,y; 1 2 )) ≤ c(1 −µ) < c. Hadamard manifolds and geodesic spaces are nonlinear examples of convex b-metric spaces while uniformly convex Banach b-metric spaces and CAT(0) spaces are examples of uniformly convex b-metric spaces [11]. Definition 1.5. [36] Let K be a subset of a b-metric space (X,d) and {xn} be a bounded sequence in X. For x ∈ X, we take r (x,{xn}) = limn→∞ sup d(x,xn). Then (1) r ({xn}) = inf {r (x,{xn}) : x ∈ K} is said to be the asymptotic radius of {xn} with respect to K ⊆ X, (2) For any z ∈ K, the set A ({xn}) = {x ∈ X : r (x,{xn}) ≤ r (z,{xn})} is said to be the asymptotic centre of {xn} with respect to K ⊆ X. A sequence {xn} ∆-converges to x if A ({un}) = {x} for every subsequence {un} of {xn}, that is, x is the unique asymptotic centre for every subsequence {un} of {xn}. It is denoted by ∆ − limn→∞xn = x. Equivalently, A sequence {xn} in X is said to ∆-converges to a point x ∈ X if lim sup k d ( xnk,x ) ≤ lim sup k d ( xnk,y ) for every subsequence {xnk} of {xn} and every y ∈ X [18]. Int. J. Anal. Appl. 18 (1) (2020) 37 It can be seen that ordinary convergence implies ∆-convergence. However, the converse is not true. Example 1.1. † Let X be the set of positive rational numbers and define d : X ×X −→ [0,∞) by d(x,y) =   1, x 6= y 0, x = y Then (X,d) is a b-metric space. Consider the sequence {xn} where xn = 1n , n ∈ N. Then evidently, {xn} is not convergent but ∆-convergent to x for every x ∈ X. Because, given any x and y in X, we can choose k large enough with xnk 6= x and xnk 6= y so that lim sup k ( d ( xnk,x ) −d ( xnk,y )) ≤ 1 − 1 = 0. Definition 1.6. [12] A subset K of a b-metric space (X,d) is said to be Chebychev if for every x ∈ X there exists y ∈ K such that d(x,y) < d(k,x) for all k ∈ K and y 6= k. If K is a Chebychev subset of a b-metric space X, then the nearest point projection P : X −→ K is defined by sending x to y. As observed in [12], this notion of nearest point projection for Chebychev sets is in accordance with that of orthogonal projection onto a subspace of the Euclidean space. It was shown in [11] that every closed and convex subset of a uniformly convex b-metric space is Chebychev. Lemma 1.1. [12] Let K be a nonempty, closed and convex subset of a complete uniformly convex metric space (X,W,d). Then every bounded sequence {xn} in K has a unique asymptotic centre in K. Lemma 1.2. [12] Let K be a nonempty, closed and convex subset of a complete uniformly convex metric space (X,W,d). Let {xn} be a bounded sequence in K such that A ({xn}) = {y} and r ({xn}) = ρ. If {ym} is another sequence in K such that limm→∞r (ym,{xn}) = ρ, then limm→∞ym = y. Lemma 1.3. [12] Let (X,W,d) be a uniformly convex metric space and {αn} a sequence in [b,c] ⊂ (0, 1). Suppose that the sequences {xn} and {yn} in X are such that lim n→∞ sup d(xn,w) ≤ c, lim n→∞ sup d(yn,w) ≤ c and lim n→∞ sup d (W(xn,yn; αn),w) ≤ c for some w ∈ X and some c ≥ 0. Then limn→∞d(xn,yn) = 0. †This example is adapted from https://math.stackexchange.com/a/3370319/445538 Int. J. Anal. Appl. 18 (1) (2020) 38 The proofs of the above lemmas are independent of the property – the triangle inequality of the metric d. Thus these results also holds true for the corresponding b-metric spaces. Let K be a non-empty subset of a b-metric space (X,d) and ∆ be the diagonal of the Cartesian product K ×K, i.e., ∆ = {(x,x) : x ∈ K}. Let G be a reflexive digraph (directed graph) with the set V (G) of its vertices coinciding with K, and the set E(G) of its edges containing all loops, i.e., E(G) ⊇ ∆. Assuming G has no parallel edges, we identify the graph G with the pair (V (G),E(G)). Definition 1.7. [40] The conversion of a graph G is the graph obtained from G by reversing the direction of the edges, denoted by G−1, that is, E(G−1) = {(x,y) ∈ X ×X : (y,x) ∈ G} . A directed graph/digraph G = (V (G),E(G)) is said to be transitive if for every x,y,z ∈ V (G) with (x,y), (y,z) ∈ E(G), we have (x,z) ∈ E(G). Definition 1.8. Let (X,d) be a b-metric space with coefficient s ≥ 1. A mapping S : X −→ X is said to be a G-contractive type mapping if S is edge-preserving ((x,y) ∈ E(G) implies (Sx,Sy) ∈ E(G)) and d (Sx,Sy) ≤ k ( d(x,y) + d(y,Sy) ) (1.4) for some k < 1 and for all x,y ∈ X. Taking y = Sx in the above relation, we get d(Sx,S2x) ≤ k ( d(x,Sx) + d(Sx,S2x) ) that is, d(Sx,S2x) ≤ k 1 −k d(x,Sx) = k′d(x,Sx) for some k′ > 0. If y = w0 ∈ F, then we have d (Sx,w0) ≤ kd(x,w0), where F = F(S) is the set of fixed points of S. 2. Main results For a b-metric space (X,d) with graph and a non-empty closed convex subset K of X, we introduce the extended S-iteration scheme given below. For x0 ∈ K, Int. J. Anal. Appl. 18 (1) (2020) 39   xn+1 = W (S2S1xn,S3S1yn,αn) , yn = W (S1xn,S3zn,βn) zn = W (S2S1xn,S3S1xn,γn) (2.1) where {αn}, {βn} and {γn} are real sequences in [0, 1] and Si : X −→ X is a G-contractive type mapping on K for i = 1, 2, 3. The existence of a fixed point for contractive type mappings given by (1.4) is known from various existing literatures (for example, refer to [11]). In this section, we prove a result on ∆-convergence and strong convergence of the iteration scheme given by (2.1) in a closed convex subset of a uniformly convex b-metric space. Let F = ⋂3 i=1 F(Si), where F(Si) are the sets of fixed points of Si. Lemma 2.1. Let w0 ∈ F be such that (x0,w0), (w0,x0) ∈ E(G). Then (xn,w0), (w0,xn), (yn,w0), (w0,yn), (zn,w0), (w0,zn), (xn,yn), (yn,zn) and (xn,xn+1) are in E(G). Proof. We will prove by induction. Since S1, S2 and S3 are edge-preserving and E(G) is convex, using (2.1) we have (x0,w0) ∈ E(G) =⇒ (W(S2S1x0,S3S1x0; γ0),w0) = (z0,w0) ∈ E(G) and (y0,w0) = (W(S1x0,z0; β0),w0) ∈ E(G). Since (x0,w0), (y0,w0) ∈ E(G), we have (x1,w0) = (W(S2S1x0,S3S1y0; α0),w0) ∈ E(G) and therefore, (z1,w0) = (W(S2S1x1,S3S1x1; γ0),w0) ∈ E(G). Similarly, (y1,w0) = (W(S1x1,z1; β1),w0) ∈ E(G). Now we assume that (xk,w0) ∈ E(G) for some positive integer k. Then by the same argument as before, (zk,w0) = (W(S2S1xk,S3S1xk; γk),w0) ∈ E(G), (yk,w0) = (W(S1xk,zk; βk),w0) ∈ E(G) and (xk+1,w0) = (W(S2S1xk,S3S1yk; αk),w0) ∈ E(G). This implies (xk+1,w0) ∈ E(G) which in turn gives (yk+1,w0) and (zk+1,w0) are in E(G). Therefore, (xn+1,w0), (yn+1,w0) and (zn+1,w0) are in E(G) for all n ∈ N. In a similar way, we can show that (w0,xn), (w0,yn) and (w0,zn) are in E(G) if (w0,x0) ∈ E(G). Int. J. Anal. Appl. 18 (1) (2020) 40 Finally, the transitivity of E(G) implies (xn,yn), (yn,zn) and (xn,xn+1) are also in E(G). � Lemma 2.2. If (X,d) is a convex b-metric space and (x0,w0), (w0,x0) ∈ E(G) for arbitrary x0 ∈ X and w0 ∈ F , then d(xn+1,w0) ≤ kd(xn,w0) for all n and hence lim n→∞ d(xn,w0) = 0. Proof. If w0 ∈ F, then by Lemma 2.1, (xn,w0), (yn,w0), (zn,w0) ∈ E(G). Let x′n = S1xn. Since S1, S2 and S3 are G-contractive type mappings, we have d (Ax,Aw0) ≤ k ( d(x,w0) + d(w0,Aw0) ) = kd(x,w0) for some k < 1 and A = S1, S2 or S3. Now, d(xn+1,w0) = d (W(S2S1xn,S3S1yn,αn),w0) ≤ αnd(S2S1xn,w0) + (1 −αn)d(S3S1yn,w0) ≤ kαnd(S1xn,w0) + k(1 −αn)d(S1yn,w0) = kαnd(S1xn,w0) + k(1 −αn)d (W(S1xn,zn; βn),w0) ≤ kαnd(S1xn,w0) + k(1 −αn)βnd(S1xn,w0) + k(1 −αn)(1 −βn)d(W(S2S1xn,S3S1xn; γn),w0) ≤ k (αn + βn −αnβn) d(S1xn,w0) + k(1 −αn)(1 −βn)( γnd(S2S1xn,w0) + (1 −γn)d(S3S1xn,w0) ) ≤ k (αn + βn −αnβn) d(S1xn,w0) + k(1 −αn)(1 −βn)( γnkd(S1xn,w0) + (1 −γn)kd(S1xn,w0) ) ≤ k ( αn + βn −αnβn + (1 −αn)(1 −βn) ) d(S1xn,w0) = kd(S1xn,w0) ≤ kd(xn,w0) for all n ∈ N. Thus the sequence {d(xn,w0)} of positive numbers is monotonically decreasing and hence limn→∞d(xn,w0) exists. In fact, since d(xn+1,w0) ≤ kd(xn,w0) for all n ≥ 0, we have d(xn+1,w0) ≤ knd(x0,w0). This proves the assertion. � Lemma 2.3. If X is a convex b-metric space and (x0,w0), (w0,x0) ∈ E(G) for arbitrary x0 ∈ X and w0 ∈ F , then lim n→∞ d(S1xn,xn) = lim n→∞ d(S2xn,xn) = lim n→∞ d(S3xn,xn) = 0. Int. J. Anal. Appl. 18 (1) (2020) 41 Proof. From Lemma 2.2, we see that lim n→∞ d(xn,w0) = 0. (2.2) Then using (1.4) we have d(Axn,xn) ≤ sd(Axn,w0) + sd(w0,xn) ≤ sk ( d(xn,w0) + d(w0,Aw0) ) + sd(xn,w0) ≤ s(k + 1)d(xn,w0) where A = S1, S2 or S3. In the limiting case, we have lim n→∞ d(Axn,xn) = 0. � We now prove a result on ∆-convergence in convex b-metric spaces following the method used in [12]. Theorem 2.1. Let K be a nonempty closed convex subset of a uniformly convex and complete b-metric space X with a continuous convex structure W and, S1,S2,S3 : K −→ K be continuous G-contractive type mappings on K with F 6= ∅. If (x0,w0), (w0,x0) ∈ E(G) for arbitrary x0 ∈ K and some w0 ∈ F , then the sequence {x′n = S1xn} given by (2.1) ∆-converges to an element of F . Proof. In Lemma 2.2, it is shown that limn→∞d(xn,w0) exists, which in turn shows that the sequence {xn} is bounded. Therefore by Lemma 1.1, A ({xn}) = {x}. Let {vn} be any subsequence of {xn} such that A ({vn}) = {v}. As in Theorem 2.4 of [12], we can show that v ∈ K. By Lemma 2.3, lim n→∞ d(S1vn,vn) = lim n→∞ d(S2vn,vn) = lim n→∞ d(S3vn,vn) = 0. Define um = T mv (T = Si, i = 1, 2 or 3) and we observe that d (um,vn) ≤ sd (Tmv,Tmvn) + m∑ j=1 sjd ( Tm−jvn,T m−j+1vn ) ≤ skd ( Tm−1v,Tm−1vn ) + skd ( Tm−1vn,T mvn ) + m∑ j=1 sjd ( Tm−jvn,T m−j+1vn ) Int. J. Anal. Appl. 18 (1) (2020) 42 ≤ sk2d ( Tm−2v,Tm−2vn ) + skd ( Tm−1vn,T mvn ) + sk2d ( Tm−2vn,T m−1vn ) + m∑ j=1 sjd ( Tm−jvn,T m−j+1vn ) ... ≤ skmd (v,vn) + m∑ j=1 (skj + sj)d ( Tm−jvn,T m−j+1vn ) ≤ d (v,vn) + d (Tvn,vn) m∑ j=1 (skj + sj)k j 1 where k1 = k 1−k > 0. Hence, r (um,{vn}) ≤ lim n→∞ sup d (um,vn) ≤ lim n→∞ sup d (v,vn) ≤ r (v,{vn}) , which shows that |r (um,{vn}) −r (v,{vn})| −→ 0 as m →∞. Now, from Lemma 1.2, limm→∞um = limm→∞T mv = v. K being closed, limm→∞T mv = v ∈ K, and limm→∞Tm+1v = Tv, which implies Tv = v. Therefore, by Lemma 2.2 (since v ∈ F) limn→∞d (v,xn) exists. Now, as in Theorem 2.4 of [12], it directly follows that x = v. Thus, x is the unique asymptotic centre for any subsequence {vn} of {xn}, showing that {xn} ∆-converges to x. � In [37], Shahzad & Al-Dubiban stated a condition called Condition (B) and proved a strong convergence theorem for nonexpansive mappings in Banach spaces. We restate the condition in a b-metric setting and prove a strong convergence theorem. The mappings S1,S2,S3 : K −→ K with F = F(S1) ∩F(S2) ∩F(S3) 6= ∅ are said to satisfy Condition (B) if there is a non decreasing function f : [0,∞) −→ [0,∞) with f(0) = 0 and f(x) > 0 for all x > 0 such that for all x ∈ K, max{d(S1x,x),d(S2x,x),d(S3x,x)}≥ f (d(x,F)) . Theorem 2.2. Let K be a nonempty closed convex subset of a uniformly convex and complete b-metric space X with continuous convex structure W and, S1,S2,S3 : K −→ K be G-contractive type mappings on K satisfying F 6= ∅. Let (x0,w0), (w0,x0) ∈ E(G) for arbitrary x0 ∈ X and w0 ∈ F . If S1,S2 and S3 satisfy condition (B), then the sequence {xn} given by (2.1) converges strongly to an element of F . Int. J. Anal. Appl. 18 (1) (2020) 43 Proof. Let w ∈ F. From Lemma 2.2, we get that {xn} is a bounded seequence and hence limn→∞d(xn,w) exists. Also, we have d(xn+1,w) < d(xn,w) for all n ≥ 1, from which we get that d(xn+1,F) ≤ d(xn,F) for all n ≥ 1. By the same argument as in Lemma 2.2, we conclude that limn→∞d(xn,F) exists. By Lemma 2.3, we have limn→∞d(Sixn,xn) = 0, where i = 1, 2, 3. Since S1, S2 and S3 satisfy conddition (B), we get lim n→∞ f (d(xn,F)) = 0 and hence lim n→∞ d(xn,F) = 0. So, there exists a subsequence {xnk} of {xn} and a sequence {wk} in F satisfying d (xnk,wk) ≤ 2 −k. Setting nk+1 = nk + j for some j ≥ 1, we have d ( xnk+1,wk ) ≤ d (xnk+j−1,wk) ≤ d (xnk,wk) ≤ 1 2k , using which we get d(wk+1,wk) ≤ s ( d ( wk+1,xnk+1 ) + d ( xnk+1,wk )) ≤ s ( 1 2k+1 + 1 2k ) = s 2k+1 . Thus {wk} is a Cauchy sequence in F. Since F is closed, there exists w∗ ∈ F such that limk→∞wk = w∗. Thus, limk→∞xnk = w ∗. As limn→∞d (xn,w ∗) exists and equals 0 by Lemma 2.2, the result follows. � 3. Numerical example In this section, we present an example with its numerical experiment in support of our results. We also make a comparison of the rate of convergence of the iteration scheme (2.1) to that of the one given in [40]. In 1976, Rhoades [35] gave a comparison between two iterations {xn} and {zn}, both converging to a fixed point p of a mapping T : K −→ K by saying {xn} converge faster than {zn} if d(xn,p) ≤ d(zn,p), n ≥ 1, where K is a non-empty closed and convex subset of a complete metric space. Int. J. Anal. Appl. 18 (1) (2020) 44 In numerical analysis, the order of convergence of a real sequence {αn} converging to α is studied using the well known method mentioned below (refer to [4]). Let {αn} be a real sequence which converges to α with αn 6= α for all n ∈ N. If lim n→∞ d(αn+1,α) [d(αn,α)] µ = λ for some positive constants λ and µ, then {αn} is said to converge to α of the order µ, with asymptotic error constant λ. For λ < 1, if µ = 1 the sequence is linearly convergent and if µ = 2, the sequence is quadratically convergent. In 2002, using the above method of comparison, Berinde [3] compared the rate of convergence between two iteration schemes as given below. Let {αn} and {βn} be sequences of positive real numbers converging to α and β, respectively. Suppose that lim n→∞ d(αn,α) d(βn,β) = l. (i). If l = 0, then the sequence {αn} is said to converge to α faster than that of the sequence {βn} to {β}. (ii). If 0 < l < ∞, then the sequences {αn} and {βn} are said to have the same rate of convergence. For a nonempty convex subset K of a complete b-metric space X with a self map T : K −→ K, if {xn} and {un} are two iterations both of which converge to a fixed point p of T , then {xn} converges faster than {un} to p if lim n→∞ d(xn,p) d(un,p) = 0. We are now in a position to give an example for our main results and compare the rate of convergence of the studied iteration scheme against the modified S-iteration scheme. In the cases when the b-metric d is induced by the norm ‖.‖X, the mapping W : X2 × [0, 1] −→ X such that W(x,y,t) = (1 − t)x + ty defines a convex structure on X. The iteration (2.1) then takes the following form. For x0 ∈ K,  xn+1 = (1 −αn)S2S1xn + αnS3S1yn, yn = (1 −βn)S1xn + βnS3zn, zn = (1 −γn)S2S1xn + γnS3S1xn (3.1) where {αn}, {βn} and {γn} are real sequences in [0, 1] and Si : X −→ X is a G-contractive type mapping on K for i = 1, 2, 3. Int. J. Anal. Appl. 18 (1) (2020) 45 Example 3.1. Let X = R and K = [0, 2]. Let G = (V (G),E(G)) be a directed graph defined by V (G) = K and (x,y) ∈ E(G) if and only if 1 ≤ x,y ≤ 7 4 and x,y ∈ Q. We consider the mappings P,Q,R : K −→ K given by Px = xlog x Qx = 1 3 arcsin(x− 1) + 1 and Rx = √ x for all x ∈ K. To show that P , Q and R are G-contractive type mappings, it is enough to show that they are G-contraction mappings on [0, 2]. Now, to show that Px = xlog x is a contraction mapping on [1, 2], we note that by Mean value theorem, |P(x) −P(y)| |x−y| ≤ c where c = max{P ′x : x ∈ [1, 2]} with P ′x = 2xlog x−1 log x. Since P ′(x) < 7 10 for all 1 ≤ x ≤ 2, we see that P is a contraction on [1, 2] and hence a G-contraction mapping. Similarly, Q′x = 1 3 1√ 1−(x−1)2 ≤ 4 3 √ 7 for all 1 ≤ x ≤ 7 4 and R′x = 1 2 1√ x ≤ 1 2 for all x ≥ 1 implies that Q and R are G-contraction mappings on [0, 2]. Their common fixed point here being x = 1. Consider the real sequences {an}, {bn} and {cn} in [0, 1], where an = n + 1 5n + 3 , bn = n + 4 10n + 7 and cn = n + 2 2n + 3 . Let {xn} be a sequence generated by the extended S-iteration (2.1) with x0 = 1.5 and S1 = P , S2 = Q, S3 = R and αn = an, βn = bn, γn = cn as defined above. Let {un} be a sequence generated by the modified S-iteration (1.2) with u0 = 1.5 and using S1 = P , S2 = Q with λn = an and µn = bn. The numerical observations for the error estimates and the rate of convergence for these two iteration schemes are shown in Tables 1 & 2 below. Int. J. Anal. Appl. 18 (1) (2020) 46 n Modified S-iteration Extended S-iteration un |un −un−1| xn |xn −xn−1| 1 1.15489 0.345105 1.04139 0.458612 2 1.02536 0.129532 1.00041 0.0409766 3 1.00201 0.023352 1.00000 0.000411667 4 1.00012 0.00188701 1.00000 4.34451×10−8 5 1.00001 0.000116509 1.00000 4.44089×10−16 6 1.00000 7.01765×10−6 1.00000 0.00000 7 1.00000 4.22157×10−7 1.00000 0.00000 Table 1. Numerical errors of modified S-iteration and extended S-iteration schemes n Modified S- Extended S- Rate of Convergence un xn |un −1| |xn −1| |xn−1||un−1| 1 1.15489 1.04139 0.154895 0.0413883 0.267203 2 1.02536 1.00041 0.025363 0.000411711 0.0162327 3 1.00201 1.00000 0.00201099 4.34451×10−8 0.0000216038 4 1.00012 1.00000 0.000123976 4.44089×10−16 3.58206×10−12 5 1.00001 1.00000 7.46682×10−6 0.00000 0.000000 6 1.00000 1.00000 4.49177×10−7 0.00000 0.000000 7 1.00000 1.00000 2.70198×10−8 0.00000 0.000000 Table 2. Rate of Convergence Int. J. Anal. Appl. 18 (1) (2020) 47 From Tables 1 & 2, it is evident that the sequence of iterates {un} and {xn} both converge to 1 ∈ F . We also observe that |xn − 1| ≤ |un − 1| and limn→∞ |xn−1| |un−1| = 0, so the seqeuence of iterates {xn} converges faster than {un}, generated by the Modified S-iteration (see Figure 1). æ æ æ æ æ à à à à à 0 1 2 3 4 5 0.0 0.1 0.2 0.3 0.4 0.5 n er ro rs à Extended S- æ Modified S- Figure 1. Comparison of error estimates of the Modified S-iteration and the Extended S-iteration schemes Conclusion In this paper, we have introduced an extended S-iteration scheme for G-contractive type mappings and proved ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph. An example is also given to compare the conver- gence rate between the studied iteration and the modified S-iteration. The iteration considered in this paper may as well be studied for other contractive type mappings and its rate of convergence can be compared with existing iteration schemes. The application of our results for solving constrained optimization problem is also a scope for further study. 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 (1) (2020) 48 References [1] A. A. 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