CUBO A Mathematical Journal Vol.20, No¯ 01, (41–64). March 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000100041 Common Fixed Point Results in C∗-Algebra Valued b-Metric Spaces Via Digraphs Sushanta Kumar Mohanta Department of Mathematics, West Bengal State University,, Barasat, 24 Parganas (North), West Bengal, Kolkata 700126, India. smwbes@yahoo.in ABSTRACT We discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on a C∗-algebra valued b-metric space endowed with a graph. Our results extend and supplement several recent results in the literature. Strength of hypotheses made in the first result have been weighted through illustrative examples. RESUMEN Discutimos la existencia y unicidad de puntos de coincidencia y puntos fijos comumes para un par de aplicaciones definidas en un b-espacio métrico a valores en una álgebra C∗ dotado de un grafo en śı mismo. Nuestros resultados extienden y suplementan diversos resultados recientes en la literatura. La fuerza de las hipótesis impuestas al primer resultado se evalúa a través de ejemplos ilustrativos. Keywords and Phrases: C∗-algebra, C∗-algebra valued b-metric, directed graph, C∗-algebra valued G-contraction, common fixed point. 2010 AMS Mathematics Subject Classification: 54H25, 47H10. http://dx.doi.org/10.4067/S0719-06462018000100041 Ignacio Castillo Ignacio Castillo Ignacio Castillo Ignacio Castillo Ignacio Castillo 42 Sushanta Kumar Mohanta CUBO 20, 1 (2018) 1 Introduction In 1922 [5], Polish mathematician S. Banach proved a very important result regarding a contrac- tion mapping, known as the Banach contraction principle. This fundamental principle was largely applied in many branches of mathematics. Several authors generalized this interesting theorem in different ways(see [1, 2, 6, 13, 18, 25, 26, 27]). In this context, Bakhtin [4] and Czerwik [11] developed the notion of b-metric spaces and proved some fixed point theorems for single-valued and multi-valued mappings in the setting of b-metric spaces. In 2014, Z. Ma et.al.[22] introduced the concept of C∗-algebra valued metric spaces by using the set of all positive elements of an unital C∗-algebra instead of the set of real numbers. In [21], the authors introduced another new concept, known as C∗-algebra valued b-metric spaces as a generalization of C∗-algebra valued metric spaces and b-metric spaces. In recent investigations, the study of fixed point theory endowed with a graph plays an im- portant role in many aspects. In 2005, Echenique [15] studied fixed point theory by using graphs. After that, Espinola and Kirk [16] applied fixed point results in graph theory. Recently, combining fixed point theory and graph theory, a series of articles(see [3, 8, 9, 10, 20, 24] and references therein) have been dedicated to the improvement of fixed point theory. The idea of common fixed point was initially given by Junck [19]. In fact, the author introduced the concept of weak compatibility and obtained a common fixed point result. Several authors have obtained coincidence points and common fixed points for various classes of mappings on a metric space by using this concept. Motivated by some recent works on the extension of Banach contraction principle to metric spaces with a graph, we reformulated some important common fixed point results in metric spaces to C∗-algebra valued b-metric spaces endowed with a graph. As some consequences of this study, we deduce several related results in fixed point theory. Finally, some examples are provided to illustrate the results. 2 Some basic concepts We begin with some basic notations, definitions and properties of C∗-algebras. Let A be an unital algebra with the unit I. An involution on A is a conjugate linear map a 7→ a∗ on A such that a∗∗ = a and (ab)∗ = b∗a∗ for all a, b ∈ A. The pair (A, ∗) is called a ∗-algebra. A Banach ∗-algebra is a ∗-algebra A together with a complete submultiplicative norm such that ‖ a∗ ‖=‖ a ‖ for all a ∈ A. A C∗-algebra is a Banach ∗-algebra such that ‖ a∗a ‖=‖ a ‖2 for all a ∈ A. Let H be a Hilbert space and B(H), the set of all bounded linear operators on H. Then, under the norm topology, B(H) is a C∗-algebra. Throughout this discussion, by A we always denote an unital C∗-algebra with the unit I and CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 43 the zero element θ. Set Ah = {x ∈ A : x = x ∗}. We call an element x ∈ A a positive element, denote it by x � θ, if x ∈ Ah and σ(x) ⊂ [0, ∞), where σ(x) is the spectrum of x. Using positive elements, one can define a partial ordering � on Ah as follows: x � y if and only if y − x � θ. We shall write x ≺ y if x � y and x 6= y. From now on, by A+, we denote the set {x ∈ A : x � θ} and by A ′ , we denote the set {a ∈ A : ab = ba, ∀b ∈ A}. Lemma 2.1. [14, 23] Suppose that A is an unital C∗-algebra with a unit I. (i) For any x ∈ A+, we have x � I ⇔‖ x ‖≤ 1. (ii) If a ∈ A+ with ‖ a ‖< 1 2 , then I − a is invertible and ‖ a(I − a)−1 ‖< 1. (iii) Suppose that a, b ∈ A with a, b � θ and ab = ba, then ab � θ. (iv) Let a ∈ A ′ , if b, c ∈ A with b � c � θ, and I − a ∈ A ′ + is an invertible operator, then (I − a)−1b � (I − a)−1c. Remark 2.2. It is worth mentioning that x � y ⇒‖ x ‖≤‖ y ‖ for x, y ∈ A+. In fact, it follows from Lemma 2.1 (i). Definition 2.3. [22] Let X be a nonempty set. Suppose the mapping d : X × X → A satisfies: (i) θ � d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, y) � d(x, z) + d(z, y) for all x, y, z ∈ X. Then d is called a C∗-algebra valued metric on X and (X, A, d) is called a C∗-algebra valued metric space. Definition 2.4. [4] Let X be a nonempty set and s ≥ 1 be a given real number. A function d : X × X → R+ is said to be a b-metric on X if the following conditions hold: (i) d(x, y) = 0 if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, y) ≤ s (d(x, z) + d(z, y)) for all x, y, z ∈ X. The pair (X, d) is called a b-metric space. 44 Sushanta Kumar Mohanta CUBO 20, 1 (2018) Definition 2.5. [21] Let X be a nonempty set and A ∈ A ′ + such that A � I. Suppose the mapping d : X × X → A satisfies: (i) θ � d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, y) � A (d(x, z) + d(z, y)) for all x, y, z ∈ X. Then d is called a C∗-algebra valued b-metric on X and (X, A, d) is called a C∗-algebra valued b-metric space. It seems important to note that if A = C, A = 1, then the C∗-algebra valued b-metric spaces are just the ordinary metric spaces. Moreover, it is obvious that C∗-algebra valued b-metric spaces generalize the concepts of C∗-algebra valued metric spaces and b-metric spaces. Definition 2.6. [26] Let (X, A, d) be a C∗-algebra valued b-metric space, x ∈ X and (xn) be a sequence in X. Then (i) (xn) converges to x with respect to A if for any ǫ > 0 there is n0 such that for all n > n0, ‖ d(xn, x) ‖≤ ǫ. We denote it by lim n→∞ xn = x or xn → x(n → ∞). (ii) (xn) is Cauchy with respect to A if for any ǫ > 0 there is n0 such that for all n, m > n0, ‖ d(xn, xm) ‖≤ ǫ. (iii) (X, A, d) is a complete C∗-algebra valued b-metric space if every Cauchy sequence with respect to A is convergent. Example 2.7. If X is a Banach space, then (X, A, d) is a complete C∗-algebra valued b-metric space with A = 2p−1I if we set d(x, y) =‖ x − y ‖p I where p > 1 is a real number. But (X, A, d) is not a C∗-algebra valued metric space because if X = R, then | x − y |p≤| x − z |p + | z − y |p is impossible for all x > z > y. Definition 2.8. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I. We call a mapping f : X → X a C∗-algebra valued contraction mapping on X if there exists B ∈ A with ‖ B ‖2< 1 ‖A‖ such that d(fx, fy) � B∗ d(x, y)B for all x, y ∈ X. Definition 2.9. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I. A mapping f : X → X is called a C∗-algebra valued Fisher contraction if there exists B ∈ A ′ + with ‖ BA ‖< 1 ‖A‖+1 such that d(fx, fy) � B [d(fx, y) + d(fy, x)] for all x, y ∈ X. CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 45 Definition 2.10. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I. A mapping f : X → X is called a C∗-algebra valued Kannan operator if there exists B ∈ A ′ + with ‖ B ‖< 1 ‖A‖+1 such that d(fx, fy) � B [d(fx, x) + d(fy, y)] for all x, y ∈ X. Definition 2.11. [2] Let T and S be self mappings of a set X. If y = Tx = Sx for some x in X, then x is called a coincidence point of T and S and y is called a point of coincidence of T and S. Definition 2.12. [19] The mappings T, S : X → X are weakly compatible, if for every x ∈ X, the following holds: T(Sx) = S(Tx) whenever Sx = Tx. Proposition 2.13. [2] Let S and T be weakly compatible selfmaps of a nonempty set X. If S and T have a unique point of coincidence y = Sx = Tx, then y is the unique common fixed point of S and T. Definition 2.14. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I. A mapping f : X → X is called C∗-algebra valued expansive if there exists B ∈ A with 0 <‖ B ‖2< 1 ‖A‖ such that B∗d(fx, fy)B � d(x, y) for all x, y ∈ X. We next review some basic notions in graph theory. Let (X, A, d) be a C∗-algebra valued b-metric space. Let G be a directed graph (digraph) with a set of vertices V(G) = X and a set of edges E(G) contains all the loops, i.e., E(G) ⊇ ∆, where ∆ = {(x, x) : x ∈ X}. We also assume that G has no parallel edges and so we can identify G with the pair (V(G), E(G)). G may be considered as a weighted graph by assigning to each edge the distance between its vertices. By G−1 we denote the graph obtained from G by reversing the direction of edges i.e., E(G−1) = {(x, y) ∈ X × X : (y, x) ∈ E(G)}. Let G̃ denote the undirected graph obtained from G by ignoring the direction of edges. Actually, it will be more convenient for us to treat G̃ as a directed graph for which the set of its edges is symmetric. Under this convention, E(G̃) = E(G) ∪ E(G−1). Our graph theory notations and terminology are standard and can be found in all graph theory books, like [7, 12, 17]. If x, y are vertices of the digraph G, then a path in G from x to y of length n (n ∈ N) is a sequence (xi) n i=0 of n + 1 vertices such that x0 = x, xn = y and (xi−1, xi) ∈ E(G) for i = 1, 2, · · · , n. A graph G is connected if there is a path between any two vertices of G. G is weakly connected if G̃ is connected. 46 Sushanta Kumar Mohanta CUBO 20, 1 (2018) Definition 2.15. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I and let G = (V(G), E(G)) be a graph. A mapping f : X → X is called a C∗-algebra valued G- contraction if there exists a B ∈ A with ‖ B ‖2< 1 ‖A‖ such that d(fx, fy) � B∗d(x, y)B, for all x, y ∈ X with (x, y) ∈ E(G). Any C∗-algebra valued contraction mapping on X is a G0-contraction, where G0 is the complete graph defined by (X, X × X). But it is worth mentioning that a C∗-algebra valued G-contraction need not be a C∗-algebra valued contraction (see Remark 3.23). Definition 2.16. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I and let G = (V(G), E(G)) be a graph. A mapping f : X → X is called C∗-algebra valued Fisher G-contraction if there exists B ∈ A ′ + with ‖ BA ‖< 1 ‖A‖+1 such that d(fx, fy) � B [d(fx, y) + d(fy, x)] for all x, y ∈ X with (x, y) ∈ E(G). It is easy to observe that a C∗-algebra valued Fisher contraction is a C∗-algebra valued Fisher G0-contraction. But it is important to note that a C ∗-algebra valued Fisher G-contraction need not be a C∗-algebra valued Fisher contraction. The following example supports the above remark. Example 2.17. Let X = [0, ∞) and B(H) be the set of all bounded linear operators on a Hilbert space H. Define d : X × X → B(H) by d(x, y) =| x − y |2 I for all x, y ∈ X. Then (X, B(H), d) is a C∗-algebra valued b-metric space with the coefficient A = 2I. Let G be a digraph such that V(G) = X and E(G) = ∆ ∪ {(3tx, 3t(x + 1)) : x ∈ X with x ≥ 2, t = 0, 1, 2, · · · }. Let f : X → X be defined by fx = 3x for all x ∈ X. For x = 3tz, y = 3t(z + 1), z ≥ 2, we have d(fx, fy) = d ( 3t+1z, 3t+1(z + 1) ) = 32t+2I � 9 58 32t(8z2 + 8z + 10)I = B [ d ( 3t+1z, 3t(z + 1) ) + d ( 3t+1(z + 1), 3tz )] = B [d(fx, y) + d(fy, x)], where B = 9 58 I ∈ B(H) ′ + with ‖ BA ‖< 1 ‖A‖+1 . Thus, f is a C∗-algebra valued Fisher G-contraction. We now verify that f is not a C∗-algebra valued Fisher contraction. In fact, if x = 3, y = 0, CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 47 then for any arbitrary B ∈ B(H) ′ + with ‖ BA ‖< 1 ‖A‖+1 = 1 3 (which implies 3BA ≺ I), we have B [d(fx, y) + d(fy, x)] = B [d(f3, 0) + d(f0, 3)] = 90BI = 45BA = 5 27 (3BA)(81I) ≺ 81I = d(fx, fy). Definition 2.18. Let (X, A, d) be a C∗-algebra valued b-metric space with the coefficient A � I and let G = (V(G), E(G)) be a graph. A mapping f : X → X is called C∗-algebra valued G-Kannan if there exists B ∈ A ′ + with ‖ B ‖< 1 ‖A‖+1 such that d(fx, fy) � B [d(fx, x) + d(fy, y)] for all x, y ∈ X with (x, y) ∈ E(G). Note that any C∗-algebra valued Kannan operator is C∗-algebra valued G0-Kannan. However, a C∗-algebra valued G-Kannan operator need not be a C∗-algebra valued Kannan operator (see Remark 3.28). Remark 2.19. If f is a C∗-algebra valued G-contraction(resp., G-Kannan or Fisher G-contraction), then f is both a C∗-algebra valued G−1-contraction(resp., G−1-Kannan or Fisher G−1-contraction) and a C∗-algebra valued G̃-contraction(resp., G̃-Kannan or Fisher G̃-contraction). 3 Main Results In this section we always assume that (X, A, d) is a C∗-algebra valued b-metric space with the coefficient A � I and G is a directed graph such that V(G) = X and E(G) ⊇ ∆. Let f, g : X → X be such that f(X) ⊆ g(X). If x0 ∈ X is arbitrary, then there exists an element x1 ∈ X such that fx0 = gx1, since f(X) ⊆ g(X). Proceeding in this way, we can construct a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · ·. Definition 3.1. Let (X, A, d) be a C∗-algebra valued b-metric space endowed with a graph G and f, g : X → X be such that f(X) ⊆ g(X). We define Cgf the set of all elements x0 of X such that (gxn, gxm) ∈ E(G̃) for m, n = 0, 1, 2, · · · and for every sequence (gxn) such that gxn = fxn−1. If g = I, the identity map on X, then obviously Cgf becomes Cf which is the collection of all elements x of X such that (fnx, fmx) ∈ E(G̃) for m, n = 0, 1, 2, · · · . 48 Sushanta Kumar Mohanta CUBO 20, 1 (2018) Theorem 3.2. Let (X, A, d) be a C∗-algebra valued b-metric space endowed with a graph G and the mappings f, g : X → X be such that d(fx, fy) � B∗ d(gx, gy) B (3.1) for all x, y ∈ X with (gx, gy) ∈ E(G̃), where B ∈ A and ‖ B ‖2< 1 ‖A‖ . Suppose f(X) ⊆ g(X) and g(X) is a complete subspace of X with the following property: (∗) If (gxn) is a sequence in X such that gxn → x and (gxn, gxn+1) ∈ E(G̃) for all n ≥ 1, then there exists a subsequence (gxni) of (gxn) such that (gxni, x) ∈ E(G̃) for all i ≥ 1. Then f and g have a point of coincidence in X if Cgf 6= ∅. Moreover, f and g have a unique point of coincidence in X if the graph G has the following property: (∗∗) If x, y are points of coincidence of f and g in X, then (x, y) ∈ E(G̃). Furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in X. Proof. Suppose that Cgf 6= ∅. We choose an x0 ∈ Cgf and keep it fixed. Since f(X) ⊆ g(X), there exists a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · · and (gxn, gxm) ∈ E(G̃) for m, n = 0, 1, 2, · · · . It is a well known fact that in a C∗-algebra A, if a, b ∈ A+ and a � b, then for any x ∈ A both x∗ax and x∗bx are positive elements and x∗ax � x∗bx[23]. For any n ∈ N, we have by using condition (3.1) that d(gxn, gxn+1) = d(fxn−1, fxn) � B ∗d(gxn−1, gxn)B. (3.2) By repeated use of condition (3.2), we get d(gxn, gxn+1) � (B ∗ ) nd(gx0, gx1)B n = (Bn)∗B0B n, (3.3) for all n ∈ N, where B0 = d(gx0, gx1) ∈ A+. CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 49 For any m, n ∈ N with m > n, we have by using condition (3.3) that d(gxn, gxm) � A[d(gxn, gxn+1) + d(gxn+1, gxm)] � Ad(gxn, gxn+1) + A 2d(gxn+1, gxn+2) + · · · +Am−n−1d(gxm−2, gxm−1) + A m−n−1d(gxm−1, gxm) � A(B∗)nB0B n + A2(B∗)n+1B0B n+1 + A3(B∗)n+2B0B n+2 + · · · +Am−n−1(B∗)m−2B0B m−2 + Am−n−1(B∗)m−1B0B m−1 � m−n−1∑ k=1 Ak(B∗)n+k−1B0B n+k−1 + Am−n(B∗)m−1B0B m−1 = m−n∑ k=1 Ak(B∗)n+k−1B0B n+k−1 � m−n∑ k=1 ‖ Ak(B∗)n+k−1B0B n+k−1 ‖ I � ‖ B0 ‖ m−n∑ k=1 ‖ A ‖k ‖ B ‖2(n+k−1) I = ‖ B0 ‖ ‖ B ‖ 2n ‖ A ‖ m−n∑ k=1 ( ‖ A ‖ ‖ B ‖2 )k−1 I � ‖ B0 ‖ ‖ B ‖ 2n ‖ A ‖ 1− ‖ A ‖ ‖ B ‖2 I, since ‖ B ‖2< 1 ‖ A ‖ → θ as n → ∞. Therefore, (gxn) is a Cauchy sequence with respect to A. Since g(X) is complete, there exists an u ∈ g(X) such that lim n→∞ gxn = u = gv for some v ∈ X. As x0 ∈ Cgf, it follows that (gxn, gxn+1) ∈ E(G̃) for all n ≥ 0, and so by property (∗), there exists a subsequence (gxni) of (gxn) such that (gxni, gv) ∈ E(G̃) for all i ≥ 1. Using condition (3.1), we have d(fv, gv) � A[d(fv, fxni) + d(fxni, gv)] � AB∗d(gv, gxni)B + Ad(gxni+1, gv) → θ as i → ∞. This implies that d(fv, gv) = θ and hence fv = gv = u. Therefore, u is a point of coincidence of f and g. The next is to show that the point of coincidence is unique. Assume that there is another point of coincidence u∗ in X such that fx = gx = u∗ for some x ∈ X. By property (∗∗), we have 50 Sushanta Kumar Mohanta CUBO 20, 1 (2018) (u, u∗) ∈ E(G̃). Then, d(u, u∗) = d(fv, fx) � B∗d(gv, gx)B = B∗d(u, u∗)B, which implies that, ‖ d(u, u∗) ‖ ≤ ‖ B∗d(u, u∗)B ‖ ≤ ‖ B∗ ‖‖ d(u, u∗) ‖‖ B ‖ = ‖ B ‖2‖ d(u, u∗) ‖ . Since ‖ B ‖2< 1 ‖A‖ ≤ 1, it follows that d(u, u∗) = θ i.e., u = u∗. Therefore, f and g have a unique point of coincidence in X. If f and g are weakly compatible, then by Proposition 2.13, f and g have a unique common fixed point in X. The following corollary gives fixed point of Banach G-contraction in C∗-algebra valued b- metric spaces. Corollary 3.3. Let (X, A, d) be a complete C∗-algebra valued b-metric space endowed with a graph G and the mapping f : X → X be such that d(fx, fy) � B∗d(x, y)B (3.4) for all x, y ∈ X with (x, y) ∈ E(G̃), where B ∈ A with ‖ B ‖2< 1 ‖A‖ . Suppose (X, A, d, G) has the following property: (∗)́ If (xn) is a sequence in X such that xn → x and (xn, xn+1) ∈ E(G̃) for all n ≥ 1, then there exists a subsequence (xni) of (xn) such that (xni, x) ∈ E(G̃) for all i ≥ 1. Then f has a fixed point in X if Cf 6= ∅. Moreover, f has a unique fixed point in X if the graph G has the following property: (∗ ∗ )́ If x, y are fixed points of f in X, then (x, y) ∈ E(G̃). Proof. The proof can be obtained from Theorem 3.2 by considering g = I, the identity map on X. Corollary 3.4. Let (X, A, d) be a C∗-algebra valued b-metric space and the mappings f, g : X → X be such that (3.1) holds for all x, y ∈ X, where B ∈ A with ‖ B ‖2< 1 ‖A‖ . If f(X) ⊆ g(X) and CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 51 g(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X. Proof. The proof follows from Theorem 3.2 by taking G = G0, where G0 is the complete graph (X, X × X). The following corollary is analogue of Banach Contraction Principle. Corollary 3.5. Let (X, A, d) be a complete C∗-algebra valued b-metric space and the mapping f : X → X be such that (3.4) holds for all x, y ∈ X, where B ∈ A with ‖ B ‖2< 1 ‖A‖ . Then f has a unique fixed point u in X and fnx → u for all x ∈ X. Proof. It follows from Theorem 3.2 by putting G = G0 and g = I. Remark 3.6. We observe that Banach contraction theorem in a complete metric space can be obtained from Corollary 3.5 by taking A = C, A = 1. Thus, Theorem 3.2 is a generalization of Banach contraction theorem in metric spaces to C∗-algebra valued b-metric spaces. From Theorem 3.2, we obtain the following corollary concerning the fixed point of expansive mapping in C∗-algebra valued b-metric spaces. Corollary 3.7. Let (X, A, d) be a complete C∗-algebra valued b-metric space and let g : X → X be an onto mapping satisfying B∗d(gx, gy)B � d(x, y) for all x, y ∈ X, where B ∈ A with ‖ B ‖2< 1 ‖A‖ . Then g has a unique fixed point in X. Proof. The conclusion of the corollary follows from Theorem 3.2 by taking G = G0 and f = I. Corollary 3.8. Let (X, A, d) be a complete C∗-algebra valued b-metric space endowed with a partial ordering ⊑ and the mapping f : X → X be such that (3.4) holds for all x, y ∈ X with x ⊑ y or, y ⊑ x, where B ∈ A and ‖ B ‖2< 1 ‖A‖ . Suppose (X, A, d, ⊑) has the following property: (†) If (xn) is a sequence in X such that xn → x and xn, xn+1 are comparable for all n ≥ 1, then there exists a subsequence (xni) of (xn) such that xni, x are comparable for all i ≥ 1. If there exists x0 ∈ X such that f nx0, f mx0 are comparable for m, n = 0, 1, 2, · · · , then f has a fixed point in X. Moreover, f has a unique fixed point in X if the following property holds: (††) If x, y are fixed points of f in X, then x, y are comparable. Proof. The proof can be obtained from Theorem 3.2 by taking g = I and G = G2, where the graph G2 is defined by E(G2) = {(x, y) ∈ X × X : x ⊑ y or y ⊑ x}. 52 Sushanta Kumar Mohanta CUBO 20, 1 (2018) Theorem 3.9. Let (X, A, d) be a C∗-algebra valued b-metric space endowed with a graph G and the mappings f, g : X → X be such that d(fx, fy) � B [d(fx, gy) + d(fy, gx)] (3.5) for all x, y ∈ X with (gx, gy) ∈ E(G̃), where B ∈ A ′ + and ‖ BA ‖< 1 ‖A‖+1 . Suppose f(X) ⊆ g(X) and g(X) is a complete subspace of X with the property (∗). Then f and g have a point of coincidence in X if Cgf 6= ∅. Moreover, f and g have a unique point of coincidence in X if the graph G has the property (∗∗). Furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in X. Proof. It follows from condition (3.5) that B(d(fx, gy) + d(fy, gx)) is a positive element. Suppose that Cgf 6= ∅. We choose an x0 ∈ Cgf and keep it fixed. We can construct a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · · . Evidently, (gxn, gxm) ∈ E(G̃) for m, n = 0, 1, 2, · · · . For any n ∈ N, we have by using condition (3.5) and Lemma 2.1(iii) that d(gxn, gxn+1) = d(fxn−1, fxn) � B[d(fxn−1, gxn) + d(fxn, gxn−1)] = B[d(fxn−1, fxn−1) + d(fxn, fxn−2)] � BA[d(fxn, fxn−1) + d(fxn−1, fxn−2)] = BA d(gxn+1, gxn) + BA d(gxn, gxn−1)] which implies that, (I − BA)d(gxn, gxn+1) � BAd(gxn, gxn−1). (3.6) Now, A, B ∈ A ′ + implies that BA ∈ A ′ +. Since ‖ BA ‖< 1 2 , by Lemma 2.1, it follows that (I − BA) is invertible and ‖ BA(I − BA)−1 ‖=‖ (I − BA)−1BA ‖< 1. Moreover, by Lemma 2.1, BA � I i.e., I − BA � θ. Since BA ∈ A ′ +, we have (I − BA) ∈ A ′ +. Furthermore, (I − BA) −1 ∈ A ′ +. By using Lemma 2.1(iv), it follows from (3.6) that d(gxn, gxn+1) � (I − BA) −1BA d(gxn, gxn−1) = td(gxn−1, gxn), (3.7) where t = (I − BA)−1BA ∈ A ′ +. By repeated use of condition (3.7), we get d(gxn, gxn+1) � t nd(gx0, gx1) = t nB0, (3.8) for all n ∈ N, where B0 = d(gx0, gx1) ∈ A+. CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 53 We now prove that if ‖ BA ‖< 1 ‖A‖+1 , then ‖ t ‖< 1 ‖A‖ . We have, ‖ t ‖ = ‖ (I − BA)−1BA ‖ ≤ ‖ (I − BA)−1 ‖‖ BA ‖ ≤ 1 1− ‖ BA ‖ ‖ BA ‖ < 1 ‖ A ‖ , since ‖ BA ‖< 1 ‖ A ‖ +1 . For any m, n ∈ N with m > n, we have by using condition (3.8) that d(gxn, gxm) � A[d(gxn, gxn+1) + d(gxn+1, gxm)] � Ad(gxn, gxn+1) + A 2d(gxn+1, gxn+2) + · · · +Am−n−1d(gxm−2, gxm−1) + A m−n−1d(gxm−1, gxm) � AtnB0 + A 2tn+1B0 + A 3tn+2B0 + · · · +Am−n−1tm−2B0 + A m−n−1tm−1B0 � m−n∑ k=1 Aktn+k−1B0, since A � I and A ∈ A ′ + � m−n∑ k=1 ‖ Aktn+k−1B0 ‖ I � ‖ B0 ‖ ‖ A ‖ ‖ t ‖ n m−n∑ k=1 (‖ A ‖ ‖ t ‖) k−1 I � ‖ B0 ‖ ‖ A ‖ ‖ t ‖ n 1 1− ‖ A ‖ ‖ t ‖ I → θ as n → ∞. Therefore, (gxn) is a Cauchy sequence with respect to A. As g(X) is complete, there exists an u ∈ g(X) such that lim n→∞ gxn = u = gv for some v ∈ X. By property (∗), there exists a subsequence (gxni) of (gxn) such that (gxni, gv) ∈ E(G̃) for all i ≥ 1. Using condition (3.5), we have d(fv, gv) � A[d(fv, fxni) + d(fxni, gv)] � AB[d(fv, gxni) + d(fxni, gv)] + Ad(gxni+1, gv) � ABA[d(fv, gv) + d(gv, gxni)] + ABd(gxni+1, gv) + Ad(gxni+1, gv) which implies that, (I − BA2)d(fv, gv) � BA2d(gv, gxni) + ABd(gxni+1, gv) + Ad(gxni+1, gv). 54 Sushanta Kumar Mohanta CUBO 20, 1 (2018) Since ‖ BA2 ‖< ‖A‖ ‖A‖+1 < 1, we have (I − BA2)−1 exists. By using Lemma 2.1, it follows that d(fv, gv) � (I − BA2)−1BA2d(gv, gxni) + (I − BA 2)−1ABd(gxni+1, gv) +(I − BA2)−1Ad(gxni+1, gv) → θ as i → ∞. This implies that d(fv, gv) = θ i.e., fv = gv = u and hence u is a point of coincidence of f and g. Finally, to prove the uniqueness of point of coincidence, suppose that there is another point of coincidence u∗ in X such that fx = gx = u∗ for some x ∈ X. By property (∗∗), we have (u, u∗) ∈ E(G̃). Then, d(u, u∗) = d(fv, fx) � B[d(fv, gx) + d(fx, gv)] = B[d(u, u∗) + d(u, u∗)] � AB[d(u, u∗) + d(u, u∗)] which implies that, d(u, u∗) � (I − AB)−1 AB d(u, u∗). So, it must be the case that ‖ d(u, u∗) ‖ ≤ ‖ (I − AB)−1AB d(u, u∗) ‖ ≤ ‖ (I − AB)−1AB ‖ ‖ d(u, u∗) ‖ . Since ‖ (I − AB)−1AB ‖< 1, we have ‖ d(u, u∗) ‖= 0 i.e., u = u∗. Therefore, f and g have a unique point of coincidence in X. If f and g are weakly compatible, then by Proposition 2.13, f and g have a unique common fixed point in X. Corollary 3.10. Let (X, A, d) be a complete C∗-algebra valued b-metric space endowed with a graph G and the mapping f : X → X be such that d(fx, fy) � B [d(fx, y) + d(fy, x)] (3.9) for all x, y ∈ X with (x, y) ∈ E(G̃), where B ∈ A ′ + and ‖ BA ‖< 1 ‖A‖+1 . Suppose (X, A, d, G) has the property (∗)́. Then f has a fixed point in X if Cf 6= ∅. Moreover, f has a unique fixed point in X if the graph G has the property (∗ ∗ )́. Proof. The proof can be obtained from Theorem 3.9 by putting g = I. CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 55 Corollary 3.11. Let (X, A, d) be a C∗-algebra valued b-metric space and the mappings f, g : X → X be such that (3.5) holds for all x, y ∈ X, where B ∈ A ′ + and ‖ BA ‖< 1 ‖A‖+1 . If f(X) ⊆ g(X) and g(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X. Proof. The proof can be obtained from Theorem 3.9 by taking G = G0. Corollary 3.12. Let (X, A, d) be a complete C∗-algebra valued b-metric space and the mapping f : X → X be such that (3.9) holds for all x, y ∈ X, where B ∈ A ′ + with ‖ BA ‖< 1 ‖A‖+1 . Then f has a unique fixed point in X. Proof. The proof follows from Theorem 3.9 by taking G = G0 and g = I. Remark 3.13. We observe that Brian Fisher’s theorem in a complete metric space can be obtained from Corollary 3.12 by taking A = C, A = 1. Thus, Theorem 3.9 is a generalization of Brian Fisher’s theorem in metric spaces to C∗-algebra valued b-metric spaces. Corollary 3.14. Let (X, A, d) be a complete C∗-algebra valued b-metric space endowed with a partial ordering ⊑ and the mapping f : X → X be such that (3.9) holds for all x, y ∈ X with x ⊑ y or, y ⊑ x, where B ∈ A ′ + with ‖ BA ‖< 1 ‖A‖+1 . Suppose (X, A, d, ⊑) has the property (†). If there exists x0 ∈ X such that f nx0, f mx0 are comparable for m, n = 0, 1, 2, · · · , then f has a fixed point in X. Moreover, f has a unique fixed point in X if the property (††) holds. Proof. The proof can be obtained from Theorem 3.9 by taking G = G2 and g = I. Theorem 3.15. Let (X, A, d) be a C∗-algebra valued b-metric space endowed with a graph G and the mappings f, g : X → X be such that d(fx, fy) � B [d(fx, gx) + d(fy, gy)] (3.10) for all x, y ∈ X with (gx, gy) ∈ E(G̃), where B ∈ A ′ + and ‖ B ‖< 1 ‖A‖+1 . Suppose f(X) ⊆ g(X) and g(X) is a complete subspace of X with the property (∗). Then f and g have a point of coincidence in X if Cgf 6= ∅. Moreover, f and g have a unique point of coincidence in X if the graph G has the property (∗∗). Furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in X. Proof. We observe that B(d(fx, gx) + d(fy, gy)) is a positive element. Suppose that Cgf 6= ∅. We choose an x0 ∈ Cgf and keep it fixed. We can construct a sequence (gxn) such that gxn = fxn−1, n = 1, 2, 3, · · · . Evidently, (gxn, gxm) ∈ E(G̃) for m, n = 0, 1, 2, · · · . 56 Sushanta Kumar Mohanta CUBO 20, 1 (2018) For any n ∈ N, we have by using condition (3.10) that d(gxn, gxn+1) = d(fxn−1, fxn) � B[d(fxn−1, gxn−1) + d(fxn, gxn)] = B d(gxn, gxn−1) + B d(gxn, gxn+1) which implies that, (I − B)d(gxn, gxn+1) � Bd(gxn, gxn−1). (3.11) Since B ∈ A ′ + and ‖ B ‖< 1 2 , by Lemma 2.1, it follows that B � I and (I − B) is invertible with ‖ B(I−B)−1 ‖=‖ (I−B)−1B ‖< 1. Furthermore, (I−B), (I−B)−1 ∈ A ′ + and so, (I−B) −1B ∈ A ′ +. Again, by using Lemma 2.1(iv), it follows from condition (3.11) that d(gxn, gxn+1) � (I − B) −1B d(gxn, gxn−1) = td(gxn−1, gxn), (3.12) where t = (I − B)−1B ∈ A ′ +. By repeated use of condition (3.12), we get d(gxn, gxn+1) � t nd(gx0, gx1) = t nB0, (3.13) for all n ∈ N, where B0 = d(gx0, gx1) ∈ A+. We now prove that if ‖ B ‖< 1 ‖A‖+1 , then ‖ t ‖< 1 ‖A‖ . We have, ‖ t ‖ = ‖ (I − B)−1B ‖ ≤ ‖ (I − B)−1 ‖‖ B ‖ ≤ 1 1− ‖ B ‖ ‖ B ‖ < 1 ‖ A ‖ , since ‖ B ‖< 1 ‖ A ‖ +1 . CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 57 For any m, n ∈ N with m > n, we have by using condition (3.13) that d(gxn, gxm) � A[d(gxn, gxn+1) + d(gxn+1, gxm)] � Ad(gxn, gxn+1) + A 2d(gxn+1, gxn+2) + · · · +Am−n−1d(gxm−2, gxm−1) + A m−n−1d(gxm−1, gxm) � AtnB0 + A 2tn+1B0 + A 3tn+2B0 + · · · +Am−n−1tm−2B0 + A m−n−1tm−1B0 � m−n∑ k=1 Aktn+k−1B0, since A � I and A ∈ A ′ + � m−n∑ k=1 ‖ Aktn+k−1B0 ‖ I � ‖ B0 ‖‖ A ‖‖ t ‖ n m−n∑ k=1 (‖ A ‖‖ t ‖)k−1I � ‖ B0 ‖‖ A ‖‖ t ‖ n 1 1− ‖ A ‖‖ t ‖ I → θ as n → ∞. Therefore, (gxn) is a Cauchy sequence with respect to A. By completeness of g(X), there exists an u ∈ g(X) such that lim n→∞ gxn = u = gv for some v ∈ X . By property (∗), there exists a subsequence (gxni) of (gxn) such that (gxni, gv) ∈ E(G̃) for all i ≥ 1. Using condition (3.10), we have d(fv, gv) � A[d(fv, fxni) + d(fxni, gv)] � AB[d(fv, gv) + d(fxni, gxni)] + Ad(gxni+1, gv) which implies that, (I − AB)d(fv, gv) � ABd(gxni+1, gxni) + Ad(gxni+1, gv). Since ‖ AB ‖< ‖A‖ ‖A‖+1 < 1, we have (I − AB)−1 exists and (I − AB) ∈ A ′ +. By using Lemma 2.1, it follows that d(fv, gv) � (I − AB)−1ABd(gxni+1, gxni) + (I − AB) −1Ad(gxni+1, gv). Then, ‖ d(fv, gv) ‖ ≤ ‖ (I − AB)−1AB ‖ ‖ d(gxni+1, gxni) ‖ + ‖ (I − AB)−1A ‖ ‖ d(gxni+1, gv) ‖ ≤ ‖ (I − AB)−1AB ‖ ‖ t ‖ni ‖ B0 ‖ + ‖ (I − AB)−1A ‖ ‖ d(gxni+1, gv) ‖ → 0 as i → ∞. 58 Sushanta Kumar Mohanta CUBO 20, 1 (2018) This implies that d(fv, gv) = θ i.e., fv = gv = u and hence u is a point of coincidence of f and g. Finally, to prove the uniqueness of point of coincidence, suppose that there is another point of coincidence u∗ in X such that fx = gx = u∗ for some x ∈ X. By property (∗∗), we have (u, u∗) ∈ E(G̃). Then, d(u, u∗) = d(fv, fx) � B[d(fv, gv) + d(fx, gx)] = θ which implies that, d(u, u∗) = θ i.e., u = u∗. Therefore, f and g have a unique point of coincidence in X. If f and g are weakly compatible, then by Proposition 2.13, f and g have a unique common fixed point in X. Corollary 3.16. Let (X, A, d) be a complete C∗-algebra valued b-metric space endowed with a graph G and the mapping f : X → X be such that d(fx, fy) � B [d(fx, x) + d(fy, y)] (3.14) for all x, y ∈ X with (x, y) ∈ E(G̃), where B ∈ A ′ + and ‖ B ‖< 1 ‖A‖+1 . Suppose (X, A, d, G) has the property (∗)́. Then f has a fixed point in X if Cf 6= ∅. Moreover, f has a unique fixed point in X if the graph G has the property (∗ ∗ )́. Proof. The proof can be obtained from Theorem 3.15 by putting g = I. Corollary 3.17. Let (X, A, d) be a C∗-algebra valued b-metric space and the mappings f, g : X → X be such that (3.10) holds for all x, y ∈ X, where B ∈ A ′ + and ‖ B ‖< 1 ‖A‖+1 . If f(X) ⊆ g(X) and g(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X. Proof. The proof can be obtained from Theorem 3.15 by taking G = G0. Corollary 3.18. Let (X, A, d) be a complete C∗-algebra valued b-metric space and the mapping f : X → X be such that (3.14) holds for all x, y ∈ X, where B ∈ A ′ + with ‖ B ‖< 1 ‖A‖+1 . Then f has a unique fixed point in X. Proof. The proof follows from Theorem 3.15 by taking G = G0 and g = I. CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 59 Remark 3.19. We observe that Kannan’s fixed point theorem in a complete metric space can be obtained from Corollary 3.18 by taking A = C, A = 1. Thus, Theorem 3.15 is a generalization of Kannan’s fixed point theorem in metric spaces to C∗-algebra valued b-metric spaces. Corollary 3.20. Let (X, A, d) be a complete C∗-algebra valued b-metric space endowed with a partial ordering ⊑ and the mapping f : X → X be such that (3.14) holds for all x, y ∈ X with x ⊑ y or, y ⊑ x, where B ∈ A ′ + with ‖ B ‖< 1 ‖A‖+1 . Suppose (X, A, d, ⊑) has the property (†). If there exists x0 ∈ X such that f nx0, f mx0 are comparable for m, n = 0, 1, 2, · · · , then f has a fixed point in X. Moreover, f has a unique fixed point in X if the property (††) holds. Proof. The proof can be obtained from Theorem 3.15 by taking G = G2 and g = I. We furnish some examples in favour of our results. Example 3.21. Let X = R and B(H) be the set of all bounded linear operators on a Hilbert space H. Define d : X × X → B(H) by d(x, y) =| x − y |3 I for all x, y ∈ X, where I is the identity operator on H. Then (X, B(H), d) is a complete C∗-algebra valued b-metric space with the coefficient A = 4I. Let G be a digraph such that V(G) = X and E(G) = ∆∪{( 1 n , 0) : n = 1, 2, 3 · · · }. Let f, g : X → X be defined by fx = x 5 , if x 6= 4 5 = 1, if x = 4 5 and gx = 2x for all x ∈ X. Obviously, f(X) ⊆ g(X) = X. If x = 0, y = 1 2n , n = 1, 2, 3, · · · , then gx = 0, gy = 1 n and so (gx, gy) ∈ E(G̃). For x = 0, y = 1 2n , we have d(fx, fy) = d ( 0, 1 10n ) = 1 103.n3 I ≺ 1 25n3 I = 1 25 d(gx, gy) = B∗ d(gx, gy)B, where B = 1 5 I ∈ B(H). 60 Sushanta Kumar Mohanta CUBO 20, 1 (2018) Therefore, d(fx, fy) � B∗ d(gx, gy)B for all x, y ∈ X with (gx, gy) ∈ E(G̃), where B ∈ B(H) and ‖ B ‖2< 1 ‖A‖ . We can verify that 0 ∈ Cgf. In fact, gxn = fxn−1, n = 1, 2, 3, · · · gives that gx1 = f0 = 0 ⇒ x1 = 0 and so gx2 = fx1 = 0 ⇒ x2 = 0. Proceeding in this way, we get gxn = 0 for n = 0, 1, 2, · · · and hence (gxn, gxm) = (0, 0) ∈ E(G̃) for m, n = 0, 1, 2, · · · . Also, any sequence (gxn) with the property (gxn, gxn+1) ∈ E(G̃) must be either a constant sequence or a sequence of the following form gxn = 0, if n is odd = 1 n , if n is even where the words ’odd’ and ’even’ are interchangeable. Consequently it follows that property (∗) holds. Furthermore, f and g are weakly compatible. Thus, we have all the conditions of Theorem 3.2 and 0 is the unique common fixed point of f and g in X. Remark 3.22. It is worth mentioning that weak compatibility condition in Theorem 3.2 cannot be relaxed. In Example 3.21, if we take gx = 2x − 9 for all x ∈ X instead of gx = 2x, then 5 ∈ Cgf and f(5) = g(5) = 1 but g(f(5)) 6= f(g(5)) i.e., f and g are not weakly compatible. However, all other conditions of Theorem 3.2 are satisfied. We observe that 1 is the unique point of coincidence of f and g without being any common fixed point. Remark 3.23. In Example 3.21, f is a C∗-algebra valued G-contraction but it is not a C∗- algebra valued contraction. In fact, for x = 4 5 , y = 0, we have d(fx, fy) = d(1, 0) = I = 125 64 . 64 125 I = 125 64 d(x, y) ≻ B∗ d(x, y) B, for any B ∈ B(H) with ‖ B ‖2< 1 ‖A‖ . This implies that f is not a C∗-algebra valued contraction. The following example shows that property (∗) is necessary in Theorem 3.2. Example 3.24. Let X = [0, ∞) and B(H) be the set of all bounded linear operators on a Hilbert space H. Define d : X×X → B(H) by d(x, y) =| x−y |3 I for all x, y ∈ X, where I is the identity op- erator on H. Then (X, B(H), d) is a complete C∗-algebra valued b-metric space with the coefficient CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 61 A = 4I. Let G be a digraph such that V(G) = X and E(G) = ∆∪{(x, y) : (x, y) ∈ (0, 1]×(0, 1], x ≥ y}. Let f, g : X → X be defined by fx = x 6 , if x 6= 0 = 1, if x = 0 and gx = x 2 for all x ∈ X. Obviously, f(X) ⊆ g(X) = X. For x, y ∈ X with (gx, gy) ∈ E(G̃), we have d (fx, fy) = 1 27 d (gx, gy) � 1 9 d (gx, gy) = B∗ d(gx, gy) B, where B = 1 3 I ∈ B(H) with ‖ B ‖2< 1 ‖A‖ . We see that f and g have no point of coincidence in X. We now verify that the property (∗) does not hold. In fact, (gxn) is a sequence in X with gxn → 0 and (gxn, gxn+1) ∈ E(G̃) for all n ∈ N where xn = 2 n . But there exists no subsequence (gxni) of (gxn) such that (gxni, 0) ∈ E(G̃). Example 3.25. Let X = R and B(H) be the set of all bounded linear operators on a Hilbert space H. Choose a positive operator T ∈ B(H). Define d : X×X → B(H) by d(x, y) =| x−y |5 T for all x, y ∈ X. Then (X, B(H), d) is a complete C∗-algebra valued b-metric space with the coefficient A = 16I. Let f, g : X → X be defined by fx = 2, if x 6= 5 = 3, if x = 5 and gx = 3x − 4 for all x ∈ X. Obviously, f(X) ⊆ g(X) = X. Let G be a digraph such that V(G) = X and E(G) = ∆ ∪ {(2, 3), (3, 5)}. If x = 2, y = 7 3 , then gx = 2, gy = 3 and so (gx, gy) ∈ E(G̃). Again, if x = 7 3 , y = 3, then gx = 3, gy = 5 and so (gx, gy) ∈ E(G̃). It is easy to verify that condition (3.5) of Theorem 3.9 holds for all x, y ∈ X with (gx, gy) ∈ E(G̃). Furthermore, 2 ∈ Cgf i.e., Cgf 6= ∅, f and g are weakly compatible, and (X, B(H), d, G) has the property (∗). Thus, all the conditions of Theorem 3.9 are satisfied and 2 is the unique common fixed point of f and g in X. Remark 3.26. It is observed that in Example 3.25, f is not a Fisher G-contraction. In fact, 62 Sushanta Kumar Mohanta CUBO 20, 1 (2018) for x = 3, y = 5, we have B [d(fx, y) + d(fy, x)] = B [d(2, 5) + d(3, 3)] = 243BT = 243 16 BAT = 243 16 × 17 17BAT ≺ T = d(fx, fy), for any B ∈ B(H) ′ + with ‖ BA ‖< 1 ‖A‖+1 . This implies that f is not a Fisher G-contraction. The following example supports our Theorem 3.15. Example 3.27. Let X = [0, ∞) and B(H) be the set of all bounded linear operators on a Hilbert space H. Choose a positive operator T ∈ B(H). Define d : X×X → B(H) by d(x, y) =| x−y |2 T for all x, y ∈ X. Then (X, B(H), d) is a complete C∗-algebra valued b-metric space with the coefficient A = 2I. Let G be a digraph such that V(G) = X and E(G) = ∆ ∪ {(4tx, 4t(x + 1)) : x ∈ X with x ≥ 2, t = 0, 1, 2, · · · }. Let f, g : X → X be defined by fx = 4x and gx = 16x for all x ∈ X. Clearly, f(X) = g(X) = X. If x = 4t−2z, y = 4t−2(z + 1), then gx = 4tz, gy = 4t(z + 1) and so (gx, gy) ∈ E(G̃) for all z ≥ 2. For x = 4t−2z, y = 4t−2(z + 1), z ≥ 2 with B = 1 117 I, we have d(fx, fy) = d ( 4t−1z, 4t−1(z + 1) ) = 42t−2T � 1 117 42t−2(18z2 + 18z + 9)T = 1 117 [ d ( 4t−1z, 4tz ) + d ( 4t−1(z + 1), 4t(z + 1) )] = B [d(fx, gx) + d(fy, gy)]. Thus, condition (3.10) is satisfied for all x, y ∈ X with (gx, gy) ∈ E(G̃). It is easy to verify that 0 ∈ Cgf. Also, any sequence (gxn) with gxn → x and (gxn, gxn+1) ∈ E(G̃) must be a constant sequence and hence property (∗) holds. Furthermore, f and g are weakly compatible. Thus, we have all the conditions of Theorem 3.15 and 0 is the unique common fixed point of f and g in X. Remark 3.28. It is easy to observe that in Example 3.27, f is a C∗-algebra valued G-Kannan operator with B = 16 117 I. But f is not a C∗-algebra valued Kannan operator because, if x = 4, y = 0, CUBO 20, 1 (2018) Common Fixed Point Results in C∗-Algebra . . . 63 then for any arbitrary B ∈ B(H) ′ + with ‖ B ‖< 1 ‖A‖+1 = 1 3 (which implies 3B ≺ I), we have B [d(fx, x) + d(fy, y)] = B [d(f4, 4) + d(f0, 0)] = 144BT = 144 3 × 256 (3B)(256T) ≺ 256T = d(fx, fy). References [1] A. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64, 2014, 941-960. [2] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341, 2008, 416-420. [3] M. R. Alfuraidan, M. A. 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Mohanta, Some fixed point theorems using wt-distance in b-metric spaces, Fasciculi Mathematici, no. 54, 2015, 125-140. [26] J. J. Nieto and R. Rodŕiguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, Englosh Ser., 2007, 2205-2212. [27] D. Reem, S. Reich, A. J. Zaslavski, Two results in metric fixed point theory, J. Fixed Point Theory Appl., 1, 2007, 149-157. Introduction Some basic concepts Main Results