J. Fixed Point Theory Appl. 87:02)1( https://doi.org/10.1007/s11784-018-0566-x c Springer International Publishing AG,  part of Springer Nature 2018

Journal of Fixed Point Theory and Applications

Some fixed point results in D ∗-quasimetric spaces Ion Marian Olaru and Adrian Nicolae Branga Abstract. In this paper, we prove some coincidence and common fixed point theorems for self-maps satisfying contractive conditions in a D∗ quasimetric space. Mathematics Subject Classification. Primary 47H10, 54H25; Secondary 55M20. Keywords. D∗ -quasimetric space, common fixed point theorems, coincidently commuting self-maps, point of coincidence.

1. Introduction In 1984, Dhage [3] introduced the notion of D-metric space. He proved the existence of a unique fixed point of a self-map satisfying a contractive condition in a complete and bounded D-metric space. Dealing with D-metric spaces, Ahmad et al. [2], Dhage [4–8], Rhoades [10] and others made a significant contribution to the fixed point theory of D-metric spaces. The examples given in Dhage [3] are the only examples in a D-metric space and its generalizations. In 2007, Shaban Sedghi et al [11] modified the D-metric space and defined the D∗ -metric space. Aage [1] proved some fixed point results for a self-map satisfying a contractive condition in a D∗ -metric space framework. In this paper, we will give some common fixed point theorems for a self-map satisfying contractive conditions in a complete D∗ -quasimetric space.

2. D ∗ -quasimetric space In this section, we will give the definition of the D∗ -quasimetric space, the convergence and the Cauchy property of the sequences in a D∗ -quasimetric space and will prove some fundamental properties of sequences defined on such space. The concept of the generalized quasimetric space is defined as follows:

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Definition 2.1. Let X be a nonempty set. A generalized quasimetric (or D∗ quasimetric) on X is a function D∗ : X × X × X → R+ that satisfies the following condition, for all x, y, z, t ∈ X: (D1∗ ) D∗ (x, y, z) = 0 if and only if x = y = z; (D2∗ ) D∗ (x, y, z) = D∗ (p{x, y, z}) (symmetry), where p is a permutation function; (D3∗ ) D∗ (x, y, z) ≤ a[D∗ (x, y, t) + D∗ (t, z, z)], where a ≥ 1 is a given real number. The pair (X, D∗ ) is called a generalized quasimetric (or D∗ -quasimetric) space. Example. For any metric space (X, d), the function D∗ : X × X × X → R+ defined by D∗ (x, y, z) = θ(d(x, y) + d(y, z) + d(z, x)), where θ > 0, is a D∗ quasimetric on X. Let x, y, z, t ∈ X be arbitrary elements, θ > 0 and a ≥ 1. Based on the metric properties, we have d(y, z) ≤ d(y, t) + d(t, z) and d(z, x) ≤ d(z, t) + d(t, x). Consequently, a[D∗ (x, y, t) + D∗ (t, z, z)] − D∗ (x, y, z) = a[θ(d(x, y) + d(y, t) + d(t, x)) + θ(d(t, z) + d(z, z) + d(z, t))] − θ(d(x, y) + d(y, z) + d(z, x)) = aθ(d(x, y) + d(y, t) + d(t, x) + 2d(t, z)) − θ(d(x, y) + d(y, z) + d(z, x)) ≥ ≥ aθ(d(x, y) + d(y, t) + d(t, x) + 2d(t, z)) − θ(d(x, y) + d(y, t) + d(t, z) + d(z, t) + d(t, x)) = (a − 1)θ(d(x, y) + d(y, t) + d(t, x) + 2d(t, z)) ≥ 0. Example. Let A = {a, b, c} ⊂ R \ [0, 1], a = b = c, and α, β, γ, δ > 0 such that α + β ≥ γ, β + γ ≥ α, γ + α ≥ β, 2δ ≥ max{α, β, γ}. Define the set X = A ∪ [0, 1] and the function d : X × X → R+ , ⎧ 0, x=y ⎪ ⎪ ⎪ ⎪ α, x = a, y = b or x = b, y = a ⎪ ⎪ ⎨ β, x = b, y = c or x = c, y = b d(x, y) = γ, x = c, y = a or x = a, y = c ⎪ ⎪ ⎪ ⎪ δ, x ∈ A, y ∈ [0, 1] or x ∈ [0, 1], y ∈ A ⎪ ⎪ ⎩ |x − y|, otherwise. It is easy to prove that d is a metric on space X. According to the previous example, the function D∗ : X × X × X → R+ , D∗ (x, y, z) = θ(d(x, y) + d(y, z) + d(z, x)), where θ > 0, is a D∗ -quasimetric on X. Definitions 2.2. (a) A sequence (xn )n∈N ⊂ X converges to x ∈ X if and only if, for each ε > 0 there exists N ∈ N such that, for all n ≥ N , we have D∗ (xn , xn , x) < ε. (b) A sequence (xn )n∈N ⊂ X is called a Cauchy sequence if, for each ε > 0, there exists N ∈ N such that D∗ (xn , xn , xm ) < ε, for each n, m ≥ N . The D∗ -quasimetric space (X, D∗ ) is said to be complete if every Cauchy sequence is convergent.

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Lemma 2.3. (i) A sequence (xn )n∈N ⊂ X converges to x ∈ X if and only if, for each ε > 0, there exists N ∈ N such that, for all n ≥ N , we have D∗ (xn , x, x) < ε. (ii) A sequence (xn )n∈N ⊂ X converges to x ∈ X if and only if, for each ε > 0, there exists N ∈ N such that, for all n, m ≥ N , we have D∗ (xn , xm , x) < ε. (iii) A sequence (xn )n∈N ⊂ X is a Cauchy sequence if and only if, for each ε > 0, there exists N ∈ N such that D∗ (xn , xm , xp ) < ε, for each n, m, p ≥ N . Proof. (i) Let us suppose that (xn )n∈N converges to x ∈ X and let ε > 0. Then, there is N ∈ N such that D∗ (xn , xn , x) < aε , for all n ≥ N . The properties (D1∗ ) − (D3∗ ) of the D∗ -quasimetric imply that D∗ (xn , x, x) ≤ aD∗ (xn , xn , x) < ε, for all n ≥ N . Conversely, let us suppose that there is N ∈ N such that D∗ (xn , x, x) < aε , for all n ≥ N . Similarly, we get D∗ (xn , xn , x) ≤ aD∗ (xn , x, x) < ε, for all n ≥ N . (ii) Let us suppose that (xn )n∈N converges to x ∈ X and let ε > 0. From (i), there is N ∈ N such that, for all n ≥ N , we have D∗ (xn , x, x) < ε ∗ ∗ ∗ 2a . Then, D (xn , xm , x) ≤ a[D (xn , x, x) + D (x, xm , xm )] < ε, for all n, m ≥ N . On the contrary, we set n = m. (iii) Let us suppose that (xn )n∈N is a Cauchy sequence. Then, there is ε , for each n, m ≥ N . It follows N ∈ N such that D∗ (xn , xn , xm ) < 2a ∗ ∗ that D (xn , xm , xp ) ≤ a[D (xn , xm , x) + D∗ (x, xp , xp )] < ε, for each n, m, p ≥ N . Conversely, we set p = m.  Lemma 2.4. Let (X, D∗ ) be a D∗ -quasimetric space and (xn )n∈N ⊂ X a sequence. (i) if xn converges to x, then x is unique; (ii) if xn converges to x, then xn is a Cauchy sequence. Proof. (i) Let us suppose that xn converges to x, respectively, y, and x = y. ε Then, for each ε > 0 there exists n1 , n2 ∈ N such that D∗ (x, x, xn ) ≤ 2a ε ∗ (for n ≥ n1 ) and D (y, y, xn ) ≤ 2a (for n ≥ n2 ). Let us consider n0 = max{n1 , n2 }. Then, for every n ≥ n0 , we have D∗ (x, x, y) ≤ a[D∗ (x, x, xn ) + D∗ (xn , y, y)] < ε. Hence, D∗ (x, x, y) = 0 and, consequently, x = y. (ii) Since xn converges to x, it follows that, for each ε > 0, there exists ε . Then, for every n, m ≥ n0 , we n0 ∈ N such that D∗ (xn , xn , x) < 2a ∗ ∗ have D (xn , xn , xm ) ≤ a[D (xn , xn , x)+ D∗ (x, xm , xm )] < ε. Therefore,  the sequence xn is a Cauchy sequence.

3. Results In this section, we obtain several coincidence and common fixed point theorems for mappings defined on a D∗ -quasimetric space (X, D∗ ).

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Definition 3.1. Let f and g be self-maps of a set X. If w = f x = gx for some x ∈ X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. Jungck [9] said that a pair of self-mappings are weakly compatible (or weakly commuting) if they commute at their coincidence points. In the following, we will use the terminology coincidently commuting, which is more meaningful. Proposition 3.2. [9] Let f and g be coincidently commuting self-maps on a set X. If f and g have a unique point of coincidence w = f x = gx, then w is the unique common fixed point of f and g. Theorem 3.3. Let (X, D∗ ) be a D∗ -quasimetric space and let S, T : X → X be two mappings which satisfy the following conditions: (i) T (X) ⊂ S(X); (ii) T (X) or S(X) is complete; (iii) there exists α, β, γ, δ ≥ 0, a(α + β + γ + δ) < 1, such that D∗ (T x, T y, T z) ≤ αD∗ (Sx, Sy, Sz) + βD∗ (Sx, T x, T x) +γD∗ (Sy, T y, T y) + δD∗ (Sz, T z, T z),

f orall x, y, z ∈ X.

Then, S and T have a unique point of coincidence in X. Moreover, if S and T are coincidently commuting, then S and T have a unique common fixed point. Proof. Let x0 ∈ X be arbitrary in X. Choose a point x1 ∈ X such that T x0 = Sx1 . Continuing this process, having chosen xn ∈ X, we obtain xn+1 ∈ X such that Sxn = T xn−1 . Then, from (iii), we have D∗ (Sxn , Sxn+1 , Sxn+1 ) = D∗ (T xn−1 , T xn , T xn ) ≤ αD∗ (Sxn−1 , Sxn , Sxn ) + βD∗ (Sxn−1 , T xn−1 , T xn−1 ) +γD∗ (Sxn , T xn , T xn ) + δD∗ (Sxn , T xn , T xn ) = αD∗ (Sxn−1 , Sxn , Sxn ) + βD∗ (Sxn−1 , Sxn , Sxn ) +γD∗ (Sxn , Sxn+1 , Sxn+1 ) + δD∗ (Sxn , Sxn+1 , Sxn+1 ) = (α + β)D∗ (Sxn−1 , Sxn , Sxn ) + (γ + δ)D∗ (Sxn , Sxn+1 , Sxn+1 ). This implies that D∗ (Sxn , Sxn+1 , Sxn+1 ) ≤ qD∗ (Sxn−1 , Sxn , Sxn ), where q =

α+β 1−(γ+δ) ∗

∈ [0, 1). From the above relation, we obtain

D (Sxn , Sxn+1 , Sxn+1 ) ≤ q n D∗ (Sx0 , Sx1 , Sx1 )

and

(3.1)

(3.2) D∗ (Sxn , Sxn , Sxn+1 ) ≤ aq n D∗ (Sx0 , Sx1 , Sx1 ). Then, from all n ∈ N, by using the properties (D1∗ ) − (D3∗ ) of the D∗ quasimetric, we get D∗ (Sxn , Sxn , Sxn+2 ) ≤ aD∗ (Sxn , Sxn , Sxn+1 ) + aD∗ (Sxn+1 , Sxn+2 , Sxn+2 ) ≤ aD∗ (Sxn , Sxn , Sxn+1 ) + a2 D∗ (Sxn+1 , Sxn+1 , Sxn+2 ),

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D∗ (Sxn , Sxn , Sxn+3 ) ≤ a[D∗ (Sxn , Sxn , Sxn+1 ) + D∗ (Sxn+1 , Sxn+3 , Sxn+3 )] = aD∗ (Sxn , Sxn , Sxn+1 ) + aD∗ (Sxn+3 , Sxn+3 , Sxn+1 ) ≤ aD∗ (Sxn , Sxn , Sxn+1 ) + a2 [D∗ (Sxn+3 , Sxn+3 , Sxn+2 ) + D∗ (Sxn+2 , Sxn+1 , Sxn+1 )] = aD∗ (Sxn , Sxn , Sxn+1 ) + a2 D∗ (Sxn+1 , Sxn+1 , Sxn+2 ) + a2 D∗ (Sxn+2 , Sxn+3 , Sxn+3 ) ≤ aD∗ (Sxn , Sxn , Sxn+1 ) + a2 D∗ (Sxn+1 , Sxn+1 , Sxn+2 ) + a3 D∗ (Sxn+2 , Sxn+2 , Sxn+3 ). By similar arguments, for all n ∈ N and p ≥ 1, we have D∗ (Sxn , Sxn , Sxn+p ) ≤ aD∗ (Sxn , Sxn , Sxn+1 ) + · · · + ap D∗ (Sxn+p−1 , Sxn+p−1 , Sxn+p ). Hence, D∗ (Sxn , Sxn , Sxn+p ) ≤

a2 q n ∗ D (Sx0 , Sx1 , Sx1 ), 1 − aq

which implies that (Sxn )n∈N is a Cauchy sequence. We distinguish two cases: Case 1 S(X) is complete. Then, there exists u ∈ S(X) such that Sxn → u as n → ∞. Consequently, we can find p ∈ X such that Sp = u. Case 2 T (X) is complete. Taking into account that Sxn = T xn−1 , it follows that there exists u ∈ T (X) ⊂ S(X) such that Sxn → u as n → ∞. Consequently, we can find p ∈ X such that Sp = u. We claim that T p = u, whence D∗ (T p, u, u) ≤ aD∗ (T p, T p, u) ≤ a2 [D∗ (T p, T p, T xn ) + D∗ (T xn , u, u)] ≤ a2 [αD∗ (Sp, Sp, Sxn ) + βD∗ (Sp, T p, T p) + γD∗ (Sp, T p, T p) +δD∗ (Sxn , T xn , T xn )] + a2 D∗ (Sxn+1 , u, u) = a2 αD∗ (u, u, Sxn ) + a2 βD∗ (u, T p, T p) + a2 γD∗ (u, T p, T p) + a2 δD∗ (Sxn , Sxn+1 , Sxn+1 ) + a2 D∗ (Sxn+1 , u, u). From the above relation, we get D∗ (T p, T p, u) a [αD∗ (u, u, Sxn ) + δD∗ (Sxn , Sxn+1 , Sxn+1 ) ≤ 1 − a(β + γ) +D∗ (Sxn+1 , u, u)]. Therefore, D∗ (T p, T p, u) = 0, so T p = Sp = u. Next, we prove that S and T have a unique point of coincidence. For this, we assume that there exists a point q ∈ X such that Sq = T q = u1 . Then,

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D∗ (T p, T p, T q) ≤ αD∗ (Sp, Sp, Sq) + βD∗ (Sp, T p, T p) + γD∗ (Sp, T p, T p) +δD∗ (Sq, T q, T q) = αD∗ (T p, T p, T q) + βD∗ (T p, T p, T p) + γD∗ (T p, T p, T p) +δD∗ (T q, T q, T q) = αD∗ (T p, T p, T q). From the above inequality, we obtain that D∗ (T p, T p, T q) = 0 and consequently, u = u1 . Moreover, if S and T are coincidently commuting, then, via Proposition 3.2, it follows that u is the unique common fixed point of S and T.  Theorem 3.4. Let (X, D∗ ) be a complete D∗ -quasimetric space and let T : X → X be a mapping which satisfies the following condition: (C) there exists α, β, γ, δ ≥ 0, a(α + β + γ + δ) < 1, such that D∗ (T x, T y, T z) ≤ αD∗ (x, y, z) + βD∗ (x, T x, T x) +γD∗ (y, T y, T y) + δD∗ (z, T z, T z),

f orall x, y, z ∈ X.

Then, T has a unique fixed point in X. 

Proof. Follows from Theorem 3.3, for S = 1X .

Theorem 3.5. Let (X, D∗ ) be a D∗ -quasimetric space and let S, T : X → X be two mappings which satisfy the following conditions: (i) T (X) ⊂ S(X); (ii) T (X) or S(X) is complete; (iii) there exists α, β, γ ≥ 0, a2 (2α + 2β + 2γ) < 1, such that D∗ (T x, T y, T z) ≤ α[D∗ (Sx, T y, T y) + D∗ (Sy, T x, T x)] +β[D∗ (Sy, T z, T z) + D∗ (Sz, T y, T y)] +γ[D∗ (Sx, T z, T z) + D∗ (Sz, T x, T x)],

f orall x, y, z ∈ X.

Then, S and T have a unique point of coincidence in X. Moreover, if S and T are coincidently commuting, then S and T have a unique common fixed point. Proof. Let x0 ∈ X be arbitrary in X. Choose a point x1 ∈ X such that T x0 = Sx1 . Continuing this process, having chosen xn ∈ X, we obtain xn+1 ∈ X such that Sxn = T xn−1 . Then, from (iii), we have D∗ (Sxn , Sxn+1 , Sxn+1 ) = D∗ (T xn−1 , T xn , T xn ) ≤ α[D∗ (Sxn−1 , T xn , T xn ) + D∗ (Sxn , T xn−1 , T xn−1 )] +β[D∗ (Sxn , T xn , T xn ) + D∗ (Sxn , T xn , T xn )] +γ[D∗ (Sxn−1 , T xn , T xn ) + D∗ (Sxn , T xn−1 , T xn−1 )] = (α + γ)D∗ (Sxn−1 , Sxn+1 , Sxn+1 ) + 2βD∗ (Sxn , Sxn+1 , Sxn+1 ) ≤ a(α + γ)[D∗ (Sxn+1 , Sxn+1 , Sxn ) + D∗ (Sxn , Sxn−1 , Sxn−1 )]

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+2βD∗ (Sxn , Sxn+1 , Sxn+1 ) = a(α + γ)D∗ (Sxn+1 , Sxn+1 , Sxn ) + a(α + γ)D∗ (Sxn , Sxn−1 , Sxn−1 ) +2βD∗ (Sxn , Sxn+1 , Sxn+1 ) ≤ a(α + γ)D∗ (Sxn+1 , Sxn+1 , Sxn ) + a2 (α + γ)D∗ (Sxn−1 , Sxn , Sxn ) +2βD∗ (Sxn , Sxn+1 , Sxn+1 ). Hence, D∗ (Sxn , Sxn+1 , Sxn+1 ) ≤ qD∗ (Sxn−1 , Sxn , Sxn ), 2

a (α+γ) ∈ [0, 1). Analogous with the proof of Theorem 3.3, we where 1−[a(α+γ)+2β] can show that there exists p ∈ X and u ∈ X such that Sxn → u as n → ∞ and Sp = u. We claim that T p = u. Indeed, from the next relations

D∗ (T p, T p, u) ≤ a[D∗ (T p, T p, T xn ) + D∗ (T xn , u, u)] = aD∗ (T p, T p, T xn ) + aD∗ (Sxn+1 , u, u) ≤ aα[D∗ (Sp, T p, T p) + D∗ (Sp, T p, T p)] +aβ[D∗ (Sp, T xn , T xn ) + D∗ (Sxn , T p, T p)] +aγ[D∗ (Sp, T xn , T xn ) + D∗ (Sxn , T p, T p)] + aD∗ (Sxn+1 , u, u) = 2aαD∗ (u, T p, T p) + aβD∗ (u, Sxn+1 , Sxn+1 ) + aβD∗ (Sxn , T p, T p) +aγD∗ (u, Sxn+1 , Sxn+1 ) + aγD∗ (Sxn , T p, T p) + aD∗ (Sxn+1 , u, u) ≤ 2aαD∗ (u, T p, T p) + aβD∗ (u, Sxn+1 , Sxn+1 ) +a2 β[D∗ (T p, T p, u) + D∗ (u, Sxn , Sxn )] + aγD∗ (u, Sxn+1 , Sxn+1 ) +a2 γ[D∗ (T p, T p, u) + D∗ (u, Sxn , Sxn )] + aD∗ (Sxn+1 , u, u), we obtain 1 [aβD∗ (u, Sxn+1 , Sxn+1 ) 1 − 2aα − a2 β − a2 γ +aγD∗ (u, Sxn+1 , Sxn+1 ) + a2 βD∗ (u, Sxn , Sxn )

D∗ (T p, T p, u) ≤

+a2 γD∗ (u, Sxn , Sxn ) + aD∗ (Sxn+1 , u, u)]. From the above inequality, it follows that D∗ (T p, T p, u) = 0 and, consequently, T p = u. Next, we prove that S and T have a unique point of coincidence. For this, we assume that there exists a point q ∈ X such that Sq = T q = u1 . Then, D∗ (T p, T p, T q) ≤ α[D∗ (Sp, T p, T p) + D∗ (Sp, T p, T p)] + β[D∗ (Sp, T q, T q) +D∗ (Sq, T p, T p)] + γ[D∗ (Sp, T q, T q) + D∗ (Sq, T p, T p)] = α[D∗ (T p, T p, T p) + D∗ (T p, T p, T p)] + β[D∗ (T p, T q, T q) +D∗ (T q, T p, T p)] + γ[D∗ (T p, T q, T q) + D∗ (T q, T p, T p)] ≤ (1 + a)(β + γ)D∗ (T p, T p, T q) ≤ 2a(β + γ) ≤ a2 (2α + 2β + 2γ). This implies that D∗ (T p, T p, T q) = 0, so u = u1 . Thus, u is the unique point of coincidence of S and T . Moreover, if S and T are coincidently commuting, then, via Proposition 3.2, it follows that u is the unique common fixed point of S and T . 

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Theorem 3.6. Let (X, D∗ ) be a complete D∗ -quasimetric space and let T : X → X be a mapping satisfying the following condition: (C) there exists α, β, γ > 0 such that a2 (2α + 2β + 2γ) < 1 and D∗ (T x, T y, T z) ≤ α[D∗ (x, T y, T y) + D∗ (y, T x, T x)] +β[D∗ (y, T z, T z) + D∗ (z, T y, T y)] + γ[D∗ (x, T z, T z) +D∗ (z, T x, T x)], f orall x, y, z ∈ X. Then, T has a unique fixed point in X. Proof. Follows from Theorem 3.4, for S = 1X .



Theorem 3.7. Let (X, D∗ ) be a D∗ -quasimetric space and let S, T : X → X be two mappings which satisfy the following conditions: (i) T (X) ⊂ S(X); (ii) T (X) or S(X) is complete; (iii) there exists α, β ≥ 0, α(1 + a + a2 ) + aβ < 1, such that D∗ (T x, T y, T y) ≤ α[D∗ (Sy, T y, T y) + D∗ (Sx, T y, T y)] +βD∗ (Sy, T x, T x), f orall x, y ∈ X. Then, S and T have a unique point of coincidence in X. Moreover, if S and T are coincidently commuting, then S and T have a unique common fixed point. Proof. Let x0 ∈ X be arbitrary in X. Choose a point x1 ∈ X such that T x0 = Sx1 . Continuing this process, having chosen xn ∈ X, we obtain xn+1 ∈ X such that Sxn = T xn−1 . Then, from (iii), we have D∗ (Sxn , Sxn+1 , Sxn+1 ) = D∗ (T xn−1 , T xn , T xn ) ≤ α[D∗ (Sxn , T xn , T xn ) + D∗ (Sxn−1 , T xn , T xn )] + βD∗ (Sxn , T xn−1 , T xn−1 ) = α[D∗ (Sxn , Sxn+1 , Sxn+1 ) + D∗ (Sxn−1 , Sxn+1 , Sxn+1 )] ≤ αD∗ (Sxn , Sxn+1 , Sxn+1 ) + aαD∗ (Sxn , Sxn+1 , Sxn+1 ) + aαD∗ (Sxn , Sxn−1 , Sxn−1 ) ≤ αD∗ (Sxn , Sxn+1 , Sxn+1 ) + aαD∗ (Sxn , Sxn+1 , Sxn+1 ) + a2 αD∗ (Sxn−1 , Sxn , Sxn ). Hence, D∗ (Sxn , Sxn+1 , Sxn+1 ) ≤ qD∗ (Sxn−1 , Sxn , Sxn ) ≤ · · · ≤ q n D∗ (Sx0 , Sx1 , Sx1 ), 2

a α where q = 1−(α+aα) . Analogous with the proof of Theorem 3.3, we show that there exists p ∈ X and u ∈ X such that Sxn → u and Sp = u. We claim that T p = u. Indeed, since

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D∗ (T p, T p, u) ≤ a[D∗ (T p, T p, T xn ) + D∗ (T xn , u, u)] = aD∗ (T xn , T p, T p) + aD∗ (Sxn+1 , u, u) = a{α[D∗ (Sp, T p, T p) + D∗ (Sxn , T p, T p)] + βD∗ (Sp, T xn , T xn )} + aD∗ (Sxn+1 , u, u) = aαD∗ (u, T p, T p) + aαD∗ (Sxn , T p, T p) + aβD∗ (u, Sxn+1 , Sxn+1 ) + aD∗ (Sxn+1 , u, u) ≤ aαD∗ (u, T p, T p) + a2 αD∗ (u, Sxn , Sxn ) + a2 αD∗ (T p, T p, u) + aβD∗ (u, Sxn+1 , Sxn+1 ) + aD∗ (Sxn+1 , u, u), we obtain 1 [a2 αD∗ (u, Sxn , Sxn ) 1 − aα − a2 α +aβD∗ (u, Sxn+1 , Sxn+1 ) + aD∗ (Sxn+1 , u, u)].

D∗ (T p, T p, u) ≤

Hence, D∗ (T p, T p, u) = 0, i.e., T p = u. We show that S and T have a unique point of coincidence. We assume that there exists a point q ∈ X such that Sq = T q = u1 . Then, D∗ (T q, T p, T p) ≤ α[D∗ (Sp, T p, T p) + D∗ (Sq, T p, T p)] + βD∗ (Sp, T q, T q) ≤ αD∗ (T q, T p, T p) + aβD∗ (T p, T p, T q) = (α + aβ)D∗ (T p, T p, T q). It follows that D∗ (T p, T p, T q) = 0 and, consequently, u = u1 . If S and T are coincidently commuting, then, via Proposition 3.2, we get that u is the unique common fixed point of S and T .  Theorem 3.8. Let (X, D∗ ) be a complete D∗ -quasimetric space and let T : X → X be a mapping satisfying the following condition: (C) there exists α, β ≥ 0, α(1 + a + a2 ) + aβ < 1, such that D∗ (T x, T y, T y) ≤ α[D∗ (y, T y, T y) + D∗ (x, T y, T y)] +βD∗ (y, T x, T x), f orall x, y ∈ X. Then, T has a unique fixed point in X. Proof. The proof uses Theorem 3.7, replacing S with the identity map.



Acknowledgements The authors thank the anonymous referee and the editor for their valuable comments and suggestions which improved greatly the quality of this paper.

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