Geraghty type contraction mappings on Branciari b-metric spaces

In this paper fixed points of α-admissible contraction mappings of Geraghty type defined on Branciari bmetric spaces are studied. Existence and uniqueness theorems for these types of mappings of are proved. Some consequences of these theorems are given and specific examples are presented.


Introduction and preliminaries
Branciari metric spaces are among the recent generalizations of metric spaces and have been defined by Branciari [3].The main feature of these spaces is the replacement of the triangular inequality by a rectangular inequality.The Branciari metric spaces are also referred to as rectangular or generalized metric spaces.Another recent generalization of the metric spaces called b-metric spaces has been introduced by Czerwik [4] and Bakhtin [2].The difference between metric and b-metric shows itself in the triangle inequality which contains a constant s ≥ 1. Combining these two concepts, George et.al.[5] defined Branciari b-metric spaces.This new metric space is also referred to as rectangular b-metric spaces.Several articles related with this new metric space have been published recently [5,11,7].
In this paper we discuss the problem of existence and uniqueness of fixed points for contraction mappings of Geraghty type defined on Branciari b-metric spaces.
We first introduce the basic notions used throughout the paper.Branciari metric spaces are defined as follows [3].
Definition 1.1.[3] Let X be a nonempty set and let d : X × X → [0, +∞) be a function such that for all x, y ∈ X and all distinct u, v ∈ X each of which is different from x and y, the following conditions are satisfied: (1.) d(x, y) = 0 if and only if x = y; (2.) d(x, y) = d(y, x); (3.) d(x, y) ≤ d(x, u) + d(u, v) + d(v, y).The map d is called a Branciari metric and the pair (X, d) is called a Branciari metric space.
Definition 1.2.[2,4] Let X be a nonempty set and let d : X × X → [0, +∞) be a mapping satisfying the following conditions for all x, y, z ∈ X: (1.) d(x, y) = 0 if and only if x = y; (2.) d(x, y) = d(y, x); Then the mapping d is called a b-metric and the pair (X, d) is called a b-metric space with a constant s ≥ 1.
Combination of the Branciari and b-metric spaces results in the following definition of the Branciari b-metric spaces.Definition 1.3.[5] Let X be a nonempty set and let d : X × X → [0, +∞) be a function such that for all x, y ∈ X and all distinct u, v ∈ X each of which is different from x and y, the following conditions are satisfied: (1.) d(x, y) = 0 if and only if x = y; (2.) d(x, y) = d(y, x); The map d is called a Branciari b-metric and the pair (X, d) is called a Branciari b-metric space with a constant s ≥ 1.
Convergent sequence, Cauchy sequence, completeness and continuity on Branciari b-metric space are defined as follows.
Definition 1.4.[5] Let (X, d) be a Branciari b-metric space, {x n } be a sequence in X and x ∈ X.Then 1.A sequence {x n } ⊂ X is said to converge to a point x ∈ X if, for every ε > 0 there exists n 0 ∈ N such that d(x n , x) < ε for all n > n 0 .The convergence is also represented as 2. A sequence {x n } ⊂ X is said to be a Cauchy sequence if, for every ε > 0 there exists n 0 ∈ N such that d(x n , x n+p ) < ε for all n > n 0 , p > 0 or equivalently, if lim n→∞ d(x n , x n+p ) = 0 for all p > 0. 3. (X, d) is said to be a complete Branciari b-metric space if every Cauchy sequence in X converges to some x ∈ X. 4. A mapping T : X → X on is said to be continuous with respect to the Branciari b-metric d if, for any sequence {x n } ⊂ X which converges to some x ∈ X, that is lim One should be careful when working with the Branciari and Branciari b-metric spaces due to some of their properties listed below.2. If T is the collection of all subsets Y of X such that for each y ∈ Y there exist r > 0 with B r (y) ⊆ Y, then T defines a topology for (X, d), which is not necessarily Hausdorff.3. The limit of a convergent sequence {x n } ∈ X is not necessarily unique.4. A convergent sequence in X is not necessarily a Cauchy sequence.5. Branciari or Branciari b-metric is not necessarily continuous.
All these drawbacks are illustrated in the following example inspired by [5].
It is not difficult to see that the function d(x, y) is not a metric, not a b-metric, not a Branciary metric but only a Branciari b-metric with s = 2.It is also clear that and that is, the sequence { 1 2n } has two different limits, the numbers 0 and 3.
In addition, the sequence { 1 2n } is convergent, but not a Cauchy sequence because Finally, note that the open set Therefore, when working on Branciari metric space, we need the following property stated in proved in [10].
Proposition 1.7.[10] Let {x n } be a Cauchy sequence in a Branciari metric space (X, d) such that lim n→∞ d(x n , x) = 0, where x ∈ X.Then lim n→∞ d(x n , y) = d(x, y), for all y ∈ X.In particular, the sequence {x n } does not converge to y if y = x.
Remark 1.8.The Proposition 1.7 is valid if we replace Branciari metric space by a Branciari b-metric space.

Geraghty type contraction mappings have been introduced by Geraghty [6] who defined a class F of functions
and with the help of these functions defined contraction mappings in the following manner.
Let (X, d) be a metric space and let T : X → X be a mapping satisfying for all x, y ∈ X and some function β ∈ F.He proved the existence and uniqueness of fixed points of such contractions on metric spaces.
In the context of b-metric spaces, Geraghty type contractions have been modified as follows [7].Let F s be the class of functions for all x, y ∈ X and some function β ∈ F s .
As examples of functions from the class F s we can give the following functions.

The function
for some s ≥ 1is in the class F s .
The function Finally, we recall the concept of α-admissible mappings defined by Samet et al [12].
Definition 1.10.A mapping T : X → X is called α-admissible if for all x, y ∈ X we have where α : X × X → [0, ∞) is a given function.

Geraghty contractions on Branciari b-metric spaces
In many recent publications on fixed point on b-metric, quasi b-metric, Branciari b-metric, b-metric like spaces etc., the authors modify the contractive condition and the auxiliary functions involved in these conditions by taking into account the constant s ≥ 1 of the space.In this sense, the Banach contractive condition on b-metric and related spaces becomes d(T x, T y) ≤ kd(x, y), for all x, y ∈ X where 0 < k < 1 s .In this paper, we deal with contractions of Geraghty type on Branciari b-metric spaces.Definition 2.1.Let (X, d) be a Branciari b-metric space with a constant s ≥ 1 and let α : X × X → [0, ∞) and β ∈ F s be two given functions.A generalized Geraghty type α-admissible contractive mapping T : X → X is of type (I) if it is α-admissible and satisfies α(x, y)d(T x, T y) ≤ β (M (x, y)) M (x, y), for all x, y ∈ X, ( where M (x, y) = max{d(x, y), d(x, T x), d(y, T y)}.
We will first prove an existence theorem for fixed point of the class of contractive mappings given in Definition 2.1.
Theorem 2.2.Let (X, d) be a complete Branciari b-metric space with a constant s ≥ 1 and α : X × X → [0, ∞) and β ∈ F s be two given functions.Let T : X → X be a continuous α-admissible mapping satisfying where M (x, y) = max{d(x, y), d(x, T x), d(y, T y)}.
Proof.Choosing x 0 ∈ X such that α(x 0 , T x 0 ) ≥ 1 and α(x 0 , T 2 x 0 ) ≥ 1 we define the sequence {x n } as Suppose that x n = x n+1 for all n ≥ 0. Otherwise, for some k ∈ N we would have x k would be a fixed point of T and the proof would be completed.Since T is α-admissible, from α(x 0 , T x 0 ) ≥ 1 we have and inductively, α(x n , x n+1 ) ≥ 1, for all n ∈ N.
(2.3) Also, from the condition α(x 0 , T 2 x 0 ) ≥ 1 we have and hence, α(x n , x n+2 ) ≥ 1, for all n ∈ N. (2.4) We define the sequences {d n } and {e n } as We will prove that both the sequence {d n } and {e n } converge to 0, that is, Regarding (2.3) and the fact that 0 ≤ β(t) < 1 s , the contractive condition (2.23) with x = x n and y = x n+1 becomes for all n ≥ 1, where ) for some n ≥ 1.Then we have which is not possible.Therefore, for all (2.8) In other words, the sequence {d n } = {d(x n−1 , x n )} is positive and decreasing and hence, converges to some d ≥ 0. If we take limit as n → ∞ in (2.8) we obtain This implies lim n→∞ β(d n ) = 1 s and hence, by (1.2), On the other hand, we observe that repeated application of (2.8) leads to (2.11) Now, taking into account (2.4), we substitute x = x n−1 and x = x n+1 in (2.23).This yields for all n ≥ 1, where which contradicts the assumption l > 0. Hence, l = 0, and then we have Next, we will prove that x n = x m for all n = m.Assume that x n = x m for some m, n ∈ N with n = m.By the initial assumption, we have d(x n , x n+1 ) > 0 for each n ∈ N. Without loss of generality we may take m > n + 1.The assumption Recalling the inequality (2.7) we have where because of (2.8).Then we have, for all m > n + 1. Continuing the process we conclude, which contradicts the assumption x n = x m for some m = n.Therefore, our initial assumption is incorrect and we should have x n = x m for all m = n.Now we will prove that {x n } is a Cauchy sequence, that is, Notice that (2.19) holds for k = 1 and k = 2 due to (2.10) and (2.15).Therefore, we assume that k ≥ 3. We consider separately the cases with odd and even k ∈ N. Case 1.Let k = 2m + 1 where m ≥ 1.We have x l = x s for all l = s and x l = x l+1 for all l ≥ 0, so that we can apply repeatedly the condition 3. in Definition 1.3 which implies Then, by the inequality (2.11) we conclude .
Letting n → ∞ in the last inequality we obtain Again, repeated application of the inequality 3. in Definition 1.3 yields By the inequality in (2.11), we have (2.21) From (2.15) we have lim n→∞ s m−1 d(x n+2m−2 , x n+2m ) = 0 and hence, letting n → ∞ in (2.21) we deduce As a result, for any k ∈ N, we have lim that is, the sequence {x n } converges to T u.Then the Proposition 1.7 implies that T u = u, that is, u is a fixed point of T .
Adding an additional condition to the statement of the Theorem 2.2, we can prove the uniqueness of the fixed point.
Theorem 2.3.Let all the conditions of Theorem 2.2 hold.Assume that for every pair x and y of fixed points of T , α(x, y) ≥ 1.Then the fixed point of the mapping T is unique.
Proof.Since the existence of a fixed point is already proved in Theorem 2.2, we need to prove only the uniqueness.Assume that the map T has two distinct fixed points, say x, y ∈ X, such that x = y, or d(x, y) > 0. We put these two points in the contractive condition (2.23) and use the fact that α(x, y) ≥ 1 which gives where, M (x, y) = max{d(x, y), d(T x, x), d(T y, y)} = d(x, y).
This implies which is a contradiction and hence, d(x, y) = 0, or, x = y.This completes the proof of the uniqueness.
In the next theorem we replace the continuity of the mapping T by the so-called α-regularity of the Branciari b-metric space.
(ii) For any sequence {x n } ⊂ X such that lim n→∞ d(x n , x) = 0 and satisfying α(x n , x n+1 ) ≥ 1 for all n ∈ N, we have α(x n , x) ≥ 1 for all n ∈ N. (iii) For every pair x and y of fixed points of T , α(x, y) ≥ 1.
Then T has a unique fixed point.
Proof.Taking x 0 ∈ X as the element satisfying the condition (i), we construct the sequence {x n } as usual, that is, The convergence of this sequence can be shown exactly as in the proof of Theorem 2.2.Let u be the limit of {x n }, that is, lim n→∞ d(x n , u) = 0.
We will show that u is a fixed point of T .For the sequence {x n } which converges to u we have from (2.3) that α(x n , x n+1 ) ≥ 1 for all n ∈ N 0 .Then, the condition (ii) in the statement of the theorem implies that α(x n , u) ≥ 1, for all n ∈ N 0 .
We write the contractive inequality (2.23) for x n and u, that is, where M (x n , u) = max{d(x n , u), d(x n , x n+1 ), d(u, T u)}.
Since the sequence {x n } is Cauchy and lim n→∞ d(x n , u) = 0, by the Proposition 1.7 we have,

.25)
On the other hand, This yields d(u, T u) = 0, hence, u is a fixed point of T .We skip the uniqueness proof since it is identical to the proof of Theorem 2.3.
We next define another class of Geraghty type mappings on Branciari b-metric spaces.Remark 2.6.For all x, y ∈ X the relation d(x, y) ≤ N (x, y) ≤ M (x, y) holds.
An existence-uniqueness theorem for the class of contraction mappings introduced in Definition 2.5 is stated below.We observe that the proof of this theorem is trivial once we take into account the Remark 2.6.
Theorem 2.7.Let (X, d) be a complete Branciari b-metric space with a constant s ≥ 1 and let α : X × X → [0, ∞) and β ∈ F s be two given functions.Let T : X → X be an α-admissible mapping satisfying α(x, y)d(T x, T y) ≤ β (N (x, y)) N (x, y), for all x, y ∈ X,
(ii) Either T is continuous or, for any sequence {x n } ⊂ X with lim n→∞ d(x n , x) = 0 and α(x n , x n+1 ) ≥ 1 for all n ∈ N, we have α(x n , x) ≥ 1 for all n ∈ N. (iii) For every pair x and y of fixed points of T , α(x, y) ≥ 1.
Then T has a unique fixed point.

Remark 1 . 5 .
Let (X, d) be a Branciari or Branciari b-metric space.1.If we denote an open ball of radius r centered at x ∈ X as B r (x, r) = {y ∈ X : |d(x, y) < r}, such an open ball in (X, d) is not always an open set.
n→∞ d(x n , x n+k ) = 0, that is, the sequence {x n } is a Cauchy sequence in (X, d).Since (X, d) is a complete Branciari b-metric space, there exists u ∈ X such that lim n→∞ d(x n , u) = 0. (2.22) Since T is a continuous mapping, then, from (2.22) we have lim n→∞ d(T x n , T u) = lim n→∞ d(x n+1 , T u) = 0,
that is, either e n or d n .From the inequality (2.12) we have that is, the sequence {max{e n , d n }} is non increasing and hence, it converges to some l ≥ 0. Assume that l > 0. Taking into account (2.10) we obtain l = lim n→∞ max{e n , d n } = max{lim n→∞ e n , lim n→∞ d n } = max{lim n→∞ e n , 0} = lim n→∞ e n .