Interpolative Ćirić-Reich-Rus Type Contractions via the Branciari Distance

In this paper, we initiate the concept of interpolative Ćirić-Reich-Rus type contractions via the Branciari distance and prove some related fixed points results for such mappings. Moreover, an example is provided to show the useability of our obtained results.

Then T has a unique fixed point.
More information concerning Kannan fixed point theorem can be found in the early paper by Reich [4].Denote by Fix(T) the set of fixed points of a self-mapping T on a non-empty set X.In 2018, Karapınar [5] considered Theorem 1 concerning interpolation theory.The main result in [5] via an interpolative Kannan type contraction is Theorem 2 ([5]).Let (X, ρ) be a complete metric space.Suppose that the self-mapping T : X → X is such that where λ ∈ [0, 1) and α ∈ (0, 1), for all ξ, η ∈ X\Fix(T) with Fix(T) = {u ∈ X : Tu = u}.Then T possesses a unique fixed point in X.
If T : X → X satisfies (1), T is said to be an interpolative Kannan type contraction.Very recently, the authors in [6] pointed out a gap in [5], that is, the fixed point in Theorem 2 may be not unique.For more other details, see ( [7,8]).On the other hand, the known fixed point of Reich [9] is stated as follows.
Theorem 3. Let (X, ρ) be a complete metric space.If T : X → X is such that for all ξ, η ∈ X, where λ ∈ 0, 1  3 , then T possess a unique fixed point.
Note that this result was proved independently also by Ćirić and Rus.For this reason, whenever we mention Reich type contractions, we shall say " Ćirić-Reich-Rus type contractions." On the other hand, the concept of a Branciari distance space has been introduced by Brianciari [10] where the triangular inequality is replaced by a quadrilateral one.For some known fixed point results in this setting, we may refer to [11][12][13][14][15][16][17][18][19][20][21].In the sequel, N will represent the set of all positive integer numbers.First, we recall some basic concepts and notations on Branciari distance (rectangular metric) spaces.Definition 1.Let X be a non-empty set.Let d : X × X → [0, ∞) be a function such that for all ξ, η ∈ X and all distinct points u, v ∈ X, each distinct from ξ and η: Then d is called a Branciari distance and the pair (X, d) is called a Branciari distance space.
Notice that in some sources, Branciari distance is called as "a rectangular metric" or "a generalized metric".On the other hand, it was reported in [22] that the topology of standard metric and Branciari distance are not comparable.Definition 2. Let (X, d) be a Branciari distance space and {ξ n } be a sequence in X.
(ii) A sequence {ξ n } is said to be Cauchy if for every ε > 0, there exists a positive integer N = N(ε) such that d(ξ n , ξ m ) < ε for all n, m > N.
(iii) We say that (X, d) is complete if each Cauchy sequence in X is convergent.Lemma 1.Let (X, d) be a Branciari distance space.We say that a mapping T : X → X is continuous at u ∈ X, if we have Tξ n → Tu, (in other words, lim n→∞ d(Tξ n , Tu) = 0,) for any sequence {ξ n } in X converges to u ∈ X, that is, ξ n → u.
The following proposition is useful in the sequel.

Proposition 1 ([23]
). Suppose that {ξ n } is a Cauchy sequence in a Branciari distance space such that where u, z ∈ X.Then u = z.
In this paper, using the Branciari distance, we initiate the notion of interpolative Ćirić-Reich-Rus type contractions.We also present an example illustrating our approach.

Main Results
We start this section by introducing the notion of interpolative Ćirić-Reich-Rus type contractions.Definition 3. Let (X, d) be a Branciari distance space.A self-mapping T on X is called an interpolative Ćirić-Reich-Rus type contraction, if there are λ ∈ [0, 1) and positive reals α, β with α for all ξ, η ∈ X\Fix(T).
Theorem 4. Let T : X → X be an interpolative Ćirić-Reich-Rus type contraction on a complete Branciari distance space (X, p), then T has a fixed point in X.
Proof.We take an arbitrary point ξ 0 ∈ (X, p).Consider {ξ n } by ξ n = T n (ξ 0 ) for each positive integer n.If there exists n 0 such that ξ n 0 = ξ n 0 +1 , then ξ n 0 is a fixed point of T. It completes the proof.
Throughout the proof, we assume that ξ n = ξ n+1 for each n ≥ 0.
Step 1: We shall prove that By substituting the values ξ = ξ n and η = ξ n−1 in (3), we find that We derive So, we conclude that That is, {d (ξ n−1 , ξ n )} is a non-increasing sequence with non-negative terms.Eventually, there is a nonnegative constant such that lim n→∞ d (ξ n−1 , ξ n ) = .Note that ≥ 0. Indeed, from (6), we deduce that Regarding λ < 1, and by taking n → ∞ in the inequality (8), we deduce that = 0.
Step 2: We shall also show that Using ( 3), ( 7) and the quadrilateral inequality, we have We deduce that Therefore, Letting n → ∞ in (10) and using (4), we get (9), which completes the proof of step 2.
Step 3: We shall prove that ξ n = ξ m for all n = m.Suppose that ξ n = ξ m for some n > m, so we have ξ n+1 = Tξ n = Tξ m = ξ m+1 .By continuing in this direction, we obtain ξ n+k = ξ m+k for all k ∈ N. By ( 5) and ( 7), we have which is a contradiction.Thus, in that follows, we can assume that ξ n = ξ m for all n = m.
We conclude that {ξ n } is a Cauchy sequence in (X, d).Since (X, d) is complete, there exists ξ ∈ X such that We shall show that ξ is a fixed point of T. We argue by contradiction by assuming that ξ = Tξ.Recall that ξ n = Tξ n for each n ≥ 0. By letting ξ = ξ n and η = ξ in (3), we determine that Letting n → ∞ in the inequality (12), we find lim n→∞ d(ξ n , Tξ) = 0.By Proposition 1, we conclude that Tξ = ξ, which contradicts our last assumption.Thus ξ = Tξ, and so ξ is a fixed point of T.
The following example illustrates Theorem 4.
Theorem 5. Let T : X → X be an interpolative Kannan type contraction on a complete Branciari distance space (X, p), then T has a fixed point in X.
We skip the proof since it is similar to the proof of Theorem 4.