A Common Fixed-Point Theorem for Iterative Contraction of Seghal Type

In this paper, we consider a common fixed-point theorem with a contractive iterative at a point in the setting of complete dislocated b-metric space that was initiated by Seghal. We shall consider an example and application in fractional differential equations to support the given results.


Introduction and Preliminaries
It is quite natural to consider the distance of a thing to itself to be 0, which seems also very reasonable.For instance, let us consider the set of all infinite sequences endowed with a metric d such that d(x, y) = 1  2 s for x = (x i ) i∈N , and y = (y i ) i∈N where s := |i ∈ N : x i = y i |.It is evident that s is infinity in the case of x = y and hence d(x, x) = 0. On the other hand, in computer science, infinite sequences are not useful because of time restriction.On the contrary, finite sequences are more useful and reasonable in programming.Using the finite sequences to infinite sequence by keeping the definition of the metric stable, we shall get a very interesting scenario.More precisely, for a finite sequence, for example for x = (x 1 , • • • , x 7 ), the self-distance of x to itself is not 0. Indeed, here s = 7 and self-distance is 1  2 7 .On account of such motivation, the notion of dislocated metric was proposed by Hitzler [1] by claiming that self-distance may not be 0. Definition 1. Suppose that X is not empty.A dislocated metric is a function δ : X × X → [0, ∞) such that for all ς, κ, ∈ X : (δ1) δ(ς, κ) = 0 ⇒ ς = κ, (δ2) δ(ς, κ) = δ(κ, ς), (δ3) δ(ς, κ) ≤ δ(ς, ) + δ( , κ).
The pair of the letters (X , δ) represent a dislocated metric space, in short DMS.Another extension of metric is a b−metric which has been introduced by Czerwik [2], see also e.g., [3,4].Definition 2. Suppose that X is not empty and s ≥ 1 is given.A b-metric is a function d : X × X → [0, ∞) such that for all ς, κ, ∈ X : The pair of letters (X , d) is called a b-metric space, in short b-MS.Notice that in some paper, this spaces was called quasi-metric space, see e.g., [5,6].
In what follows, we shall consider the unification of the above-mentioned notions: Definition 3. Suppose that X is not empty and s ≥ The pair (X , δ d , s) is said to be a dislocated b-metric space, in short b-DMS.
It is obvious that b-metric spaces are b-DMS, but conversely this is not true.
For more examples see e.g., [7][8][9][10][11][12].The topology of dislocated b-metric space (X , δ d , s) was generated by the family of open balls exists and is finite.In addition, if the following limit exists and is finite we say that the sequence {ς n } is Cauchy.Moreover, if lim n→∞ δ d (ς n , ς m ) = 0, then we say that {ς n } is a 0-Cauchy sequence.
Moreover, a b-DMS (X , δ d , s) is said to be 0-complete if for each 0-Cauchy sequence {ς n } converges to a point ς ∈ X so that L = 0 in (2).
Let (X , δ d , s) be a b-DMS.A mapping f : X → X is continuous if { f ς n } converges to f ς for any sequence {ς n } in X converges to ς ∈ X .Proposition 1. [7] Let (X , δ d , s) be a b-DMS and {ς n } be a sequence in X such that We need the following definitions from [6,13] in our main results.
We denote by Φ c the family of c-comparison functions.
Remark 2. Any c-comparison function is a comparison function.
We denote by Φ b the family of b-comparison functions.

Remark 3. Any b-comparison function is a comparison function.
Let Ψ be the family of functions In what follows, we shall mention one of the interesting extensions of the Banach contraction principle [14] that was given by Seghal [15]: Theorem 1. ( [15]) Let (M, d) be a complete metric space, T a continuous self-mapping of M that satisfies the condition that there exists a real number q, 0 < q < 1 such that for each v ∈ M there exists a positive integer m(v) such that for each w ∈ M, Then T has a unique fixed point in M.
In this paper, we shall investigate the fixed point of a certain mapping with a contractive iterate at a point in the setting of dislocated b-metric space.Such fixed-point results were introduced by Seghal [15] and continued by many others; see e.g., [16,17].Furthermore, we shall consider an application to support the obtained result.

Main Results
In this section, we prove some new fixed-point results in the setting of b Theorem 2. Let U, V be two self-mappings on a complete b-MS (X , δ, s).Suppose that for any ς, κ ∈ X there exist positive integers p(ς), q(κ), and that there exist ψ ∈ Ψ and an upper semicontinuous ϕ ∈ Φ b such that Then the pair of the functions U, V has exactly one fixed point ς * .
Corollary 1.Let U, V be two self-mappings on a complete b-MS (X , δ, s).Suppose that there exist 0 ≤ c < 1 s and k ≥ 1 such that for all ς, κ ∈ X there exist positive integers p(ς), q(κ) such that then the pair of the mappings U, V possesses a common fixed point ς * .
Corollary 2. Let U be a self-mapping on a complete b-MS (X , δ, s).Suppose that for any ς, κ ∈ X there exist positive integer p(ς) and there exist ψ ∈ Ψ and upper semicontinuous ϕ ∈ Φ b such that then the map U has a unique fixed point ς * .Now we take the same idea in the context of b-DMS.
Theorem 3. Let (X , δ d , s) be a 0-complete b-DMS and U, V : (X , δ d , s) → (X , δ d , s) be two functions.Let the function ϕ ∈ Φ b .Suppose that for all ς, κ ∈ X we can find the positive integers p(ς), q(κ) such that Then the pair of the functions U, V has exactly one fixed point ς * .
Proof.Consider a point ς 0 ∈ X and as in above theorem we shall define the sequence {ς n } in X as follows: Denoting p i−1 = p(ς 2i−1 ) and q i = q(ς 2i ), for any i ∈ N, we can write ς 2i = U p i−1 ς 2i−1 and ς 2i+1 = V q i ς 2i .As we have seen in Theorem 2, the first purpose is to show that the sequence {ς n } is Cauchy.For this, let us get in (23) ς = ς 2i−1 and κ = ς 2i .We have, and then two situations can be considerate.

Application
Let 0 < γ be a real number and ς : [1, ∞) → R be a function.Throughout this part, we consider that [γ] represents the integer part of real number γ and by log(•) we denote log e (•).
The Hadamard derivative of fractional order γ for ς is defined by The Hadamard fractional integral of order γ for ς is given by provided the integral exists.
Starting from [18], where the problems involving Hadamard-type fractional derivatives are studied, we discuss here the existence of a solution for the following system of fractional functional differential equations with initial values: where the functions ξ, η : [1,    for all ς, κ ∈ X .We conclude that for any ς, κ ∈ X taking p(ς) = q(κ) = 2 and c = λ (log t) γ Γ(γ+1) r all presumptions of Corollary 1 are verified and the maps U and V have exactly one common fixed point on X , so the system (36) has a unique common solution in [1, t].

Proposition 3 .
If ϕ is a comparison function then: