Admissible Hybrid Z-Contractions in b-Metric Spaces

In this manuscript, we introduce a new notion, admissible hybrid Z -contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the context of complete b-metric space. The given result not only unifies the several existing results in the literature, but also extends and improves them. We express some consequences of our main theorem by using variant examples of simulation functions. As applications, the well-posedness and the Ulam–Hyers stability of the fixed point problem are also studied.


Introduction
Metric fixed point theory can be settled in the intersection of two disciplines; (nonlinear) functional analysis and topology. From the fixed point researchers' aspect, the first application of the metric fixed point theory is on the solution of differential equations. However, according to the point of view of researchers in applied mathematics, metric fixed point theory is a tool in the solution of a first-order ordinary differential equation with an initial value. Indeed, fixed point theory appears, firstly, in the paper of Liouville in 1837, and, later, in the paper of Picard in 1890. In the paper of Picard, the method of the successive approaches was used to investigate the existence of the solution. In 1922, Banach reported the first metric fixed point result in the setting of complete norm space that would be called Banach space later. Examined enough and carefully, we realized that Banach's theorem is the abstraction of the successive approaches. The characterization of the nominated fixed point theorem of Banach, in the complete metric space, was reported by Caccioppoli in 1931. This can be accepted as the first generalization of Banach's theorem. After this, a huge number of papers, on the generalization and extension of Banach's fixed point theorem, has been released.
Extensions and generalizations of Banach's theorem are based on two elements: by changing the structure (abstract space) and by changing the conditions on the considered mappings. The immediate examples of these new structures are partial metric space, quasi-metric space, semi-metric space, b-metric space, etc. Among all of these, we shall consider the b-metric that is the most interesting and most general form of the distance. The notion of b-metric has been discovered by several authors, such as Bourbaki [1], Bakhtin [2], and Czerwik [3], in different periods of time. Roughly speaking, b-metric space is derived from metric space by relaxing the triangle inequality.
1. d is symmetric, that is, d(x, y) = d(y, x) for all x, y, 2. d is self-distance, that is, d(x, y) = 0 if and only if x = y, 3. d provides s-weighted triangle inequality, that is d(x, z) ≤ s[d(x, y) + d(y, z)], for all x, y, z ∈ X.
In this case, the triple (X, d, s) is called a b-metric space with constant s.
It is evident that the notions of b-metric and standard metric coincide in case of s = 1. For more details on b-metric spaces, see, e.g., [8][9][10][11] and corresponding references therein.
In what follows, we express the following immediate interesting examples of b-metric space to indicate the richness of this abstract space. Example 1. Let S be any set that has more than three elements. Suppose that S 1 , S 2 are the subsets of S such that S 1 ∩ S 2 = ∅ and S = S 1 ∪ S 2 Let s ≥ 1 be arbitrary. Consider the functional d : otherwise.
It is obvious that (X, d, s) forms a b-metric space.
Another simple, but interesting example is the following: is a b-metric on R with s = 2. Clearly, the first two conditions are satisfied. For the third condition, we have Thus, (X, d, 2) is a b-metric space.

Example 4 ([8])
. Let B be a Banach space with the zero vector 0 B . Suppose that P be a cone whose interior is non-empty. Suppose also that forms a partial order with respect to P. For a non-empty set S, we consider the functional d : for all a, b, c ∈ S. Then, δ is said to be a cone metric (or, Banach-valued metric). Furthermore, the pair (S, δ) is called a cone metric space (or Banach-valued metric space). Let E be a Banach space and P be a normal cone in E with the coefficient of normality denoted by K. Let D : X × X → [0, ∞) be defined by D(x, y) = ||d(x, y)||, where d : X × X → E is a cone metric space. Then, (X, D, K) forms a b-metric space.
Then, γ forms a comparison function.
Then, γ forms a comparison function.

Remark 1.
Note that γ in Example 7 is also c-comparison function. On the other hand, β in Example 8 is not a c-comparison function.
It is evident that the c-comparison function is not useful to work in the setting of b-metric space due to the third axiom, s-weighted triangle inequality. In the setting of b-metric space, we should involve the b-metric constant "s" in our analysis. That is why the b-comparison function was suggested by Berinde [10]. Indeed, the idea is so simple. In order to investigate fixed point results in the class of b-metric spaces, the notion of c-comparison function was extended to the b-comparison function by involving the b-metric constant "s".
The following lemma is very important in the proof of our results. (1) the series ∞ ∑ k=0 s k ϕ k (t) converges for any t ∈ [0, ∞); , is increasing and continuous at 0.

Remark 2.
Due to the Lemma 1.2., any b-comparison function is a comparison function.
Very recently, an interesting auxiliary function, to unify the different type contraction, was defined by Khojasteh [5] under the name of simulation function.
A self-mapping f , defined on a metric space (X, d), is called a Z-contraction with respect to ζ ∈ Z [5], if The following is the main results of [5]: Theorem 1. Every Z-contraction on a complete metric space has a unique fixed point.
As it is mentioned above, the immediate example ζ(t, s) := ks − t implies the outstanding Banach contraction mapping principle. where 1+d(x,y) where R q f (x, y) is as above.

Existence and Uniqueness Results
Theorem 2. Let (X, d) be a complete b-metric space with constant s ≥ 1 and let f : X → X be an admissible hybrid Z-contraction. Suppose also that: Then, f has a fixed point.
Proof. Let x 0 ∈ X be an arbitrary point. Starting from here, we recursively construct the sequence (x n ) n∈N , as x n = f n (x 0 ) for all n ∈ N. Supposing that there exists some m ∈ N such that f x m = x m+1 = x m , we find that x m is a fixed point of f and the proof is finished. Thus, we can presume, from now on, that x n = x n−1 for any n ∈ N. Under the assumption (i), f is an admissible hybrid Z-contraction, if we consider in (6) x = x n−1 and y = x n , we get which is equivalent to Taking into account that f is triangular α-orbital admissible, from (ii) and Lemma 1.3., we have α(x n−1 , x n ) ≥ 1. In this way, the above inequality becomes Suppose that d(x n−1 , x n ) ≤ d(x n , x n+1 ). Since ϕ is a nondecreasing function, Equation (9) can be estimated as follows: on account of the fact that ϕ(t) < t, we find which is a contradiction. Therefore, for every n ∈ N, we have Now let m, p ∈ N such that p > m. Using the triangle inequality and (10), we have Since ϕ is a b-comparison function, the series which tells us that (x n ) n∈N is a Cauchy sequence on a complete b-metric space, so there exists x * ∈ X such that lim n→∞ d(x n x * ) = 0.
We shall prove that x * is a fixed point of f . If f is continuous, (due to assumption (iii)) Suppose now that f 2 is continuous. It follows that This is a contradiction, so that f (x * ) = x * . Case 2. For the case q = 0, if we consider x = x n−1 and y = x n , we have Employing the triangle inequality, we have Using the following inequality and, from (6), Supposing that d(x n−1 , x n ) ≤ d(x n , x n+1 ), since ϕ is a nondecreasing function, we have which is a contradiction. Then, from (14), inductively, we obtain By using the same arguments as the case q > 0, we shall easily obtain that (x n ) n∈N is a Cauchy sequence in a complete metric space and thus there exists x * such that lim n→∞ x n = x * . We claim that x * is a fixed point of f . Under the assumption that f is continuous, we have and together with the uniqueness of limit, f (x * ) = x * . In addition, if f 2 is continuous, as in case 1, we have that f 2 (x * ) = x * and suppose that f (x * ) = x * . Then, we get which is a contradiction.

Theorem 3. In the hypothesis of Theorem 2, if we assume supplementary that
for any x * , y * ∈ Fix f (X), then the fixed point of f is unique.
Proof. Let y * ∈ X be another fixed point of f . Suppose that x * = y * . In the case that q > 0, using (6), we have: which is a contradiction.
In the case that q = 0, if we suppose that x * = y * , then we obtain that 0 < d(x * , y * ) < 0, which is a contradiction.
Thus, x * = y * , so that f possesses exactly one fixed point.
We can easily observe that: If x = 0 and y = 2, then if we consider q = 2, λ 1 = λ 2 = λ 3 = λ 4 = λ 5 = 1 5 , then we have Hence, In all other cases, α(x, y) = 0 and Thus, we obtain that f is an admissible hybrid Z-contraction which satisfies the assumptions of Theorem 2 and then x = 1 2 is the fixed point of f .

Remark 3.
If, in the above example, we consider f (x) , then f is not continuous, but f 2 (x) = 1 3 and for x = 1 Furthermore, we suppose that: (i) f is triangular α-orbital admissible; (ii) there exists x 0 ∈ X such that α(x 0 , f (x 0 )) ≥ 1; (iii) either, f is continuous or (iv) f 2 is continuous and α( f x, x) ≥ 1 for any x ∈ Fix f 2 (X).
Then, f has a unique fixed point.
Suppose that there exists a function φ ∈ Φ, such that Furthermore, we suppose that Then, f has a unique fixed point.
Furthermore, we suppose that Then, f has a unique fixed point. The following example is derived from [5,14,15].

Well Posedness and Ulam-Hyers Stability
Considered as a type of data dependence, the notion of Ulam stability was started by Ulam [19,20] and developed by Hyers [21], Rassias [22], etc. In this section, we investigate the general Ulam type stability in sense of a fixed point problem as well the well posedness of the fixed point problem. Suppose that f : X → X is a self-mapping on a b-metric space (X, d) with the constant s > 1 and let us consider the following fixed point problem: (ii) If (x n ) n∈N is a sequence such that d (x n , f (x n )) → 0, as n → ∞, then x n → x * , as n → ∞.
From here, we obtain Letting n → ∞ in the above inequality and keeping in mind that lim n→∞ d(x n , f (x n )) = 0, we obtain lim n→∞ d(x n , x * ) = 0, that is, the fixed point Equation (19) is well-posed.
is increasing, continuous in 0 and ρ(0) = 0, such that for each ε > 0 and for each y * ∈ X, which satisfy the inequality d(y, f (y)) ≤ ε, there exists a solution x * of the fixed point problem (19) such that If there exists c > 0 such that ρ(t) := c · t, for each t ∈ R + , then the fixed point problem (19) is said to be Ulam-Hyers stable.

Conclusions
In this paper, we unify, extend, and improve several existing fixed point theorems by introducing the notion of admissible hybrid Z-contraction in the setting of complete b-metric spaces. Consequently, all presented results valid in the setting of complete metric space by letting s = 1. On the other hand, unifying several existing results in the literature, we have used admissible mappings, simulation functions, and hybrid contractions. We need to underline the fact that, by setting admissible function α in a proper way, one can get several new consequences of the existence results in the setting of (i) standard metric space, (ii) metric space endowed a partial order on it, and (iii) cyclic contraction. One can easily get these consequences by using the techniques in [4]. Furthermore, for the different examples of simulation functions (as we showed in Theorems 5 and 6), one can get more new corollaries. Lastly, by regarding hybrid contraction approaches, one can get several more consequences, by following the techniques in [21,[24][25][26].
Besides expressing a more generalized result in the setting of a complete b-metric space, we also present some applications for our obtained results. In particular, we shall consider the well-posedness and the Ulam-Hyers stability of the fixed point problem. We note that the word 'hybrid' has been used in different ways, in particular, in applicable nonlinear fields, see, e.g., [27,28].
Author Contributions: Writing-original draft, I.C.C.; Writing-review and editing, E.K. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.