On the Power of Simulation and Admissible Functions in Metric Fixed Point Theory

The crucial notion of this research is the simulation function which is defined by Khojasteh et al. [1]. After that, Argoubi et al. [2] relaxed the conditions of the notion of simulation function a little bit to guarantee that the considered set is nonempty. In this manuscript, we respond to the question, how do we guarantee the existence of fixed points of the new contraction defined by the help of the admissible function and the simulation function in the frame of complete b-metric spaces? The presented main theorem of the paper covers and unifies a huge number of published results on the topic in the related literature.


Introduction
The crucial notion of this research is the simulation function which is defined by Khojasteh et al. [1].After that, Argoubi et al. [2] relaxed the conditions of the notion of simulation function a little bit to guarantee that the considered set is nonempty.
In this manuscript, we respond to the question, how do we guarantee the existence of fixed points of the new contraction defined by the help of the admissible function and the simulation function in the frame of complete -metric spaces?The presented main theorem of the paper covers and unifies a huge number of published results on the topic in the related literature.
We shall use the letter S to indicate the class of all simulation functions  : R + 0 × R + 0 → R. It is obvious from the axiom (S 2 ) that  (, ) < 0 for every  > 0. (2) Note that the condition (0, 0) = 0 in the original definition of the simulation function is removed in Definition 1. Indeed, this condition gives a contradiction when one takes  =  in the first condition (S 1 ).For further detail on the discussion, see, for example, [2].
Throughout the paper, we shall use R + 0 to represent nonnegative real numbers.
The following example [1,3,4] shall be helpful to illustrate the worth of the notion of simulation function.
Example 2. Suppose that Φ denotes the set of all continuous functions  : R + 0 → R + 0 such that () = 0 if, and only if,  = 0.The following functions  1 − 6 form a simulation function.
where  ∈ Φ and it is upper semicontinuous.
In 1993 Czerwik [5] proposed a more general frame for the notion of standard metric, so called a -metric.Definition 3.For  ̸ = 0, let   :  ×  → R + 0 be a function satisfying the following conditions: (1)   (, ) = 0 if and only if  = .
Here,   is called a -metric.Further, the triple (,   , ) is called a -metric space.
For the special case of  = 1, the notion of -metric turns into the standard metric.Consequently, the notion of -metric is more general than the standard metric.
For the sake of completeness, we recollect standard but interesting three examples of -metric spaces; see, for example, [6,7] Example 5.For a fixed  ∈ (0, 1), consider We introduce the corresponding distance functions as for each ,  ∈ .(10) Then, (,   , ) forms a -metric space with the constant  = 2 1/ .Example 6. Suppose that  is a Banach space with the zero vector 0  of .Take  as a cone in  such that int() ̸ = 0 and further, ⪯ is partial ordering with respect to .For a nonempty set , we define a mapping  :  ×  →  as follows: (1) 0 ⪯ (, ) for each ,  ∈ .Then, the mapping  is called cone metric on .Moreover, the pair (, ) is said to be a cone metric space.
If a normal cone  in  is normal with the normality constant , then, the mapping  :  ×  → R + 0 , defined by   (, ) = ‖(, )‖, forms a -metric space where the function  :  ×  →  is a cone metric.Moreover, the triple (,   , ) forms a -metric space with the constant  fl  ≥ 1.
Suppose that (,   , ) is a -metric space.A selfmapping  on  is said to be a S-contraction with respect to  [1], if the following inequality is fulfilled: On account of ( 2 ), we derive that Taking ( 12) into account, we find that  cannot be an isometry whenever  is a S-contraction.Moreover, if  is a Scontraction in the setting of -metric space with a fixed point, then the desired fixed point is necessarily unique.
Theorem 7. In a complete -metric space, each S-contraction has a unique fixed point.
This theorem can be stated also as follows: each Scontraction yields a Picard sequence that converges to a unique fixed point.
For a family Ψ := { : R + 0 → R + 0 }, if the following two conditions are fulfilled, (i) each function  ∈ Ψ is nondecreasing; (ii) there exist  ∈ (0, 1) and  0 ∈ N and a convergent series of nonnegative terms ∑ ∞ =1 V  such that for any  ∈ R + and for  ≥  0 we have Here, Ψ is called the class of ()-comparison functions (see [8]).For a  ∈ Ψ the notation   indicates the th iteration of the function .The following lemma is recollected from [8].
Lemma 8.For a  ∈ Ψ, we have (ii) for any  ∈ R + , the inequalities () <  are fulfilled; (iii) each auxiliary function  is continuous at 0; Berinde [9] characterized ()-comparison functions to use for the contraction mappings in the setting of -metric spaces, as follows.
Popescu [11] introduces the notion of the -orbital admissible as follows.
Definition 12 (see [11]).Suppose that  is a self-mapping over a nonempty set  and  :  ×  → R + 0 is a function.The mapping  is called an -orbital admissible if the following implication is provided: We should mention the notion of the -orbital admissible [11,12] inspired from the notion of the -orbital admissible notion defined in [13,14].
In this paper, by combining the notion of the simulation function together with the admissible functions, we shall consider a new type contractive mapping in the frame of complete -metric spaces.Accordingly, our results improve and extend the main results in [15] in twofold: first, we investigate the existence and uniqueness of a fixed point in -metric spaces instead of standard metric space.Secondly, we extend the condition ((, )  (, ), (  (, )) ≥ 0 for each , ∈  by adding an auxiliary function  into account.Consequently, we investigate the existence and uniqueness of a fixed point in the new extended condition ((, )  (, ), (  (, ))) ≥ 0 for each ,  ∈ .We illustrate that the class of the new contractive mapping covers several well-known contractive mappings.

Main Results
We start this section by defining the ( − )-type Scontraction which is a generalization of the notion of Scontraction.Definition 13.Let  be a nonempty set  ≥ 1 and  :  ×  → [0, ∞) be function.Suppose that  is a self-mapping defined over a -metric space (,   , ).The self-mapping  is called an ( − )-type S-contraction with respect to  if there are  ∈ S and  ∈ Φ such that  ( (, )   (, ) ,  (  (, ))) ≥ 0 for each ,  ∈ .(16) Before stating our main theorem, we shall give lemmas that have a crucial role in the proof of the main result.Lemma 14.Let  be a nonempty set.Suppose that  :  ×  → R + 0 is a function and  :  →  is an orbital admissible mapping.If there exists  0 ∈  such that ( 0 ,  0 ) ≥ 1 and   =  −1 for  = 0, 1, . .., then, we have Proof.On account of the assumptions of the theorem, there exists  0 ∈  such that ( 0 ,  0 ) ≥ 1. Owing to the fact that  is -orbital admissible, we find By iterating the above inequality, we derive that for each  = 0, 1, . . . .
Theorem 15.Let  be a nonempty set,  ≥ 1, and  :  ×  → R + 0 be a function.Suppose that a continuous self-mapping  over a complete -metric space (,   , ) is -orbital admissible.Suppose also the mapping  forms an ( − )-type S-contraction with respect to .If there exists  0 ∈  such that ( 0 ,  0 ) ≥ 1, then there exists  ∈  such that  = .
In the next step, we shall show that the constructive sequence {  } is Cauchy.By iteration on the inequality (22), we derive that From (26) and using the triangular inequality, for each  ≥ 1, we have The precedent inequality is which yields that {  } is a Cauchy sequence in (,   , ).Since (,   , ) is complete, there exists  ∈  such that lim Since  is continuous, we obtain from (29) that lim Combining the uniqueness of the limit together with (29) amd (30), we find that  forms a fixed point of ; that is,  = .
The continuity condition can be relaxed in Theorem 15 by replacing a suitable condition like the given below.Definition 16.Let  ≥ 1.We say that a -metric space (,   , ) is regular if {  } is a sequence in  such that (  ,  +1 ) ≥ 1 for each  and   →  ∈  as  → ∞; then there is a subsequence { () } of {  } such that ( () , ) ≥ 1 for each .
By removing the continuity condition from the main result, Theorem 15 is possible.But, in this case, we should add the "regularity" condition which is mentioned in Definition 16.

Theorem 17.
Let  be a nonempty set,  ≥ 1, and  :  ×  → R + 0 be a function.Suppose that (,   , ) is regular and a self-mapping  on a complete -metric space (,   , ) is orbital admissible.Suppose also the mapping  forms an ( − )-type S-contraction with respect to .If there exists  0 ∈  such that ( 0 ,  0 ) ≥ 1, then there exists  ∈  such that  = .
Note that, in Theorems 15 and 17, we observe only the existence of the fixed point of the given operator.As a next step, we shall investigate the uniqueness of the obtained fixed point.Let Fix() represent the set of all fixed points of operator .For this purpose, we need the following additional condition: (U) (, ) ≥ 1 for each ,  ∈ Fix().
Theorem 18.Under the assumption of additional condition (U), the obtained fixed point  of the operator  defined in Theorem 15 (resp., Theorem 17) turns to be unique fixed point.
Proof.Let  be an -orbital admissible S-contraction with respect to .Regarding Theorem 15 or Theorem 17, we guarantee the existence of a fixed point of the mapping ; namely,  = .Suppose  is not the unique fixed point of ; thus, there exists  with  ̸ = .So, we have   (, ) > 0. Regarding the condition (U), the definition of  yields that  ( (, )   (, ) ,  (  (, ))) ≥ 0.
which is a contradiction.Thus,  is the unique fixed point of .
Then there exists  ∈  such that  = .Moreover, if the condition (U) is fulfilled, then we guarantee that the obtained fixed point  of  is unique.
Then there exists  ∈  such that  = .Moreover, if the condition (U) is fulfilled, then we guarantee that the obtained fixed point  of  is unique.
By letting (, ) = 1 in Theorems 19 and 20 we get the main results of Theorems 1 and 2 of Czerwik [5].Notice that, in this case, conditions (i)-(iii) of Theorems 19 and 20 are fulfilled trivially.

Consequences in the Setting of Standard Metric Space.
In this section, we consider the results in the setting of standard metric.Thus, we consider  = 1 throughout this section.We shall show that a number of existing fixed point results in the literature are the simple consequence of our main results.In particular, by taking Example 2 into consideration, we can list many well-known results as a consequence of our main results.
If  ∈ Ψ and we define   (, ) =  () −  for each ,  ∈ R + 0 , then  BW is a simulation function (cf.Example 2 (v)).First, we derive the very interesting recent results of Samet et al. [13] as a corollary of Theorem 18.
Theorem 21.Theorems 2.1 and 2.2 in [13] are consequences of the following.
As is well known, the main theorem in [13] covers several fixed point results, including the pioneer fixed point theorem of Banach.Moreover, as is shown in [13,16], several fixed point theorems in different settings (in the sense of partially ordered set, in the sense of cyclic mapping, etc.) can be concluded from Theorems 2.1 and 2.2 in [13] by setting (, ) in a proper way.
Notice also that one can express the main result of Khojasteh et al. [1] as a straight consequence of our main result.
Proof.It is sufficient to take (, ) = 1 for each ,  ∈  in Theorem 21.
It is obvious that all presented results in [1] follow from our main result.

Conclusion
It is very easy to see that one can list a further outcome of our main results by letting the mappings , , ,  in a suitable way like in Example 2.More precisely, by following the techniques in [13,16] one can easily derive a number of well-known fixed point results in the distinct settings (such as in the frame of cyclic contraction and in the setting of partially ordered set endowed with a metric).We prefer not to list all consequences due to our concerns on the length of the paper.This paper can be also considered as a continuation of the recent paper [17].