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

Abstract

We investigate the existence and uniqueness of certain operators which form a new contractive condition via the combining of the notions of admissible function and simulation function contained in the context of complete -metric spaces. The given results not only unify but also generalize a number of existing results on the topic in the corresponding literature.

1. 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.

Definition 1 (see [2], cf. [1]). Let be a mapping that satisfies the following inequality and the condition below: for each .if are sequences in such that , thenWe shall use the letter to indicate the class of all simulation functions . It is obvious from the axiom that

Note that the condition 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 . For further detail on the discussion, see, for example, [2].

Throughout the paper, we shall use 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 denotes the set of all continuous functions such that if, and only if, . The following functions form a simulation function.  (i) and , where for each  (ii) Let be two continuous functions with respect to each variable and the inequality holds for each Then, (iii) for each  (iv) For each , where with for each . (v) For each  where and it is upper semicontinuous. (vi) For each , where is a self-mapping on with the following properties: (1) exists, (2) for every , .

For the further attracted simulation function examples see, for example, [1, 3, 4].

In 1993 Czerwik [5] proposed a more general frame for the notion of standard metric, so called a -metric.

Definition 3. For , let be a function satisfying the following conditions: (1) if and only if .(2) for each .(3) for each , where . Here, is called a -metric. Further, the triple is called a -metric space.

For the special case of , 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] and the related references therein.

Example 4. Let . For all , we define a function asThen, is a -metric on real numbers. The first two axioms are fulfilled in a straightforward way. The last axiom is satisfied for :

Example 5. For a fixed , consider We introduce the corresponding distance functions as Then, forms a -metric space with the constant .

Example 6. Suppose that is a Banach space with the zero vector of . Take as a cone in such that and further, is partial ordering with respect to . For a nonempty set , we define a mapping as follows: for each . if and only if ., for each . 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 , defined by , forms a -metric space where the function is a cone metric. Moreover, the triple forms a -metric space with the constant .

Suppose that is a -metric space. A self-mapping on is said to be a -contraction with respect to [1], if the following inequality is fulfilled: On account of , we derive thatTaking (12) into account, we find that cannot be an isometry whenever is a -contraction. Moreover, if is a -contraction 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 -contraction has a unique fixed point.

This theorem can be stated also as follows: each -contraction yields a Picard sequence that converges to a unique fixed point.

For a family , if the following two conditions are fulfilled,(i)each function is nondecreasing;(ii)there exist and and a convergent series of nonnegative terms such that for any and for we have Here, is called the class of -comparison functions (see [8]). For a the notation indicates the th iteration of the function . The following lemma is recollected from [8].

Lemma 8. For a , we have (i)for each , the sequence converges to as ;(ii)for any , the inequalities are fulfilled;(iii)each auxiliary function is continuous at ;(iv)for any , the series is convergent.

Berinde [9] characterized -comparison functions to use for the contraction mappings in the setting of -metric spaces, as follows.

Definition 9. Fix a real number . An increasing function is said to be -comparison if there exist a convergent nonnegative series ,  , and such that for any and for ,

Lemma 10 (see [10]). Let be a -comparison function. Define a self-mapping on as : (1)For any , the series is convergent.(2)The function is increasing and continuous at .

Remark 11. It is obvious that each -comparison function is a comparison function. Consequently, on account of Lemma 8, we deduce that any -comparison function satisfies the inequality .

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 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 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 for each We illustrate that the class of the new contractive mapping covers several well-known contractive mappings.

2. Main Results

We start this section by defining the -type -contraction which is a generalization of the notion of -contraction.

Definition 13. Let be a nonempty set and be function. Suppose that is a self-mapping defined over a -metric space . The self-mapping is called an -type -contraction with respect to if there are and such that

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 is a function and is an -orbital admissible mapping. If there exists such that and for , then, we have

Proof. On account of the assumptions of the theorem, there exists such that . Owing to the fact that is -orbital admissible, we find By iterating the above inequality, we derive that

Theorem 15. Let be a nonempty set, , and be a function. Suppose that a continuous self-mapping over a complete -metric space is -orbital admissible. Suppose also the mapping forms an -type -contraction with respect to . If there exists such that , then there exists such that .

Proof. Based on the assumption, there exists such that . We shall construct an iterative sequence in by setting for each . By Lemma 14, we have (17); that is, Taking (16) and (17) into account, for each , we derive thatAccordingly, from (16) and (21) we conclude thatHence, we conclude that the constructive sequence is bounded from below by zero, and moreover, it is nondecreasing. Hereby, there exists such that We shall indicate thatSuppose, on the contrary, that On account of inequality (22), we find thatTaking and together with the condition , we derive thata contradiction. Consequently, we have
In the next step, we shall show that the constructive sequence is Cauchy. By iteration on the inequality (22), we derive thatFrom (26) and using the triangular inequality, for each , we haveThe precedent inequality is which yields that is a Cauchy sequence in . Since is complete, there exists such thatSince is continuous, we obtain from (29) thatCombining 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 . We say that a -metric space is regular if is a sequence in such that for each and as ; then there is a subsequence of such that 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, , and 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 with respect to . If there exists such that , then there exists such that .

Proof. By repeating the steps in the proof of Theorem 15, we guarantee that the sequence defined by for each converges for some . From (17) and regularity of the metric, there exists a subsequence of such that for each . By implementing (16), for each , we get thatwhich leads toTaking in inequality (32), we deduce that ; that is, .

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 represent the set of all fixed points of operator . For this purpose, we need the following additional condition: for each .

Theorem 18. Under the assumption of additional condition , 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 -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 . Regarding the condition , the definition of yields thatDue to definition of the auxiliary function , the inequality above implies that which is a contradiction. Thus, is the unique fixed point of .

3. Consequences

3.1. Consequences in the Setting of -Metric Space

Consider a mapping related with an -orbital admissible -contraction with respect to , namely ; that is, . For a function , we setIt is straightforward that is a simulation function. Combining the observations above, we have which is equivalent towhere

Thus, the above sample of simulation function together with Theorem 18 yields the following result.

Theorem 19. Let be a complete -metric space and let be defined aswhere and Suppose that (i) is -orbital admissible;(ii)there exists such that ;(iii)either is continuous or is regular. Then there exists such that . Moreover, if the condition is fulfilled, then we guarantee that the obtained fixed point of is unique.

Letting with , Theorem 19 implies the following.

Theorem 20. Let be a complete -metric space and let be defined as where , , and Suppose that (i) is -orbital admissible;(ii)there exists such that ;(iii)either is continuous or is regular. Then there exists such that . Moreover, if the condition is fulfilled, then we guarantee that the obtained fixed point of is unique.

By letting in Theorems 19 and 20 we get the main results of Theorems and of Czerwik [5]. Notice that, in this case, conditions (i)–(iii) of Theorems 19 and 20 are fulfilled trivially.

3.2. Consequences in the Setting of Standard Metric Space

In this section, we consider the results in the setting of standard metric. Thus, we consider 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 and we define then 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 and in [13] are consequences of the following.

Proof. Taking in Theorem 18, we derive that We skip the details.

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 and 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.

Theorem 22. Theorem 18 yields Theorem 7.

Proof. It is sufficient to take for each in Theorem 21.

It is obvious that all presented results in [1] follow from our main result.

4. 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].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.