Abstract

In this article, we introduce Reich type contractions and ()-contractions in the class of controlled metric spaces and establish some new related fixed point theorems. Our results are generalizations of some known results of literature. Some examples and certain consequences are given to illustrate significance of presented results.

1. Introduction and Preliminaries

In 1993, Czerwik [1] reintroduced a new class of generalized metric spaces, called as -metric spaces, as generalizations of metric spaces.

Definition 1 ([1]). Let be a nonempty set and . A function is said to be a -metric if for all ,
(b1) iff
(b2) for all
(b3)
The pair is then called a -metric space. Subsequently, many fixed point results on such spaces were given (see [2–7]).
Kamran et al. [8] initiated the concept of extended -metric spaces.

Definition 2. Let be a nonempty set and be a function. A function is called an extended -metric if for all , (i) iff (ii)(iii)The pair is called an extended -metric space.
Very recently, a new kind of a generalized -metric space was introduced by Mlaiki et al. [9].

Definition 3. Let be a nonempty set and be a function. A function is called a controlled metric if for all , (i) iff (ii)(iii)The pair is called a controlled metric space (see also [10]).
The Cauchy and convergent sequences in controlled metric type spaces are defined in this way.

Definition 4 ([9]). Let be a controlled metric space and be a sequence in Then, (i)The sequence converges to some in ; if for every there exists such that for all . In this case, we write (ii)The sequence is Cauchy; if for every there exists such that for all (iii)The controlled metric space is called complete if every Cauchy sequence is convergent

Definition 5 ([9]). Let be a controlled metric space. Let and (i)The open ball is(ii)The mapping is said to be continuous at ; if for all there exists such that Very recently, Wardowski [11] introduced a new type of contractions, called -contractions and established some new related fixed point theorems in the context of complete metric spaces.

Definition 6. Let be a function satisfying
(F₁) is strictly increasing, that is, for all such that implies that .
(F₂) For every sequence of positive real numbers, and are equivalent.
(F₃) There is so that .
Let be the set of above functions satisfying ()-() (to be consistent with Wardowski [11]). A self-mapping on the metric space is said to be an -contraction if there are a function satisfying ()-() and a constant so that for all .

Theorem 7 [11]. Let be a complete metric space and be an -contraction, then admits a unique fixed point.
The authors in [11] manifested that a Banach contraction is a specific case of -contractions, while there are many -contractions which need not be a Banach contraction. For more details, we refer the readers to ([12–22]).

In this paper, we first define Reich [23, 24] and ()-contractions in the setting of controlled metric spaces and prove some new fixed point results. We also provide some examples to illustrate significance of the established results.

2. Results on Reich Type Contractions

Theorem 8. Let be a complete controlled metric space. Let be so that there are with , for all For take Assume that Suppose that and exist, are finite, and for every , then possesses a unique fixed point.

Proof. The considered sequence verifies for all Obviously, if there exists for which then , and the proof is finished. Thus, we suppose that for every Thus, by (1), we have which implies that Thus, we have For all , we have This implies that Let Consider We have In view of condition (4) and the ratio test, we ensure that the series converges. Thus, exists. Hence, the real sequence is Cauchy.
Now, using (9), we get Above, we used . Letting in (13), we obtain Thus, the sequence is Cauchy in the complete controlled metric space . So, there is some so that that is, as Now, we will prove that is a fixed point of By (3) and condition (iii), we get Taking the limit as and using (5, 6) and the fact that and exist, are finite, we obtain that Suppose that , having in mind that , so It is a contradiction. This yields that . The uniqueness of the fixed point follows easily. It completes the proof.

Example 9. Consider . Take the controlled metric defined as where is symmetric such that Given as Consider and . Take , then and for all . Clearly, (4) is satisfied. On the other hand, note that (3) holds for all . All other hypotheses of Theorem 8 are verified, and so has a unique fixed point, which is .

Example 10. Let . Consider the controlled metric type defined as where for . Take . Consider and . Take ; so, (4) is satisfied. Also, (3) holds. All conditions in Theorem 8 are fulfilled, and so, there is a unique fixed point, which is .

Corollary 11 (see. [9]). Let be a complete controlled metric space. Let be that there are and for all For take Assume that
(36)
Suppose that and exist, are finite, and for every , then possesses a unique fixed point.

Proof. Taking in Theorem 8.

3. Results on ()-Contractions

In 2012, Samet et al. [25] initiated the notion of -admissible mappings and proved some related fixed point results in the context of complete metric spaces.

Definition 12 ([25]). Let be a nonempty set, and be a given function. A self-mapping on is called -admissible if

Definition 13. Let be a controlled metric space. A mapping is said to be an ()-contraction if there are some , and so that for all with

Theorem 14. Let be a complete controlled metric space. Let be an ()-contraction so that (i) is -admissible(ii)There is so that (iii) is continuous(iv)For define the Picard sequence such that Assume that and exist and are finite, for every , then possesses a unique fixed point.