Related Fixed Point Theorems via General Approach of Simulations Functions

Abstract

In this work, we extend and complement some results in view of general and wider structures, such as metric spaces. By considering existing classes of contractions and simulating functions with a solid impact in database results of fixed point theory, we introduce a new general class of simulating functions, called as simulation functions, and also types of contractions in a more general framework. This approach covers, extends, and unifies several published works in the early and late literature.

1. Introduction

Some of the significant generalizations of metric fixed point theory are related with the well-known Banach Contraction Principle [1] and classical contractions such as Boyd and Wong, Geraghty, Browder, and Ciric. In recent years, the theory of fixed points has attracted widespread attention and has been rapidly growing. It was massively studied by many researchers giving new results by using classes of implicit functions defining new and large contractive conditions. Recently, Khojasteh et al. [2] presented the notion of contractions involving a new class of simulation functions that has been used and improved by many authors in various spaces, see [3–30]. Authors in [19] proposed new notion simulation functions and established the type of contractions.

Inspired by the above works, in this paper we introduce a new class of general type of simulation functions, defined in the setting of metric-like spaces. This class generalizes further and complements some results given in the framework of metric spaces.

2. Preliminaries

Definition 1 (see [6]). Let be a nonempty set and be a given real number. A mapping is called a metric-like if for all the following conditions are satisfied:The pair is called a metric-like space.
In a metric-like space , if and , then however, the converse need not be true, and may be positive for .

Definition 2 (see [6]). Let be a metric-like space with parameter and let be any sequence in and . Then, we have the following:(a) is said to be convergent to if (b) is said to be a Cauchy sequence in if exists and is finite(c)The pair is called a complete metric-like space if, for every Cauchy sequence in , there is such that

Lemma 1 (see [6, 29, 30]). Let and be two sequences in that converge to and , respectively. Then, we have

In particular,

Also, for each , the above inequality becomes

In particular, if, , then

Lemma 2 (see [23]). Let be a sequence in the-metric-like space with parameter , such that

If , then there are and two sequences of natural numbers , with (positive integers) such that

Note: in the continuous section of the paper, we will use (resp. ) to denote that the space with parameter is complete (resp. noncomplete).

3. Main Results

Let be a -metric-like space and represent the collection of continuous functions with the following properties:

Definition 3. A function is a simulation function if there are and a coefficient so that : for all  : If are sequences in such that and , then

Remark 1.  If in the definition above we take , then we obtain the definition of a simulation function. If we take as the identity function, then we get a definition of an simulation function. If we take and , then we get the definition of a simulation function.We denote by the set of all simulation functions. In the following example, we give such a kind of functions.

Example 1. Let be defined by(1) for all where (2) for all where is such that for all (3) for all where are continuous and is increasing such that for all (4) for all where is a C-class function where is continuous such that for all For a self-mapping , we denote by the following:

Theorem 1. Let be a self-map on a metric-like space with parameter . Suppose that there is such that

for all , where , is defined as in (8), then the self-map has a unique fixed point in .

Proof. Let be an arbitrary element. Define a sequence in such that .
If for some that is, and ; therefore, is a fixed point of . Thus, suppose that for all Considering the set , we haveSincewe obtain using (10),By the supposition and (12), we get Assume that .
Then, applying condition (9) and property , we have for all That is, a contradiction. Therefore,From (9) and using (14), we obtainIn view of property of , the above inequality gives for all . Hence, is a decreasing sequence of nonnegative reals, so there is so that . Also, by (14),Suppose that , then . By property , we havewhich is a contradiction. Therefore, . Hence,Next, we show that . Suppose, to the contrary, that is, , then by Lemma 2, there are and sequences and of positive integers with such thatFrom the definition of , we haveBy the upper limit in (20) and keeping in mind (18–19), we obtainAlso, from condition , we havewhich by property of impliesBy taking upper limit on both sides of (23) in view of (19) and (21), it follows thatwhich contradicts . Thus, and the sequence is Cauchy in . So, there is , such thatFor elements and , we considerBy Lemma 1 together with (18) and (25), it follows by passing in the upper limit of (26):Now, using the condition, we havewhich impliesTaking the limit superior in (29) and by Lemma 1 and inequality (27), we obtainBy (30), it follows that , and so .
Suppose are two different fixed points of . By (14), we have (and also ). Since , one writesFrom condition (9) and property , we havewhich is a contradiction. Therefore, and . Thus, there is a unique fixed point of .

Example 2. Let with the metric-like . Define as .
Also, we take the functions and , (where ) for all , where are continuous and is increasing such that for all
The pair is a metric-like space with coefficient We claim that the mapping satisfies the contraction type condition (8):

Case1. For , we haveAnd . Then,

Case 2. For , we noteAnd . Then,

Case 3. we noteThen,Here, is the unique fixed point of .
Some applications of Theorem 1 are the following corollaries.

Corollary 1. Let be a mapping on a metric-like space . Suppose that there are and such thatfor all , where is defined as in (8). Then, the self-map has a unique fixed point in .

Proof. In Theorem 1, take into account the function for all

Corollary 2. Letbe a mapping on a metric-like space. Suppose that there are , a lower semicontinuous function with iff and such thatfor all , where is defined as in (8). Then, the self-map admits a unique fixed point in .

Proof. In Theorem 1, take into account the function for all

Corollary 3. Let be a mapping on a metric-like space . Suppose that there are , and such thatfor all , where is defined as in (8). Then, the self-map has a unique fixed point in .

Proof. In Theorem 1, take into account the function for all and .

Corollary 4. Let be a mapping on a metric-like space . Suppose that there are , , and continuous with for , such thatfor all , where is defined as in (8). Then, the self-map has a unique fixed point in .

Proof. In Theorem 1, take into account the function for all .

Corollary 5. Let be a mapping on a metric-like space . Suppose that there are , , a C-class function and a continuous function, such thatfor all , where is defined as in (8). Then, the self-map has a unique fixed point in .

Proof. In Theorem 1, take into account the function for all , where is a C-class function.

Remark 2. Corollary 5 is much wider because condition (43) includes many other contractive conditions.

Corollary 6. Let be a mapping on a metric-like space . Suppose that there exist a function with for all and some constant such thatfor all , where is defined as in (8). Then, the self-map has a unique fixed point in .

Proof. In Theorem 1, take into account the function for all and take (it corresponds to Theorem 3.16 in [23]).
In the following result, we include two mappings and in the set

Theorem 2. Let be a metric-like space and be two given mappings. Suppose that there exists such thatfor all , where is denoted by (45); then, the mappings and have a unique common fixed point in .

Proof. Let be an arbitrary element. Define a sequence in such that and .
Let for some . SinceThen, by (46) and , we haveBy property , we get , that is, . We deduce that and . Hence, is a common fixed point of and .
Assume the general case that for all , thenIf for some then (49) impliesFrom (50), applying , (46), and , we haveThat is a contradiction. So, we have for all Hence, is a decreasing sequence of nonnegative reals, so there is so thatAssume that ; then, by applying , we havea contradiction. Therefore,Now, we prove that . It is enough to prove that On the contrary, assume that . Then, from Lemma 2, there are and two subsequences and of positive integers, with , such thatFrom (45), we noteHence, by (54)–(56), and Lemma 2, we haveBy (46) and using properties , , we havewhich leads toHence, by (55), (57), and (58) and taking the upper limit, we obtainwhich implies that , a contradiction with . It remains that ; therefore, is a Cauchy sequence in . Since is a complete b-metric-like space, there is such that is convergent to that is,Also, the subsequences are convergent, soConsiderTaking the limit superior in (63) and applying Lemma 1 and (62), it followsFrom condition (46),which impliesTaking the upper limit as and using Lemma 1 and (64), we have
, that is, and is a fixed point of . Similarly, we can get and so is a common fixed point for mappings and . Suppose are two different common fixed points of and such that . Then,From , , (67), and (46), we havewhich contradicts the supposition . Hence, and the common fixed point is unique.

Corollary 7. Let be two self-mappings given in a metric-like space . Suppose that there exist and with for all such thatfor all , where is defined as in (45).

Then, the self-mappings and have a unique common fixed point in .

Proof. In Theorem 2, take the simulation function for all .

Remark 3. The above theorem reduces to a one mapping if we put Further corollaries can be stated for , either by taking the function as an identity function or by taking different functions as listed in Corollary 1–6.

4. Conclusion

 In this work, we established common fixed point results for one and two mappings on a metric-like space which overcomes and unifies classical and previous results developed in papers [19–28]. The considered set of generalized contractive mappings contains the families of many contractions as a proper subset. We remark based on Example which are functions of class used by many researchers and taken as a special case of simulation functions. By using additional set of functions , , coefficient , and parameter , the rich class of simulation functions make it possible to collect, extend, and complement previously existing results related to a variety types of contractions. In terms of simulating functions, many classical and still recent contractions take a simple form as not including other additional symbols and long formulas. This wide approach reflects a wide work and an unifying power for more general theorems made on the theory of fixed points.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript. All contributed equally to the writing of this paper.

Acknowledgments

The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.