Abstract

In this paper, we firstly introduce a new notion of inverse class functions which extends the notion of inverse class functions introduced by Saleem et al., 2018. Secondly, some common fixed point theorems are stated under some compatible conditions such as weak semicompatible of type , weak semicompatibility, and conditional semicompatibility in metric spaces. Moreover, we introduce a new kind of compatibility called compatibility which is weaker than property and also present a common fixed point theorem in metric spaces via inverse class functions. Some examples are provided to support our results.

1. Introduction and Preliminaries

As a follow-up work of A.H. Ansariâ€™s research on fixed point (or common fixed point) theory via auxiliary C-class functions, very recently, Saleem et al. [1] introduced the new concept of inverse C-class functions and obtained some corresponding fixed point theorems under certain weak compatibility assumption via inverse C-class functions. In 1976, Jungck [2] defined the concept of commutative maps and initiated the study of the existence of a common fixed point of such maps in metric spaces. After which, Sessa [3] introduced the weak version of commuting maps called weak commuting maps. Next, Jungck [4, 5] provided some generalizations of weak commuting maps by providing the notions of compatible maps and compatible maps of type . Minor relaxations of compatible of type are introduced by Pathak and Khan [6], which are well known as compatible and compatible (see [6], for more details).

Singh et al. [7] proposed the notion of compatibility of type by making a minor modification of compatibility of type . By splitting the concept of compatibility of type , Singh et al. [7] also gave some relaxations of compatibility type of which are known as compatibility of type and compatibility of type .

Definition 1. (see [7]). Two self-maps and of a metric space are said to be compatible of type , if , whenever is a sequence in such that , for some . Similarly, two self-maps and of a metric space are said to be compatible of type , if , whenever is a sequence in such that , for some .
It is easy to see that compatibility of type implies both compatibility of type and compatibility of type ; however, compatibility or compatibility of type does not imply compatibility of type .
In 1994, Pant [8] introduced the following definition.

Definition 2. (see [8]). Two self-maps and defined on a metric space are said to be weakly commuting, if there exists a real number such that , for all .
Note that ; then, and are weakly commuting.
In 1997, Pathak et al. [9] introduced the notions of weak commuting of type and weak commuting of type as follows.

Definition 3. (see [9]). Two self-maps and of a metric space are said to be weakly commuting of type , if there exists a real number such that , for all .

Definition 4. (see [9]). Two self-maps and of a metric space are said to be weakly commuting of type , if there exists a real number such that , for all .
It is noted that compatible maps and are also weakly commuting of type and weakly commuting of type . Moreover, we can find suitable examples which show that weakly commuting mappings and weakly commuting of type (or ) are independent concepts (see examples of [9, 10]).
In 2008, Gopal et al. [10] introduced the notions of absorbing and absorbing stated as follows.

Definition 5. (see [10]). Let and be two self-maps of a metric space ; then, is said to be -absorbing, if there exists a real number such that for all . Similarly, let and be two self-maps of a metric space ; then, is said to be -absorbing, if there exists a real number such that for all .
Jungck and Rhoades [11], in 1998, introduced the concept of weak compatibility which is weaker than the concept of compatibility.
Another generalization of compatible maps called semicompatible maps was firstly introduced by Cho et al. [12] under the setting of topological spaces in which a pair of self-maps are called to be semicompatible if condition implies that ; for sequence in and , whenever , , and then , as , hold. However, Singh and Jain [13] redefined this concept by using condition only stated as follows.

Definition 6. (see [13]). A pair of self-maps of a metric space is said to be semicompatible, if holds whenever is a sequence in such that for some .
It follows that if is semicompatible and , then . It is also noted that if the pair is semicompatible, then it is weak compatible; however, the converse is not true. Further, the semicompatibility of the pair does not imply the semicompatibility of the pair (see Example 3.2 in [13]).
Now, we make a minor modification of semicompatibility to introduce the notion of semicompatible of type as follows.

Definition 7. A pair of self-maps of a metric space is said to be semicompatible of type , if and hold whenever is a sequence in such that for some .
It is obvious that semicompatibility of type implies semicompatibility; however, the converse is not true.
Recently, Saluja et al. [14, 15] introduced the weak semicompatible maps and conditional semicompatible maps and obtained corresponding fixed point theorems (see [14â€“16], for more details).

Definition 8. (see [14]). A pair of self-maps of a metric space is said to be weakly semicompatible, if or , whenever is a sequence in such that for some .

Definition 9. (see [15]). A pair of self-maps of a metric space is said to be conditionally semicompatible; if whenever the set of sequences satisfying is nonempty, then there exists at least a sequence satisfying such that and .
It is obvious that semicompatibility of type implies weak semicompatibility. From the definition itself, it is clear that if a pair of self-maps is semicompatible of type , then it is necessarily conditionally semicompatible; however, the conditionally semicompatible maps are not necessarily semicompatible of type .

Example 1. Let and be the usual metric on . Define as follows:Let us consider the sequence ; we haveHowever, if we take , we have thatThus, the pair is conditional semicompatible.
Finally, we introduce a new kind of compatibility of a pair of self-maps called compatible firstly proposed by Jain et al. [17] as follows.

Definition 10. Let be a self-map defined on satisfying for some sequence and . Then, a pair of self-maps defined on is called compatible if .

Example 2. Let , , , and . Take . Since with and , then pair is compatible. However, and .
It is obvious that compatibility of a pair self-maps implies . property of a pair of self-maps by taking self-map as an identity map.
Let , , , and (identity function on ). Take . Here, . Hence, pair self-maps satisfy property.
Now, we introduce one more example of compatibility including four maps as follows.

Example 3. If with the usual metric. Define byChoose , where when , then , , , and .
Since and , then the pair is compatible. Next, since and , then the pair is compatible. Further, since and , then the pair is compatible. Finally, since and , then the pair is compatible.
Ansari, in 2014, firstly [18], introduced the concept of class functions and proved some fixed point theorems via class functions (see [19, 20] for more details).

Definition 11. (see [18]). A mapping is called a class function if it is continuous and the following axioms hold:(1) for all (2) implies that or Denote the family of class functions by .

Example 4. (see [18]). The following functions are elements of , for all :(1), implies (2), for some , implies (3), for some , implies or (4), for some , implies or (5), for , implies (6), , for , implies (7), for , implies or (8), implies (9), where is continuous, implies (10), implies (11), implies , where is a continuous function such that if and only if (12), implies , where is a continuous function such that for all (13), implies (14), implies (15), implies , where is a continuous function such that and , for Afterward, by the motivation of class functions, Saleem et al. [1, 21] introduced a new notion of inverse class functions as follows.

Definition 12. (see [1]). A mapping is called an inverse class function if it is continuous and the following axioms hold:(1) for all (2) implies that or Denote the family of inverse class functions by .

Example 5. (see [1]). The following functions are elements of , for all :(1) implies (2), for some implies (3), for some , implies or (4), for some , implies (5), implies , where is an upper semicontinuous function such that and , for Motivated by the above definition, we now define inverse class functions as follows.

Definition 13. A mapping is called an inverse class function if it is continuous and the following axioms hold:(1) for all and some (2) implies that or Denote the family of inverse class functions by . Every inverse class function and inverse class function are equivalent when ; however, an inverse class function may not be an inverse class function.

Example 6. A mapping is defined by for all . Then, clearly, is an inverse class function for , but it is not an inverse class function.

Example 7. The following functions are elements of , for all :(1) implies for some and (2) implies for and some (3) implies or for and some (4) implies for some and some (5), implies , where is an upper semicontinuous function such that and , for and some

Definition 14. Let denote the class of functions which satisfy the following conditions:(a) is continuous and increasing with (b)

Definition 15 (see [22]). A function is said to be ultra-altering distance function if is nondecreasing and continuous; , for all and . Denote the class of ultra-altering distance functions by .

Lemma 1. Every sequence in metric space will be Cauchy if there exists such that , for all .
The aim of this presented paper is to provide some common fixed point theorems under several compatible conditions mentioned above via inverse class functions, which extend, generalize, and improve the existing results in the literature. Some examples are provided to illustrate the validity of our results.

2. Main Results

Theorem 1. Let be a complete metric space and a let pair of self-maps be semicompatible, satisfying the following assumptions:â€‰ â€‰ Here, and , for all . Moreover, , with , , , , , for some and , . If the pair is compatible of type , then and have a unique common fixed point in .

Proof. Let be any point in . Since , there exists such thatContinuing this way, we can construct a sequence in satisfyingBy the assumption of , we haveBy the monotonicity of , we haveAgain, from the triangle inequality, that is, , we havewhich further yields thatSince , , then ; from Lemma 1, it follows that sequence is a Cauchy sequence. Since is complete, there exists a point such thatNow, we will show that is a common fixed point of and .
Since the pair is semicompatible, we haveSince and are compatible of type , it follows thatNow, by the definition of and assumption , we obtainTaking the limit as in above inequality, it follows thatwhich implies thatAgain, it follows from the definition of and assumption thatTaking the limit as in above inequality, we conclude thatSince and , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that , that is, is a common fixed point of and .
Next, we will prove the uniqueness of common fixed point of and in .
Suppose that is another common fixed point of and , that is .
From above argument, it may be concluded thatSince and , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that .

Remark 1. (i)The conclusion of Theorem 1 still holds under the assumptions of semicompatibility of the pair and compatibility of type .(ii)If the inequality in assumption is replaced bythe conclusion still holds.
Here is an illustrated example to support the validity of Theorem 1 as follows.

Example 8. Let be a usual metric space. We define and on as follows:To verify that the pair is semicompatible as well as compatible of type , we take any sequence ; then, and . Define by , with and , for all . Then, for all , we have