Semi-compatible mappings and common ﬁxed point theorems of an implicit relation via inverse C − class functions

: In this paper, we prove some common ﬁxed point theorems by exploring a new kind of generalized semi-compatibility and an implicit relation via inverse C − class functions. The results generalize, extend and improve the main results of [19,21–23]. Moreover, some examples are given to illustrate the validity of our results.


Introduction and preliminaries
The study of metric fixed point theory is playing an important role in linear and nonlinear analysis. In 1922, Stephen Banach [1] laid the foundation of metric fixed point theory and gave a very fruitful concept of contraction mapping. Since then, many researchers studied that field in many directions. It was indeed a turning point in fixed point theory when Sessa [2] introduced the notion of weak commutativity. Later, this concept was executed by several researchers in considerable amounts. Further, the generalization of weak commutativity came to exist in 1986 when Jungck [3] firstly introduced compatible mappings. Definitely, this research had opened some new directions in fixed point theory for many researchers. Later, in 1996 Jungck generalized his own concept by new class of compatible mappings, named weak compatible mappings [4], and through various examples he had shown that each of these generalizations of commutativity are proper extensions of previous definitions. Abbas eal. [5] pointed out that weakly compatible maps remains a minimal commutativity condition for the existence of unique common fixed of contractive type maps. In the last few decades many generalizations came to exist like compatible mapping of type (A) [6], compatible mapping of type (B) [7], compatible mapping of type (C) [8], compatible mapping of type (P) [9], semi-compatible mappings [10], weak semi-compatible mappings [11], conditional semi-compatible mappings [12], faintly compatible mappings [13], occasionally weakly compatible mappings [14][15][16] and other types of mappings [17,18]. In 2011, Singh et al. [19] gave brief discussion of various types of mappings as compatible mappings of type (A), type (B), type (C) and type (P) and compared these mappings with compatible mappings of type (E). He introduced new concepts of S −compatible mappings of type (E) and S −reciprocal continuous mappings by splitting the concepts of compatible mappings of type (E) and reciprocal continuous mappings [20], and moreover, obtained some common fixed point theorems for non-continuous self-mappings on metric spaces. Recently, Ansari et al. [21] used the concept of compatibility of type (E) and reciprocal continuity and obtained some fixed point results by using an implicit relation via C−class functions.
In this paper, we introduce a new concept of semi-compatible mappings and establish some common fixed point results by using an implicit relation introduced by Djoudi [22,23] via inverse C−class functions on metric spaces.
Throughout the paper, we will denote by N, R, R + and N 0 the set of natural numbers (positive integers), real numbers, positive real numbers and N ∪ {0}, respectively. whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. Definition 1.2.
[19] A pair of self-mappings ( f, g) on a metric space (X, d) is said to be f −compatible of type (E), if lim n→+∞ f f x n = lim n→+∞ f gx n = gt, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. Definition 1.3.
[19] A pair of self-mappings ( f, g) on a metric space (X, d) is said to be g−compatible of type (E), if lim n→+∞ ggx n = lim n→+∞ g f x n = f t, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. It is easy to see that the compatibility of type (E) implies both f − and g− compatibility of type (E), however the f − or g−compatibility of type (E) do not imply the compatibility of type (E) (See Example 2.10 [19]). Definition 1.4.
[10] A pair of self-mappings ( f, g) on a metric space (X, d) is said to be semicompatible, if lim n→+∞ f gx n = gt, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X.
A simple but genuine question rises: "Does semi-compatibility of ( f, g) imply the semi-compatibility of (g, f )?" That is lim n→+∞ g f x n = f t, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. Actually they are two different notions. We give the following example to verify it. Example 1.1. Let X = [0, +∞) endowed with usual metric d. Define a self-mappings f, g on X as follows: , and g = I X (the identity mapping). If we consider the sequence x n = 1 2 − ε n , where ε n > 0, ε n → 0 as n → +∞, then lim In the following we do a modification of the definition of semi-compatibility. Definition 1.5. A pair ( f, g) of self-mappings on a metric space (X, d) is said to be semi-compatible of type (A), if lim n→+∞ f gx n = gt and lim n→+∞ g f x n = f t, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. We give an example to demonstrate it as follows. Example 1.2. Let X = [1, +∞) endowed with the usual metric d and f, g : X → X be the selfmappings defined by x ∈ [3, +∞) .
If we consider the sequence x n = 1 + ε n , where ε n > 0, ε n → 0 as n → +∞, then lim Definition 1.6. A pair ( f, g) of self-mappings on a metric space (X, d) is said to be f −semi-compatible, if lim n→+∞ f gx n = gt, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. Definition 1.7. A pair ( f, g) of self-mappings on a metric space (X, d) is said to be g−semi-compatible, if lim n→+∞ g f x n = f t, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. It is obvious from the above definitions that semi-compatibility of type (A) implies f −semi-compatibility and g−semi-compatibility of a pair ( f, g), however the converse is not true. Moreover, f −semi-compatibility and g−semi-compatibility coincide with the semi-compatibility of the pair ( f, g) and semi-compatibility of the pair (g, f ) introduced by Singh et al. [10], respectively. Definition 1.8. [20] A pair ( f, g) of self-mappings on a metric space (X, d) is said to be reciprocal continuous, if lim n→+∞ f gx n = f t and lim n→+∞ g f x n = gt, whenever {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. We introduce some definitions by splitting the concept of reciprocal continuity of a pair ( f, g) of self-mappings as follows. Definition 1.9. A pair ( f, g) of self-mappings on a metric space (X, d) is said to be f −reciprocal continuous, if lim n→+∞ f gx n = f t, whenever {x n } is a sequence in X such that lim The notion of f −reciprocal continuity or g−reciprocal continuity coincides with the concept of weak reciprocal continuity introduced by Pant et al. [25].
It is obvious that f −semi-compatibility and g−semi-compatibility are independent notions with respect to f −reciprocal continuity and g−reciprocal continuity, respectively. It is noticed that compatibility of type (E) implies semi-compatibility of type (A) but implication is not reversible.
We now provide two examples to verify above discussion and also show the comparison between semi-compatible mappings of type (A) and reciprocal continuous mappings (compatible mappings of type (E)). .
Then the pair of mappings ( f, g) is semi-compatible of type (A) and reciprocal continuous. However, the pair ( f, g) is not compatible mapping of type (E).
It is easy to see that x n = 1 + ε n , where ε n > 0, ε n → 0 as n → +∞, is the only sequences which satisfy the conditions lim n→+∞ f x n = lim n→+∞ gx n = 1.
The other sentences follow from the following relations , .
and lim Therefore the pair of mappings ( f, g) is not only semi-compatible of type (A), but it is also reciprocal continuous, even g−compatible of type (E). However, it is not compatible of type (E).
According to the previous examples and Singh [19], we have the following proposition. Proposition 1.1. Let f and g be self-mappings on a metric space (X, d). Suppose that {x n } is a sequence in X such that lim n→+∞ f x n = lim n→+∞ gx n = t, for some t ∈ X. If one of the following conditions is satisfied:

Then
(a) f t = gt and (b) if there exists u ∈ X such that f u = gu = t, then f gu = g f u.
Proof. Follows immediately. Remark 1.1. By the above, it follows that each of condition of Proposition 1.1 implies the weak compatibility of pair ( f, g), introduced by Jungck in [4], however, the inverse is not applicable.
Definition 1.11. [26] A continuous function F : [0, +∞) × [0, +∞) → R is called an inverse C−class function, if for every s, t ∈ [0, +∞), the following conditions hold: We will denote by C inv the class of all inverse C−class functions. In the following we will provide some examples (for further details, one should refer [26]).
Example 1.5. The following functions F : is called an ultra-altering distance if ϕ is continuous, and ϕ(0) = 0, ϕ(t) > 0, t > 0. We denote by Φ u the set of all ultra-altering distance functions.
An implicit relation, introduced by Djoudi [22,23], is stated as follows. Let G be the set of all continuous functions G(t 1 , . . . , t 6 ) : → R satisfying the following conditions: We now provide some examples of G ∈ G (for more details, one can refer Djoudi [22,23]). Example 1.6. Let G(t 1 , . . . , t 6 ) : where c, d, e ≥ 0, a > 0 and b > a + c. We generalize the implicit relation of Djoudi [22,23] by using the inverse C−class functions. Let G c be the set of all continuous functions G(t 1 , . . . , t 6 ) : R + × R + × · · · × R + 6 → R satisfying the following conditions: (G 1 ) : G is non decreasing in variables t 5 and t 6 .
It is easy to obtain that G ⊆ G c .
Define F ∈ C inv by F(s, t) = hs and ϕ ∈ Φ u by ϕ(t) = t, for all t ≥ 0.
Using (G b ), we get Au = t and S u = Au = t. Next, we also claim that Bv = t. To see this, note that by condition (ii) with x = x 2n , y = v, we have where b−c a+c > 1 such that 0 < c < b−a 2 and a, d > 0. Define F ∈ C inv as F(s, t) = hs with h ∈ (1, +∞) and ϕ ∈ Φ u . For all x, y ∈ [1, +∞), we have Choosing a = 1, b = 4, c = 1 and d = 2, one has 0 < c < b−a 2 and Therefore all the conditions in Theorem 2.1 are satisfied and 1 is the common fixed point of A, B, S and T . Example 2.2. Let X = [1, +∞) and x, y ∈ X with usual metric d. We define maps A, B, S , T : X → X such that, It is obvious from the example that mappings S and T are surjective. On taking sequence {x n } = 1 + ε n , where ε n → 0 as n → +∞, then it is easy to show that pair (A, S ) is A−Semi compatible and A−reciprocal continuous. Also pair (B, T ) is B−semi compatible and B−reciprocal continuous. Now we define G(t 1 , . . . , t 6 ) : R + × R + × · · · × R + 6 → R as in Example 1.11: where h ∈ (1, +∞).
Define F ∈ C inv and F(s, t) = hs for all h ∈ (1, +∞) and ϕ ∈ Φ u . Now for all x, y ∈ [1, 2), we have which shows that (h 2 + 1)d(Ax, By) ≤ max{d(S x, T y), d(Ax, By), d(Ax, S x), d(By, T y), d(By, S x) + d(Ax, T y)}. Thus, Now for all x, y ∈ [2, +∞), we have Taking h = Theorem 2.2. Let G ∈ G c and let {A i } i∈N 0 and {B i } i∈N 0 be two sequences of self-mappings on a complete metric space (X, d) satisfying condition (ii) of Theorem 2.1. Assume that, for every n ∈ N 0 , the following properties are satisfied: Then {A i } i∈N 0 and {B i } i∈N 0 have a unique common fixed point.
Proof. Fix k ∈ N 0 . From hypothesis, we deduce that A 2k , A 2k+1 , B 2k and B 2k+1 satisfy the inequality for all x, y ∈ X, where G ∈ G c . Therefore all the conditions of Theorem 2.1 are satisfied. So A 2n , A 2n+1 , B 2n and B 2n+1 have a common fixed point in X.
which contradicts (G 3 ). Therefore t = t and so the sequences of maps {A i } i∈N 0 and {B i } i∈N 0 have a unique fixed point. Remark 2.2. According to [19,Remark 3.9], one can verify the followings: (i) Theorems 2.1 improves Theorems 3.7, 3.8 in [19] by exploring new kind of semi-compatibility in lieu of responding compatibility of type (E).
Proof. Take a function G ∈ G c as in Example 1.9 with d = 0. We have G(d(S x, T y), d(Ax, By), d(Ax, S x), d(By, T y), d(By, S x), d(Ax, T y)) = d(S x, T y) − [ad p (Ax, By) + bd p (Ax, S x) + cd p (By, T y)] 1 p ≥ 0.
The conclusion follows from Theorem 2.1. Remark 2.4. Corollary 2.2 is an improved result of Corollary 1 in [22] and Corollary 4.1 in [23] in the following aspects: (i) compatibility of type (B) in [22] and weak compatibility in [23] are replaced by semi-compatibility and reciprocal continuity, (ii) the requirement of continuity of mappings in [22] are relaxed.
Theorem 2.5. Let G ∈ G c and let A and S be two self-mappings on a complete metric space (X, d) satisfying the following conditions:

Conclusion
Based on the notions of semi-compatibility and reciprocal continuity of a pair of self-mappings ( f, g), we introduce some new types of a pair of self-mappings ( f, g), called semi-compatibility of type (A), f −semi-compatibility of type (A), g−semi-compatibility of type (A), f −reciprocal continuity and g−reciprocal continuity, which are extensions of the corresponding notions. Some valid examples are set up to demonstrate the comparisons between these conceptions. Moreover, by using the inverse C−class functions, we provide a new kind of implicit relations G c which is a generalization of the implicit relations G introduced by Djoudi. The achievement of this paper is to extend and improve the results of [19,[21][22][23] by using general implicit relations, weakening compatibility and dropping continuity.