Common fixed points for pairs of mappings with variable contractive parameters

In this paper, we establish some common fixed point results for a new class of pair of contractions mappings having functions as contractive parameters, and satisfying certain commutative properties.


Introduction and preliminaries
The metric fixed point theory has a vast literature, the Banach-Caccioppoli's contraction principle is one of the most outstanding result in this theory. Since its appearance, several generalizations of this result have been appeared in the literature. In 1976, Jungck [13] generalize this principle by considering two commutating mappings and proved a common fixed point theorem for these mappings. Afterwards, the commutative property of the mappings assumed by Jungck has been relaxed by introducing "weak" alternative notions as weakly commutativity, (non-) compatible and R-weakly commutating among others, which allowed to generalize different classes of well-known contractive type of mappings.
Here, we are going to establish existence and uniqueness results of common fixed points for a pair of contractive-type of mappings whose contractive parameters are non-constants and its contractive's inequality is controlled by altering distance functions. To attain our goals, we will assume that the mappings under consideration are occasionally weakly compatible. Also, alternatively, we will assume that the pair of mappings satisfies the so-called property (E.A.).
We would like to recall that in 1984, M.S. Khan et al [16] introduced the notion of altering distance functions. Since then, it has been used to solve several problems in the metric fixed point theory (see, e.g., [11,12,18,19,20,21,25]). Definition 1.1. A function ψ : R + −→ R + := [0, +∞) is called an altering distance function if the following properties are satisfied: By Ψ we are going to denote the set of all the altering distance functions.
By Φ is denoted the set of all mappings φ : R + −→ R + satisfying the following conditions: A relation between these two classes of functions Ψ and Φ is given in the following result ( [2,22]): In order to establish our results the following notions will be needed: A pair of selfmappings (S, T ) on a metric space (M, d) is said compatible [14] if and only if lim n→∞ d(T Sx n , ST x n ) = 0, whenever (x n ) n ⊂ M is such that lim n→∞ Sx n = lim n→∞ T x n = t for some t ∈ M. A pair of mappings (S, T ) is said to be noncompatible [1] if there exits at least one sequence (x n ) n ⊂ M such that lim n→∞ Sx n = lim n→∞ T x n = t for some t ∈ M, but lim n→∞ d(ST x n , T Sx n ) is either nonzero or non-existent. A pair of selfmappings (S, T ) is said to satisfy the property (E.A.), [1], if there exists a sequence (x n ) n ⊂ M such that for some t ∈ M.
A point x ∈ M is called a coincidence point (CP) of S and T if Sx = T x. The set of coincidence points of S and T will be denoted by C(S, T ). If x ∈ C(S, T ), then w = Sx = T x is called a point of coincidence (POC) of S and T .
Finally, a pair of mappings (S, T ) is said to be occasionally weakly compatible (OWC), [5], if there exists x ∈ C(S, T ) such that ST x = T Sx. [15] proved that if a pair of OWC maps (S, T ) has a unique POC, then it has a unique common fixed point, and M. Abbas and B.E. Rhoades [3] asserted that the property (E.A.) implies OWC.

Remark 1. In 2006, G. Jungck and B.E. Rhoades
G.U.R. Babu and G.N. Alemyehu in [7] proved that every pair of noncompatible selfmaps on a metric space satisfies the property (E.A.), but its converse is not true. Also, they showed that the property (E.A.) and OWC are independent conditions. The following lemma due to G.U. Babu and P.P. Sailaja in [8] will be useful in the sequel.
If (x n ) is not a Cauchy sequence in M, then there exist an ε 0 > 0 and sequences of integers positive (m(k)) and (n(k)) with

The class of pairs of mappings with nonconstant contractive parameters
In order to introduce the class of mappings which will be the focus of study of this paper, as in [17], we are going to use the functions α, β, γ : Now, we introduce the following class of pair of contraction-type of mappings.

Remark 2.
In addition, we can consider a class of pair of mappings satisfying the following inequality contraction of integral type: for all x, y ∈ M, where ψ ∈ Ψ, ϕ ∈ Φ and α, β, γ are functions satisfying (2.1). Notice that this class can be rewrite as for all x, y ∈ M, where ψ 0 ∈ Ψ is the function defined in the Lemma 1.1. Thus, we have that all the conclusions given for ψ−(α, β, γ)-contraction pairs are valid for pair of mappings satisfying the inequality contraction (2.3).
(2) (y n ) ⊂ M is a Cauchy sequence in M.
Proof. To prove (1), let x 0 ∈ M be an arbitrary point. Since S(M) ⊂ T (M), then there exists x 1 ∈ M such that Sx 0 = T x 1 . By continuing this process inductively we obtain a sequence (x n ) in M such that It follows that Therefore, we obtain from which, together with (2.1), we conclude that Since ψ ∈ Ψ, then (d(T x n , T x n+1 )) is a monotone decreasing sequence of non negative real numbers which converges to a ≥ 0. Thus, lim n→∞ d(T x n , T x n+1 ) = a ≥ 0.
We want to prove that a ≡ 0. We are going to assume that a > 0. Using the continuity of ψ and conditions (2.1) we have which is a contradiction. So, a = 0 and therefore, To prove (2), we are going to suppose that (y n ) ⊂ T (M) is not a Cauchy sequence. Then, from Lemma 1.2 there exists ε > 0 and sequences (m(k)) and (n(k)) with m(k) ≥ n(k) > k such that In this way we have which is a contradiction, hence (T x n ) is a Cauchy sequence in M.

On the existence and uniqueness of common fixed points
In this section we prove our main results concerning to the existence and uniqueness of common fixed points for a ψ − (α, β, γ)-contraction pair of mappings without continuity requirement. In comparison with classical results in this theory, the commutativity property in this case is reduced to the existence of points of coincidence and the completeness of the space is reduced to natural conditions.  and thus we can find u ∈ M such that T u = z. Now, we are going to assume that Su = z. Then, ψ(d(Sx n+1 , Su)) ≤ α(d(T x n+1 , T u))ψ(d(T x n+1 , T u)) + β(d(T x n+1 , T u))ψ(Sx n+1 , T x n+1 ) + γ(T x n+1 , T u)ψ(Su, T u).
Letting n → ∞, we obtain ψ(d(Su, z)) ≤ lim sup n→∞ α(d(T x n+1 , T u))ψ(d(z, T u)) which is a contradiction, therefore Su = z, hence z is a POC of S and T . From the Proposition 2.1 we conclude that z is the unique POC.
On the other hand, since the pair (S, T ) is OWC, then it has a unique common fixed point (see Remark 1).

Conclusions and examples
Notice that our results extend several classes of well known contractive type of mappings, including various classes of contractive mappings of the integral type. Even more, the mappings T and S considered here are not necessarily continuous, so in this way our results are more general compared with other results in this line of research.
Next, we are going to show some examples in support of our results. and γ(t) = 1 4 for all x ∈ R + and ψ : R + −→ R + given by the formula ψ(t) = t 2 , t ∈ R + .
Notice that ψ ∈ Ψ and the functions α, β, γ satisfy the conditions (2.1), also note that S(M) ⊂ T (M) and T (M) is a complete subspace of M. Moreover, is not difficult to show that the pair (S, T ) is a ψ − (α, β, γ)-contraction pair. Besides C(S, T ) = {0} and ST 0 = T S0 = 0, which mean that (S, T ) is OWC. Then, the Theorem 3.1 guarantee that w = 0 is the unique common fixed point of S and T . and T x = x 2 for all x ∈ M. Let α, β, γ : R + −→ [0, 1) defined as follow: , for all t ∈ R + .
Let ψ : R + −→ R + defined by ψ(t) = t 2 , t ∈ R + . Then, the pair (S, T ) is a ψ − (α, β, γ)-contraction pair satisfying the hypotheses of Theorem 3.1, thus w = 0 is the unique POC and moreover the unique common fixed point of S and T .