No . 2 ( 2017 ) , 103 – 114 A general fixed point theorem for two hybrid pairs of mappings satisfying a mixed implicit relation and applications

The purpose of this paper is to extend Theorem 3.2 [16] for two hybrid pairs of mappings satisfying a mixed implicit relation and a new type of common limit range property without weak compatibility. As applications, some fixed point results for pairs of mappings satisfying contractive conditions of integral type and φ-contractive maps are obtained.

In 2011, Sintunavarat and Kumam [20] introduced the notion of common limit range property for single-valued mappings.
Imdad et al. [4] established common limit range property for a hybrid pair of mappings and obtained some fixed point results in symmetric spaces.
Quite recently, Imdad et al. [5] introduced the notion of joint common limit range property for two pairs of hybrid mappings.
The study of fixed points for mappings satisfying a contractive condition of integral type is introduced by Branciari [1].
It is proved in [15] that the study of fixed points of single-valued mappings and set-valued mappings satisfying integral condition is reduced to the study of fixed points for mappings involving altering distances.
Several classical fixed point theorems have been unified considering a general condition by an implicit relation in [9], [10] and in other papers.Recently, the method is used in the study of fixed points in metric spaces, symmetric spaces, quasi-metric spaces, b-metric spaces, ultra-metric spaces, Hilbert spaces, reflexive spaces, compact metric spaces, in two and three metric spaces, for single-valued mappings, hybrid pairs of mappings and set-valued mappings.
Quite recently, the method is used in the study of fixed points for mappings satisfying contractive/extensive conditions of integral type, in fuzzy metric spaces, probabilistic metric spaces, intuitionistic metric spaces, G -metric spaces, G p -metric spaces, and partial metric spaces.
With this method the proofs of existence of fixed points are more simple.Also, the method allows the study of local and global properties of fixed point structures.
In [11]- [13] and in other papers, the study of fixed points for hybrid pairs of mappings and set-valued mappings satisfying implicit relations is introduced.
A general fixed point theorem for a hybrid pair of mappings with common limit range property satisfying an implicit relation is proved in [2].

Preliminaries
Let (X, d) be a metric space.We denote by CL (X) the family of all closed sets of X and by H the Hausdorff-Pompeiu metric, i.e.
Definition 2.1.Let f : X → X be a single valued mapping and let F : X → 2 X be a multi-valued mapping.
1) A point x ∈ X is said to be a coincidence point of f and The set of all coincidence points of f and F is denoted by

Definition 2.2 ([3]
).Let f : X → X and F : X → 2 X be.The mapping f is said to be coincidentally idempotent with respect to F if f x ∈ F x implies f x = f f x, that is, f is idempotent at coincidence points of f and F .

Definition 2.3 ([4]
).Let (X, d) be a metric space, f : X → X and F : X → CL (X).Then, (f, F ) has a common limit range property if there exists a sequence {x n } in X such that for some u ∈ X and A ∈ CL (X).

Definition 2.4 ([14]
).Let A, S and T be self mappings of a metric space (X, d).The pair (A, S) is said to satisfy common limit range property with respect to T , denoted CLR (A,S),T if there exists a sequence {x n } in X such that lim for some t ∈ S (X) ∩ T (X).

Definition 2.5 ([5]
).Let (X, d) be a metric space, f, g : X → X and F, G : X → CL (X).Then, the pairs (f, F ) and (g, G) are said to have joint common limit range property, denoted (JCLR)-property, if there exist two sequences {x n } and {y n } in X and A, B ∈ CL (X) such that Now we introduce a new type of common limit range property.
Definition 2.6.Let (X, d) be a metric space, A : X → CL(X) and S, T : X → X.Then (A, S) satisfy a common limit range property with respect to T , denoted CLR (A,S)T -property, if there exists a sequence {x n } in X such that Example 2.1.Let X = [0, ∞) be a metric space with the usual metric, and Remark 2.1.Let (X, d) be a metric space, A, B : X → CL (X) and S, T : X → X.If (A, S) and (B, T ) satisfy (JCLR) -property, then (A, S) and T satisfy CLR (A,S)T -property.

Implicit relations
Definition 3.1.Let F M be the set of all lower semi-continuous functions  The purpose of this paper is to extend Theorem 3.1 for two hybrid pairs of mappings satisfying a mixed implicit relation and a new type of common limit property without weak compatibility.As applications, some fixed point results for mappings satisfying contractive conditions of integral type and ϕ-contractive maps.
On the other hand, z ∈ S(X).Hence, there exists v ∈ X such that z = Sv.By (4.1) for x = v and y = u we obtain Since z ∈ Bu, then d (z, Av) ≤ H (Av, Bu) which implies by (ψ 1 ) that ψ (d (z, A)) ≤ ψ(H (Av, Bu)).By (F 1 ) we have Moreover, a) If S is coincidentally idempotent with respect to A, then Sz = SSz = Sv = z and z is a fixed point of S. By (4.1) for x = z and y = u we obtain By (F 1 ) we have Hence, a contradiction of (F 3 ).Hence, d (z, Az) = 0 which implies Sz = z ∈ Az.Therefore z is a common fixed point of A and S. b) If T is coincidentally idempotent with respect to B, then T z = T T u = T u = z and z is a fixed point of T .By (4.1) for x = v and y = z we have Hence, a contradiction of (F 3 ).Hence, d (z, Bz) = 0 which implies T z = z ∈ Bz and z is a common fixed point of B and T .c) If the conditions of a) and b) hold, then z is a common fixed point of A, B, S and T .
If ψ(t) = t by Theorem 4.1 we obtain Theorem 4.2.Let (X, d) be a metric space, A, B : X → CL(X) and S, T : X → X such that for all x, y ∈ X  ).Let (X, d) be a metric space, c ∈ (0, 1) and f : X → X such that for all x, y ∈ X where h : [0, ∞) → [0, ∞) is a Lebesgue measurable mapping which is summable (i.e. with finite integral) on each compact subset of [0, ∞), such that ε 0 h(t)dt > 0, for each ε > 0.Then, f has a unique fixed point z ∈ X such that for all x ∈ X, z = lim n→∞ f n x.Some fixed point results for mappings satisfying contractive conditions of integral type are obtained in [15] and in other papers.Lemma 5.1 ([15]).Let h : [0, ∞) → [0, ∞) be as in Theorem 5.1.Then ψ (t) = t 0 h(x)dx is an altering distance.

( 4 . 1 )
F ψ (H (Ax, By)) , ψ (d (Sx, T y)) , ψ (d (Sx, Ax)) , ψ (d (T y, By)) , ψ (d (Sx, By)) , ψ(d (T y, Ax)) + + G ψ(d (Sx, T y)), ψ (d (Sx, Ax)) , ψ (d (T y, By)) , ψ (d (Sx, By)) , ψ (d (T y, Ax)) ≤ 0 for all x, y ∈ X, some F ∈ F M , G ∈ G M and ψ is an altering distance.If (A, S) and T satisfy CLR (A,S)T -property, then 1) C(A, S) = ∅, 2) C(B, T ) = ∅.Moreover, a) if S is coincidentally idempotent with respect to A, then S and A have a common fixed point, b) if T is coincidentally idempotent with respect to B, then T and B have a common fixed point, c) if the conditions of a) and b) hold, then S, T, A and B have a common fixed point.Proof.Since (A, S) and T satisfy CLR (A,S)T -property, there exists a sequence {x n } in X such that lim n→∞ Sx n = z, lim n→∞ Ax n = D, D ∈ CL (X) and z ∈ D ∩ S(X) ∩ T (X).

(4. 2 )
F H (Ax, By) , d (Sx, T y) , d (Sx, Ax) , d (T y, By) , d (Sx, By) , d (T y, Ax) + G (dSx, T y), d (Sx, Ax) , d (T y, By) , d (Sx, By) , d (T y, Ax) ≤ 0 for some F ∈ F M and G ∈ G M .If (A, S) and T satisfy CLR (A,S)T -property, then 1) C(A, S) = ∅, 2) C(B, T ) = ∅.Moreover, a) if S is coincidentally idempotent with respect to A, then S and A have a common fixed point, b) if T is coincidentally idempotent with respect to B, then T and B have a common fixed point, c) if the conditions of a) and b) hold, then A, B, S and T have a common fixed point.5.Applications5.1.Fixed points for hybrid pairs of mappings satisfying contractive conditions of integral type.In[1], Branciari established the following theorem, which opened the way to the study of fixed points for mappings satisfying a contractive condition of integral type.