Common Fixed Point Theorems for Hybrid Pairs of Mappings using Implicit Relations

In this paper, we prove coincidence and fixed point theorems for two hybrid pairs of single valued and multivalued mappings on noncomplete metric spaces. We prove our results without using conditions of compatibility and continuity of mappings. We also give an example to validate our result.


Introduction
Jungck [5] generalized the concept of weak commutativity introduced by Sessa [24]. He defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that the weakly commuting maps are compatible but the converse is not true. The concept of weakly commuting mappings was extended by Kaneko [6] for multivalued set up and extended the result of Jungck [4]. Kaneko and Sessa [7] extended the concept of compatible mappings for multivalued mappings and generalized the result of Kubiak [26].
Nadlar [17] published a paper on multivalued mappings. Since then, the fixed point theory for single valued and multivalued mappings has been studied extensively and applied to diverse problems. This theory provides techniques for solving a variety of applied problems in mathematical science and engineering [19,23]. A number of generalization of Nadlar's results have appeared.
The study of noncompatible mappings was initiated by Pant [13][14][15][16]. He introduced R−weakly commutativity of mappings and compared R−weak commutativity and weak compatibility for single valued mappings. He showed that both R−weak commutativity and weak compatibility for single valued mappings are equivalent at the coincidence points. This idea of R−weak commutativity was independently extended to the settings of single valued and multivalued mappings by Shahzad and Kamran [12], Singh and Mishra [22].
Recently, Aamri and Moutawakil [10] defined a property (E.A) for self maps which contains the class of noncompatible maps. They obtained some fixed point theorems for such mappings using property (E.A) under strict contractive conditions.
Kamran [25] extended property (E.A) in the settings of single valued and multivalued mappings and generalized the notion of (IT )-commutativity for such pairs. He introduced the notion of T −weakly commuting map and showed that for hybrid pairs of mappings, (IT )−commuting at coincidence points implies T −weakly commuting but the converse is not true. He also showed that for single valued mappings T −weak commutativity at the coincidence points is equivalent to the weak compatibility.
We have observed that proving fixed point theorems using an implicit relation is a good idea since it covers several contractive conditions rather than one contractive condition (see [11]).
In this paper, we prove coincidence point and fixed point theorems for hybrid pairs of single valued and multivalued mappings on noncomplete metric spaces. The mappings involved in our results are noncompatible and discontinuous. We also give an example to validate our main result. We improve and extend the result of Kamran [25].

Preliminaries
Let (X, d) be a metric space and suppose that CB (X) denotes the set of non-empty, closed and bounded subsets of X.
For A, B ∈ CB (X), we denote It is well known that (CB (X) , H) is a metric space with the distance function H. Moreover, (CB (X) , H) is complete in the event that (X, d) is complete.

Definition 1 ( [6]
). Let (X, d) be a metric space, F : X → CB (X) and T : X → X. Then the pair {F, T } is said to be weakly commuting if for

Definition 2 ( [7]
). Let (X, d) be a metric space, F : X → CB (X) and T : X → X. Then the pair {F, T } is said to be compatible if and only if

Definition 3 ( [10]
). The maps f : X → X and g : X → X are said to satisfy the property (E.A) if there exists a sequence {x n } in X such that lim n→∞ f x n = lim n→∞ gx n = t ∈ X.

Definition 4 ( [25]
). The maps f : X → X and T : X → CB (X) are said to satisfy the property (E.A) if there exists a sequence {x n } in X such that lim n→∞ f x n = t ∈ A = lim n→∞ T x n for some t ∈ X and A ∈ CB (X).
Then lim Therefore, f and F satisfy property (E.A), but they are not compatible.
]. Therefore f and F are not (IT ) −commuting. Also note that f and F are not weakly compatible.
Then lim Therefore, f and F satisfy property (E.A) and the hybrid pair

Main Results
Theorem 1. Let (X, d) be a metric space. Let f, g : X → X and F, G : X → CB (X) be mappings satisfying the following conditions: where ϕ ∈ Φ, (1.3) the pairs {G, f } and {F, g} satisfy property (E.A), then (i) there exists u, v ∈ X such that f u ∈ Gu and gv ∈ F v.

Further if
(1.4) f is G−weakly commuting and g is F −weakly commuting at their coincidence point, then (ii) if f u = gv = z ∈ X, then f z ∈ Gz and gz ∈ F z, (iii) if f z = gz then f z = gz ∈ F z ∩ Gz, (iv) if f z = gz = z, then z is a common fixed point of f, g, F and G.
Proof. Since the pair {F, g} satisfies property (E.A), there is a sequence {x n } in X such that lim n→∞ gx n = t ∈ A = lim n→∞ F x n for some t ∈ X and A ∈ CB (X).
Since F (X) is closed, t ∈ F (X). Therefore by (1.1) there exists u ∈ X such that t = f u.
By Using (ϕ a ), we get D (f u, Gu) ≤ 0, which is a contradiction. Hence f u ∈ Gu.
Similarly, since the pair {G, f } satisfies property (E.A), there is a sequence {y n } in X such that lim n→∞ f y n = q ∈ M = lim n→∞ Gy n for some q ∈ X and M ∈ CB (X).
Since G (X) is closed, q ∈ G (X). Therefore by (1.1), there exists v ∈ X such that q = gv.
By Since gv ∈ M , it follows from the definition of Hausdorff metric that This proves (i). Now since f is G−weakly commuting at u ∈ X and g is F −weakly commuting at v ∈ X, we have f f u ∈ Gf u and ggv ∈ F gv.
If f u = gv = z ∈ X, then f z ∈ Gz and gz ∈ F z.
If f z = gz, then f z = gz ∈ F z ∩ Gz. If f z = gz = z, then z is the common fixed point of f, g, F and G. This proves the Theorem.
Theorem 1 and the Examples 3-8 imply the following: Corollary 1. Let (X, d) be a metric space. Let f, g : X → X and F, G : X → CB (X) be mappings satisfying the conditions (1.1), (1.3), (1.4) and the following: , Then f, g, F and G have a common fixed point.
for all x, y ∈ X, where α ∈ (0, 1) Then f, g, F and G have a common fixed point.  for all x, y ∈ X, where α, β ∈ (0, 1) , α + β < 1 and p is an integer such that p ≥ 1. Then f, g, F and G have a common fixed point. for all x, y ∈ X, where a ∈ (0, 1). Then f, g, F and G have a common fixed point. − α max {d p (f y, gx) − D p (f y, F x) , D p (gx, Gy)} 1 + (αD p (gx, Gy) + βD p (f y, Gy)) (d p (f y, gx) − D p (f y, F x)) for all x, y ∈ X, where β ∈ (0, 1), α ≥ 1 and p is an integer such that p ≥ 1. Then f, g, F and G have a common fixed point.

Conclusion.
To prove common fixed point theorems for two hybrid pairs of mappings using the property (E.A), the condition f u = gv = z ∈ X of Theorem 1 is essential. If we drop the property (E.A) and assume (X, d) as complete metric space, then the Theorem 1 can be proved without using f u = gv = z ∈ X.