In this paper, we deﬁne the tangential property and the generalized **coincidence** property for a pair of **set**-**valued** and **single**-**valued** **mappings** and use it to prove some **coupled** co- incidence and **common** ﬁxed **point** **theorems** for a hybrid pair of **mappings** without appeal to the completeness of the underlying **space**.

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Very recently, some wonderful research on ﬁxed **point** theory in a **metric** **space** endowed with a graph has been carried out by Alfuraidan [, ] and Alfuraidan and Khamsi []. Again, the study of **coupled** and **common** ﬁxed **point** **theorems** remain a well motivated area of research in ﬁxed **point** theory due to their applications in a wide variety of prob- lems. For example, applications of **coupled** ﬁxed points for binary **mappings** were studied by Bhaskar and Lakshmikantham []. They have used such ﬁxed **point** results to prove the existence and uniqueness of solution for a periodic boundary value problem. Recently, Chifu and Petrusel [] have developed some **coupled** ﬁxed **point** results in **metric** **space** en- dowed with a directed graph to prove the existence of a continuous solution for a system of Fredholm and Volterra integral equations.

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uang and Zhang [8] generalized the notion of **metric** **space** by replacing the **set** of real numbers by ordered Banach **space** and defined cone **metric** **space** and extended Banach type **fixed** **point** **theorems** for contractive type **mappings**. Subsequently, some other authors [1,4,5,7,10,12,13,14,15,17] studied properties of cone **metric** spaces and **fixed** points results of **mappings** satisfying contractive type condition in cone **metric** spaces. Recently Beg, Azam and Arshad [6], introduced and studied topological vector **space**(TVS) **valued** cone **metric** spaces which is bigger than that of introduced by Huang and Zhang [8]. TVS **valued** cone **metric** spaces were further considered by some other authors in [3,9,11,16,18]. In this paper we obtain **common** **fixed** points of a pair of **mappings** satisfying a generalized contractive type condition without the assumption of normality in TVS-**valued** cone **metric** spaces. Our results improve and generalize some significant recent results.

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In , Zadeh [] introduced the concept of **fuzzy** sets. Then many authors gave the im- portant contribution to development of the theory of **fuzzy** sets and applications. George and Veeramani [, ] gave the concept of a **fuzzy** **metric** **space** and deﬁned a Hausdorﬀ topology on this **fuzzy** **metric** **space**, which have very important applications in quantum particle physics, particularly, in connection with both string and E-inﬁnity theory.

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A nonempty **set** X endowed with a vector-**valued** **metric** d is called a generalized **metric** **space** in the sense of Perov (in short, a generalized **metric** **space**) and it will be denoted by (X, d). The usual notions of analysis (such as convergent sequence, Cauchy sequence, completeness, open subset, closed **set**, open and closed ball, etc.) are deﬁned similarly to the case of **metric** spaces.

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The study of **fixed** **point** **theorems**, involving four **single**-**valued** maps, began with the assumption that all of the maps are commuted. Sessa [6] weakened the condition of commutativity to that of pairwise weakly commuting. Jungck generalized the notion of weak commutativity to that of pairwise compatible [3] and then pairwise weakly compatible maps [4]. Jungck and Rhoades [5] proved some **common** **fixed** **point** **theorems** on the concept of occasionally weakly compatible maps.

in respective **fuzzy** **metric** spaces and proved **fuzzy** contraction ﬁxed **point** **theorems** under diﬀerent hypotheses. For instance, Mihet assumed that the **space** under consideration is an M-complete non-Archimedean KM-**space**. Moreover, he posed an open question whether this ﬁxed **point** theorem holds if the non-Archimedean **fuzzy** **metric** **space** is replaced by a **fuzzy** **metric** **space**. Vetro [] introduced a notion of weak non-Archimedean **fuzzy** **metric** **space** and proved **common** ﬁxed **point** results for a pair of generalized contractive-type **mappings**. Wang [] gave a positive answer for the open question.

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Bhaskar and Lakshmikantham [3] introduced the concept of **coupled** **fixed** **point** of a mapping F from X ×X to X and established some **coupled** **fixed** **point** **theorems** in par- tially ordered sets. As an application, they studied the existence and uniqueness of solution for a periodic boundary value problem associated with a first order ordinary differential equation. Ć iri ć et al. [4] proved **coupled** **common** **fixed** **point** **theorems** for **mappings** satisfying nonlinear contractive conditions in partially ordered complete **metric** spaces and generalized the results given in [3]. Sabetghadam et al. [5] employed these concepts to obtain **coupled** **fixed** **point** in the frame work of cone **metric** spaces. Lakshmikantham and Ćirić [4] introduced the concepts of **coupled** **coincidence** and **coupled** **common** **fixed** **point** for **mappings** satisfying nonlinear contractive conditions in partially ordered complete **metric** spaces. The study of **fixed** points for multi-**valued** contractions **mappings** using the Hausdorff **metric** was initiated by Nadler [1] and Markin [6]. Later, an interesting and rich **fixed** **point** theory for such maps was devel- oped which has found applications in control theory, convex optimization, differential inclusion and economics (see [7] and references therein). Klim and Wardowski [8] also obtained existence of **fixed** **point** for **set**-**valued** contractions in complete **metric** spaces. Dhage [9,10] established hybrid **fixed** **point** **theorems** and gave some applications (see also [11]). Hong in his recent study [12] proved hybrid **fixed** **point** **theorems** involving multi-**valued** operators which satisfy weakly generalized contractive conditions in ordered complete **metric** spaces. The study of **coincidence** **point** and **common** **fixed** points of hybrid pair of **mappings** in Banach spaces and **metric** spaces is interesting and well developed. For applications of hybrid **fixed** **point** theory we refer to [13-16]. For a survey of **fixed** **point** theory and coincidences of multimaps, their applications and related results, we refer to [16-22].

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In 1970, Covitz Nadler’s (see [6]) gave the following results “Multi-**valued** contraction **mappings** generalized **metric** spaces” using this result. H. E. Kunze et, al (see [3]) introduce an iterative method involving projections that guarantees convergence, from any starting **point** to a **point** the **set** of all **fixed** points of a multifunction operator T. The results [3] were generalized by Dubey [16]. Especially, Nadler’s. Jr. [7] gave a generalization of Banach’s contraction principle to the case of **set**-**valued** maps in **metric** spaces. Recently, Huang and Zhang [1] introduced the concept of cone **metric** **space** by replacing the **set** of real numbers by an ordered Banach **space** and obtain some **fixed** **point** **theorems** for **mappings** satisfying different contractive conditions. Subsequently, the results [1] were generalized and studied the existence of **common** **fixed** points of a pair of self **mappings** satisfying a contractive type condition in the frame work of normal cone **metric** spaces, see for instance [2], [4], [5],[9] and [11]. The authors [10, 14] introduced the concept of multi-**valued** contractions in cone **metric** spaces and using the notion of normal cones, obtained **fixed** **point** **theorems** for such **mappings**. As we know, most of known cones are normal with normal constant . Further, the author [12] and [13] proved two results, **fixed** points and **common** **fixed** points of multifunction on cone **metric** spaces. These results also generalized by Dubey and Narayan [17].In this paper, we prove **common** **fixed** **point** **theorems** for pair of multi-**valued** maps in cone **metric** spaces with normal constant K=1, which generalize and extend the results of [1], [8] and [15].

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The notion of b-**metric** **space** was introduced by Bakhtin in[1]. Since then, many actions generalized the b-**metric** spaces(see [2], [3], [4]). Recently, Ma and Jiang[5] introduced the concept of a C ∗ algebras-**valued** b-**metric** spaces,and they obtained the basic **fixed** **point** theo- rems for self-map with contractive condition in C ∗ algebras-**valued** b-**metric** spaces. In 2016, Kamranetal [6] also introduced the concept of this **space**, and generalized the Banach contrac- tion principle on this **space**.

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Then there exist x ∈ X such that x Fx, x, that is, F admits a unique **fixed** **point** in X. Let φt kt, where 0 < k < 1, the following by Lemma 1, we get the following. Corollary 2 see 6. Let a ∗ b ≥ ab for all a, b ∈ 0, 1 and X, M, ∗ be a complete **fuzzy** **metric** **space** such that M has n-property. Let F : X × X → X and g : X → X be two functions such that

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Since Zadeh [1] introduced the concept of **fuzzy** sets, many authors have extensively developed the theory of **fuzzy** sets and applications. George and Veeramani [2, 3] gave the concept of **fuzzy** **metric** **space** and defined a Hausdorff topology on this **fuzzy** **metric** **space** which have very important applications in quantum particle physics particularly in connection with both string and infinity theory.

Theorem 2.9. Let (X, d) be a complex **valued** **metric** **space**, F : X × X → X and g : X → X be two **mappings** which satisfy all the conditions of Theorem 2.5. If F and g are w−compatible, then F and g have unique **common** **coupled** **fixed** **point**. Moreover, **common** **fixed** **point** of F and g has the form (u, u) for some u ∈ X.

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One of the main pillar in the study of **fixed** **point** theory is Banach Contraction priciple which was done by Banach in 1922. **Fixed** In 2011 Akbar Azam et al., 2011 introduced the concept of complex **valued** **metric** **space**. The concept of **coupled** **fixed** **point** was first introduced by Bhaskar and Laxikantham in 2006. Recently some researchers prove some **coupled** **fixed** **point** **theorems** in complex **valued** **metric** **space** in (Kang et al., 2013;

Throughout, unless otherwise stated, (X,d) is a **metric** **space**, K(X) is the collec- tion of all nonempty compact subsets of X , CL (X) is the collection of all nonempty closed subsets of X , H is the extended Hausdorﬀ **metric** on CL (X) , F is a map- ping from X into CL (X) , f , S are self-maps on X , I is the identity map on X , for any self-map h on X , (h) = {hx : x ∈ X}, R + is the **set** of all nonnegative real numbers, N is the **set** of all positive integers, Ω : (R + ) 5 → R + is monotonically in-

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Theorem 1.1 [1] Let (X , d) be a complete **metric** **space**, and F : X → CB(X) is a multivalued mapping, where CB(X) is the **set** of all nonempty closed bounded subsets of X . Assume that there exists α ∈ [0, 1) such that H(Fx, Fy) ≤ α d(x, y) for all x, y ∈ X . Then F has a **fixed** **point**. The Nadler’s **fixed** **point** theorem has been generalized in many ways. One generalization of Nadler’s **fixed** **point** theorem was given by Reich in 1972 [3], which was followed with a relaxed condition by Mizoguchi and Takahashi in 1989 [4] where they used the concept of M T −function ( R −function).

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directly from () and the fact that F(x) is closed for each x ∈ X. We now recall some deﬁnitions of continuity for **set**-**valued** **mappings** (see [] for more details). For our purpose, let X and Y be **metric** spaces (with no ambiguity, their metrics will be denoted by the same symbol ‘d’). A **set**-**valued** mapping F : X → Y is said to be

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E. Ozgur [15] presented the notion of complex **valued** G b -**metric** **space**. In 2006, Bhaskar et al. [5] introduced the notion of **coupled** **fixed** **point** and proved some **fixed** **point** results in this context. Similarly, we introduced the notion of **coupled** **fixed** **point** for a mapping in complex **valued** G b -**metric** spaces.

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In this paper, we prove a fixed point theorem for set valued directional contraction mappings.. see definition below..[r]