In this paper, we introduce the concept of ω - compatibility and weakly commutativity for hybrid pair of mappings F : X × X × X × → 2^X and g : X → X and establish a common tripled fixed point theorem under generalized nonlinear contraction. An example is also given to validate our result. We improve, extend and generalize various known results. Let ( X, d ) be a metric space and CB ( X ) be the set of all nonempty closed bounded subsets of X . Let D ( x, A ) denote the distance from x to A ⊂ X and H denote the Hausdorff metric induced by d , that is, and for all A, B ∈ CB(X). Markin [23] initiated the study of fixed points for multivalued contractions and non-expansive maps using the Hausdorff metric. Fixed points existence for various multivalued contractive mappings has been studied by several authors under different conditions. For details, we refer the reader to [1 , 2 , 12 , 13 , 14 , 15 , 16 , 18 , 19 , 20 , 21 , 25 , 26 , 27] and the reference therein. Multivalued maps theory has application in control theory, convex optimization, differential equations and economics. Bhaskar and Lakshmikantham [10] , established some coupled fixed point theorems and apply these to study the existence and uniqueness of solution for periodic boundary value problems. Lakshmikantham and Ciric [22] proved coupled coinci-dence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Bhaskar and Lakshmikantham [10] . Berinde and Borcut [8] introduced the concept of tripled fixed point for single valued mappings in partially ordered metric spaces. In [8] , Berinde and Borcut established the existence of tripled fixed point of single-valued mappings in partially ordered metric spaces. For more details on tripled fixed point theory, we also refer the reader to [3 , 4 , 5 , 6 , 7 , 9 , 11] : Samet and Vetro [24] introduced the notion of fixed point of N order in case of single-valued mappings. In particular for N=3 (tripled case), we have the following definition: Definition 1.1 ( [24] ). Let X be a non-empty set and F : X × X × X → X be a given mapping. An element ( x, y, z ) ∈ X × X × X is called a tripled fixed point of the mapping F if F(x, y, z) = x, F(y, z, x) = y and F(z, x, y) = z. In this paper, we prove a common tripled fixed point for hybrid pair of mappings under generalized nonlinear contraction. We improve, extend and generalize the results of Ding, Li and Radenovic [17] and Abbas, Ciric, Damjanovic and Khan [2] . The effectiveness of the present work is validated with the help of suitable example. First we introduce the following: Definition 2.1. Let X be a nonempty set, F : X × X × X → 2 ^X (a collection of all nonempty subsets of X ) and g be a self-map on X . An element ( x, y, z )∈ X × X × X is called We denote the set of tripled coincidence points of mappings F and g by C ( F, g ). Note that if ( x, y, z ) ∈ C ( F, g ), then ( y, z, x ) and ( z, x, y ) are also in C ( F, g ). Deginition 2.2 . Let F : X × X × X→ 2^X be a multivalued mapping and g be a self-map on X . The hybrid pair { F, g } is called w-compatible if g ( F ( x, y, z )) ⊆ F ( gx, gy, gz ) whenever ( x, y, z ) ∈ C ( F, g ). Deginition 2.3 . Let F : X × X × X→ 2^X be a multivalued mapping and g be a self-map on X . The mapping g is called F-weakly commuting at some point (x, y, z) ∈ X ³ if g ² x ∈ F ( gx, gy, gz ), g ² y ∈ F ( gy, gz, gx ) and g ² z ∈ F ( gz, gx, gy ). Lemma 2.1 . Let ( X, d ) be a metric space. Then, for each a ∈ X and B ∈ CB ( X ), there is b ₀ ∈ B such that D ( a, B ) = d (a , b ₀ ), where D ( a, B ) = inf _b∈B d ( a, b ). Proof. Let a ∈ X and B ∈ CB ( X ). Since the function d is continuous. Thus, by the closedness of B , there exists b ₀ ∈ B such that inf _{b ∈B} d ( a, b ) = d ( a, b ₀ ), that is, D ( a,B ) = d ( a, b ₀ ). It is clear that φ(t ) < t for each t > 0: In fact, if φ ( t ₀ ) ≥ t ₀ for some t ₀ > 0. then, since φ is non-decreasing, φ ⁿ ( t ₀ ) ≥ t ₀ for all n ∈ , which contradicts with lim _n→∞ φ ⁿ ( t ₀ ) = 0. In addition, it is easy to see that φ (0) = 0. □ Theorem 2.1 . Let ( X , d ) be a metric space. Assume F : X × X × X → CB ( X ) and g : X → X be two mappings satisfying for all x, y, z, u, v, w ∈ X, where φ ∈ Φ. Furthermore assume that F ( X×X×X ) ⊆ g ( X ) and g ( X ) is a complete subset of X. Then F and g have a tripled coincidence point. Moreover, F and g have a common tripled fixed point, if one of the following conditions holds. Proof. Let x₀, y₀, z₀ ∈ X be arbitrary. Then F ( x₀, y₀, z₀ ), F ( y₀, z₀, x₀ ) and F ( z₀, x₀, y₀ ) are well defined. Choose gx₁ ∈ F ( x₀, y₀, z₀ ), gy ₁ ∈ F( y₀, z₀, x₀ ) and gz ₁ ∈ F ( z₀, x₀, y₀ ), because F ( X×X×X ) ⊆ g ( X ). Since F . X×X×X → CB ( X ), therefore by Lemma 2.1, there exist u ₁ ∈ F ( x ₁ , y ₁ , z ₁ ), u ₂ ∈ F ( y ₁ , z ₁ , x ₁ ) and u ₃ ∈ F ( z ₁ , x ₁ , y ₁ ) such that Since F ( X×X×X ) ⊆ g ( X ), there exist x ₂ , y ₂ , z ₂ ∈ X such that u ₁ = gx ² , u ₂ = gy ₂ and u ₃ = gz ₂ , Thus Continuing this process, we obtain sequences { x_n }, { y_n } and { z_n } in X such that for all n ∈ , we have gx _n+1 ∈ F ( x_n, y_n, z_n ), gy _n+1 ∈ F ( y_n, z_n, x_n ) and gz _n+1 ∈ F ( z_n, x_n, y_n ) such that Thus, Similarly Combining (2,2), (2,3) and (2,4), we get Thus If we suppose that then by (2, 5), ( i_φ ) and ( ii_φ ), we have which is a contradiction. Thus, we must have Hence by (2,5), we have for all n ∈ , Thus where δ= max {d(gx₀, gx₁), d(gy₀, gy₁), d(gz₀, gz₁)} . Without loss of generality, one can assume that max max { d ( gx ₀ , gx ₁ ), d ( gy ₀ , gy ₁ ), d ( gz ₀ , gz ₁ )} ≠ In fact, if this is not true, then gx ₀ = gx ₁ ∈ F ( x ₀ , y ₀ , z ₀ ), gy ₀ = gy ₁ ∈ F ( y ₀ , z ₀ , x ₀ ) and gz ₀ = gz ₁ ∈ F ( z ₀ , x ₀ , y ₀ ) that ( x ₀ , y ₀ , z ₀ ) s a tripled coincidence point of F and g . Thus, for m, n ∈ with m > n , by triangle inequality and (2,6), we get which implies, by ( ii_φ ), that { gx_n } is a Cauchy sequence in g ( X ). Similarly we obtain that { gy_n } and { gz_n } are Cauchy sequences in g ( X ). Since g ( X ) is complete, there exist x, y, z ∈ X such that Now, since gx _n+1 ∈ F ( x_n, y_n, z_n ), gy _n+1 ∈ F ( y_n, z_n, x_n ) and gz _n+1 ∈ F ( z_n, x_n, y_n ), therefore by using condition (2,1), we get where Since lim _n→∞ gx_n = gx , lim _n→∞ gy_n = gy and lim _n→∞ gz_n = gz there exists n ₀ ∈ such that for all n > n ₀ , Combining this with (2,8), (2,9) and (2,10), we get for all n > n ₀ , Now, we claim that If this is not true, then Thus, by (2,11), ( i_φ ) and ( ii_φ ), we get for all n > n ₀ , Thus Letting n → ∞ in (2:13), by using (2,7), we obtain which is a contradiction. So (2, 12) holds. Thus, it follows that gx∈ F(x, y, z), gy∈ F(y, z, x) and gz∈ F(z, x, y), that is, ( x, y, z ) is a tripled coincidence point of F and g . Hence C ( F, g ) is nonempty. Suppose now that (a) holds. Assume that for some ( x, y, z ) ∈ C ( F, g ), where u, v, w ∈ X . Since g is continuous at u, v and w . We have, by (2,14), that u,v and w are fixed points of g , that is, As F and g are w -compatible, so for all n ≥ 1, Now, by using (2,1) and (2,16), we obtain where By (2,14) and (2,15), there exists n ₀ ∈ such that for all n > n ₀ , Combining this with (2,17), we get for all n > n ₀ , Now, we claim that If this is not true, then Thus, by (2,18), ( i_φ ) and ( ii_φ ), we get for all n > n ₀ , On taking limit as n → ∞ in (2, 20), by using (2, 14) and (2, 15), we get which is a contradiction. So (2, 19) holds. Thus, it follows that Now, from (2, 15) and (2, 21), we have that is, ( u, v, w ) is a common tripled fixed point of F and g . Suppose now that ( b ) holds. Assume that for some ( x, y, z ) ∈ C ( F, g ), g is F- weakly commuting, that is, g ² x ∈ F ( gx, gy, gz ), g ² y ∈ F ( gy, gz, gx ), g ² z ∈ F ( gz, gx, gy ), and g ² x = gx , g ² y = gy , g ² z = gz . Thus gx = g ² x ∈ F ( gx, gy, gz ), gy = g ² y ∈ F ( gy, gz, gx ) and gz = g ² z ∈ F ( gz, gx, gy ), that is ( gx, gy, gz ) is a common tripled fixed point of F and g . Suppose now that ( c ) holds. Assume that for some ( x, y, z ) ∈ C ( F, g ) and for some u, v, w ∈ X , lim _n→∞ gⁿu = x , lim _n→∞ gⁿv = y and lim _n→∞ gⁿw = z . Since g is continuous at x, y and z . We have that x, y and z are fixed point of g , that is, gx = x , gy = y and gz = z . Since ( x, y, z ) ∈ C( F, g ), therefore, we obtain x = gx ∈ F(x, y, z), y = gy ∈ F(y, z, x) and z = gz ∈ F(z, x, y) , Finally, suppose that ( d ) holds. Let g ( C ( F, g )) = {( x, x, x )}. Then { x } = { gx } = F( x, x, x ): Hence ( x, x, x ) is tripled fixed point of F and g . □ Example 2.1. Suppose that X = [0, 1], equipped with the metric d : X ×X →[0, +∞) defined by d ( x, y ) = max{ x, y } and d ( x, x ) = 0 for all x, y ∈ X . Let F : X×X×X → CB ( X ) be defined as and g : X → X be defined as g(x) = x ² , for all x ∈ X . Define φ : [0, ∞ ] → [0, ∞ ] by Now, for all x, y, z, u, v, w ∈ X with x, y, z, u, v, w ∈ [0, 1), we have Case (a) If x² + y² + z² = u² + v² + w², then Case ( b ) If x ² + y ² + z ² ≠ u ² + v ² + w ² with x ² + y ² + z ² < u ² + v ² + w ² , then Similarly, we obtain the same result for u ² + v ² + w ² < x ² + y ² + z ² . Thus the contractive condition (2,1) is satisfied for all x, y, z, u, v, w ∈ X with x, y, z, u, v, w ∈ [0, 1). Again for all x, y, z, u, v, w ∈ X with x, y, z , ∈ [0, 1) and u, v, w = 1, we have Thus the contractive condition (2:1) is satisfied for all x, y, z, u, v, w ∈ X with x, y z ∈ [0, 1) and u, v, w = 1, Similarly, we can see that the contractive condition (2,1) is satisfied for all x, y, z, u, v, w ∈ X with x, y, z, u, v, w = 1. Hence, the hybrid pair {F, g} satisfies the contractive condition (2,1), for all x, y, z, u, v, w ∈ X . In addition, all the other conditions of Theorem 2.1 are satisfied and z = (0, 0, 0) is a common tripled fixed point of hybrid pair { F, g } . The function F : X×X×X → CB ( X ) involved in this example is not continuous at the point (1, 1, 1) ∈ X×X×X . Remark 2.1 . We improve, extend and generalize the result of Ding, Li and Radenovic [17] in the following sense: If we put g = I ( I is the identity mapping) in Theorem 2.1, then we have the following result: Corollary 2.2 . Let ( X, d ) be a complete metric space, F : X × X× X → CB ( X) be a mapping satisfying for all x, y, z, u, v, w ∈ X , where φ ∈ Φ. Then F has a tripled fixed point. If we put φ ( t ) = kt where 0 < k < 1 in Theorem 2.1, then we have the following result: Corollary 2.3 . Let ( X, d ) be a metric space. Assume F : X × X× X → CB ( X ) and g : X → X be two mapping satisfying for all x, y, z, u, v, w ∈ X where 0 < k < 1. Furthermore assume that F ( X ×X×X ) ⊆ g ( X ) and g ( X ) is a complete subset of X. Then F and g have a tripled coincidence point. Moreover, F and g have a common tripled fixed point, if one of the following conditions holds. If we put g = I ( I is the identity mapping) in Corollary 2.3, then we have the following result: Corollary 2.4 . Let ( X, d ) be a complete metric space, F : X × X × X → CB(X) be a mapping satisfying for all x, y, z, u, v, w ∈ X . Then F has a tripled fixed point.