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TRIPLED FIXED POINT THEOREM FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION
TRIPLED FIXED POINT THEOREM FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Jan, 21(1): 23-38
Copyright © 2014, Korean Society of Mathematical Education
  • Received : August 02, 2013
  • Accepted : November 25, 2013
  • Published : January 30, 2014
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About the Authors
BHAVANA DESHPANDE
DEPARTMENT OF MATHEMATICS, GOVT. ARTS & SCIENCE P.G. COLLEGE, RATLAM- 457001 (M.P.) INDIAEmail address:bhavnadeshpande@yahoo.com
SUSHIL SHARMA
DEPARTMENT OF MATHEMATICS, GOVT. P. G. MADHAV SCIENCE COLLEGE, UJJAIN (MP), INDIAEmail address:sksharma2005@yahoo.com
AMRISH HANDA
DEPARTMENT OF MATHEMATICS, GOVT. P. G. ARTS AND SCIENCE COLLEGE, RATLAM-457001 (MP), INDIAEmail address:amrishhanda83@gmail.com

Abstract
In this paper, we introduce the concept of ω - compatibility and weakly commutativity for hybrid pair of mappings F : X × X × X × → 2X 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.
Keywords
1. INTRODUCTION AND PRELIMINARIES
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, BCB(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.
2. MAIN RESULTS
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
  • (1)a tripled fixed pointofFifx∈F(x, y, z),y∈F(y, z, x) and z ∈F(z, x, y).
  • (2)a tripled coincidence pointof hybrid pair {F, g} ifg(x) ∈F(x, y, z),g(y) ∈F(y, z, x) andg(z) ∈F(z, x, y).
  • (3)a common tripled fixed pointof hybrid pair {F, g} ifx=g(x) ∈F(x, y, z),y=g(y) ∈F(y, z, x) andz=g(z) ∈F(z, x, y).
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→ 2X 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→ 2X 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 ).
  • Let Φ denote the set of all functionsφ: [0, ∞) → [0, ∞) satisfying
  • (iφ)φis non-decreasing,
  • (iiφ) limn→ ∞φn(t) = 0 for allt> 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
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, 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
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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.
  • (a)F and g are w-compatible.limn→∞gnx= u, limn→∞gny=v andlimn→∞gnz=w for some(x, y, z) ∈C(F, g)and for some u, v, w∈Xand g is continuous at u, v and w.
  • (b)g is F-weakly commuting for some(x, y, z) ∈C(F, g)and gx, gy and gz are fixed points of g, that is, g2x = gx, g2y = gy and g2z = gz.
  • (c)g is continuous at x, y and z.limn→∞gnu = x, limn→∞gnv = y andlimn→∞gnw = z for some(x, y, z) ∈C(F, g) and for someu, v, w∈X:
  • (d)g(C(g, F))is singleton subset of C(g, F):
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
  • d(gx1,u1) ≤H(F(x0,y0,z0),F(x1,y1,z1)),
  • d(gy1,u2) ≤H(F(y0,z0,x0),F(y1,z1,x1)),
  • d(gz1,u3) ≤H(F(z0,x0,y0),F(z1,x1,y1)).
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
  • d(gx1,gx2) ≤H(F(x0,y0,z0),F(x1,y1,z1)),
  • d(gy1,gy2) ≤H(F(y0,z0,x0),F(y1,z1,x1)),
  • d(gz1,gz2) ≤H(F(z0,x0,y0),F(z1,x1,y1)).
Continuing this process, we obtain sequences { xn }, { yn } and { zn } in X such that for all n
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, we have gx n+1 F ( xn, yn, zn ), gy n+1 F ( yn, zn, xn ) and gz n+1 F ( zn, xn, yn ) such that
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Thus,
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Similarly
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Combining (2,2), (2,3) and (2,4), we get
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Thus
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If we suppose that
then by (2, 5), ( iφ ) and ( iiφ ), we have
which is a contradiction. Thus, we must have
Hence by (2,5), we have for all n
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,
Thus
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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
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with m > n , by triangle inequality and (2,6), we get
which implies, by ( iiφ ), that { gxn } is a Cauchy sequence in g ( X ). Similarly we obtain that { gyn } and { gzn } are Cauchy sequences in g ( X ). Since g ( X ) is complete, there exist x, y, z X such that
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Now, since gx n+1 F ( xn, yn, zn ), gy n+1 F ( yn, zn, xn ) and gz n+1 F ( zn, xn, yn ), therefore by using condition (2,1), we get
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where
Since lim n→∞ gxn = gx , lim n→∞ gyn = gy and lim n→∞ gzn = gz there exists n 0
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such that for all n > n 0 ,
Combining this with (2,8), (2,9) and (2,10), we get for all n > n 0 ,
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Now, we claim that
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If this is not true, then
Thus, by (2,11), ( iφ ) and ( iiφ ), we get for all n > n 0 ,
Thus
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Letting n → ∞ in (2:13), by using (2,7), we obtain
which is a contradiction. So (2, 12) holds. Thus, it follows that gxF(x, y, z), gyF(y, z, x) and gzF(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 ),
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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,
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As F and g are w -compatible, so for all n ≥ 1,
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Now, by using (2,1) and (2,16), we obtain
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where
By (2,14) and (2,15), there exists n 0
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such that for all n > n 0 ,
Combining this with (2,17), we get for all n > n 0 ,
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Now, we claim that
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If this is not true, then
Thus, by (2,18), ( iφ ) and ( iiφ ), we get for all n > n 0 ,
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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
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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
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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
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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:
  • (i) We prove our result in the settings of multivalued mapping and for hybrid pair of mappings while Ding, Li and Radenovic[17]proved result for single valued mappings.
  • (ii) We prove tripled coincidence and common tripled fixed point theorem while Ding, Li and Radenovic[17]proved coupled coincidence and common coupled fixed point theorems.
  • (iii) To prove the result we consider non complete metric space and the space is also not partially ordered.
  • (iv) The mappingF : X ×X ×X→CB(X) is discontinuous and not satisfying mixed g-monotone property.
  • (v) The functionφ: [0, ∞) → [0, ∞) involved in our theorem and example is discontinuous.
  • (vi) Our proof is simple and different from the other results in the existing literature.
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.
  • (a)F and g are w-compatible. limn→∞gnx=u, limn→∞gny=vandlimn→∞gnz=wfor some (x, y, z)∈C(F, g)and for some u, v, w ∈ X and g is con-tinuous at u, v and w.
  • (b)g is F-weakly commuting for some (x, y, z) ∈ C(F, g) and gx, gy and gz are fixed points of g, that is, g2x = gx, g2y = gyandg2z = gz.
  • (c)g is continuous at x, y and z. limn→∞gnu=x, limn→∞gnv=yandlimn→∞gnw=z for som (x, y, z ) ∈ C(F, g) and for some u, v, w ∈ X.
  • (d)g(C(g, F)) is singleton subset of C(g, F).
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.
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