Common tripled fixed point results in cone metric type ...

Rend. Circ. Mat. Palermo DOI 10.1007/s12215-014-0158-6

Common tripled fixed point results in cone metric type spaces Ghasem Soleimani Rad · Hassen Aydi · Poom Kumam · Hamidreza Rahimi

Received: 23 April 2014 / Accepted: 19 May 2014 © Springer-Verlag Italia 2014

Abstract In this paper, we prove some common tripled fixed point and tripled coincidence point results for contractive conditions in a cone metric type space. Our results extend, unify and generalize well-known results in the literature, in particular the recent results of Aydi et al. (Fixed Point Theory Appl 2012:134, 2012). Some examples are also presented to validate our obtained results and new concepts. Keywords Cone metric type space · Tripled coincidence point · Common tripled fixed point · W -compatible mappings Mathematics Subject Classification (2000)

54H25 · 47H10

1 Introduction and preliminaries In 1922, Banach proved the famous contraction mapping principle [4]. Afterward, other authors considered various definitions of contractive mappings and obtained several fixed G. Soleimani Rad (B) Young Researchers and Elite club, Central Tehran Branch, Islamic Azad University, Tehran, Iran e-mail: [email protected]; [email protected] H. Aydi Department of Mathematics, College of Education of Jubail, Industrial Jubail, Dammam University, Dammam 31961, Saudi Arabia e-mail: [email protected] P. Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Khru, Bangkok 10140, Thailand e-mail: [email protected] G. Soleimani Rad · H. Rahimi Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, P.O. Box 13185/768, Tehran, Iran e-mail: [email protected]

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point results and a survey of above is considered by Rhoades [36]. In 1976, Jungck [19] proved a common fixed point theorem for two commuting mappings. This theorem has many applications but it requires the continuity of one of the two mappings. In 1996, Jungck [18] defined a pair of self-mappings to be weakly compatible if they commute at their coincidence points. Then, Jungck and Rhoades [20] proved some common fixed point results for noncommuting and compatible mappings in metric spaces. On the other hand, the symmetric space as metric-like spaces lacking the triangle inequality was introduced in 1931 by Wilson [40]. In the sequel, a new type of spaces which they called metric type spaces (or b-metric spaces) are defined by Boriceanu [7] and Khamsi and Hussain [22,23]. Also, Jovanović et al. [17], Rahimi and Soleimani Rad [30,31], Bota et al. [8], Pavlović et al. [26] and Hussain et al. [15] generalized and unified some fixed point theorems of metric spaces by considering metric type spaces. The cone metric space was reintroduced in 2007 by Huang and Zhang [14] and several fixed and common fixed point results in cone metric spaces were proved, see for example [29,30,35] and the references contained therein. Recently, analogously with definition of a ´ metric type space, Radenović and Kadelburg [27], Cvetkovi´ c et al. [10], Rahimi et al. [33] considered cone metric type spaces (or cone b-metric spaces) and proved several fixed and common fixed point theorems. In 2006, Bhaskar and Lakshmikantham [6] defined the concept of a coupled fixed point. Afterwards, many authors obtained some coupled fixed point results with applications in nonlinear and differential equations (see [5,9,24,37] and references therein). In 2011, Abbas et al. [1] defined the concept of w-compatible mappings and obtained a common coupled fixed point for mappings satisfying a contractive condition in cone metric spaces. Recently, Aydi et al. [3] introduced the concepts of w-compatible ˜ mappings and generalized the results in [1]. Very recently, Aydi et al. [2] introduced the notion of common tripled fixed and tripled coincidence points to prove some common tripled fixed and tripled coincidence point results in cone metric spaces. On the other hand, other authors published several new generalizations of fixed point theorems and n-tuple fixed point results in metric type spaces and cone metric type spaces (see the following papers: Hussain et al. [16], Huang and Xu [13], Rahimi and Soleimani Rad [32], Parvaneh et al. [25], Shi and Xu [39], Fadail and Ahmad [12], Rezaei Roshan et al. [34], Yingtaweesittikul [41] and Rahimi et al. [28]). We recall the following definitions and concepts. Definition 1.1 [11,14] Let E be a real Banach space and P a subset of E. Then P is called a cone if and only if (a) P is closed, non-empty and P = {θ }; (b) a, b ∈ R, a, b ≥ 0, x, y ∈ P implies ax + by ∈ P; (c) if x ∈ P and −x ∈ P, then x = θ . Given a cone P ⊂ E, a partial ordering with respect to P is defined by x y ⇐⇒ y − x ∈ P. We write x ≺ y to mean x y and x = y. Also, we write x y if and only if y − x ∈ int P (where int P is the interior of P). If int P = ∅, the cone P is called solid. A cone P is named normal if there exists a number K > 0 such that, for all x, y ∈ E, θ x y ⇒ x ≤ K y . The least positive number satisfying the above inequality is called the normal constant of P. Definition 1.2 [14] Let X be a nonempty set and the mapping d : X × X → E satisfies (d1) θ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;

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(d2) d(x, y) = d(y, x) for all x, y ∈ X ; (d3) d(x, z) d(x, y) + d(y, z) for all x, y, z ∈ X . Then, d is called a cone metric on X and (X, d) is called a cone metric space. Definition 1.3 [17,23] Let X be a nonempty set, and K ≥ 1 be a real number. Suppose the mapping Dm : X × X → [0, ∞) satisfies (D1) Dm (x, y) = 0 if and only if x = y; (D2) Dm (x, y) = Dm (y, x) for all x, y ∈ X ; (D3) Dm (x, z) ≤ K (Dm (x, y) + Dm (y, z)) for all x, y, z ∈ X . (X, Dm , K ) is called a metric type space. Obviously, for K = 1, metric type space is a metric space. Example 1.4 Let X = C([0, T ], R) be the set of real continuous functions defined on [0, T ]. Also, let d : X × X → [0, ∞) be defined by d(x, y) = supt∈[0,T ] |x(t) − y(t)|2 . Then (X, d) is a metric type space with the constant K = 2. Definition 1.5 [10,23] Let X be a nonempty set, K ≥ 1 be a real number and E a real Banach space with cone P. Suppose that the mapping d : X × X → E satisfies (cd1) θ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y; (cd2) d(x, y) = d(y, x) for all x, y ∈ X ; (cd3) d(x, z) K (d(x, y) + d(y, z)) for all x, y, z ∈ X . (X, d, K ) is called a cone metric type space. Obviously, for K = 1, cone metric type space is a cone metric space. Example 1.6 [10] Consider the space L p of all real function f (t) such that |x(t)| p dt < ∞. Let X = L p , E = R, P = { f ∈ E| f ≥ 0} ⊂ R and d : X × X → E such that d( f, g) =

⎧ 1 ⎨ ⎩

| f (t) − g(t)| p dt

⎫ 1p ⎬ ⎭

,

t ∈ [0, 1].

0

Then (X, d) is a cone metric type space with K = 2 p−1 . Lemma 1.7 [10] Let (X, d, K ) be a cone metric type space over ordered real Banach space E. Then the following properties are often used, particularly when dealing with cone metric type spaces in which the cone need not be normal. (P1 ) (P2 ) (P3 ) (P4 )

If u v and v w, then u w. If θ u c for each c ∈ int P, then u = θ . If u λu where u ∈ P and 0 ≤ λ < 1, then u = θ . Let xn → θ in E and θ c. Then, there exists a positive integer n 0 such that xn c for each n > n 0 .

There exists some difference between cone metric type spaces with cone metric spaces. For more details about a cone metric type space and its properties, see [10,15,27]. In this paper, we prove some tripled fixed point theorems on cone metric type spaces. It is worth mentioning that our results do not rely on normality condition on cones involved therein. Recently, Karapinar and Du [21] investigated the answer to the question whether the given results generalize the existing ones or are equivalent to them. But we use the new definition from Samet and Vetro’s

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work [38]. Very recently, many papers are devoted to present different results to ensure the existence and uniqueness of coupled, tripled, quadrupled and multidimensional fixed points. Thus, some other authors are proving that these results can be reduced to their corresponding unidimensional versions. But, in next section, the proof of the Corollary 2.9 proved the inversion of previous papers are true. Therefore, our results extend, unify and generalize well-known results in the literature. An illustrative example is also given to validate our results.

2 Main results For simplicity, denote X × X × X by X 3 , where X is a non-empty set. At the first, we consider the following concept of Samet and Vetro [38]. Definition 2.1 [38] An element (x, y, z) ∈ X 3 is called a tripled fixed point of the mapping F : X 3 → X if x = F(x, y, z),

y = F(y, z, x), z = F(z, x, y).

Aydi et al. [2] introduced also the following. Definition 2.2 [2] An element (x, y, z) ∈ X 3 is called (i) a tripled coincidence point of mappings F : X 3 → X and g : X → X if gx = F(x, y, z), gy = F(y, z, x), gz = F(z, x, y). In this case (gx, gy, gz) is called a tripled point of coincidence; (ii) a common tripled fixed point of mappings F : X 3 → X and g : X → X if x = gx = F(x, y, z),

y = gy = F(y, z, x), z = gz = F(z, x, y).

Definition 2.3 [2] Mappings F : X 3 → X and g : X → X are called W -compatible if F(gx, gy, gz) = g(F(x, y, z)) whenever gx = F(x, y, z), gy = F(y, z, x), gz = F(z, x, y). Our first main result on cone metric type spaces is as follows: Theorem 2.4 Let (X, d, K ) be a cone metric type space with K ≥ 1 and P a solid cone. Suppose that the mappings F : X 3 → X and g : X → X satisfy the following contractive condition for all x, y, z, u, v, w ∈ X : d(F(x, y, z), F(u, v, w)) α1 d(F(x, y, z), gx) + α2 d(F(y, z, x), gy) + α3 d(F(z, x, y), gz) + α4 d(F(u, v, w), gu) + α5 d(F(v, w, u), gv) + α6 d(F(w, u, v), gw) + α7 d(F(u, v, w), gx) + α8 d(F(v, w, u), gy) + α9 d(F(w, u, v), gz) + α10 d(F(x, y, z), gu) + α11 d(F(y, z, x), gv) + α12 d(F(z, x, y), gw) + α13 d(gx, gu) + α14 d(gy, gv) + α15 d(gz, gw),

(1)

where αi for i = 1, 2, . . . , 15 are nonnegative constants with (K + 1)

6

i=1

123

αi + (K 2 + K )

12

i=7

αi + 2K

15

i=13

αi < 2.

(2)


If F(X 3 ) ⊆ g(X ) and g(X ) is a complete subset of X , then F and g have a tripled coincidence point in X . Moreover, if F and g are W -compatible, then F and g have a unique common tripled fixed point. Also, such common tripled fixed point of F and g is of the form (v, v, v) for some v ∈ X . Proof Let x0 , y0 , z 0 ∈ X and set ⎧ ⎨ gx1 = F(x0 , y0 , z 0 ) gy1 = F(y0 , z 0 , x0 ) ⎩ gz 1 = F(z 0 , x0 , y0 )

···

⎧ ⎨ gxn+1 = F(xn , yn , z n ) gyn+1 = F(yn , z n , xn ) ⎩ gz n+1 = F(z n , xn , yn ).

This can be done because of F(X 3 ) ⊆ g(X ). Now, according to (1), we have d(gxn , gxn+1 ) = d(F(xn−1 , yn−1 , z n−1 ), F(xn , yn , z n )) α1 d(F(xn−1 , yn−1 , z n−1 ), gxn−1 ) + α2 d(F(yn−1 , z n−1 , xn−1 ), gyn−1 ) + α3 d(F(z n−1 , xn−1 , yn−1 ), gz n−1 ) + α4 d(F(xn , yn , z n ), gxn ) + α5 d(F(yn , z n , xn ), gyn ) + α6 d(F(z n , xn , yn ), gz n ) + α7 d(F(xn , yn , z n ), gxn−1 ) + α8 d(F(yn , z n , xn ), gyn−1 ) + α9 d(F(z n , xn , yn ), gz n−1 ) + α10 d(F(xn−1 , yn−1 , z n−1 ), gxn ) + α11 d(F(yn−1 , z n−1 , xn−1 ), gyn ) + α12 d(F(z n−1 , xn−1 , yn−1 ), gz n ) + α13 d(gxn−1 , gxn ) + α14 d(gyn−1 , gyn ) + α15 d(gz n−1 , gz n ) = α1 d(gxn , gxn−1 ) + α2 d(gyn , gyn−1 ) + α3 d(gz n , gz n−1 ) + α4 d(gxn+1 , gxn ) + α5 d(gyn+1 , gyn ) + α6 d(gz n+1 , gz n ) + α7 d(gxn+1 , gxn−1 ) + α8 d(gyn+1 , gyn−1 ) + α9 d(gz n+1 , gz n−1 ) + α13 d(gxn−1 , gxn ) + α14 d(gyn−1 , gyn ) + α15 d(gz n−1 , gz n ). It follows (1 − α4 − K α7 )d(gxn , gxn+1 ) (α1 + K α7 + α13 )d(gxn−1 , gxn ) + (α2 + K α8 + α14 )d(gyn−1 , gyn ) + (α3 + K α9 + α15 )d(gz n−1 , gz n ) + (α5 + K α8 )d(gyn+1 , gyn ) + (α6 + K α9 )d(gz n+1 , gz n ).

(3)

Similarly, we obtain (1 − α4 − K α7 )d(gyn , gyn+1 ) (α1 + K α7 + α13 )d(gyn−1 , gyn ) + (α2 + K α8 + α14 )d(gz n−1 , gz n ) + (α3 + K α9 + α15 )d(gxn−1 , gxn ) + (α5 + K α8 )d(gz n+1 , gz n ) + (α6 + K α9 )d(gxn+1 , gxn ),

(4)

and (1 − α4 − K α7 )d(gz n , gz n+1 ) (α1 + K α7 + α13 )d(gz n−1 , gz n ) + (α2 + K α8 + α14 )d(gxn−1 , gxn ) + (α3 + K α9 + α15 )d(gyn−1 , gyn )

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+ (α5 + K α8 )d(gxn+1 , gxn ) + (α6 + K α9 )d(gyn+1 , gyn ).

(5)

Because of the symmetry in (1), we have (1 − α1 − K α10 )d(gxn+1 , gxn ) (α4 + K α10 + α13 )d(gxn , gxn−1 ) + (α5 + K α11 + α14 )d(gyn−1 , gyn ) + (α6 + K α12 + α15 )d(gz n−1 , gz n ) + (α2 + K α11 )d(gyn+1 , gyn ) + (α3 + K α12 )d(gz n+1 , gz n ),

(6)

and (1 − α1 − K α10 )d(gyn+1 , gyn ) (α4 + K α10 + α13 )d(gyn , gyn−1 ) + (α5 + K α11 + α14 )d(gz n−1 , gz n ) + (α6 + K α12 + α15 )d(gxn−1 , gxn ) + (α2 + K α11 )d(gz n+1 , gz n ) + (α3 + K α12 )d(gxn+1 , gxn ),

(7)

and (1 − α1 − K α10 )d(gz n+1 , gz n ) (α4 + K α10 + α13 )d(gz n , gz n−1 ) + (α5 + K α11 + α14 )d(gxn−1 , gxn ) + (α6 + K α12 + α15 )d(gyn−1 , gyn ) + (α2 + K α11 )d(gxn+1 , gxn ) + (α3 + K α12 )d(gyn+1 , gyn ).

(8)

Let Dn = d(gxn , gxn+1 ) + d(gyn , gyn+1 ) + d(gz n , gz n+1 ). Now, adding (3)–(5) and (6)–(8), we have (1 − α4 − α5 − α6 − K (α7 + α8 + α9 ))Dn (α1 + α2 + α3 + K (α7 + α8 + α9 ) + α13 + α14 + α15 )Dn−1 ,

(9)

and (1 − α1 − α2 − α3 − K (α10 + α11 + α12 ))Dn (α4 + α5 + α6 + K (α10 + α11 + α12 ) + α13 + α14 + α15 )Dn−1 .

(10)

Ultimately, adding (9) and (10), we have 6 6 12 12 15

αi − K αi Dn αi + K αi + 2 αi Dn−1 . 2− i=1

i=7

i=1

i=7

i=13

Thus, for all n, θ Dn λDn−1 λ2 Dn−2 · · · λn D0 , where

6 λ=

123

i=1 αi

2−

15 + 2 i=13 αi 1 < .

12 K i=1 αi − K i=7 αi

+K

6

12

i=7 αi

(11)


If D0 = θ then (x0 , y0 , z 0 ) is a tripled coincidence point of F and g. Now, let θ ≺ D0 . If m > n, we have d(gxn , gxm ) K [d(gxn , gxn+1 ) + d(gxn+1 , gxm )] K d(gxn , gxn+1 ) + K 2 [d(gxn+1 , gxn+2 ) + d(gxn+2 , gxm )] · · · K d(gxn , gxn+1 ) + K 2 d(gxn+1 , gxn+2 ) + · · · + K m−n−1 d(gxm−2 , gxm−1 ) + K m−n d(gxm−1 , gxm ),

(12)

and d(gyn , gym ) K d(gyn , gyn+1 ) + K 2 d(gyn+1 , gyn+2 ) + · · · + K m−n−1 d(gym−2 , gym−1 ) + K m−n d(gym−1 , gym ),

(13)

and d(gz n , gz m ) K d(gz n , gz n+1 ) + K 2 d(gz n+1 , gz n+2 ) + · · · + K m−n−1 d(gz m−2 , gz m−1 ) + K m−n d(gz m−1 , gz m ).

(14)

Adding (12)–(14) and using (11) and the fact that λ < 1/K , we have d(gxn , gxm ) + d(gyn , gym ) + d(gz n , gz m ) K Dn + K 2 Dn+1 + · · · + K m−n Dm−1 (K λn + K 2 λn+1 + · · · + K m−n λm−1 )D0 K λn D0 → θ as n → ∞. 1 − Kλ Now, by (P1 ) and (P4 ), it follows that for every c ∈ int P there exists a positive integer N such that d(gxn , gxm ) + d(gyn , gym ) + d(gz n , gz m ) c. for every m > n > N , so {gxn }, {gyn } and {gz n } are Cauchy sequences in X . Since g(X ) is a complete subset of the cone metric type space X , there exist x, y, z ∈ X such that gx n → gx, gyn → gy and gz n → gz as n → ∞. Now, we prove that F(x, y, z) = gx, F(y, z, x) = gy and F(z, x, y) = gz. Note that d(F(x, y, z), gx) K [d(F(x, y, z), F(xn , yn , z n )) + d(F(xn , yn , z n ), gx)] = K [d(F(x, y, z), F(xn , yn , z n )) + d(gxn+1 , gx)].

(15)

Using (1), we have d(F(x, y, z), F(xn , yn , z n )) α1 d(F(x, y, z), gx) + α2 d(F(y, z, x), gy) + α3 d(F(z, x, y), gz) + α4 d(gxn+1 , gxn ) + α5 d(gyn+1 , gyn ) + α6 d(gz n+1 , gz n ) + α7 d(gxn+1 , gx) + α8 d(gyn+1 , gy) + α9 d(gz n+1 , gz) + α10 d(F(x, y, z), gxn ) + α11 d(F(y, z, x), gyn ) + α12 d(F(z, x, y), gz n ) + α13 d(gx, gxn ) + α14 d(gy, gyn ) + α15 d(gz, gz n ). (16) Now, consider (15) and (16), we obtain (1 − K α1 − K 2 α10 )d(F(x, y, z), gx) − (K α2 + K 2 α11 )d(F(y, z, x), gy)

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− (K α3 + K 2 α12 )d(F(z, x, y), gz) K α4 d(gxn+1 , gxn ) + α5 d(gyn+1 , gyn ) + α6 d(gz n+1 , gz n ) + (1 + α7 )d(gxn+1 , gx) + α8 d(gyn+1 , gy) + α9 d(gz n+1 , gz) + (K α10 + α13 )d(gx, gxn ) + (K α11 + α14 )d(gy, gyn ) (17) + (K α12 + α15 )d(gz, gz n ) . Similarly, we have (1 − K α1 − K 2 α10 )d(F(y, z, x), gy) − (K α2 + K 2 α11 )d(F(z, x, y), gz) − (K α3 + K 2 α12 )d(F(x, y, z), gx) K α4 d(gyn+1 , gyn ) + α5 d(gz n+1 , gz n ) + α6 d(gxn+1 , gxn ) + (1 + α7 )d(gyn+1 , gy) + α8 d(gz n+1 , gz) + α9 d(gxn+1 , x z) + (K α10 + α13 )d(gy, gyn )

+ (K α11 + α14 )d(gz, gz n ) + (K α12 + α15 )d(gx, gxn ) ,

(18)

and (1 − K α1 − K 2 α10 )d(F(z, x, y), gz) − (K α2 + K 2 α11 )d(F(x, y, z), gx) − (K α3 + K 2 α12 )d(F(y, z, x), gy) K α4 d(gz n+1 , gz n ) + α5 d(gxn+1 , gxn ) + α6 d(gyn+1 , gyn ) + (1 + α7 )d(gz n+1 , gz) + α8 d(gxn+1 , gx) + α9 d(gyn+1 , gy) + (K α10 + α13 )d(gz, gz n ) + (K α11 + α14 )d(gx, gxn ) (19) + (K α12 + α15 )d(gy, gyn ) . Now, set φ = d(F(z, x, y), gz) + d(F(x, y, z), gx) + d(F(y, z, x), gy), ψn = d(gxn , gx) + d(gyn , gy) + d(gz n , gz). Adding (17)–(19), we obtain 1 − K (α1 + α2 + α3 ) − K 2 (α10 + α11 + α12 ) φ K (α4 + α5 + α6 )Dn + K (1 + α7 + α8 + α9 )ψn+1 + (K (α13 + α14 + α15 ) + K 2 (α10 + α11 + α12 ))ψn . Therefore, φ

B C D Dn + ψn+1 + ψn , A A A

where A = 1 − K (α1 + α2 + α3 ) − K 2 (α10 + α11 + α12 ),

B = K (α4 + α5 + α6 ),

D = K (α13 + α14 + α15 ) + K 2 (α10 + α11 + α12 ),

C = K (1 + α7 + α8 + α9 ).

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Since gxn → gx, gyn → gy and gz n → gz, by using Lemma 1.7 we have d(F(x, y, z), gx) = θ, d(F(y, z, x), gy) = θ, d(F(z, x, y), gz) = θ, which implies that F(x, y, z) = gx,

F(y, z, x) = gy,

F(z, x, y) = gz.

Thus, (x, y, z) is a tripled coincidence point of the mappings F and g. Now, we prove that the tripled point of coincidence is unique. Suppose that (x, y, z), (x , y , z ) ∈ X 3 such that ⎧ ⎧ ⎨ gx = F(x , y , z ) ⎨ gx = F(x, y, z) gy = F(y, z, x) · · · gy = F(y , z , x ) ⎩ ⎩ gz = F(z, x, y) gz = F(z , x , y ). From (1), we obtain d(gx, gx ) = d(F(x, y, z), F(x , y , z )) (α7 + α10 + α13 )d(gx , gx) + (α8 + α11 + α14 )d(gy , gy) + (α9 + α12 + α15 )d(gz , gz),

(20)

d(gy, gy ) = d(F(y, z, x), F(y , z , x )) (α7 + α10 + α13 )d(gy , gy) + (α8 + α11 + α14 )d(gz , gz) + (α9 + α12 + α15 )d(gx , gx),

(21)

and d(gz, gz ) = d(F(z, x, y), F(z , x , y )) (α7 + α10 + α13 )d(gz , gz) + (α8 + α11 + α14 )d(gx , gx) + (α9 + α12 + α15 )d(gy , gy).

(22)

Adding (20)–(22), we have d(gx, gx ) + d(gy, gy ) + d(gz, gz )

15

[d(gx, gx ) + d(gy, gy ) + d(gz, gz )]. i=7

Thus, d(gx, gx ) + d(gy, gy ) + d(gz, gz ) = θ, which implies that gx = gx ,

gy = gy ,

gz = gz .

gx = gy ,

gy = gz ,

gz = gx .

Similarly, we can obtain

From two above equalities, it follows that gx = gy = gz, that is, the tripled point of coincidence of F and g is unique. Now, let v = gx. Since mappings F and g are W compatible, we have gv = g(gx) = g(F(x, y, z)) = F(gx, gy, gz) = F(v, v, v). Therefore, (gv, gv, gv) is a tripled point of coincidence of mappings F and g, and also we have (v, v, v) is a tripled coincidence point of mappings F and g. The uniqueness of the

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tripled point of coincidence implies that gv = v. Therefore v = gv = F(v, v, v). Hence, (v, v, v) is the unique common tripled fixed point of F and g. This completes the proof. Example 2.5 Let X = [0, 1]. Take E = CR1 [0, 1] endowed with order induced by P = {φ ∈ E : φ(t) ≥ 0 for t ∈ [0, 1]}. The mapping d : X × X → E is defined by d(x, y)(t) = |x − y|2 et . In this case, (X, d) is a complete cone metric type space but it is not a cone metric space since the triangle inequality is not satisfied. Starting with Minkowski inequality, we get |x − z|2 2(|x − y|2 + |y − z|2 ). Here K = 2. Define the mappings F : X 3 → X and g : X → X by gx =

x , 2

F(x, y, z) =

1 1 1 x + y + z. 7 5 9

Then F and g satisfy the contractive condition (2) with αi = 0 for i = 1, 2, . . . , 12, α13 = 8/49, α14 = 8/25 and α15 = 8/81, that is, d(F(x, y, z), F(x , y , z ))(t) = |F(x, y, z) − F(x , y , z )|2 et 1 1 1 1 2 t 1 1 = x + y + z − x + y + z e 7 5 9 7 5 9 2 2 2 x 2 y x y − − 2 + 7 2 2 5 2 2 2 z z 2 t + e − 9 2 2 =

8 8 8 d(gx, gx ) + d(gy, gy ) + d(gz, gz ) 49 25 81

for all x, y, z, x , y , z . Moreover, F and g are W -compatible. All conditions of Theorem 2.4 are satisfied. According to Theorem 2.4, F and g have a unique common tripled fixed point. In this example, (0, 0, 0) is the unique common tripled fixed point of F and g. As consequences of Theorem 2.4, we have the following corollaries. Corollary 2.6 Let (X, d, K ) be a cone metric type space with K ≥ 1 and P a solid cone. Suppose that the mappings F : X 3 → X and g : X → X satisfy the following contractive condition for all x, y, z, u, v, w ∈ X : d(F(x, y, z), F(u, v, w)) μd(gx, gu) + νd(gy, gv) + τ d(gz, gw), where μ, ν, τ are nonnegative constants with μ+ν+τ