Quadrupled fixed point results in abstract metric spaces

Comp. Appl. Math.

DOI 10.1007/s40314-013-0088-5

Quadrupled fixed point results in abstract metric spaces

Hamidreza Rahimi · Stojan Radenovic ·

Ghasem Soleimani Rad · Poom Kumam

Received: 25 May 2013 / Revised: 18 August 2013 / Accepted: 13 October 2013

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013

Abstract In this paper we consider some new definitions about quadrupled fixed point in

abstract metric spaces and obtain some new fixed point results in this field. These results

unify, extend and generalize well-known comparable results in the existing literature. We

also provide some examples and applications to support our results.

Keywords Abstract metric space · Quadrupled fixed point · T-contraction ·

Quadrupled coincidence point · Common quadrupled fixed point · W-compatible mapping

Mathematics Subject Classification (2000) 47H10 · 54H25 · 46J10 · 34A34

Communicated by Marko Rojas-Medar.

H. Rahimi · G. Soleimani Rad

Department of Mathematics, Faculty of Science, Central Tehran Branch,

Islamic Azad University, P. O. Box 13185/768, Tehran, Iran

e-mail: [email protected]

S. Radenovic

Faculty of Mechanical Engineering, University of Belgrade,

Kraljice Marije 16, 11120 Beograd, Serbia


G. Soleimani Rad (B)

Young Researchers and Elite club, Central Tehran Branch,

Islamic Azad University, Tehran, Iran

e-mail: [email protected]; [email protected]

P. Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s

University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand


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H. Rahimi et al.

1 Introduction and preliminaries

The Banach contraction principle is the most celebrated fixed point theorem Banach (1922).

Afterward, other people considered various definitions of contractive mappings and proved

several fixed point theorems (see, for example, Hardy and Rogers 1973; Rhoades 1977).

Fixed point theory is an important and useful tool for different branches of mathematical

analysis. Also, it may be discussed an essential subject of nonlinear analysis.

Recently, non-convex analysis has found some applications in optimization theory, and so

there have been some investigations about non-convex analysis, especially ordered normed

spaces and topical functions. Fixed point theory in K-metric and K-normed spaces was

developed by Perov (1964), Mukhamadiev and Stetsenko (1969) (also, see Zabrejko 1997).

Huang and Zhang (2007) have introduced the concept of the cone metric space, replacing the

set of real numbers by an ordered Banach space and proved some fixed point results. Then,

several fixed point results on cone metric spaces were introduced in Abbas and Jungck (2008);

Abbas and Rhoades (2009); Radojevic et al. (2011); Rezapour and Hamlbarani (2008) and

references therein. On the other hand, the concept of T-contractions in a metric space was

studied in Moradi (2011); Morales and Rojas (2009). In the sequel, other authors Filipovic

et al. (2011); Morales and Rojas (2010); Rahimi et al. (2013); Rahimi and Soleimani Rad

(2012) obtained some fixed point results under T-contractions on cone metric spaces.

Bhaskar and Lakshmikantham (2006) defined the concept of coupled fixed point in par-

tially ordered metric spaces. Then, other authors obtained some coupled fixed point theorems

with application in nonlinear and differential equations (see Berinde 2011; Cho et al. 2012;

Lakshmikanthama and Ciric 2009; Nashine and Shatanawi 2011; Sabetghadam et al. 2009;

Samet and Vetro 2010). In 2010, Abbas et al. (2010) introduced the concept of w-compatible

mappings and obtained a coupled coincidence point and a coupled point of coincidence

for mappings satisfying a contractive condition in cone metric spaces. On the other hand,

Karapinar and Luong (2012) introduced the notion of quadrupled fixed point.

In this paper we define some new notions in quadrupled fixed point theory and prove

some fixed and common fixed point results on abstract metric spaces. It is worth mentioning

that our results do not rely on normality condition on cones involved therein. Our theorems

extend, unify and generalize well-known results in the literature. Some illustrative examples

and applications are given to validate our results. For this, we start with some important

definitions.

Definition 1.1 (Deimling 1985; Huang and Zhang 2007), 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 if and

only if y − x ∈ P. We write x ≺ y to mean x � y and x �= y and 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 called normal if there exists a number K > 0 such that, for all x, y ∈ E , θ � x � y

implies that ‖x‖ ≤ K‖y‖. The least positive number satisfying the above inequality is called

the normal constant of P .

Definition 1.2 Let P ⊂ E be a cone, � a partial ordering with respect to P and X a nonempty

set. Suppose that the mapping d : X × X → E satisfies

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Quadrupled fixed point results

(d1) θ � d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;

(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 Huang and Zhang (2007) or K-metric Zabrejko (1997) on X

and (X, d) is called a cone metric space Huang and Zhang (2007) or K-metric space Zabrejko

(1997).

The concept of a K-metric space is more general than that of a metric space, because each

metric space is a K-metric space where E = R and P = [0,∞).

Definition 1.3 (Filipovic et al. 2011), Let (X, d) be a K-metric space, {xn} a sequence in X

and x ∈ X . Then

(i) {xn} converges to x if, for every c ∈ E with θ ≪ c there exists an n0 ∈ N such that

d(xn, x) ≪ c for all n > n0. We denote this by limn→∞ d(xn, x) = θ ;

(ii) {xn} is called a Cauchy sequence if, for every c ∈ E with θ ≪ c there exists an n0 ∈ N

such that d(xn, xm) ≪ c for all m, n > n0;

(iii) a K-metric space X is said to be complete if every Cauchy sequence in X is convergent

in X .

Lemma 1.4 (Filipovic et al. 2011). Let (X, d) be a K-metric space over an ordered real

Banach space E. Then the following properties are often used.

(P1) If x � y and y ≪ z, then x ≪ z.

(P2) If θ � x ≪ c for each c ∈ int P, then x = θ .

(P3) If x � λx where x ∈ P and 0 ≤ λ < 1, then x = θ .

(P4) Let xn → θ in E and θ ≪ c. Then there exists a positive integer n0 such that xn ≪ c

for each n > n0.

Definition 1.5 (Filipovic et al. 2011), Let (X, d) be a K-metric space, P a solid cone and

S : X → X . Then

(i) S is said to be sequentially convergent if we have, for every sequence {xn}, if {Sxn} is

convergent, then {xn} also is convergent.

(ii) S is said to be subsequentially convergent if, for every sequence {xn} that {Sxn} is

convergent, {xn} has a convergent subsequence.

(iii) S is said to be continuous if limn→∞ xn = x implies that limn→∞ Sxn = Sx , for all

{xn} in X .

Theorem 1.6 (Rahimi et al. 2013; Rahimi and Soleimani Rad 2013). Let (X, d)be a complete

K-metric space, P a solid cone and T : X → X a continuous and one to one mapping.

Moreover, let the mapping f be a map of X satisfying

d(T f x, T f y) � αd(T x, T y) + β[d(T x, T f x) + d(T y, T f y)]

+γ [d(T x, T f y) + d(T y, T f x)],

for all x, y ∈ X, where α, β, γ ≥ 0 with α + 2β + 2γ < 1. Then

(1) For each x0 ∈ X, {T f n x0} is a Cauchy sequence.

(2) There exists a zx0 ∈ X such that limn→∞ T f n x0 = zx0 .

(3) If T is subsequentially convergent, then { f n x0} has a convergent subsequence.

(4) There exists a unique wx0 ∈ X such that f wx0 = wx0 .

(5) If T is sequentially convergent, then, for each x0 ∈ X, { f n x0} converges to wx0 .

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Example 1.7 (Morales and Rojas 2010; Rahimi et al. 2013). Let X = [0, 1], E = C1R[0, 1]

with the norm ‖ f ‖ = ‖ f ‖∞ + ‖ f ′‖∞, P = { f ∈ E | f ≥ 0} and d(x, y) = |x − y|2t where

2t ∈ P ⊂ E . Moreover, suppose that T x = x2 and f x = x/2, which map the set X into X .

(X, d) is a cone metric space with non-normal cone, T is a one to one, continuous mapping,

and f is not a Kannan contraction; indeed, for x = 0 and y = 1 there is not k from [0, 1/2)

such that Kannan contraction holds. All of the conditions of Theorem 1.6 are satisfied with

α = γ = 0, β = 13

. Thus, x = 0 is the unique fixed point of f .

2 Main results

For simplicity, denote X × X × X × X by X4, where X is a non-empty set.

Definition 2.1 An element (x, y, z, u) ∈ X4 is called a quadrupled fixed point of a given

mapping F : X4 → X if x = F(x, y, z, u), y = F(y, z, u, x), z = F(z, u, x, y) and

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

The main results of this work are divided into two separate parts. In the first part, we

define the notion of T-contraction in context quadrupled fixed point theory and prove some

related results on abstract metric spaces. In the second part, we introduce the concepts of w-

compatible mappings. Based on this notion, a quadrupled coincidence point and a common

quadrupled fixed point for mappings F : X4 → X and g : X → X are obtained, where

(X, d) is a abstract metric space. It is worth mentioning that all of our results do not depend

on normality condition on cones involved herein.

2.1 Quadrupled fixed point results for T-contractions

Definition 2.2 Let (X, d) be a K-metric space and T : X → X be a mapping. A mapping

F : X4 → X is said to be a T -contraction, if there exist α, β, γ, δ ≥ 0 with α+β+γ +δ < 1

such that for all x, y, z, u, x∗, y∗, z∗, u∗ ∈ X ,

d(TF(x, y, z, u), TF(x∗, y∗, z∗, u∗)) � αd(T x, T x∗) + βd(T y, T y∗)

+ γ d(T z, T z∗) + δd(T u, T u∗).

Theorem 2.3 Suppose that (X, d) is a complete K-metric space, P is a solid cone, and

T : X → X is a continuous and one to one mapping. Moreover, let F : X4 → X be a

mapping satisfying

d(TF(x, y, z, u), TF(x∗, y∗, z∗, u∗)) � αd(T x, T x∗) + βd(T y, T y∗)

+ γ d(T z, T z∗) + δd(T u, T u∗) (1)

for all x, y, z, u, x∗, y∗, z∗, u∗ ∈ X, where α, β, γ, δ ≥ 0 with

α + β + γ + δ < 1. (2)

Then

(t1) For each x0, y0, z0, u0 ∈ X,

{TFn(x0, y0, z0, u0)}, {TFn(y0, z0, u0, x0)},

{TFn(z0, u0, x0, y0)}, {TFn(u0, x0, y0, z0)}

are Cauchy sequences.

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(t2) There exist vx0 , vy0 , vz0 , vu0 ∈ X such that

limn→∞

TFn(x0, y0, z0, u0) = vx0 , limn→∞ TFn(y0, z0, u0, x0) = vy0 ,

limn→∞

TFn(z0, u0, x0, y0) = vz0 , limn→∞ TFn(u0, x0, y0, z0) = vu0 .

(t3) If T is subsequentially convergent, then

{TFn(x0, y0, z0, u0)}, {TFn(y0, z0, u0, x0)}

{TFn(z0, u0, x0, y0)}, {TFn(u0, x0, y0, z0)}

have a convergent subsequence.

(t4) There exist unique wx0 , wy0 , wz0 , wu0 ∈ X such that

F(wx0 , wy0 , wz0 , wu0) = wx0 , F(wy0 , wz0 , wu0 , wx0) = wy0 ,

F(wz0 , wu0 , wx0 , wy0) = wz0 , F(wu0 , wx0 , wy0 , wz0) = wu0 ;

that is, F has a unique quadruple fixed point.

(t5) If T is sequentially convergent, then, for each x0, y0, z0, u0 ∈ X, the sequence

{TFn(x0, y0, z0, u0)} converges to wx0 ∈ X, the sequence {TFn(y0, z0, u0, x0)} converges

to wy0 ∈ X, the sequence {TFn(z0, u0, x0, y0)} converges to wz0 ∈ X and the sequence

{TFn(u0, x0, y0, z0)} converges to wu0 ∈ X.

Proof Let x0, y0, z0, u0 ∈ X and set

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

x1 = F(x0, y0, z0, u0)

y1 = F(y0, z0, u0, x0)

z1 = F(z0, u0, x0, y0)

u1 = F(u0, x0, y0, z0)

· · ·

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

xn+1 = F(xn, yn, zn, un) = Fn+1(x0, y0, z0, u0)

yn+1 = F(yn, zn, un, xn) = Fn+1(y0, z0, u0, x0)

zn+1 = F(zn, un, xn, yn) = Fn+1(z0, u0, x0, y0)

un+1 = F(un, xn, yn, zn) = Fn+1(u0, x0, y0, z0),

for n = 0, 1, . . .. Now, according to (1), we have

d(T xn, T xn+1) = d(TF(xn−1, yn−1, zn−1, un−1), TF(xn, yn, zn, un))

� αd(T xn−1, T xn) + βd(T yn−1, T yn)

+ γ d(T zn−1, T zn) + δd(T un−1, T un). (3)

Similarly, we obtain

d(T yn, T yn+1) � αd(T yn−1, T yn) + βd(T zn−1, T zn)

+ γ d(T un−1, T un) + δd(T xn−1, T xn) (4)

and

d(T zn, T zn+1) � αd(T zn−1, T zn) + βd(T un−1, T un)

+ γ d(T xn−1, T xn) + δd(T yn−1, T yn) (5)

and

d(T un, T un+1) � αd(T un−1, T un) + βd(T xn−1, T xn)

+ γ d(T yn−1, T yn) + δd(T zn−1, T zn). (6)

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Let dn = d(T xn, T xn+1)+ d(T yn, T yn+1)+ d(T zn, T zn+1)+ d(T un, T un+1). Adding up

(3), (4), (5) and (6), we get

dn � λ(

d(T xn−1, T xn) + d(T yn−1, T yn) + d(T zn−1, zn) + d(T un−1, T un))

= λdn−1,

where λ = α + β + γ + δ < 1 by (2). Thus, for all n,

θ � dn � λdn−1 � λ2dn−2 � · · · � λnd0. (7)

If d0 = θ then (x0, y0, z0, u0) is a quadruple fixed point of F . Now, let d0 ≻ θ . If m > n,

we have

d(T xn, T xm) � d(T xn, T xn+1) + d(T xn+1, T xn+2) + · · · + d(T xm−1, T xm), (8)

d(T yn, T ym) � d(T yn, T yn+1) + d(T yn+1, T yn+2) + · · · + d(T ym−1, T ym), (9)

d(T zn, T zm) � d(T zn, T zn+1) + d(T zn+1, T zn+2) + · · · + d(T zm−1, T zm), (10)

d(T un, T um) � d(T un, T un+1) + d(T un+1, T un+2) + · · · + d(T um−1, T um). (11)

Adding up (8), (9), (10) and (11) and using (7). Since λ < 1, we have

dm,n � dn + dn+1 + · · · + dm−1

� (λn + λn+1 + · · · + λm−1)d0

�λn

1 − λd0 → θ as n → ∞,

where

dm,n = d(T xn, T xm) + d(T yn, T ym) + d(T zn, T zm) + d(T un, T um).

Now, by (P1) and (P4), it follows that for every c ∈ int P there exist positive integer N such

that dm,n ≪ c for every m > n > N . Thus {T xn}, {T yn}, {T zn} and {T un} are Cauchy

sequences in X . Since X is a complete K-metric space, there exist vx0 , vy0 , vz0 , vu0 ∈ X

such that

limn→∞

T Fn(x0, y0, z0, u0) = vx0 , limn→∞ T Fn(y0, z0, u0, x0) = vy0 ,

limn→∞

T Fn(z0, u0, x0, y0) = vz0 , limn→∞ T Fn(u0, x0, y0, z0) = vu0 . (12)

Now, if T is subsequentially convergent, then Fn(x0, y0, z0, u0), Fn(y0, z0, u0, x0),

Fn(z0, u0, x0, y0) and Fn(u0, x0, y0, z0) have convergent subsequences. Thus, there exist

wx0 , wy0 , wz0 , wu0 ∈ X and the sequences {xni}, {yni

}, {zni} and {uni

} such that

limi→∞

Fni (x0, y0, z0, u0) = wx0 , limi→∞ Fni (y0, z0, u0, x0) = wy0 ,

limi→∞

Fni (z0, u0, x0, y0) = wz0 , limi→∞ Fni (u0, x0, y0, z0) = wu0 .

Because of the continuity of T , we have

limi→∞

T Fni (x0, y0, z0, u0) = T wx0 , limi→∞ T Fni (y0, z0, u0, x0) = T wy0 ,

limi→∞

T Fni (z0, u0, x0, y0) = T wz0 , limi→∞ T Fni (u0, x0, y0, z0) = T wu0 . (13)

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Now, by (12) and (13), we conclude that

T wx0 = vx0 , T wy0 = vy0 , T wz0 = vz0 , T wu0 = vu0 .

On the other hand, from (d3) and (1), we have

d(TF(wx0 , wy0 , wz0 , wu0), T wx0) � d(TF(wx0 , wy0 , wz0 , wu0), TF(xni, yni

, zni, uni

))

+ d(T xni +1, T wx0)

� αd(T wx0 , T xni) + βd(T wy0 , T yni

)

+ γ d(T wz0 , T zni) + δd(T wu0 , T uni

)

+ d(T xni +1, T wx0).

By applying Lemma 1.4, we can obtain d(TF(wx0 , wy0 , wz0 , wu0), T wx0) = θ . Hence

TF(wx0 , wy0 , wz0 , wu0) = T wx0 . Since T is one to one, F(wx0 , wy0 , wz0 , wu0) = wx0 .

Analogously, we have F(wy0 , wz0 , wu0 , wx0) = wy0 , F(wz0 , wu0 , wx0 , wy0) = wz0 and

F(wu0 , wx0 , wy0 , wz0) = wu0 . Therefore, (wx0 , wy0 , wz0 , wu0) is a quadrupled fixed point

of F . Now, if (w∗x0

, w∗y0

, w∗z0

, w∗u0

) is another quadrupled fixed point of F , then

d(T wx0 , T w∗x0

) = d(TF(wx0 , wy0 , wz0 , wu0), TF(w∗x0

, w∗y0

, w∗z0

, w∗u0

))

� αd(T wx0 , T w∗x0

) + βd(T wy0 , T w∗y0

)

+ γ d(T wz0 , T w∗z0

) + δd(T wu0 , T w∗u0

). (14)

Similarly, we obtain

d(T wy0 , T w∗y0

) = d(TF(wy0 , wz0 , wu0 , wx0), TF(w∗y0

, w∗z0

, w∗u0

, w∗x0

))

� αd(T wy0 , w∗y0

) + βd(T wz0 , T w∗z0

)

+ γ d(T wu0 , T w∗u0

) + δd(T wx0 , T w∗x0

) (15)

and

d(T wz0 , T w∗z0

) = d(TF(wz0 , wu0 , wx0 , wy0), TF(w∗z0

, w∗u0

, w∗x0

, w∗y0

))

� αd(T wz0 , T w∗z0

) + βd(T wu0 , T w∗u0

)

+ γ d(T wx0 , T w∗x0

) + δd(T wy0 , T w∗y0

) (16)

and

d(T wu0 , T w∗u0

) = d(TF(wu0 , wx0 , wy0 , wz0), TF(w∗u0

, w∗x0

, w∗y0

, w∗z0

))

� αd(T wu0 , T w∗u0

) + βd(T wx0 , T w∗x0

)

+ γ d(T wy0 , T w∗y0

) + δd(T wz0 , T w∗z0

). (17)

Adding up (14), (15), (16) and (17), we obtain

dw,w∗ � λdw,w∗ , (18)

where

dw,w∗ = d(T wx0 , T w∗x0

) + d(T wy0 , T w∗y0

) + d(T wz0 , T w∗z0

) + d(T wu0 , T w∗u0

).

Since λ = α + β + γ + δ < 1, (18) follows that dw,w∗ = θ . Hence,

d(T wx0 , T w∗x0

) = d(T wy0 , T w∗y0

) = d(T wz0 , T w∗z0

) = d(T wu0 , T w∗u0

) = θ.

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Thus,

T wx0 = T w∗x0

, T wy0 = T w∗y0

, T wz0 = T w∗z0

, T wu0 = T w∗u0

.

Since T is one to one, we have (wx0 , wy0 , wz0 , wu0) = (w∗x0

, w∗y0

, w∗z0

, w∗u0

). Ultimately,

if T is sequentially convergent, then we can replace n by ni . Thus, we have

limn→∞

T Fn(x0, y0, z0, u0) = wx0 , limn→∞ T Fn(y0, z0, u0, x0) = wy0

limn→∞

T Fn(z0, u0, x0, y0) = wz0 , limn→∞ T Fn(u0, x0, y0, z0) = wu0 .

This completes the proof of Theorem 2.3. ⊓⊔

Corollary 2.4 Suppose that (X, d) is a complete K-metric space, P is a solid cone, and

T : X → X is a continuous and one to one mapping. Moreover, let F : X4 → X be a

mapping satisfying

d(T F(x, y, z, u), T F(x∗, y∗, z∗, u∗)) �k

4

[

d(T x, T x∗) + d(T y, T y∗)

+ d(T z, T z∗) + d(T u, T u∗)]

for all x, y, z, u, x∗, y∗, z∗, u∗ ∈ X, where k ∈ [0, 1). Then, the results of Theorem 2.3 hold.

Proof In (1), set α = β = γ = δ = k/4. ⊓⊔

Corollary 2.5 Suppose that (X, d) is a complete K-metric space and P is a solid cone.

Moreover, let F : X4 → X be a mapping satisfying

d(F(x, y, z, u), F(x∗, y∗, z∗, u∗)) � αd(x, x∗) + βd(y, y∗) + γ d(z, z∗) + δd(u, u∗)

for all x, y, z, u, x∗, y∗, z∗, u∗ ∈ X, where α, β, γ, δ ≥ 0 with

α + β + γ + δ < 1.

Then, F has a unique quadruple fixed point.

Proof In Theorem 2.3, take T = Ix , where Ix is an identity map. ⊓⊔

Corollary 2.6 Suppose that (X, d) is a complete K-metric space and P is a solid cone.

Moreover, let F : X4 → X be a mapping satisfying

d(

F(x, y, z, u), F(x∗, y∗, z∗, u∗))

�k

4

[

d(x, x∗) + d(y, y∗) + d(z, z∗) + d(u, u∗)]

(19)

for all x, y, z, u, x∗, y∗, z∗, u∗ ∈ X, where k ∈ [0, 1). Then, F has a unique quadruple fixed

point.

Proof In Corollary 2.4, take T = Ix , where Ix is an identity map. ⊓⊔

2.2 Common quadrupled fixed point results

We start by recalling the following definitions:

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Definition 2.7 (Bhaskar and Lakshmikantham 2006; Lakshmikanthama and Ciric 2009) An

element (x, y) ∈ X × X is said to be

(i) a coupled fixed point of the mapping F : X × X → X if F(x, y) = x and F(y, x) = y;

(ii) a coupled coincidence point of the mappings F : X × X → X and g : X → X if

F(x, y) = g(x) and F(y, x) = g(y), and (gx, gy) is called coupled point of coinci-

dence;

(iii) a coupled common fixed point of the mappings F : X × X → X and g : X → X if

F(x, y) = g(x) = x and F(y, x) = g(y) = y.

Definition 2.8 (Abbas et al. 2010) The mappings F : X ×X → X and g : X → X are called

w-compatible if g(F(x, y)) = F(gx, gy) whenever g(x) = F(x, y) and g(y) = F(y, x).

Now, we consider the following definitions.

Definition 2.9 An element (x, y, z, u) ∈ X4 is called

(i) a quadrupled coincidence point of mappings F : X4 → X and g : X → X if

gx = F(x, y, z, u), gy = F(y, z, u, x)

gz = F(z, u, x, y), gu = F(u, x, y, z).

In this case (gx, gy, gz, gu) is called a quadrupled point of coincidence;

(ii) a common quadrupled fixed point of mappings F : X4 → X and g : X → X if

x = gx = F(x, y, z, u), y = gy = F(y, z, u, x)

z = gz = F(z, u, x, y), u = gu = F(u, x, y, z).

Definition 2.10 Mappings F : X4 → X and g : X → X are called w-compatible if

F(gx, gy, gz, gu) = g(F(x, y, z, u))

whenever

gx = F(x, y, z, u), gy = F(y, z, u, x)

gz = F(z, u, x, y), gu = F(u, x, y, z).

Example 2.11 Let X = R. Define F : X4 → X and g : X → X as follows:

F(x, y, z, u) =x + y + z + u

8and gx =

x

4

for all x, y, z, u ∈ X . It is easy to show that (x, y, z, u) is a quadrupled coincidence point of

F and g if and only if x = y = z = u = 0. Moreover, (0, 0, 0, 0) is a common quadrupled

fixed point of F and g. Also, F and g are w-compatible.

Now, we introduce our main result:

Theorem 2.12 Let (X, d) be a K-metric space and P a solid cone. Suppose that the map-

pings F : X4 → X and g : X → X satisfy the following contractive condition for all

x, y, z, u, x∗, y∗, z∗, u∗ ∈ X:

d(F(x, y, z, u), F(x∗, y∗, z∗, u∗)) � α1d(gx, gx∗) + α2d(gy, gy∗)

+α3d(gz, gz∗) + α4d(gu, gu∗), (20)

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H. Rahimi et al.

where αi for i = 1, 2, 3, 4 are nonnegative constants with

α1 + α2 + α3 + α4 < 1. (21)

If F(X4) ⊂ g(X) and g(X) is complete subset of X, then F and g have a quadrupled

coincidence point in X. Moreover, if F and g are w-compatible, then F and g have a unique

common quadrupled fixed point. Also, common quadrupled fixed point of F and g is of the

form (v, v, v, v) for some v ∈ X.

Proof Let x0, y0, z0, u0 ∈ X and set

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

gx1 = F(x0, y0, z0, u0)

gy1 = F(y0, z0, u0, x0)

gz1 = F(z0, u0, x0, y0)

gu1 = F(u0, x0, y0, z0)

· · ·

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

gxn+1 = F(xn, yn, zn, un)

gyn+1 = F(yn, zn, un, xn)

gzn+1 = F(zn, un, xn, yn)

gun+1 = F(un, xn, yn, zn)

This can be done because F(X4) ⊂ g(X). Now, according to (20), we have

d(gxn+1, gxn) = d(F(xn, yn, zn, un), F(xn−1, yn−1, zn−1, un−1))

� α1d(gxn, gxn−1) + α2d(gyn, gyn−1)

+α3d(gzn, gzn−1) + α4d(gun, gun−1). (22)

Similarly, we have

d(gyn+1, gyn) � α1d(gyn, gyn−1) + α2d(gzn, gzn−1)

+α3d(gun, gun−1) + α4d(gxn, gxn−1) (23)

and

d(gzn+1, gzn) � α1d(gzn, gzn−1) + α2d(gun, gun−1)

+α3d(gxn, gxn−1) + α4d(gyn, gyn−1) (24)

and

d(gun+1, gun) � α1d(gun, gun−1) + α2d(gxn, gxn−1)

+α3d(gyn, gyn−1) + α4d(gzn, gzn−1). (25)

Let

dn = d(gxn+1, gxn) + d(gyn+1, gyn) + d(gzn+1, gzn) + d(gun+1, gun).

Then, adding (22) to (25), we have

dn � (α1 + α2 + α3 + α4)dn−1. (26)

If we replace the first component and the second component, then we will obtain (26). Thus,

for all n,

θ � dn � λdn−1 � λ2dn−2 � · · · � λnd0, (27)

where λ = α1+α2+α3+α4 < 1. If d0 = θ , then (x0, y0, z0, u0) is a quadrupled coincidence

point of F and g. Now, let d0 ≻ θ . If m > n, we have

d(gxn, gxm) � d(gxn, gxn+1) + d(gxn+1, gxn+2) + · · · + d(gxm−1, gxm), (28)

d(gyn, gym) � d(gyn, gyn+1) + d(gyn+1, gyn+2) + · · · + d(gym−1, gym), (29)

123


d(gzn, gzm) � d(gzn, gzn+1) + d(gzn+1, gzn+2) + · · · + d(gzm−1, gzm), (30)

d(gun, gum) � d(gun, gun+1) + d(gun+1, gun+2) + · · · + d(gum−1, gum). (31)

Adding (28) to (31) and using (27). Since λ < 1, we have

dm,n � dn + dn+1 + · · · + dm−1

� (λn + λn+1 + · · · + λm−1)d0

�λn

1 − λd0 → θ as n → ∞,

where

dm,n = d(gxn, gxm) + d(gyn, gym) + d(gzn, gzm) + d(gun, gum).

Now, by (P1) and (P4), it follows that for every c ∈ int P there exist positive integer

N such that dm,n ≪ c for every m > n > N . Thus {gxn}, {gyn}, {gzn} and {gun} are

Cauchy sequences in g(X). Since g(X) is a complete subset of K-metric space, there exist

x, y, z, u ∈ X such that

gxn → gx, gyn → gy, gzn → gz, gun → gu as n → ∞.

Now, we prove that F(x, y, z, u) = gx , F(y, z, u, x) = gy, F(z, u, x, y) = gz,

F(u, x, y, z) = gu. From (d3) and (20), we have

d(F(x, y, z, u), gx) � d(F(x, y, z, u), gxn+1) + d(gxn+1, gx)

= d(F(x, y, z, u), F(xn, yn, zn, un)) + d(gxn+1, gx)

� α1d(gx, gxn) + α2d(gyn, gy) + α3d(gz, gzn)

+α4d(gu, gun) + d(gxn+1, gx).

Since gxn → gx , gyn → gy, gzn → gz and gun → gu, by using Lemma 1.4 we have

d(F(x, y, z, u), gx) = θ ; that is, F(x, y, z, u) = gx . Similarly, we can get F(y, z, u, x) =

gy, F(z, u, x, y) = gz and F(u, x, y, z) = gu. Therefore, (x, y, z, u) ia a quadrupled coin-

cidence point of the mappings F and g. Now, we prove that quadrupled point of coincidence

is unique. Suppose that (x, y, z, u), (x ′, y′, z′.u′) ∈ X4 such that

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

gx = F(x, y, z, u)

gy = F(y, z, u, x)

gz = F(z, u, x, y)

gu = F(u, x, y, z)

· · ·

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

gx ′ = F(x ′, y′, z′, u′)

gy′ = F(y′, z′, u′, x ′)

gz′ = F(z′, u′, x ′, y′)

gu′ = F(u′, x ′, y′, z′)

From (20), we obtain

d(gx, gx ′) = d(F(x, y, z, u), F(x ′, y′, z′, u′))

� α1d(gx, gx ′) + α2d(gy, gy′)

α3d(gz, gz′) + α4d(gu, gu′). (32)

Similarly, we have

d(gy, gy′) � α1d(gy, gy′) + α2d(gz, gz′)

α3d(gu, gu′) + α4d(gx, gx ′) (33)

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H. Rahimi et al.

and

d(gz, gz′) � α1d(gz, gz′) + α2d(gu, gu′)

α3d(gx, gx ′) + α4d(gy, gy′) (34)

and

d(gu, gu′) � α1d(gu, gu′) + α2d(gx, gx ′)

α3d(gy, gy′) + α4d(gz, gz′). (35)

Adding up (32), (33), (34) and (35), we have

d ′ � λd ′,

where

d ′ = d(gx, gx ′) + d(gy, gy′) + d(gz, gz′) + d(gu, gu′).

Since λ = α1 + α2 + α3 + α4 < 1, using Lemma 1.4, we have

d(gx, gx ′) + d(gy, gy′) + d(gz, gz′) + d(gu, gu′) = θ.

It follows that gx = gx ′, gy = gy′, gz = gz′ and gu = gu′. Similarly, we have gx = gy′,

gy = gz′, gz = gu′ and gu = gx ′. Thus gx = gy = gz = gu and (gx, gx, gx, gx)

is unique quadrupled point of coincidence of F and g. Now, let g(x) = v. Then we have

v = gx = F(x, y, z, u). By w-compatibility of F and g, we have

g(v) = g(g(x)) = g(F(x, y, z, u)) = F(gx, gx, gx, gx) = F(v, v, v, v).

Thus (gv, gv, gv, gv) is quadrupled point of coincidence of F and g. Therefore v = gv =

F(v, v, v, v). Consequently (v, v, v, v) is unique quadrupled common fixed point of F and

g. This completes the proof. ⊓⊔

3 Examples and applications

In this section we consider some examples and applications using the results proved in Sect. 2

Example 3.1 Consider an integral equation

x(t)=

T∫

0

l(t, s)[ f (s, x(s))+g(s, x(s))+h(s, x(s))+k(s, x(s))] ds+v(t), t ∈ [0, T ].

(36)

Let X = C([0, T ], R) be the set of continuous functions defined on [0, T ] endowed with the

metric given by

d(u, v) = supt∈[0,T ]

|u(t) − v(t)| for all u, v ∈ X.

In the following, we study the existence and uniqueness of solution of above integral equation.

Assume that the following conditions hold:

(i) l ∈ C([0, T ] × [0, T ], R) such that supt,s∈[0,T ] |l(t, s)| = M <1

T;

(ii) v ∈ C([0, T ], R);

123


(iii) f, g, h, k ∈ C([0, T ] × R, R);

(iv) for all xi , yi , zi , wi ∈ C([0, T ], R), where i = 1, 2, and t ∈ [0, T ] we have

N (t) ≤1

4

(

|x1(t) − x2(t)| + |y1(t) − y2(t)|

+ |z1(t) − z2(t)| + |w1(t) − w2(t)|)

,

where

N (t) = | f (t, x1(t)) − f (t, x2(t))| + |g(t, y1(t)) − g(t, y2(t))|

+|h(t, z1(t)) − h(t, z2(t))| + |k(t, w1(t)) − k(t, w2(t))|.

Then, the integral equation (36) has a unique solution.

Consider the mapping

F : C([0, T ], R) × C([0, T ], R) × C([0, T ], R) × C([0, T ], R) → C([0, T ], R)

defined by

F(x, y, z, w)(t) =

T∫

0

l(t, s)[ f (s, x(s)) + g(s, y(s)) + h(s, z(s)) + k(s, w(s))] ds + v(t),

where t ∈ [0, T ]. Obviously (x, y, z, w) is a solution of (36) if and only if (x, y, z, w) is

a quadrupled fixed point of F . To establish the existence of such a point, we will consider

Corollary 2.6. Indeed, by condition (iv), we have

|F(x1, y1, z1, w1)(t) − F(x2, y2, z2, w2)(t)|

≤

T∫

0

|l(t, s)|1

4(|x1(s) − x2(s)| + |y1(s) − y2(s)|

+|z1(s) − z2(s)| + |w1(s) − w2(s)|) ds

≤1

4

⎛

⎝

T∫

0

|l(t, s)| ds

⎞

⎠ (d(x1, x2) + d(y1, y2) + d(z1, z2) + d(w1, w2)),

for all xi , yi , zi ∈ C([0, T ], R), where i = 1, 2 and t ∈ [0, T ]. Now, using (i), it follows that

d(F(x1, y1, z1, w1), F(x2, y2, z2, w2))

≤MT

4(d(x1, x2) + d(y1, y2) + d(z1, z2) + d(w1, w2)),

for all xi , yi , zi , wi ∈ C([0, T ], R), where i = 1, 2. Then, condition (19) of Corollary 2.6

is satisfied with k = MT < 1. Hence, applying Corollary 2.6, we obtain the existence of a

unique quadrupled fixed point of F , that is, the integral equation (36) has a unique solution.

Now, we give an example to illustrate the validity of the Theorem 2.12.

Example 3.2 Let X = [0,∞). Take E = C1R[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|2t . In this case (X, d) is a complete K-metric space with a cone having non-empty

interior. Define the mappings F : X4 → X and g : X → X by

gx =x

3, F(x, y, z, u) =

x − y + z − u

18.

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H. Rahimi et al.

Then F and g satisfy the contractive condition (21) with αi = 1/6; that is,

d(F(x, y, z, u), F(x ′, y′, z′, u′))(t)=|F(x, y, z, u) − F(x ′, y′, z′, u′)|2t

=1

18|(x − y + z − u) − (x ′ − y′ + z′ − u′)|2t

≤1

6

[

d(gx, gx ′)+d(gy, gy′)+d(gz, gz′)+d(gu, gu′)]

,

for all x, y, z, u, x ′, y′, z′, u′. Moreover, F and g are w-compatible. All conditions of Theo-

rem 2.12 are satisfied. According to Theorem 2.12, F and g have a unique common quadru-

pled fixed point. In this example, (0, 0, 0, 0) is the unique common quadrupled fixed points

of F and g.

Acknowledgments The authors are grateful to the associate editor and two referees for their accurate reading

and their helpful suggestions. Also, the first and the third authors would like to thank the Young Researchers

and Elite club, Central Tehran Branch of Islamic Azad University, for financial support.

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