CUBO A Mathematical Journal
Vol.14, No¯ 03, (115–127). October 2012

A unique common coupled fixed point theorem for four
maps under ψ - φ contractive condition in partial metric

spaces

K.P.R.Rao

Department of Mathematics,

Nagarjuna Nagar522 510,

Acharya Nagarjuna Univertsity

Guntur District,Andhra

Pradesh,India

kprrao2004@yahoo.com

G.N.V.Kishore

Department of Mathematics,

Swarnandhra Institute of

Engineering and Technology,

West Godavari District, Andhra

Pradesh, India

kishore.apr2@gmail.com

Nguyen Van Luong

Department of Natural

Sciences,

Hong Duc University,

Thanh Hoa , Viet Nam

luonghdu@gmail.com

ABSTRACT

In this paper, we obtain a unique coupled common fixed point theorem for four maps

in partial metric spaces.

RESUMEN

En este art́ıculo obtenemos un teorema del punto fijo clásico acoplado único para cuatro

aplicaciones en espacios métricos parciales.

Keywords and Phrases: Partial metric, weakly compatible maps, complete space.

2010 AMS Mathematics Subject Classification: 54H25, 47H10.



116 K.P.R.Rao , G.N.V.Kishore and Nguyen Van Luong CUBO
14, 3 (2012)

1 Introduction and Preliminaries

The notion of partial metric space was introduced by S.G.Matthews [13] as a part of the study

of denotational semantics of data flow networks. In fact, it is widely recognized that partial metric

spaces play an important role in constructing models in the theory of computation ([6-10, 14-16],

etc).

S.G.Matthews [13], Sandra Oltra and Oscar Valero[11] and Salvador

Romaguera [12] and I.Altun, Ferhan Sola, Hakan Simsek [1], T. Abdeljawad, E. Karapinar, K. Tas

[3], E. Karapinar, I.M. Erhan [5] proved fixed point theorems in partial metric spaces for a single

map and a pair of maps.

Regarding the concept of coupled fixed points introduced by Bhaskar and Lakshmikantham

[17], in [4], Aydi proved some coupled fixed point theorems for the mappings satisfying contractive

conditions in partial metric spaces. In this paper, we obtain a unique common coupled fixed point

theorem for four self mappings satisfying a ψ − φ contractive condition in partial metric spaces.

Our result is inspired by the results of Luong and Thuan [18].

First we recall some definitions and lemmas of partial metric spaces.

Definition 1.1. [13]. A partial metric on a nonempty set X is a function p : X × X → R+ such that
for all x,y,z ∈ X:

(p1) x = y ⇔ p(x,x) = p(x,y) = p(y,y),
(p2) p(x,x) ≤ p(x,y),p(y,y) ≤ p(x,y),

(p3) p(x,y) = p(y,x),

(p4) p(x,y) ≤ p(x,z) + p(z,y) − p(z,z).

(X,p) is called a partial metric space.

It is clear that p(x,y) = 0 implies x = y from (p1) and (p2).

But if x = y, p(x,y) may not be zero. A basic example of a partial metric space is the pair (R+,p),

where p(x,y) = max{x,y} for all x,y ∈ R+.

Each partial metric p on X generates τ0 topology τp on X which has a base the family of open p

- balls {Bp(x,ε)|x ∈ X,ε > 0} for all x ∈ X and ε > 0, where Bp(x,ε) = {y ∈ X/p(x,y) < p(x,x)+ε}

for all x ∈ X and ε > 0.

If p is a partial metric on X, then the function ps : X×X → R+ given by ps(x,y) = 2p(x,y) −
p(x,x) − p(y,y) is a metric on X.

Definition 1.2. [13]. Let (X, p) be a partial metric space.

(i) A sequence {xn} in (X,p) is said to converge to a point x ∈ X if and only if p(x,x) =

lim
n→∞

p(x,xn).

(ii) A sequence {xn} in (X,p) is said to be Cauchy sequence if lim
n,m→∞

p(xn,xm) exists and is finite

.

(iii) (X,p) is said to be complete if every Cauchy sequence {xn} in X converges, w.r.to τp, to a

point x ∈ X such that p(x,x) = lim
n,m→∞

p(xn,xm).



CUBO
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A unique common coupled fixed point theorem ... 117

Lemma 1.1. [13]. Let (X,p) be a partial metric space.

(a) {xn} is a Cauchy sequence in (X,p) if and only if it is a Cauchy sequence in the metric space

(X,ps).

(b) (X,p) is complete if and only if the metric space (X,ps) is complete.

Furthermore, lim
n→∞

ps(xn,x) = 0 if and only if

p(x,x) = lim
n→∞

p(xn,x) = lim
n,m→∞

p(xn,xm).

Remark 1.2. Let (X,p) be a partial metric space. If {xn} converges to x in (X,p), then

lim
n→∞

p(xn,y) ≤ p(x,y), ∀ y ∈ X.

Proof. Since {xn} converges to x we have p(x,x) = lim
n→∞

p(xn,x).

Now p(xn,y) ≤ p(xn,x) + p(x,y) − p(x,x).

Letting n → ∞,
lim

n→∞
p(xn,y) ≤ lim

n→∞
p(xn,x) + p(x,y) − p(x,x).

Thus lim
n→∞

p(xn,y) ≤ p(x,y).

Definition 1.3. [17]. An element (x,y) ∈ X × X is called a coupled fixed point of mapping F :

X × X → X if x = F(x,y) and y = F(y,x).

Definition 1.4. [2]. An element (x,y) ∈ X × X is called

(g1) a coupled coincident point of mappings F : X × X → X and f : X → X if fx = F(x,y) and
fy = F(y,x).

(g2) a common coupled fixed point of mappings F : X × X → X and f : X → X if x = fx = F(x,y)
and y = fy = F(y,x).

Definition 1.5. [2]. The mappings F : X × X → X and f : X → X are called w - compatible if
f(F(x,y)) = F(fx,fy) and f(F(y,x)) = F(fy,fx) whenever fx = F(x,y) and fy = F(y,x).

Using concept of coupled fixed points, Luong and Thuan in [18] proved some coupled fixed

point theorems for a mapping F : X × X → X satisfying the following contractive condition in the
partially ordered metric spaces (X,d,≤)

ψ(d(F(x,y),F(u,v))) ≤
1

2
ψ(d(x,u) + d(y,v)) − φ

(

d(x,u) + d(y,v)

2

)

for all x,y,u,v ∈ X with x ≥ u and y ≤ v, with φ ∈ Φ and ψ ∈ Ψ, where Ψ denotes the set of all

functions ψ : [0,∞) → [0,∞) satisfying

(ψ1) ψ is continuous and non-decreasing,

(ψ2) ψ(t) = 0 if and only if t = 0,

(ψ3) ψ(t + s) ≤ ψ(t) + ψ(s), for all t,s ∈ [0,∞),

while Φ denotes the set of all functions φ : [0,∞) → [0,∞) satisfying

(φ1) limt→rφ(t) > 0 for all r > 0.



118 K.P.R.Rao , G.N.V.Kishore and Nguyen Van Luong CUBO
14, 3 (2012)

(φ2) limt→0+φ(t) = 0.

From (φ1), it is clear that φ(t) > 0 for all t > 0.

Now we prove our main result.

2 Main Result

Theorem 1. Let (X,p) be a partial metric space and let f,g : X → X and F,G : X × X → X be such
that

(i) For all x,y,u,v ∈ X,

ψ(p(F(x,y),G(u,v))) ≤
1

2
ψ(p(fx,gu) + p(fy,gv)) − φ(p(fx,gu) + p(fy,gv)) ,

where ψ ∈ Ψ and φ ∈ Φ,

(ii) F(X × X) ⊆ g(X),G(X × X) ⊆ f(X),

(iii) either f(X) or g(X) is a complete subspace of X and

(iv) the pairs (F,f) and (G,g) are w - compatible.

Then F,G,f and g have a unique common coupled fixed point in X × X. Moreover, the common

coupled fixed point of F,G,f and g have the form (u,u).

Proof. Let x0,y0 be arbitrary points in X.

From(ii), there exist sequences {xn}, {yn}, {zn} and {wn} in X such that

F(x2n,y2n) = gx2n+1 = z2n,

F(y2n,x2n) = gy2n+1 = w2n,

G(x2n+1,y2n+1) = fx2n+2 = z2n+1

and

G(y2n+1,x2n+1) = fy2n+2 = w2n+1,

n = 0,1,2, .......

We have

ψ(p(z2n+1,z2n)) = ψ(p(F(x2n,y2n),G(x2n+1,y2n+1))

≤
1

2
ψ(p(z2n,z2n−1) + p(w2n,w2n−1))

−φ(p(z2n,z2n−1) + p(w2n,w2n−1)) (2.1)

Similarly,

ψ(p(w2n+1,w2n)) ≤
1

2
ψ(p(z2n,z2n−1) + p(w2n,w2n−1))

−φ(p(z2n,z2n−1) + p(w2n,w2n−1)) (2.2)



CUBO
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A unique common coupled fixed point theorem ... 119

From (2.1), (2.2) and (ψ3), we have

ψ(p(z2n+1,z2n) + p(w2n+1,w2n)) ≤ ψ(p(z2n+1,z2n)) + ψ(p(w2n+1,w2n))

≤ ψ(p(z2n,z2n−1) + p(w2n,w2n−1))

−2φ(p(z2n,z2n−1) + p(w2n,w2n−1)) (2.3)

≤ ψ(p(z2n,z2n−1) + p(w2n,w2n−1)) .

Since ψ is non - decreasing, we have

p(z2n+1,z2n) + p(w2n+1,w2n) ≤ p(z2n,z2n−1) + p(w2n,w2n−1).

Similarly, we can show that

p(z2n,z2n−1) + p(w2n,w2n−1) ≤ p(z2n−1,z2n−2) + p(w2n−1,w2n−2).

Thus

p(zn+1,zn) + p(wn+1,wn) ≤ p(zn,zn−1) + p(wn,wn−1).

Put δn = p(zn+1,zn) + p(wn+1,wn).Then we have

δn ≤ δn−1,n = 1,2,3, ...

Thus {δn} is a non - increasing sequence of non- negitive real numbers and must converge to a real

number, say, δ ≥ 0.

Suppose δ > 0.

Letting n → ∞ in (2.3) and using the properties of ψ and φ, we get

ψ(δ) ≤ ψ(δ) − 2 lim
δ2n→δ

φ(δ2n) < ψ(δ)

which is a contradiction. Hence δ = 0.

Thus

lim
n→∞

[p(zn+1,zn) + p(wn+1,wn)] = 0 (2.4)

Hence from (p2),

lim
n→∞

[p(zn,zn) + p(wn,wn)] = 0 (2.5)

From (2.4) and (2.5) we have that

lim
n→∞

ps(zn+1,zn) = 0 (2.6)

and

lim
n→∞

ps(wn+1,wn) = 0 (2.7)

Now we prove that {z2n} and {w2n} are Cauchy sequences.

On contrary, suppose that {z2n} or {w2n} is not Cauchy.This implies that p
s(z2m,z2n) 6→ 0 or



120 K.P.R.Rao , G.N.V.Kishore and Nguyen Van Luong CUBO
14, 3 (2012)

ps(w2m,w2n) 6→ 0 as n,m → ∞.
Consequently

max{ps(z2m,z2n),p
s
(w2m,w2n)} 6→ 0 as n,m → ∞.

Then there exist an ǫ > 0 and monotone increasing sequences of natural numbers {2mk} and {2nk}

such that nk > mk,

max{ps(z2mk,z2nk),p
s(w2mk,w2nk)} ≥ ǫ (2.8)

and

max{ps(z2mk,z2nk−2),p
s(w2m,w2nk−2)} < ǫ. (2.9)

From (2.8) and (2.9), we have

ǫ ≤ max{ps(z2mk,z2nk),p
s
(w2mk,w2nk)}

≤ max{ps(z2mk,z2nk−2),p
s
(w2mk,w2nk−2)}

+ max{ps(z2nk−2,z2nk−1),p
s(w2nk−2,w2nk−1)}

+ max{ps(z2nk−1,z2nk),p
s(w2nk−1,w2nk)}

< ǫ + max{ps(z2nk−2,z2nk−1),p
s(w2nk−2,w2nk−1)}

+ max{ps(z2nk−1,z2nk),p
s(w2nk−1,w2nk)}.

Letting k → ∞ and using (2.6) and (2.7) we have

lim
k→∞

max{ps(z2mk,z2nk),p
s(w2mk,w2nk)} = ǫ. (2.10)

Also,

ǫ ≤ max{ps(z2mk,z2nk),p
s
(w2mk,w2nk)}

≤ max{ps(z2mk,z2mk−1),p
s(w2mk,w2mk−1)}

+ max{ps(z2mk−1,z2nk),p
s(w2mk−1,w2nk)} (2.11)

≤ max{ps(z2mk,z2mk−1),p
s(w2mk,w2mk−1)}

+ max{ps(z2mk−1,z2mk),p
s(w2mk−1,w2mk)}

+ max{ps(z2mk,z2nk),p
s(w2mk,w2nk)}

= 2max{ps(z2mk,z2mk−1),p
s
(w2mk,w2mk−1)}

+ max{ps(z2mk,z2nk),p
s
(w2mk,w2nk)}.

Letting k → ∞ and using (2.6), (2.7), (2.10) and (2.11), we have

lim
k→∞

max{ps(z2mk−1,z2nk),p
s(w2mk−1,w2nk)} = ǫ. (2.12)

On other hand we have

max {ps(z2mk,z2nk),p
s(w2mk,w2nk)} ≤ max {p

s(z2mk,z2nk+1),p
s(w2mk,w2nk+1)}

+ max {ps(z2nk+1,z2nk),p
s(w2nk+1,w2nk)}



CUBO
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A unique common coupled fixed point theorem ... 121

Letting k → ∞ and using (2.5),(2.6) and (2.7), we have
ǫ ≤ lim

k→∞
max{ps(z2mk,z2nk+1),p

s(w2mk,w2nk+1)} + 0

≤ lim
k→∞

max

{
2p(z2mk,z2nk+1) − p(z2mk,z2mk) − p(z2nk+1,z2nk+1),

2p(w2mk,w2nk+1) − p(w2mk,w2mk) − p(w2nk+1,w2nk+1)

}

≤ 2 lim
k→∞

max{p(z2mk,z2nk+1),p(w2mk,w2nk+1)}

Thus,

ǫ

2
≤ lim

k→∞
max{p(z2mk,z2nk+1),p(w2mk,w2nk+1)}

By the properties of ψ

ψ
(

ǫ

2

)

≤ ψ

(

lim
k→∞

max{p(z2mk,z2nk+1),p(w2mk,w2nk+1)}

)

= lim
k→∞

max{ψ(p(z2mk,z2nk+1)),ψ(p(w2mk,w2nk+1))} (2.13)

Now

ψ(p(z2mk,z2nk+1)) = ψ(p(F(x2mk,y2mk),G(x2nk+1,y2nk+1)))

≤
1

2
ψ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

−φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

≤
1

2
[ψ(p(z2mk−1,z2nk)) + ψ(p(w2mk−1,w2nk))]

−φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

≤ max{ψ(p(z2mk−1,z2nk)),ψ(p(w2mk−1,w2nk))}

−φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

= ψ(max{p(z2mk−1,z2nk),p(w2mk−1,w2nk)})

−φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

Similarly

ψ(p(w2mk,w2nk+1)) ≤ ψ(max{p(z2mk−1,z2nk),p(w2mk−1,w2nk)})

−φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk)) .



122 K.P.R.Rao , G.N.V.Kishore and Nguyen Van Luong CUBO
14, 3 (2012)

Hence from (2.13),(2.5) and (2.12), we have

ψ
(

ǫ

2

)

≤ lim
k→∞

{
ψ(max{p(z2mk−1,z2nk),p(w2mk−1,w2nk)})

−φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

}

≤ lim
k→∞

ψ













max






1
2

(

ps(z2mk−1,z2nk) + p(z2mk−1,z2mk−1)

+p(z2nk,z2nk)

)

,

1
2

(

ps(w2mk−1,w2nk) + p(w2mk−1,w2mk−1)

+p(w2nk,w2nk)

)


















− lim
k→∞

φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

= ψ
(

ǫ

2

)

− lim
k→∞

φ(p(z2mk−1,z2nk) + p(w2mk−1,w2nk))

= ψ
(

ǫ

2

)

− lim
k→∞

φ









1

2









ps(z2mk−1,z2nk) + p(z2mk−1,z2mk−1)

+p(z2nk,z2nk) + p
s(w2mk−1,w2nk)

+p(w2mk−1,w2mk−1) + p(w2nk,w2nk)

















= ψ
(

ǫ

2

)

− lim
t→ ǫ

2

φ(t) ,

where
ǫ

2
= lim

k→∞

1

2









ps(z2mk−1,z2nk) + p(z2mk−1,z2mk−1)

+p(z2nk,z2nk) + p
s(w2mk−1,w2nk)

+p(w2mk−1,w2mk−1) + p(w2nk,w2nk)









< ψ
(

ǫ

2

)

,

which is a contradiction. Hence {z2n} and {w2n} are Cauchy sequences in the metric space (X,p
s).

Letting n,m → ∞ in

|ps(z2n+1,z2m+1) − p
s(z2n,z2m)| ≤ p

s(z2n+1,z2n) + p
s(z2m+1,z2m).

we get

lim
n→∞

ps(z2n+1,z2m+1) = 0.

Letting n,m → ∞ in

|ps(w2n+1,w2m+1) − p
s(w2n,w2m)| ≤ p

s(w2n+1,w2n) + p
s(w2m+1,w2m)

we get

lim
n→∞

ps(w2n+1,w2m+1) = 0.

Thus {z2n+1} and {w2n+1} are Cauchy sequences in the metric space (X,p
s).

Hence {zn} and {wn} are Cauchy sequences in the metric space (X,p
s).

Hence we have that lim
n→∞

ps(zn,zm) = 0 = lim
n→∞

ps(wn,wm).

Now from definition of ps and from (2.5) we have

lim
n→∞

p(zn,zm) = 0 (2.14)



CUBO
14, 3 (2012)

A unique common coupled fixed point theorem ... 123

and

lim
n→∞

p(wn,wm) = 0. (2.15)

Suppose f(X) is complete. Since {z2n+1} ⊆ f(X) and {w2n+1} ⊆ f(X) are Cauchy sequences in the

complete metric space (f(X),ps), it follows that the sequences {z2n+1} and {w2n+1} are convergent

in (f(X),ps).Thus

lim
n→∞

ps(z2n+1,u) = 0

and

lim
n→∞

ps(w2n+1,v) = 0

for some u and v in f(X).

Since u,v ∈ f(X). there exist s,t ∈ X such that u = fs and v = ft.

Since {zn} and {wn} are Cauchy sequences in X and {z2n+1} → u and {w2n+1} → v, it follows that
{z2n} → u and {w2n} → v.
From Lemma 1.1, we have

p(u,u) = lim
n→∞

p(z2n,u) = lim
n→∞

p(z2n+1,u) = lim
n, m→∞

p(zn,zm) (2.16)

and

p(v,v) = lim
n→∞

p(w2n,v) = lim
n→∞

p(w2n+1,v) = lim
n, m→∞

p(wn,wm) (2.17)

From (2.16), (2.17), (2.14) and (2.15) we have

p(u,u) = 0 = p(v,v). (2.18)

Now,

p(F(s,t),u) ≤ p(F(s,t),z2n+1) + p(z2n+1,u) − p(z2n+1,z2n+1)

≤ p(F(s,t),G(x2n+1,y2n+1)) + p(z2n+1,u).

Therefore,

ψ(p(F(s,t),u)) ≤ ψ(p(F(s,t),G(x2n+1,y2n+1)) + p(z2n+1,u))

≤ ψ(p(F(s,t),G(x2n+1,y2n+1))) + ψ(p(z2n+1,u)), from (ψ3)

≤
1

2
ψ(p(u,z2n) + p(v,w2n)) −

φ(p(u,z2n) + p(v,w2n)) + ψ(p(z2n+1,u)).

Letting n → ∞ and using (2.16), (2.17), (2.18) and (φ2), (ψ1) we get ψ(p(F(s,t),u)) ≤ 0. Hence
F(s,t) = u = fs (by (ψ2)).

Similarly, we have F(t,s) = v = ft.

Since the pair (F,f) is w - compatible, we have fu = F(u,v) and fv = F(v,u). Suppose that fu 6= u

or fv 6= v.

ps(fu,z2n) = 2p(fu,z2n) − p(fu,fu) − p(z2n,z2n).



124 K.P.R.Rao , G.N.V.Kishore and Nguyen Van Luong CUBO
14, 3 (2012)

Letting n → ∞, we get

ps(fu,u) = 2 lim
n→∞

p(fu,z2n) − p(fu,fu) − 0, from (2.5)

or

2p(fu,u) − p(fu,fu) − p(u,u) = 2 lim
n→∞

p(fu,z2n) − p(fu,fu)

or

p(fu,u) = lim
n→∞

p(fu,z2n), from (2.18).

Similarly, we have p(fv,v) = lim
n→∞

p(fv,w2n).Thus

lim
n→∞

[p(fv,z2n) + p(fv,w2n)] = p(fu,u) + p(fv,v) > 0 (2.19)

We have

p(fu,u) ≤ p(fu,z2n+1) + p(z2n+1,u) − p(z2n+1,z2n+1)

≤ p(F(u,v),G(x2n+1,y2n+1)) + p(z2n+1,u).

Thus,

ψ(p(fu,u)) ≤ ψ(p(F(u,v),G(x2n+1,y2n+1)) + ψ(p(z2n+1,u)), from (ψ3)

≤
1

2
ψ(p(fu,z2n) + p(fv,w2n))

−φ(p(fu,z2n) + p(fv,w2n)) + ψ(p(z2n+1,u)).

Similarly, we have

ψ(p(fv,v)) ≤
1

2
ψ(p(fu,z2n) + p(fv,w2n))

−φ(p(fu,z2n) + p(fv,w2n)) + ψ(p(w2n+1,v)).

Hence

ψ(p(fu,u) + p(fv,v)) ≤ ψ(p(fu,u)) + ψ(p(fv,v)), from (ψ3)

≤ ψ(p(fu,z2n) + p(fv,w2n))

−2φ(p(fu,z2n) + p(fv,w2n))

+ψ(p(z2n+1,u)) + ψ(p(w2n+1,v)).

Letting n → ∞ and using (2.19),(φ1),(2.16),(2.17) and (ψ1), we get

ψ(p(fu,u) + p(fv,v)) < ψ(p(fu,u) + p(fv,v)).

It is a contradiction. Hence fu = u and fv = v. Thus

F(u,v) = fu = u and F(v,u) = fv = v. (2.20)



CUBO
14, 3 (2012)

A unique common coupled fixed point theorem ... 125

Since F(X × X) ⊆ g(X), there exist a,b ∈ X such that u = F(u,v) = ga and v = F(v,u) = gb.

ψ(p(u,G(a,b))) = ψ(p(F(u,v),G(a,b)))

≤
1

2
ψ(p(u,u) + p(v,v)) − φ(p(u,u) + p(v,v))

=
1

2
ψ(0) − φ(0), ( from (2.18))

≤ 0, (since ψ(0) = 0 and φ(0) ≥ 0).

Hence ψ(p(u,G(a,b))) = 0, which implies that G(a,b) = u = ga.

Similarly, we have G(b,a) = v = gb.

Since the pair (G,g) is w - compatible, we have gu = G(u,v) and gv = G(v,u). Suppose gu 6= u

or gv 6= v. We have

ψ(p(u,gu)) = ψ(p(F(u,v),G(u,v)))

≤
1

2
ψ(p(u,gu) + p(v,gv)) − φ(p(u,gu) + p(v,gv))

and

ψ(p(v,gv)) = ψ(p(F(v,u),G(v,u)))

≤
1

2
ψ(p(u,gu) + p(v,gv)) − φ(p(u,gu) + p(v,gv)) .

Hence

ψ(p(u,gu) + p(v,gv)) ≤ ψ(p(u,gu)) + ψ(p(v,gv))

≤ ψ(p(u,gu) + p(v,gv)) − 2φ(p(u,gu) + p(v,gv))

< ψ(p(u,gu) + p(v,gv)) (since φ(t) > 0 ∀ t > 0).

Hence gu = u and gv = v.Thus,

u = gu = G(u,v) and v = gv = G(v,u) (2.21)

From (2.20) and (2.21), it follows that (u,v) is a common coupled fixed point of F,G,f and g.

Let (u∗,v∗) be another common coupled fixed point of F,G,f and g. We have

ψ(p(u,u∗) + p(v,v∗)) ≤ ψ(p(u,u∗)) + ψ(p(v,v∗))

≤ ψ(p(F(u,v),G(u∗,v∗))) + ψ(p(F(v,u),G(v∗,u∗)))

≤
1

2
ψ(p(u,u∗) + p(v,v∗)) − φ(p(u,u∗) + p(v,v∗))

+
1

2
ψ(p(u,u∗) + p(v,v∗)) − φ(p(u,u∗) + p(v,v∗))

= ψ(p(u,u∗) + p(v,v∗)) − 2φ(p(u,u∗) + p(v,v∗))

< ψ(p(u,u∗) + p(v,v∗)),



126 K.P.R.Rao , G.N.V.Kishore and Nguyen Van Luong CUBO
14, 3 (2012)

which is a contradiction. Hence (u,v) is the unique common coupled fixed point of F,G,f and g.

Now we will show that u = v. Suppose u 6= v.

ψ(p(u,v)) = ψ(p(F(u,v),G(u,v)))

≤
1

2
ψ(p(u,v) + p(v,u)) − φ(p(u,v) + p(v,u))

≤ ψ(p(u,v)) − φ(p(u,v))

< ψ(p(u,v)).

Hence u = v.

Thus u = fu = F(u,u) = G(u,u) = gu, that is, the common coupled fixed point of F,G,f and g

has the form (u,u).

Received: June 2011. Revised: February 2012.

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