on coupled common fixed points for mixed weakly monotone maps in partially ordereds-metric spaces

Dung Fixed Point Theory and Applications 2013, 2013:48http://www.fixedpointtheoryandapplications.com/content/2013/1/48

RESEARCH Open Access

On coupled common fixed points for mixedweakly monotone maps in partially orderedS-metric spacesNguyen Van Dung*

*Correspondence:[email protected];[email protected] of Mathematics, DongThap University, Cao Lanh, DongThap, Vietnam

AbstractIn this paper, we use the notion of a mixed weakly monotone pair of maps of Gordji etal. (Fixed Point Theory Appl. 2012:95, 2012) to state a coupled common fixed pointtheorem for maps on partially ordered S-metric spaces. This result generalizes themain results of Gordji et al. (Fixed Point Theory Appl. 2012:95, 2012), Bhaskar,Lakshmikantham (Nonlinear Anal. 65(7):1379-1393, 2006), Kadelburg et al. (Comput.Math. Appl. 59:3148-3159, 2010) into the structure of S-metric spaces.

1 Introduction and preliminariesThere aremany generalizedmetric spaces such as -metric spaces [],G-metric spaces [],D*-metric spaces [], partial metric spaces [] and cone metric spaces []. These notionshave been investigated bymany authors and various versions of fixed point theorems havebeen stated in [–] recently. In [], Sedghi, Shobe and Aliouche have introduced thenotion of an S-metric space and proved that this notion is a generalization of a G-metricspace and a D*-metric space. Also, they have proved some properties of S-metric spacesand some fixed point theorems for a self-map on an S-metric space. An interesting workthat naturally rises is to transport certain results in metric spaces and known generalizedmetric spaces to S-metric spaces. In this way, some results have been obtained in [–].In [], Gordji et al. have introduced the concept of a mixed weakly monotone pair

of maps and proved some coupled common fixed point theorems for a contractive-typemaps with themixed weakly monotone property in partially orderedmetric spaces. Theseresults give rise to stating coupled common fixed point theorems for maps with the mixedweakly monotone property in partially ordered S-metric spaces.In this paper, we use the notion of a mixed weakly monotone pair of maps to state a

coupled common fixed point theorem for maps on partially ordered S-metric spaces. Thisresult generalizes the main results of [, , ] into the structure of S-metric spaces.First we recall some notions, lemmas and examples which will be useful later.

Definition . [, Definition .] LetX be a nonempty set. An S-metric onX is a functionS : X –→ [,∞) that satisfies the following conditions for all x, y, z,a ∈ X:. S(x, y, z) = if and only if x = y = z.. S(x, y, z) ≤ S(x,x,a) + S(y, y,a) + S(z, z,a).

The pair (X,S) is called an S-metric space.

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The following is an intuitive geometric example for S-metric spaces.

Example . [, Example .] Let X =R and d be an ordinary metric on X. Put

S(x, y, z) = d(x, y) + d(x, z) + d(y, z)

for all x, y, z ∈ R, that is, S is the perimeter of the triangle given by x, y, z. Then S is an

S-metric on X.

Lemma . [, Lemma .] Let (X,S) be an S-metric space. Then S(x,x, y) = S(y, y,x) forall x, y ∈ X.

The following lemma is a direct consequence of Definition . and Lemma ..

Lemma . [, Lemma .] Let (X,S) be an S-metric space. Then

S(x,x, z)≤ S(x,x, y) + S(y, y, z)

and

S(x,x, z)≤ S(x,x, y) + S(z, z, y)

for all x, y, z ∈ X.

Definition . [, Definition .] Let (X,S) be an S-metric space.. A sequence {xn} ⊂ X is said to converge to x ∈ X if S(xn,xn,x) → as n→ ∞. That is,

for each ε > , there exists n ∈N such that for all n≥ n we have S(xn,xn,x) < ε. Wewrite xn → x for brevity.

. A sequence {xn} ⊂ X is called a Cauchy sequence if S(xn,xn,xm) → as n,m → ∞.That is, for each ε > , there exists n ∈N such that for all n,m ≥ n we haveS(xn,xn,xm) < ε.

. The S-metric space (X,S) is said to be complete if every Cauchy sequence is aconvergent sequence.

From [, Examples on p.] we have the following.

Example .. Let R be a real line. Then

S(x, y, z) = |x – z| + |y – z|

for all x, y, z ∈R is an S-metric on R. This S-metric is called the usual S-metric on R.Furthermore, the usual S-metric space R is complete.

. Let Y be a nonempty subset of R. Then

S(x, y, z) = |x – z| + |y – z|

for all x, y, z ∈ Y is an S-metric on Y . Furthermore, if Y is a closed subset of the usualmetric space R, then the S-metric space Y is complete.



Lemma . [, Lemma .] Let (X,S) be an S-metric space. If xn → x and yn → y, thenS(xn,xn, yn) → S(x,x, y).

Definition . [] Let (X,S) be an S-metric space. For r > and x ∈ X, we define theopen ball BS(x, r) and the closed ball BS[x, r] with center x and radius r as follows:

BS(x, r) ={y ∈ X : S(y, y,x) < r

},

BS[x, r] ={y ∈ X : S(y, y,x) ≤ r

}.

The topology induced by the S-metric or the S-metric topology is the topology generatedby the base of all open balls in X.

Lemma . Let {xn} be a sequence in X. Then xn → x in the S-metric space (X,S) if andonly if xn → x in the S-metric topological space X .

Proof It is a direct consequence of Definition .() and Definition .. �

The following lemma shows that every metric space is an S-metric space.

Lemma . Let (X,d) be a metric space. Then we have. Sd(x, y, z) = d(x, z) + d(y, z) for all x, y, z ∈ X is an S-metric on X .. xn → x in (X,d) if and only if xn → x in (X,Sd).. {xn} is Cauchy in (X,d) if and only if {xn} is Cauchy in (X,Sd).. (X,d) is complete if and only if (X,Sd) is complete.

Proof. See [, Example (), p.].. xn → x in (X,d) if and only if d(xn,x) → , if and only if

Sd(xn,xn,x) = d(xn,x) → ,

that is, xn → x in (X,Sd).. {xn} is Cauchy in (X,d) if and only if d(xn,xm) → as n,m → ∞, if and only if

Sd(xn,xn,xm) = d(xn,xm)→

as n,m → ∞, that is, {xn} is Cauchy in (X,Sd).. It is a direct consequence of () and (). �

The following example proves that the inverse implication of Lemma . does not hold.

Example . Let X = R and S(x, y, z) = |y + z – x| + |y – z| for all x, y, z ∈ X. By [, Ex-ample (), p.], (X,S) is an S-metric space. We will prove that there does not exist anymetric d such that S(x, y, z) = d(x, z) + d(y, z) for all x, y, z ∈ X. Indeed, suppose to the con-trary that there exists a metric d with S(x, y, z) = d(x, z) + d(y, z) for all x, y, z ∈ X. Thend(x, z) =

S(x,x, z) = |x – z| and d(x, y) = S(x, y, y) = |x – y| for all x, y, z ∈ X. It is a contra-diction.



Lemma . [, p.] Let (X,d) be a metric space. Then X × X is a metric space with themetric Dd given by

Dd((x, y), (u, v)

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

for all x, y,u, v ∈ X.

Lemma . Let (X,S) be an S-metric space. Then X × X is an S-metric space with theS-metric D given by

D((x, y), (u, v), (z,w)

)= S(x,u, z) + S(y, v,w)

for all x, y,u, v, z,w ∈ X.

Proof For all x, y,u, v, z,w ∈ X, we have D((x, y), (u, v), (z,w)) ∈ [,∞) and

D((x, y), (u, v), (z,w)

)= if and only if S(x,u, z) + S(y, v,w) =

if and only if x = u = z, y = v = w, that is, (x, y) = (u, v) = (z,w); and

D((x, y), (u, v), (z,w)

)= S(x,u, z) + S(y, v,w)

≤ S(x,x,a) + S(u,u,a) + S(z, z,a) + S(y, y,b) + S(v, v,b) + S(w,w,b)

=D((x, y), (x, y), (a,b)

)+D

((u, v), (u, v), (a,b)

)+D

((z,w), (z,w), (a,b)

).

By the above, D is an S-metric on X ×X. �

Remark . Let (X,d) be a metric space. By using Lemma . with S = Sd , we get

D((x, y), (x, y), (u, v)

)= Sd(x,x,u) + Sd(y, y, v) =

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

)= Dd

((x, y), (u, v)

)

for all x, y,u, v ∈ X.

Lemma . [, p.] Let (X,�) be a partially ordered set. Then X × X is a partially or-dered set with the partial order � defined by

(x, y) � (u, v) if and only if x� u, v � y.

Remark . Let X be a subset of R with the usual order. For each (x,x), (y, y) ∈ X ×X, put z = max{x, y} and z = min{x, y}, then (x,x) � (z, z) and (y, y) � (z, z).Therefore, for each (x,x), (y, y) ∈ X ×X, there exists (z, z) ∈ X ×X that is comparableto (x,x) and (y, y).

Definition . [,Definition .] Let (X,�) be a partially ordered set and f , g : X×X –→X be two maps. We say that a pair (f , g) has themixed weakly monotone property on X if,



for all x, y ∈ X, we have

x � f (x, y), f (y,x)� y implies f (x, y)� g(f (x, y), f (y,x)

), g

(f (y,x), f (x, y)

) � f (y,x)

and

x � g(x, y), g(y,x)� y implies g(x, y) � f(g(x, y), g(y,x)

), f

(g(y,x), g(x, y)

) � g(y,x).

Example . [, Example .] Let f , g :R×R –→R be two functions given by

f (x, y) = x – y, g(x, y) = x – y.

Then the pair (f , g) has the mixed weakly monotone property.

Example . [, Example .] Let f , g :R×R –→ R be two functions given by

f (x, y) = x – y + , g(x, y) = x – y.

Then f and g have the mixed monotone property but the pair (f , g) does not have themixed weakly monotone property.

Remark . [, Remark .] Let (X,�) be a partially ordered set; f : X × X –→ X be amap with the mixed monotone property on X. Then for all n ∈N, the pair (f n, f n) has themixed weakly monotone property on X.

2 Main resultsTheorem . Let (X,�,S) be a partially ordered S-metric space; f , g : X ×X –→ X be twomaps such that. X is complete;. The pair (f , g) has the mixed weakly monotone property on X ;

x � f (x, y), f (y,x) � y or x � g(x, y), g(y,x) � y for some x, y ∈ X ;. There exist p,q, r, s ≥ satisfying p + q + r + s < and

S(f (x, y), f (x, y), g(u, v)

)

≤ pD

((x, y), (x, y), (u, v)

)+qD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+rD

((u, v), (u, v),

(g(u, v), g(v,u)

))+sD

((x, y), (x, y),

(g(u, v), g(v,u)

))

+sD

((u, v), (u, v),

(f (x, y), f (y,x)

))(.)

for all x, y,u, v ∈ X with x � u and y v where D is defined as in Lemma .;. f or g is continuous or X has the following property:

(a) If {xn} is an increasing sequence with xn → x, then xn � x for all n ∈N;(b) If {xn} is an decreasing sequence with xn → x, then x� xn for all n ∈N.

Then f and g have a coupled common fixed point in X.



Proof First we note that the roles of f and g can be interchanged in the assumptions.We need only prove the case x � f (x, y) and f (y,x) � y, the case x � g(x, y) andg(y,x) � y is proved similarly by interchanging the roles of f and g .Step . We construct two Cauchy sequences in X.Put x = f (x, y), y = f (y,x). Since (f , g) has the mixed weakly monotone property, we

have

x = f (x, y)� g(f (x, y), f (y,x)

)= g(x, y)

and

y = f (y,x) g(f (y,x), f (x, y)

)= g(y,x).

Put x = g(x, y), y = g(y,x). Then we have

x = g(x, y) � f(g(x, y), g(y,x)

)= f (x, y)

and

y = g(y,x) f(g(y,x), g(x, y)

)= f (y,x).

Continuously, for all n ∈N, we put

xn+ = f (xn, yn), yn+ = f (yn,xn),

xn+ = g(xn+, yn+), yn+ = g(yn+,xn+)(.)

that satisfy

x � x � · · · � xn � · · · and y y · · · yn · · · . (.)

We will prove that {xn} and {yn} are two Cauchy sequences. For all n ∈ N, it followsfrom (.) that

S(xn+,xn+,xn+)

= S(f (xn, yn), f (xn, yn), g(xn+, yn+)

)

≤ pD

((xn, yn), (xn, yn), (xn+, yn+)

)

+qD

((xn, yn), (xn, yn),

(f (xn, yn), f (yn,xn)

))

+rD

((xn+, yn+), (xn+, yn+),

(g(xn+, yn+), g(yn+,xn+)

))

+sD

((xn, yn), (xn, yn),


))

+sD

((xn+, yn+), (xn+, yn+),

(f (xn, yn), f (yn,xn)

)).



By using (.) we get

S(xn+,xn+,xn+) ≤ pD

((xn, yn), (xn, yn), (xn+, yn+)

)

+qD

((xn, yn), (xn, yn), (xn+, yn+)

)

+rD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+sD

((xn, yn), (xn, yn), (xn+, yn+)

)

+sD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

=p + q

D((xn, yn), (xn, yn), (xn+, yn+)

)

+rD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+sD

((xn, yn), (xn, yn), (xn+, yn+)

)

≤ p + q + s

D((xn, yn), (xn, yn), (xn+, yn+)

)

+r + s

D((xn+, yn+), (xn+, yn+), (xn+, yn+)

).

That is,

S(xn+,xn+,xn+)

≤ p + q + s

(S(xn,xn,xn+) + S(yn, yn, yn+)

)

+r + s

(S(xn+,xn+,xn+) + S(yn+, yn+, yn+)

). (.)

Analogously to (.), we have

S(yn+, yn+, yn+)

≤ p + q + s

(S(yn, yn, yn+) + S(xn,xn,xn+)

)

+r + s

(S(yn+, yn+, yn+) + S(xn+,xn+,xn+)

). (.)

It follows from (.) and (.) that

S(xn+,xn+,xn+) + S(yn+, yn+, yn+)

≤ p + q + s – (r + s)


). (.)

For all n ∈ N, by interchanging the roles of f and g and using (.) again, we have

S(xn+,xn+,xn+)

= S(g(xn+, yn+), g(xn+, yn+), f (xn+, yn+)

)

≤ pD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)



+qD

((xn+, yn+), (xn+, yn+),


))

+rD

((xn+, yn+), (xn+, yn+),

(f (xn+, yn+), f (yn+,xn+)

))

+sD

((xn+, yn+), (xn+, yn+),

(f (xn+, yn+), f (yn+,xn+)

))

+sD

((xn+, yn+), (xn+, yn+),


)).

By using (.) we get

S(xn+,xn+,xn+) ≤ pD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+qD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+rD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+sD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+sD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

=p + q

D((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+rD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+sD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

≤ p + q + s

D((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+r + s

D((xn+, yn+), (xn+, yn+), (xn+, yn+)

).

That is,

S(xn+,xn+,xn+)

≤ p + q + s


)

+r + s


). (.)

Analogously to (.), we have

S(yn+, yn+, yn+)

≤ p + q + s


)

+r + s


). (.)

It follows from (.) and (.) that


≤ p + q + s – (r + s)


). (.)



For all n ∈N, (.) and (.) combine to give


≤ p + q + s – (r + s)


)

≤(

p + q + s – (r + s)

)(S(xn,xn,xn+) + S(yn, yn, yn+)

). (.)

Now we have


≤ p + q + s – (r + s)


)

≤(

p + q + s – (r + s)

)(S(xn–,xn–,xn–) + S(yn–, yn–, yn–)

)· · ·

≤(

p + q + s – (r + s)

)n+(S(x,x,x) + S(y, y, y)

)(.)

and


≤(

p + q + s – (r + s)

)(S(xn,xn,xn+) + S(yn, yn, yn+)

)

≤(

p + q + s – (r + s)

)(S(xn–,xn–,xn–) + S(yn–, yn–, yn–)

)· · ·

≤(

p + q + s – (r + s)

)n+(S(x,x,x) + S(y, y, y)

). (.)

For all n,m ∈N with n≤ m, by using Lemma . and (.), (.), we have

S(xn+,xn+,xm+) + S(yn+, yn+, ym+)

≤ (S(xn+,xn+,xn+) + S(yn+, yn+, yn+)

)+

(S(xn+,xn+,xm+) + S(yn+, yn+, ym+)

)≤ (

S(xn+,xn+,xn+) + S(yn+, yn+, yn+))

+(S(xn+,xn+,xn+) + S(yn+, yn+, yn+)

)+ · · · + (

S(xm–,xm–,xm) + S(ym–, ym–, ym))

+(S(xm,xm,xm+) + S(ym, ym, ym+)

)≤ (

S(xn+,xn+,xn+) + S(yn+, yn+, yn+))

+ · · · + (S(xm,xm,xm+) + S(ym, ym, ym+)

)

≤[(

p + q + s – (r + s)

)n++ · · · +

(p + q + s – (r + s)

)m]



× (S(x,x,x) + S(y, y, y)

)

≤ ( p+q+s–(r+s) )

n+

– p+q+s–(r+s)

(S(x,x,x) + S(y, y, y)

).

Similarly, we have

S(xn,xn,xm+) + S(yn, yn, ym+)

≤[(

p + q + s – (r + s)

)n+ · · · +

(p + q + s – (r + s)

)m](S(x,x,x) + S(y, y, y)

)

≤ ( p+q+s–(r+s) )

n

– p+q+s–(r+s)

(S(x,x,x) + S(y, y, y)

)

and

S(xn,xn,xm) + S(yn, yn, ym)

≤[(

p + q + s – (r + s)

)n+ · · · +

(p + q + s – (r + s)

)m–](S(x,x,x) + S(y, y, y)

)

≤ ( p+q+s–(r+s) )

n

– p+q+s–(r+s)

(S(x,x,x) + S(y, y, y)

)

and

S(xn+,xn+,xm) + S(yn+, yn+, ym)

≤[(

p + q + s – (r + s)

)n++ · · · +

(p + q + s – (r + s)

)m–](S(x,x,x) + S(y, y, y)

)

≤ ( p+q+s–(r+s) )

n+

– p+q+s–(r+s)

(S(x,x,x) + S(y, y, y)

).

Hence, for all n,m ∈N with n≤ m, it follows that

S(xn,xn,xm) + S(yn, yn, ym) ≤( p+q+s–(r+s) )

n

– p+q+s–(r+s)

(S(x,x,x) + S(y, y, y)

).

Since ≤ p+q+s–(r+s) < , taking the limit as n,m → ∞, we get

limn,m→∞

(S(xn,xn,xm) + S(yn, yn, ym)

)= .

It implies that

limn,m→∞S(xn,xn,xm) = lim

n,m→∞S(yn, yn, ym) = .

Therefore, {xn} and {yn} are two Cauchy sequences in X. Since X is complete, there existx, y ∈ X such that xn → x and yn → y in X as n→ ∞.Step . We prove that (x, y) is a coupled common fixed point of f and g . We consider the

following two cases.



Case .. f is continuous. We have

x = limn→∞xn+ = lim

n→∞ f (xn, yn) = f(

limn→∞xn, lim

n→∞ yn)= f (x, y)

and

y = limn→∞ yn+ = lim

n→∞ f (yn,xn) = f(

limn→∞ yn, lim

n→∞xn)= f (y,x).

Now using (.) we have

S(f (x, y), f (x, y), g(x, y)

)+ S

(f (y,x), f (y,x), g(y,x)

)

≤ pD

((x, y), (x, y), (x, y)

)+qD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+rD

((x, y), (x, y),

(g(x, y), g(y,x)

))+sD

((x, y), (x, y),

(g(x, y), g(y,x)

))

+sD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+pD

((y,x), (y,x), (y,x)

)+qD

((y,x), (y,x),

(f (y,x), f (x, y)

))

+rD

((y,x), (y,x),

(g(y,x), g(x, y)

))+sD

((y,x), (y,x),

(g(y,x), g(x, y)

))

+sD

((y,x), (y,x),

(f (y,x), f (x, y)

))

=pD

((x, y), (x, y), (x, y)

)+qD

((x, y), (x, y), (x, y)

)

+rD

((x, y), (x, y),

(g(x, y), g(y,x)

))+sD

((x, y), (x, y),

(g(x, y), g(y,x)

))

+sD

((x, y), (x, y), (x, y)

)

+pD

((y,x), (y,x), (y,x)

)+qD

((y,x), (y,x), (y,x)

)

+rD

((y,x), (y,x),

(g(y,x), g(x, y)

))+sD

((y,x), (y,x),

(g(y,x), g(x, y)

))

+sD

((y,x), (y,x), (y,x)

)

=rD

((x, y), (x, y),

(g(x, y), g(y,x)

))+sD

((x, y), (x, y),

(g(x, y), g(y,x)

))

+rD

((y,x), (y,x),

(g(y,x), g(x, y)

))+sD

((y,x), (y,x),

(g(y,x), g(x, y)

)).

Therefore,

S(f (x, y), f (x, y), g(x, y)

)+ S

(f (y,x), f (y,x), g(y,x)

)≤ (r + s)

(S(x,x, g(x, y)

)+ S

(y, y, g(y,x)

)).

That is,

S(x,x, g(x, y)

)+ S

(y, y, g(y,x)

) ≤ (r + s)(S(x,x, g(x, y)

)+ S

(y, y, g(y,x)

)).



Since ≤ r+s < , we get S(x,x, g(x, y)) = S(y, y, g(y,x)) = . That is, g(x, y) = x and g(y,x) = y.Therefore, (x, y) is a coupled common fixed point of f and g .Case .. g is continuous. We can also prove that (x, y) is a coupled common fixed point

of f and g similarly as in Case ..Case .. X satisfies two assumptions (a) and (b). Then by (.) we get xn � x and y� yn

for all n ∈N. By using Lemma . and Lemma ., we have

D((x, y), (x, y),

(f (x, y), f (y,x)

))≤ D

((x, y), (x, y), (xn+, yn+)

)+D

((xn+, yn+), (xn+, yn+),

(f (x, y), f (y,x)

))= D

((x, y), (x, y), (xn+, yn+)

)+D

((g(xn+, yn+), g(yn+,xn+)

),


),(f (x, y), f (y,x)

))≤ D

((x, y), (x, y), (xn+, yn+)

)+ S

(g(xn+, yn+), g(xn+, yn+), f (x, y)

)+ S

(g(yn+,xn+), g(yn+,xn+), f (y,x)

)= S(x,x,xn+) + S(y, y, yn+) + S

(g(xn+, yn+), g(xn+, yn+), f (x, y)

)+ S

(f (y,x), f (y,x), g(yn+,xn+)

). (.)

By interchanging the roles of f and g and using (.), we have

S(g(xn+, yn+), g(xn+, yn+), f (x, y)

)

≤ pD

((xn+, yn+), (xn+, yn+), (x, y)

)

+qD

((xn+, yn+), (xn+, yn+),


))

+rD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+sD

((xn+, yn+), (xn+, yn+),

(f (x, y), f (y,x)

))

+sD

((x, y), (x, y),


))

=pD

((xn+, yn+), (xn+, yn+), (x, y)

)

+qD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)

+rD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+sD

((xn+, yn+), (xn+, yn+),

(f (x, y), f (y,x)

))

+sD

((x, y), (x, y), (xn+, yn+)

). (.)

Again, by using (.), we have

S(f (y,x), f (y,x), g(yn+,xn+)

)

≤ pD

((y,x), (y,x), (yn+,xn+)

)+qD

((y,x), (y,x),

(f (y,x), f (x, y)

))



+rD

((yn+,xn+), (yn+,xn+),

(g(yn+,xn+), g(xn+, yn+)

))

+sD

((y,x), (y,x),

(g(yn+,xn+), g(xn+, yn+)

))

+sD

((yn+,xn+), (yn+,xn+),

(f (y,x), f (x, y)

))

=pD

((y,x), (y,x), (yn+,xn+)

)+qD

((y,x), (y,x),

(f (y,x), f (x, y)

))

+rD

((yn+,xn+), (yn+,xn+), (yn+,xn+)

)+sD

((y,x), (y,x), (yn+,xn+)

)

+sD

((yn+,xn+), (yn+,xn+),

(f (y,x), f (x, y)

)). (.)

It follows from (.), (.) and (.) that

D((x, y), (x, y),

(f (x, y), f (y,x)

))

≤ S(x,x,xn+) + S(y, y, yn+) +pD

((xn+, yn+), (xn+, yn+), (x, y)

)

+qD

((xn+, yn+), (xn+, yn+), (xn+, yn+)

)+rD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+sD

((xn+, yn+), (xn+, yn+),

(f (x, y), f (y,x)

))+sD

((x, y), (x, y), (xn+, yn+)

)

+pD

((y,x), (y,x), (yn+,xn+)

)+qD

((y,x), (y,x),

(f (y,x), f (x, y)

))

+rD

((yn+,xn+), (yn+,xn+), (yn+,xn+)

)+sD

((y,x), (y,x), (yn+,xn+)

)

+sD

((yn+,xn+), (yn+,xn+),

(f (y,x), f (x, y)

)). (.)

By using Lemma . and taking the limit as n→ ∞ in (.), we have

D((x, y), (x, y),

(f (x, y), f (y,x)

))

≤ S(x,x,x) + S(y, y, y) +pD

((x, y), (x, y), (x, y)

)+qD

((x, y), (x, y), (x, y)

)

+rD

((x, y), (x, y),

(f (x, y), f (y,x)

))+sD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+sD

((x, y), (x, y), (x, y)

)+pD

((y,x), (y,x), (y,x)

)

+qD

((y,x), (y,x),

(f (y,x), f (x, y)

))+rD

((y,x), (y,x), (y,x)

)

+sD

((y,x), (y,x), (y,x)

)+sD

((y,x), (y,x),

(f (y,x), f (x, y)

))

=r + s

D((x, y), (x, y),

(f (x, y), f (y,x)

))+q + s

D((y,x), (y,x),

(f (y,x), f (x, y)

)). (.)

It implies that

S(x,x, f (x, y)

)+ S

(y, y, f (y,x)

)

≤ r + s

(S(x,x, f (x, y)

)+

(y, y, f (y,x)

))+q + s

(S(y, y, f (y,x)

)+ S

(x,x, f (x, y)

))

=r + q + s

(S(x,x, f (x, y)

)+

(y, y, f (y,x)

)). (.)



Since r+q+s < , we have S(x,x, f (x, y))+S(y, y, f (y,x)) = , that is, f (x, y) = x and f (y,x) = y.

Similarly, one can show that g(x, y) = x and g(y,x) = y. This proves that (x, y) is a coupledcommon fixed point of f and g . �

From Theorem ., we have following corollaries.

Corollary . [, Theorems . and .] Let (X,�,d) be a partially orderedmetric space;f , g : X ×X –→ X be two maps such that. X is complete;. The pair (f , g) has the mixed weakly monotone property on X ; x � f (x, y),

f (y,x) � y or x � g(x, y), g(y,x) � y for some x, y ∈ X ;. There exist p,q, r, s ≥ satisfying p + q + r + s < and

d(f (x, y), g(u, v)

)

≤ pDd

((x, y), (u, v)

)+qDd

((x, y),

(f (x, y), f (y,x)

))

+rDd

((u, v),

(g(u, v), g(v,u)

))+sDd

((x, y),

(g(u, v), g(v,u)

))

+sDd

((u, v),

(f (x, y), f (y,x)

))(.)

for all x, y,u, v ∈ X with x � u and y v, where Dd is defined as in Lemma .;. f or g is continuous or X has the following property:


Then f and g have a coupled common fixed point in X.

Proof It is a direct consequence of Lemma ., Remark . and Theorem .. �

For similar results of the following for maps on metric spaces and cone metric spaces,the readers may refer to [, Theorems ., ., . and .] and [, Theorem .].

Corollary . Let (X,�,S) be a partially ordered S-metric space and f : X ×X –→ X be amap such that. X is complete;. f has the mixed monotone property on X ; x � f (x, y) and f (y,x) � y for some

x, y ∈ X ;. There exist p,q, r, s ≥ satisfying p + q + r + s < and

S(f (x, y), f (x, y), f (u, v)

)

≤ pD

((x, y), (x, y), (u, v)

)+qD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+rD

((u, v), (u, v),

(f (u, v), f (v,u)

))+sD

((x, y), (x, y),

(f (u, v), f (v,u)

))

+sD

((u, v), (u, v),

(f (x, y), f (y,x)

))(.)

for all x, y,u, v ∈ X with x � u and y v;. f is continuous or X has the following property:




Then f has a coupled fixed point in X.

Proof By choosing g = f in Theorem . and using Remark ., we get the conclusion. �

Corollary . Let (X,�,S) be a partially ordered S-metric space and f : X ×X –→ X be amap such that. X is complete;. f has the mixed monotone property on X ; x � f (x, y) and f (y,x) � y for some

x, y ∈ X ;. There exists k ∈ [, ) satisfying

S(f (x, y), f (x, y), f (u, v)

) ≤ k(S(x,x,u) + S(y, y, v)

)(.)

for all x, y,u, v ∈ X with x � u and y v;. f is continuous or X has the following property:


Then f has a coupled fixed point in X.

Proof By choosing g = f and p = k, q = r = s = in Theorem . and using Remark ., weget the conclusion. �

Corollary . Assume that X is a totally ordered set in addition to the hypotheses of The-orem .; in particular, Corollary ., Corollary .. Then f and g have a unique coupledcommon fixed point (x, y) and x = y.

Proof By Theorem ., f and g have a coupled common fixed point (x, y). Let (z, t) beanother coupled commonfixed point of f and g .Without loss of generality, wemay assumethat (x, y)� (z, t). Then by (.) and Lemma ., we have

D((x, y), (x, y), (z, t)

)= S(x,x, z) + S(y, y, t)

= S(f (x, y), f (x, y), g(z, t)

)+ S

(f (y,x), f (y,x), g(t, z)

)

≤ pD

((x, y), (x, y), (z, t)

)+qD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+rD

((z, t), (z, t),

(g(z, t), g(t, z)

))+sD

((x, y), (x, y),

(g(z, t), g(t, z)

))

+sD

((z, t), (z, t),

(f (x, y), f (y,x)

))+pD

((y,x), (y,x), (t, z)

)

+qD

((y,x), (y,x),

(f (y,x), f (x, y)

))+rD

((t, z), (t, z),

(g(t, z), g(z, t)

))

+sD

((y,x), (y,x),

(g(t, z), g(z, t)

))+sD

((t, z), (t, z),

(f (y,x), f (x, y)

))

=pD

((x, y), (x, y), (z, t)

)+qD

((x, y), (x, y), (x, y)

)+rD

((z, t), (z, t), (z, t)

)



+sD

((x, y), (x, y), (z, t)

)+sD

((z, t), (z, t), (x, y)

)+pD

((y,x), (y,x), (t, z)

)

+qD

((y,x), (y,x), (y,x)

)+rD

((t, z), (t, z), (t, z)

)+sD

((y,x), (y,x), (t, z)

)

+sD

((t, z), (t, z), (y,x)

)

=pD

((x, y), (x, y), (z, t)

)+sD

((x, y), (x, y), (z, t)

)+sD

((z, t), (z, t), (x, y)

)

+pD

((y,x), (y,x), (t, z)

)+sD

((y,x), (y,x), (t, z)

)+sD

((t, z), (t, z), (y,x)

)

=p + s

(D

((x, y), (x, y), (z, t)

)+D

((y,x), (y,x), (t, z)

))

= (p + s)(S(x,x, z) + S(y, y, t)

).

Since p + s < , we have S(x,x, z) + S(y, y, t) = . Then x = z and y = t. This proves that thecoupled common fixed point of f and g is unique.Moreover, by using (.) and Lemma . again, we get

S(x,x, y) = S(f (x, y), f (x, y), g(y,x)

)

≤ pD

((x, y), (x, y), (y,x)

)+qD

((x, y), (x, y),

(f (x, y), f (y,x)

))

+rD

((y,x), (y,x),

(g(y,x), g(x, y)

))+sD

((x, y), (x, y),

(g(y,x), g(x, y)

))

+sD

((y,x), (y,x),

(f (x, y), f (y,x)

))

=pD

((x, y), (x, y), (y,x)

)+sD

((x, y), (x, y), (y,x)

)+sD

((y,x), (y,x), (x, y)

)

=p + s

D((x, y), (x, y), (y,x)

)= (p + s)S(x,x, y).

Since p + s < , we get S(x,x, y) = , that is, x = y. �

Finally, we give an example to demonstrate the validity of the above results.

Example . Let X =R with the S-metric as in Example . and the usual order ≤. ThenX is a totally ordered, complete S-metric space. For all x, y ∈ X, put

f (x, y) = g(x, y) = x – y +

.

Then the pair (f , g) has the mixed weakly monotone property and

S(f (x, y), f (x, y), g(u, v)

)=

∣∣f (x, y) – g(u, v)∣∣

= ∣∣∣∣x – y +

–u – v +

∣∣∣∣≤

|x – u| +

|y – v|

≤

(|x – u| + |y – v|).



Then the contraction (.) is satisfied with p = and q = r = s = . Note that other as-

sumptions of Corollary . are also satisfied and (, ) is the unique common fixed pointof f and g .

Competing interestsThe author declares that they have no competing interests.

AcknowledgementsThe author would like to thank the referees for their valuable comments.

Received: 31 July 2012 Accepted: 21 February 2013 Published: 8 March 2013

References1. Gähler, S: Über die uniformisierbarkeit 2-metrischer Räume. Math. Nachr. 28, 235-244 (1965)2. Mustafa, Z, Sims, B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7(2), 289-297 (2006)3. Sedghi, S, Shobe, N, Zhou, H: A common fixed point theorem in D*-metric spaces. Fixed Point Theory Appl. 2007,

Article ID 27906 (2007)4. Bukatin, M, Kopperman, R, Matthews, S, Pajoohesh, H: Partial metric spaces. Am. Math. Mon. 116, 708-718 (2009)5. Huang, LG, Zhang, X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332,

1468-1476 (2007)6. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear

Anal. 65(7), 1379-1393 (2006)7. Sintunavarat, W, Radenovic, S, Golubovic, Z, Kumam, P: Coupled fixed point theorems for F-invariant set and

applications. Appl. Math. Inf. Sci. 7(1), 247-255 (2013)8. Abbas, M, Nazir, T, Radenovic, S: Common coupled fixed points of generalized contractive mappings in partially

ordered metric spaces. Positivity (2013). doi:10.1007/s11117-012-0219-z9. Kadelburg, Z, Nashine, HK, Radenovic, S: Coupled fixed point results under tvs-cone metric and w-cone-distance.

Bull. Math. Anal. Appl. 2(2), 51-63 (2012)10. Kadelburg, Z, Radenovic, S: Common coupled fixed point results in partially ordered Gmetric spaces. Adv. Fixed Point

Theory 2, 29-36 (2012)11. Radenovic, S, Pantelic, S, Salimi, P, Vujakovic, J: A note on some tripled coincidence point results in G-metric spaces.

Int. J. Math. Sci. Eng. App. 6(VI), 23-28 (2012)12. Rajic, VC, Radenovic, S: A note on tripled fixed point of w-compatible mappings in tvs-cone metric spaces. Thai J.

Math. (2013, accepted paper)13. Nashine, HK, Kadelburg, Z, Radenovic, S: Coupled common fixed point theorems for w*-compatible mappings in

ordered cone metric spaces. Appl. Math. Comput. 218, 5422-5432 (2012)14. Jleli, M, Rajic, VC, Samet, B, Vetro, C: Fixed point theorems on ordered metric spaces and applications to nonlinear

elastic beam equations. J. Fixed Point Theory Appl. (2012). doi:10.1007/s11784-012-0081-415. Golubovic, Z, Kadelburg, Z, Radenovic, S: Coupled coincidence points of mappings in ordered partial metric spaces.

Abstr. Appl. Anal. 2012, Article ID 192581 (2012)16. Doric, D, Kadelburg, Z, Radenovic, S: Coupled fixed point results for mappings without mixed monotone property.

Appl. Math. Lett. 25, 1803-1808 (2012)17. Ding, HS, Li, L, Radenovic, S: Coupled coincidence point theorems for generalized nonlinear contraction in partially

ordered metric spaces. Fixed Point Theory Appl. 2012, 96 (2012)18. Altun, I, Acar, O: Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces. Topol.

Appl. 159, 2642-2648 (2012)19. Aydi, H, Karapinar, E: A Meir-Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory

Appl. 2012, 26 (2012)20. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric

spaces. Nonlinear Anal. 74, 18 (2011)21. Berinde, V, Vetro, F: Common fixed points of mappings satisfying implicit contractive conditions. Fixed Point Theory

Appl. 2012, 105 (2012)22. Choudhurya, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible

mappings. Nonlinear Anal. 73, 2524-2531 (2010)23. Lakshmikantham, V, Ciric, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric

spaces. Nonlinear Anal. 70(12), 4341-4349 (2009)24. Sedghi, S, Shobe, N, Aliouche, A: A generalization of fixed point theorem in S-metric spaces. Mat. Vesn. 64(3), 258-266

(2012)25. An, TV, Dung, NV: Two fixed point theorems in S-metric spaces. Preprint (2012)26. Sedghi, S, Dung, NV: Fixed point theorems on S-metric spaces. Mat. Vesnik (2012, accepted paper)27. Gordji, ME, Ramezani, M, Cho, YJ, Akbartabar, E: Coupled common fixed point theorems for mixed weakly monotone

mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2012, 95 (2012)28. Kadelburg, Z, Pavlovic, M, Radenovic, S: Common fixed point theorems for ordered contractions and

quasicontractions in ordered cone metric spaces. Comput. Math. Appl. 59, 3148-3159 (2010)

doi:10.1186/1687-1812-2013-48Cite this article as: Dung: On coupled common fixed points for mixed weakly monotone maps in partially orderedS-metric spaces. Fixed Point Theory and Applications 2013 2013:48.


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on coupled common fixed points for mixed weakly monotone maps in partially ordereds-metric spaces

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