suzuki type unique common tripled fixed point theorem for … · some fixed point theorems in...

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391

Volume 6 Issue 2, February 2017

www.ijsr.net Licensed Under Creative Commons Attribution CC BY

Suzuki Type Unique Common Tripled Fixed Point

Theorem for Four Maps under Ψ - Φ Contractive

Condition in Partial Metric Spaces

V. M. L. Hima Bindu1, G. N. V. Kishore

2

1, 2Department of Mathematics, K L University, Vaddeswaram, Guntur - 522 502, Andhra Pradesh, India

Abstract: In this paper, we obtain a Suzuki type unique common tripled fixed point theorem for four maps in partial metric spaces. Keywords and phrases: Partial metric, weakly compatible maps, Suzuki type contraction, Triple fixed point

2000 Mathematics Subject Classification: 54H25, 47H10, 54E50

1. Introduction

The notion of a partial metric space was introduced by

Matthews [9] as a part of the study of denotational semantics

of data ow networks. In fact, it is widely recognized that

partial metric spaces play an important role in constructing

models in the theory of computation and domain theory in

computer science.

Matthews [9] and Romaguera [11] and Altun et al. [2] proved

some fixed point theorems in partial metric spaces for a

single map. For more works on fixed, common fixed point

theorems in partial metric spaces, we refer [1-12]). The aim

of this paper is to study Suzuki type unique common tripled

fixed point theorem for four maps satisfying a ψ - ϕ

contractive condition in partial metric spaces.

First we give the following theorem of Suzuki [15].

Theorem 1.1. (See [15]): Let (X, d) be a complete metric

space and let T be a mapping on X. Define a non - increasing

function θ from [0, 1) onto ]1,2

1( by

.12

1

1

1

,2

1

2

151

,2

1501

)(2

rifr

rifr

r

rif

r

Assume that there exists r ∈ [0, 1) such that

),(),( implies ),(),()( yxrdTyTxdyxdTxxdr

for all ., Xyx

Then there exists a unique fixed point z of T .

Moreover zxT n

n

lim for all .Xx

First we recall some basic definitions and lemmas which play

crucial role in the theory of partial metric spaces.

Definition 1.4. (See [1,9]) A partial metric on a nonempty

set X is a function RXXp : such that for all

:,, Xzyx

).,(),(),(),()(

),,(),()(

),,(),(),,(),()(

),,(),(),()(

4

3

2

1

zzpyzpzxpyxpp

xypyxpp

yxpyypyxpxxpp

yypyxpxxpyxp

The pair ),( PX is called a partial metric space (PMS).

If p is a partial metric on X, then the function RXXp s : given by

),,(),(),(2),( yypxxpyxpyxp s (1.1)

is a metric on X.

Example 1.5. (See e.g. [9]) Consider X = [0, ∞) with

yxyxp ,max),( .Then (X, p) is a partial metric space.

It is clear that p is not a (usual) metric. Note that in this case

.),( yxyxp s

Example 1.6. Let X = b a R, ba, :, ba and define

p( [a,b],[c,d] ) = max{b,d} – min{a,c}. Then (X,p) is a

partial metric space.

We now state some basic topological notions (such as

convergence, completeness, continuity) on partial metric

spaces (see e.g. [1, 2, 7, 8, 9].)

Definition 1.7.

(i) A sequence nx in the PMS (X, p) converges to the

limit x if and only if

).,(lim),( nn

xxpxxp

(ii) A sequence nx in the PMS (X, p) is called a Cauchy

sequence if

),(lim,

mnmn

xxp

exists and is finite.

(iii) A PMS (X, p) is called complete if every Cauchy

Paper ID: NOV164622 DOI: 10.21275/NOV164622 75





sequence nx in X converges with respect to τ p , to a

point Xx such that ).,(lim),(,

nmmn

xxpxxp

The following lemma is one of the basic results in PMS([1, 2,

7, 8, 9]).

Lemma 1.8.

(i) A sequence nx is a Cauchy sequence in the PMS (X,

p) if and only if it is a Cauchy sequence in the metric space

(X,sp ).

(ii) A PMS (X, p) is complete if and only if the metric space

),( spX is complete.

Moreover ).,(lim),(lim),(0),(lim,

nmmn

nn

n

s

nxxpxxpxxpxxp

Next, we give two simple lemmas which will be used in the

proof of our main result. For the proofs we refer to [1].

Lemma 1.9.

Assume zxn as n → ∞ in a PMS (X, p) such

that 0),( zzp . Then ),(lim),( yxpyzp nn

for

every .Xy

Lemma 1.10. Let (X, p) be a PMS. Then

(A) If p(x, y) = 0 then x = y,

(B) If x y, then p(x, y) > 0.

Remark 1.11. If x = y, p(x, y) may not be 0.

Definition 1.12. Let Ψ be the set of all altering distance

functions ,0,0:

such that

(i) is continuous and non - decreasing.

(ii) (t) = 0 if and only if t = 0.

Definition 1.13. Let be the set of all functions

,0,0: such that

(i) is continuous .

(ii) (t) = 0 if and only if t = 0.

Defnition 1.14. [14] An element (x, y,z) X X X is

called

(i) a tripled coincident point of F : X X X X and f :

X X if fx = F(x, y, z), fy = F(y, z, x) and fz = F(z, x, y),

(ii) a common tripled fixed point of F : X X X X

and f : X X if

x = fx = F(x, y, z), y = fy = F(y, z, x) and z = fz = F(z, x, y).

Defnition 1.15. [14] The mappings F : X X X X and

f : X X are called w - compatible if f(F(x, y,z)) = F(fx,

fy,fz), f(F(y, z,x)) = F(fy, fz, fx) and f(F(z, x,y)) = F(fz,fx,fy)

whenever fx = F(x, y, z), fy = F(y, z, x) and fz = F(z, x, y).

Now we prove our main result.

2. Main Result

Theorem 2.1. Let (X, p) be a partial metric space and let S,

T: X X X X and f, g : X → X be mappings

satisfying

(2.1.1) ),(),,(),,(max))},,(,()),,,(,(min{2

1gwfzpgvfypgufxpwvuTgupzyxSfxp

or

),(),,(),,(max))},,(,()),,,(,(min{2

1gwfzpgvfypgufxpuwvTgvpxzySfyp

or

),(),,(),,(max))},,(,()),,,(,(min{2

1gwfzpgvfypgufxpvuwTgwpyxzSfzp

implies that

)),,,,,,(()),,,,,(())),,(),,,((( wvuzyxMwvuzyxMwvuTzyxSp

for all x, y , z ,u , v, w in X, where and are functions and

,

)),,(,()),,(,(2

1

,)),,(,()),,(,(2

1

,)),,(,()),,(,(2

1

)),,,(,()),,,(,(

)),,,(,()),,,(,(

)),,,(,()),,,(,(

),,(gv),p(fy,gu),p(fx,

max ),,,,,(

yxzSgwpvuwTfzp

xzySgvpuwvTfyp

zyxSgupwvuTfxp

vuwTgwpuwvTgvp

wvuTgupyxzSfzp

xzySfypzyxSfxp

gwfzp

wvuzyxM






(2.1.2) S(X X X) g(X), T(X X X) f(X),

(2.1.3) either f(X) or g(X) is a complete subspaces of X,

(2.1.4) the pairs (f, S) and (g, T) are weakly compatible.

Then S, T, f and g have a unique common tripled fixed point

of the form XXX ,, .

Proof: Let x0, y0, z0 X be arbitrary points in X. From

(2.1.2), there exist sequences of {xn} , {yn} , {zn}, {un} ,{vn}

,{wn} in X such that

S(x2n, y2n , z2n )= gx2n+1 = u2n,

S(y2n, z2n , x2n )= gy2n+1 = v2n,

S(z2n, x2n , y2n )= gz2n+1 = w2n,

T(x2n+1, y2n+1, z2n+1)= fx2n+2 = u2n+1,

T(y2n+1, z2n+1, x2n+1)= fy2n+2 = v2n+1,

T(z2n+1, x2n+1, y2n+1)= fz2n+2 = w2n+1, n = 0, 1, 2, · · ·

For simplification we denote

.),(),,(),,(max 111 nnnnnnn wwpvvpuupR

Clearly

.),(),,(),,(max)),,(,(

)),,,(,(min

2

1122122122

12121212

2222

nnnnnn

nnnn

nnnngzfzpgyfypgxfxp

zyxTgxp

zyxSfxp

From (2.1.1), we

)).,,,,,((

)),,,,,(())),,(),,,(((

121212222

121212222121212222

nnnnnn

nnnnnnnnnnnn

zyxzyxM

zyxzyxMzyxTzyxSp

But

.),(),,(max

)(,),(),(2

1 ),(),(

2

1

122212

4122212221212

nnnn

nnnnnnnn

uupuup

pfromuupuupuupuup

Similarly,

.),(),,(max),(),(2

1122212221212 nnnnnnnn vvpvvpvvpvvp

and

.),(),,(max),(),(2

1122212221212 nnnnnnnn wwpwwpwwpwwp

Hence

.,max

),(),,(),,(

),,(),,(),,(max),,,,,(

212

122122122

212212212

121212222

nn

nnnnnn

nnnnnn

nnnnnn

RR

wwpvvpuup

wwpvvpuupzyxzyxM

Thus

).,(max),(max)),(( 212212122 nnnnnn RRRRuup

Similarly,

),(max),(max)),(( 212212122 nnnnnn RRRRvvp

and

).,(max),(max)),(( 212212122 nnnnnn RRRRwwp

Now,






).,(max),(max

))},(()),,(()),,((max{

)),(),,(),,((max)(

212212

122122122

1221221222

nnnn

nnnnnn

nnnnnnn

RRRR

wwpvvpuup

wwpvvpuupR

We have the following two cases.

Case (a) : If R 2n is maximum in the right hand side, we obtain

).()()( 222 nnn RRR

It follows that 0)( 2 nR so that R2n = 0:

Thus

. and ,

.0),( 0),(,0),(

.0),(),,(),,(max

122122122

122122122

122122122

nnnnnn

nnnnnn

nnnnnn

wwvvuuThus

wwpandvvpuup

wwpvvpuup

Now we claim that . , 122212221222 nnnnnn wwandvvuu

Clearly

.),(),,(),,(max)),,(,(

)),,,(,(min

2

1122212221222

12121212

22222222

nnnnnn

nnnn

nnnngzfzpgyfypgxfxp

zyxTgxp

zyxSfxp

From (2.1.1),we get

)).,,,,,((

)),,,,,(())),,(),,,(((

121212222222

121212222222121212222222

nnnnnn

nnnnnnnnnnnn

zyxzyxM

zyxzyxMzyxTzyxSp

But

).(),,(

),(),(2

1 ),(),(

2

1

21222

121212221212222

pfromuup

uupuupuupuup

nn

nnnnnnnn

Similarly,

),(),(),(2

112221212222 nnnnnn vvpvvpvvp

and

).,(),(),(2

112221212222 nnnnnn wwpwwpwwp

Hence

).(,),,,,,( 212121212222222 pfromRzyxzyxM nnnnnnn

Thus

).()()),(( 12121222 nnnn RRuup

Similarly,

)()()),(( 12121222 nnnn RRvvp

and

).()()),(( 12121222 nnnn RRwwp

Now,






).()(

))},(()),,(()),,((max{

)),(),,(),,((max)(

1212

122212221222

12221222122212

nn

nnnnnn

nnnnnnn

RR

wwpvvpuup

wwpvvpuupR

It follows that .0)( 12 nR So that .012 nR

Hence 1222 nn uu , 1222 nn vv and .1222 nn ww

Continuing in this way, we can conclude that un = un+k, vn = vn+k and wn = wn+k for all k > 0.

Thus, nu , nv and nw are Cauchy sequences.

Case(b): If R2n-1 is maximum, then

).(

)()()(

12

12122

n

nnn

R

RRR

(1)

Since is monotone increasing, we have R2n R2n-1.

Similarly R2n+1 R2n.

Continuing in this way we can conclude that nR is non -

increasing sequence of non - negative real numbers and must

converges to a real number r 0 say.

Suppose r > 0.

Letting n in (1), we get

),()()()( rrrr

a contradiction.

Hence r = 0. Thus

.0),(),,(),,(maxlim 111

nnnnnnn

wwpvvpuup

.0),(lim),(lim),(lim 111

nnn

nnn

nnn

wwpvvpuup

(2)

Hence from (p2), we get

.0),(lim),(lim),(lim

nnn

nnn

nnn

wwpvvpuup

(3)

From (2) and (3), and by definition of ps , we get

.0),(lim 1

nn

s

nuup (4)

.0),(lim 1

nn

s

nvvp (5)

and

.0),(lim 1

nn

s

nwwp

(6)

Now we prove that {u2n},{v2n} and {w2n} is a Cauchy

sequence in (X, ps). On contrary suppose that {u2n} or {v2n}

or {w2n} is not Cauchy.

There exist an > 0 and monotone increasing sequences of

natural numbers {2mk} and {2nk} such that nk > mk >k,

),(),,(),,(max 222222 kkkkkk nm

s

nm

s

nm

s wwpvvpuup

(7)

and

.),(),,(),,(max222 222222

kkkkkk nm

s

nm

s

nm

s wwpvvpuup

(8)

From (7) and (8), we obtain

.),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

222222

222222

222222

222222

222222

222222

111

121212

111

121212

222

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nm

s

nm

s

nm

s

nm

s

nm

s

nm

s

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

Letting k and then using (4), (5) and (6) ,we get

.),(),,(),,(maxlim 222222 kkkkkk nm

s

nm

s

nm

s

kwwpvvpuup (9)

Hence from definition of ps, we have

.

),(),(),(2

),,(),(),(2

),,(),(),(2

maxlim

222222

222222

222222

kkkkkk

kkkkkk

kkkkkk

nnmmnm

nnmmnm

nnmmnm

k

wwpwwpwwp

vvpvvpvvp

uupuupuup

By using (3) , we have






.2

),(),,(),,(maxlim 222222

kkkkkk nmnmnmk

wwpvvpuup (10)

From (7), we have

.),(),,(),,(max

),(),,(),,(max2

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

(11) ),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

222222

222222

222222

222222

222222

222222

222222

222222

111

111

111

111

111

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

nm

s

nm

s

nm

s

mm

s

mm

s

mm

s

nm

s

nm

s

nm

s

mm

s

mm

s

mm

s

mm

s

mm

s

mm

s

nm

s

nm

s

nm

s

mm

s

mm

s

mm

s

nm

s

nm

s

nm

s

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

Letting k ,using (4),( 5),( 6),(9) and (11), we have

.),(),,(),,(maxlim 222222 111

kkkkkk nm

s

nm

s

nm

s

kwwpvvpuup (12)

Hence, we have

.2

),(),,(),,(maxlim 222222 111

kkkkkk nmnmnmk

wwpvvpuup (13)

Again from (7), we have

. ),(),,(),,(max2

),(),,(),,(max

),(),,(),,(max2

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

(14) ),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

222222

222222

222222

222222

222222

222222

222222

222222

222222

222222

222222

222222

111

111

111

111

111

111

111

111111

111

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

nn

s

nn

s

nn

s

nm

s

nm

s

nm

s

mm

s

mm

s

mm

s

nn

s

nn

s

nn

s

nn

s

nn

s

nn

s

nm

s

nm

s

nm

s

mm

s

mm

s

mm

s

mm

s

mm

s

mm

s

nn

s

nn

s

nn

s

nm

s

nm

s

nm

s

mm

s

mm

s

mm

s

nm

s

nm

s

nm

s

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

wwpvvpuup

Letting k and then using (4), (5), (6), (9) and (14), we have

.),(),,(),,(maxlim111111 222222

kkkkkk nm

s

nm

s

nm

s

kwwpvvpuup (15)

Hence, we have

.2

),(),,(),,(maxlim111111 222222

kkkkkk nmnmnmk

wwpvvpuup (16)

Again from (7), we have






),(),,(),,(max

),(),,(),,(max

),(),,(),,(max

222222

222222

222222

111

111

kkkkkk

kkkkkk

kkkkkk

nn

s

nn

s

nn

s

nm

s

nm

s

nm

s

nm

s

nm

s

nm

s

wwpvvpuup

wwpvvpuup

wwpvvpuup

Letting k , using (4),(5) and (6), we have

).3(,),(),,(),,(maxlim2

),(),(),(2

),,(),(),(2

),,(),(),(2

maxlim

0),(),,(),,(maxlim

111

111

111

111

111

222222

222222

222222

222222

222222

fromwwpvvpuup

wwpwwpwwp

vvpvvpvvp

uupuupuup

wwpvvpuup

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkk

nmnmnmk

nnmmnm

nnmmnm

nnmmnm

k

nm

s

nm

s

nm

s

k

Thus,

.),(),,(),,(maxlim2 111 222222

kkkkkk nmnmnmk

wwpvvpuup

By the properties of ψ,

.),(,),(,),(maxlim

),(),,(),,(maxlim2

111

111

222222

222222

kkkkkk

kkkkkk

nmnmnmk

nmnmnmk

wwpvvpuup

wwpvvpuup

(17)

Now , we show that

.

),(

),,(

),,(

max),(),,(min2

1

22

22

22

2222

1

1

1

11

kk

kk

kk

kkkk

nm

nm

nm

nnmm

wwp

vvp

uup

uupuup

On contrary suppose that

.),(),,(min2

1

),(

),,(

),,(

max11

1

1

1

2222

22

22

22

kkkk

kk

kk

kk

nnmm

nm

nm

nm

uupuup

wwp

vvp

uup

as k , in above we get < 1.

It is a contradiction.

Hence

.),(),,(),,(max),(),,(min2

12222222222 11111 kkkkkkkkkk nmnmnmnnmm wwpvvpuupuupuup






From (2.1.1), we have

),(),(2

1,),(),(

2

1

,),(),(2

1),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

max-

),(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

max

)),,,,,(()),,,,,((

))),,(),,,((()),((

22222222

2222212

22222222

22222222

222222

222222212

22222222

22222222

222222222222

22222222

111111

111

1111

1111

111

111111

1111

1111

111111

1111

kkkkkkkk

kkkk

kkkkkkkk

kkkkkkkk

kkkkkk

kkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkkkkkk

kkkkkkkk

nmnmnmnm

nmnmnn

nnnnmmmm

mmnmnmnm

nmnmnm

nmnmnmnn

nnnnmmmm

mmnmmmnm

nnnmmmnnnmmm

nnnmmmnm

wwpwwpvvpvvp

uupuupwwp

vvpuupwwpvvp

uupwwpvvpuup

wwpvvpuup

wwpvvpuupwwp

vvpuupwwpvvp

uupwwpvvpuup

zyxzyxMzyxzyxM

zyxTzyxSpuup

since is increasing.

Similarly, we have

),(),(2

1,),(),(

2

1

,),(),(2

1),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

max

),(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

max

))),,(),,,((()),((

22222222

2222212

22222222

22222222

222222

222222212

22222222

22222222

22222222

111111

111

1111

1111

111

111111

1111

1111

1111

kkkkkkkk

kkkk

kkkkkkkk

kkkkkkkk

kkkkkk

kkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkk

nmnmnmnm

nmnmnn

nnnnmmmm

mmnmnmnm

nmnmnm

nmnmnmnn

nnnnmmmm

mmnmmmnm

nnnmmmnm

wwpwwpvvpvvp

uupuupwwp

vvpuupwwpvvp

uupwwpvvpuup

wwpvvpuup

wwpvvpuupwwp

vvpuupwwpvvp

uupwwpvvpuup

zyxTzyxSpvvp

and

),(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

max

))),,(),,,((()),((

222222

222222212

22222222

22222222

22222222

111

111111

1111

1111

1111

kkkkkk

kkkkkk

kkkkkkkk

kkkkkkkk

kkkkkkkk

nmnmnm

nmnmnmnn

nnnnmmmm

mmnmmmnm

nnnmmmnm

wwpvvpuup

wwpvvpuupwwp

vvpuupwwpvvp

uupwwpvvpuup

zyxTzyxSpwwp






),(),(2

1,),(),(

2

1

,),(),(2

1),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

max

22222222

2222212

22222222

22222222

111111

111

1111

1111

kkkkkkkk

kkkk

kkkkkkkk

kkkkkkkk

nmnmnmnm

nmnmnn

nnnnmmmm

mmnmnmnm

wwpwwpvvpvvp

uupuupwwp

vvpuupwwpvvp

uupwwpvvpuup

Hence from (17), we have

),(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

maxlim2

222222

222222212

22222222

22222222

111

111111

1111

1111

kkkkkk

kkkkkk

kkkkkkkk

kkkkkkkk

nmnmnm

nmnmnmnn

nnnnmmmm

mmnmmmnm

k

wwpvvpuup

wwpvvpuupwwp

vvpuupwwpvvp

uupwwpvvpuup

),(2

),(),(2

1,),(),(

2

1

,),(),(2

1),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

maxlim_

22222222

2222212

22222222

22222222

111111

111

1111

1111

t

wwpwwpvvpvvp

uupuupwwp

vvpuupwwpvvp

uupwwpvvpuup

kkkkkkkk

kkkk

kkkkkkkk

kkkkkkkk

nmnmnmnm

nmnmnn

nnnnmmmm

mmnmnmnm

k

.0 0)( ince ,2

0

),(),(2

1,),(),(

2

1

,),(),(2

1),,(

),,(),,(),,(),,(

),,(),,(),,(),,(

maxlim

22222222

2222212

22222222

22222222

111111

111

1111

1111

tforts

wwpwwpvvpvvp

uupuupwwp

vvpuupwwpvvp

uupwwpvvpuup

t

kkkkkkkk

kkkk

kkkkkkkk

kkkkkkkk

nmnmnmnm

nmnmnn

nnnnmmmm

mmnmnmnm

k

Is a contradiction.

Hence {u2n},{v2n} and {w2n} are Cauchy sequences in the metric space (X, ps).

Letting n,m in

),(),(),(),( 122212221212 mm

s

nn

s

mn

s

mn

s uupuupuupuup

we have

.0),(lim 1212,

mn

s

mnuup

Similarly we have 0),(lim 1212,

mn

s

mnvvp

and

.0),(lim 1212

,

mn

s

mnwwp

Thus{u2n+1},{v2n+1} and {w2n+1} are Cauchy sequences in the metric space (X, ps).

Hence {un},{vn} and {wn} are Cauchy sequences in the metric space (X, ps).

Hence, we have .0),(lim),(lim),(lim,,,

mn

s

mnmn

s

mnmn

s

mnwwpvvpuup

Now, from the definition of ps and from (3), we obtain






.0),(lim,

mnmn

uup (18)

.0),(lim,

mnmn

vvp (19)

and

.0),(lim,

mnmn

wwp (20)

Suppose g(X) is complete.

Since u2n = S(x2n , y2n, z2n)= gx2n+1, it follows {u2n} g(X) , {v2n} g(X) and {w2n} g(X) are Cauchy sequence in the

complete metric space (g(X), ps), it follows that {u2n},{v2n} and {w2n} are convergent in (g(X), p

s).

Thus ,0),(lim 2

n

s

nup

,0),(lim 2

n

s

nvp

and

,0),(lim 2

n

s

nwp for some , and g(X).

Since ,, g(X), there exist s, t, r X such that = gs , = gt and = gr.

Since {un},{vn} and {wn} are Cauchy sequences in X and {u2n} , {v2n}

and {w2n} , it follows that {u2n+1} , {v2n+1} and {w2n+1} .

From Lemma 1:5 (b), we have

(21) ).,(lim),(lim),(lim),(

,212 mn

mnn

nn

nuupupupp

(22) ),(lim),(lim),(lim),(,

212 mnmn

nn

nn

vvpvpvpp

and

(23) ).,(lim),(lim),(lim),(,

212 mnmn

nn

nn

wwpwpwpp

From (23),(22), (21), (20), (19) and (18) we obtain

(24) .0),(lim),(lim),( 212

nn

nn

upupp

(25) 0),(lim),(lim),( 212

nn

nn

vpvpp

and

(26) .0),(lim),(lim),( 212

nn

nn

wpwpp

Now we claim that, for each n 1, atleast one of the following assertions holds.

),( ),,(, ),(max),(2

1121212212 nnnnn wpvpupuup

or

),( ),,(, ),(max),(2


Suppose to the contrary that

),( ),,(, ),(max),(2


and

),( ),,(, ),(max),(2







.

),(

),(),(2

1

),p(-),( ),(),(

212

122212

212212

nn

nnnn

nnnn

uup

uupuup

upupuup

is a contradiction .

Hence claim holds.

Sub case(a) :

Claim : Show that ).,,( ),,( ),,,( tsrTandsrtTrtsT

Suppose ),( ),,(, ),(max),(2


.

Suppose ).,,( ),,(or ),,( tsrTorsrtTrtsT

From (2.1.1), we get

.

)),,(,()),,(,(2

1

,)),,(,()),,(,(2

1

,)),,(,()),,(,(2

1

)),,,(,()),,,(,(

)),,,(,()),,,(,(

)),,,(,()),,,(,(

),,(),,p(f),,p(f

mx

)),,(,()),,,(,()),,,(,(

)),,,(,()),,,(,()),,,(,(

)),,,(,()),,,(,(

)),,,(,()),,,(,(

)),,,(,()),,,(,(

),,(),,p(f),,p(f

max

))),,(),,,(((

2222

2222

2222

2222

22222222

222

222222222

222

2222

22222222

222

222

nnnn

nnnn

nnnn

nnnn

nnnnnnnn

nnn

nnnnnnnnn

nnn

nnnn

nnnnnnnn

nnn

nnn

yxzSptsrTfzp

xzySpsrtTfyp

zyxSprtsTfxp

tsrTpsrtTp

rtsTpyxzSfzp

xzySfypzyxSfxp

fzpyx

yxzSpxzySpzyxSp

tsrTfzpsrtTfyprtsTfxp

tsrTpsrtTp

rtsTpyxzSfzp

xzySfypzyxSfxp

fzpyx

rtsTzyxSp

Letting n .

.)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max))),,(,((

tsrTpsrtTprtsTp

tsrTpsrtTprtsTprtsTp

Similarly ,

)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max))),,(,((

tsrTpsrtTprtsTp

tsrTpsrtTprtsTpsrtTp

and

)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max))),,(,((

tsrTpsrtTprtsTp

tsrTpsrtTprtsTptsrTp

Now,






.)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max

))),,(,(())),,,(,(())),,,(,((max

))),,(,()),,,(,()),,,(,((max

tsrTpsrtTprtsTp

tsrTpsrtTprtsTp

tsrTpsrtTprtsTp

tsrTpsrtTprtsTp

It follows that

.0)),,(,()),,,(,()),,,(,(max tsrTpsrtTprtsTp

Therefore .),,( ),,(,),,( grtsrTandgtsrtTgsrtsT

Since (g, T) is weakly compatible, we have

.),,( ),,(,),,( gTandgTgT

We claim that

.),,( ),,(,),,( TandTT

.),,( ),,( ),,(

thatsuppose us

TorTorT

Let

.),(),,(),,(max)),,(,()),,,(,(min2

1 Since 2222222 gfzpgfypgfxpTgpzyxSfxp nnnnnnn


)),,(,()),,,(,()),,,(,(

)),,,(,()),,,(,()),,,(,(

)),,,(,()),,,(,(

)),,,(,()),,,(,(

)),,,(,()),,,(,(

),,(),g,p(f),g,p(f

max

))),,(),,,(((

222222222

222

2222

22222222

222

222

nnnnnnnnn

nnn

nnnn

nnnnnnnn

nnn

nnn

yxzSgpxzySgpzyxSgp

TfzpTfypTfxp

TgpTgp

TgpyxzSfzp

xzySfypzyxSfxp

gfzpyx

TzyxSp

)),,(,()),,(,(2

1,)),,(,()),,(,(

2

1

,)),,(,()),,(,(2

1)),,,(,(

)),,,(,()),,,(,()),,,(,()),,,(,(

)),,,(,(),,(),g,p(f),g,p(f

max

22222222

2222

22222222

2222222

nnnnnnnn

nnnn

nnnnnnnn

nnnnnnn

yxzSgpTfzpxzySgpTfyp

zyxSgpTfxpTgp

TgpTgpyxzSfzpxzySfyp

zyxSfxpgfzpyx

Letting n .

.)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max))),,(,((

TpTpTp

TpTpTpTp

Similarly,

)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max))),,(,((

TpTpTp

TpTpTpTp

and

)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max))),,(,((

TpTpTp

TpTpTpTp

Now,






.)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max

))),,(,(())),,,(,(())),,,(,((max

)),,(,()),,,(,()),,,(,(max

TpTpTp

TpTpTp

TpTpTp

TpTpTp

It follows that.

0)),,(,()),,,(,()),,,(,(max TpTpTp .

Hence

.),,(and ),,(,),,( gTgTgT (27)

Since T(X × X× X) f(X), then there exist a,b,c X such that

.),,(and ),,(,),,( fcTfbTfaT

Claim: S(a, b,c) = fa = , S(b, c,a) = fb = and S(c, a, b) = fc = .

Suppose that S(a, b,c) or S(b, c,a) or S(c, a, b) .

Clearly,

.),(),,(),,(max)),,(,()),,,(,(min2

1 gfcpgfbpgfapTgpcbaSfap


).26()25(),24(,)),,(,()),,,(,()),,,(,(max

)),,(,(),(2

1,)),,(,(),(

2

1

,)),,(,(),(2

1),,(

),,(),,()),,,(,()),,,(,(

)),,,(,(),,(),,(),,(

max),,,,,(

)).,,,,,((

)),,,,,((

)),,(),,,((())),,,(((

andfrombacSpacbSpcbaSp

bacSppacbSpp

cbaSppp

ppbacSpacbSp

cbaSpppp

cbaM

cbaM

cbaM

TcbaSpfacbaSp

Hence

).)),,(,()),,,(,()),,,(,((max

))),,(,()),,,(,()),,,(,((max ))),,,(((

bacSpacbSpcbaSp

bacSpacbSpcbaSpcbaSp

Similarly,

))),,(,()),,,(,()),,,(,((max

))),,(,()),,,(,()),,,(,((max ))),,,(((

bacSpacbSpcbaSp

bacSpacbSpcbaSpacbSp

and

).)),,(,()),,,(,()),,,(,((max

))),,(,()),,,(,()),,,(,((max ))),,,(((

bacSpacbSpcbaSp

bacSpacbSpcbaSpbacSp

Now,

.)),,(,()),,,(,()),,,(,(max

)),,(,()),,,(,()),,,(,(max

))),,(,(())),,,(,(())),,,(,((max

)),,(,()),,,(,()),,,(,(max

bacSpacbSpcbaSp

bacSpacbSpcbaSp

bacSpacbSpcbaSp

bacSpacbSpcbaSp

It follows that

.0)),,(,()),,,(,()),,,(,(max bacSpacbSpcbaSp

Hence .),,( ,),,(,),,( fcbacSandfbacbSfacbaS

Since (S,f) is weakly compatible.

Thus .),,( ,),,(,),,( fSandfSfS






Suppose .),,(or ),,( ),,( SSorS

Clearly,

.),(),,(),,(max)),,(,()),,,(,(min2

1 grfpgtfpgsfprtsTgspSfp


)).,,,,,((

)),,,,,((

)),,(),,,((()),((

rtsM

rtsM

rtsTSpfp

).),(),,(),,((max

)),(),,(),,((max)),((

fpfpfp

fpfpfpfp

Hence

Similarly,

)),(),,(),,((max

)),(),,(),,((max)),((

fpfpfp

fpfpfpfp

and

).),(),,(),,((max

)),(),,(),,((max)),((

fpfpfp

fpfpfpfp

Now,

).),(),,(),,((max

)),(),,(),,((max

)),(()),,(()),,((max),(),,(),,(max

fpfpfp

fpfpfp

fpfpfpfpfpfp

It follows that.

.0),(),,(),,(max fpfpfp

Therefore

.),,( ),,(,),,( fSandfSfS (28)

From (27) and (28) , ),,( is common tripled fixed point of S, T, f and g. Suppose ),,( is another common

tripled fixed point of S, T, f and g.

Clearly

.

),(

),,(

),,(

max)),,(,()),,,(,(min2

1

gfp

gfp

gfp

TgpSfp







)).,,,,,((

)),,,,,((

)),,(),,,((()),((

M

M

TSpp

Hence

).),(),,(),,((max

)),(),,(),,((max)),((

ppp

pppp

Similarly,

)),(),,(),,((max

)),(),,(),,((max)),((

ppp

pppp

and

).),(),,(),,((max

)),(),,(),,((max)),((

ppp

pppp

Now,

).),(),,(),,((max

)),(),,(),,((max

)),(()),,(()),,((max)),(),,(),,((max

ppp

ppp

pppppp

It follows that.

.0),(),,(),,(max ppp

Therefore

. , and

Therefore ),,( is a unique common tripled fixed point of S, T, f and g.

Now we claim that . , and

Suppose .or or

Clearly

.

),(

),,(

),,(

max)),,(,()),,,(,(min2

1

gfp

gfp

gfp

TgpSfp


)).,,,,,((

)),,,,,((

)),,(),,,((()),((

M

M

TSpp






).),(),,(),,((max

)),(),,(),,((max)),((

ppp

pppp

Similarly,

)),(),,(),,((max

)),(),,(),,((max)),((

ppp

pppp

and

).),(),,(),,((max

)),(),,(),,((max)),((

ppp

pppp

Now,

).),(),,(),,((max

)),(),,(),,((max

)),(()),,(()),,((max)),(),,(),,((max

ppp

ppp

pppppp

It follows that,

.0),(),,(),,(max ppp

Hence . and ,

Therefore ),,( is a unique common tripled fixed point of S, T, f and g. Sub case (b): There exist a unique common

tripled fixed point of S, T, f and g when ),( ),,(, ),(max),(2

1222212 nnnnn wpvpupuup holds.

References

[1] Abdeljawad T, Karapınar E, Tas K., Existence and

uniqueness of a common fixed point on partial metric

spaces, Appl. Math. Lett. 2011, 24 (11), 1894 - 1899.

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extensions of Banach’s contraction principle to partial

metric spaces, Appl. Math. Letters ,

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Applications 1994. vol. 728, pp. 183-197, Annals of the

New York Academy of Sciences, 1994.

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condition in partial metric spaces. Bulletin of

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[11] Romaguera S. A Kirk type characterization of

completeness for partial metric spaces. Fixed Point

Theory and Applications 2010, Article ID 493298, 6

pages.

[12] Valero O. On Banach fixed point theorems for partial

metric spaces. Applied General Topology 2005, 6 (2),

229 - 240.

[13] Berinde, V, Borcut, M Tripled fixed point theorems for

contractive type mappings in partially ordered metric

spaces.

[14] Nonlinear Anal. 74(15), 4889-4897 (2011).

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unique common 3-tupled fixed point theorem for

contractions in partial metric spaces, Mathematica

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