Fixed Point Results of Contractive Mappings by Altering Distances and C Class Functions in b Dislocated Metric Spaces

ISSN Online: 2333-9721 ISSN Print: 2333-9705

DOI: 10.4236/oalib.1104657 Jul. 19, 2018 1 Open Access Library Journal

Fixed Point Results of Contractive Mappings by

Altering Distances and C-Class Functions in

b-Dislocated Metric Spaces

Jani Dine

1

, Kastriot Zoto

1

, Arslan H. Ansari

2

1_{Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, Albania} 2_{Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran}

Abstract

In this work, we recall definition of functions called as C-class and use the concepts of dislocated metric, b-dislocated metric, altering distance function. We prove some coincidence, fixed and common fixed point results for two pairs of weakly compatible mappings under f −

(

ψ φ, ,s

)

_-contractive

condi-tions and contractive condicondi-tions depended on another function T. Our theo-rems extend and generalize related results in the literature.

Subject Areas

Functional Analysis, Mathematical Analysis

Keywords

(

, ,

)

f − ψ φ s Contraction, C-Class Functions, bd-Dislocated Metric Space,

Coincidence Point, Common Fixed, Altering Distance Functions, Weakly Compatible Maps

1. Introduction

The study of metric fixed point theory in dislocated metric spaces was consi-dered by P. Hitzler and A. K. Seda in [1] who introduced this metric as a genera-lization of usual metric, and generalized the Banach contraction principle on this space. Since then a lot of papers have been written on this topic treating the problem of existence and uniqueness of fixed points for mappings satisfying dif-ferent contractive conditions, see [2]-[14]. N. Hussain et al. in [15] introduced the b-dislocated metric spaces associated with some topological aspects and properties. These spaces can be seen as generalizations of dislocated metric

How to cite this paper: Dine, J., Zoto, K. and Ansari, A.H. (2018) Fixed Point Results of Contractive Mappings by Altering Distances and C-Class Functions in b-Dislocated Metric Spaces. Open Access Library Journal, 5: e4657.

https://doi.org/10.4236/oalib.1104657

Received: May 13, 2018 Accepted: July 16, 2018 Published: July 19, 2018

Copyright © 2018 by authors and Open Access Library Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/oalib.1104657 2 Open Access Library Journal

spaces and also as generalization of b-metric space introduced by Bakhtin in [16]

and extensively used by Czerwik in [8]. Recently, there are many papers on exis-tence and uniqueness of fixed point and common fixed point for one, two or more mappings under different types of contractive conditions in the setting of dislocated spaces and b-dislocated metric spaces.

Since altering distance functions were introduced by Khan et al. in [17], the study of the existence of fixed points of contractive maps in metric spaces and generalized metric spaceshas a lot of interest for many authors which are based on this category of functions (see [17]-[23]). In September 2014, a class of func-tions called as C-class is presented by A. H. Ansari, see in [24] [25] and is im-portant, see example 2.15.

The present paper is organized in two sections. Using concepts mentioned above, in the first section, we develop some coincidence and common fixed point theorems (existence and uniqueness) for two pairs of weakly compatible map-pings in the framework of bd-dislocated metric space, using weak generalized

(

, ,

)

f − ψ φ s _{contractive conditions. In the second section, we prove common}

fixed point theorems for a pair of mappings using generalized f −

(

ψ φ, ,s

)

contractive condition and the concept of T-contractions. The related results ge-neralize and improve various theorems in recent literature.

2. Preliminaries

Consistent with [1] and [15], the following definitions, notations, basic lemma and remarks will be needed in the sequel.

Definition 2.1 [1] Let X be a nonempty set and a mapping d X Xl: × →

[

0,∞

)

is called a dislocated metric (or simply dl-metric) if the following conditions

hold for any x y z X, , ∈ _: 1) If d x y_l

(

,

)

=0, then x y=

2) d x yl

(

,

)

=d y xl

(

,

)

3) d x y_l

(

,

)

≤d x z_l

( )

, +d z y_l

( )

,

The pair

(

X d, l

)

is called a dislocated metric space (or d-metric space for

short). Note that for x y= , d x y_l

(

,

)

may not be 0.

Definition 2.2 [15] Let X be a nonempty set and a mapping b X Xd: × →

[

0,∞

)

is called a b-dislocated metric (or simply bd-dislocated metric) if the following

conditions hold for any x y z X, , ∈ _and s≥1_:

1) If b x y_d

(

,

)

=0, then x y=

2) b x yd

(

,

)

=b y xd

(

,

)

3) b x yd

( )

, ≤s b x z b z y d

( )

, + d

( )

, 

The pair

(

X b, d

)

is called a b-dislocated metric space. And the class of

b-dislocated metric space is larger than that of dislocated metric spaces, since a

b-dislocated metric is a dislocated metric when s=1_.

Example 2.3 If X R= _{, then} d x y_l

(

,

)

= x+ y _{defines a dislocated metric}

on X.

DOI: 10.4236/oalib.1104657 3 Open Access Library Journal

1) a Cauchy sequence if, for given ε >0_{, there exists} n₀∈N _{such that for} all m n n, ≥ 0, we have or _{n m}lim_{, →∞}d x xl

(

n, m

)

=0;

2) convergent with respect to dl if there exists x X∈ such that

(

,

)

0

l n

d x x → as n→ ∞. In this case, x is called the limit of

( )

xn and we

write xn→x.

A dl-metric space X is called complete if every Cauchy sequence in X

con-verges to a point in X.

In [15], it was shown that each bd-metric on X generates a topology τbd

whose base is the family of open bd-balls B xbd

( )

,ε =

{

y X b x y∈ : d

( )

, <ε

}

.

Also in [15], there are presentedsome topological properties of bd-metric spaces.

Definition 2.5 [15] Let

(

X b, _d

)

be a bd-metric space, and

{ }

xn be a

se-quence of points in X. A point x X∈ _{is said to be the limit of the sequence}

{ }

xn if lim_n→∞b x xd

(

n,

)

=0 and we say that the sequence

{ }

xn is bd

-convergent to x and denote it by xn→x as n→ ∞.

The limit of a bd-convergent sequence in a bd-metric space is unique ([15],

Proposition 1.27).

Definition 2.6 [15] A sequence

{ }

xn in a bd-metric space

(

X b, d

)

is

called a bd-Cauchy sequence if, given ε >0, there exists n0∈N such that for

all n m n, > 0 , we have b x xd

(

n, m

)

<ε or _{n m,}lim_→∞b x xd

(

n, m

)

=0 . Every bd

-convergent sequence in a bd-metric space is a bd-Cauchy sequence.

Remark 2.7 The sequence

{ }

x

n in a bd-metric space

(

X b, d

)

is called a d

b -Cauchy sequence if _{n m,}lim_→∞b x xd

(

n, n p+

)

=0 for all p N∈ ∗.

Definition 2.8 [15] A bd-metric space

(

X b, d

)

is called complete if every d

b -Cauchy sequence in X is bd-convergent.

Example 2.9 If X R₌ +_∪

{ }

0 _{, then}

( ) (

_,

)

2

d

b x y = x y+ defines a b-dislocated metric on X _{with parameter} s=2_.

Example 2.10 Let X R₌ +_∪

{ }

0 _{and any constant}

α

_>

₀

_{. Define function} :

l

d X X_{× →}R+ _by

(

_,

)

(

)

l

d x y =α x y+ . Then, the pair

(

X d, l

)

is a dislocated

metric space.

If Fx Sx= _{for some} x X∈ _{, then x is called the coincidence point of F and S.}

Furthermore, if the mappings commute at each coincidence point, then such mappings are called weakly compatible [4].

Definition 2.11 [17] The altering distances functions ψ _and ϕ _{are defined}

as

[

)

[

)

( )

{

ψ : 0, 0, / is continuous,nondecreasing,andψ ψ t 0 ift 0

}

Ψ = ∞ → ∞ = =

[

)

[

)

( )

{

φ: 0, 0, / is lower semicontinuous,andφ φ t 0 ift 0

}

Φ = ∞ → ∞ = =

The following lemmas are used to prove our results.

Lemma 2.12 Let

(

X b, d

)

be a b-dislocated metric space with parameter s≥1.

Then

1) If b x yd

(

,

)

=0 then b x xd

( )

, =b y yd

(

,

)

=0;

DOI: 10.4236/oalib.1104657 4 Open Access Library Journal

(

)

(

1 1

)

lim d n, n lim d n , n 0

n→∞b x x =n→∞b x+ x+ = ;

3) If x y≠ , then b x yd

(

,

)

>0;

Proof. It is clear.

Lemma 2.13 [15] Let

(

X b, d

)

be a b-dislocated metric space with parameter

1

s≥ _{. Suppose that}

{ }

x_n and

{ }

yn are bd-convergent to x y X, ∈ ,

respec-tively. Then we have

( )

(

)

(

)

2

( )

2

1 _{, lim inf , lim sup , ,}

d _n d n n _n d n n d

b x y b x y b x y s b x y

s ≤ →∞ ≤ →∞ ≤

In particular, if b x y_d

(

,

)

=0_{, then we have} lim_n→∞b x yd

(

n, n

)

= =0 b x yd

( )

, .

Moreover, for each z X∈ _{, we have}

( )

(

)

(

)

( )

1 _{, lim inf , lim sup , ,}

d _n d n _n d n d

b x z b x z b x z sb x z

s ≤ →∞ ≤ →∞ ≤

In particular, if b x z_d

( )

, =0, then we have lim_n→∞b x zd

(

n,

)

= =0 b x zd

( )

, .

Definition 2.14. [24] [25] We say that a function f: 0,

[

∞ →

)

2 R is called a

C-class function if it is continuous and satisfies the following properties.

( )

( )

[

)

( )

1) ,

2) , 0 or 0 for all , 0,

3) 0,0 0

f s t s

f s t s s t s t

f

≤

= ⇒ = = ∈ ∞

=

We denote C-class functions as C.

Example 2.15 [24] [25] The following functions f : 0,

[

∞ →

)

2 R are ele-ments of C, for all s t, ∈

[

0,∞

)

:

1) f s t

( )

, = −s t f s t,

( )

, = ⇒ =s t 0

2)

( )

, ,

( )

, 0

1

s t

f s t f s t s t

t

−

= = ⇒ =

+

3)

( )

, ,

( )

, 0

1

st

f s t f s t s s

t

= = ⇒ =

+

4)

( )

, ,

( )

, 0 or 0

1

s

f s t f s t s s t

t

= = ⇒ = =

+

5)

( )

, log , 1,

( )

, 0 or 0

1

s

t a

f s t a f s t s s t

t

+

= > = ⇒ = =

+

For t=1, we have

( )

,1 ln1 , , ,1

( )

0

2 s

a

f s = + a e f s> = ⇒ =s s

6) f s t

( ) (

, = s k+

)

1+1t −k k, >1,f s t

( )

, = ⇒ =s t 0

7) f s t

( )

, =slog_{a t}+ a a, >1,f s t

( )

, = ⇒ =s s 0 ort=0

8) f s t

( )

, =ms,0< <m 1;f s t

( )

, = ⇒ =s s 0

DOI: 10.4236/oalib.1104657 5 Open Access Library Journal

3. Main Results

Before we give the main result we denote with letter N x y

(

,

)

the following set

( )

, max

{

d

(

,

) (

, d ,

) (

, d ,

) (

, d ,

) (

, d ,

)

}

N x y = b Sx Ty b Fx Sx b Gy Ty b Sx Gy b Fx Ty (3.1.1)

for all x y X, ∈ .

Motivated by the works of [15] [21]-[29] we extend the concept of

(

ψ φ,

)

-weakly contractive maps to four maps in a b-dislocated metric space, giving the following definition.

Definition 3.1 Let F G S T, , , be four self maps of a b-dislocated metric space

(

X b, _d

)

with parameter s≥1_{. If there exists} ψ∈ Ψ, ϕ∈Φ and f C∈ such that

(

)

(

₂ 4 _,

)

(

(

( )

_,

)

_,

(

( )

_,

)

)

d

s b Fx Gy f N x y N x y

ψ ≤ ψ φ _(A)

for all x y X, ∈ _{, where} N x y

(

,

)

is defined as in (3.1.1) then F G S, , and T

are said to satisfy a generalized f −

(

ψ φ, ,s

)

weakly contractive condition. Theorem 3.2 Let

(

X b, _d

)

be a b-dislocated metric space with parameter

1

s≥ _and S T F G X, , , : →X are self-mappings such that (a) F X

( )

⊂T X

( )

,

( )

( )

G X ⊂S X _{and satisfy generalized} f −

(

ψ φ, ,s

)

_{weakly contractive}

condi-tion. If one of F X S X T X

( ) ( ) ( )

, , or G X

( )

is a complete subspace of X, then

(

F S,

)

and

(

G T,

)

have a point of coincidence in X. Moreover if sup-pose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , ,

have a unique common fixed point.

Proof. Let x0 be an arbitrary point in X. Since F X

( )

⊂T X

( )

we can

choose x X1∈ such that y0= fx0=Tx1. And since G X

( )

⊂S X

( )

corres-ponding to x X1∈ we can choose x2∈X such that y Gx Sx1= 1= 2.

Contin-uing the same process we obtain sequences

{ }

xn and

{ }

yn in X such that:

2n 2n 2 1n ; 2 1n 2 1n 2 2n

y =Fx =Tx + y + =Gx + =Sx + for all n=0,1,2,

We consider following steps:

Step 1. If y2n=y2 1n+ (that means b y yd

(

2n, 2 1n+

)

=0) for some n, then 2 1n 2 1n

Gx + =Tx + . Hence x2 1n+ is a coincidence point of G and T. Using definition

of N x y

(

,

)

and lemma 2.12 we have,

(

)

(

) (

) (

)

{

(

) (

)

}

(

) (

) (

)

{

(

) (

)

}

2 2 2 1

2 2 2 1 2 2 2 2 2 1 2 1

2 2 2 1 2 2 2 1

2 1 2 2 2 2 1 2 1 2

2 1 2 1 2 2 2

,

max , , , , , ,

, , ,

max , , , , , ,

, , ,

n n

d n n d n n d n n

d n n d n n

d n n d n n d n n

d n n d n n

N x x

b Sx Tx b Fx Sx b Gx Tx

b Sx Gx b Fx Tx

b y y b y y b y y

b y y b y y

+ +

+ + + + + +

+ + + +

+ + + +

+ + +

=

=

(

) (

) (

)

{

(

)

(

)

(

)

}

(

)

(

)

{

}

(

)

2 1 2 2 2 2 1 2 1 2

2 1 2 2 2 2 1 2 1 2

2 2 2 1 2 2 2 1

2 2 2 1

max , , , , , ,

2 , , , ,

max 0, , ,0,0, ,

,

d n n d n n d n n

d n n d n n d n n

d n n d n n

d n n

b y y b y y b y y

sb y y s b y y b y y

b y y sb y y

sb y y

+ + + +

+ + + +

+ + + + + +

≤

 + 

 

DOI: 10.4236/oalib.1104657 6 Open Access Library Journal

Thus N x

(

2 2n+ ,x2 1n+

)

≤sb yd

(

2 2n+ ,y2 1n+

)

(3.2.1)

Using condition (A) and property of C-class, we have that

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

2 1 2 2 2 1 2 2

2 2 2 1 4

2 2 2 1

2 2 2 1 2 2 2 1

2 2 2 1

, ,

2 ,

2 ,

, , ,

,

d n n d n n

d n n

d n n

n n n n

n n

sb y y sb Gx Fx

sb Fx Gx

s b Fx Gx

f N x x N x x

N x x

ψ ψ ψ ψ ψ φ ψ + + + + + + + + + + + + + + = = ≤ ≤ ≤

By property of ψ _{we have,}

(

2 2, 2 1

)

(

2 2, 2 1

)

d n n n n

sb y + y + ≤N x + x + (3.2.2)

As a result we get,

(

2 2n , 2 1n

)

d

(

2 1n , 2 2n

)

N x + x + =sb y + y + .

Again from contractive condition of theorem have,

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

2 2 2 1 2 2 2 1

4

2 2 2 1

2 2 2 1 2 2 2 1

2 1 2 2 2 1 2 2

2 1 2 2

, ,

2 ,

, , ,

, , ,

,

d n n d n n

d n n

n n n n

d n n d n n

d n n

sb y y sb Fx Gx

s b Fx Gx

f N x x N x x

f sb y y sb y y

sb y y

ψ ψ ψ ψ φ ψ φ ψ + + + + + + + + + + + + + + + + = ≤ ≤ = ≤

The inequality above implies,

(

)

(

)

(

(

)

)

(

d 2 1n , 2 2n , d 2 1n , 2 2n

)

(

d

(

2 1n , 2 2n

)

)

f ψ sb y ₊ y ₊ φ sb y ₊ y ₊ =ψ sb y ₊ y ₊

By property of function f of C-class we obtain

(

)

(

sb yd 2 1n ,y2 2n

)

0

ψ + + = or φ

(

sb yd

(

2 1n+,y2 2n+

)

)

=0.

And also by property of ψ φ, _{we get} b y_d

₍

_{2 1n}+,y_{2 2n}+

₎

=0 so that y2 1n+ =y2 2n+

and then Fx2 2n+ =Sx2 2n+ . Also x2 2n+ is a coincidence point of F and S.

Step 2. Suppose y2n≠y2 1n+ that means b y yd

(

2n, 2 1n+

)

>0 for all n by

condi-tion (3.1.1) we have:

(

)

{

(

) (

) (

)

(

) (

)

}

(

) (

) (

)

{

(

) (

)

}

2 2 1 2 2 1 2 2 2 1 2 1

2 2 1 2 2 1

2 1 2 2 2 1 2 1 2

2 1 2 1 2 2

, max , , , , ,

, , ,

max , , , , ,

, , ,

n n d n n d n n d n n

d n n d n n

d n n d n n d n n

d n n d n n

N x x b Sx Tx b Fx Sx b Gx Tx

b Sx Gx b Fx Tx

b y y b y y b y y

b y y b y y

+ + + + + + − − + − + = =

(

) (

) (

)

{

(

)

(

)

(

)

}

(

)

(

)

{

}

2 1 2 2 2 1 2 1 2

2 1 2 2 2 1 2 2 1

2 2 1 2 1 2

max , , , , ,

, , ,2 ,

max 2 , ,2 ,

d n n d n n d n n

d n n d n n d n n

d n n d n n

b y y b y y b y y

s b y y b y y sb y y

sb y y sb y y

− − + − + + + − ≤  +    ≤

DOI: 10.4236/oalib.1104657 7 Open Access Library Journal

Also from condition of theorem we have:

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

4

2 2 1 2 2 1

4

2 2 1

2 2 1 2 2 1

2 2 1

2 , 2 ,

2 ,

, , ,

,

d n n d n n

d n n

n n n n

n n

sb y y s b y y

s b Fx Gx

f N x x N x x

N x x

ψ ψ

ψ

ψ φ

ψ

+ +

+

+ +

+

≤

=

≤ ≤

By property of function ψ _{we have}

(

2 2 1

)

(

2 2 1

)

2sb y y_{d n}, _n+ ≤N x x_n, _n+ (3.2.4)

From (3.2.3) and (3.2.4) we get

(

2n, 2 1n

)

2 d

(

2n, 2 1n

)

N x x + = sb y y + (3.2.5)

Also from condition of theorem and (3.2.5) we have,

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

4

2 2 1 2 2 1

4

2 2 1

2 2 1 2 2 1

2 2 1 2 2 1

2 2 1

2 , 2 ,

2 ,

, , ,

2 , , 2 ,

2 ,

d n n d n n

d n n

n n n n

d n n d n n

d n n

sb y y s b y y

s b Fx Gx

f N x x N x x

f sb y y sb y y

sb y y

ψ ψ

ψ

ψ φ

ψ φ

ψ

+ +

+

+ +

+ +

+

≤

=

≤

= ≤

The above inequality implies:

(

)

(

)

(

(

(

)

)

(

(

)

)

)

(

)

(

)

2 2 1 2 2 1 2 2 1

2 2 1

2 , 2 , , 2 ,

2 ,

d n n d n n d n n

d n n

sb y y f sb y y sb y y

sb y y

ψ ψ φ

ψ

+ + +

+

≤ ≤

which means

(

)

(

)

(

(

)

)

(

2 d 2n, 2 1n , 2 d 2n, 2 1n

)

(

2 d

(

2n, 2 1n

)

)

f ψ sb y y ₊ φ sb y y ₊ =ψ sb y y ₊ .

From property of C-class we obtain

(

)

(

2sb y yd 2n, 2 1n

)

0 or

(

2sb y yd

(

2n, 2 1n

)

)

0

ψ + = φ + = .

So we have b y yd

(

2n, 2 1n+

)

=0 that is a contradiction since we suppose

(

2 , 2 1

)

0

d n n

b y y + > .

So we have b y yd

(

2n, 2 1n+

)

≤b yd

(

2 1n−,y2n

)

.

In a similar way as above we have b yd

(

2 1n+,y2 2n+

)

≤b y yd

(

2n, 2 1n+

)

. As a result

the sequence

{

b y yd

(

n, n+1

)

}

is non increasing and bounded below. And so there

exists l≥0 such that,

(

1

)

(

1

)

lim d n, n lim n, n

n→∞b y y+ =n→∞N x x+ =l.

Suppose that l>0. Since ψ _{is continuous and} φ is lower semi continuous we have:

( )

l lim_n

(

N x x

(

n, n1

)

)

and

( )

l lim inf_n

(

N x x

(

n, n1

)

)

ψ ψ ₊ φ φ ₊

→∞ →∞

= ≤ _(3.2.6)

DOI: 10.4236/oalib.1104657 8 Open Access Library Journal

(

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

4

2 2 2 1 2 2 2 1

2 2 2 1 2 2 2 1

2 2 2 1

2 , 2 ,

, , ,

,

d n n d n n

n n n n

n n

sb y y s b y y

f N x x N x x

N x x

ψ ψ

ψ φ

ψ

+ + + +

+ + + +

+ +

≤

≤ ≤

(3.2.7)

taking the upper limit as n→ ∞ in (3.2.7) and using (3.2.6) we have that,

( )

(

( ) ( )

)

( )

( ) ( )

(

)

( )

( )

( )

2 2 , 2 2

2 , 2 2

2 0 or 2 0.

sl f sl sl sl

f sl sl sl

sl sl

ψ ψ φ ψ

ψ φ ψ

ψ φ

≤ ≤

⇒ =

⇒ = =

(3.2.8)

From (3.2.8) and property of ψ _{we get} l=0 that is a contradiction. Hence

(

1

)

(

1

)

lim d n, n lim n, n 0

n→∞b y y+ =n→∞N x x+ = (3.2.9)

Now we prove that

{ }

yn is a bd-Cauchy sequence. Assume the contrary that

{ }

y2n is not a bd-Cauchy sequence. Then there exists ε >0 for which we can

find subsequences

{ }

y2mk and

{ }

y2nk of

{ }

y2n so that nk is the smallest

index for which 2nk >2mk >k, that

(

2 k, 2k

)

d m n

b y y ≥ε _(3.1.10)

and b yd

(

2mk,y2nk−2

)

<ε (3.2.11)

From property c) of definition 2.2 we have:

(

)

(

)

(

)

(

)

(

)

(

)

2 2 2 2 2 2 2 2

2 2

2 2 2 2 2 2 1 2 1 2

, , ,

, , ,

k k k k k k

k k k k k k

d m n d m n d n n

d m n d n n d n n

b y y sb y y sb y y

sb y y s b y y s b y y

ε − −

− − − −

≤ ≤ +

≤ + + (3.2.12)

Taking the upper limit as k→ ∞ in (3.2.12) and using result (3.129) and (3.2.11), we get

(

2 2

)

lim sup d m_k, n_k

k b y y s

ε ε

→∞

≤ ≤ (3.2.13)

Also we have

(

2 k, 2k

)

(

2 k, 2k 1

)

(

2k 1, 2k

)

d m n d m n d n n

b y y sb y y sb y y

ε≤ ≤ − + − (3.2.14)

Hence taking the upper limit in above inequality, we obtain

(

2 2 1

)

limsup d m_k, n_k

k b y y

s

ε

− →∞

≤ (3.2.15)

Again from property c) of definition 2.2, we have

(

2 k, 2 k 1

)

(

2 k, 2k

)

(

2 k, 2k 1

)

d m n d m n d n n

b y y − ≤sb y y +sb y y − (3.2.16)

Thus from 3.2.9; 3.2.15 we have

(

)

2

2 2 1

lim sup d m_k, n_k

k→∞ b y y − ≤εs (3.2.17)

As a result,

(

)

2

2 2 1

lim sup d m_k, n_k

k b y y s

s

ε _ε

− →∞

DOI: 10.4236/oalib.1104657 9 Open Access Library Journal

Similarly,

(

2 k, 2k

)

(

2 k, 2 k 1

)

(

2 k 1, 2k

)

d m n d m m d m n

b y y sb y y sb y y

ε≤ ≤ ₊ + _{+ (3.2.19)}

Taking the upper limit in (3.2.19) and using 3.2.9, we get

(

2 1 2

)

limsup d m_k , n_k

k b y y

s

ε

+ →∞

≤ (3.2.20)

Similarly,

(

)

(

)

(

)

(

)

(

)

(

)

2 2 2 2 1 2 1 2

2 2

2 2 1 2 1 2 1 2 1 2

, , ,

, , ,

k k k k k k

k k k k k k

d m n d m m d m n

d m m d m n d n n

b y y sb y y sb y y

sb y y s b y y s b y y

ε _{+ +}

+ + − −

≤ ≤ +

≤ + +

Taking the upper limit in above inequality and using (3.2.9), we have

(

2 1 2 1

)

2 limsup d mk , nk

k b y y

s

ε

+ − →∞

≤ (3.2.21)

Also,

(

2 k 1, 2k 1

)

(

2 k 1, 2 k

)

(

2 k, 2k 1

)

d m n d m m d m n

b y + y − ≤sb y + y +sb y y −

Taking the upper limit and using 3.2.9; 3.2.18 we get

(

)

3

2 1 2 1

limsup d m_k , n_k

k→∞ b y + y − εs

≤ _(3.2.22)

So, by (3.2.21) and (3.2.22) we have

(

)

3

2 1 2 1

2 limsup d mk , nk

k b y y s

s

ε _ε

+ − →∞

≤ ≤ (3.2.23)

According to the set (3.1.1) we have:

(

)

{

(

) (

) (

)

(

) (

)

}

(

) (

) (

)

{

(

) (

)

}

2 2 1 2 2 1 2 2 2 1 2 1

2 2 1 2 2 1

2 1 2 2 2 1 2 1 2

2 1 2 1 2 2

, max , , , , , ,

, , ,

max , , , , , ,

, , ,

k k k k k k k k

k k k k

k k k k k k

k k k k

n m d n m d n n d m m

d n m d n m

d n m d n n d m m

d n m d n m

N x x b Sx Tx b Fx Sx b Gx Tx

b Sx Gx b Fx Tx

b y y b y y b y y

b y y b y y

+ + + +

+ +

− − +

− +

=

=

(3.2.24) Taking the upper limit in (3.2.24) and using results 3.2.9; 3.2.18; 3.2.13; 3.2.23 we get

(

)

(

) (

) (

)

{

(

) (

)

}

{

}

2 2 1

2 1 2 2 2 1 2 1 2

2 1 2 1 2 2

2 3

lim sup ,

limsup max , , , , , ,

, , ,

max ,0,0, ,

k k

k k k k k k

k k k k

n m k

d n m d n n d m m

k

d n m d n m

N x x

b y y b y y b y y

b y y b y y

s s s

ε ε ε

+ →∞

− − +

→∞

− +

=

≤

(3.2.25)

Similarly, we can show,

(

)

3

2 2 1

2

min _s, ,_s liminf_k N xn_k,xm_k s

ε _ε ε _ε

+ →∞

 _{ ≤ ≤}

 

  (3.2.26)

DOI: 10.4236/oalib.1104657 10 Open Access Library Journal

(

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

4 4

2 2 1 2 2 1

2 2 1 2 2 1

2 2 1

2 , 2 ,

, , ,

,

k k k k

k k k k

k k

d n m d n m

n m n m

n m

s b y y s b Fx Gx

f N x x N x x

N x x

ψ ψ

ψ φ

ψ

+ +

+ +

+

≤

≤

≤

(3.2.27)

Taking the upper limit as k→ ∞ _{in (3.2.27) and using 3.2.25; 3.2.26, we}

ob-tain

(

)

(

)

(

)

(

)

(

(

)

)

( ) ( )

(

)

( )

3 4 4

2 2 1

4

2 2 1

2 2 1 2 2 1

3 3 3

2 2 2 limsup ,

2 limsup ,

limsup , , liminf ,

,

k k

k k

k k k k

d n m

k

d n m

k

n m _k n m

k

s s s b y y

s

s b Fx Gx

f N x x N x x

f s s s

ε

ψ ε ψ ψ

ψ

ψ φ

ψ ε φ ε ψ ε

+ →∞

+ →∞

+ _→∞ +

→∞

   

= _{ ≤  }

 

 

 

=  

 

   

≤ _{   }

 

 

≤ ≤

From this inequality and since ψ _{is non decreasing follows that}

3 3 3

2εs ≤εs ⇔εs = ⇔ =0 ε 0_.

That is a contradiction since we supposed ε >0_{. Thus}

_{{ }}

y_n is a bd-Cauchy

sequence in b-dislocated metric space

(

X b, _d

)

. Also the subsequences

{

fx2n

}

,

{

Tx2 1n+

}

,

{

Gx2 1n+

}

,

{

Sx2 2n+

}

are bd-Cauchy. Let we suppose that F X

( )

is a

complete subspace of X, since the subsequence

{

fx2n

}

is bd-Cauchy then there

exists z F X∈

( )

such that lim_n→∞y2n=_nlim_→∞Fx2n =z. Then we have,

2 1 2 2 1 2 2

lim n lim n lim n lim n

n→∞Tx + =n→∞Fx =n→∞Gx + =n→∞Sx + =z. (3.2.28)

Since z F X∈

( )

⊂T X

( )

, then there exists y X∈ _{such that} Ty z= _. Ac-cording to (3.1.1) consider

(

)

(

) (

) (

) (

) (

)

{

}

(

) (

) (

) (

) (

)

{

}

2

2 2 2 2 2

2 1 2 2 1 2 1 2

,

max , , , , , , , , ,

max , , , , , , , , ,

n

d n d n n d d n d n

d n d n n d d n d n

N x y

b Sx Ty b Fx Sx b Gy Ty b Sx Gy b Fx Ty

b y − z b y y − b Gy z b y − Gy b y z

= =

(3.2.29) Taking the upper limit and using lemma 2.13, result (3.2.9) and (3.2.28) we obtain

(

2

)

(

)

lim sup n, d ,

n→∞ N x y ≤sb Gy z (3.2.30)

Using contractive condition (A) of theorem we have,

(

)

(

)

(

(

(

)

)

(

(

)

)

)

(

)

(

)

4

2 2 2

2

2 , , , ,

,

d n n n

n

s b Fx Gy f N x y N x y

N x y

ψ ψ φ

ψ ≤

≤ (3.2.31)

Taking the upper limit in (3.2.31) and using (3.2.30) we get

(

)

(

₂ 3 _,

)

₂ 41

(

_,

)

(

(

_,

)

)

d d d

s b z Gy s b z Gy sb Gy z

s

ψ =ψ_ _≤ψ

 

DOI: 10.4236/oalib.1104657 11 Open Access Library Journal

of coincidence of the pair

(

G T,

)

.

Similarly we can show that Fv Sv z= = _{, so v is a point of coincidence of the}

pair

(

F S,

)

. Therefore we have

Gy Ty Fv Sv z= = = = _{. (3.2.32)} Let show that z is a unique point of coincidence of pairs

(

F S,

)

and

(

G T,

)

. Suppose that exists another point z1∈X such that Gy Ty1= 1=Fv Sv1= 1=z1.

We consider,

(

)

{

(

) (

) (

) (

) (

)

}

(

) ( ) (

) (

) (

)

{

}

(

)

1 1 1 1 1 1

1 1 1 1 1

1

, max , , , , , , , , ,

max , , , , , , , , ,

2 ,

d d d d d

d d d d d

d

N z z b Sz Tz b Fz Sz b Gz Tz b Sz Gz b Fz Tz

b z z b z z b z z b z z b z z sb z z

= = ≤

Using contractive condition of theorem we have,

(

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

)

(

)

4 4

1 1

1 1

1

1

2 , 2 ,

, , ,

,

2 ,

d d

d

s b z z s b Fz Gz

f N z z N z z

N z z sb z z

ψ ψ

ψ φ

ψ ψ

=

≤ ≤ ≤

The inequality above implies that b z zd

(

, 1

)

=0 so z z= 1 that means the

point of coincidence is unique.

Let prove that z is a common fixed point. By the weak compatibility of the pairs

(

F S,

)

and

(

G T,

)

have: Sz SFv FSv Fz= = = _and Gz GTy TGy Tz= = = _. From condition of theorem we have,

(

)

(

)

(

(

)

)

(

(

)

)

( )

(

)

(

( )

)

(

)

( )

(

)

4 4

2 , 2 , 2 ,

, , ,

,

d d d

sb Fz Gz s b Fz Gz s b Fz Gz

f N z z N z z

N z z

ψ ψ ψ

ψ φ

ψ

≤ =

≤ ≤

(3.2.33)

This inequality implies 2sb Fz Gzd

(

,

)

≤N z z

( )

, .

And

( )

{

(

) (

) (

) (

) (

)

}

(

) (

) (

) (

) (

)

{

}

(

)

, max , , , , , , , , ,

max , , , , , , , , ,

2 ,

d d d d d

d d d d d

d

N z z b Sz Tz b Fz Sz b Gz Tz b Sz Gz b Fz Tz

b Fz Gz b Fz Fz b Gz Gz b Fz Gz b Fz Gz sb Fz Gz

= = ≤

(3.2.34) Again from (3.2.33) and (3.2.34) we get, N z z

( )

, =2sb Fz Gz_d

(

,

)

.

By property of functions ψ φ; and C-class, we have

(

)

(

)

(

(

(

)

)

(

(

)

)

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

(

)

(

)

2 , 2 , , 2 ,

2 ,

2 , , 2 , 2 ,

2 , 0 or 2 , 0

2 , 0 , 0

d d d

d

d d d

d d

d d

sb Fz Gz f sb Fz Gz sb Fz Gz

sb Fz Gz

f sb Fz Gz sb Fz Gz sb Fz Gz

sb Fz Gz sb Fz Gz

sb Fz Gz b Fz Gz

ψ ψ φ

ψ

ψ φ ψ

ψ φ

≤ ≤

⇒ =

⇒ = =

⇒ = ⇒ =

DOI: 10.4236/oalib.1104657 12 Open Access Library Journal

So we obtained b Fz Gz_d

(

,

)

=0_{, that iz} Fz Gz= _{. Therefore} Fz Gz Sz Tz= = = _.

Let we prove that z is a fixed point of F. Again we consider

(

)

(

)

(

(

)

)

(

(

)

)

( )

(

)

(

( )

)

(

)

( )

(

)

4 4

2 , 2 , 2 ,

, , ,

,

d d d

sb Fz z s b Fz z s b Fz Gz

f N z z N z z

N z z

ψ ψ ψ

ψ φ

ψ

≤ =

≤ ≤

By property of ψ _follows

(

)

( )

2sb Fz z_d , ≤N z z, _(3.2.36)

where

( )

{

(

) (

) (

) (

) (

)

}

(

) (

) ( ) (

) (

)

{

}

(

)

, max , , , , , , , , ,

max , , , , , , , , ,

2 ,

d d d d d

d d d d d

d

N z z b Sz Tz b Fz Sz b Gz Tz b Sz Gz b Fz Tz

b Fz z b Fz Fz b z z b Fz z b Fz z sb Fz z

= = ≤

(3.2.37) From (3.2.36), (3.2.37) we get

( )

, 2 d

(

,

)

N z z = sb Fz z (3.2.38)

In similar way as in (3.2.35) using (3.2.38), property of C-class and functions

;

ψ φ _{we obtain,} b Fz zd

(

,

)

=0 and Fz z= . Hence z is a common fixed

point.

Uniqueness. Let we prove that the fixed point is unique. If suppose that u and z

are two common fixed points of F, G, S, T then from condition (b) we have,

( )

(

)

(

( )

)

(

(

)

)

( )

(

)

(

( )

)

(

)

( )

(

)

4 4

2 , 2 , 2 ,

, , ,

,

d d d

sb u z s b u z s b Fu Gz

f N u z N u z

N u z

ψ ψ ψ

ψ φ

ψ

≤ =

≤ ≤

By property of ψ _{we get} 2sb u zd

( )

, ≤N u z

( )

, . Also we have,

( )

{

(

) (

) (

) (

) (

)

}

( ) ( ) ( ) ( ) ( )

{

}

( )

, max , , , , , , , , ,

max , , , , , , , , ,

2 ,

d d d d d

d d d d d

d

N u z b Su Tz b Fu Su b Gz Tz b Su Gz b Fu Tz

b u z b u u b z z b u z b u z sb u z

= = ≤

So, N u z

( )

, =2sb u z_d

( )

,

and

( )

(

)

(

(

( )

)

(

( )

)

)

(

( )

)

( )

(

)

(

( )

)

2 , 2 , , 2 , 2 ,

2 , 0 or 2 , 0

d d d d

d d

sb u z f sb u z sb u z sb u z

sb u z sb u z

ψ ψ φ ψ

ψ φ

≤ ≤

⇒ = =

As a result b u z_d

( )

, =0 _{and so} u z= _.

The following is corollary of theorem 3.2 which is taken for parameter s=1

in a dislocated metric space.

Corollary 3.3 Let

(

X d, l

)

be a dislocated metric space and

, , , :

DOI: 10.4236/oalib.1104657 13 Open Access Library Journal

( )

( )

F X ⊂T X and exists ψ∈ Ψ, φ∈Φ and f C∈ _{such that satisfy the}

condition

(

)

(

2d Fx Gyl ,

)

f

(

(

N x y

( )

,

)

,

(

N x y

( )

,

)

)

ψ ≤ ψ φ _(A)

for all x y X, ∈ _{, where} N x y

(

,

)

is defined as in (3.1.0). If one of

( ) ( ) ( )

, ,

F X S X T X or G X

( )

is a complete subspace of X, then

(

F S,

)

and

(

G T,

)

have a point of coincidence in X. Moreover if suppose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , , have a unique common fixed point.

Now we give an example to support our Theorem 3.2.

Example 3.4 Let X =

[

0,∞

)

and b x yd

( ) (

, = x y+

)

2. Then the pair

(

X b, d

)

is a b-dislocated metric space with parameter s=2_{. We define the functions} , ,

F G S and T as follows: 2 , , ,

5 2 30 24

x x x x

Sx= Tx= Fx= Gx= . The pairs

(

S F,

)

and

(

T G,

)

are weakly compatible, functions F G S, , and T are continuous and F X

( )

⊂T X G X

( ) ( )

, ⊂S X

( )

We have,

(

)

(

)

(

)

(

)

(

)

4

2 2

2 2 2

2

2 ,

32

32 , 32

30 24 30 24 36 5 4

4 5 4 5

8 2 2 2

9 400 9 100 9 5 2

1 2 1 _, 1

4 5 2 4 4

d

d

d

s b Fx Gy

x y x y x y

b

x y x y x y

x y _{b Sx Ty}

ψ

ψ ψ ψ

ψ ψ ψ

ψ ψ

   

      

= _{  }= _{ } + _ _= _{ } + _ _

     

  _{   }

 +   +     

   

= _{ }= _{ }= _{ } + _ _

 

 

   

 _{ }  _{ }

≤ _{ } + _ _= _{ }=

   

 

(

)

( )

( )

( )

(

( )

)

(

( )

)

( )

(

)

(

( )

)

(

)

1

, ,

4 3

, , , ,

4

, , ,

d

b Sx Ty M x y

M x y M x y M x y M x y

f M x y M x y

ψ φ

ψ φ

≤

= − = −

=

where f s t

( )

, = −s t; ψ

( )

t =t and

( )

3

4

t t

φ = _{, for all} x y X, ∈ _.

Thus all conditions of theorem 3.2 are satisfied and x=0 _{is the unique}

common fixed point of S T F, , and G.

In a similar way as in Theorem 3.2, the following theorem can be proved. Theorem 3.5 Let

(

X b, d

)

be a complete b-dislocated metric space with

pa-rameter s≥1 _and S T F G X, , , : →X are self-mappings such that (a)

( )

( ) ( )

,

( )

F X ⊂T X G X ⊂S X and satisfy generalized f −

(

ψ φ, ,s

)

weakly contractive condition. If one of F X S X T X

( ) ( ) ( )

, , or G X

( )

is closed, then

(

F S,

)

and

(

G T,

)

have a point of coincidence in X. Moreover if suppose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , , have a unique common fixed point.

DOI: 10.4236/oalib.1104657 14 Open Access Library Journal

Corollary 3.6 Let

(

X b, _d

)

be a bd-dislocated metric space with parameter 1

s≥ _and S T F G X, , , : →X are self-mappings where (a)

( )

( ) ( )

,

( )

F X ⊂T X G X ⊂S X _{and exists} ψ∈ Ψ_, φ∈Φ _{such that satisfies} the condition

(

)

(

₂ 4 _,

)

(

( )

_,

)

(

( )

_,

)

d

s b Fx Gy N x y N x y

ψ ≤ψ −φ

for all x y X, ∈ , where N x y

(

,

)

is defined as in (3.1.0). If one of

( ) ( ) ( )

, ,

F X S X T X or G X

( )

is a complete subspace of X, then

(

F S,

)

and

(

G T,

)

have a point of coincidence in X. Moreover if suppose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , , have a unique common fixed point.

Proof. If we take in Theorem 3.2 the function f as f s t

( )

, = −s t _{then we get}

the corollary.

Corollary 3.7 Let

(

X b, _d

)

be a bd-dislocated metric space with parameter 1

s≥ _and S T F G X, , , : →X are self-mappings where (a)

( )

( ) ( )

,

( )

F X ⊂T X G X ⊂S X _{and exists} ψ∈ Ψ_, φ∈Φ _{such that satisfies} the condition

(

)

(

₂ 4 _,

)

(

( )

,

₍

)

_{( )}

(

( )

₎

,

)

1 ,

d

N x y N x y

s b Fx Gy

N x y

ψ φ

ψ

φ

− ≤

+

for all x y X, ∈ , where N x y

(

,

)

is defined as in (3.1.0). If one of

( ) ( ) ( )

, ,

F X S X T X or G X

( )

is a complete subspace of X, then

(

F S,

)

and

(

G T,

)

have a point of coincidence in X. Moreover if suppose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , , have a unique common fixed point.

Proof. This corollary is obtained from Theorem 3.2 if we take as f the function

( )

, 1

s t f s t

t

− =

+ .

Corollary 3.8 Let

(

X b, _d

)

be a bd-dislocated metric space with parameter 1

s≥ _and S T F G X, , , : →X are self-mappings where (a)

( )

( ) ( )

,

( )

F X ⊂T X G X ⊂S X and exists ψ∈ Ψ, φ∈Φ such that satisfies the condition

(

)

(

₂ 4 _,

)

(

₍

( )

_{( )}

,

)

₎

1 ,

d

N x y s b Fx Gy

N x y

ψ ψ

φ

≤ +

for all x y X, ∈ _{, where} N x y

(

,

)

is defined as in (3.1.0). If one of

( ) ( ) ( )

, ,

F X S X T X or G X

( )

is a complete subspace of X, then

(

F S,

)

_and

(

G T,

)

have a point of coincidence in X. Moreover if suppose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , , have a unique common fixed point.

Proof. If we take in Theorem 3.2 the function f as

( )

, 1

s f s t

t

=

+ then we get

the corollary.

DOI: 10.4236/oalib.1104657 15 Open Access Library Journal 1

s≥ _and S T F G X, , , : →X are self-mappings where (a)

( )

( ) ( )

,

( )

F X ⊂T X G X ⊂S X and exists ψ∈ Ψ_, φ∈Φ _{such that satisfies} the condition

(

)

(

₂ 4 _,

)

(

( )

,

₍

)

_{( )}

(

( )

₎

,

)

1 ,

d

N x y N x y

s b Fx Gy

N x y

ψ φ

ψ

φ

≤ +

for all x y X, ∈ _{, where} N x y

(

,

)

is defined as in (3.1.0). If one of

( ) ( ) ( )

, ,

F X S X T X or G X

( )

is a complete subspace of X, then

(

F S,

)

and

(

G T,

)

have a point of coincidence in X. Moreover if suppose that

(

F S,

)

and

(

G T,

)

are weakly compatible pairs, then F G S T, , , have a unique common fixed point.

Proof. If we take in Theorem 3.2 the function f as

( )

, 1

st f s t

t

=

+ then we get

the corollary.

Remark 3.10 As a consequence of theorem 3.2 and all corollaries for taking 1) the parameter s=1_.

2) the parameter s=1 _and G F= _and T S= _.

3) functions f from the set C and taking ψ

( )

t =t and φ

( ) (

t = −1 k t

)

.

We can establish many other corollaries in the setting of dislocated and

b-dislocated metric spaces.

4) Our results unify, generalize, and extend several ones obtained earlier in a lot of papers concerning b-metric, dislocated and b-dislocated metric spaces (as in references [13] [15] [25] [26] [30] [31]).

In this section, we use the notion of T-contractions introduced by Beiranvad

et al. in [3] as a new class of contractive mappings, by generalizing the contrac-tive condition in terms of another function. These contractions have been used by many authors. In this direction in order to generalize some other well-known results as in [32] [33] [34] we extend the notion of f −

(

ψ φ, ,s

)

generalized weak contractions in the context of T-contractions, giving the following theo-rem.

Theorem 3.11 Let

(

X b, _d

)

be a complete b-dislocated metric space with pa-rameter s≥1 _and T X: →X _{be an injective, continuous and sequentially} convergent mapping. Let F G X, : →X be self-mappings and if exist ψ∈ Ψ,

φ∈Φ _and f C∈ _{such that}

(

)

(

₂ 4 _,

)

(

(

(

_,

)

)

_,

(

(

_,

)

)

)

d

s b TFx TGy f N Tx Ty N Tx Ty

ψ ≤ ψ φ _(B)

for all x y X, ∈ , where

(

)

(

) (

) (

) (

)

(

)

,

, ,

max , , , , , ,

4

d d

d d d

N Tx Ty

b Tx TGy b Ty TFx b Tx Ty b Tx TFx b Ty TGy

s

+

 

= _{ }

 

then F G, have a unique common fixed point.

Proof. We divide the proof into two parts as follows.

DOI: 10.4236/oalib.1104657 16 Open Access Library Journal

common fixed point of F G, is unique.

Let u X∈ _{be a fixed point of F. If} b Fu Gud

(

,

)

=0 then, follows that

u Fu Gu= = _{and so u is a fixed point of G. If we suppose that} b Fu Gu_d

(

,

)

>0_,

we evaluate N Tu Tu

(

,

)

as;

(

)

(

) (

) (

) (

)

(

)

(

) (

) (

) (

)

(

)

(

)

,

, ,

max , , , , , ,

2

, ,

max , , , , , ,

4

2 ,

d d

d d d

d d

d d d

d

N Tu Tu

b Tu TGu b Tu TFu b Tu Tu b Tu TFu b Tu TGu

s b Tu TGu b Tu Tu b Tu Tu b Tu Tu b Tu TGu

s sb Tu TGu

+

 

= _{ }

 

+

 

=  

 

≤

So we have

(

,

)

2 _d

(

,

)

N Tu Tu ≤ sb Tu TGu (3.11.1)

Then by contractive condition (B), we have

(

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

2

2 , 2 ,

, , ,

,

d d

sb Tu TGu s b TFu TGu

f N Tu Tu N Tu Tu

N Tu Tu

ψ ψ

ψ φ

ψ

≤

≤ ≤

By property of ψ we have

(

)

(

)

2sb Tu TGud , ≤N Tu Tu, (3.11.2)

Hence from (3.11.1) and (3.11.2) follows N Tu Tu

(

,

)

=2sb Tu TGu_d

(

,

)

_.

Again

(

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

2

2 , 2 ,

, , ,

2 , , 2 ,

2 ,

d d

d d

d

sb Tu TGu s b TFu TGu

f N Tu Tu N Tu Tu

f sb Tu TGu sb Tu TGu

sb Tu TGu

ψ ψ

ψ φ

ψ φ

ψ ≤

≤

= ≤

(

)

(

)

(

(

)

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

2 , , 2 , 2 ,

2 , 0 or 2 , 0

d d d

d d

f sb Tu TGu sb Tu TGu sb Tu TGu

sb Tu TGu sb Tu TGu

ψ φ ψ

ψ φ

⇒ =

⇒ = =

By property of ψ φ, we get b Tu TGud

(

,

)

=0 that is Tu TGu= and by

in-jectivity of T follows u Gu= _.

Thus u is a fixed point of G. Similarly we can prove the other implication.

Second part. We prove that the function F has a fixed point. We define two eterative sequences

{ }

xn as x2 1n+ = fx2n and x2 2n+ =gx2 1n+ , and

{ }

yn as

n n

y =Tx _{for each} n=0,1,2,.

If for some n, we have x2 1n+ =x2n then x2n=x2 1n+ = fx2n and x2n is a fixed

point of F and by the first part x2n is a fixed point of G and the proof is

com-pleted.

Now, we assume that x2 1n+ ≠x2n for all n, and since T is injective we have 2 1n 2n