Common Fixed Points for Compatible Mappings of Type (A)

(1)

of the

MALAYSIAN MATHEMATICAL SOCIETY

Common Fixed Points for Compatible Mappings of Type (A)

1ZEQING LIU,¹FENGRONG ZHANG AND ²JIANFENG MAO

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029 People’s Republic of China

2Department of Mathematics, Xianning Teachers’College, Xianning, Hubei 437005 People’s Republic of China

Abstract. In this paper we give criteria for the existence of a common fixed point of a pair of mappings in 2-metric spaces and establish common fixed and coincidence point theorems for certain classes of contractive type mappings. The results presented here generalize, improve and unify a number of fixed point theorems given by Cho [1], Imdad et al. [5], Khan and Fisher [11], Kubiak [14], Murthy et al. [17], Rhoades [25], Singh et al. [34]and others.

1. Introduction

Gähler [4] introduced the concept of 2-metric spaces. A 2-metric space is a set X with a function d :X ×X × X →

[

⁰^,^∞

)

satisfying the following conditions:

(G1) for two distinct points x,y∈X, there exists a point z∈ X such that ,

0 ) , ,

(x y z ≠

d

(G2) d(x,y,z) =0 if at least two of x,y,z are equal, (G3) d(x,y,z) = d(x,z,y) = d(y,z,x),

(G4) d(x,y,z) ≤ d(x,y,u)+d(x,u,z)+d(u,y,z) for all x,y,z,u∈X.

It has been shown by Gähler [4] that a 2-metricd is a continuous function of any one of its three arguments but it need not be continuous in two arguments. If it is continuous in two arguments, then it is continuous in all three arguments. A 2-metricdwhich is continuous in all of its arguments will be called continuous.

Iséki [7], for the first time, established a fixed point theorem in 2-metric spaces.

Since then a quite number of authors ([1]-[3], [5]-[36]) have extended and generalized the result of Iséki and various other results involving contractive and expansive type mappings. Especially, Murthy et al. [17] introduced the concepts of compatible mappings and compatible mappings of type (A) in 2-metric spaces, derived some relations between these mappings and proved common fixed point theorems for compatible mappings of type (A) in 2-metric spaces.

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On the other hand, Cho [1], Constantin [2], Khan and Fisher[11] and Kubiak [14]

established some necessary and sufficient conditions which guarantee the existence of a common fixed point for a pair of continuous mappings in 2-metric spaces.

In this paper we establish criteria for the existence of a common fixed point of a pair of mappings in 2-metric spaces and establish common fixed and coincidence point theorems for certain classes of contractive type mappings. The results presented here generalize, improve and unify the corresponding results of Cho [1], Imdad et al. [5], Khan and Fisher [11], Kubiak [14], Murthy et al. [17], Rhoades [25], Singh et al. [34]

and others.

2. Preliminaries

Throughout this paper, Nandωdenote the sets of positive and nonnegative integers, respectively. Let R⁺ =[0,∞) and

{

^: ^:

( )

⁵ ^satisfies^conditions^(a1)^and^(a2)

}

^,

1

+

+ →

= ϕ ϕ R R

Φ

{

^: ^:

( )

¹¹ ^satisfies^conditions^(a1)^and^(a3)

}

^,

2

+

+ →

= ϕ ϕ R R

Φ

where conditions (a1), (a2) and (a3) are as follows:

(a1) ϕ is upper semicontinuous, nondecreasing in each coordinate variable, (a2) b(t)= max

{

ϕ(t,0,0,t,t),ϕ(t,t,t,2t,0),ϕ(t,t,t,0,2t)

}

< t for all t > 0, (a3) c(t) = max

{

ϕ(t,t,t,0,2t,t,0,2t,0,2t,0),ϕ(t,0,0,t,t,0,0,0,0,0,t),

ϕ(t,t,t,2t,0,t,2t,0,2t,0,0)

}

< t for allt > 0.

Lemma 2.1. [37] For every t > 0 ,c(t)<t if and only if ( )= 0,

∞

→ c t lim ⁿ

n where cⁿ

denotes the n-times composition of c .

Definition 2.1. A sequence {x_n}_n_∈_N in a 2-metric space (X,d) is said to be convergent to a point x∈X if ( , , ) =0

∞

→ d x x a

lim _n

n for all a∈X. The point x is called the limit of the sequence {x_n}_n_∈_Nin X .

Definition 2.2. A sequence {x_n}_n_∈_N in a 2-metric space (X,d) is said to be a Cauchy

sequence if ( , , ) 0

, =

∞

→ d x x a

lim _m _n

n

m for all a∈X.

Definition 2.3. A 2-metric space(X,d) is said to be complete if every Cauchy sequence inX is convergent.

(3)

Note that, in a 2-metric space (X,d), a convergent sequence need not be a Cauchy sequence, but every convergent sequence is a Cauchy sequence when the 2-metric d is continuous on X ([19]).

Definition 2.4. Let f and g be mappings from a 2-metric spaces (X,d) into itself.

f and g are said to be compatible if

(

^, ^,

)

⁼ ⁰

∞

→ d fgx gfx a

lim _n _n

n

for all a∈X, whenever {x_n}_n_∈_N ⊂ X such that lim fx lim gx_n t

n n

n = =

∞

→

∞

→ for some

t∈X; f and g are said to be compatible of type (A) if

(

^, ^,

)

⁼ _→_∞

(

^, ^,

)

⁼⁰

∞

→ d fgx ggx a lim d gfx ffx a

lim _n _n

n n n n

for all a∈X, whenever {x_n}_n_∈_N ⊂ X such that lim fx lim gx_n t

n n

n = =

∞

→

∞

→ for some

. X t∈

Definition 2.5. A mapping f from a 2-metric space (X,d) into itself is said to be continuous at x∈X if for every sequence {x_n}_n_∈_N ⊂ X such that

0 ) , ,

( =

∞

→ d x x a

lim _n

n for all a∈X, ( , , )= 0.

∞

→ d fx fx a

lim _n

n f is called continuous on

X if it is so at all points of X.

Lemma 2.2. [1] Let f and g be compatible mappings of type (A) from a 2-metric spaces(X,d) into itself. If ft = gtfor some t∈X, then fgt = ggt = gft = fft.

Lemma 2.3. [1] Let f and g be compatible mappings of type (A) from a 2-metric spaces (X,d) into itself. If f is continuous at some t∈X and if

, t gx lim fx im

l _n

n n

n = =

∞

→

∞

→ then lim gfx_n ft.

n =

∞

→

Lemma 2.4. [1] Let f and g be compatible mappings from a 2-metric spaces )

,

(X d into itself. If f and g are continuous, then they are compatible of type (A).

3. Characterizations of common fixed points

Our main results are as follows:

Theorem 3.1. Let (X,d) be a complete 2-metric space with d continuous on X and let h and t be two mappings from ^X into itself. Then the following conditions are equivalent:

(4)

(1) h and t have a common fixed point;

(2) there exist r∈(0,1) ,f :X →t(X) and g :X →h(X) such that

(b1) the pairs f,h and g,t are compatible of type (A), (b2) one of f,g,h and t is continuous,

(b3) d

(

fx,gy,a

)

≤ rmax

{

d

(

hx,ty,a

) (

,d hx, fx,a

) (

,d ty,gy,a

)

, ₂¹

[

^d

(

^hx^,^gy^,^a

)

⁺^d

(

^ty^, ^fx^,^a

) ] }

^{for all}^x^,^y^,^a^∈^X^;

(3) there exist ϕ∈Φ₁, f :X →t(X) and g :X →h(X) satisfying conditions (b1), (b2) and (b4):

(b4) ^d

(

^fx^,^gy^,^a

)

^≤^ϕ

(

^d

(

^hx^,^ty^,^a

) (

^,^d ^hx^,^fx^,^a

) (

^,^d ^ty^,^gy^,^a

) (

^,^d ^hx^,^gy^,^a

) (

^,^d ^ty^,^fx^,^a

) )

for all x y a, , ∈X;

(4) there exist ϕ∈Φ₂, f :X →t(X) and g :X →h(X) satisfying conditions (b1), (b2) and (b5):

d²

(

fx,gy,a

)

≤ ϕ

(

d²

(

hx,ty,a

) (

,d hx,ty,a

) (

d hx, fx,a

)

, d

(

hx,ty,a

) (

d ty,gy,a

)

,

(b5)

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

(

^ty ^gy ^a

) (

^d ^ty ^fx ^a

) (

^d ^hx ^gy ^a

) (

^d ^ty ^fx ^a

))

d

a gy hx d a gy ty d

a fx ty d a fx hx d a gy hx d a fx hx d

a gy ty d a fx hx d

a fx ty d a ty hx d a gy hx d a ty hx d

, , , , , , , , ,

, , , ,

,

, , , , , , , , ,

,

, , , , ,

, , , , , , , , , ,

for all x y a, , ∈X;

Proof. (1)⇒(2) and (4). Let z be a common fixed point of h and t. Define )

(

:X t X

f → and g :X →h(X) by fx=gx=z for allx∈X. Then (b1) and (b2) hold. For each r∈(0,1) and ϕ ∈Φ₂,(b3) and (b5) also hold.

) 3 ( ) 2

( ⇒ Take^ϕ

(

^u^,^v^,^w^,^x^,^y

)

⁼ ^r^max

{

^u^,^v^,^w^,₂¹

(

^x⁺ ^y

) }

^{for all}^{u v w x y}^{, , , ,} ^∈^R⁺^.

Then ϕ ∈Φ₁ and (b3) implies (b4).

) 1 ( ) 3

( ⇒ By using the method of Cho [1], we can similarly show that (3) implies (1).

) 1 ( ) 4

( ⇒ Let x0 be an arbitrary point in X. Since f(X)⊂t(X) and ,

) ( )

(X h X

g ⊂ there exist sequences {x_n}_n_∈_ω and {y_n}_n_∈_ω in X satisfying

1 2 2 2 1 2 2 1 2

2_n = tx _n₊ = fx _n, y _n₊ = hx _n₊ = gx _n₊

y for all n∈ω. Define

(5)

(

^y ^y ^a

)

d a

d_n( ) = _n, _n₊₁, for all a∈X and n∈ω. We claim that for any i, j,k∈ω

(

y_i,y_j,y_k

)

=0.

d (3.1) Suppose that d₂_n(y₂_n₊₂)> 0. Using (b5), we have

(

, ,

) (

2

(

2 2, 2 1, 2

)

,

2 1 2 2 2 2

n n n n

n

n gx y d hx tx y

fx

d ₊ ₊ ≤ϕ ₊ ₊

d

(

hx₂_n₊₂,tx₂_n₊₁,y₂_n

) (

d hx₂_n₊₂,fx₂_n₊₂,y₂_n

)

,

( ) ( )

(

^,^, ^,^,

) (

^,^, ^,^,

))

^, ^,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

2 2 2 1 2 2 1 2 2 2

2 2 2 1 2 2 1 2 1 2

2 1 2 2 2 2 1 2 1 2

2 2 2 1 2 2 2 2 2 2

2 1 2 2 2 2 2 2 2 2

2 1 2 1 2 2 2 2 2 2

2 2 2 1 2 2 1 2 2 2

2 1 2 2 2 2 1 2 2 2

2 1 2 1 2 2 1 2 2 2

n n n n n n

y fx tx d y gx hx d

y fx tx d y gx tx d

y gx hx d y gx tx d

y fx tx d y fx hx d

y gx hx d y fx hx d

y gx tx d y fx hx d

y fx tx d y tx hx d

y gx hx d y tx hx d

y gx tx d y tx hx d

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

which implies that

( ) (

⁰^,⁰^,⁰^,⁰^,⁰^,⁰^,⁰^,⁰^,⁰^,⁰^,⁰

) ( ( ) ) (

2 2

)

^,

2 2 2 2 2 2 2

2 2

2_n y _n+ ≤ ≤ c d _n y _n+ <d _n y _n+

d ϕ

which is a contradiction. Henced₂_n(y₂_n₊₂) =0. Similarly, we have d₂_n₊₁(y₂_n₊₃)= 0. Consequently, d_n(y_n₊₂)= 0for all n∈ω. Note that

(

y ,y ₂,a

)

d

(

y ₂

)

d

( )

a d ₁

( )

a d

( )

a d ₁

( )

a.

d _n _n₊ ≤ _n _n₊ + _n + _n₊ = _n + _n₊ (3.2)

By (b5) and (3.2) we have

(6)

( ) ( )

( )

( ( ) ( )

( ) ( )

(

^, ^,

) (

^, ^,

))

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , , ,

,

, , ,

, ,

2 2 1 2 1

2 2 2

2 2 1 2 1

2 1 2

1 2 2 2 1

2 1 2

2 2 1 2 2

2 2 2

1 2 2 2 2

2 2 2

1 2 1 2 2

2 2 2

2 2 1 2 1

2 2 2

1 2 2 2 1

2 2 2

1 2 1 2 1

2 2 2

2 2 2 2 1

2 2 2

1 2 2 2 2

1 2 2 2 2 2

1 2

a fx tx d a gx hx d

a fx tx d a gx tx d

a gx hx d a gx tx d

a fx tx d a fx hx d

a gx hx d a fx hx d

a gx tx d a fx hx d

a fx tx d a tx hx d

a gx hx d a tx hx d

a gx tx d a tx hx d

a fx hx d a tx hx d

a tx hx d

a gx fx d a d

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

+

+ + +

≤

= ϕ

( [ ]

[ ]

( ) [

⁽ ⁾ ⁽ ⁾

]

^,⁰

)

^.

, 0 , ) ( )

( ) (

, 0 , ) ( , ) ( )

( ) ( , 0

, ) ( , ) ( ) ( , ) (

1 2 2

2

1 2 2

1 2

2 1 2 1 2 2

2

2 2 1 2 2 2 2

a d a d a d

a d a d a d a d

n n

n

n n

n

n n

+ + +

+ +

+

+ + +

=ϕ

Suppose that d₂_n₊₁(a)> d₂_n(a). Then

(

⁽ ⁾^, ⁽ ⁾^, ⁽ ^),⁰^,² ⁽ ⁾^,

)

( ₂ ₁² ₂ ₁² ₂ ₁² ₂ ₁²

2 1

2 a d a d a d a d a

d _n₊ ≤ϕ _n₊ _n₊ _n₊ _n₊

)

0 , ) ( 2 , 0 , ) ( 2 , 0 , )

( ₂ ₁² ₂ ₁²

2 1

2 a d a d a

d _n₊ _n₊ _n₊

(

⁽ ⁾

)

2 1²⁽ ⁾^, 2

1

2 a d a

d

c _n₊ < _n₊

≤

which is a contradiction. Hence d₂_n₊₁(a) ≤d₂_n(a) and so d₂_n₊₁²(a)≤ c(d₂_n²(a)). Similarly, we have d₂_n(a) ≤d₂_n₋₁(a) and d₂_n²(a) ≤c(d₂_n₋₁²(a)). That is, for all

N n∈

(

⁽ ⁾

)

^.

) ( ), ( )

( ₁² ²

1 a d a d a c d a

d_n₊ ≤ _n _n₊ ≤ _n

Hence for all n∈N

(

⁽ ⁾

) (

⁽ ⁾

) (

⁽ ⁾

)

^.

)

( ² ² ₁² ¹ ₀²

2

1 a c d a c d a c d a

d_n₊ ≤ _n ≤ _n₋ ≤L≤ ⁿ⁺ (3.3)

It follows from (3.3) and Lemma 2.1 that 0 )

1( →

+ a

d_n as n → ∞. (3.4)

(7)

Let n,m be in .ω If n ≥ m, then 0 =d_m(y_m)≥ d_n(y_m); if n < m, then

. 0 ) ( )

( ) (

) ( ) ( ) (

) ( )

( )

( ) (

1 2

1

1 1

1 1 1

1

=

≤

+ +

≤

+ +

≤

+

−

+

−

+

−

n n m

n m

n

n n n n m

n

n m n m m

n m n

y d y

d y

d

y d y d y

d

y d y d y

d y d

L Thus, for any n,m∈ω

. 0 )

( _m =

n y

d (3.5) For all i,j,k∈ω, we may, without loss of generality, assume that i < j. It follows from (3.5) that

. 0 ) ( )

, , (

) , , ( ) , , (

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

1 1

2 1

1

=

≤

=

+ +

≤

−

+ +

+

k j k j j

k j i k

j i

k j i k

i j i k j i

y d y y y d

y y y d y y y d

y y y d y d y d y y y d

L Therefore (3.1) holds.

In order to show that {y_n}_n_∈_ω is a Cauchy sequence, by (3.4), it is sufficient to show that {y₂_n}_n_∈_ω is a Cauchy sequence. Suppose that {y₂_n}_n_∈_ω is not a Cauchy sequence. Then there exist ε > 0 and a∈ X such that for each even integer 2k, there are even integers2m(k) and 2n(k) with 2m(k) >2n(k) >2k and

. ) , ,

(y₂ ₍ ₎ y₂ ₍ ₎ a ≥ε d _m_k _n _k

For each even integer2k,let 2m(k) be the least even integer exceeding 2n(k) satisfying the above inequality, so that

(

y2 ( )₋2^,y2 ( )^,a

)

≤ε^,

d _m_k _n_k d

(

y2_m(_k)^,y2_n(_k)^,a

)

> ε^. (3.6) For each even integer 2k, by (3.1) and (3.6) we have

ε < ^d

(

^y₂m₍k₎,^y₂n₍k₎,^a

)

( ) ( )

( )

( ) ( )

(

^y ^y ^a

)

d

a y

y d y

y y

d

y y y d

a y

y d a y y

d

k m k m

k m k

m k

m k m k m

k m k n k m

k m k m k

n k m

, ,

) ( 2 1 ) ( 2

1 ) ( 2 2 ) ( 2 1

) ( 2 ) ( 2 2 ) ( 2

2 ) ( 2 ) ( 2 ) ( 2

2 ) ( 2 ) ( 2 )

( 2 2 ) ( 2

−

+

+ +

≤ +

+

≤

ε

=ε +d₂_m₍_k₎₋₂(a)+d₂_m₍_k₎₋₁(a)

(8)

which implies that

(

^, ^,

)

^.

lim ₂ ₍ ₎ ₂ ₍ ₎ =ε

∞

→ d y _m_k y _n _k a

k (3.7) It follows from (3.7) that

(

^y ^y ^a

) (

^d ^y ^y ^a

)

d _n_k , _m_k , _n_k , _m_k ,

0 < ₂ ₍ ₎ ₂ ₍ ₎ − ₂ ₍ ₎ ₂ ₍ ₎₋₂

≤^d

(

^y₂m₍k₎₋₂,^y₂m₍k₎,^a

)

≤d₂_m₍_k₎₋₂(a)+d₂_m₍_k₎₋₁(a). In view of (3.5) and (3.7) we immediately obtain that

(

^, ^,

)

^.

lim ₂ ₍ ₎ ₂ ₍ ₎₋₂ =ε

∞

→ d y _n_k y _m_k a

k (3.8) Note that

(

y2n(k)^,y2m(k) 1^,a

) (

d y2n(k)^,y2m(k)^,a

)

d2m(k) 1⁽a⁾ d2m(k) 1

(

y2n(k)

)

^,

d ₋ − ≤ ₋ + ₋

d

(

y2n(k)₊1^,y2m(k)^,a

) (

−d y2n(k)^,y2m(k)^,a

)

≤d2n(k)⁽a⁾+d2n(k)

(

y2m(k)

)

^,

d

(

y2_n(_k)+1^,y2_m(_k)−1^,a

) (

−d y2_n(_k)^,y2_m(_k)−1^,a

)

≤ d2_n(_k)⁽a⁾+d2_n(_k)

(

y2_m(_k)−1

)

^.

It is easy to see that

(

y y a

)

d

(

y y a

)

d nk mk

k k m k

klim→∞ 2n( ), 2 ( )−1, = lim→∞ 2 ( )+1, 2 ( ),

(

, ,

)

.

lim ₂ ₍ ₎ ₁ ₂ ₍ ₎ ₁ =ε

= ₊ ₋

∞

→ d y _n_k y _m_k a

k (3.9)

It follows from (b5) that

( ) ( )

( )

( ( ) ( )

( ) ( )

(

^, ^,

) (

^, ^,

)

^,

, , ,

, ,

, , ,

, ,

, , ,

, ,

, , ,

, ,

, , ,

, ,

, , ,

, ,

, , ,

, ,

) ( 2 1 ) ( 2 )

( 2 ) ( 2

1 ) ( 2 ) ( 2 )

( 2 ) ( 2

1 ) ( 2 1 ) ( 2 )

( 2 ) ( 2

) ( 2 1 ) ( 2 1

) ( 2 ) ( 2

1 ) ( 2 ) ( 2 1

) ( 2 ) ( 2

1 ) ( 2 1 ) ( 2 1

) ( 2 ) ( 2

) ( 2 ) ( 2 1

) ( 2 ) ( 2

1 ) ( 2 ) ( 2 2

1 ) ( 2 ) ( 2 1

) ( 2 ) ( 2

a fx tx

d a fx hx

d

a gx

hx d a fx hx

d

a gx

tx d a fx hx

d

a fx tx

d a tx

hx d

a gx

hx d a tx

hx d

a gx

tx d a tx

hx d

a fx hx

d a tx

hx d

a tx

hx d

a gx

fx d a y

y d

k m k

n k

m k m

k n k m k

m k m

k n k

n k

m k m

k m k

n k

n k m

k n k m k

n k m

k n k

n k

n k m

k m k m k

n k m

k n k m

k n k m k

n k m

+

+ + +

+ +

+ + + + +

≤

= ϕ

(9)

( ) ( )

(

^hx ^gx ^a

) (

^d^tx ^fx ^a

))

d

a fx tx

d a gx

tx d

a gx

hx d a gx

tx d

k m k

n k

n k m

k m k

n k

n

k n k m k

n k

n

, ,

, , ,

, ,

, , ,

, ,

) ( 2 1 ) ( 2 1

) ( 2 ) ( 2

) ( 2 1 ) ( 2 1

) ( 2 1 ) ( 2

1 ) ( 2 ) ( 2 1

) ( 2 1 ) ( 2

+ +

+

+ +

+

( )

( ( )

( )

(

^, ^,

) (

^, ^,

)

^,

, ) ( ,

,

, ) ( ,

, ,

1 ) ( 2 1 ) ( 2 ) ( 2 1 ) ( 2

) ( 2 ) ( 2 1 ) ( 2

1 ) ( 2 ) ( 2 1 ) ( 2 )

( 2 1 ) ( 2 2

a y

y d a y y

d

a d a y y

d

a d

a y y

d a y y

d

k n k m k

n k m

k n k n k m

k m k

n k m k

n k m

+

−

=ϕ −

d

(

y2_m(_k)₋1^,y2_n(_k)^,a

) (

d y2(_k)^,y2_m(_k)^,a

)

^,

( )

( ) (

, ,

)

,

, , , )

(

, , ,

) (

, ) ( ) (

1 ) ( 2 1 ) ( 2 )

( 2

) ( 2 ) ( 2 1

) ( 2

1 ) ( 2 1 ) ( 2 1

) ( 2

) ( 2 1 ) ( 2

a y

y d a d

a y y d a d

a y

y d a d

a d a d

k n k m k

n

k m k n k

m

k n k m k

m

k n k

m

+

−

+

−

( ) (

_{( )}

) ( )

(

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

))

^, ^, ^, ^, ^,

1 ) ( 2 1 ) ( 2 2

) ( 2 )

( 2

a y y d

a y

y d a y y d a d

k m k n

k n k m k

m k n k

n − +

Letting k → ∞, by (3.9), (3.7) and (3.5) we have

(

²^,⁰^,⁰^, ²^, ²^,⁰^,⁰^,⁰^,⁰^,⁰^, ²

) ( )

² ²^,

2 ϕ ε ε ε ε ε ε

ε ≤ ≤c <

which is a contradiction. Therefore {y₂_n}_n_∈_ω is a Cauchy sequence inX. It follows from completeness of (X,d)that {y_n}_n_∈_ω converges to a point u∈X .

Now, suppose that t is continuous. Since g and t are compatible of type (A) and

ω

∈ + n

gx n }

{ ₂ ₁ and {tx₂_n₊₁}_n_∈_ω converge to the point ,u by Lemma 2.3 we get that tu

ttx

gtx₂_n₊₁, ₂_n₊₁ → as n→ ∞. In virtue of (b5), we have

( ) ( ( ) ( ) ( )

( ) ( )

( ) ( ))

.

, , , ,

, , , , ,

,

2 1 2 1

2 2

2 1 2 1

2 1 2

1 2 2 1

2 1 2

2 1 2 2

2

1 2 2 2

2

1 2 1 2 2

2

2 1 2 1

2 2

1 2 2 1

2 2

1 2 1 2 1

2 2

2 2 1

2 2 2

1 2 2

,a ,fx ttx d ,a ,gtx hx d

, ,a ,fx ttx d ,a ,gtx ttx d

, ,a ,gtx hx d ,a ,gtx ttx d

, ,a ,fx ttx d ,a ,fx hx d

, ,a ,gtx hx d ,a ,fx hx d

, ,a ,gtx ttx d ,a ,fx hx d

, ,a ,fx ttx d ,a ,ttx hx d

, ,a ,gtx hx d ,a ,ttx hx d

, ,a ,gtx ttx d ,a ,ttx hx d

a fx hx d a ttx hx d a ttx hx d a

gtx fx d

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n

n n n

n n

n

+ +

+

+ +

+

+ +

+ + +

+ +

+ ≤ϕ