International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 2 (2015), 79-86
http://www.etamaths.com

FIXED POINT THEOREMS FOR α−ψ−QUASI CONTRACTIVE
MAPPINGS IN METRIC-LIKE SPACES

VILDAN OZTURK

Abstract. In this paper, we give fixed point theorems for α − ψ−quasi
contractions and α−ψ−p−quasi contractions in complete metric-like spaces.

1. Introduction and preliminaries

Fixed point theory became one of the most interesting area of research in the last
fifty years. Many authors studied contractive type mappings on a complete metric
space which are generalizations of Banach contraction principles. Recently, Samet
et al. [17] introduced the notion of α − ψ contractive mappings and established
some fixed point theorems in complete metric spaces. Later some other authors
generalized α−ψ contractions ([5-7][9-14],[18]).

In last years, many generalizations of the concept of metric spaces are defined
and some fixed point theorems was proved in these spaces.In particular, in 1994,
Matthews introduced the notion of a partial metric space as a part of the study
of denotational semantics of dataflow networks and showed that the Banach con-
traction principle can be generalized to the partial metric context for applications
in program verification ([15]). Later on, many researchers studied fixed point
theorems in partial metric spaces ([1],[2],[8],[16],[20]). Recently, Amini-Harandi
generalized the partial metric spaces by introducing the metric-like spaces and
proved some fixed point theorems in such spaces ([3]). After authors gived some
fixed point theorems in metric-like spaces ([19]).

In this paper, we introduce the notion of α−ψ−quasi contractive mappings in
complete metric-like spaces and in last parts we give α−ψ−p−quasi contraction
in metric like spaces. Our results are generalisations of the many existing results
in the literature.

First we give some definitions and facts about metric-like spaces.

Definition 1. ([3]) A mapping σ : X × X → R+ , where X is a nonempty set,
is said to be metric-like on X if for any x,y,z ∈ X, the following three conditions
hold true:

2010 Mathematics Subject Classification. 47H10.
Key words and phrases. α − ψ−quasi contraction, α − ψ − p−quasi contraction fixed point,

metric-like spaces.

c©2015 Authors retain the copyrights of their papers, and all open
access articles are distributed under the terms of the Creative Commons Attribution License.

79



80 OZTURK

(σ1) σ (x,y) = 0 ⇒ x = y;
(σ2) σ (x,y) = σ (y,x) ,
(σ3) σ (x,z) ≤ σ (x,y) + σ (y,z) .
The pair (X,σ) is called a metric-like space.Then a metric-like on X satisfies

all of the conditions of a metric except that σ (x,x) may be positive for x ∈ X.
Each metric-like σ on X generates a topology τσ on X whose base is the family
of open σ−balls

Bσ (x,ε) = {y ∈ X : |σ (x,y) −σ (x,x)| < ε}
for all x ∈ X and ε > 0.

Then the sequence {xn} in the metric-like space (X,σ) converges to a point
x ∈ X if and only if lim

n→∞
σ (xn,x) = σ (x,x) .

Let (X,σ) and (Y,τ) be metric-like spaces and let f : X → Y be a continuous
mapping. Then

lim
n→∞

xn = x =⇒ lim
n→∞

f(xn) = f(x).

A sequence {xn}
∞
n=0 of elements of X is called σ−Cauchy if limn,m→∞ σ (xn,xm)

exists and is finite.The metric-like space (X,σ) is called complete if for each
σ−Cauchy sequence {xn}

∞
n=0 , there is some x ∈ X such that

lim
n→∞

σ (xn,x) = σ (x,x) = lim
n,m→∞

σ (xn,xm) .

Every partial metric space is a metric-like space. Below we give another example
of a metric-like space.

Example 1. ([3]) Let X = {0, 1} , and let

σ(x,y) =

{
2, if x = y = 0
1, otherwise

Then (X,σ) is a metric-like space, but since σ (0, 0) � σ (0, 1) , then (X,σ) is not
a partial metric space.

2. Fixed Point Results For α−ψ Contractive Mappings

Denote by Ψ the family of nondecreasing functions ψ : [0,∞) → [0,∞) such
that limn→∞ψ

n (t) = 0 for all t > 0.

Lemma 1. If ψ ∈ Ψ, then the following are satisfied.

(a) ψ (t) < t for all t > 0
(b) ψ (0) = 0
(c) ψ is right continuous at t = 0.

Remark 1. (a) If ψ : [0,∞) → [0,∞) is nondecreasing such that
∞∑
n=1

ψn (t) < ∞

for each t > 0, then ψ ∈ Ψ.

(b) If ψ : [0,∞) → [0,∞) is upper semicontinuous such that ψ (t) < t for all
t > 0, then limn→∞ψ

n (t) = 0 for all t > 0.



FIXED POINT THEOREMS 81

Definition 2. ([16])Let T : X → X and α : X ×X → [0,∞) . We say that T is
α-admissible if

x,y ∈ X, α (x,y) ≥ 1 ⇒ α (Tx,Ty) ≥ 1.

Definition 3. Let (X,σ) be a complete metric-like space and T : X → X be a
given mapping. We say that T is an α − ψ−quasi contractive mapping if there
exist α : X ×X → [0,∞) and ψ ∈ Ψ such that

(1) α (x,y) σ (Tx,Ty) ≤ ψ(M(x,y))

for all x,y ∈ X where

M(x,y) = max{σ (x,y) ,σ (x,Tx) ,σ (y,Ty) ,σ (x,Ty) ,σ (y,Tx) ,σ (x,x) ,σ (y,y)} .

Theorem 2. Let (X,σ) be a complete metric-like space and T : X → X be an
α − ψ−quasi contractive mapping. Assume that there exists x0 ∈ X such that
O (x0,∞) = {Tnx0 : n = 0, 1, 2...} is bounded and

(i) α
(
Tix0,T

jx0
)
≥ 1 for all i,j ≥ 0 with i < j,

(ii) T is σ−continuous or

limn→∞ inf α (T
nx0,x) ≥ 1 for any cluster point x of {Tnx0} .

Then T has a fixed point.

Proof. Let x0 ∈ X be such that O (x0,∞) = {Tnx0 : n = 0, 1, 2...} is bounded
and α

(
Tix0,T

jx0
)
≥ 1 for all i,j ≥ 0 with i < j. Define the sequence {xn} in X

by xn+1 = Txn for all n ∈ N∪{0} .
If xn = xn+1 for some n ∈ N, then x∗ = xn is a fixed point of T.
Assume that xn 6= xn+1 for all n ∈ N∪{0} .
Now we shall show {xn} is a σ−Cauchy sequence.
Let δ (xn) = diam ({Txn,Txn+1, ...}) for n = 0, 1, 2, .... Since δ (xn) ≤ δ (x0)

and δ (x0) < ∞
We assert that for n = 0, 1, 2, ...

(2) δ (xn) ≤ ψn (δ (x0)) .

For n = 0, (2) holds. Suppose that (2) holds for n = k. We will show that ( 2)
holds when n = k + 1. Let Txr−1,Txs−1 ∈ {Txk,Txk+1, ...} for any r,s ≥ k + 1.
Then

σ (xr,xs) = σ (Txr−1,Txs−1)

≤ α (xr−1,xs−1) σ (Txr−1,Txs−1)
≤ ψ(M (xr,xs))(3)



82 OZTURK

where

M (xr,xs) = max




σ (xr−1,xs−1) ,σ (xr−1,Txr−1) ,σ (xs−1,Txs−1) ,
σ (xr−1,Txs−1) ,σ (xs−1,Txr−1) ,
σ (xr−1,xr−1) ,σ (xs−1,xs−1)




= max




σ (Txr,Txs) ,σ (Txr,Txr+1) ,σ (Txs,Txs+1)
,σ (Txr,Txs+1) ,σ (Txs,Txr+1) ,

σ (Txr,Txr) ,σ (Txs,Txs)




≤ δ (xk) .

Then by (3),

σ (xr,xs) = σ (Txr−1,Txs−1) ≤ ψ(δ (xk))

≤ ψ
(
ψk (δ (x0))

)
= ψk+1 (δ (x0)) .

Thus (2) is proved for n = 0, 1, 2, ....
Hence from (2) we have limn→∞δ (xn) = 0. Thus {xn} is a σ−Cauchy sequence

in (X,σ) . By the completeness of X , there exists z ∈ X such that limn→∞xn = z,
that is,

(4) lim
n→∞

σ (xn,z) = σ (z,z) = lim
n,m→∞

σ (xn,xm) = 0.

If T is σ−continuous,

lim
n→∞

σ (Txn,Tz) = lim
n→∞

σ (xn+1,Tz) = σ (z,Tz) = 0.

This proves z is a fixed point.
If limn→∞ inf α (T

nx0,x) ≥ 1 for any cluster point x of {Tnx0} , there exists
n0 ∈ N such that α (xn,z) ≥ 1, for all n > n0. Thus,

σ (xn+1,Tz) ≤ σ (Txn,Tz)
≤ α (xn,z) σ (Txn,Tz)
≤ ψ (M (xn,z))(5)

where

M (xn,z) = max

{
σ (xn,z) ,σ (xn,Txn) ,σ (z,Tz) ,σ (xn,Tz) ,

σ (z,Txn) ,σ (xn,xn) ,σ (z,z)

}
If σ (z,Tz) > 0, using upper semicontinuity of ψ,

σ (z,Tz) = limn→∞ sup σ (xn+1,Tz)

≤ limn→∞ sup ψ(M (xn,z))
≤ ψ (σ(z,Tz))
< σ(z,Tz)

which is a contradiction. Thus, we obtain σ (Tz,z) = 0.So Tz = z. �



FIXED POINT THEOREMS 83

Example 2. Let X = {0, 1, 2} . Define σ : X ×X → R+ as follows:

σ (0, 0) = 0 σ(1, 1) = 3 σ(2, 2) = 1

σ (0, 1) = σ (1, 0) = 7 σ (0, 2) = σ (2, 0) = 3

σ (1, 2) = σ (2, 1) = 4.

Then (X,σ) is a complete metric-like space. Define the mapping T : X → X by

T0 = 0, T1 = 2 and T2 = 0

and α : X ×X → [0,∞) by

α (x,y) =

{
1
4
, if (x,y) 6= (0, 0)

1, (x,y) = (0, 0)

Then T is an α− ψ-quasi contractive mapping with ψ (t) = t
1+t

.

Moreover, there exists x0 ∈ X such that α
(
Tix0,T

jx0
)
≥ 1, for all i,j ≥ 0

with i < j. So for x0 = 0, we have

α
(
Ti0,Tj0

)
= α (0, 0) = 1.

Obviously (1) is satisfied for all x,y ∈ X.
All hypotheses of Theorem 2 are satisfied. Consequently T has a fixed point.

And x0 = 0 is fixed point of T.
Taking in Theorem 2, α (x,y) = 1 for all x,y ∈ X,we obtain immediately the

following corollaries.

Corollary 3. Let (X,σ) be a complete metric-like space and T : X → X be a
given mapping. Suppose that there exists a function ψ ∈ Ψ such that

σ (Tx,Ty) ≤ ψ(M(x,y))

where

M(x,y) = max{σ (x,y) ,σ (x,Tx) ,σ (y,Ty) ,σ (x,Ty) ,σ (y,Tx) ,σ (x,x) ,σ (y,y)}

for all x,y ∈ X. Then T has a unique fixed point.

Corollary 4. Let (X,σ) be a complete metric-like space and T : X → X be a
given mapping. Suppose that there exists a constant c ∈ (0, 1) such that

σ (Tx,Ty) ≤ cM(x,y)

where

M(x,y) = max{σ (x,y) ,σ (x,Tx) ,σ (y,Ty) ,σ (x,Ty) ,σ (y,Tx) ,σ (x,x) ,σ (y,y)}

for all x,y ∈ X. Then T has a unique fixed point.



84 OZTURK

3. Fixed Point Results For α−ψ −p−Quasi Contractive Mappings

In this section we give α−ψ −p−quasi contraction in consideration of Amini-
Harandi [4].

Definition 4. Let (X,σ) be a complete metric-like space and T : X → X be a
given mapping. We say that T is an α−ψ−p−quasi contractive mapping if there
exist α : X ×X → [0,∞) and ψ ∈ Ψ such that

(6) α (x,y) σ (Tpx,Tpy) ≤ ψ(M(x,y))

for all x,y ∈ X where

M(x,y) = max
{
σ
(
Tiu,Tjv

)
: u,v ∈{x,y} , 0 ≤ i,j ≤ p and i + j < 2p

}
.

Theorem 5. Let (X,σ) be a complete metric-like space and let p ∈ N. Suppose
that T : X → X be an α−ψ−p−quasi contractive map. Assume that there exists
x0 ∈ X such that O (x0,∞) = {Tnx0 : n = 0, 1, 2...} is bounded and

(i) there exists x0 ∈ X such that α
(
Tix0,T

jx0
)
≥ 1 for all i,j ∈ N∪{0} ,

(ii) Tm : X → X is σ−continuous for some m ∈ N.
Then T has a fixed point.

Proof. Let x0 ∈ X be such that O (x,∞) = {x,Tx,...} is bounded and α
(
Tix0,T

jx0
)
≥

1 for all i,j ∈ N∪{0} . Define the sequence {xn} in X by xn+1 = Txn for all n ≥ 0.
Let n be a positive integer with n ≥ p, and let i,j ∈ {p,p + 1, ...,n} . Since T is
an α−ψ −p−quasi contractive map, then

σ
(
Tix,Tjx

)
= σ

(
TpTi−px,TpTj−px

)
≤ α

(
Ti−px,Tj−px

)
σ
(
TpTi−px,TpTj−px

)
≤ ψ{max

{
σ
(
TkTi−px,T lTj−px

)
: 0 ≤ k,l ≤ p and k + l < 2p

}
}

≤ ψ(δ [O (x,n)])(7)

Hence by lemma 1 (a) ,

σ
(
Tix,Tjx

)
< δ [O (x,n)] .

Thus for sufficiently large n ∈ N there exist positive integers k,l with k < p and
p ≤ l ≤ n such that

σ
(
Tkx,T lx

)
= δ [O (x,n)] .

we show that {Tnx} is a σ−Cauchy sequence. Without loss of generality assume
that p ≤ n < m.Then, from (7)

σ (Tnx,Tmx) = σ
(
TpTn−px,Tm−n+pTn−px

)
≤ α

(
Tn−px,Tm−px

)
σ
(
TpTn−px,Tm−n+pTn−px

)
≤ ψ

(
δ
[
O
(
Tn−px,m−n + p

)])
.

by (7), there exists positive integers k1 and l1 with k1 < p and p ≤ l1 ≤ m−n + p
such that

δ
[
O
(
Tn−px,m−n + p

)]
= σ

(
Tk1Tn−px,T l1Tn−px

)
.



FIXED POINT THEOREMS 85

Similarly we have

σ
(
Tk1Tn−px,T l1Tn−px

)
= σ

(
Tk1+pTn−2px,T l1Tn−px

)
≤ ψ(δ

[
O
(
Tn−2px,m−n + 2p

)]
).

Thus,

σ (Tnx,Tmx) ≤ ψ(δ
[
O
(
Tn−px,m−n + p

)]
) ≤ ψ2(δ

[
O
(
Tn−2px,m−n + 2p

)]
).

Proceeding in this manner, we obtain

σ (Tnx,Tmx) ≤ ψ[
n
p ]
(
δ

[
O

(
Tn−[

n
p ]px,m−n +

[
n

p

]
p

)])
≤ ψ[

n
p ] (δ [O (x,m + p)]) .

Hence

σ (Tnx,Tmx) ≤ ψ[
n
p ] (δ [O (x,∞)]) .

By definition of ψ, limn→∞σ (T
nx,Tmx) = 0. Hence we conclude that {Tnx} is

a σ−Cauchy sequence. By the completeness of X, there is some u ∈ X such that

lim
n→∞

σ (Tnx,u) = lim
n→∞

σ (Tnx,Tmx) = σ (u,u) = 0 for each x ∈ X.

Now we show that Tu = u. By the continuity of Tm,

lim
n→∞

σ
(
Tm+nx,Tu

)
= σ (u,Tu) = 0.

Hence, u = Tu. �

Example 3. Let X = [0,∞) and σ : X ×X → [0,∞) be defined

σ (x,y) =

{
0, x = y

max{x,y} , otherwise
.

Then, (X,σ) is a complete metric-like space. Let Q and Q′denote respectively
the set of rational numbers and irrational numbers. Let T : [0,∞) → [0,∞) and
α : X ×X → [0,∞) be defined by

T (x) =

{√
2, x ∈ Q√
3, otherwise

α (x,y) =

{
1, x ∈ Q′

0, otherwise

Then T is an α− ψ-2−contractive mapping with ψ (t) = t
1+t

.

Then T2 (x) =
√

3 for each x ∈ X. Moreover T is discontinuous and T2 is
continuous. Then all conditions of Theorem 5 are satisfied. And x =

√
3 is fixed

point of T.



86 OZTURK

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Department Of Mathematics, Faculty Of Science And Arts, Artvin Coruh Univer-

sity, Artvin, Turkey