International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 1 (2015), 16-21
http://www.etamaths.com

A NOTE ON FIXED POINT THEORY FOR CYCLIC WEAKER

MEIR–KEELER FUNCTION IN COMPLETE METRIC SPACES

STOJAN RADENOVIĆ

Abstract. In this paper we consider, discuss, improve and complement recent

fixed points results for so-called cyclical weaker Meir-Keeler functions, estab-

lished by Chi-Ming Chen [Chi-Ming Chen, Fixed point theory for the cyclic
weaker Meir-Keeler function in complete metric spaces, Fixed Point Theory

Appl., 2012, 2012:17]. In fact, we prove that weaker Meir-Keeler notion is

superfluous in results.

1. Introduction and preliminaries

The Banach contraction principle [1] has various applications in many branches
of applied science. It ensures the existence and uniqueness of fixed point of a
contraction on a complete metric space. After this interesting principle, several
authors generalized it by introducing the various contractions on metric spaces
(see, e.g., [2]-[14]). Rhoades [19], in his work compare several contractions defined
on metric spaces.

Cyclic representations and cyclic contractions were introduced by Kirk et al. [9]
and further used by several authors to obtain various interesting and significant
fixed point results (see, e.g., [2], [3], [8], [11],[12], [13], [14], [16]-[18]). However, we
have proved ([16]-[18]) the following result:
• If some ordinary fixed point theorem in the setting of complete metric spaces

has a true cyclic-type extension, then these both theorems are equivalent.
In this paper we prove the similar things. Namely, we consider, discuss, improve

and complement recent fixed points results for so-called cyclical weaker Meir-Keeler
functions, established by Chi-Ming Chen in [4]. In fact, we prove that weaker Meir-
Keeler notion introduced in [4], is superfluous in results.

It is well known that a function ψ : [0, +∞) → [0, +∞) is said to be a Meir-
Keeler function if for each η > 0, there exists δ > 0 such that for t ∈ [0, +∞) with
η ≤ t < η + δ, we have ψ (t) < η. Chi-Ming Chen introduced weaker Meir-Keeler
function:

Definition 1.1. [4] The function ψ : [0, +∞) → [0, +∞) is said to be a weaker
Meir-Keeler function for each η > 0, there exists δ > 0 such that for t ∈ [0, +∞)
with η ≤ t < η + δ, there exists n0 ∈ N such that ψn0 (t) < η.

Also in [4], the author assume the following conditions for a weaker Meir-Keeler
function ψ : [0, +∞) → [0, +∞) :

2010 Mathematics Subject Classification. 47H10, 54H25.
Key words and phrases. Fixed point theory; weaker Meir-Keeler function; cyclic type-

contraction; Cauchy sequence.

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.

16



FIXED POINT THEORY FOR CYCLIC WEAKER MEIR–KEELER FUNCTION 17

(ψ1) ψ (t) > 0 for t > 0 and ψ (0) = 0;
(ψ2) for all t ∈ [0,∞),{ψ

n (t)}n∈N is decreasing;
(ψ3) for tn ∈ [0,∞), we have that:
(a) if limn→∞ tn = γ > 0, then limn→∞ψ (tn) < γ, and
(b) if limn→∞ tn = 0, then limn→∞ψ (tn) = 0.
Chi-Ming Chen in [4] suppose that ϕ : [0, +∞) → [0, +∞) is a non-decreasing

and continuous function satisfying:
(ϕ1) ϕ (t) > 0 for t > 0 and ϕ (0) = 0;
(ϕ2) ϕ is subadditive, that is, for every µ1,µ2 ∈ [0, +∞),ϕ (µ1 + µ2) ≤ ϕ (µ1) +

ϕ (µ2) ;
(ϕ3)for all t ∈ (0,∞) , limn→∞ tn = 0 if and only if limn→∞ϕ (tn) = 0.
Author state the notion of cyclic weaker (ψ �ϕ)−contraction as follows:
Definition 1.2. [4] Let (X,d) be a metric space, m ∈ N, A1, ...,Am be nonempty

subsets of X and X = ∪mi=1Ai. An operator f : X → X is called a cyclic weaker
(ψ �ϕ)−contraction if:

(i) X = ∪mi=1Ai is a cyclic representation of X with respect to f;
(ii) for any x ∈ Ai,y ∈ Ai+1, i ∈{1, 2, ...,m} ,

(1.1) ϕ (d (fx,fy)) ≤ ψ (ϕ (d (x,y))) ,

where Am+1 = A1.
In [4] author proved the following:
Theorem 1.3. Let (X,d) be a complete metric space, m ∈ N,A1, ...,Am be

nonempty closed subsets of X and X = ∪mi=1Ai. Let f : X → X be a cyclic weaker
(ψ �ϕ)−contraction. Then, f has a unique fixed point z ∈∩mi=1Ai.:

The cyclic weaker (ψ,ϕ)−contraction is defined in [4]:
Definition 1.4. Let ψ : [0,∞) → [0,∞) be a weaker Meir-Keeler function

satisfying conditions (ψ1) , (ψ2) and (ψ3) . Also, let ϕ : [0,∞) → [0,∞) be a non-
decreasing and continuous function satisfying (ϕ1) .

Definition 1.5. Let (X,d) be a metric space, m ∈ N, A1, ...,Am be nonempty
subsets of X and X = ∪mi=1Ai. An operator f : X → X is called a cyclic weaker
(ψ,ϕ)−contraction if:

(i) X = ∪mi=1Ai is a cyclic representation of X with respect to f;
(ii) for any x ∈ Ai,y ∈ Ai+1, i ∈{1, 2, ...,m} ,

(1.2) d (fx,fy) ≤ ψ (d (x,y)) −ϕ (d (x,y)) ,

where Am+1 = A1.
In [4] author proved the following result for this type of operator:
Theorem 1.6. Let (X,d) be a complete metric space, m ∈ N, A1, ...,Am be

nonempty closed subsets of X and X = ∪mi=1Ai. Let f : X → X be a cyclic weaker
(ψ,ϕ)−contraction. Then, f has a unique fixed point z ∈∩mi=1Ai.

Here we will use the following (new, useful and very significant) result for the
proofs of cyclic-type results (see also [15]-[18]):

Lemma 1.7. Let (X,d) be a metric space, f : X → X be a mapping and let
X = ∪pi=1Ai be a cyclic representation of X w.r.t. f. Assume that

(1.3) lim
n→∞

d (xn,xn+1) = 0,

where xn+1 = fxn,x1 ∈ A1. If {xn} is not a Cauchy sequence then there exist
ε > 0 and two sequences {m (k)} and {n (k)} of positive integers such that the



18 RADENOVIĆ

following sequences tend to ε+ when k →∞ :
(1.4) d

(
xm(k)−j(k),xn(k)

)
, d
(
xm(k)−j(k)+1,xn(k)

)
, d
(
xm(k)−j(k),xn(k)+1

)
,

d
(
xm(k)−j(k)+1,xn(k)+1

)
, where j (k) ∈{1, 2, ...,p} is chosen so that n (k)−m (k)+

j (k) ≡ 1 (mod p) , for each k ∈ N.

2. Main results

In this section, first of all, we announce the following remarks:
(a) Author in [4] has not the assumption that the function ψ is a non-decreasing.

However, from the proof of both Theorems follows that he use this fact (page 3, lines
22-25; page 6, lines 15-18).

(b) Further, from (ψ2) and (ψ3) , (a) we follows that ψ
n (t) → 0 (as n →∞) for

all t ∈ [0,∞).
Proof. Indeed, there exists limn→∞ψ

n (t) = γ ≥ 0. If γ > 0, then
(2.1) γ = lim

n→
ψn+1 (t) = lim

n→∞
ψ (ψn (t)) < γ

(by (ψ3) , (a)). A contradiction. �
(c) Since, must non-decreasing and ψn (t) ↓ 0 as n → ∞ for all t ∈ [0,∞) we

easy obtain that ψ (t) < t for t > 0.
(d) Further, we have that d (xn+1,xn) → 0 (as n →∞)without using the notion

of a weaker Meir-Keeler function. That is, lines 26-33 on page 3 are superfluous.
(e) Now, according to Lemma 1.7. one can obtain much shorter proof of Theorem

1.3. Namely, we do not use the property (ϕ2) of the function ϕ.
Proof. Indeed, putting x = xm(k)−j(k),y = xn(k) in (1.1) we obtain a contra-

diction:

(2.2) ϕ
(
d
(
fxm(k)−j(k),,fxn(k)

))
≤ ψ

(
ϕ
(
d
(
xm(k)−j(k),,xn(k)

)))
that is.,

(2.3) ϕ
(
d
(
xm(k)−j(k)+1,xn(k)+1

))
≤ ψ

(
ϕ
(
d
(
xm(k)−j(k),xn(k)

)))
.

Now, passing to limit as k →∞ and using the properties of ϕ and ψ, follows
(2.4) ϕ (ε) ≤ lim

k→∞
ψ
(
ϕ
(
d
(
xm(k)−j(k),xn(k)

)))
< ϕ (ε) .

Hence, {xn} is a Cauchy sequence. �
(e’) Similarly, putting x = xm(k)−j(k),y = xn(k) in (1.2) we obtain again a

contradiction:
(2.5)
d
(
xm(k)−j(k)+1,xn(k)+1

)
≤ ψ

(
d
(
xm(k)−j(k),xn(k)

))
−ϕ

(
d
(
xm(k)−j(k),xn(k)

))
.

Letting to limit as k →∞ and using again the properties of ϕ and ψ, we have
(2.6) ε ≤ lim

k→∞
ψ
(
d
(
xm(k)−j(k),xn(k)

))
−ϕ (ε) < ε−ϕ (ε) .

This means that {xn} is a Cauchy sequence. �
By the same method as in [16]-[18] one can prove the following two results:
Theorem 2.1. Theorem 1.3. is a equivalent with the following:
• Let (X,d) be a complete metric space and let f : X → X be a weaker

(ψ �ϕ)−contraction, that is.,
(2.7) ϕ (d (fx,fy)) ≤ ψ (ϕ (d (x,y))) ,
for all x,y ∈ X. Then, f has a unique fixed point z ∈ X.



FIXED POINT THEORY FOR CYCLIC WEAKER MEIR–KEELER FUNCTION 19

Theorem 2.2. Theorem 1.6. is a equivalent with the following:
• Let (X,d) be a complete metric space and let f : X → X be a weaker

(ψ,ϕ)−contraction, that is.,
(2.8) d (fx,fy) ≤ ψ ((d (x,y))) −ϕ (d (x,y)) ,
for all x,y ∈ X. Then, f has a unique fixed point z ∈ X.

Conclusion: In all previous results, that is in Theorems 3 and 4 of [4] it is
sufficient that the functions ψ and ϕ satisfy the following conditions:

1. ψ : [0,∞) → [0,∞) is a non-decreasing function satisfying (ψ1) , (ψ2)and
(ψ3) ;

2. ϕ : [0, +∞) → [0, +∞) is a non-decreasing and continuous function satisfying
(ϕ1) and (ϕ3) .

Hence, without weaker Meir-Keeler property for ψ as well as without the subad-
ditivity for ϕ.

In the sequel we announce the following two results generalizing Theorems 1.3.
and 1.6. above, that is., Theorems 3 and 4 from [4]. Firstly, we define:

Definition 2.3. Let (X,d) be a metric space, m ∈ N, A1, ...,Am be nonempty
subsets of X and X = ∪mi=1Ai. An operator f : X → X is called a cyclic generalized
(ψ �ϕ)−contraction (resp. cyclic generalized (ψ,ϕ)−contraction) if:

(i) X = ∪mi=1Ai is a cyclic representation of X with respect to f;
(ii) for any x ∈ Ai,y ∈ Ai+1, i ∈{1, 2, ...,m} ,

(2.9) ϕ (d (fx,fy)) ≤ ψ (ϕ (M (x,y))) ,
where Am+1 = A1

(2.10) (resp. d (fx,fy) ≤ ψ (M (x,y)) −ϕ (M (x,y))),

where M (x,y) = max
{
d (x,y) ,d (x,fx) ,d (y,fy) ,

d(x,fy)+d(y,fx)
2

}
(iii) ψ,ϕ : [0,∞) → [0,∞) are functions satisfying 1. and 2. from above

Conclusion.
Theorem 2.4. Let (X,d) be a complete metric space, m ∈ N,A1, ...,Am be

nonempty closed subsets of X and X = ∪mi=1Ai. Let f : X → X be a cyclic gener-
alized (ψ �ϕ)−contraction (resp. cyclic generalized (ψ,ϕ)−contraction). Then, f
has a unique fixed point z ∈∩mi=1Ai.

Proof. Given x0 ∈ X and let xn+1 = fxn, for n ∈ {0, 1, ..} . Picard sequence.
If there exists n0 ∈ {0, 1, ...} such that xn0+1 = xn0, then we finished the proof.
Therefore, let xn+1 6= xn for all n ∈{0, 1, ...} . It is clear, that for any n ∈{1, 2, ...}
there exists in ∈{1, 2, ...,m} such that xn−1 ∈ Ain and xn ∈ Ain+1. Since f : X →
X is a cyclic generalized (ψ �ϕ)−contraction, we have that for all n ∈{0, 1, ...}
(2.11) ϕ (d (xn,xn+1)) = ϕ (d (fxn−1,fxn)) ≤ ψ (ϕ (M (xn−1,xn))) ,
where

M (xn−1,xn) = max

{
d (xn−1,xn) ,d (xn−1,xn) ,d (xn,xn+1) ,

d (xn−1,xn+1) + d (xn,xn)

2

}
(2.12)

= max

{
d (xn−1,xn) ,d (xn,xn+1) ,

d (xn−1,xn+1)

2

}
≤ max{d (xn−1,xn) ,d (xn,xn+1)} .

If d (xn,xn+1) > d (xn−1,xn) then from (2.11) follows (because ψ (t) < t,t > 0):

(2.13) ϕ (d (xn,xn+1)) ≤ ψ (ϕ (d (xn,xn+1))) < ϕ (d (xn,xn+1)) .



20 RADENOVIĆ

A contradiction.
Therefore, for all n ∈{0, 1, ...} we obtain (because ψ is nondecreasing):

(2.14) ϕ (d (xn,xn+1)) ≤ ψ (ϕ (d (xn−1,xn))) .
That is., we have that

(2.15) ϕ (d (xn,xn+1)) ≤ ψ (ϕ (d (xn−1,xn))) ≤ ... ≤ ψn (ϕ (d (x0,x1))) .
Hence, ϕ (d (xn,xn+1)) → 0, i.e., d (xn,xn+1) → 0 as n →∞.

Next, we claim that {xn} is a Cauchy sequence. If this is not case, then according
to Lemma 1.7. by putting in x = xm(k)−j(k),y = xn(k) in (2.9) we have:

(2.16) ϕ
(
d
(
xm(k)−j(k)+1,xn(k)+1

))
≤ ψ

(
ϕ
(
M
(
xm(k)−j(k),xn(k)

)))
,

where

(2.17)

M
(
xm(k)−j(k),xn(k)

)
= max

{
d
(
xm(k)−j(k),xn(k)

)
,d
(
xm(k)−j(k),xm(k)−j(k)+1

)
,d
(
xn(k),xn(k)+1

)
,

d
(
xm(k)−j(k),xn(k)+1

)
+ d

(
xm(k)−j(k)+1,xn(k)

)
2

}
.

First of all, we have

(2.18) lim
k→∞

M
(
xm(k)−j(k),xn(k)

)
= max

{
ε, 0, 0,

ε + ε

2

}
= ε,

that is.,

(2.19) lim
k→∞

ϕ
(
M
(
xm(k)−j(k),xn(k)

))
= ϕ

(
lim
k→∞

M
(
xm(k)−j(k),xn(k)

))
= ϕ (ε) .

Further from (2.17) as well as by the properties of the functions ψ and ϕ follows:
(2.20)
0 < ϕ (ε) ≤ lim

k→∞
ψ
(
ϕ
(
M
(
xm(k)−j(k),xn(k)

)))
< lim

k→∞
M
(
xm(k)−j(k),xn(k)

)
= ϕ (ε) .

A contradiction.
Hence {xn} is a Cauchy sequence.
The rest of the proof is further as in any of papers [16]-[18].
The proof for the case of cyclic generalized (ψ,ϕ)−contraction is very similar.

�
Finaly, we announce the following important and significant remark regarding

several proofs that Picard sequence {xn} is a Cauchy:
Remark 2.5. Using our Lemma 1.7. we can obtain much shorter proofs that

Picard sequence xn+1 = fxn, in each of the papers [2], [3], [6], [7], [9], [12], [11] and
[13] is a Cauchy. For this, it is sufficient putting x = xm(k)−j(k),y = xn(k) in the
contractive condition of cyclic type theorem in each of the papers.

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FIXED POINT THEORY FOR CYCLIC WEAKER MEIR–KEELER FUNCTION 21

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Faculty of Mathematics and Information Technology, Teacher Education, Dong
Thap University, Cao Lanch City, Dong Thap Province, Viet Nam