376 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 
 

Fixed Point Results and Application of Cyclic Contractive Maps in b-

Metric Spaces 

 

 Abbas Karim Nahi1*      and  Salwa Salman Abed2  
1,2Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of   

Baghdad, Baghdad, Iraq. 

*Corresponding Author. 

 

          Received:10 November 2023               Accepted:16 January 2024                     Published: 20 April 2025 
 
 

Abstract   

     One of the generalizations of the usual metric function is the b-metric, which provides 

researchers with a broader field for deriving numerous results and applications related to fixed 

point theory. The aim of this paper is to develop three new fixed point principles in the complete 

b-metric space (£,𝜌) when 𝜌 is a continuous function in two variables. Here there are three 

directions to prove the existence and uniqueness of fixed points. First, we derive a result in 

terms of Branciari’s theorem by combining integral contractive conditions with the notion of a 

cyclic map. Second, we apply the notion of cyclic representation to maps satisfying general 

weak conditions, including a changing distance function, to simulate the content of  Boyd and 

Wong's theorem. Using this result, an application to the existence and uniqueness of the 

solution of an integral equation is given. Finally, an implicit relation with a changing distance 

function is used to construct a cyclic contractive map. Some examples are also presented to 

analyze and illustrate the main results. 

Keywords: Alternating distance functions, Complete b-metric spaces, Contractive conditions, 

Cyclic representation, Fixed points. 

 

1. Introduction 

In 2003, Cyclic contraction was first introduced by Kirk et al. (1), who introduced results 

dealing with mappings of the type 𝑓: £𝑖 → £𝑖+1, 𝑖 =  1, 2, · · ·, 𝑝 +  1, with £𝑖 = £𝑖+1, where 

the contractive hypotheses are restricted to pairs (𝑎, 𝑏) ∈ £𝑖 × £𝑖+1. 

 Extensions of the Banach’s theorem and an extension of the Caristi theorem were proved. In 

addition, results related to non-expansive mappings in a Banach space were included. Several 

authors have contributed to research on fixed points in various cases for cyclic contractions  (2-

9). Backhtin (10) presented a definition of b-metric by replacing the triangle inequality as the 

following  

Definition1.1:  Let £ be a nonempty set and 𝑞 ≥  1. A function 𝜌 ∶  £ ×  £ →  𝑅+ is said to 

be a b-metric on £ if (10):  

1. 𝜌(𝑎, 𝑏)  =  0 if and only if 𝑎 =  𝑏;  

doi.org/10.30526/38.2.3828 

 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
mailto:https://orcid.org/0009-0006-3158-1348
mailto:https://orcid.org/0000-0002-0581-253X
mailto:salwa.s.a@ihcoedu.uobaghdad.edu.iq
mailto:abbas.nahi2103m@ihcoedu.uobaghdad.edu.iq


IHJPAS. 2025,38(2) 

377 
 

2. 𝜌(𝑎, 𝑏)  =  𝜌(𝑏, 𝑎) for all 𝑎, 𝑏 ∈  £;  

3.  𝜌(𝑎, 𝑏)  ≤  𝑞(𝜌(𝑎, 𝑐)  + 𝜌(𝑐, 𝑏)) for all 𝑎, 𝑏, 𝑐 ∈  £.     

The pair (£, 𝜌) is called a b-metric space. 

Definition 1.2:  Let (£, 𝜌) be a b-metric space, 𝑎 ∈  £ and (𝑎𝑛) be a sequence in £. Then  

1. (𝑎𝑛)  converges to 𝑎 if and only if lim
𝑛→∞

𝜌(𝑎𝑛, 𝑎) =  0. We denote this by 𝑙𝑖𝑚
𝑛→∞

𝑎𝑛  = 𝑎 or 

𝑎𝑛  →  𝑎( as 𝑛 → ∞). 

2. (𝑎𝑛)   is Cauchy if and only if lim
𝑛,𝑚→∞

𝜌(𝑎𝑛, 𝑎𝑚)  =  0.  

3. (£, 𝜌) is complete if and only if every Cauchy sequence in  £ is convergent. 

Remark 1.3:  In a b-metric space (£, 𝜌), the following assertions hold(11,12): 

1. A convergent sequence has a unique limit. 

2. Each convergent sequence is Cauchy.  

3. In general, a b-metric is not continuous. 

As in the usual metric space )1(, we reform the following definition: 

Definition1.4:  Let {£𝑖}𝑖=1
𝑛 be a nonempty subsets of a b-metric space (£, 𝜌), £ = ⋃ £𝑖

𝑛
𝑖=1  and 

𝑓: £ → £  such that  

1) 𝑓(£1) ⊂ £2, … , 𝑓(£𝑛−1) ⊂ £𝑛, 𝑓(£𝑛) ⊂ £1 for 1 ≤ 𝑖 ≤ 𝑛; 

2) ∃𝑘 ∈ (0,1) such that 𝜌(𝑓𝑎, 𝑓𝑏) ≤  𝑘 𝜌(𝑎, 𝑏) ∀𝑎 ∈ £𝑖 , 𝑏 ∈ £𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 

Then 𝑓 is the cyclic contraction map and £ is cyclic representation w.r.t., 𝑓. 

In the field of fixed point theory for cyclicity, see (13-20) 

Example1.5: Let  £ = [−1,1], 𝜌(𝑎, 𝑏) = |𝑎 − 𝑏|2 is b-metric with 𝑠 = 2 and  £1  =

[−1,0], £2 = [0,1], £3 = [−1,0], £4 = [0,1], £5 = [−1,0], £6 = [0,1]. So,  £ = ⋃ £𝑖
6
𝑖=1 . 

Define 𝑓:⋃ £𝑖
6
𝑖=1  → ⋃ £𝑖

6
𝑖=1  such that 𝑓𝑎 = −

𝑎

2+𝑎
, ∀𝑎 ∈  ⋃ £𝑖

6
𝑖=1 . Here £ is cyclic 

representation w.r.t.,𝑓.  (𝑓(£1) ⊂ £2, 𝑓(£2) ⊂ £3, 𝑓(£3) ⊂ £4, and 𝑓(£4) ⊂ £5, 𝑓(£5) ⊂

£6, 𝑓(£6) ⊂ £1) and 𝑓is cyclic contraction with constant 0 < 𝑘 =  
1

2
, where 𝑎 ∈ £𝑖 𝑏 ∈ £𝑖+1.  

Definition 1.6:  A point 𝑎 is called a fixed point of a map 𝑓: £ → £  if 𝑓(𝑎) = 𝑎 (21). 

An example, 𝑎 = 0 is a fixed point of 𝑓 in the previous example. 

This work includes four main fixed point theorems based on different cyclic contractive 

conditions. Here, (£, ρ) denote to complete b-metric space where ρ is continuous. 

 

2. Materials and Methods 

     In this section, there are three axes; the first one is  

Fixed point for cyclic contractive maps with integral condition 

Not that, A Lebesgue-integrable function ¥: [0,1) → [0,1) is called summable if ∫¥(𝑟)𝑑𝑟 <

∞ (22). 

Theorem 2.1: Let (£, 𝜌) be b-metric space, 𝑘 ∈ (0,1), and let 𝑓: £ →  £ be a map such that for 

each 𝑎, 𝑏 ∈ £ 

∫ ¥(𝑟)
𝜌(𝑓𝑎,   𝑓𝑏)

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)𝑑𝑟

𝜌(𝑎,   𝑏)

0
, ∀𝑎 ∈ £𝑖, 𝑏 ∈ £𝑖+1                                                 (1) 

where, ¥ is summable on each compact subset of [0,∞), nonnegative and for any 𝜀 >

0, ∫ ¥(𝑟)
ε

0
𝑑𝑟 > 0. Then ∃! 𝑐 ∈ ⋂ £𝑖

𝑛
𝑖=1 , moreover, ∀𝑎 ∈ £,  lim

𝑚→∞
𝑓𝑚(𝑎) = 𝑐. 

Proof:  Let 𝑎0 ∈  £ = ⋃ £𝑖
𝑛
𝑖=1  and consider 𝑎𝑚+1 = 𝑓𝑎𝑚 for each 𝑚 ∈ 𝑁⋃{0}, so for any 𝑚 ∈

𝑁⋃{0},  ∃𝑖𝑚 ∈ {1,2, … , 𝑛} such that 𝑎𝑚 ∈ £𝑖𝑚  and 𝑎𝑚+1 ∈ £𝑖𝑚+1.  

If 𝑎𝑚0
= 𝑎𝑚0+1 for some 𝑚0 then, since 𝑎𝑚0+1 = 𝑓𝑎𝑚0

= 𝑎𝑚0
 this mean 𝑎𝑚0

 is a fixed point 



IHJPAS. 2025,38(2) 

378 
 

of  𝑓. Thus, assume that 𝑎𝑚 ≠ 𝑎𝑚+1 for all 𝑚 ∈ 𝑁⋃{0}. So, 

∫ ¥(𝑟)
𝜌(𝑓𝑎𝑚,   𝑓𝑎𝑚+1)

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)

𝜌(𝑎𝑚, 𝑎𝑚+1)

0
𝑑𝑟.                                                                (2) 

By repeating the inequality (2) m times, it follows directly 

∫ ¥(𝑟)
𝜌(𝑓𝑎𝑚,   𝑓𝑎𝑚+1)

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)

𝜌(𝑎𝑚, 𝑎𝑚+1)

0
𝑑𝑟 = 𝑘𝑚 ∫ ¥(𝑟)

𝜌(𝑎0,   𝑓𝑎0)

0
𝑑𝑟. 

As consequence, since 𝑘 ∈ (0,1), obtaining a monotone decreasing sequence 

(∫ ¥(𝑟)
𝜌(𝑓𝑎𝑚,   𝑓𝑎𝑚+1)

0
𝑑𝑟) = (∫ ¥(𝑟)

𝜌(𝑎𝑚+1, 𝑎𝑚+2)

0
𝑑𝑟) which has lower bound is 0. We have 

𝜌(𝑓𝑎𝑚, 𝑓𝑎𝑚+1) → 0 as𝑚 →  ∞. By properties of real sequence, (∫ ¥(𝑟)
𝜌(𝑓𝑎𝑚,   𝑓𝑎𝑚+1)

0
𝑑𝑟) 

convergences 𝜀 ≥ 0 such that lim
𝑚→∞

∫ ¥(𝑟)
𝜌(𝑎𝑚+1,𝑎𝑚+2)

0
𝑑𝑟 = 𝜀.  Suppose that   𝜀 > 0, it is 

enough to assume lim
𝑚→∞

𝑠𝑢𝑝 𝜌(𝑎𝑚+1, 𝑎𝑚+2) = 𝜀 > 0. Then there exists a 𝑢𝜀𝜖𝑁 and a sequence 

(𝑓𝑎𝑚𝑢
)𝑢≥𝑢𝜀 such that  𝜌(𝑓𝑎𝑚𝑢

, 𝑓𝑎𝑚𝑢+1) → 𝜀 > 0  as  𝑢 → ∞   and  𝜌(𝑓𝑎𝑚𝑢
, 𝑓𝑎𝑚𝑢+1) ≥

𝜀

2
. 

For each 𝑢 ≥ 𝑢𝜀, by 
𝜀

2
≤ lim

𝑚→∞
𝜌(𝑓𝑎𝑚𝑢

, 𝑓𝑎𝑚𝑢+1) = lim
𝑚→∞

sup 𝜌(𝑓𝑎𝑚𝑢
, 𝑓𝑎𝑚𝑢+1) = 𝜀 > 0, 

this is true only if 𝜀 = 0. The next step is proving that for each 𝑎0 ∈ £, (𝑎𝑚) is a Cauchy 

sequence. For 𝑗 >  𝑛 define  £𝑗 = £𝑖 if  𝑗 =  𝑖 mod 𝑛. 

Claim I: for all 𝜀 > 0 there exist 𝑚 ∈ 𝑁 such that for all 𝑗, 𝑖 ≥ 𝑚, 𝑗 − 𝑖 ≡ 1(mod 𝑛) 

then 𝜌(𝑎𝑗, 𝑎𝑖) < 𝜀. 

Suppose that there exists 𝜀 > 0 such that for each 𝑚 ∈ 𝑁, one can find 𝑗 > 𝑖 > 𝑚 with 𝑗 − 𝑖 ≡

1(mod 𝑛) satisfying 𝜌(𝑎𝑗, 𝑎𝑖) ≥ 𝜀. Clearly, 𝜌(𝑎𝑚, 𝑎𝑚+1) < 𝜀.  

Now, take 𝑚 ≥ 2(mod 𝑛). Then, corresponding to 𝑖 ≥ 𝑚 use can choose 𝑗 in such a way that 

it is the smallest integer with 𝑗 > 𝑖 satisfying 𝑗 − 𝑖 ≡ 1(mod 𝑛) and 𝜌(𝑎𝑗 , 𝑎𝑖) ≥ 𝜀. Therefore, 

𝜌(𝑎𝑗−𝑛, 𝑎𝑖) ≤ 𝜀. By triangular inequality  

𝜀 ≤ 𝜌(𝑎𝑗 , 𝑎𝑖) ≤ 𝑞(𝜌(𝑎𝑗 , 𝑎𝑖−𝑛) + ∑ 𝜌(𝑎𝑖−𝑘, 𝑎𝑖−𝑘+1)
𝑛
𝑘=1 )  

≤ 𝑞(∑ 𝜌(𝑎𝑖−𝑘, 𝑎𝑖−𝑘+1)
𝑛
𝑘=1 ) + 𝑞 𝜀 → 𝑞 𝜀  as  𝑛 → ∞. 

Again, by triangular inequality 

𝜀 ≤ 𝜌(𝑎𝑗 , 𝑎𝑖) ≤ 𝑞 𝜌(𝑎𝑗 , 𝑎𝑗+1) + 𝑞𝜌(𝑎𝑗+1, 𝑎𝑖+1) + 𝑞𝜌(𝑎𝑖+1, 𝑎𝑖) → 𝑞 ε    as 𝑗, 𝑖 → ∞, 

we get             

  ∫ ¥(𝑟)
𝜌(𝑓𝑎𝑗+1,𝑓𝑎𝑖+1)

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)

𝜌(𝑎𝑗+1,𝑎𝑖+1)

0
𝑑𝑟.                                                                (3) 

Letting 𝑗, 𝑖 → ∞ implies to ∫ ¥(𝑟)
𝑞𝜀

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)

𝑞𝜀

0
𝑑𝑟, which is a contradiction. Therefore, 

the (Claim I) is proved. 

Now, to prove (𝑎𝑚) is Cauchy sequence in (£, 𝜌). Fix ε > 0. By the claim, ∃𝑚0 such that if 

𝑗, 𝑖 ≥ 𝑚0 with 𝑗 − 𝑖 ≡ 1(mod 𝑛)  𝜌(𝑎𝑗 , 𝑎𝑖) ≤
ε

𝑛
.                                          

Since lim
𝑚→∞

𝜌(𝑎𝑚, 𝑎𝑚+1) = 0, ∃𝑚1 ∈ 𝑁 such that 𝜌(𝑎𝑚, 𝑎𝑚+1) ≤
𝜀

𝑛
, ∀𝑚 ≥ 𝑚1 

 Suppose 𝑐, 𝑣 ≥ 𝑚𝑎𝑥{𝑚0, 𝑚1} and 𝑐 > 𝑣. Then there exists ℎ ∈ {1,2, … . . , 𝑛} such that 𝑣 −

𝑐 ≡ ℎ(mod 𝑛). Therefore, 𝑣 − 𝑐 + 𝑟 ≡ 1(mod 𝑛) for 𝑟 = 𝑛 − ℎ + 1. So, getting 

𝜌(𝑎𝑐, 𝑎𝑣) ≤ 𝑞(𝜌(𝑎𝑐, 𝑎𝑣+𝑟) + 𝜌(𝑎𝑣+𝑟 , 𝑎𝑣)). 

≤ 𝑞 𝜌(𝑎𝑐, 𝑎𝑣+𝑟) + 𝑞
2 𝜌(𝑎𝑣+𝑟 , 𝑎𝑣+𝑟−1) + 𝑞

3𝜌(𝑎𝑣+𝑟−1, 𝑎𝑣+𝑟−2) + ⋯+ 𝑞
𝑟𝜌(𝑎𝑣+1, 𝑎𝑣) 

By 𝜌(𝑎𝑗 , 𝑎𝑖) ≤
ε

𝑛
 and 𝜌(𝑎𝑚, 𝑎𝑚+1) ≤

𝜀

𝑛
 and from the last inequality, 

𝜌(𝑎𝑐, 𝑎𝑣) ≤ 𝑞
𝜀

𝑛
+ 𝑞2

𝜀

𝑛
+ 𝑞3

𝜀

𝑛
+⋯+ 𝑞𝑟

𝜀

𝑛
≤ 𝑞

𝜀

𝑛
(
1

1−𝑞
) → 0, as 𝑛 → ∞ 

This proves that (𝑎𝑚) is the Cauchy sequence. The completeness of (£, 𝜌) implies to exist 𝑐 ∈



IHJPAS. 2025,38(2) 

379 
 

£ such that lim
𝑚→∞

𝑎𝑚 = 𝑐.  

To prove 𝑐 is a fixed point for 𝑓. Since £ = ⋃ £𝑖
𝑛
𝑖=1  is a cyclic representation of £ w.r.t., 𝑓, the 

sequence (𝑎𝑚) has  infinite terms in each £𝑖𝑚for 𝑖𝑚  ∈ {1,2, . . . , 𝑛}. Closeness of  £𝑖𝑚 for 𝑖𝑚  ∈

{1,2, . . . , 𝑛} implies to  𝑐 ∈ ⋂ £𝑖
𝑛
𝑖=1 . Suppose that 𝑐 ∈ £𝑖 and  𝑓𝑐 ∈ £𝑖+1. Since (£, 𝜌) is 

complete, there exists appoint  𝑐 ∈ £ = ⋃ £𝑖
𝑛
𝑖=1   such that 𝑐 = lim

 𝑚→∞
𝑓𝑎𝑚    

0 < 𝜌(𝑐, 𝑓𝑐) ≤ 𝑞 𝜌(𝑐, 𝑎𝑚+1) + 𝑞 𝜌(𝑓𝑎𝑚, 𝑓𝑐)  → 0 as  𝑚 → ∞. 

Indeed both 𝜌(𝑐, 𝑓𝑎𝑚𝑟
) and 𝜌(𝑓𝑎𝑚𝑟

, 𝑓𝑐) converge to 0 as  𝑚 → ∞, for the first one it is 

obvious, while for the second one we have 

∫ ¥(𝑟)
𝜌(𝑓𝑎𝑚,𝑓𝑐)

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)

𝜌(𝑎𝑚,𝑐)

0
𝑑𝑟  → 0  as  𝑚 → ∞. 

Now, if  𝜌(𝑓𝑎𝑚, 𝑓𝑐) does not converge to 0 as 𝑚 → ∞, then there exists a subsequence 

(𝑎𝑚𝑟
)𝑟∈𝑁 of (𝑎𝑚) with 𝑎𝑚𝑟

∈ £𝑖−1 such that  𝜌(𝑓𝑎𝑚𝑟
, 𝑓𝑐) ≥ 𝜀 for a certain 𝜀 > 0, we have 

the following contradiction 0 < ∫ ¥(𝑟)
𝜀

0
𝑑𝑟 ≤ ∫ ¥(𝑟)

𝜌(𝑓𝑎𝑚𝑟 ,𝑓𝑐)

0
𝑑𝑟 → 0 as 𝑟 → ∞. This means 

that 𝜌(𝑐, 𝑓𝑐) ≤ 0 thus, 𝜌(𝑐, 𝑓𝑐) = 0, 𝑐 is the fixed point of 𝑓.  

For the uniqueness of fixed point 𝑐. Assume a fixed point 𝑤 of 𝑓 differs from 𝑐, i.e., 𝑓𝑤 = 𝑤. 

The cyclic character of 𝑓 and 𝑐, 𝑤 ∈ £ = ⋃ £𝑖
𝑛
𝑖=1  are fixed points of 𝑓 implying that 𝑐, 𝑤 ∈

⋂ £𝑖
𝑛
𝑖=1 . By (1), we obtain 

∫ ¥(𝑟)
𝜌(𝑐,𝑤)

0
𝑑𝑟 = ∫ ¥(𝑟)

𝜌(𝑓𝑐,𝑓𝑤)

0
𝑑𝑟 ≤ 𝑘 ∫ ¥(𝑟)

𝜌(𝑐,𝑤)

0
𝑑𝑟, 

which is contradiction, consequently, 𝑐 = 𝑤 for each 𝑎 ∈ £, lim
𝑚→∞

𝑓𝑚𝑎 = 𝑐. 

Example 2.2 :Let £ = [−1, 1] and 𝜌(𝑎, 𝑏) = |𝑎 − 𝑏|2 is b-metric with 𝑞 = 2. Suppose £1 =

[−1, 0], £2 = [0, 1] and £ = ⋃ £𝑖
2
𝑖=1 . Define 𝑓:⋃ £𝑖

2
𝑖=1  → ⋃ £𝑖

2
𝑖=1  such that (𝑎) =

−𝑎

2
 ∀𝑎. So, 

𝑓(£1) ⊂ £2, 𝑓(£2) ⊂ £1 and 𝑓 is contraction of integral type with constant 𝑘 =
1

2
   ∈ (0, 1) 

and ¥(𝑟) =
𝑟

3
, for 𝑎 ∈ £1, 𝑏 ∈ £2 

∫
𝑟

3

𝜌(𝑓𝑎,𝑓𝑏)

0
𝑑𝑟 = ∫

𝑟

3

1

2
|𝑏−𝑎|2

0
𝑑𝑟 ≤

1

2
∫

𝑟

3
𝑑𝑟

𝜌(𝑎,𝑏)

0
. 

Hence 𝑓 satisfies the hypothesis of (Theorem 2.1), which has a unique fixed point at 0. 

Fixed point for general cyclic(∅ − 𝜓) weak contractive maps 

Recall the following two definitions: 

Definition 2.3: Let 𝜓 is function where ψ: [0,∞) → [0,∞) is called altering distance function 

if satisfies(23). 

1. 𝜓 is monotone increasing and lower semi-continuous; 

2. 𝜓(𝑟) = 0 if and only if  𝑟 = 0. 

As in usual metric spaces )4(, below, we reform many concepts in a b-metric space 

Definition 2.4: Let (£, 𝜌) be a b-metric space, 𝑛 a positive integer £1, £2, …, £𝑛 nonempty 

closed subsets of £ and £ = ⋃ £𝑖
𝑛
𝑖=1 . An operator 𝑓 ∶ £ → £ is said to be a cyclic weak (∅ −

𝜓)-contraction if 

1) £ = ⋃ £𝑖
𝑛
𝑖=1  is a cyclic representation of £ w.r.t., 𝑓.  

2) ∅(𝜌(𝑓𝑎, 𝑓𝑏))  ≤  ∅(𝜌(𝑎, 𝑏)) −  𝜓(𝜌(𝑎, 𝑏)), for any 𝑎 ∈ £𝑖, 𝑏 ∈ £𝑖+1, 𝑖 =  1, 2, … , 𝑛, 

where £𝑛+1  =  £1 and ∅,𝜓: [0,∞) →  [0,∞) is a non-decreasing and continuous function 

satisfying ∅(𝑟) > 0, 𝜓(𝑟) > 0 for 𝑟 ∈  (0,∞) and ∅(0) =  0, 𝜓(0)  =  0. 

Theorem 2.5: Let 𝑓 be a self-map of  (£, 𝜌) satisfies 

∅(𝜌(𝑓𝑎, 𝑓𝑏)) ≤ ∅(𝑀(𝑎, 𝑏)) − 𝜓(𝑁(𝑎, 𝑏)), ∀𝑎 ∈ £𝑖, 𝑏 ∈ £𝑖+1                                            (4) 

where 



IHJPAS. 2025,38(2) 

380 
 

𝑀(𝑎, 𝑏) = 𝑡 max{𝜌(𝑎, 𝑏), 𝜌(𝑎, 𝑓𝑎), 𝜌(𝑏, 𝑓𝑏), 𝜌(𝑎, 𝑓𝑏), 𝜌(𝑏, 𝑓𝑎)}, 

and 

𝑁(𝑎, 𝑏) = 𝑡 min{𝜌(𝑎, 𝑏), 𝜌(𝑎, 𝑓𝑎), 𝜌(𝑏, 𝑓𝑏), 𝜌(𝑎, 𝑓𝑏), 𝜌(𝑏, 𝑓𝑎)}, 

 where 𝑡 ∈ (0,1), and 𝜓, ∅: [0,∞) → [0,∞) are altering distance functions. Then ∃ 𝑐 ∈

⋂ £𝑖
𝑛
𝑖=1 , 𝑐 is a unique fixed point of 𝑓. 

Proof: Let 𝑎0 ∈ ⋃ £𝑖
𝑛
𝑖=1  and consider 𝑎𝑚+1 = 𝑓𝑎𝑚 for each 𝑚 ∈ 𝑁⋃{0}, so for any 𝑚 ∈

𝑁⋃{0}, ∃𝑖𝑚 ∈ {1, 2, … , 𝑛} such that 𝑎𝑚 ∈ £𝑖𝑚  and 𝑎𝑚+1 ∈ £𝑖𝑚+1.  

If 𝑎𝑚0
= 𝑎𝑚0+1 for some 𝑚0 then, since 𝑎𝑚0+1 = 𝑓𝑚0𝑎0 = 𝑎𝑚0

 this means 𝑎𝑚0
 is fixed point 

of  𝑓. Thus, assume that 𝑎𝑚 ≠ 𝑎𝑚+1 for all 𝑚 ∈ 𝑁⋃{0}. getting 

∅(𝜌(𝑎𝑚, 𝑎𝑚+1)) ≤ ∅(𝑀(𝑎𝑚−1, 𝑎𝑚)) − 𝜓(𝑁(𝑎𝑚−1, 𝑎𝑚)),                                                 (5)  

where  

𝑀(𝑎𝑚−1, 𝑎𝑚) = 𝑡 max {𝜌(𝑎𝑚−1, 𝑎𝑚), 𝜌(𝑎𝑚, 𝑎𝑚+1), 𝜌(𝑎𝑚−1, 𝑎𝑚+1) , 𝜌(𝑎𝑚, 𝑎𝑚)}, 

and 

𝑁(𝑎𝑚−1, 𝑎𝑚) = 𝑡min  {𝜌(𝑎𝑚−1, 𝑎𝑚), 𝜌(𝑎𝑚, 𝑎𝑚+1), 𝜌(𝑎𝑚−1, 𝑎𝑚+1), 0} = 0. 

By triangular inequality, so 

𝑀(𝑎𝑚−1, 𝑎𝑚) = 𝑡 max  {𝜌(𝑎𝑚−1, 𝑎𝑚), 𝜌(𝑎𝑚, 𝑎𝑚+1), 𝑞 𝜌(𝑎𝑚−1, 𝑎𝑚) + 𝑞 𝜌(𝑎𝑚, 𝑎𝑚+1), 0}, 

if 𝜌(𝑎𝑚−1, 𝑎𝑚) < 𝜌(𝑎𝑚, 𝑎𝑚+1), then 𝑀(𝑎𝑚−1, 𝑎𝑚) ≤ 𝑡 2 𝑞 𝜌(𝑎𝑚, 𝑎𝑚+1). And by (5), we 

obtain  ∅(𝜌(𝑎𝑚, 𝑎𝑚+1)) ≤ ∅(𝑡 2 𝑞 𝜌(𝑎𝑚, 𝑎𝑚+1)). Since ∅ is altering distance function, 

subsequently 𝜌(𝑎𝑚, 𝑎𝑚+1) ≤ 𝑡 2 𝑞 𝜌(𝑎𝑚, 𝑎𝑚+1), this not true for all 𝑡 ∈ (0,1), as result  

𝑀(𝑎𝑚−1, 𝑎𝑚) = 𝑡 2 𝑞 𝜌(𝑎𝑚−1, 𝑎𝑚) and  𝑁(𝑎𝑚−1, 𝑎𝑚) = 0.                                                (6) 

Now put (6) in (5) 

∅(𝜌(𝑎𝑚, 𝑎𝑚+1)) ≤ ∅(𝑡 2 𝑞 𝜌(𝑎𝑚−1, 𝑎𝑚)).                                                                           (7) 

Since ∅ is altering distance function, as a result  𝜌(𝑎𝑚, 𝑎𝑚+1) ≤ 𝑡 2 𝑞 𝜌(𝑎𝑚−1, 𝑎𝑚). Thus for 

all 𝑚 ∈ 𝑁⋃{0} we have a monotone decreasing sequence (𝜌(𝑎𝑚, 𝑎𝑚+1)) = 

(𝜌(𝑓𝑎𝑚−1, 𝑓𝑎𝑚)). By properties of real sequence, (𝜌(𝑎𝑚, 𝑎𝑚+1)) there exist 𝜀 ≥ 0 such that 

lim
𝑚→∞

𝜌(𝑎𝑚, 𝑎𝑚+1) = 𝜀. On letting 𝑚 → ∞ in (7), obtaining ∅(𝜀) ≤ ∅(𝑡 2 𝑞 𝜀). Assume that 

𝜀 ≠ 0. So since ∅ is altering distance function, getting 𝜀 ≤ 𝑡 2 𝑞 𝜀 < ε, which a contradiction. 

Hence lim
𝑚→∞

𝜌(𝑎𝑚, 𝑎𝑚+1) = 0. 

For 𝑗 >  𝑛 define £𝑗 = £𝑖 if 𝑗 =  𝑖 mod 𝑛.  

Claim I: for all 𝜀 > 0 there exist 𝑚 ∈ 𝑁 such that for all 𝑗, 𝑖 ≥ 𝑚, 𝑗 − 𝑖 ≡ 1(mod 𝑛) then 

 𝜌(𝑎𝑗, 𝑎𝑖) < 𝜀. 

Suppose that there exists 𝜀 > 0 such that for each 𝑚 ∈ 𝑁 we can find 𝑗 > 𝑖 > 𝑚  with 𝑗 − 𝑖 ≡

1(mod 𝑛) satisfying 𝜌(𝑎𝑖, 𝑎𝑗) ≥ 𝜀.  

Now, take 𝑚 ≥ 2(mod 𝑛). Then, corresponding to 𝑖 ≥ 𝑚 use can choose 𝑗 in such a way that 

it is the smallest integer with 𝑗 > 𝑖 satisfying 𝑗 − 𝑖 ≡ 1(mod 𝑛) and 𝜌(𝑎𝑗 , 𝑎𝑖) ≥ 𝜀. Therefore, 

𝜌(𝑎𝑗−𝑛, 𝑎𝑖) ≤ 𝜀. By triangular inequality  

𝜀 ≤ 𝜌(𝑎𝑗 , 𝑎𝑖) ≤ 𝑞 𝜌(𝑎𝑗 , 𝑎𝑗−𝑛) + 𝑞 ∑ 𝜌(𝑎𝑖−𝑘, 𝑎𝑖−𝑘+1)
𝑛
𝑘=1  ≤ 𝑞 ∑ 𝜌(𝑎𝑖−𝑘, 𝑎𝑖−𝑘+1)

𝑛
𝑘=1 + 𝑞𝜀. 

Taking 𝑛 → ∞ and since lim
𝑚→∞

𝜌(𝑎𝑚, 𝑎𝑚+1) = 0, we obtain lim
𝑖,𝑗→∞

𝜌(𝑎𝑗 , 𝑎𝑖) = 𝑞𝜀. 

Again, by triangular inequality 

𝜀 ≤ 𝜌(𝑎𝑗 , 𝑎𝑖) ≤ 2𝑞𝜌(𝑎𝑗+1, 𝑎𝑗) + 𝑞𝜌(𝑎𝑗 , 𝑎𝑖) + 2𝑞𝜌(𝑎𝑖+1, 𝑎𝑖). 

Letting 𝑗, 𝑖 → ∞ and since lim
𝑚→∞

𝜌(𝑎𝑚, 𝑎𝑚+1) = 0, so lim
𝑖,𝑗→∞

𝜌(𝑎𝑗+1, 𝑎𝑖+1) = 𝑞𝜀. 



IHJPAS. 2025,38(2) 

381 
 

Since 𝑎𝑖, 𝑎𝑗 belong to different sets £𝑖 and £𝑖+1, and using (2.4) 

∅(𝜌(𝑓𝑎𝑗 , 𝑓𝑎𝑖)) ≤ ∅ (𝑀(𝑎𝑗 , 𝑎𝑖)) − 𝜓 (𝑁(𝑎𝑗 , 𝑎𝑖)),                                                              (8)  

where  

𝑀(𝑎𝑗, 𝑎𝑖) = 𝑡 max {𝜌(𝑎𝑗 , 𝑎𝑖), 𝜌(𝑎𝑗 , 𝑎𝑗+1), 𝜌(𝑎𝑖, 𝑎𝑖+1), 𝜌(𝑎𝑗 , 𝑎𝑖+1), 𝜌(𝑎𝑖, 𝑎𝑗+1)} 

𝑁(𝑎𝑗, 𝑎𝑖) = 𝑡 min {𝜌(𝑎𝑗 , 𝑎𝑖), 𝜌(𝑎𝑗, 𝑎𝑗+1), 𝜌(𝑎𝑖, 𝑎𝑖+1), 𝜌(𝑎𝑗, 𝑎𝑖+1), 𝜌(𝑎𝑖, 𝑎𝑗+1)}. 

 Since  lim
𝑖,𝑗→∞

𝜌(𝑎𝑗 , 𝑎𝑖) = 𝑞𝜀, lim
𝑖,𝑗→∞

𝜌(𝑎𝑗+1, 𝑎𝑖+1) = 𝑞𝜀, and by the triangle inequality 

𝜌(𝑎𝑖+1, 𝑎𝑗+1) ≤ 𝑞𝜌(𝑎𝑖+1, 𝑎𝑖) + 𝑞𝜌(𝑎𝑖, 𝑎𝑗) + 𝑞𝜌(𝑎𝑗 , 𝑎𝑗+1) → 𝑞𝜀  as 𝑖, 𝑗 → ∞. 

Again by the triangle inequality 

𝜌(𝑎𝑗, 𝑎𝑖+1) ≤ 𝑞𝜌(𝑎𝑗 , 𝑎𝑗+1) + 𝑞𝜌(𝑎𝑗+1, 𝑎𝑖+1) → 𝑞𝜀 as 𝑖, 𝑗 → ∞. 

Again by the triangle inequality 

𝜌(𝑎𝑖, 𝑎𝑗+1) ≤ 𝑞𝜌(𝑎𝑖, 𝑎𝑖+1) + 𝑞𝜌(𝑎𝑖+1, 𝑎𝑗+1) → 𝑞𝜀 as 𝑖, 𝑗 → ∞. 

Letting 𝑖, 𝑗 → ∞, we have 𝑀(𝑎𝑗 , 𝑎𝑖) = 𝑡 max {𝑞𝜀, 0,0, 𝑞𝜀, 𝑞𝜀} → 𝑡𝑞𝜀 and 𝑁(𝑎𝑗 , 𝑎𝑖) → 0 and by 

the inequality (8), we have ∅(𝜀) ≤ ∅(𝑡𝑞𝜀) − 𝜓(0). 

Since ∅ is altering distance function and 𝑡 ∈ (0,1), then  𝜀 ≤ 𝑡𝑞𝜀 which a contradiction. 

Therefore, the claim (I) is held. 

Now, we will prove (𝑎𝑚) is a Cauchy sequence in (£, 𝜌). Fix ε > 0. By the claim, ∃𝑚0 such 

that if 𝑗, 𝑖 ≥ 𝑚0 with 𝑗 − 𝑖 ≡ 1(mod𝑛) such that 𝜌(𝑎𝑗, 𝑎𝑖) ≤
ε

2
.  

Since lim
𝑚→∞

𝜌(𝑎𝑚, 𝑎𝑚+1) = 0, also ∃𝑚1 ∈ 𝑁 such that𝜌(𝑎𝑚, 𝑎𝑚+1) ≤
𝜀

2𝑛
 , ∀𝑚 ≥ 𝑚1.   

Suppose 𝑐, 𝑣 ≥ max{𝑚0,𝑚1} and 𝑐 > 𝑣. Then there exists ℎ ∈ {1, 2, … . . , 𝑛} such that 𝑣 − 𝑐 ≡

ℎ(mod 𝑛). Therefore, 𝑣 − 𝑐 + 𝑟 ≡ 1(mod 𝑛) for 𝑟 = 𝑛 − ℎ + 1. So, we have 

𝜌(𝑎𝑐, 𝑎𝑣) ≤ 𝑞 𝜌(𝑎𝑐, 𝑎𝑣+𝑟) + 𝑞
2 𝜌(𝑎𝑣+𝑟 , 𝑎𝑣+𝑟−1) + 𝑞

3𝜌(𝑎𝑣+𝑟−1, 𝑎𝑣+𝑟−2) + 

+𝑞𝑟𝜌(𝑎𝑣+1, 𝑎𝑣)                                                                                                                      (9)  

By 𝜌(𝑎𝑗 , 𝑎𝑖) ≤
ε

2
 and 𝜌(𝑎𝑚, 𝑎𝑚+1) ≤

𝜀

2𝑛
 and from (9), 𝜌(𝑎𝑐, 𝑎𝑣) ≤ 𝑞

𝜀

𝑛
(
1

1−𝑞
) → 0, as 𝑛 → ∞ 

This proves that (𝑎𝑚) is a Cauchy sequence. The completeness of (£, 𝜌) implies to exists 𝑐 ∈

£ such that lim
𝑚→∞

𝑎𝑚 = 𝑐.  

Now, to prove 𝑐 is a fixed point for 𝑓. Since £ = ⋃ £𝑖
𝑛
𝑖=1  is a cyclic representation of £ w.r.t., 

𝑓, the sequence (𝑎𝑚) has  infinite terms in each £𝑖𝑚 for 𝑖𝑚  ∈ {1, 2, . . . , 𝑛}. Closeness of £𝑖𝑚 

for 𝑖𝑚  ∈ {1, 2, . . . , 𝑛} implies to  𝑐 ∈ ⋂ £𝑖
𝑛
𝑖=1 . Suppose that 𝑐 ∈ £𝑖 and 𝑓𝑐 ∈ £𝑖+1  and take a 

subsequence (𝑎𝑚𝑟
)𝑟∈𝑁 of (𝑎𝑚) with 𝑎𝑚𝑟

∈ £𝑖−1 and take 𝑎 = 𝑐, 𝑏 = 𝑎𝑚𝑟
 in (4)  

∅(𝜌(𝑓𝑐, 𝑓𝑎𝑚𝑟
)) ≤ ∅(𝑀(𝑎𝑚𝑟

, 𝑐)) − 𝜓 (𝑁(𝑎𝑚𝑟
, 𝑐))                                                          (10) 

where  

𝑀(𝑎𝑚𝑟
, 𝑐) = 𝑡 max {𝜌(𝑎𝑚𝑟

, 𝑐), 𝜌(𝑎𝑚𝑟
, 𝑓𝑎𝑚𝑟

), 𝜌(𝑐, 𝑓𝑐), 𝜌(𝑎𝑚𝑟
, 𝑓𝑐), 𝜌(𝑐, 𝑓𝑎𝑚𝑟

)}, 

𝑁(𝑎𝑚𝑟
, 𝑐) = 𝑡 min {𝜌(𝑎𝑚𝑟

, 𝑐), 𝜌(𝑎𝑚𝑟
, 𝑓𝑎𝑚𝑟

), 𝜌(𝑐, 𝑓𝑐), 𝜌(𝑎𝑚𝑟
, 𝑓𝑐), 𝜌(𝑐, 𝑓𝑎𝑚𝑟

)}. 

Taking 𝑟 → ∞, hence  

𝑀(𝑐, 𝑎𝑚𝑟
) = 𝑡 𝜌(𝑐, 𝑓𝑐). And (𝑐, 𝑎𝑚𝑟

) = 0, using (10) subsequently  

∅(𝜌(𝑓𝑐, 𝑐)) ≤ ∅(𝑡 𝜌(𝑐, 𝑓𝑐)) − 𝜓(0) ≤  ∅(𝑡 𝜌(𝑐, 𝑓𝑐)). 

Since ∅ is altering distance function, then 𝜌(𝑓𝑐, 𝑐) < 𝑡 𝜌(𝑐, 𝑓𝑐), which a contradiction because 

𝑡 ∈ (0,1), hence 𝑓𝑐 = 𝑐. Thus 𝑐 is a fixed point of 𝑓. 

For the uniqueness, suppose that there are two distinct points 𝑐, 𝑤 with 𝑐 and 𝑤 fixed points of 

𝑓. The cyclic character of 𝑓 and the fact that 𝑐, 𝑤 ∈ £ = ⋃ £𝑖
𝑛
𝑖=1  are fixed points of 𝑓 imply 



IHJPAS. 2025,38(2) 

382 
 

that 𝑐, 𝑤 ∈ ⋂ £𝑖
𝑛
𝑖=1 . Using (4), we can obtain 

∅(𝜌(𝑓𝑐, 𝑓𝑤)) ≤ ∅(𝑀(𝑐, 𝑤)) − 𝜓(𝑁(𝑐, 𝑤)), 

Where, 𝑀(𝑐,𝑤) = 𝑡 𝜌(𝑐, 𝑤) and 𝑁(𝑐, 𝑤)} = 0. Hence ∅(𝜌(𝑐, 𝑤)) = ∅(𝜌(𝑓𝑐, 𝑓𝑤)) ≤

∅(𝑡 𝜌(𝑐, 𝑤)). And since ∅ is altering distance function, we obtain 𝜌(𝑐, 𝑤) ≤ 𝑞 𝜌(𝑐, 𝑤) which 

a is contradiction since it is not true for all 𝑡 ∈ (0,1), then  𝑐 = 𝑤. Hence 𝑓 has a unique fixed 

point in £ and 𝑐 ∈ ⋂ £𝑖
𝑛
𝑖=1 . 

Example 2.6: Let  £ = {6, 7, 8, 9, 10}  with  𝜌: £ × £ → [0,∞)  defined by 

𝜌(𝑎, 𝑏) =

{
 

 
0, 𝑖𝑓 𝑎 = 𝑏

6, 𝑖𝑓 𝑎 ≠ 𝑏, 𝑎, 𝑏 ∈ {6,7,8,9} 

17, 𝑖𝑓 𝑎, 𝑏 ∈ {9,10} and 𝑎 ≠ 𝑏

40, 𝑖𝑓𝑎 ∈ {6,7,8} and 𝑏 = 10 (or 𝑏 ∈ {6,7,8} and 𝑎 = 10) 

. 

Since all Cauchy sequences in £ are constant. Therefore, are convergent. Then (£, 𝜌) is 

complete b-metric space with 𝑞 = 2. And £1 = {6,8,10} and £2 = {6,7,9}, £ = ⋃ £𝑖
2
𝑖=1 . Define 

𝑓:⋃ £𝑖
2
𝑖=1  → ⋃ £𝑖

2
𝑖=1  such that 𝑓(𝑎) = 6   and 𝑖𝑓 𝑎 ∈ {6,7,8,9} and 𝑓(10) = 8. So, (£1) 

⊂ £2,𝑓(£2) ⊂ £1, for 𝑎 ∈ £1, 𝑏 ∈ £2, and take 𝑡 =
1

4
. Let ∅,𝜓: [0,∞) → [0,∞) such that  

∅(𝑟) =
𝑟

4
 and 𝜓(𝑟) =

𝑟

2
. Then ∅ and 𝜓 are altering distance functions. It is easy to check 

condition (4) holds with fixed point 𝑎 = 6. 

An application for solving integral equations. Consider the integral equation (24,25)  

𝑤(𝑡) = ∫ 𝑄(𝑡, 𝑟)𝑇(𝑟, 𝑤(𝑟))
𝐽

0
𝑑𝑟   for all 𝑡 ∈ [0, 𝐽],                                                               (11)  

where 𝐽 > 0, 𝑇: [0, 𝐽] × 𝑅 → 𝑅 and 𝑄: [0, 𝐽] × [0, 𝐽] → [0,∞) are continuous functions. 

In this section, we look for a nonnegative solution to (11) in £ = 𝐶([0, 𝐽], 𝑅) by (Theorem 2.5). 

Let £ = 𝐶[0, 𝐽] be the set of real valued continuous functions on [0, 𝐽], where [0, 𝐽] is a closed 

and bounded interval in 𝑅. For 𝑝 > 1, define 𝜌: [0, 𝐽] × [0, 𝐽] → 𝑅 by   

𝜌(𝑤, 𝑣) = max
𝑡𝜖[0,𝐽]

|𝑤(𝑡) − 𝑣(𝑡)|𝑝, for all 𝑤, 𝑣 ∈ £. 

Therefore, (£ ,𝜌)is a complete b-metric space with 𝑞 = 2𝑝−1.  Let 𝛼, 𝛽 ∈ £ and 𝛼0, 𝛽0 ∈ 𝑅 

such that 

𝛼0 ≤ α(𝑡) ≤ 𝛽(𝑡) ≤  𝛽0,   ∀ 𝑡 ∈ [0, 𝐽].                                                                                 (12) 

Suppose that for all 𝑡 ∈ [0, 𝐽], we have  

𝛼(𝑡) ≤ ∫ 𝑄(𝑡, 𝑟)𝑇(𝑟, 𝛽(𝑟))
𝐽

0
𝑑𝑟,                                                                                            (13) 

and  

𝛽(𝑡) ≥ ∫ 𝑄(𝑡, 𝑟)𝑇(𝑟, 𝛼(𝑟))
𝐽

0
𝑑𝑟.                                                                                            (14) 

We suppose that ∀ 𝑟 ∈ [0, 𝐽], 𝑇(𝑟, . )be a decreasing function, that  

𝑎, 𝑏 ∈ 𝑅,   𝑎 ≥ 𝑏 then 𝑇(𝑟, 𝑎) ≤ 𝑇(𝑟, 𝑏).                                                                               (15) 

Assume that 𝑘 > 0 is such that  

𝑘(max
𝑡∈[0,𝐽]

∫ 𝑄(𝑡, 𝑟)
𝐽

0
𝑑𝑟) < 1.                                                                                                    (16) 

Define a map 𝑓: £ → £ by  𝑓𝑤(𝑡) = ∫ 𝑄(𝑡, 𝑟)𝑇(𝑟, 𝑤(𝑟))
𝐽

0
𝑑𝑟, for all 𝑡 ∈ [0, 𝐽]. 

Suppose that ∀𝑟 ∈ [0, 𝐽] and 𝑎, 𝑏 ∈ £ with (𝑎(𝑟) ≤ 𝛼0 and 𝑏(𝑟) ≤ 𝛽0) or vice versa, 

0 ≤ [𝑇(𝑞, 𝑎(𝑟)) − 𝑇(𝑟, 𝑏(𝑟))] ≤ 𝑘 max {|𝑎(𝑟) − 𝑏(𝑟)|𝑝, 0, |𝑏(𝑟) − 𝑓𝑏(𝑟)|𝑝, 

|𝑎(𝑟) − 𝑓𝑏(𝑟)|𝑝, |𝑏(𝑟) − 𝑎(𝑟)|𝑝})
1

𝑝                                                                                      (17) 

Theorem 2.7: Under the assumptions (12)-(17), the integral equation (11) has a solution in the 

set {𝑤 ∈ 𝐶([0, 𝐽]): 𝛼 ≤ 𝑤 ≤ 𝛽}. 



IHJPAS. 2025,38(2) 

383 
 

Proof : We omit the details because the proof steps are classic with some minor differences 

due to the specificity of the b-metric space. 

Fixed points by implicit conditions 

The following list of implicit functions under various conditions (26). Let Ω be the set of all 

real continuous functions 𝑀:𝑅+
6 → 𝑅, satisfying the following conditions: 

𝑀1) is non-increasing in variables 𝑟2, 𝑟3, 𝑟4, 𝑟5, 𝑟6  

𝑀2) there exists a right continuous function  𝐴: [0,∞) → [0,∞), 𝐴(0) = 0,𝐴(𝑟) < 𝑟, for 𝑟 >

0 such that for 𝑐 ≥ 0 𝑀(𝑐, 𝑢, 𝑐, 𝑢, 0, 𝑐 + 𝑢) ≤ 0 or 𝑀(𝑐, 𝑢, 0,0, 𝑢, 𝑢) ≤ 0, implies 𝑐 ≤ 𝐴(𝑢).  

𝑀3) 𝑀(𝑐, 0, 𝑐, 0,0, 𝑐) > 0 and 𝑀(𝑐, 𝑐, 0,0, 𝑐, 𝑐) > 0 ∀ 𝑐 > 0. 

Also, let  𝛹 ≔ the set of functions 𝜓: [0,∞) → [0,∞) such that: 

i) 𝜓 is monotone increasing and continuous; 

ii) 𝜓(𝑟) = 0 if and only if  𝑟 = 0; 

iii) 𝜓 is subadditive, i.e.,  ∀𝑟1, 𝑟2 ∈ [0,+∞),𝜓(𝑟1 + 𝑟2) = 𝜓(𝑟1) + 𝜓(𝑟2). 

Lemma 2.8: Let 𝐴: [0,∞) → [0,∞) be a right continuous function such that 𝐴(𝑟) < 𝑟, for 𝑟 >

0.  (27( . Then lim
𝑚→∞

𝐴𝑚(𝑟) = 0, where 𝐴𝑚≔𝑚 times repeated composition of 𝐴. 

Theorem 2.9: If 𝑀 ∈ 𝛺 exists and £ =  ⋃ £𝑖
𝑛
𝑖=1  is a cyclic representation of £ w.r.t., 𝑓: £ → £. 

If  for any (𝑎, 𝑏) ∈ £𝑖 × £𝑖+1, 𝑖 = 1,2, … , 𝑛    

𝑀(𝜓(𝜌(𝑓𝑎, 𝑓𝑏)), 𝜓(𝜌(𝑎, 𝑏)), 𝜓(𝜌(𝑎, 𝑓𝑎)), 𝜓(𝜌(𝑏, 𝑓𝑏)), 𝜓(𝜌(𝑎, 𝑓𝑏)), 𝜓(𝜌(𝑏, 𝑓𝑎))) ≤ 0 

(18), and 𝜓 ∈ 𝛹. ∃!  𝑐 ∈ ⋂ £𝑖
𝑛
𝑖=1 , 𝑐 is a unique fixed point. Moreover, 𝑙𝑖𝑚

𝑚→∞
𝑓𝑚(𝑎) = 𝑐, for any 

𝑎 ∈ £. 

Proof: Let 𝑎0 ∈ ⋃ £𝑖
𝑛
𝑖=1  and 𝑎𝑚 define by 𝑎𝑚+1 = 𝑓𝑎𝑚. So for 𝑚 ≥ 0, ∃𝑖𝑚 ∈ {1,2, … 𝑛} such 

that 𝑎𝑚−1 ∈ £𝑖𝑚  and 𝑎𝑚 ∈ £𝑖𝑚+1. If 𝑎𝑚0
= 𝑎𝑚0−1for some 𝑚0, 𝑎𝑚0

= 𝑓𝑎𝑚0−1 = 𝑎𝑚0−1 then 

𝑎𝑚0
is fixed point of 𝑓. Thus, suppose that 𝑎𝑚 ≠ 𝑎𝑚−1, for all 𝑚 ∈ 𝑁⋃{0}. By using (2.18) 

therefore    

𝑀(𝜓(𝜌(𝑎𝑚+1, 𝑎𝑚)), 𝜓(𝜌(𝑎𝑚, 𝑎𝑚−1)), 𝜓(𝜌(𝑎𝑚, 𝑎𝑚+1)), 𝜓(𝜌(𝑎𝑚−1, 𝑎𝑚)), 

𝜓(𝜌(𝑎𝑚, 𝑎𝑚)), 𝜓(𝜌(𝑎𝑚−1, 𝑎𝑚+1))) ≤ 0. 

And since 𝜓(𝑟) = 0 if and only if 𝑟 = 0, also by using triangle inequality and since 𝜓 is 

subadditive, therefore   

𝑀(𝜓(𝜌(𝑎𝑚+1, 𝑎𝑚)), 𝜓(𝜌(𝑎𝑚, 𝑎𝑚−1)), 𝜓(𝜌(𝑎𝑚, 𝑎𝑚+1)), 𝜓(𝜌(𝑎𝑚−1, 𝑎𝑚)),0,  

𝜓(𝑞𝜌(𝑎𝑚−1, 𝑎𝑚)) + 𝜓(𝑞𝜌(𝑎𝑚, 𝑎𝑚+1))) ≤ 0. 

And from 𝑀2, there exists a right continuous function A: [0,∞) → [0,∞), A(0) = 0, A(𝑟) <

𝑟, for 𝑟 > 0, such that for all  𝑚 ∈ 𝑁⋃{0} 

𝜓(𝜌(𝑎𝑚+1, 𝑎𝑚) ≤ 𝐴( 𝜓(𝜌(𝑎𝑚, 𝑎𝑚−1))). 

If we use this procedure, we get 

𝜓(𝜌(𝑎𝑚+1, 𝑎𝑚) ≤ 𝐴( 𝜓(𝜌(𝑎𝑚, 𝑎𝑚−1))) ≤ ⋯ ≤ A𝑚  (𝜓(𝜌(𝑎1, 𝑎0)).                                   (19) 

And by (Lemma 8) and continuity of 𝜓, subsequently     

𝑙𝑖𝑚
𝑚→∞

𝜓( 𝜌(𝑎𝑚+1, 𝑎𝑚)) = 0 = 𝜓( 𝑙𝑖𝑚
𝑚→∞

 𝜌(𝑎𝑚+1, 𝑎𝑚)). Since 𝜓(𝑟) = 0 if and only if  𝑟 = 0, 

therefore   

𝑙𝑖𝑚
𝑚→∞

𝜌(𝑎𝑚+1, 𝑎𝑚) = 0.                                                                                                          (20) 

To prove for each 𝑎0 ∈ £, (𝑎𝑚) is a Cauchy sequence. Assume it is false. Then we can find a 

𝜀 > 0 and  {𝑝𝑟}, {𝑑𝑟}, 𝑑𝑟 > 𝑝𝑟 ≥ 𝑟 where {𝑝𝑟}, {𝑑𝑟} two subsequences of integers with                                   

𝜓(𝜌(𝑎𝑝𝑟 , 𝑎𝑑𝑟)) ≥ 𝜀 for 𝑛 ∈ {1,2, … }.                                                                                    (21) 

We also assume 



IHJPAS. 2025,38(2) 

384 
 

𝜓(𝜌(𝑎𝑝𝑟 , 𝑎𝑑𝑟−1)) < 𝜀.                                                                                                           (22( 

By selecting 𝑑𝑟 to be the least number surpassing 𝑝𝑟 for which inequality (21) holds, now by 

(19) and (21), (22),  and since 𝜓 is subadditive, getting   

𝜀 ≤ 𝜓(𝜌(𝑎𝑝𝑟 , 𝑎𝑑𝑟−1)) ≤ 𝜓(𝑞𝜌(𝑎𝑝𝑟 , 𝑎𝑑𝑟−1)) + 𝜓(𝑞𝜌(𝑎𝑑𝑟−1, 𝑎𝑝𝑟)) 

≤ 𝑞𝜀 + 𝐴𝑑𝑟−1𝜓(𝑞𝜌(𝑎0, 𝑎1)).                                                                                                  (23) 

And so 

 lim
𝑟→∞

𝜓(𝜌(𝑎𝑑𝑟−1, 𝑎𝑝𝑟)) = 𝑞𝜀.                                                                                                 (24) 

On the other hand, ∀𝑟, ∃𝑖𝑟 ∈ {1,2, … , 𝑛} such that 𝑑𝑟 − 𝑝𝑟 + 𝑖𝑟 ≡ 1(mod 𝑛). Then 𝑎𝑝𝑟−𝑖𝑟(for 

𝑟 large enough, 𝑝𝑟 > 𝑖𝑟) and 𝑎𝑑𝑟belong to different sets £𝑖 and £𝑖+1 for 𝑖 ∈ {1,2, … , 𝑛}. By the 

triangle inequality, also 𝜓 is subadditive, obtaining                                                                           

𝜓(𝜌(𝑎𝑑𝑟 , 𝑎𝑝𝑟)) ≤ 𝜓(𝑞𝜌( 𝑎𝑑𝑟 , 𝑎𝑑𝑟−𝑖𝑟) + 𝑞𝜌(𝑎𝑑𝑟−𝑖𝑟 , 𝑎𝑝𝑟)) 

≤ 𝜓(𝑞𝜌(𝑎𝑑𝑟 , 𝑎𝑑𝑟−𝑖𝑟)) + 𝜓(𝑞𝜌(𝑎𝑑𝑟−𝑖𝑟 , 𝑎𝑝𝑟)). 

Now taking 𝑟 → ∞, by (19) and from (24), as a results 

𝜓(𝜌(𝑎𝑝𝑟 , 𝑎𝑑𝑟)) ≤ 𝐴
𝑑𝑟−𝑖𝑟𝜓(𝑞𝜌(𝑎0, 𝑎1)) + 𝑞𝜀. 

And so, 

lim
𝑟→∞

𝜓(𝜌(𝑎𝑑𝑟 , 𝑎𝑝𝑟)) = 𝑞𝜀.                                                                                                            (25) 

By using (20), so 

Lim
𝑟→∞

𝜓(𝜌(𝑎𝑑𝑟+1, 𝑎𝑑𝑟)) = 0, Lim
𝑟→∞

𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1, 𝑎𝑝𝑟−𝑖𝑟)) = 0.                                              (26)   

And by using the triangle inequality, then 

𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑑𝑟)) ≤ 𝜓(𝑞𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑝𝑟)) + 𝜓(𝑞𝜌( 𝑎𝑝𝑟 , 𝑎𝑑𝑟)). 

Letting 𝑟 → ∞ in the last inequality and using (2.19) and (2.25), consequently 

𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑑𝑟)) ≤ 𝐴𝑝𝑟−𝑖𝑟𝜓(𝑞𝜌(𝑎0, 𝑎1)) + 𝑞𝜀 

lim
𝑟→∞

𝜌 (𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑑𝑟)) = 𝑞𝜀.                                                                                                  (27) 

Again, by using the triangle inequality, obtaining 

𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑞𝑟+1)) ≤ 𝜓(𝑞𝜌( 𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑑𝑟)) + 𝜓(𝑞𝜌(𝑎𝑑𝑟 , 𝑎𝑑𝑟+1)). 

Letting 𝑟 → ∞ in the last inequality and using (26) and (27), therefore   

lim
𝑟→∞

𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑑𝑟+1)) = 𝑞𝜀.                                                                                              (28) 

And in the same way, as a results 

𝜓(𝜌(𝑎𝑑𝑟 , 𝑎𝑝𝑟−𝑖𝑟+1)) ≤ 𝜓(𝑞𝜌(𝑎𝑑𝑟 , 𝑎𝑝𝑟−𝑖𝑟)) + 𝜓(𝑞𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑝𝑟−𝑖𝑟+1)). 

Letting 𝑟 → ∞ and using (27) and (26), getting 

lim
𝑟→∞

𝜓(𝜌(𝑎𝑑𝑟 , 𝑎𝑝𝑟−𝑖𝑟+1)) = 𝑞𝜀.                                                                                             (29) 

Again, by using the triangle inequality, therefore   

𝜓(𝜌( 𝑎𝑝𝑟−𝑖𝑟+1, 𝑎𝑑𝑟+1)) ≤ 𝜓(𝑞𝜌( 𝑎𝑝𝑟−𝑖𝑟+1, 𝑎𝑑𝑟)) + 𝜓(𝑞𝜌(𝑎𝑑𝑟 , 𝑎𝑑𝑟+1)). 

Letting 𝑟 → ∞ in the last inequality and using (2.26) and (2.29), having 

Lim
𝑟→∞

𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1, 𝑎𝑑𝑟+1)) = 𝑞𝜀.                                                                                          (30) 

Using (2.18) for 𝑎 = 𝑎𝑝𝑟−𝑖𝑟 and 𝑏 = 𝑎𝑑𝑟 , subsequently 

𝑀(𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1, 𝑎𝑑𝑟+1)), 𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑑𝑟)), 𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟 , 𝑎𝑝𝑟−𝑖𝑟+1)), 



IHJPAS. 2025,38(2) 

385 
 

𝜓(𝜌(𝑎𝑑𝑟 , 𝑎𝑑𝑟+1)), 𝜓(𝜌(𝑎𝑑𝑟+1, 𝑎𝑝𝑟−𝑖𝑟)), 𝜓(𝜌(𝑎𝑑𝑟 , 𝑎𝑝𝑟−𝑖𝑟+1))) ≤ 0. 

Letting 𝑟 → ∞ , and using (30), (27), (26), (28), (29), then, by continuity of 𝑀 and 𝜓(𝑟) = 0 

if and only if  𝑟 = 0, 𝑀(𝑞𝜀, 𝑞𝜀, 0,0, 𝑞𝜀, 𝑞𝜀) ≤ 0, a contradiction with 𝑀3. Thus, (𝑎𝑚) is Cauchy 

sequence in (£, 𝜌). Now to prove that 𝑐 is fixed point of 𝑓. In fact  𝑓𝑎𝑚 → 𝑐 and since £ =

⋃ £𝑖
𝑛
𝑖=1  is cyclic representation of £ w.r.t., 𝑓, the sequence (𝑎𝑚) has infinite terms in each 

£𝑖𝑚for 𝑖𝑚 ∈ {1,2, … , 𝑛}. Considering that £𝑖𝑚 is closed for 𝑖𝑚 ∈ {1,2, … , 𝑛} we have 𝑐 ∈

⋂ £𝑖
𝑛
𝑖=1 . 

Suppose that 𝑐 ∈ £𝑖 and 𝑓𝑐 ∈ £𝑖+1, and take a subsequence (𝑎𝑚𝑟
)
𝑟∈𝑁

of (𝑎𝑚) with 𝑎𝑚𝑟
∈ £𝑖−1, 

using (2.18), take 𝑎 = 𝑐 and 𝑏 = 𝑎𝑚𝑟
, as result 

𝑀(𝜓(𝜌(𝑓𝑐, 𝑓𝑎𝑚𝑟
)), 𝜓(𝜌(𝑐, 𝑎𝑚𝑟

)), 𝜓(𝜌(𝑐, 𝑓𝑐)), 𝜓(𝜌(𝑎𝑚𝑟
, 𝑓𝑎𝑚𝑟

)), 

𝜓(𝜌(𝑐, 𝑓𝑎𝑚𝑟
)), 𝜓(𝜌(𝑎𝑚𝑟

, 𝑓𝑐))) ≤ 0. 

Taking 𝑟 → ∞, hence 

𝑀(𝜓(𝜌(𝑓𝑐, 𝑐)), 𝜓(𝜌(𝑐, 𝑐)), 𝜓(𝜌(𝑐, 𝑓𝑐)), 𝜓(𝜌(𝑐, 𝑐)), 𝜓(𝜌(𝑐, 𝑐)), 𝜓(𝜌(𝑐, 𝑓𝑐))) ≤ 0,                      

and 𝜓(𝑟) = 0 if and only if  𝑟 = 0, then 𝑀(𝜓(𝜌(𝑓𝑐, 𝑐)),0, 𝜓(𝜌(𝑐, 𝑓𝑐)),  0,0, 𝜓(𝜌(𝑐, 𝑓𝑐))) ≤

0, which is contradiction to 𝑀3. Thus  𝜓(𝜌(𝑓𝑐, 𝑐)) = 0 then 𝜌(𝑓𝑐, 𝑐) = 0, then 𝑓𝑐 = 𝑐. We 

obtain 𝑐 is a fixed point of 𝑓. Suppose that there are two distinct points 𝑐, 𝑤 with 𝑐 and 𝑤 are 

two fixed points of 𝑓. The cyclic nature of 𝑓, also, the fact 𝑐, 𝑤 ∈ £ = ⋃ £𝑖
𝑛
𝑖=1  are fixed points 

of 𝑓 imply that 𝑐, 𝑤 ∈ ⋂ £𝑖
𝑛
𝑖=1  and by (18), obtaining     

𝑀(𝜓(𝜌(𝑓𝑐, 𝑓𝑤)),𝜓(𝜌(𝑐, 𝑤)), 𝜓(𝜌(𝑐, 𝑓𝑐)), 𝜓(𝜌(𝑤, 𝑓𝑤)),𝜓(𝜌(𝑐, 𝑓𝑤)),𝜓(𝜌(𝑤, 𝑓𝑐))) ≤ 0. 

Since 𝑐, 𝑤 are fixed points of 𝑓, 𝜓(𝑟) = 0 if and only if 𝑟 = 0, we get    

𝑀(𝜓(𝜌(𝑐, 𝑤)), 𝜓(𝜌(𝑐, 𝑤)),0,0, 𝜓(𝜌(𝑐, 𝑓𝑤)),𝜓(𝜌(𝑤, 𝑐))) ≤ 0, 

which is a contradiction to 𝑀3, then 𝜓(𝜌(𝑐, 𝑤)) = 0, hence 𝜌(𝑐, 𝑤) = 0, that is,  𝑐 = 𝑤. Hence 

𝑓 has a unique fixed point in £ and 𝑐 ∈ ⋂ £𝑖
𝑛
𝑖=1 . 

Remark 2.10: It is worth noting, it is worth noting that we can obtain good results by including 

the concept of cyclicity in  cases of )28,29(. 

 

3. Discussion 

     This work is classified within the field depending on the classification 2010 MSC: 47H09, 

47H10.  Our study of this topic is the first in Iraq (to the best of our knowledge) in the field of 

fixed points for cyclic maps, and it is taken from a master’s thesis by researcher Abbas Karim 

Nahi. The paper included new results in the field of integral contractions in the b-metric spaces 

of the cyclic type, as well as new generalizations of the results of other researchers in the case 

of the b-metric space. In the future, we would like to study the results in )30( in the case of 

cyclic maps. 

 

4. Conclusions 

      In this  paper,  new theorems were established to find fixed points. First, by merging integral 

contractive conditions with the concept of cyclic map, and second, by applying the concept of 

cyclic representation with respect to maps satisfying general weak conditions, including a 

changing distance function. And third, by merging the changing distance function with the sub 

additive to find theorems in the field of fixed points through the concept of cyclic representation 

for maps satisfying an implicit relation including a changing distance function. 

 

 

 



IHJPAS. 2025,38(2) 

386 
 

Acknowledgments 

       I would like to thank the previous researchers whose results have played an important role 

in obtaining our current results. Especially Professors W. A. Kirk, P.S. Srinivasan and P. 

Veeramani. 

 

Conflict of Interest  

       The authors declare no conflict of interest. 

 

Funding 

      This research received no external funding. 

 

References 

1. Kirk WA, Srinivasan PS, Veeramani P. Fixed points for mappings satisfying cyclical contractive 

conditions. Fixed Point Theory. 2003;4(1):79-89. 

2. Asem V, Singh YM, Khan MS, Sessa S. On (α, p)-Cyclic Contractions and Related Fixed Point 

Theorems. Symmetry. 2023;15(10):1826. http://doi.org/10.3390/sym15101826 

3. Konar P, Bhandari SK, Chandok S, Mukheimer A. Multivalued weak cyclic δ-contraction 

mappings. Journal of Inequalities and Applications. 2020;2020(1):1-

11. http://doi.org/10.1186/s13660-020-02373-1 

4. Karapinar E, Sadarangani K. Fixed point theory for cyclic (ϕ-ψ)-contractions. Fixed Point Theory 

and Applications. 2011;2011(1):1-8. http://doi.org/10.1186/1687-1812-2011-1 

5. Hussain N, Parvaneh V, Roshan JR, Kadelburg Z. Fixed points of cyclic weakly (ψ, φ, L, A, B)-

contractive mappings in ordered b-metric spaces with applications. Fixed Point Theory and 

Applications. 2013;2013(1):1-18. http://doi.org/10.1186/1687-1812-2013-1 

6. Hiranmoy G, Karapınar E, Kanta Dey L. Best proximity point results for contractive and cyclic 

contractive type mappings. Numerical Functional Analysis and Optimization. 2021;42(7):849-

864. http://doi.org/10.1080/01630563.2021.1929578 

7. Chaipunya P, Cho YJ, Sintunavarat W, Kumam P. Fixed point and common fixed point theorems 

for cyclic quasi-contractions in metric and ultrametric spaces. Advances in Pure Mathematics. 

2012;2(6):401-407. http://doi.org/10.4236/apm.2012.26059 

8. Abou Bakr, Sahar MA. Cyclic G‐Ω-Weak Contraction-Weak Nonexpansive Mappings and Some 

Fixed Point Theorems in Metric Spaces. Abstract and Applied Analysis. 2021;2021:1-9. 

9. Karapınar E, Romaguera S, Taş K. Fixed points for cyclic orbital generalized contractions on 

complete metric spaces. Open Mathematics. 2013;11(3):552-560. http://doi.org/10.2478/s11533-

013-0245-5 

10. Bakhtin I. The contraction mapping principle in quasimetric spaces. Functional Analysis. 

1989;30:26-37. 

11. Al-Bundi SS. Iterated function system in metric spaces. Bol Soc Paran Mat. 2022;40:1-10. 

12. Abed SS. Fixed point principles in general b-metric Spaces and b-Menger probabilistic 

spaces. Journal of AL-Qadisiyah for Computer Science and Mathematics. 2018;10(2):42. 

13. Karapınar E. Fixed point theory for cyclic weak ϕ-contraction. Applied Mathematics Letters. 

2011;24(6):822-825. http://doi.org/10.1016/j.aml.2011.01.021 

14. Hammad HA, De la Sen M. Solution of nonlinear integral equation via fixed point of cyclic α L ψ-

rational contraction mappings in metric-like spaces. Bulletin of the Brazilian Mathematical Society. 

2020;51(1):81-105. http://doi.org/10.1007/s00574-019-00162-1 

15. Jabbar HA, Kadhim SN, Abed SS. Best approximation in b-modular spaces. Baghdad Science 

Journal. 2023. http://doi.org/10.21123/bsj.2023.8230 

16. Qawaqneh H, Noorani M, Shatanawi W, Alsamir H. Common fixed point theorems for generalized 

geraghty (α, ψ, ϕ)-quasi contraction type mapping in partially ordered metric-like spaces. Axioms. 

2018;7(4):74. http://doi.org/10.3390/axioms7040074 

http://doi.org/10.3390/sym15101826
http://doi.org/10.1186/s13660-020-02373-1
http://doi.org/10.1186/1687-1812-2011-1
http://doi.org/10.1186/1687-1812-2013-1
http://doi.org/10.1080/01630563.2021.1929578
http://doi.org/10.4236/apm.2012.26059
http://doi.org/10.2478/s11533-013-0245-5
http://doi.org/10.2478/s11533-013-0245-5
http://doi.org/10.1016/j.aml.2011.01.021
http://doi.org/10.1007/s00574-019-00162-1
http://doi.org/10.21123/bsj.2023.8230
http://doi.org/10.3390/axioms7040074


IHJPAS. 2025,38(2) 

387 
 

17. Mustafa Z, Mustafa Z, Khan SU, Jaradat MMM, Arshad M, Jaradat HM. Fixed point results of F-

rational cyclic contractive mappings on 0-complete partial metric spaces. Forum. 2018;40:394-409. 

18. Păcurar M, Rus IA. Fixed point theory of cyclic operators. Journal of Fixed Point Theory and 

Applications. 2022;24(4):79. 

19. Theivaraman R, Srinivasan P, Radenovic S, Park C. New approximate fixed point results for various 

cyclic contraction operators on E-metric spaces. Journal of the Korean Society for Industrial and 

Applied Mathematics. 2023;27(3):160-179. 

20. Kumari PS, Panthi D. Cyclic contractions and fixed point theorems on various generating 

spaces. Fixed Point Theory and Applications. 2015;2015(1):1-17. http://doi.org/10.1186/s13663-

015-0365-7 

21. Hussein Luaibi H, Abed SS. Fixed point theorems in general metric space with an 

application. Baghdad Science Journal. 2021;18(1 Suppl.):0812. 

22. Branciari A. A fixed point theorem for mappings satisfying a general contractive condition of 

integral type. International Journal of Mathematics and Mathematical Sciences. 2002;29:531-

536. http://doi.org/10.1155/S0161171202007524 

23. Khan MS, Swaleh M, Sessa S. Fixed point theorems by altering distances between the 

points. Bulletin of the Australian Mathematical Society. 1984;30(1):1-

9. http://doi.org/10.1017/S0004972700001659 

24. Nieto JJ, Rodríguez-López R. Contractive mapping theorems in partially ordered sets and 

applications to ordinary differential equations. Order. 2005;22(3):223-

239. http://doi.org/10.1007/s11083-005-9018-5 

25. Rabaiah A, Tallafha A, Shatanawi W. Common fixed point results for mappings under nonlinear 

contraction of cyclic form in b-metric spaces. Nonlinear Functional Analysis and Applications. 

2021;26(2):289-301. 

26. Nashine HK, Kadelburg Z, Kumam P. Implicit-relation-type cyclic contractive mappings and 

applications to integral equations. Abstract and Applied Analysis. 2012;2012:1-15. 

27. Matkowski J. Fixed point theorems for mappings with a contractive iterate at a point. Proceedings 

of the American Mathematical Society. 1977;62(2):344-348. http://doi.org/10.2307/2041931 

28. Abed SS, Abed AN. Convergence and Stability of Iterative Scheme for a Monotone Total 

Asymptotically Non-expansive Mapping. Iraqi Journal of Science. 2022;63(1):241-250. 

29. Abed SS, Taresh NS. On stability of iterative sequences with error. Mathematics. 

2019;7(8):765. http://doi.org/10.3390/math7080765 

30. Monje ZAM, Ahmed BA. A study of stability of first-order delay differential equations using fixed 

point theorem Banach. Iraqi Journal of Science. 2019;60(12):2719-2724 

 

http://doi.org/10.1186/s13663-015-0365-7
http://doi.org/10.1186/s13663-015-0365-7
http://doi.org/10.1155/S0161171202007524
http://doi.org/10.1017/S0004972700001659
http://doi.org/10.1007/s11083-005-9018-5
http://doi.org/10.2307/2041931
http://doi.org/10.3390/math7080765