IHJPAS. 36 (4) 2023 

367 

This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 
 

 

 

 

 

 

 

*Corresponding Author: mrs_zena.hussein@yahoo.com 

 

 
  

 

 

Abctract 

       Based on the needs of the scientific community, researchers tended to find new iterative 

schemes or develop previous iterative schemes that would help researchers reach the fixed point 

with fewer steps and with stability, will be define in this paper the Multi_Iplicit Four-Step 

Iterative (MIFSI) which is development to four-step impicit fixed point iterative, to develop the 

aforementioned iterative scheme, we will use a finite set of projective functions ,nonexpansive 

function and finite set from a new functions called generalized quasi like contractive which is an 

amalgamation of quasi contractive function and contractive like function , by the last function 

and a set of sequential organized steps, we will be able to prove the existence of the fixed point(f-

point) of the MIFSI and Fur-Step Iterative(FSI), furthermore, we found MIFSI faster than FSI. 

On the other hand, the stability of the new iterative is proved. 

 

Keywords: Fixed point, implicit iterative schemes, convergent, projective function, stability, 

contractive function, four-step iterative. 

  

1. Introduction and preliminary notes  

         The fixed-point theory is a key component in the solution of numerous issues in a wide 

range of scientific disciplines. Takahashi who originally introduced the idea of convexity in 

metric spaces[1]. Metric spaces that are convex are more generalized. Fixed point theorems in 

doi.org/10.30526/36.4.3164 

Article history: Received 28 November 2022, Accepted 18 January 2023, Published in October 2023 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

An Analytical Study of the Convergence and Stability of the New Four-Step 

Iterative Schemes 

 
Zena Hussein Maibed *  

Department of Mathematics, College of Education 

for Pure Sciences Ibn AL-Haitham, University of 

Baghdad, Baghdad, Iraq. 

 

 

    Omar Mohammed Abbas Joodi                                            

Department of Mathematics, College of Education 

for Pure Sciences Ibn AL-Haitham, University of 

Baghdad, Baghdad, Iraq. 

 

 Shrooq Bahjat Smeein  

Department of Information Section Mathematics, 

University of Technology and Applied Sciences-Muscat, 

Sultanate of Oman. 

 

https://creativecommons.org/licenses/by/4.0/
mailto:mrs_zena.hussein@yahoo.com
mailto:mrs_zena.hussein@yahoo.com
mailto:Omar.Mohammed1203a@ihcoedu.uobaghdad.edu.iq
mailto:shrooq.bahjat@utas.edu.om


IHJPAS. 36 (4) 2023 

368 
 

convex metric spaces (CMS) have been studied by numerous researchers, including Ciric [2], 

Shimiz, and Takahashi [3], and many more. 

Definition 1.1:[1]: Let 𝒳: 𝑆 × 𝑆 × [0,1] → 𝑆 is namely convex structure when the condition is 

hold: 

𝑑(𝑡, 𝒳(ᶉ, 𝑠, ℷ)) ≤ ℷ𝑑(𝑡, ᶉ) + (1 − ℷ)𝑑(𝑡, 𝑠)                             (1.1) 

When 𝑆 is metric space, say about the metric space and convex structure 𝒳 which is denoted by 

(𝑆, 𝑑, ℷ). Let 𝑈 be a nonempty, closed subset of metric space S, say 𝑈 is closed convex subset if 

𝒳(ᶉ, 𝑠, ℷ) ∈ 𝑈 for all ᶉ, 𝑠 ∈ 𝑈 and ℷ ∈ [0,1].₮: 𝑆 → 𝑆 be self mapping, while some CMS cannot 

be embedded into normed spaces, all normed spaces are inherently CMS. 

Example1.1. Let 𝑆 = {(𝓇1, 𝓇2, 𝓇3) ∈ 𝑅3: 𝓇1, 𝓇2, 𝓇3 > 0}. For r = (𝓇1, 𝓇2, 𝓇3), 

s = (𝑠1, 𝑠2, 𝑠3) ∈ S  𝑡 = (𝑡1, 𝑡2, 𝑡3),  and 𝛼, 𝛽, 𝛾 ∈ 𝐼 = [0,1]  with 𝛼 + 𝛽 + 𝛾 = 1 , we define a 

mapping 𝒳: 𝑆3 × 𝐼3 → 𝑆 by : 

𝒳(ᶉ, 𝑠, 𝑡; 𝛼, 𝛽, 𝛾) = (𝛼𝓇1 + 𝛽𝑠1 + 𝛾𝑡1, 𝛼𝓇2 + 𝛽𝑠2 + 𝛾𝑡2, 𝛼𝓇3 + 𝛽𝑠3 + 𝛾𝑡3). 

And define a metric 𝑑: 𝑆 × 𝑆 → [0, ∞) by : 

𝑑(ᶉ, 𝑠) = |𝓇1𝑠1 + 𝓇2𝑠2 + 𝓇3𝑠3|. 

Then, it is clear that (𝑆, 𝑑, 𝒳) is CMS, but not normed space. 

When the Banach principle could not be applied, Mann [4] developed the Mann iterative 

technique, recognized as one-step iterative, so as to demonstrate that the series tends toward the 

f- points. As a follow-up to Mann's iterative technique, Ishikawa [5] developed a new iterative 

process recognized as two-step iterative to attain the convergence of a Lipschitzian 

pseudocontractive operator in 1974, using the techniques of solution iteration and the auxiliary 

principle.Noor [6] developed the Noor iterative scheme and also recognized the three-step 

iterative scheme to investigate the approximation of the solutions to the inclusions of variations 

in Hilbert spaces. Using contractive-like operators, Asaduzzaman studied a four-step fixed-point 

iterative technique and its convergence in real Banach space [7]. 

The set for all f-points of ₮ denoted by𝐹(₮) = {ᶉ ∈ 𝑈;  ₮ᶉ = ᶉ} can be written as the f-point 

equation: ₮ᶉ = ᶉ for a wide variety of physical equations. We solve this problem by selecting an 

initial value, ᶉ
0
, and solving it iteratively. The iterative definition of the sequence {ᶉ

𝑛
}𝑛=0

∞  leads 

to an increasing f-point in the constant 𝑈, or a mapping 𝑔 from 𝑈 on to 𝑈 nonexpansive if the 

next statement holds for any ᶉ, 𝑠 ∈ 𝑈 

𝑑(𝑔ᶉ, 𝑔𝑠) ≤ 𝑑(ᶉ, 𝑠)                                                    (1.4) 

Numerous writers have presented and explored alternative iteration strategies for approximating 

fixed points for diverse classes of contractive circumstances and studied the convergence, rate of 

convergence and stability (see[8–23]). 

The function Ƿ𝑈: 𝑆 → 𝑈  is called the projective metric function, such that 𝑑(ᶉ, Ƿ𝑈(ᶉ)) ≤

𝑚𝑖𝑛𝑠∈𝑈𝑑(ᶉ, 𝑠) for all ᶉ ∈ 𝑆 and 𝑠 ∈ 𝑈. There are three main parts to this paper: first, a study of 

the convergence of each of MIFSI and FSI; second, a study of the acceleration between the two 

previous iterative schemes and third, a proof of stability for the new iterative. 

 

Definition 1.2 [14]: Any mapping₮ is called quasi-contractive if there exists 𝜛 ≥ 0 and 𝑝 ∈

(0,1) such that  

𝑑(₮ᶉ, ₮𝑠) ≤ 𝜛𝑑(ᶉ, ₮ᶉ) + 𝑝𝑑(ᶉ, 𝑠)     ∀ᶉ, 𝑠 ∈ 𝑈                             (1.5) 

 



IHJPAS. 36 (4) 2023 

369 
 

Definition1.3 [15]: Any mapping₮ is called contractive-like mapping if the statement below is 

true: 

𝑑(₮ᶉ, ₮𝑠) ≤ ∅(𝑑(ᶉ, ₮ᶉ)) + 𝜔𝑑(ᶉ, 𝑠)        ∀ᶉ, 𝑠 ∈ 𝑈                          (1.6) 

Where ∅: 𝑅+ → 𝑅+ strictly increasing with ∅(0) = 0 and 𝜔 ∈ (0,1). 

 

 

Definition1.4 [7]: The FSI is defined as follows:  

ᶉ
𝑛

= (1 − 𝑎𝑛)ᶉ
𝑛−1

+ 𝑎𝑛₮𝑠𝑛 

𝑠𝑛 = (1 − 𝑏𝑛)ᶉ
𝑛−1

+ 𝑏𝑛₮𝑡𝑛 

𝑡𝑛 = (1 − 𝑐𝑛)ᶉ
𝑛−1

+ 𝑐𝑛₮𝑣𝑛 

𝑣𝑛 = (1 − 𝑞𝑛)ᶉ
𝑛−1

+ 𝑞𝑛₮ᶉ
𝑛−1

             𝑛 ≥ 1                   (1.7) 

{𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛},nd {𝑞𝑛} real sequences in [0,1]. 

Definition1.5 [16]: Let (𝑆, 𝑑) be a metric space, then the sequence {ᶉ
𝑛

}𝑛=0
∞  convergence to 𝑓 ∈

𝑆 . If for every 𝜖 > 0, there exists 𝑘 ∈ 𝑁, such that 𝑑(ᶉ
𝑛

, 𝑓) < 𝜖 , for every 𝑛 > 𝑘 , and write 

lim
𝑛→∞

ᶉ
𝑛

= 𝑓 or write ᶉ
𝑛

→ 𝑓. 

Definition 1.6 [17]: Let {ᶉ
𝑛

}
𝑛=0

∞
 and {𝑠𝑛}𝑛=0

∞  be two sequences lies in 𝑅 converge to ᶉ and 𝑠, 

respectively, such that ℓ = lim
𝑛→∞

|ᶉ𝑛−ᶉ|

|𝑠𝑛−𝑠|
 : 

1- If ℓ = 0 then {ᶉ
𝑛

}
𝑛=0

∞
 is converge to ᶉ faster than {𝑠𝑛}𝑛=0

∞  converge to 𝑠. 

2- If 0 < ℓ < ∞ then {ᶉ
𝑛

}
𝑛=0

∞
 and {𝑠𝑛}𝑛=0

∞  have the same rate. 

Some authors have defined new iterative schemes in different spaces and functions,and 

demonstrated their acceleration compared to the currently leading iterative schemes, see [18–21]. 

 

Definition1.7 [22]: suppose (𝑆, 𝑑, 𝒳)  are a CMS and ₮: 𝑆 → 𝑆  self-mapping, 𝑓 ∈ 𝐹(₮) . Let 

{ᶉ
𝑛

}𝑛=0
∞ ⊂ 𝑆  be the sequence produced by an iterative method of hiring ₮ with the definition 

given by:  

ᶉ
𝑛+1

= ƒ₮,𝑎𝑛

ᶉ𝑛  , n=0,1,2,…                                                  (1.8) 

Some functions ƒ₮,𝛼𝑛

ᶉ𝑛  have a convex structure with 𝑎𝑛 ∈ [0,1]  and ᶉ
0

∈ 𝑆  the initial 

approximation , ᶉ
𝑛

→ 𝑓 . Let {𝑠𝑛}𝑛=0
∞ ⊂ 𝑆   and 𝜖𝑛 = 𝑑(𝑠𝑛+1, ƒ₮,𝑎𝑛

𝑠𝑛 ), 𝑛 = 0,1,2, … . say ᶉ
𝑛+1

=

ƒ₮,𝑎𝑛

ᶉ𝑛  is ₮-stable if and only if lim
𝑛→∞

𝜖𝑛 = 0 implies lim
𝑛→∞

𝑠𝑛 = 𝑓 

Lemma 1.1[17]: If 𝜎 ∈ 𝑅 , 0 ≤ 𝜎 < 1  and {𝜖𝑛}𝑛=0
∞  is a sequence of positive numbers and  

lim
𝑛→∞

𝜖𝑛 = 0, {𝑔𝑛}𝑛=0
∞  any sequence of positive number satisfying:  

𝑔𝑛+1 ≤ 𝜎𝑔𝑛 + 𝜖𝑛      n=0,1,2,…                                      (1.9) 

Then lim
𝑛→∞

𝑔𝑛 = 0. 

In this paper, we denote that Ƿ0𝑈, Ƿ1𝑈, Ƿ2𝑈, … , and Ƿ𝑘𝑈 are finite projective metric functions, 

𝑔 nonexpansive function. We represent for f-point of ₮, Ƿ𝑈,  𝑔  by 𝐹(₮), (Ƿ𝑈)and  𝐹(𝑔) 

respectively, 𝑓 is common f-point if 𝑓 ∈ 𝐹(₮) ∩ 𝐹(𝑔) ∩ 𝐹(Ƿ𝑈), and represent for the set of all 

common f-point by 𝐶𝐹(₮, Ƿ𝑈, 𝑔). 

 

2. Main Results  



IHJPAS. 36 (4) 2023 

370 
 

In this section we define MIFSI and study the convergence, rate of converge and  stability with 

respect to generalized quasi like contractive. 

Definition 2.1 Any mapping₮ is called generalized quasi like contractive mapping if  

𝑑(₮ᶉ, ₮𝑠) ≤ ℷ𝑑(ᶉ, 𝑠) + 𝜉∅(𝑑(₮ᶉ, ᶉ)) + 𝜂 min {𝑑(₮ᶉ, ᶉ), 𝑑(₮𝑠, 𝑠)} ,∀ᶉ, 𝑠 ∈ 𝑈        (2.1) 

Where ∅: 𝑅+ → 𝑅+ with ∅(0) = 0 , ℷ ∈ [0,1] and 𝜂, 𝜉 ≥ 0.  

Definition 2.2 The multi_implicit four-step iterative (MIFS) is defined as follows: 

ᶉ
𝑛

= 𝑎𝑛0Ƿ0𝑈ᶉ𝑛−1
+ ∑ 𝑎𝑛𝑖

𝑘
𝑖=1 ₮𝑖𝑠𝑛  

𝑠𝑛 = 𝑏𝑛0𝑔𝑡𝑛 + (1 − 𝑏𝑛0)Ƿ𝑖𝑈₮0𝑡𝑛    

𝑡𝑛 = 𝑐𝑛0Ƿ0𝑈𝑣𝑛 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 Ƿ𝑖𝑈₮𝑖𝑣𝑛  

𝑣𝑛 = 𝑞𝑛0ᶉ𝑛
+ (1 − 𝑞𝑛0)₮0ᶉ

𝑛
                       𝑛 ≥ 1                 (2.2) 

Where ₮𝑖 are finite generalized quasi like contractive mapping define by (2.1) such that 

i=0,1,2,…,k. {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛} and {𝑞𝑛} real sequences in [0,1]. 

Now we state and prove the convergence and stability theorem,for MIFS iteratives . 

Theorem 2.1: Let  ₮𝑖  be a finite generalized quasi like contractive mappings for all 𝑖 =

0,1,2, … , 𝑘  and 𝑔  be a nonexpansive mapping if 𝑓 ∈ 𝐶F(₮𝑖, Ƿ𝑖𝑈, 𝑔).  Then, the MIFS {ᶉ
n

}n=0
∞  

defined by (2.2) with 𝛴(1 − 𝑎𝑛) = ∞, converges to the f-point 𝑓 of ₮𝑖:. 

Proof: Let  𝑓 ∈ 𝐶F(₮𝑖, Ƿ𝑖𝑈, 𝑔), then from {rn}n=0
∞ MIFS  

𝑑(ᶉ
𝑛

, 𝑓) = 𝑑(𝒳(Ƿ0𝑈ᶉ𝑛−1
, ₮𝑖𝑠𝑛, 𝑎𝑛𝑖), 𝑓) 

                 ≤ 𝑎𝑛0𝑑(Ƿ0𝑈ᶉ𝑛−1
, 𝑓) + ∑ 𝑎𝑛𝑖𝑑(₮𝑖𝑠𝑛, 𝑓)𝑘

𝑖=1  

                 ≤ 𝑎𝑛0𝑑(ᶉ
𝑛−1

, 𝑓) + ∑ 𝑎𝑛𝑖 (
ℷ𝑖𝑑(𝑠𝑛, 𝑓) + 𝜉𝑖∅𝑖(₮𝑖𝑓, 𝑓)

+𝜂𝑖  𝑚𝑖𝑛 {𝑑(₮𝑖𝑠𝑛, 𝑠𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
)𝑘

𝑖=1  

                 ≤ 𝑎𝑛0𝑑(ᶉ
𝑛−1

, 𝑓) + ℷ𝑖 ∑ 𝑎𝑛𝑖𝑑(𝑠𝑛, 𝑓)𝑘
𝑖=1                                                                  (2.3) 

𝑑(𝑠𝑛, 𝑓) = 𝑑(𝒳(𝑔𝑡𝑛, Ƿ0𝑈₮0𝑡𝑛, 𝑏𝑛0), 𝑓) 

                 ≤ 𝑏𝑛0𝑑(𝑔𝑡𝑛, 𝑓) + (1 − 𝑏𝑛0)𝑑(𝑓0𝑈₮0𝑡𝑛, 𝑓) 

                 ≤ 𝑏𝑛0𝑑(𝑡𝑛, 𝑓) + (1 − 𝑏𝑛0)𝑑(₮0𝑡𝑛, 𝑓) 

                 ≤ 𝑏𝑛0𝑑(𝑡𝑛, 𝑓) + (1 − 𝑏𝑛0) (
ℷ0𝑑(𝑡𝑛, 𝑓) + 𝜉0∅0(₮𝑖𝑓, 𝑓)

+𝜂0𝑚𝑖𝑛 {𝑑(₮0𝑡𝑛, 𝑡𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
) 

                ≤ (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ0)𝑑(𝑡𝑛, 𝑓)                                                                              (2.4) 

𝑑(𝑡𝑛, 𝑓) = 𝑑(𝒳(Ƿ0𝑈𝑣𝑛, Ƿ𝑖𝑈₮𝑖𝑣𝑛, 𝑐𝑛𝑖), 𝑓) 

                  ≤ 𝑐𝑛0𝑑(𝑓0𝑈𝑣𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(𝑘
𝑖=1 𝑓𝑖𝑈₮𝑖𝑣𝑛, 𝑓)  

                  ≤ 𝑐𝑛0𝑑(𝑣𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(𝑘
𝑖=1 ₮𝑖𝑣𝑛, 𝑓)  

                   ≤ 𝑐𝑛0𝑑(𝑣𝑛, 𝑓) + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 (

ℷ𝑖𝑑(𝑣𝑛, 𝑓) + 𝜉𝑖∅𝑖(₮𝑖𝑓, 𝑓)

+𝜂𝑖  𝑚𝑖𝑛 {𝑑(₮𝑖𝑣𝑛, 𝑣𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
)  

                   ≤ 𝑑(𝑣𝑛, 𝑓)(𝑐𝑛0 + ℷ𝑖 ∑ 𝑐𝑛𝑖)
𝑘
𝑖=1                                                                               (2.5) 

𝑑(𝑣𝑛, 𝑓) = 𝑑(𝒳(ᶉ
𝑛

, ₮0ᶉ
𝑛

, 𝑞𝑛0), 𝑓) 

              ≤ 𝑞𝑛0𝑑(ᶉ
𝑛

, 𝑓) + (1 − 𝑞𝑛0)𝑑(₮0ᶉ
𝑛

, 𝑓) 

                ≤ 𝑞𝑛0𝑑(ᶉ
𝑛

, 𝑓) + (1 − 𝑞𝑛0) (
ℷ0𝑑(ᶉ

𝑛
, 𝑓) + 𝜉0∅0(₮0𝑓, 𝑓)

+𝜂0 𝑚𝑖𝑛 {𝑑(₮0ᶉ
𝑛

, ᶉ
𝑛

), 𝑑(₮𝑖𝑓, 𝑓)}
) 

                ≤ (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ0)𝑑(ᶉ
𝑛

, 𝑓)                                                                              (2.6) 

Take ℷ = 𝑚𝑎𝑥 { ℷ𝑖 , 𝑖 = 0,1,2 … 𝑘}. 

From (2.3),(2.4),(2.5), and (2.6) we have : 



IHJPAS. 36 (4) 2023 

371 
 

𝑑(ᶉ𝑛, 𝑓) ≤ 𝑎𝑛0 𝑑(ᶉ
𝑛−1

, 𝑓) + [ℷ ∑ 𝑎𝑛𝑖
𝑘
𝑖=1 (

𝑞𝑛0 +
(1 − 𝑞𝑛0)ℷ

) (
𝑐𝑛0 +

ℷ ∑ 𝑐𝑛𝑖
𝑘
𝑖=1

) (
𝑏𝑛0 +

(1 − 𝑏𝑛0)ℷ
)] 𝑑(ᶉ

𝑛
, 𝑓)  

(1 − [ℷ ∑ 𝑎𝑛𝑖
𝑘
𝑖=1 (

𝑞𝑛0 +
(1 − 𝑞𝑛0)ℷ

) (
𝑐𝑛0 +

ℷ ∑ 𝑐𝑛𝑖
𝑘
𝑖=1

) (
𝑏𝑛0 +

(1 − 𝑏𝑛0)ℷ
)] 𝑑(ᶉ

𝑛
, 𝑓) ≤ 𝑎𝑛0 𝑑(ᶉ

𝑛−1
, 𝑓)  

𝑑(ᶉ
𝑛

, 𝑓) ≤
𝑎𝑛0 𝑑(ᶉ

𝑛−1
, 𝑓)

1 − [ℷ ∑ 𝑎𝑛𝑖
𝑘
𝑖=1 (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ) (𝑐𝑛0 + ℷ ∑ 𝑐𝑛𝑖)

𝑘
𝑖=1 (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)]

 

Let 
𝑅𝑛

𝑆𝑛
=

𝑎𝑛0

1−[ℷ ∑ 𝑎𝑛𝑖
𝑘
𝑖=1 (𝑞𝑛0+(1−𝑞𝑛0)ℷ) (𝑐𝑛0+ℷ ∑ 𝑐𝑛𝑖)𝑘

𝑖=1 (𝑏𝑛0+(1−𝑏𝑛0)ℷ)]
  

1 − 
𝑅𝑛

𝑆𝑛
= 1 −

𝑎𝑛0

1 − [ℷ ∑ 𝑎𝑛𝑖
𝑘
𝑖=1 (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ) (𝑐𝑛0 + ℷ ∑ 𝑐𝑛𝑖)

𝑘
𝑖=1 (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)]

 

𝑅𝑛

𝑆𝑛
≤ ℷ ∑ 𝑎𝑛𝑖

𝑘
𝑖=1 (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ) (𝑐𝑛0 + ℷ ∑ 𝑐𝑛𝑖)

𝑘
𝑖=1 (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ) + 𝑎𝑛0  

𝑑(ᶉ
𝑛

, 𝑓) ≤ [ℷ ∑ 𝑎𝑛𝑖
𝑘
𝑖=1 (

𝑞𝑛0 +
(1 − 𝑞𝑛0)ℷ

) (
𝑐𝑛0 +

ℷ ∑ 𝑐𝑛𝑖
𝑘
𝑖=1

) (
𝑏𝑛0 +

(1 − 𝑏𝑛0)ℷ
) + 𝑎𝑛0]  𝑑(ᶉ

𝑛−1
, 𝑓)  

𝑑(ᶉ
𝑛

, 𝑓) ≤ [ℷ (
1 −
𝑎𝑛0

) (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
) (

𝑐𝑛0 +

ℷ(1 − 𝑐𝑛0)
) (

𝑏𝑛0 +
(1 − 𝑏𝑛0)ℷ

) + 𝑎𝑛0]  𝑑(ᶉ
𝑛−1

, 𝑓) 

            ≤ [1 − (1 − 𝑎𝑛0) (1 − ℷ (
1 −
𝑎𝑛0

) (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
) (

𝑐𝑛0 +
(1 − 𝑐𝑛0)ℷ

) (
𝑏𝑛0 +

(1 − 𝑏𝑛0)ℷ
))] 𝑑(ᶉ

𝑛−1
, 𝑓) 

               : 

    ≤ ∏  [1 − (1 − 𝑎𝑗0) (1 − ℷ (
1 −
𝑎𝑗0

) (
𝑞𝑗0 +

(1 − 𝑞𝑗0)ℷ
) (

𝑐𝑗0 +

(1 − 𝑐𝑗0)ℷ
) (

𝑏𝑗0 +

(1 − 𝑏𝑗0)ℷ
))] 𝑑(ᶉ

0
, 𝑓)𝑛

𝑗=1   

Take limit as 𝑛 → ∞ for both side we have 𝑑(ᶉ
𝑛

, 𝑓) = 0. 

∎ 

By the same way, we can prove the four-step iterative converges to the f-point 𝑓 of ₮:. 

Now, we prove the MIFSI convergent is faster than FSI. 

 

Theorem 2.2 Let ₮𝒊 be a finite generalized quasi like contractive mapping for all 𝑖 = 0,1,2, … , 𝑘 

defined by (2.1) and 𝑔 be a nonexpansive mapping, If 𝑓 ∈ 𝐶F(₮𝑖, Ƿ𝑖𝑈, 𝑔). Then, for ᶉ0 ∈ U, the 

MIFSI {ᶉ
n

}n=0
∞  defined by (2.2) convergence faster than FSI {ᶉ

n
}n=0

∞  defined by (1.4): 

Proof: Since 𝑓 ∈ 𝐶F(₮𝑖, Ƿ𝑖𝑈, 𝑔), then from {ᶉ
n

}n=0
∞ MIFS  

𝑑(ᶉ
𝑛

, 𝑓) = 𝑑(𝒳(Ƿ0𝑈ᶉ𝑛−1
, ₮𝑖𝑠𝑛, 𝑎𝑛𝑖), 𝑓) 

                ≤ 𝑎𝑛0𝑑(Ƿ0𝑈ᶉ𝑛−1
, 𝑓) + ∑ 𝑎𝑛𝑖𝑑(₮𝑖𝑠𝑛, 𝑓)𝑘

𝑖=1  

                ≤ 𝑎𝑛0𝑑(ᶉ
𝑛−1

, 𝑓) + ℷ𝑖 ∑ 𝑎𝑛𝑖𝑑(𝑠𝑛, 𝑓)𝑘
𝑖=1                                                                  (2.3) 

𝑑(𝑠𝑛, 𝑓) = 𝑑(𝒳(𝑔𝑡𝑛, Ƿ0𝑈₮0𝑡𝑛, 𝑏𝑛0), 𝑓) 

                ≤ 𝑏𝑛0𝑑(𝑔𝑡𝑛, 𝑓) + (1 − 𝑏𝑛0)𝑑(𝑓0𝑈₮0𝑡𝑛, 𝑓) 

                ≤ (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ0)𝑑(𝑡𝑛, 𝑓)                                                                              (2.4) 

𝑑(𝑡𝑛, 𝑓) = 𝑑(𝒳(Ƿ0𝑈𝑣𝑛, Ƿ𝑖𝑈₮𝑖𝑣𝑛, 𝑐𝑛𝑖), 𝑓) 

                ≤ 𝑐𝑛0𝑑(𝑓0𝑈𝑣𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(𝑘
𝑖=1 𝑓𝑖𝑈₮𝑖𝑣𝑛, 𝑓)  

                ≤ 𝑑(𝑣𝑛, 𝑓)(𝑐𝑛0 + ℷ𝑖 ∑ 𝑐𝑛𝑖)
𝑘
𝑖=1                                                                                  (2.5) 

𝑑(𝑣𝑛, 𝑓) = 𝑑(𝒳(ᶉ
𝑛

, ₮0ᶉ𝑛 , 𝑞𝑛), 𝑓) 

              ≤ 𝑞𝑛0𝑑(ᶉ
𝑛

, 𝑓) + (1 − 𝑞𝑛0)𝑑(₮0ᶉ
𝑛

, 𝑓) 

                ≤ (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ0)𝑑(ᶉ
𝑛

, 𝑓)                                                                              (2.6) 

Take ℷ = 𝑚𝑎𝑥 { ℷ𝑖 , 𝑖 = 0,1,2 … 𝑘}. 

From (2.3),(2.4),(2.5) ,and (2.6) we have :  



IHJPAS. 36 (4) 2023 

372 
 

𝑑(ᶉ
𝑛

, 𝑓) ≤ [ℷ (
1 −
𝑎𝑛0

) (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
) (

𝑐𝑛0 +

ℷ(1 − 𝑐𝑛0)
) (

𝑏𝑛0 +
(1 − 𝑏𝑛0)ℷ

) + 𝑎𝑛0]  𝑑(ᶉ
𝑛−1

, 𝑓) 

                ≤ [1 − (
1 −
𝑎𝑛0

) (1 − ℷ (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
) (

𝑐𝑛0 +
(1 − 𝑐𝑛0)ℷ

) (
𝑏𝑛0 +

(1 − 𝑏𝑛0)ℷ
))] 𝑑(ᶉ

𝑛−1
, 𝑓) 

 

 

We can write  

[1 − (1 − 𝑎𝑛0) (1 − ℷ (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
) (

𝑐𝑛0 +
(1 − 𝑐𝑛0)ℷ

) (
𝑏𝑛0 +

(1 − 𝑏𝑛0)ℷ
))] ≤ 1 − (1 − 𝑎𝑛0)(1 − ℷ) 

𝑑(ᶉ
𝑛

, 𝑓) ≤ (1 − (1 − 𝑎𝑛0)(1 − ℷ))𝑑(ᶉ
𝑛−1

, 𝑓)          

              : 

             ∏ (1 − (1 − 𝑎𝑗0)(1 − ℷ)) 𝑑(ᶉ
0

, 𝑓)𝑛
𝑗=1                                                           (2.7) 

Now, to get the {ᶉ
n

}n=0
∞  for FSI: 

𝑑(ᶉ
𝑛

, 𝑓) = 𝑑(𝒳(ᶉ
𝑛−1

, ₮0𝑠𝑛, 𝑎𝑛0), 𝑓) 

              ≤ (1 − 𝑎𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + 𝑎𝑛0𝑑(₮0𝑠𝑛, 𝑓) 

              ≤ (1 − 𝑎𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + ℷ𝑎𝑛0𝑑(𝑠𝑛, 𝑓)                                                                  (2.8) 

𝑑(𝑠𝑛, 𝑓) = 𝑑(𝒳(ᶉ
𝑛−1

, ₮0𝑡𝑛, 𝑏𝑛0), 𝑓) 

              ≤ (1 − 𝑏𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + 𝑏𝑛0𝑑(₮0𝑡𝑛, 𝑓) 

              ≤ (1 − 𝑏𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + ℷ𝑏𝑛0𝑑(𝑡𝑛, 𝑓)                                                                   (2.9) 

𝑑(𝑡𝑛, 𝑓) = 𝑑(𝒳(ᶉ
𝑛−1

, ₮0𝑣𝑛, 𝑐𝑛0), 𝑓) 

              ≤ (1 − 𝑐𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + 𝑐𝑛0𝑑(₮0𝑣𝑛, 𝑓) 

              ≤ (1 − 𝑐𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + ℷ𝑐𝑛0𝑑(𝑣𝑛, 𝑓)                                                                 (2.10) 

𝑑(𝑣𝑛, 𝑓) = 𝑑(𝒳(ᶉ
𝑛−1

, ₮0ᶉ
𝑛−1

, 𝑞𝑛), 𝑓) 

              ≤ (1 − 𝑞𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + 𝑞𝑛0𝑑(₮0ᶉ
𝑛−1

, 𝑓) 

              ≤ (1 − 𝑞𝑛0)𝑑(ᶉ
𝑛−1

, 𝑓) + ℷ𝑞𝑛0𝑑(ᶉ
𝑛−1

, 𝑓)                                                            (2.11)    

From (2.8),(2.9),(2.10) ,and (2.11) we have:  

𝑑(ᶉ
𝑛

, 𝑓) ≤ ((1 − 𝑎𝑛0) + ℷ𝑎𝑛0((1 − 𝑏𝑛) + ℷ𝑏𝑛0((1 − 𝑐𝑛0) +

ℷ𝑐𝑛0((1 − 𝑞𝑛0) + ℷ𝑞𝑛0))))𝑑(ᶉ
𝑛−1

, 𝑓)     

             = (1 − 𝑎𝑛0(1 − ℷ(1 − 𝑏𝑛0(1 − ℷ(1 − 𝑐𝑛0) − ℷ2𝑐𝑛0(1 − ℷ) − ℷ3𝑐𝑛0𝑞𝑛0))))𝑑(ᶉ
𝑛−1

, 𝑓)     

            = (1 − 𝑎𝑛0(1 − ℷ(1 − 𝑏𝑛0(1 − ℷ(1 − 𝑐𝑛0(1 − ℷ(1 − 𝑞𝑛0(1 − ℷ))))))))𝑑(ᶉ
𝑛−1

, 𝑓)  

We can write : 

(1 − 𝑎𝑛0(1 − ℷ(1 − 𝑏𝑛0(1 − ℷ(1 − 𝑐𝑛0(1 − ℷ(1 − 𝑞𝑛0(1 − ℷ)))))))) ≤ (1 − 𝑎𝑛0(1 − ℷ)) 

𝑑(ᶉ
𝑛

, 𝑓) ≤ (1 − 𝑎𝑛0(1 − ℷ))𝑑(ᶉ
𝑛−1

, 𝑓)  

               : 

              ≤ ∏ (1 − 𝑎𝑗0(1 − ℷ)) 𝑑(ᶉ
0

, 𝑓)𝑛
𝑗=1                                                                         (2.12) 

 

Take lim
𝑛→∞

{ᶉn}n=0
∞  𝑀𝐼𝐹𝑆𝐼

{ᶉn}n=0
∞  𝐹𝑆𝐼

=
∏ (1−(1−𝑎𝑗0)(1−ℷ))𝑑(ᶉ0,𝑓)𝑛

𝑗=1

∏ (1−𝑎𝑗0(1−ℷ))𝑑(ᶉ0,𝑓)𝑛
𝑗=1

= 0. 

Then, by definition 1.6 the MIFSI converges faster than FSI to f-point when 1 − 𝑎𝑗0 > 𝑎𝑗0. 

∎ 



IHJPAS. 36 (4) 2023 

373 
 

Theorem 2.3 Let ₮𝑖 be a finite generalized quasi-like contractive mapping satisfying (2.1) 

with𝐶𝐹(₮𝑖 , Ƿ𝑖𝑈, 𝑔) ≠ ∅. Then, for ᶉ
0

∈ 𝑈, the sequence {ᶉ𝑛}𝑛=0
∞  defined by the MIFSI iterative 

(2.2) with the converging point at 𝑓 ∈ 𝐶𝐹(₮𝑖, Ƿ𝑖𝑈, 𝑔) ,is ₮𝑖-stable. 

Proof: Let {𝑙𝑛}𝑛=0
∞  ⊂ 𝑈 be an arbitrary sequence such that 𝜖𝑛 = 𝑑(𝑙𝑛, 𝒳(Ƿ0𝑈𝑙𝑛−1, ₮𝑖𝑗𝑛, 𝑎𝑛𝑖))  

where,  𝑗𝑛 = 𝒳(𝑔𝑚𝑛, Ƿ𝑖𝑈₮0𝑚𝑛, 𝑏𝑛0) , 𝑚𝑛 = 𝒳(Ƿ0𝑈𝑛𝑛, Ƿ𝑖𝑈₮𝑖𝑛𝑛, 𝑐𝑛𝑖)  , 𝑛𝑛 =

𝒳(𝑙𝑛, ₮0𝑙𝑛, 𝑞𝑛0)and 

𝑙𝑖𝑚
𝑛→∞

𝜖𝑛 = 0. 

𝑑(𝑙𝑛, 𝑓) ≤ 𝑑(𝑙𝑛, 𝒳(Ƿ0𝑈𝑙𝑛−1, ₮𝑖𝑗𝑛, 𝑎𝑛𝑖) ) + 𝑑(𝒳(Ƿ0𝑈𝑙𝑛−1, ₮𝑖𝑗𝑛, 𝑎𝑛𝑖), 𝑓) 

               ≤ 𝜖𝑛 + 𝑎𝑛0𝑑(Ƿ0𝑈𝑙𝑛−1, 𝑓) + ∑ 𝑎𝑛𝑖𝑑(₮𝑖𝑗𝑛, 𝑓)𝑘
𝑖=1   

                ≤ 𝜖𝑛+ ≤ 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓) + ∑ 𝑎𝑛𝑖 [
ℷ𝑖𝑑(𝑗𝑛, 𝑓) + 𝜉𝑖∅𝑖 (𝑑(₮𝑖𝑓, 𝑓))

+𝜂𝑖  𝑚𝑖𝑛 {𝑑(𝑗𝑛, ₮𝑖𝑗𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
]𝑘

𝑖=1   

                ≤ 𝜖𝑛+ ≤ 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓) + ∑ 𝑎𝑛𝑖ℷ𝑖𝑑(𝑗𝑛, 𝑓)𝑘
𝑖=1                                                      (2.13) 

𝑑(𝑗𝑛, 𝑓) ≤ 𝑏𝑛0𝑑(𝑔𝑚𝑛, 𝑓) + (1 − 𝑏𝑛0)𝑑(Ƿ0𝑈₮0𝑚𝑛, 𝑓) 

                ≤ 𝑏𝑛0𝑑(𝑚𝑛, 𝑓) + (1 − 𝑏𝑛0)𝑑(₮0𝑚𝑛, 𝑓) 

                ≤ 𝑏𝑛0𝑑(𝑚𝑛, 𝑓) + (1 − 𝑏𝑛0) [
ℷ0𝑑(𝑚𝑛, 𝑓) + 𝜉0∅0(𝑑(₮0𝑓, 𝑓))

+𝜂0 𝑚𝑖𝑛 {𝑑(𝑚𝑛, ₮0𝑚𝑛), 𝑑(₮0𝑓, 𝑓)}
] 

                ≤ (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ0)𝑑(𝑚𝑛, 𝑓)                                                                          (2.14) 

  𝑑(𝑚𝑛, 𝑓) ≤ 𝑐𝑛0𝑑(Ƿ0𝑈𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(Ƿ𝑖𝑈₮𝑖𝑛𝑛, 𝑓)𝑘
𝑖=1   

                ≤ 𝑐𝑛0𝑑(𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(₮𝑖𝑛𝑛, 𝑓)𝑘
𝑖=1   

                ≤ 𝑐𝑛0𝑑(𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 [

ℷ𝑖𝑑(𝑛𝑛, 𝑓) + 𝜉𝑖∅𝑖 (𝑑(₮𝑖𝑓, 𝑓))

+𝜂𝑖 𝑚𝑖𝑛 {𝑑(𝑛𝑛, ₮𝑖𝑛𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
]  

                ≤ 𝑐𝑛0𝑑(𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ𝑖𝑑(𝑛𝑛, 𝑓)  

                ≤ (𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ𝑖)𝑑(𝑛𝑛, 𝑓)                                                                              (2.15) 

𝑑(𝑛𝑛, 𝑓) ≤ 𝑞𝑛0𝑑(𝑙𝑛, 𝑓) + (1 − 𝑞𝑛0)𝑑(₮0𝑙𝑛, 𝑓) 

𝑑(𝑛𝑛, 𝑓) ≤ 𝑞𝑛0𝑑(𝑙𝑛, 𝑓) + (1 − 𝑞𝑛0) [
ℷ0𝑑(𝑙𝑛, 𝑓) + 𝜉0∅0(𝑑(₮0𝑓, 𝑓))

+𝜂0 𝑚𝑖𝑛 {𝑑(𝑙𝑛, ₮0𝐿𝑛), 𝑑(₮0𝑓, 𝑓)}
] 

𝑑(𝑛𝑛, 𝑓) ≤ (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ0)𝑑(𝑙𝑛, 𝑓)                                                                           (2.16) 

Take ℷ = 𝑚𝑎𝑥 { ℷ𝑖 , 𝑖 = 1,2, … , 𝑘}. 

From (2.13),(2.14),(2.15), and (2.16) we have . 

𝑑(𝑙𝑛, 𝑓) ≤ 𝜖𝑛+ ≤ 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓) +

∑ 𝑎𝑛𝑖ℷ(𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)(𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ)𝑑(𝑙𝑛, 𝑓)𝑘

𝑖=1   

(1 − ∑ 𝑎𝑛𝑖 ℷ (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)(
𝑐𝑛0 +

∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ

) (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
)𝑘

𝑖=1 )𝑑(𝑙𝑛, 𝑓) ≤ 𝜖𝑛 + 𝑎𝑛0𝑑 (𝑙𝑛−1, 𝑓)  

𝑑(𝑙𝑛, 𝑓) ≤
𝜖𝑛

1 − [∑ 𝑎𝑛𝑖ℷ(𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)(𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ)𝑘

𝑖=1 ]

+
𝑎𝑛0𝑑 (𝑙𝑛−1, 𝑓)

1 − [∑ 𝑎𝑛𝑖ℷ(𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)(𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ)𝑘

𝑖=1 ]
 

𝑑(𝑙𝑛, 𝑓) ≤
𝜖𝑛𝑎𝑛0

[1 − ∑ 𝑎𝑛𝑖ℷ(𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)(𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ)𝑘

𝑖=1 ]𝑎𝑛0

+
𝑎𝑛0𝑑 (𝑙𝑛−1, 𝑓)

1 − [∑ 𝑎𝑛𝑖ℷ(𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)(𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ)𝑘

𝑖=1 ]
. 



IHJPAS. 36 (4) 2023 

374 
 

But 

 
𝑎𝑛0

1−[∑ 𝑎𝑛𝑖ℷ(𝑏𝑛0+(1−𝑏𝑛0)ℷ)(𝑐𝑛0+∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0+(1−𝑞𝑛0)ℷ)𝑘

𝑖=1 ]
 ≤ 1 − (1 − 𝑎𝑛0)(1 − ℷ) 

𝑑(𝑙𝑛, 𝑓) ≤ [1 − (1 − 𝑎𝑛0)(1 − ℷ)]
𝜖𝑛

𝑎𝑛0
+ [1 − (1 − 𝑎𝑛0)(1 − ℷ)]𝑑 (𝑙𝑛−1, 𝑓)  

 But, 1 − (1 − 𝑎𝑛0)(1 − ℷ) < 1. 

since 𝑙𝑖𝑚
𝑛→∞

𝜖𝑛 = 0, and by Lemma(1.1), we have 𝑙𝑖𝑚
𝑛→∞

𝑑(𝑙𝑛, 𝑓) = 0 : 

which implies that 𝑙𝑖𝑚
𝑛→∞

𝑙𝑛 = 𝑓. 

Conversely, if 𝑙𝑖𝑚
𝑛→∞

𝑙𝑛 = 𝑓  

𝜖𝑛 = 𝑑(𝑙𝑛, 𝒳(Ƿ𝑜𝑈𝑙𝑛−1, ₮𝑖𝑗𝑛, 𝑎𝑛𝑖) 

      ≤ 𝑑(𝑙𝑛, 𝑓) + 𝑑(𝒳(Ƿ0𝑈𝑙𝑛−1, ₮𝑖𝑗𝑛, 𝑎𝑛𝑖) , 𝑓) 

      ≤ 𝑑(𝑙𝑛, 𝑓) + 𝑎𝑛0𝑑(Ƿ0𝑈𝑙𝑛−1, 𝑓) + ∑ 𝑎𝑛𝑖𝑑(₮𝑖𝑗𝑛, 𝑓)𝑘
𝑖=1   

      ≤ 𝑑(𝑙𝑛, 𝑓) + 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓) + ∑ 𝑎𝑛𝑖 [
ℷ𝑖𝑑(𝑗𝑛, 𝑓) + 𝜉𝑖∅𝑖 (𝑑(₮𝑖𝑓, 𝑓))

+𝜂𝑖 𝑚𝑖𝑛 {𝑑(𝑗𝑛, ₮𝑖𝑗𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
]𝑘

𝑖=1   

      ≤ 𝑑(𝑙𝑛, 𝑓) + 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓) + ∑ 𝑎𝑛𝑖ℷ𝑖𝑑(𝑗𝑛, 𝑓)𝑘
𝑖=1                                                          (2.17) 

𝑑(𝑗𝑛, 𝑓) ≤ 𝑏𝑛0𝑑(𝑔𝑚𝑛, 𝑓) + (1 − 𝑏𝑛0)𝑑(₮0𝑚𝑛, 𝑓) 

                ≤ 𝑏𝑛0𝑑(𝑚𝑛, 𝑓) + (1 − 𝑏𝑛0) [
ℷ0𝑑(𝑚𝑛, 𝑓) + 𝜉0∅0(𝑑(₮0𝑓, 𝑓))

+𝜂0 𝑚𝑖𝑛 {𝑑(𝑚𝑛, ₮0𝑚𝑛), 𝑑(₮0𝑓, 𝑓)}
] 

                ≤ (𝑏𝑛0 + (1 − 𝑏𝑛)ℷ0)𝑑(𝑚𝑛0, 𝑓)                                                                          (2.18) 

𝑑(𝑚𝑛, 𝑓) ≤ 𝑐𝑛0𝑑(Ƿ0𝑈𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(Ƿ𝑖𝑈₮𝑖𝑛𝑛, 𝑓)𝑘
𝑖=1   

                  ≤ 𝑐𝑛0𝑑(𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖𝑑(₮𝑖𝑛𝑛, 𝑓)𝑘
𝑖=1   

                  ≤ 𝑐𝑛0𝑑(𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 [

ℷ𝑖𝑑(𝑛𝑛, 𝑓) + 𝜉𝑖∅𝑖 (𝑑(₮𝑖𝑓, 𝑓))

+𝜂𝑖 𝑚𝑖𝑛 {𝑑(𝑛𝑛, ₮𝑖𝑛𝑛), 𝑑(₮𝑖𝑓, 𝑓)}
]  

                  ≤ 𝑐𝑛0𝑑(𝑛𝑛, 𝑓) + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ𝑖𝑑(𝑛𝑛, 𝑓)  

                  ≤ (𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ𝑖)𝑑(𝑛𝑛, 𝑓)                                                                            (2.19) 

𝑑(𝑛𝑛, 𝑓) ≤ 𝑞𝑛0𝑑(𝐿𝑛, 𝑓) + (1 − 𝑞𝑛0)𝑑(₮0𝐿𝑛, 𝑓) 

                 ≤ 𝑞𝑛0𝑑(𝐿𝑛, 𝑓) + (1 − 𝑞𝑛0) [
ℷ0𝑑(𝐿𝑛, 𝑓) + 𝜉0∅0(𝑑(₮0𝑓, 𝑓))

+𝜂0 𝑚𝑖𝑛 {𝑑(𝐿𝑛, ₮0𝐿𝑛), 𝑑(₮0𝑓, 𝑓)}
] 

                 ≤ (𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ0)𝑑(𝐿𝑛, 𝑓)                                                                          (2.20) 

Take ℷ = 𝑚𝑎𝑥 { ℷ𝑖 , 𝑖 = 1,2, … , 𝑘}. 

From (2.17),(2.18),(2.19) ,and (2.20) we have: 

𝜖𝑛 ≤ 𝑑(𝑙𝑛, 𝑓) + 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓) + (∑ 𝑎𝑛𝑖 ℷ
(𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ)

(𝑐𝑛0 + ∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ)(𝑞𝑛0 + (1 − 𝑞𝑛0)ℷ)

𝑘
𝑖=1 ) 𝑑(𝑙𝑛, 𝑓)  

𝜖𝑛 ≤ (1 + ∑ 𝑎𝑛𝑖 ℷ (𝑏𝑛0 + (1 − 𝑏𝑛0)ℷ) (
𝑐𝑛0 +

∑ 𝑐𝑛𝑖
𝑘
𝑖=1 ℷ

) (
𝑞𝑛0 +

(1 − 𝑞𝑛0)ℷ
)𝑘

𝑖=1 )𝑑(𝑙𝑛, 𝑓) + 𝑎𝑛0𝑑(𝑙𝑛−1, 𝑓).  

Take limit for two sides with 𝑙𝑖𝑚
𝑛→∞

𝑑(𝑙𝑛, 𝑓) = 0, we have 𝑙𝑖𝑚
𝑛→∞

𝜖𝑛 = 0. 

∎ 

3. Conclusion 

The results of this paper as follows: 

1. MIFSI is convergent to the f-point of generalized quasi-like contractive. 

2. MIFSI has a rate of convergence faster than FSI. 



IHJPAS. 36 (4) 2023 

375 
 

3. We proved the stability of MIFSI with generalized quasi-contractive. 

 

 

 

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