IHJPAS. 36(1)2023 

380 

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

 

   

 

 

Study Sstrong Convergence and Acceleration of New Iteration Type Three – 

Step 

 

 

 

 
 

 

 

Abstract  

I The paper aims to define a new iterative method called the three-step type in which Jungck 

resolvent CR-iteration and resolvent Jungck SP-iteration are discussed and study rate convergence 

and strong convergence in Banach space to reach the fixed point which is differentially solved of 

nonlinear equations. The studies also expanded around it to find the best solution for nonlinear 

operator equations in addition to the varying inequalities in Hilbert spaces and Banach spaces, as 

well as the use of these iterative methods to approximate the difference between algorithms and 

their images, where we examined the necessary conditions that guarantee the unity and existence 

of the solid point. Finally, the results show that resolvent CR-iteration is faster than resolvent 

Jungck SP-iteration using Jungck resolvent estimation. 

Keywords: Jungck mapping, Rate of convergence, Jungck 𝐶𝑅 –  iteration, Rsolvent Jungck 𝒮P 
– iteration.  

 

1.Introduction and Preliminary 

     In the last three decades, a lot of literature has been published on the iterative approximation of 

fixed −points for certain classes of operators, using the methods of the Mann and Ishikawa 

algorithm. Fixed−point theorems were developed for single-valued or definite-value assignments 

of metric spaces and Banach. Among the topics of fixed−point theorem, the topic of fixed−point 

approximations for assignments is important because it is useful for proving the existence of 

fixed−point assignments. However, once one knows that there is a fixed− point for some map, 

then finding the value of that fixed point is not an easy task, which is the reason of why we use 

iterative operations to calculate it. Over time, many iterative processes have been developed. These 

doi.org/10.30526/36.1.2917 

Article history: Received 30 June 2022, Accepted 21 Augest 2022, Published in January 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iqx 

 

Zena Hussein Maibed 

Department of Mathematics , College of 

Education for Pure Sciences,Ibn Al –Haitham/ 

University of  Baghdad, Baghdad, Iraq. 

mrs_ zena.hussein@yahoo.com 

 
 

Alaa M. AL- Hameedwi 

Department of Mathematics , College of 

Education for Pure Sciences,Ibn Al –Haitham/ 

University of  Baghdad, Baghdad, Iraq. 

Alaa.Mohsen1203a@ihcoedu.uobaghdad.edu.iq 

 
 

https://creativecommons.org/licenses/by/4.0/
about:blank
mailto:Alaa.Mohsen1203a@ihcoedu.uobaghdad.edu.iq


IHJPAS. 36(1)2023 
 

381 

iterative processes have been used, to solve various types of differential equations in recent years. 

The well-known Bannack deflationary theorem uses the process of Picard's algorithm to 

approximate the fixed− point [1]. Some other well-known iterative processes are Agarwal [2], 

Noor [3], SP [4], Piccard S [5], and so on. Recently, Qing and Qhou [6] extended their results to a 

random period and the Ishikawa algorithm. They gave some control conditions for the convergence 

of the Ishikawa algorithm over a random period. In 2006, [7] presented that Mann's algorithm 

converges faster than Ishikawa's algorithm. Subsequently, many studies were conducted on this 
subject see [8, 9, 10]. 

This paper aims to propose new iterative schemes for a fixed point and proves the convergence 

and speediness using resolvent ZA-Jungck mapping 

Now, let 𝒟 be a nonempty subset closed -convex of a Banach space 𝑁 ,we recall the following: 
Definition 1.1 [11]: Let {𝑣𝑛 }𝑛=0

∞  be a bounded sequence in 𝑁 for all  𝑣 ∈ 𝑁and 

𝑟(𝑣 , 〈𝑣𝑛 〉 ) =   𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝 ║𝑣𝑛–  𝑣║. Then 

1. the asymptotic radius of 〈𝑣𝑛 〉 relative to 𝒟 is given by: 
𝑟(𝒟, 〈𝑣𝑛 〉 ) =  𝑖𝑛𝑓 {𝑟(𝑣 , 〈𝑣𝑛 〉 ) ∶  𝑣 ∈ 𝒟}. 
2. The asymptotic center A(𝒟, 〈𝑣n〉) of 〈𝑣n〉 is defined as: 
𝐴 (𝒟, 〈𝓊𝑛 〉) = {𝑣 ∈  𝒟: 𝑟(𝑣, 〈𝑣𝑛 〉) = 𝑟( 𝒟, 〈𝑣𝑛 〉)}.  
Definition 1.2 [12]: Let  〈𝑦𝑛  〉, 〈𝑜𝑛 〉  be a real sequence such that 〈𝑦𝑛 〉 converges to 𝑦 , 

  〈o𝑛〉 Converges to 𝑜. And E = lim
𝑛→∞

|𝑦𝑛−𝑦|

|𝑜𝑛−𝑜|
 

1. If E = 0  ↦ the sequence 〈𝑦𝑛  〉 is converge to 𝑦 Faster then 〈𝑜𝑛 〉 converge to 𝑜 
2. if    0 < 𝐸 < ∞  ↦     〈𝑌𝑛 〉 and 〈𝑂𝑛〉  have the same rate of convergence. 
Lemma 1.3 [13]:  Let ℬ be a uniformly convex - space and 〈𝓊𝑛 〉 be any sequence such that  
0 < q ≤ 𝓊𝑛 < 1  For some q ∈ ℝ and∀ 𝑛 ≥ 1. Let 〈𝑣𝑛 〉 and 〈𝑙𝑛〉 be any two sequences of ℬ such 
that lim

𝑛→∞
sup‖𝑣𝑛‖ ≤ 𝑐, lim

𝑛→∞
sup‖𝑙𝑛‖ ≤ 𝑐and  𝑐 = lim

𝑛→∞
sup‖𝓊𝑛 𝑣𝑛 + (1 − 𝓊𝑛)𝑙𝑛‖  for some 𝑐 ≥

0. 
 𝑇ℎ𝑒𝑛 𝑙𝑖𝑚

𝑛→∞
‖𝑣𝑛 − 𝑙𝑛‖ = 0. 

2.Main Results 

In this part, we will present a new iteration and study the existence of fixed points and the 

convergence of these iterations 

Definition 2.1: A resolvent Jungck 𝒞ℛ – iteration  
Sen+1 = 𝒥𝓇n(( 1 −  an)TSwn +  an 𝒥𝓇nTTwn) , 

Swn  = ( 1 −  cn)𝒥𝓇nTSzn +  cn TTzn, 
Szn  = ( 1 −  bn)𝒥𝓇nTSen +  bn𝒥𝓇n Ten , where 𝑧0 ∈ 𝒞 , S, T commute mappings are 
on 𝒞 and  〈an〉, 〈bn〉 and 〈cn〉 are sequences in [0,1]   
Definition 2.2: A resolvent Jungck 𝒮𝒫 – iteration  
𝑆𝑢𝑛+1 = ( 1 −  𝑎)𝒥𝓇𝑛𝑇𝑆𝑥𝑛 +  𝑎𝑛 𝑇𝑥𝑛,  

𝑆𝑥𝑛  = 𝒥𝓇𝑛(( 1 −  𝑐𝑛)𝑆𝑣𝑛 +  𝑐 𝑇𝑣𝑛), 

𝑆𝑣𝑛  = ( 1 −  𝑏𝑛)𝑆𝑢𝑛 +  𝑏𝑛𝒥𝓇𝑛 𝑇𝑢𝑛, where 𝑢0 ∈ 𝒞, 〈an〉, 〈bn〉 and 〈cn〉 are sequences in [0,1] 
Definition 2.3 [14]: A self-mapping,  𝑇: 𝒟 → 𝒟 is called a resolvent 𝒵𝒜-Jungck mapping if  
‖𝑇𝑛 –  𝑇ℎ‖ ≤ 𝜓(𝒶1‖𝑆𝑛 – 𝑆ℎ‖ + 𝒶2‖𝑆𝑛 – 𝒥𝓇𝑛 ℎ‖)

+ min {‖𝒥𝓇𝑛𝑛 − 𝑇𝑛‖,
‖𝑆𝑛 − 𝒥𝓇𝑛 𝑛‖ + ‖𝒥𝓇𝑛ℎ − 𝑆ℎ‖

2
, ‖𝑇ℎ − 𝒥𝓇𝑛 ℎ‖} 

∀𝑛 , ℎ ∈ ∁, 𝒶1, 𝒶2, ∈  [0, 1]𝑎𝑛𝑑   𝒶1 +  𝒶2  ≤ 1 
Now, we give new results. 

Lemma 2.4: Let(𝑵 , ‖ . ‖ ) be a Banach - space, and 𝑇 be a resolvent 𝒵𝒜-jungck self-maping on 
𝒟 with 𝔟 ∈ [0,1).Suppose that 〈𝒮𝑈𝑛 〉 be a resolvent Jungck 𝒮𝒫 – iteration scheme in 𝒟. There 
exists p ∈ F(s) ∩ F (𝒯) ∩ A−1(0)  ≠ ∅, then   lim

n→∞
‖𝑆𝑢𝑛 – 𝓉‖ exists, for all  n ∈ N. 



IHJPAS. 36(1)2023 
 

382 

Proof: Since 𝓉 be a fixed point of 𝒮 , 𝑇 𝑎𝑛𝑑 𝐴−1(0) .  Then the following inequalities hold 
‖𝑆𝑢𝑛+1 − 𝑝‖ = ‖( 1 −  𝑎𝑛)𝒥𝓇𝑛𝑇𝑆𝑥𝑛 +  𝑎𝑛  𝑇𝑥𝑛 − 𝑝‖                    
                          ≤ ( 1 −  𝑎𝑛)‖𝒥𝓇𝑛𝑇𝑆𝑥𝑛 − 𝑝‖ + 𝑎𝑛 ‖  𝑇𝑥𝑛 − 𝑝‖ 
                          ≤ ( 1 −  𝑎𝑛)‖𝑇𝑆𝑥𝑛 − 𝑝‖ +𝑎𝑛 ‖  𝑇 𝑥𝑛 − 𝑝‖                                                                         ≤

( 1 −  𝑎𝑛)𝔟𝜓(𝒶1‖𝑆𝑆𝑥n – 𝑆𝑃‖ + 𝒶2‖𝑆𝑆𝑥n – 𝒥𝓇𝑛 𝑝‖) +   min {      ‖𝒥𝓇𝑛 𝑆𝑥n −

 𝑇𝑆𝑥𝑛‖,
‖𝑆𝑆𝑥n−𝒥𝓇𝑛𝑆𝑥n‖+‖𝒥𝓇𝑛𝑝−𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

                     +𝑎𝑛𝔟𝜓(𝒶1‖𝑆𝑥n – 𝒮𝑝‖ + 𝒶2‖𝑆𝑥n – 𝒥𝓇𝑛𝑝‖) 

+  min {‖𝒥𝓇𝑛𝑥n −  𝑇𝑥𝑛‖,
‖𝑆𝑥n − 𝒥𝓇𝑛𝑥n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖}       

                         = ( 1 −  𝑎𝑛)𝔟𝜓‖𝑆𝑥𝑛 − 𝑝‖ +  𝑎𝑛𝔟𝜓(‖𝑆𝑥n – 𝑝‖) 
                          ≤ ( 1 −  𝑎𝑛)𝔟‖𝑆𝑥𝑛 − 𝑝‖ +  𝑎𝑛𝔟‖𝑆𝑥n – 𝑃‖                        
                       ≤ (( 1 −  𝒶𝑛) +  𝑎𝑛𝔟)‖𝑆𝑥n – 𝑃‖ 

                       ≤ 𝔟‖𝑆𝑥n – 𝑝‖.                                                                                             (2.1) 
   
‖𝑆𝑥𝑛 − 𝑃‖ = ‖𝒥𝓇𝑛[( 1 −  𝑐𝑛)𝑆𝑣𝑛 +  𝑐𝑛 𝑇𝑣𝑛] − 𝑝‖ 
                      ≤ ‖( 1 −  𝑐𝑛)𝑆𝑣𝑛  +  𝑐𝑛 𝑇𝑣𝑛 − 𝑝‖  
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝓉‖ + 𝑐𝑛‖ 𝑇𝑣𝑛 − 𝑝‖        

                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝑐𝑛𝔟𝜓(𝒶1‖𝑆𝑣n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑣n – 𝒥𝓇𝑛 𝑝‖) 

+   min {    ‖𝒥𝓇𝑛𝑣n − 𝑇𝑣𝑛‖,
‖𝑆𝑣n − 𝒥𝓇𝑛 𝑣n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛 𝑝‖} 

             ≤ ( 1 −  𝒸𝑛 )‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛𝔟𝜓(𝒶1‖𝑆𝑣n – 𝑝‖ + 𝒶2‖𝑆𝑣n – 𝑝‖)     
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛 𝔟𝜓(‖𝑆𝑣n – 𝑝‖) 
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛 𝔟‖𝑆𝑣n – 𝑝‖  
                   = (( 1 −  𝑐𝑛) + 𝒸𝑛 𝔟)‖𝑆𝑣n – 𝑝‖.                                                                       (2.2) 

 
‖𝑆𝑣𝑛  − 𝑝‖ = ‖( 1 −  𝑏𝑛)𝑆𝑢𝑛 +  𝑏𝑛𝒥𝓇𝑛  𝑇𝑢𝑛 − 𝑝‖ 
                     ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 ‖𝑇𝑢𝑛 − 𝑝‖  
                     ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑃‖ + 𝑏𝑛 ‖𝑇𝑢𝑛 − 𝑃‖ 
           ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 𝔟𝜓(𝒶1‖𝑆𝑢 – 𝑆𝑝‖ + 𝒶2‖𝑆𝑢𝑛  – 𝒥𝓇𝑛𝑝‖) 

+  min {  ‖𝒥𝓇𝑛𝑢𝑛 − 𝑇𝑢𝑛 ‖  ,
‖𝑆𝑢𝑛 − 𝒥𝓇𝑛 𝑢𝑛 ‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛 𝑝‖} 

                   = ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑃‖ + 𝑏𝑛𝔟𝜓(𝒶1‖𝑆𝑢𝑛  – 𝑃‖ + 𝒶2‖𝑆𝑢𝑛  – 𝑃‖)       
                   ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 𝔟𝜓(‖𝑆𝑢𝑛  – 𝑝‖) 
                   ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝓉‖ + 𝑏𝑛𝔟‖𝑆𝑢𝑛 – 𝑝‖ 
                 = (( 1 −  𝑏𝑛) + 𝑏𝑛𝔟) ‖𝑆𝑢𝑛  – 𝑝‖                                                                          (2.3) 
We substitute (2.3) in (2.2) as follows 
‖𝑆𝑥𝑛 − 𝑝‖ ≤ (( 1 −  𝒸𝑛 ) + 𝒸𝑛 𝔟)(( 1 −  𝑏𝑛) + 𝑏𝑛𝔟)) ‖𝑆𝑢𝑛  – 𝑝‖                                        (2.4)  
We substitute (2.4) in (2.1) as follows 
‖𝒮𝑢𝑛+1 − 𝑝‖ ≤ 𝔟(1 −  𝒸𝑛(1 − 𝔟)(1 −  𝑏𝑛(1 − 𝔟)) ‖𝑆𝑢𝑛  – 𝑝‖ 
                          ≤ (1 −  𝒸𝑛 (1 − 𝔟) ‖𝑆𝑢𝑛 – 𝑝‖ 
‖𝑆𝑢𝑛+1 − 𝑝‖ ≤ ‖𝑆𝑢𝑛  – 𝑝‖ 
‖𝑆𝑢𝑛+1 − 𝑝‖ ≤ ‖𝑆𝑢𝑛  – 𝑝‖ ≤ ⋯ … … ≤ ‖𝑆𝑢0 – 𝑝‖. 
So, {‖𝑆𝑢𝑛  – 𝑝‖} is decreasing and bounded, for each p ∈ f(S) ∩ f(T) ∩ A

−1(0), this implies that 
lim
n→∞

‖𝑆𝑢𝑛  – 𝑝‖ exists.   

Theorem 2.2: Let(N , ‖ . ‖ ) be a Banach-space, and 𝑇 be a resolvent 𝒵𝒜-jungck self-mapping 
on. If 〈𝒮𝑢𝑛 〉 is resolvent Jungck 𝒮𝒫 – iteration scheme. Then  the resolvent Jungck 𝒮𝒫 
– iteration scheme converge-strongly to a unique common fixed- point of 𝑆 , 𝑇 𝑎𝑛𝑑 𝐴−1(0) 
Proof: Let   𝑃 ∈ 𝐹(𝑆) ∩ 𝐹(𝑇) ∩ 𝐴−1(0) . By Lemma. (2.1), we have 



IHJPAS. 36(1)2023 
 

383 

‖𝑆𝑢𝑛+1 − 𝑝‖ ≤ (1 −  𝒸𝑛(1 − 𝔟)  ‖𝑆𝑢𝑛  – 𝑝‖                                                                                                 
‖𝑆𝑢𝑛+1 – 𝑝‖ ≤ ∏ (1 −  𝒸𝒿 (1 − 𝔟)  ‖𝑆𝑢0 – 𝑝‖

𝑛
𝒿=0                                                                  (2.5)  

‖𝑆𝑢𝑛 – 𝑝‖ ≤ e
−( 1−𝔟) ∑  𝒸𝒿

∞
𝒿=0 ‖𝑆𝑢0 – 𝑝‖ 

 Since 𝒸𝒿 ∈ [0 , 1] and 𝔟 ∈ [0 , 1] and ∑ 𝒸𝒿
∞
𝒿=0 = ∞ 

Then e−( 1−𝔟)
∑ 𝒸𝒿

∞
𝒿=0 → 0 as n → ∞ 

from (2.5) , lim
n →∞

 ‖𝑆𝑢𝑛 – 𝑝‖ = 0  . Therefore,  〈𝑆𝑒𝑛 〉 converges strongly to 𝓉 

Now, we prove that the un𝔦que common f𝔦xed point of 𝑆 , 𝑇 𝑎𝑛𝑑 𝐴−1(0). Let there exist another 
point p∗ ∈ ∁ such that 𝑇p∗ = 𝒥𝓇𝑛 p

∗ = 𝑆p∗ =  p∗ 
0 ≤ ‖p − p∗‖ = ‖𝑇p − 𝑇 p∗‖ 
                       ≤ 𝑏𝜓(𝒶1‖𝑆p – 𝑆p

∗‖ + 𝒶2‖𝑆p – 𝒥𝓇𝑛 p
∗‖) + 

                          min {‖𝒥𝓇𝑛p − 𝒯p‖,
‖𝑆p − 𝒥𝓇𝑛  p

∗‖ + ‖𝒥𝓇𝑛 p
∗ − 𝑆p∗‖

2
, ‖𝑇p∗ − 𝒥𝓇𝑛  p

∗‖} 

                        ≤ 𝑏(𝒶1‖𝑆p – 𝑆ρ
∗‖ + 𝒶2‖𝑆p – 𝒥𝓇𝑛  p

∗‖) 
            ≤ 𝑏‖𝑆p – 𝑆p∗‖ 
                            p = p∗ 
Theorem 2.3: Let 𝒩 be a uniformly convex Banach-space, 𝒟 be a nonempty convex-closed subset 
of  N and T be a resolvent 𝒵𝒜-jungck self-mapping on 𝒟. If 〈𝒮𝑢𝑛 〉 the resolvent Jungck 𝒮𝒫 
– iteration scheme in 𝒟  then 𝐹(𝒮) ∩ 𝐹(𝑇) ∩ 𝐴−1(0) ≠ ϕ if and only if  lim

n→∞
‖T𝑢n −

𝒥𝓇nS𝑢n+1‖ = 0 .. 
Proof: Since 𝐹(𝑆) ∩ 𝐹(𝒢) ∩ 𝐴−1(0) ≠ ∅ ⟹ ∃𝑝 ∈ 𝐹(𝑆) ∩ 𝐹(𝑇) ∩ 𝐴−1(0) ⟹ 𝑝 = 𝐹(𝑆) =
𝑓(𝑡) = 𝐴−1(0). 
By Lemma (2.1) 𝑊𝑒 ℎ𝑎𝑣𝑒 , 𝑙𝑖𝑚

𝑛→∞
‖𝑆𝑢n − 𝑝‖ = 𝛿 

We get , lim
n→∞

sup‖𝒥𝓇𝑛𝑆𝑢n+1 − 𝑝‖ ≤ 𝛿                                                                              (2.6) 

Now, To prove that  lim
n→∞

sup‖𝑇𝑢n − 𝑝‖ ≤ 𝛿 

lim
n→∞

sup‖𝑇𝑢n − 𝑝‖ ≤ lim
n→∞

sup 𝔟𝜓(𝒶1‖𝑆𝑢n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑢n – 𝒥𝓇𝑛𝑝‖) 

+  min {‖𝒥𝓇𝑛𝑢n − 𝑇𝑢n‖,
‖𝑆𝑢 − 𝒥𝓇𝑛𝑢n‖ + ‖𝒥𝓇𝑛 𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

= lim
n→∞

sup𝔟𝜓(𝒶1‖𝑆𝑢n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑢n – 𝒥𝓇𝑛 𝑝‖) 

≤ lim
n→∞

sup‖𝑆𝑢n – 𝑆𝑝‖. 

= 𝛿 
𝑊𝑒 𝑔𝑒𝑡 , lim

n→∞
sup‖𝑇𝑢n − 𝑝‖ ≤ 𝛿                                                                                        (2.7)  

Now by Lemma (2.1) we have          

𝛿 = lim
n→∞

sup ‖𝑆𝑢n+1 − 𝑝‖ = lim
n→∞

sup‖( 1 −  𝑎𝑛)𝒥𝓇𝑛𝑇𝑆𝑥𝑛 +  𝑎𝑛 𝑇𝑥𝑛 − 𝑝‖                    

     ≤ lim
n→∞

sup [( 1 −  𝑎𝑛)‖𝒥𝓇𝑛𝑇𝑆𝑥𝑛 − 𝑝‖  +𝑎𝑛 ‖  𝑇𝑥𝑛 − 𝑝‖] 

      ≤ lim
n→∞

sup [( 1 − 𝑎𝑛)‖𝑇𝑆𝑥𝑛 − 𝑝‖  +𝑎𝑛 ‖   𝑇𝑥𝑛 − 𝑝‖] 

      ≤ lim
n→∞

sup [( 1 −  𝑎𝑛)𝔟𝜓(𝒶1‖𝑆𝑆𝑥n – 𝒮𝑝‖ + 𝒶2‖𝑆𝑆𝑥n – 𝒥𝓇𝑛 𝑝‖) + 

min {‖𝒥𝓇𝑛𝑆𝑥n −  𝑇𝑆𝑥𝑛‖,
‖𝑆𝑆𝑥n − 𝒥𝓇𝑛𝑆𝑥n‖ + ‖𝒥𝓇𝑛 𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

+ 𝑎𝑛𝔟𝜓(𝒶1‖𝑆𝑥n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑥n – 𝒥𝓇𝑛𝑝‖) 

+ min {‖𝒥𝓇𝑛𝑥n −  𝑇𝑥𝑛‖,
‖𝑆𝑥n − 𝒥𝓇𝑛𝑥n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖}       

       ≤ lim
n→∞

sup [( 1 −  𝑎𝑛)𝔟𝜓‖𝑆𝑥𝑛 − 𝑝‖ +  𝑎𝑛𝔟𝜓(‖𝑆𝑥n – 𝑝‖)] 

       ≤ lim
n→∞

sup [( 1 −  𝑎𝑛)𝔟‖𝑆𝑥𝑛 − 𝑝‖ +  𝑎𝑛𝔟‖𝑆𝑥n – 𝑝‖]                        

      = lim
n→∞

sup [(( 1 −  𝑎𝑛) +  𝑎𝑛𝔟)‖𝑆𝑥n – 𝑝‖] 



IHJPAS. 36(1)2023 
 

384 

       ≤ lim
n→∞

sup𝔟‖𝑆𝑥n – 𝑝‖ .                                                                                                   (2.8) 

  
‖𝑆𝑥𝑛 − 𝑃‖ = ‖𝒥𝓇𝑛[( 1 −  𝑐𝑛)𝑆𝑣𝑛 +  𝑐𝑛 𝑇𝑣𝑛] − 𝑝‖ 
                      ≤ ‖( 1 −  𝑐𝑛)𝑆𝑣𝑛  +  𝑐𝑛 𝑇𝑣𝑛 − 𝑝‖  
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝓉‖ + 𝑐𝑛‖ 𝑇𝑣𝑛 − 𝑝‖        

                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝑐𝑛𝔟𝜓(𝒶1‖𝑆𝑣n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑣n – 𝒥𝓇𝑛 𝑝‖) 

+   min {    ‖𝒥𝓇𝑛 𝑣n − 𝑇𝑣𝑛‖,
‖𝑆𝑣n − 𝒥𝓇𝑛 𝑣n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

             ≤ ( 1 −  𝒸𝑛 )‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛𝔟𝜓(𝒶1‖𝑆𝑣n – 𝑝‖ + 𝒶2‖𝑆𝑣n – 𝑝‖)     
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛 𝔟𝜓(‖𝑆𝑣n – 𝑝‖) 
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛 𝔟‖𝑆𝑣n – 𝑝‖  
                   = (( 1 −  𝑐𝑛) + 𝒸𝑛𝔟)‖𝑆𝑣n – 𝑝‖                                                                                   (2.9) 
‖𝑆𝑣𝑛  − 𝑝‖ = ‖( 1 −  𝑏𝑛)𝑆𝑢𝑛 +  𝑏𝑛𝒥𝓇𝑛  𝑇𝑢𝑛 − 𝑝‖ 
                     ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 ‖𝑇𝑢𝑛 − 𝑝‖  
                     ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑃‖ + 𝑏𝑛 ‖𝑇𝑢𝑛 − 𝑃‖ 
           ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 𝔟𝜓(𝒶1‖𝑆𝑢 – 𝑆𝑝‖ + 𝒶2‖𝑆𝑢𝑛  – 𝒥𝓇𝑛𝑝‖) 

+  min {  ‖𝒥𝓇𝑛𝑢𝑛 − 𝑇𝑢𝑛 ‖  ,
‖𝑆𝑢𝑛 − 𝒥𝓇𝑛 𝑢𝑛 ‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛 𝑝‖} 

                   = ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑃‖ + 𝑏𝑛𝔟𝜓(𝒶1‖𝑆𝑢𝑛  – 𝑃‖ + 𝒶2‖𝑆𝑢𝑛  – 𝑃‖)       
                   ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 𝔟𝜓(‖𝑆𝑢𝑛  – 𝑝‖) 
                   ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝓉‖ + 𝑏𝑛𝔟‖𝑆𝑢𝑛 – 𝑝‖ 
                 = (( 1 −  𝑏𝑛) + 𝑏𝑛𝔟) ‖𝑆𝑢𝑛  – 𝑝‖                                                                        (2.10) 
We substitute (2.10) in (2.9) as follows: 
‖𝑆𝑥𝑛 − 𝑝‖ ≤ (( 1 −  𝒸𝑛 ) + 𝒸𝑛 𝔟)(( 1 −  𝑏𝑛) + 𝑏𝑛𝔟)) ‖𝑆𝑢𝑛  – 𝑝‖                                       (2.11)  
We substitute (2.11) in (2.8) as follows: 
‖𝑆𝑢𝑛+1 − 𝑝‖ ≤ 𝑙𝑖𝑚

𝑛→∞
𝑠𝑢𝑝𝔟(1 −  𝒸𝑛(1 − 𝔟)(1 −  𝑏𝑛(1 − 𝔟)) ‖𝑆𝑢𝑛  – 𝑝‖ 

                         ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝(1 −  𝒸𝑛(1 − 𝔟) ‖𝑆𝑢𝑛  – 𝑝‖. 

lim
n→∞

sup‖𝑆𝑒𝑛+1 − 𝓉‖ ≤ lim
n→∞

sup‖𝑆𝑒𝑛  – 𝓉‖= 𝛿                                                                  (2.12) 

By (2.6), (2.7), (2.12) and use Lemma (1.3) we get  
 lim
n→∞

‖𝑇𝑒𝑛 − 𝒥𝓇𝑛𝑆𝑒n+1‖ = 0 

Now, to prove that  𝑓(T)  ≠ ∅ 
Let p ∈ A(𝒟 , 〈𝒥𝓇𝑛 𝑆𝑢n+1〉 )  ⇒  r(𝒟 , 〈𝑆𝑢n〉 ) = r(𝒯p , 〈𝒥𝓇𝑛𝑆𝑢n+1〉 )         

Now,    r(Tρ, 〈𝒥𝓇𝑛𝑆un+1〉  ) =   𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝 ║𝒥𝓇𝑛𝑆𝑢n+1–  Tρ║ ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝[║𝒥𝓇𝑛 𝑆𝑢n+1–  T𝒪n ║ +

║T𝑢n–  Tρ║] 
      ≤ 𝑙𝑖𝑚

𝑛→∞
𝑠𝑢𝑝[ 𝑏𝜓(𝒶1‖S𝑢n − ρ‖ + 𝒶2‖S𝑢n − ρ‖)

+      min {‖𝒥𝓇𝑛𝑢n − T𝑢n‖,
‖𝑆𝑢n − 𝒥𝓇𝑛 𝓎n‖ + ‖𝒥𝓇𝑛ρ − 𝑆p‖

2
, ‖T𝒫 − 𝒥𝓇𝑛 ρ‖} 

                                                  = lim
n→∞

sup 𝑏𝜓(𝒶1‖S𝑢n − ρ‖ + 𝒶2‖S𝓎n − ρ‖)] 

                                                  ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝‖S𝑢n − ρ‖ = r(ρ, 〈𝑆𝑢n〉 ) = r(𝒟 , 〈𝑆𝑢n〉 ) 

r(𝒟 , 〈𝑆𝑢n〉 ) = r(Tρ, 〈𝒥𝓇𝑛𝑆𝑢n+1〉 ) 
Tp ∈ A(𝒟 , 〈𝑆𝑢n〉 )  𝒟 a uniformly convex ⇒ A(𝒟 , 〈𝑆𝑢n〉 ) is singleton 
⇒ p =  Tp ⇒ 𝒫 ∈  𝑓(T)    ⇒  𝑓(T)  ≠ ∅ 
In the same way, we get, ρ ∈  f(T) & ρ ∈ A−1(0) 
⟹ f(T) ∩ f(S) ∩ A−1(0) ≠ ∅   



IHJPAS. 36(1)2023 
 

385 

Lemma 2.7: Let(𝑵 , ‖ . ‖ ) be a Banach-space, and 𝑇 be a resolvent 𝒵𝒜-jungck self-mapping on 
𝒟 with 𝔟 ∈ [0,1). Suppose that 〈 𝒮𝑒𝑛〉 be a resolvent Jungck 𝒞ℛ – iteration scheme in 𝒟. There 
exists p ∈ F(𝒮) ∩ F (𝒢) ∩ A−1(0)  ≠ ∅, then   lim

n→∞
‖𝒮𝑒𝑛 – 𝓉‖ exists, for all  n ∈ N. 

Proof: As the same proof of lemma 2.1 

Theorem 2.8: Let(𝑵 , ‖ . ‖ ) be a Banch-space, and 𝑇 be a resolvent 𝒵𝒜-jungck self-mapping on 
𝒟 .If 〈𝒮𝑒𝑛〉 is resolvent Jungck 𝒞ℛ – iteration scheme  . Then the resolvent Jungck  𝒞ℛ – iteration 
scheme converges-strongly to a  unique common fixed− point of  𝒮 , 𝑇 𝑎𝑛𝑑 𝐴−1 
Proof: As the same proof of Theorem 2.2 

Theorem 2.9: Let 𝑵 be a uniformly convex Banach-space, 𝒟 be a nonempty convex-closed subset 
of  𝑵 and T be a resolvent 𝒵𝒜-jungck self-maping on 𝒟. If 〈 𝒮𝑒𝑛 〉 the resolvent Jungck 𝒞ℛ 
– Iteration scheme in 𝒟  then 𝐹(𝒮) ∩ 𝐹(𝑇) ∩ 𝐴−1(0) ≠ ϕ if and only if  lim

n→∞
‖𝒢𝑒n −

𝒥𝓇n𝒮𝑒n+1‖ = 0 
Proof:      the same proof Theorem 2.3 

Theorem 2.10: Let(𝑵 , ‖ . ‖ ) be a normed−space, and 𝑇 be a resolvent 𝒵𝒜-jungck self-mapping 
on 𝒟 if 𝑃 ∈ 𝑓(𝑠) ∩ 𝑓(𝑇) ∩ 𝐴−1(0) ≠ 𝜙 where 0 < 𝔯 ≤ an, bn, cn < 1  then the resolvent Jungck 
𝒞ℛ – iteration scheme is faster the resolvent Jungck 𝒮𝒫 – iteration scheme  
Proof: Let 𝑝 ∈ 𝑓(𝒮) ∩ 𝑓(𝑇) ∩ 𝐴−1(0) 
The resolvent Jungck 𝒞ℛ – iteration scheme, we have 
‖𝑠𝑒𝑛+1   − 𝑝‖ = ‖𝒥𝓇𝑛(( 1 −  𝑎𝑛)𝑇𝒮𝑤𝑛 +  𝑎𝑛 𝒥𝓇𝑛𝑇𝑇𝑤𝑛) − 𝑝‖ 

                         ≤ ‖(1 −  𝑎𝑛  )𝑇𝒮𝑤𝑛  +  𝑎𝑛 𝑇𝑇𝑤𝑛 − 𝑝‖  
                            ≤ (1 −  𝑎𝑛  )‖𝑇𝒮𝑤n − 𝑝‖ + an‖𝑇𝑇𝑤𝑛 − 𝑝‖        
                        ≤ (1 −  𝑎𝑛  )𝔟𝜓(𝒶1‖𝒮𝒮𝑤n – 𝒮𝑝‖ + 𝑎2‖𝒮𝒮𝑤n – 𝒥𝓇𝑛𝑝‖) 

+ min {‖𝒥𝓇𝑛𝒮𝑤n − 𝑇𝒮𝑤𝑛‖,
‖𝒮𝒮𝑤n − 𝒥𝓇𝑛 𝒮𝑤n‖ + ‖𝒥𝓇𝑛𝑝 − 𝒮𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖}

+ an𝔟𝜓(𝒶1‖𝒮𝑇𝑤n – 𝒮𝑝‖ + 𝒶2‖𝒮𝑇𝑤n – 𝒥𝓇𝑛 𝑝‖) + 

 min {‖𝒥𝓇𝑛𝑇𝑤n − 𝑇𝑇𝑤𝑛‖,
‖𝒮𝑤n − 𝒥𝓇𝑛𝑚n‖ + ‖𝒥𝓇𝑛𝑝 − 𝒮𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

                     ≤ (1 − 𝑎𝑛)𝔟𝜓‖𝒮𝑤n − 𝑝‖ + an𝔟‖𝑇𝒮𝑤n − 𝑝‖                  
 ≤ (1 − 𝑎𝑛)𝔟‖𝒮𝑤n − 𝑝‖ + an𝔟𝔟𝜓(𝒶1‖𝒮𝒮𝑤n – 𝒮𝑝‖ + 𝒶2‖𝒮𝒮𝑤n – 𝒥𝓇𝑛𝑝‖) 

+    min {‖𝒥𝓇𝑛 𝒮𝑤n − 𝒢𝒮𝑤𝑛 ‖,
‖𝒮𝒮𝑤n −𝒥𝓇𝑛𝒮𝑤n‖+‖𝒥𝓇𝑛𝑝−𝒮𝑝‖

2
, ‖𝒢𝑝 − 𝒥𝓇𝑛𝑝‖}  

≤ (1 −  𝑎𝑛  )𝔟‖𝒮𝑤n − 𝑝‖ + an𝔟
2‖𝒮𝑤n − 𝑝‖                                                                 (2.13) 

≤ 𝔟(1 −  𝑎𝑛 (1 −  𝔟 )‖𝒮𝑤n − 𝑝‖  
Now, to find ‖𝒮𝑤𝑛  − 𝑝‖. 
‖𝒮𝑤𝑛  − 𝑃‖ = ‖ ( 1 −  𝒸𝑛)𝒥𝓇𝑛𝑇𝑆𝑧𝑛 +  𝒸𝑛 𝑇𝑇𝑧𝑛 − 𝑝‖ 
                        ≤ (1 −  𝒸𝑛   )‖𝑇𝑆𝑧𝑛 − 𝑝‖  + 𝒸𝑛 ‖  𝑇𝑇𝑧𝑛 − 𝑝‖ 
                        ≤ (1 − 𝒸𝑛   )‖𝑇𝑆𝑧𝑛 − 𝑝‖  + 𝒸𝑛 ‖  𝑇 𝑇𝑧𝑛 − 𝑝‖ 
                        ≤ (1 − 𝒸𝑛   )𝔟𝜓(𝒶1‖𝑆𝑆𝑧n – 𝒮𝑝‖ + 𝒶2‖𝑆𝑆𝑧n – 𝒥𝓇𝑛 𝑝‖) + 

 min {‖𝒥𝓇𝑛𝑆𝑧n −  𝑇𝑆𝑧𝑛‖,
‖𝑆𝑆𝑧n − 𝒥𝓇𝑛𝑆𝑧n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛 𝑝‖} 

+ 𝒸𝑛𝔟𝜓(𝒶1‖𝑆𝑇𝑧n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑇𝑧n – 𝒥𝓇𝑛𝑝‖) + 

 min {‖𝒥𝓇𝑛 𝑇𝑧n −  𝑇𝑇𝑧𝑛‖,
‖𝑆𝑇𝑧n − 𝒥𝓇𝑛 𝑇𝑧n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

≤ (1 −   𝒸𝑛)𝔟‖𝑆𝑧n – 𝑝‖ +  𝒸𝑛 𝔟(‖𝑇𝑆𝑧n – 𝑝‖)                      
≤ (1 −   𝒸𝑛)𝔟‖𝑆𝑧n – 𝑝‖ +  𝒸𝑛 𝔟𝔟𝜓(𝒶1‖𝑆𝑧n – 𝒮𝑝‖ + 𝒶2‖𝑆𝑧n – 𝒥𝓇𝑛𝑝‖) 

+   min {‖𝒥𝓇𝑛𝑧n −  𝒢𝑧𝑛‖,
‖𝑆𝑧n − 𝒥𝓇𝑛𝑧n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

≤ (1 −   𝒸𝑛  )𝔟‖𝑆𝑧n – 𝑝‖ +  𝒸𝑛 𝔟
2‖𝑆𝑧n – 𝑝‖ 

= 𝔟(1 −   𝒸𝑛(1 − 𝔟))‖𝑆𝑧n – 𝑝‖  .                                                                                            (2.14) 



IHJPAS. 36(1)2023 
 

386 

 
‖𝑆𝑧𝑛  − 𝑝‖ = ‖( 1 −  𝑏𝑛)𝒥𝓇𝑛𝑇𝑆𝑒𝑛 +  𝑏𝑛𝒥𝓇𝑛 𝑇𝑒𝑛𝑝‖ 
                      ≤ ( 1 −  𝑏𝑛)‖𝒥𝓇𝑛𝑇𝑆𝑒𝑛 − 𝑝‖ + 𝑏𝑛‖𝒥𝓇𝑛 𝑇𝑒𝑛 − 𝑝‖  
                      ≤ ( 1 −  𝑏𝑛)‖𝑇𝑆𝑒𝑛 − 𝑝‖ + 𝑏𝑛 ‖𝑇𝑒𝑛 − 𝑝‖ 

    ≤ ( 1 −  𝑏)𝔟𝜓(𝒶1‖𝑆𝑆𝑒𝑛 – 𝑆𝑝‖ + 𝒶2‖𝑆𝑆𝑒𝑛 – 𝒥𝓇𝑛 𝑝‖) +    min {‖𝒥𝓇𝑛𝑆𝑒𝑛 −

𝑇𝑆𝑒𝑛‖,
‖𝑆𝑆𝑒𝑛−𝒥𝓇𝑛𝑆𝑒𝑛‖+‖𝒥𝓇𝑛𝑝−𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

+𝑏𝑛𝔟𝜓(𝒶1‖𝑆𝑒𝑛 – 𝑆𝑝‖ + 𝒶2‖𝑆𝑒𝑛  – 𝒥𝓇𝑛 𝑝‖) 

+    min {‖𝒥𝓇𝑛𝑒𝑛 − 𝑇𝑒𝑛‖,
‖𝑆𝑒𝑛 − 𝒥𝓇𝑛𝑒𝑛‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖}   

                     ≤ ( 1 −  𝑏𝑛)𝔟‖𝑆𝑒𝑛 − 𝑝‖ + 𝑏𝑛 𝔟(‖𝑆𝑒𝑛 – 𝑝‖) 
                     ≤ 𝔟 ‖𝑆𝑒𝑛 – 𝑝‖ 
                  ≤ 1 ‖𝑆𝑒𝑛 – 𝑝‖                                                                                                (2.15) 
We substitute (2.15) in (2.14) as follows 

‖𝑆𝑤𝑛  − 𝑝‖ ≤ 𝔟
2(1 −   𝒸𝑛(1 − 𝔟))‖𝑆𝑒𝑛 – 𝓉‖                                                                         (2.16) 

We substitute (2.16) in (2.13) as follows 

‖𝑆𝑒𝑛+1   − 𝑝‖ ≤ 𝔟
2(1 −  𝑎𝑛(1 −  𝔟 )(1 −   𝒸𝑛(1 − 𝔟))‖𝑆𝑒𝑛 – 𝓉‖ 

                             ≤ 𝔟2(1 −   𝔯(1 − 𝔟))
2

‖𝑆𝑒𝑛 – 𝓉‖ 
                                                  . 

‖𝑆𝑒𝑛+1   − 𝓉‖ ≤ 𝔟
2𝑛(1 −  𝔯 (1 − 𝔟))

2𝑛
‖𝑆𝑒0 – 𝓉‖ 

 The resolvent Jungck 𝒮𝒫 – Iteration scheme, we have 
‖𝑆𝑢𝑛+1 − 𝑝‖ = ‖( 1 −  𝑎𝑛)𝒥𝓇𝑛𝑇𝑆𝑥𝑛 +  𝑎𝑛  𝑇𝑥𝑛 − 𝑝‖                    
                          ≤ ( 1 −  𝑎𝑛)‖𝒥𝓇𝑛𝑇𝑆𝑥𝑛 − 𝑝‖ + 𝑎𝑛 ‖  𝑇𝑥𝑛 − 𝑝‖ 
                          ≤ ( 1 −  𝑎𝑛)‖𝑇𝑆𝑥𝑛 − 𝑝‖ +𝑎𝑛 ‖  𝑇 𝑥𝑛 − 𝑝‖                                                                            ≤

( 1 −  𝑎𝑛)𝔟𝜓(𝒶1‖𝑆𝑆𝑥n – 𝑆𝑃‖ + 𝒶2‖𝑆𝑆𝑥n – 𝒥𝓇𝑛 𝑝‖) +   min {      ‖𝒥𝓇𝑛 𝑆𝑥n −

 𝑇𝑆𝑥𝑛‖,
‖𝑆𝑆𝑥n−𝒥𝓇𝑛𝑆𝑥n‖+‖𝒥𝓇𝑛𝑝−𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

                     +𝑎𝑛𝔟𝜓(𝒶1‖𝑆𝑥n – 𝒮𝑝‖ + 𝒶2‖𝑆𝑥n – 𝒥𝓇𝑛 𝑝‖) 

+  min {‖𝒥𝓇𝑛𝑥n −  𝑇𝑥𝑛‖,
‖𝑆𝑥n − 𝒥𝓇𝑛𝑥n‖ + ‖𝒥𝓇𝑛 𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖}       

                         = ( 1 −  𝑎𝑛)𝔟𝜓‖𝑆𝑥𝑛 − 𝑝‖ +  𝑎𝑛𝔟𝜓(‖𝑆𝑥n – 𝑝‖) 
                          ≤ ( 1 −  𝑎𝑛)𝔟‖𝑆𝑥𝑛 − 𝑝‖ +  𝑎𝑛𝔟‖𝑆𝑥n – 𝑃‖                        
                       ≤ (( 1 −  𝒶𝑛 ) +  𝑎𝑛𝔟)‖𝑆𝑥n – 𝑃‖ 

                       ≤ 𝔟‖𝑆𝑥n – 𝑝‖                                                                                                           (2.17) 
   
‖𝑆𝑥𝑛 − 𝑃‖ = ‖𝒥𝓇𝑛[( 1 −  𝑐𝑛)𝑆𝑣𝑛 +  𝑐𝑛 𝑇𝑣𝑛] − 𝑝‖ 
                      ≤ ‖( 1 −  𝑐𝑛)𝑆𝑣𝑛  +  𝑐𝑛 𝑇𝑣𝑛 − 𝑝‖  
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝓉‖ + 𝑐𝑛‖ 𝑇𝑣𝑛 − 𝑝‖        

                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝑐𝑛𝔟𝜓(𝒶1‖𝑆𝑣n – 𝑆𝑝‖ + 𝒶2‖𝑆𝑣n – 𝒥𝓇𝑛 𝑝‖) 

+   min {    ‖𝒥𝓇𝑛 𝑣n − 𝑇𝑣𝑛‖,
‖𝑆𝑣n − 𝒥𝓇𝑛 𝑣n‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛𝑝‖} 

             ≤ ( 1 −  𝒸𝑛 )‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛𝔟𝜓(𝒶1‖𝑆𝑣n – 𝑝‖ + 𝒶2‖𝑆𝑣n – 𝑝‖)     
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛 𝔟𝜓(‖𝑆𝑣n – 𝑝‖) 
                   ≤ ( 1 −  𝑐𝑛)‖𝑆𝑣𝑛 − 𝑝‖ + 𝒸𝑛 𝔟‖𝑆𝑣n – 𝑝‖  
                   = (( 1 −  𝑐𝑛) + 𝒸𝑛𝔟)‖𝑆𝑣n – 𝑝‖                                                                                 (2.18) 
 
‖𝑆𝑣𝑛  − 𝑝‖ = ‖( 1 −  𝑏𝑛)𝑆𝑢𝑛 +  𝑏𝑛𝒥𝓇𝑛  𝑇𝑢𝑛 − 𝑝‖ 
                     ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 ‖𝑇𝑢𝑛 − 𝑝‖  
                     ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑃‖ + 𝑏𝑛 ‖𝑇𝑢𝑛 − 𝑃‖ 



IHJPAS. 36(1)2023 
 

387 

           ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 𝔟𝜓(𝒶1‖𝑆𝑢 – 𝑆𝑝‖ + 𝒶2‖𝑆𝑢𝑛  – 𝒥𝓇𝑛𝑝‖) 

+  min {  ‖𝒥𝓇𝑛𝑢𝑛 − 𝑇𝑢𝑛‖  ,
‖𝑆𝑢𝑛 − 𝒥𝓇𝑛𝑢𝑛 ‖ + ‖𝒥𝓇𝑛𝑝 − 𝑆𝑝‖

2
, ‖𝑇𝑝 − 𝒥𝓇𝑛 𝑝‖} 

                   = ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑃‖ + 𝑏𝑛𝔟𝜓(𝒶1‖𝑆𝑢𝑛  – 𝑃‖ + 𝒶2‖𝑆𝑢𝑛  – 𝑃‖)       
                   ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝑝‖ + 𝑏𝑛 𝔟𝜓(‖𝑆𝑢𝑛  – 𝑝‖) 
                   ≤ ( 1 −  𝑏𝑛)‖𝑆𝑢𝑛 − 𝓉‖ + 𝑏𝑛𝔟‖𝑆𝑢𝑛 – 𝑝‖ 
                 = (( 1 −  𝑏𝑛 ) + 𝑏𝑛𝔟) ‖𝑆𝑢𝑛  – 𝑝‖                                                                                 (2.19) 
We substitute (2.19) in (2.18) as follows 
‖𝑆𝑥𝑛 − 𝑝‖ ≤ (( 1 −  𝒸𝑛 ) + 𝒸𝑛 𝔟)(( 1 −  𝑏𝑛) + 𝑏𝑛𝔟)) ‖𝑆𝑢𝑛  – 𝑝‖                                        (2.20)  
We substitute (2.20) in (2.17) as follows 
‖𝒮𝑢𝑛+1 − 𝑝‖ ≤ 𝔟(1 −  𝒸𝑛(1 − 𝔟)(1 −  𝑏𝑛(1 − 𝔟)) ‖𝑆𝑢𝑛  – 𝑝‖ 
                          ≤ (1 −  𝒸𝑛 (1 − 𝔟) ‖𝑆𝑢𝑛 – 𝑝‖ 
‖𝑆𝑢𝑛+1 − 𝓉‖  ≤ 𝔟

𝑛(1 − 𝔯(1 − 𝔟))2𝑛  ‖𝑆𝑒0 – 𝓉‖                         
 

‖𝑆𝑒𝑛+1   − 𝓉‖

‖𝑆𝑢𝑛+1 − 𝓉‖
≤

𝔟2𝑛(1 −  𝔯 (1 − 𝔟))
2𝑛

‖𝑆𝑒0 – 𝓉‖

𝔟𝑛(1 − 𝔯(1 − 𝔟))
2𝑛

  ‖𝑆𝑢0 – 𝓉‖
→ 0   𝑛 → ∞  

 

3. Conclusion  

     In this section, we will provide some conclusions. New iterations were performed and the 

convergence and acceleration of these iterations to the common fixed point were demonstrated. 

We also install the resolvent Jungck CR – iteration is faster than resolvent Jungck 𝒮P – iteration. 
 

References 

1. Ali ,J. ; Ali ,F., Approximation of Common Fixed Points and the Solution of Image Recovery 

Problem  J. Phys.: Conf. Ser.  2019 ,74, 4, 0–22, 2019.  

2. Bahuguna,D. ; Sharma, A., On the Convergence of a New Iterative Algorithm of Three Infinite 

Families of Generalized Nonexpansive Multi-Valued Mappings Proc. Indian Acad. Sci. Math. 

2018 Sci. 128, 4, 1–15. 

3. Garodia, Chanchal, ; Izhar Uddin., A New Fixed Point Algorithm for Finding the Solution of a 

Delay Differential Equation.  AIMS Mathematics ,2020,3182-3200, 5.4  

4.Saad Hussein ; Zena H. Maibed  ,Iterative Methods for Approximation of Fixed Points Via Like 

ContractionMapping, J. Phys.: Conf. Ser. 2021,1818 012152. 

5. Daengsaen,.j ; Khemphet Hindawi Abstr.On the Rate of Convergence of P-Iteration, SP-

Iteration, and D-Iteration Methods for Continuous Nondecreasing Functions on Closed Intervals, 

Appl. Anal. 2018, 6.  

6. Ullah ,K. ; Muhammad, New Three-Step Iteration Process and Fixed Point Approximation in 

Banach Spaces Linear Topol. Algebra. 2018, 07, 02, 87–100,.  

7. Zena. H. Maibed and Saad. S. Hussein, Some Theorems of Fixed Point Approximations by 

Iteration Processes, J. Phys. 2021, 1818 01215 

8. Zena,H .Maibed ; Ali Q Thajil, Zenali Iteration Method for Approximating Fixed Point of A 

δZA - Quasi Contractive Mappings” Haitham Journal for Pure and Applied Science Oct 2021. 

9.Zena H Maibed, Contractive Mappings Having Mixed Finite Monotone Property in Generalized 

Metric Spaces Haitham Journal for Pure and Applied Science.l. 2019, 32 (1) . 

10. Singh S. L.; Bhatnagar Charu ; Mishra S. N. Stability of Jungck-type iterative procedures, Int. 

J. Math. Sci., 2015, 19, 3035-3043.  

11.Garodia ; Izhar, A New Fixed Point Algorithm for Finding the Solution of  a Delay Differential 

Equation, AIMS Math., 2020 , 3182–3200.  



IHJPAS. 36(1)2023 
 

388 

12.Karakaya, V., Atalan, ; Y., Doğan, K. Convergence Analysis for a New Faster Iteration Method 

İstanbul Ticaret Üniversitesi Fen Bilim . 2016,15.30, 35–53.  

13.Jamil, Z. Z.  ; M. B. Abed, On A Modified SP-Iterative Scheme for approximating Fixed Point 

of A Contraction Mapping, Iraqi Journal of Scienceno  .2015,l 56, 4B, 3230-3239,June. 

14.Zena H. M ; Alaa M. AOn the Convergence of New Iteration Schemes by Resolvent ZA-Jungck 

mapping,  Journal of Interdisciplinary Mathematics to appear, 2022.