354 

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

 IHJPAS. 37 (2) 2024 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

PISSN: 1609-4042, EISSN: 2521-3407 

 

Mustafa Dawood Talak Alobadi1*  and Zena Hussein Maibed2  

1,2 Department of Mathematics, College of Education for Pure Science (Ibn-Al-Haitham), University of 

Baghdad, Baghdad, Iraq 

*Corresponding Author.   

 

 

 

Abstract   

Depending on the needs and requirements of keeping up with the scientific procession, researchers 

tend to find new recurrence schemes or develop previous recurrence schemes that will help 

researchers reach the fixed point and solution of variational inequality .The objective of this article 

is to provide novel approaches to finding a common fixed point of different types of important 

mappings and  the set of zeros of maximal monotone operators .Also, we studied the convergence 

weakly and convergence strongly of the proposed iterative method under some suitable conditions. 

To achieve this goal, we will introduce a new technical method of resolvent operators and metric 

projection using different types of function sequences, including sequence of  maximal monotone 

operators, sequence of 𝒦-strictly pseudo-contractive mappings, and sequence of nonexpansive 

mappings defined on nonempty convex-closed subset of Hilbert space . 

Keywords : Nonexpansive  Mapping , Metric Projection , Strictly Pseudo Contraction  

Mapping , Strongly Pseudo-Contractive ,  Fixed Point .  

 

1. Introduction 

Let ℋ be the Hilbert space and let 𝒦 be a convex closed subset of ℋ. The metric projection of 

ℋ onto 𝒜 is denoted by 𝒫𝕔 , while the set of fixed points of 𝒯 is denoted as ℱ(𝒯). Keep in 

mind, if a constant 𝒦 ∈ [0,1) exists, then there is a mapping 𝒯:𝒦 → ℋ that is 𝒦 - strictly 

pseudo-contractive . 

       ‖𝒯𝜔 − 𝒯𝓌‖
2 ≤ ‖𝜔 −𝓌‖2 +𝒦‖(𝐼 − 𝒯)𝜔 − (𝐼 − 𝒯)𝓌‖

2  , ∀ 𝓌 ∈ 𝒦                                      (1)      

When 𝒦 = 0, 𝒯 is nonexpansive, and when 𝒦 =1, 𝒯 is pseudo-contractive. A strong pseudo-

contractive 𝒯 is one for which there exists a positive constant 𝜁  between 0 and 1 such that 𝒯 − 𝜁𝔩  

is pseudo-contractive. Without a doubt, the 𝒦 - strictly pseudo-contractive class is a subset of the 

larger class that contains both pseudo-contractions and nonexpansive mapping. Furthermore, we 

highlight the fact that the class 𝒦 - strictly pseudo-contractive mappings is unique from the other 

classes of strictly pseudo-contractive mappings(see [1-3]).  

In [4] suggested the now-common Mann's iterative approach. Since then, researchers have 

looked at the 𝒦 - strictly pseudo-contractive mapping strategy and the standard  iterative process 

for constructing fixed points for nonexpansive mapping .When used iteratively, Mann's method 

yields the sequence 𝜔𝑛, as seen below: 

Received: 28 February 2023                     Accepted: 30 March 2023           Published: 20 April 2024 

 

 

A New Iterative Algorithms for Finite Family of Resolvent Operators 

doi.org/10.30526/37.2.3301 

https://creativecommons.org/licenses/by/4.0/
https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042
https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407
https://orcid.org/
mailto:mostafa.dawood1103a@ihcoedu.uobaghdad.edu.iq
https://orcid.org/
mailto:mrs_ zena.hussein@yahoo.com


IHJPAS. 37 (2) 2024 

 

355 
 

∀ 𝜔1 ∈ 𝒜 , 𝜔𝓃+1 = (1 − 𝜎𝓃)𝜔𝓃 + 𝜎𝓃𝒯𝜔𝓃 , 𝓃 ≥ 1                  (2)                                      

Lots of studies have been rolled out on this subject, see[5-25] . In [3] established the first 

convergence result for 𝒦 - strictly pseudo-contractive mapping self mapping in concrete Hilbert 

space. In subsequent work,  [21] considered a varying control sequence denoted by the symbol 

{𝜎𝑛} to provide a limited extension of the applicable result in[5]. Using the conditions   𝜎1 = 1,0 <

𝜎𝑛 < 1 , ∑ 𝜎𝑛
∞
𝑛=1 = ∞, and the lim

𝑛→∞
𝑠𝑢𝑝𝜎𝑛 = 𝜎 < 1 −𝒦  , he proved a convergence theorem 

using an algorithm (2). This operation was performed assuming that R's domain is compact and 

convex. This means that the compact condition on the domain of mapping 𝒯 is required to derive 

the convergence results from Rhoades's convergence theorem.  

  Recently, a weak  convergence theorem was shown [26] using an algorithm . 

𝜔𝓃+1 = 𝜎𝓃𝜔𝓃 + (1 − 𝜎𝓃)∑ 𝜁𝔦𝒯𝔦𝜔𝓃
ℕ
𝔦=1                                                                                      (3)                                                                          

Despite the examples in [27] and [28] , is nevertheless shown that this convergence is frequently 

weak. In order to get high convergence, it is necessary to make adjustments to Mann's scheme (2). 

We shall restate the modification proposed by Nakajo and Takahashi for a nonexpansive mapping 

𝒯. 

     Consider the algorithm  

{
 
 

 
 

𝜔0 ∈ ℂ 

𝒲𝓃 = 𝜎𝓃𝜔𝓃 + (1 − 𝜎𝓃)𝒜𝜔𝓃
    𝔼𝓃 = {𝓈 ∈ ℂ: ‖𝒲𝓃 − 𝓈‖ ≤ ‖𝜔𝓃 − 𝓈‖}

𝒪𝓃 = {𝓈 ∈ ℂ: 〈𝜔𝓃 − 𝓈,𝜔0 − 𝜔𝓃〉 ≥ 0}
𝜔𝑛+1 = 𝒫    𝔼𝓃⋂𝒪𝓃𝓌0

 

                                                                 (4)                                                                                

Where 𝒫ℂ stands for the metric projection from ℋ onto ℂ.The convergence of the sequence{𝜔𝓃}   

generated by algorithm (3) to a fixed point of 𝒯 is shown by Nakajo and Takahashi, under the 

condition that the control sequence {𝜎𝓃}𝓃=0
∞    is selected so that𝑠𝑢𝑝𝓃≥0𝜎𝓃 < 1  (i.e., {𝜎𝓃} is limited 

away above from 0 and 1), where T is a fixed point of the Similar large convergence findings were 

published in the works [25-32].  

 

2 . Preliminaries  

If ℋ is a real Hilbert space and that ℂ is a nonempty closed convex subset of H.Then we give some 

necessary lemmas:  

Lemma 2.1 [ 28] The following identities hold  

 (i) ‖𝜔 ∓𝓌‖2 = ‖𝜔‖2 ∓ 2〈𝜔,𝓌〉 + ‖𝜔‖2  , ∀𝜔 ,𝓌 ∈ ℋ   

(ii) ‖𝓉𝜔 + (1 − 𝓉)𝓌‖2 = 𝓉‖𝜔‖2 + (1 − 𝓉)‖𝓌‖2 − 𝓉(1 − 𝓉)‖𝜔 −𝓌‖2, ∀ 𝓉 ∈ [0,1],

∀ 𝜔 ,𝓌 ∈ ℋ  

Lemma 2.2 [28] : If  𝓏 ∈ ℋ 𝑎𝑛𝑑 𝓌 ∈ ℂ . Then 

 𝓌 = 𝒫ℂ𝜔  iff   there satisfies 〈𝜔 −𝓌 ,𝓌 − 𝓈 〉 ≥ 0 , ∀𝓈 ∈ ℂ . 

Lemma 2.3 [33] : Let {𝜔𝑛} be a sequence of nonnegative real numbers that satisfies the condition  

 𝛼𝓃+1 ≤ (1 − 𝓉𝓃)𝛼𝓃 + 𝔟𝓃 + 0(𝔢𝓃),𝓃 ≥ 1   

where {𝔢𝓃} satisfies the restrictions : 



IHJPAS. 37 (2) 2024 

 

356 
 

(i) 𝔢𝓃 → 0(𝓃 → ∞) 

(ii) ∑ 𝔟𝓃 < ∞ 
∞
𝓃=1  

(iii) ∑ 𝔢𝓃 = ∞
∞
𝓃=1  

 

3.  Main Result 

      In this  section, we introduce a new technique f-point, and prove its strong-weak convergence 

Theorem 3.1 : Let ℳ𝑖 be M.M. operator and 〈𝒯𝑖〉 be a sequence of 𝒦 - strictly pseudo-contractive 

map on ℂ. Define the Technique as:  

{

    𝜔0 = 𝜔 ∈ ℂ                                                                                     

   𝓌𝓃 = ∑ 𝒥𝑟𝓃,𝔦(𝜔𝓃)
𝒦
𝔦=0                                                                                 

     𝜔𝓃+1 = (𝔞𝓃 − 𝔟𝓃)ℱ(𝜔) + (1 − 𝔞𝓃)∑ 𝒯𝑖(𝜔𝓃) + 𝔟𝓃ℱ(𝓌𝓃)
𝒦
𝔦=0

   

Where  ℱ is nonexpansive,〈𝓇𝓃〉  be a sequence in (0,∞), such that 〈𝔞𝓃〉, 〈𝔟𝓃〉 are sequence in (0,∞] 

and 𝔞𝓃 + 𝔟𝓃 = 1 . If the following condition are satisfies :  

(i) ‖𝓌𝓃‖
2 ≤ 𝜓𝓃 + ∑ ‖𝒥𝑟𝓃,𝔦(𝜔𝓃)‖

2𝒦
𝔦=1   and ∑ (𝔟𝓃𝜓𝑛 + ∑ ‖𝜔𝓃 − 𝒯

𝔦‖
2𝒦

𝔦=1 )∞
𝑖=0 < ∞ .  

(ii) ∑ 𝔞𝓃 = ∞ , ∑ ‖𝓌𝓃‖
2 ≤ ∞  , 𝑎𝑛𝑑 lim

𝓃→∞

𝔞𝓃−𝔟𝓃

𝔞𝓃
= 0∞

𝔦=1
∞
𝑖=0  . 

(iii) ℳ𝑖
−1(0)⋂(⋂ f𝔦𝓍(𝒯𝑖)) ≠ ∅ 𝑎𝑛𝑑   𝔞𝓃 ≥ 𝔟𝓃 

∞
𝔦=1 .Then the Technique f-point 〈𝜔𝓃〉   converge to 

point ℳ𝑖
−1(0)⋂(⋂ 𝔣𝔦𝓍(𝒯𝑖)∞

𝔦=1 ).  

Proof : Let 𝔯 ∈ ℳ𝑖
−1(0)⋂(⋂ f𝔦𝓍(𝒯𝑖))    ∞

𝑖=1  

‖𝜔1 − 𝔯‖
2 = ‖(𝔞0 − 𝔟0)ℱ(𝜔) + (1 − 𝔞0)∑ 𝒯𝑖(𝜔0) + 𝔟0ℱ(𝓌0)

𝒦
𝔦=0 − 𝔯‖

2
  

                      where 𝑟 = ((𝔞0 − 𝔟0) + (1 − 𝔞0) + 𝔟0)𝑟 

≤ (𝔞0 − 𝔟0)‖ℱ(𝜔) − 𝔯‖
2+(1 − 𝔞0)∑ ‖𝒯𝑖(𝜔0) − 𝔯‖

2𝒦
𝔦=0 + 𝔟0‖ℱ(𝓌0) − ((𝔞0 − 𝔟0) +         (1 −

𝔞0) + 𝔟0)𝑟‖
2
 

≤ (𝔞0 − 𝔟0)‖ℱ(𝜔) − 𝔯‖
2 +(1 − 𝔞0)∑ ‖𝒯𝑖(𝜔0) − 𝔯‖

2𝒦
𝔦=0 + 𝔟0‖ℱ(𝓌0) − 𝔯‖

2  

But 𝒯𝑖  𝑖𝑠 𝒦 - strictly pseudo-contractive 

≤ (𝔞0 − 𝔟0)‖ℱ(𝜔) − 𝔯‖
2 + (1 − 𝔞0)∑ ‖𝜔0 − 𝔯‖

2𝒦
𝔦=0 + (1 − 𝔞0)𝒦∑ ‖𝜔0 − 𝒯

𝑖(𝜔0) − 𝔯 −
𝒦
𝔦=0

(𝒯𝑖(𝔯)‖
2
+ 𝔟0𝜓𝑛 + 𝔟0∑ ‖𝒥𝑟𝓃,𝔦(𝜔0) − 𝔯‖

2𝒦
𝔦=0   

Where 𝒦 ∈ [0,1) 

≤ (1 − 𝔟0)∑ ‖𝜔 − 𝔯‖2𝒦
𝔦=0  +(1 − 𝔞0)∑ ‖𝜔0 − 𝒯

𝑖(𝜔0)‖
2𝒦

𝔦=1 + 𝔟0𝜓𝑛 + 𝔟0∑ ‖𝜔𝑜 − 𝔯‖
2𝒦

𝔦=0  

≤ ∑ ‖𝜔 − 𝔯‖2𝒦
𝔦=0  + 𝔟0𝜓𝑛 + ∑ ‖𝜔0 − 𝒯

𝔦(𝜔0)‖
2𝒦

𝔦=1  

If 𝓃 = 𝒦, we have  

‖𝜔𝒦 − 𝔯‖
2 < ∑ ‖𝜔 − 𝔯‖2𝒦

𝑖=0  + ∑ 𝔟𝑖𝜓𝑖
𝑘−1
𝑖=0 + ∑ ‖𝜔𝑖 − 𝒯

𝔦(𝜔𝔦)‖
2𝒦−1

𝑖=1  

If   𝓃 = 𝒦 + 1 , we have  

‖𝜔𝒦+1 − 𝔯‖
2 = ‖(𝔞𝒦 − 𝔟𝒦)ℱ(𝜔) + (1 − 𝔞𝒦)∑ 𝒯𝑖(𝜔𝒦) + 𝔟𝒦ℱ(𝓌𝒦)

𝒦
𝔦=0 − 𝔯‖

2
  

≤ (𝔞𝒦 − 𝔟𝒦)‖ℱ(𝔞) − 𝔯‖
2 + (1 − 𝔞𝒦)∑ ‖ℱ 𝔦(𝓌𝒦) − 𝔯‖

2𝒦
𝑖=0 + 𝔟𝒦‖ℱ(𝓌𝒦) − 𝔯‖

2  



IHJPAS. 37 (2) 2024 

 

357 
 

‖𝜔𝒦+1 − 𝔯‖
2  ≤ (𝔞𝒦 − 𝔟𝒦)‖𝜔 − 𝔯‖

2 +(1 − 𝔞𝒦)∑ [‖𝜔𝒦 − 𝔯‖
2 +𝒦‖𝜔𝒦 − 𝒯

𝑖(𝜔𝒦) − (𝔯 −
𝒦
𝑖=0

𝒯 𝔦(𝔯)‖
2
] + 𝔟𝒦𝜓𝒦 + 𝔟𝑘 ∑ ‖𝒥𝑟𝓃,𝔦(𝜔𝒦) − 𝔯‖

2𝒦
𝔦=0  

 < (𝔞𝒦 − 𝔟𝒦)‖𝜔 − 𝔯‖
2 + (1 − 𝔞𝒦)∑ ‖𝜔𝒦 − 𝔯‖

2 +𝒦
𝑖=0 (1 − 𝔞𝒦)∑ ‖𝜔𝒦 − 𝒯

𝑖(𝜔𝒦)‖
2𝒦

𝑖=0 +

𝔟𝒦𝜓𝒦 + 𝔟𝒦 ∑ ‖𝜔𝒦 − 𝔯‖
2𝑘

𝑖=0  

< (𝔞𝒦 − 𝔟𝒦)‖𝜔 − 𝔯‖
2 + 𝔟𝒦𝜓𝑘 + ∑ ‖𝜔𝑘 − 𝒯

𝑖(𝜔𝒦)‖
2𝒦

𝔦=0 + (1 − (𝔞𝒦 − 𝔟𝒦)∑ ‖𝜔𝒦 − 𝔯‖
2𝒦

𝑖=0   

< (𝔞𝒦 − 𝔟𝒦)‖𝜔 − 𝔯‖
2 + 𝔟𝒦𝜓𝑘 + ∑ ‖𝜔𝑘 − 𝒯

𝑖(𝜔𝒦)‖
2𝒦

𝔦=0 + (1 − (𝔞𝒦 − 𝔟𝒦)∑ ‖𝜔𝒦 − 𝔯‖
2𝒦

𝑖=0 +

(1 − (𝔞𝒦 − 𝔟𝒦) ∑ 𝔟𝔦𝜓𝑖
𝒦−1
𝔦=0 + ‖𝜔𝑖 − 𝒯

𝑖(𝜔𝑖)‖
2
  

‖𝜔𝒦+1 − 𝔯‖
2 ≤ ∑ ‖𝜔 − 𝔯‖2𝒦

𝑖=0  + ∑ 𝔟𝔦𝜓𝑖
𝒦−1
𝑖=0 + ∑ ‖𝜔𝑖 − 𝒯

𝑖(𝜔𝑖)‖
2𝒦−1

𝔦=1  

But ∑  (𝔟𝑖𝜓𝑖
𝒦
𝔦=0 + ‖𝜔𝑖 − 𝒯

𝑖(𝜔𝑖)‖
2
) < ∞. 

 〈𝜔𝓃〉is bounded sequence . 

Since 〈𝒥𝑟𝓃,𝔦(𝜔𝓃)〉 also bounded then there exist subsequence 〈𝒥𝑟𝓃𝒦,𝔦
(𝜔𝑛𝒦)〉 of 〈𝒥𝑟𝓃,𝔦(𝜔𝓃)〉 

converge weakly  to 𝓋.  

Now, since 𝒩𝓇𝓃,𝔦(𝜔𝑛) =  
(𝔩−𝒥𝓇𝓃,𝔦)

(𝜔𝓃)

𝓇𝓃
 

lim
𝓃→∞

‖𝒩𝓇𝓃,𝔦(𝜔𝑛)‖ = lim
𝑛→∞

‖
𝜔𝓃−𝒥𝓇𝓃,𝔦

(𝜔𝓃)

𝑟𝑛
‖ = 0   𝑎𝑠 𝓇𝓃 → ∞  

And 𝒩𝓇𝓃,𝔦(𝜔𝑛)  ∈ ℳ𝔦(𝒥𝓇𝓃,𝔦(𝜔𝓃)) ,so, 

〈𝓏 − 𝒥𝑟𝓃𝒿,𝔦
(𝜔𝓃𝒿), 𝓏 −́  𝒩𝓇𝑛𝒿,𝑖(𝜔𝑛𝒿)〉 ≥ 0 , �́� ∈ ℳ𝑖(𝓏) 

〈𝓏 − 𝓋, 𝓏 −́  0〉 ≥ 0 ,    �́� ∈ ℳ𝑖(𝓈)      𝓃𝒿 → ∞ 

Since ℳ𝑖 be M.M. operator,so 

0 ∈ ℳ𝑖(𝓋)  ⟹ 𝓋 ∈ ℳ𝔦
−1(0) ⟹ 𝓋 ∈ 𝔣𝔦𝓍(𝒥𝓇𝓃,𝔦) . But ‖𝜔𝓃 − 𝒯

𝑖(𝜔𝓃)‖ → 0 𝑎𝑠 𝓃 → ∞ 

Therefore, 𝓋 ∈ (⋂ 𝔣𝔦𝓍(𝒯𝑖))∞
𝔦=1  

‖𝜔𝓃+1 −𝓋‖
2 = ‖(𝔞𝓃 − 𝔟𝓃)ℱ(𝜔) + (1 − 𝔞𝓃)∑ 𝒯𝑖(𝜔𝓃) + 𝔟𝑛𝒯(𝓌𝑛)

𝑘
𝑖=0 −𝓋‖

2
  

≤ (𝔞𝓃 − 𝔟𝓃)‖ℱ(𝜔) − 𝓋‖
2 +(1 − 𝔞𝓃)∑ ‖𝒯𝑖(𝜔𝓃) − 𝓋‖

2𝒦
𝑖=0 + 𝔟𝓃‖ℱ(𝓌𝑛) − 𝓋‖

2 

≤ (𝔞𝓃 − 𝔟𝓃)‖𝜔 − 𝓋‖
2 +(1 − 𝔞𝓃)∑ ‖𝜔𝓃 −𝓋‖

2𝒦
𝑖=0 + (1 − 𝔞𝓃)𝒦∑ ‖𝜔𝓃 − 𝒯

𝑖(𝜔𝓃) − (𝓋 −
𝒦
𝑖=0

𝒯𝑖(𝓋)‖
2
+ 𝔟𝓃𝜓𝓃 + 𝔟𝓃‖𝓌𝑛 −𝓋‖

2 

Hence, ‖𝜔𝑛+1 −𝓋‖
2 ≤ (1 − 𝔞𝓃)∑ ‖𝜔𝓃 −𝓋‖

2𝒦
𝑖=0 + 𝔞𝓃 (

𝔞𝓃−𝔟𝓃

𝔞𝓃
‖𝜔 − 𝓋‖2) + ‖𝓌𝓃 −𝓋‖

2 

we get, ‖𝜔𝑛 −𝓋‖ → 0 , 𝑛 → ∞.And hence the Technique f-point converge strongly   to 𝓋 in 

ℳ𝑖
−1(0)⋂(⋂ 𝔣𝔦𝓍(𝒯𝑖)) ∞

𝑖=1 .                                

Theorem (2.2) : Let ℳ𝑖 be M.M. operator, 〈𝒯𝑖〉 be a sequence of 𝒦 - strictly pseudo-contractive 

map on ℂ ,  〈ℱ𝑛〉 be a sequence of nonexpansive mapping  and 〈𝔞𝑛〉,〈𝔟𝑛〉 are sequence in  (0,∞] 

such that  𝔞𝑛 + 𝔟𝑛 = 1 and  𝔞𝑛 ≥ 𝔟𝑛  . Define the Technique f-point as:  

{

                                                                                     
   𝓌𝓃 = ∑ 𝒥𝓇𝓃,𝔦(𝜔𝓃)

𝒦
𝔦=0                                                                                 

     𝜔𝓃+1 = (𝔞𝓃 − 𝔟𝓃)ℱ(𝜔) + (1 − 𝔞𝓃)∑ 𝒯𝑖(𝜔𝓃) + 𝔟𝓃ℱ(𝓌𝓃)
𝒦
𝔦=0

   

 

 



IHJPAS. 37 (2) 2024 

 

358 
 

satisfies :  

(i) ‖𝓌𝓃‖
2 ≤ 𝜓𝓃 + ∑ ‖𝒥𝓇𝓃,𝔦(𝜔𝑛)‖

2𝒦
𝔦=1   and ∑ (𝔟𝓃𝜓𝓃 + ∑ ‖𝜔𝓃 − 𝒯

𝔦(𝜔𝓃)‖
2𝒦

𝔦=1 )∞
𝑖=0 < ∞,where 

〈𝜓𝑛〉 be a sequence in (0,∞) 

 (ii) ℳ𝑖
−1(0)⋂(⋂ f𝔦𝓍(𝒯𝑖))⋂(⋂ f𝔦𝓍(ℱ𝑛)) 

∞
𝑖=1 ≠ ∅     𝑎𝑛𝑑   𝔞𝑛 ≥ 𝔟𝑛

∞
𝑖=1  .  

Then 〈𝑃ℳ𝑖
−1(0)𝜔𝓃〉 converge strongly to point 𝓋 in ℳ𝑖

−1(0) and 

 lim
𝓃→∞

‖𝜔𝓃 −𝓋‖ = 𝔦nf { lim
𝑛→∞

‖𝜔𝓃 − 𝔯‖ , 𝔯 ∈ ℳ𝑖
−1(0)} . 

Proof : Let 𝔯 ∈ (⋂ 𝔣𝔦𝓍(𝒯 𝔦))  ⋂ℳ𝑖
−1(0)  ∞

𝔦=1  

‖𝜔𝓃+1 − 𝔯‖
2 = ‖(𝔞𝑛 − 𝔟𝑛)ℱ(𝜔) + (1 − 𝔞0)∑ 𝒯 𝔦(𝜔𝓃) + 𝔟𝓃ℱ(𝓌𝑛)

𝒦
𝔦=0 − 𝔯‖

2
  

≤ (𝔞𝓃 − 𝔟𝓃)‖𝒯(𝜔𝓃) − 𝔯‖
2 +(1 − 𝔞𝑛)∑ ‖𝒯𝑖(𝜔𝓃) − 𝔯‖

2𝒦
𝔦=0 + 𝔟𝑛‖ℱ(𝓌𝓃) − 𝔯‖

2 

‖𝜔𝓃+1 − 𝔯‖
2  ≤ (𝔞𝓃 − 𝔟𝓃)‖𝜔𝓃 − 𝔯‖

2 +(1 − 𝔞𝑛)∑ ‖𝜔𝓃 − 𝔯‖
2 + (1 − 𝔞𝓃)∑ 𝒦‖𝜔𝓃 −

𝒦
𝑖=0

𝒦
𝔦=0

𝒯𝑖(𝜔𝓃) − (𝔯 − 𝒯
𝑖(𝔯)‖

2
+ 𝔟𝑛‖𝓌𝑛 − 𝔯‖

2  , wherever  𝒦 = s𝓊p{ 𝒦𝑖 ∈ [0,1) , 𝔦 ∈ ℕ  

≤  (1 − 𝔟𝓃)∑ ‖𝜔𝓃 − 𝔯‖
2 +𝒦

𝔦=0 (1 − 𝔞𝓃)∑ 𝒦‖𝜔𝓃 − 𝒯
𝔦(𝜔𝓃)‖

2𝒦
𝔦=0 + 𝔟𝓃𝜓𝓃 + 𝔟𝑛 ∑ ‖𝒥𝓇𝓃,𝔦(𝜔𝑛) −

𝒦
𝑖=0

𝔯‖
2
 

≤ (1 − 𝔟𝓃)∑ ‖𝜔𝓃 − 𝔯‖
2 +𝒦

𝑖=0 𝔟𝓃𝜓𝓃 + ∑ ‖𝜔𝓃 − 𝒯
𝑖(𝜔𝓃)‖

2𝒦
𝔦=0 + 𝔟𝓃𝜓𝓃 + 𝔟𝓃 ∑ ‖𝜔𝓃 − 𝔯‖

2𝒦
𝔦=0   

‖𝜔𝓃+1 − 𝔯‖
2 < ∑ ‖𝜔𝓃 − 𝔯‖

2 +𝒦
𝔦=0 𝔟𝑛𝜓𝑛 + ∑ ‖𝜔𝓃 − 𝒯

𝑖(𝜔𝓃)‖
2𝒦

𝔦=0   

Since   ∑ (∞
𝔦=0 𝔟𝓃𝜓𝓃 + ‖𝜔𝓃 − 𝒯

𝔦(𝜔𝓃)‖
2
)  < ∞ 

 we get , ℛ(𝔯) = lim
𝓃→∞

‖𝜔𝓃 − 𝔯 ‖ exist .This is 〈𝜔𝓃〉 is bounded sequence . 

Put ℒ = 𝔦nf {ℛ(𝔯), 𝔯 ∈ (⋂ f𝔦𝓍(𝒯𝑖)) ⋂ℳ𝑖
−1(0)   ∞

𝔦=1 }and 𝕂 = {𝒲 ∈ (⋂ f𝔦𝓍(𝒯𝑖)) ⋂ℳ𝑖
−1(0) ∶∞

𝑖=1

ℛ(𝒲) = ℒ } 

‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖ ≤ ‖𝜔𝓃 −𝓋‖  , ∀𝓋 ∈ 𝒦  

lim
𝓃→∞

‖𝜔𝓃 −𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖ ≤ ℒ   for all  𝓃 ∈ ℕ . 

To prove that  lim
𝓃→∞

‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖ = ℒ 

Suppose that lim
𝓃→∞

‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖ < ℒ . This implies  

lim
𝓃→∞

‖𝜔𝓃 −𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

< ℒ2 .Then there exist ℴ > 0 , such that  

lim
𝓃→∞

‖𝜔𝓃 −𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

< ℒ2 − ℴ  ∀ 𝓃 ≥ 𝓂  ,𝓂 ∈ ℕ 

‖𝜔𝓃+𝒽+1 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

≤ ∑ ‖𝜔𝓃+𝒽 − 𝔯‖
2 +𝒦

𝔦=0 𝔟𝓃+𝒽𝜓𝓃+𝒽 + (∑ ‖𝜔𝓃+𝒽 −
𝒦
𝔦=0

𝒯𝑖(𝜔𝓃+𝒽)‖
2
)   

≤ [∑ ‖𝜔𝓃+𝒽−1 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

+𝒦
𝔦=0 𝔟𝓃+𝒽−1𝜓𝓃+𝒽−1 + (∑ ‖𝜔𝓃+𝒽−1 − 𝒯

𝑖(𝜔𝓃+𝒽−1)‖
2
)𝒦

𝔦=0 ] +

𝔟𝓃+𝒽𝜓𝓃+𝒽 − 𝒯
𝑖(𝜔𝓃+𝒽)  

≤ ∑ ‖𝜔𝓃+𝒽−2 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

+𝒦
𝔦=0 𝔟𝓃+𝒽−2𝜓𝓃+𝒽−2 + (∑ ‖𝜔𝓃+𝒽−2 − 𝒯

𝑖(𝜔𝓃+𝒽−2)‖
2
)𝒦

𝔦=0 +

𝔟𝓃+𝒽−1𝜓𝓃+𝒽−1 − 𝒯
𝑖(𝜔𝓃+𝒽−1)  

≤ ‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

+ ∑ (𝔟𝔦𝜓𝔦
𝓃+𝒽
𝔦=𝓃 + ‖𝜔𝔦 − 𝒯

𝔦𝜔𝔦‖
2
)  

This implies , for each 𝓃 ≥ 𝓂 ,𝒽 ∈ ℕ  the following satisfied  



IHJPAS. 37 (2) 2024 

 

359 
 

ℒ2 ≤ lim
𝓃→∞

(‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

+∑ (𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝔦𝜔𝔦‖

2
)𝓃+𝒽

𝔦=𝓃 )  

= lim
𝓃→∞

(‖𝜔𝓃+𝒽+1 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

+ ∑ (𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝔦𝜔𝔦‖

2
)𝓃+𝒽

𝔦=𝓃 )  

≤ lim
𝓃→∞

(ℒ2 − ℴ + ∑ (𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝔦𝜔𝔦‖

2
)𝓃+𝒽

𝔦=𝓃 )  

But   ∑ (𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝔦𝜔𝔦‖

2
)∞

𝔦=𝓃 < ∞ , ∀ 𝓃 ≥ 𝓂 

And  lim
𝓃→∞

‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖

2

< ℒ2 − ℴ.So, ℒ2 ≤ ℒ2 − ℴ < ℒ2 , which is a contradiction  

So , lim
𝓃→∞

‖𝜔𝓃 − 𝒫ℳ𝑖
−1(0)(𝜔𝓃)‖ = ℒ 

Now,to prove that 𝒫ℳ𝑖
−1(0)(𝜔𝓃) → 𝓋 

If not then there exists 𝔈 > 0 such that , ∀ 𝒽 ∈ ℕ ,we have  

‖𝒫ℳ𝑖
−1(0)(𝜔𝓃) − 𝓋‖ ≥ 𝔈 , for some �́� ≥ 𝒽.  If 𝔟 ≥ 0 such that 𝔟 < √ℒ2 +

𝔈2

8
 − ℒ   , 𝓀 ∈ ℕ  

𝔟 + ℒ < √ℒ2 +
𝔈2

8
   ⟹ (𝔟 + ℒ)2 < ℒ2 +

𝔈2

8
 ⟹ (𝔟 + ℒ)2 −

𝔈2

8
< ℒ2  

∑ (𝔟𝓃𝜓𝓃 + ‖𝜔𝓃 − 𝒯
𝑖𝜔𝓃‖

2
)∞

𝔦=𝓃 ≤
𝔈2

8
  

‖𝜔�́� − 𝒫ℳ𝑖
−1(0)(𝜔�́�)‖ ≤ ℒ + 𝔟  and ‖𝜔�́� −𝓋‖ ≤ ℒ + 𝔟 

‖𝜔𝓃+�́�+1 −
𝒫
ℳ𝑖
−1(0)

(𝜔�́�)+𝓋

2
‖
2

≤ ‖𝜔�́� −
𝒫
ℳ𝑖
−1(0)

(𝜔�́�)+𝓋

2
‖
2

+ ∑ (𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝔦(𝜔𝔦)‖

2
)𝓃+�́�

𝔦=�́�    

=‖
2𝜔�́�−(𝒫ℳ𝑖

−1(0)
(𝜔�́�))+𝓋

2
‖

2

+∑ (𝓃+�́�
𝔦=�́�

𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝑖(𝜔𝔦)‖

2
) 

= ‖
𝜔�́�−𝒫ℳ𝑖

−1(0)
(𝜔�́�)

2
+
𝜔�́�−𝓋

2
‖
2

+ ∑ (𝓃+�́�
𝔦=�́�

𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 − 𝒯
𝑖(𝜔𝔦)‖

2
) 

= 2 ‖
𝜔�́�−𝒫ℳ𝑖

−1(0)
(𝜔�́�)

2
‖
2

+ 2‖
𝜔�́�−𝓋

2
‖
2

− ‖
𝜔�́�−𝒫ℳ𝑖

−1(0)
(𝜔�́�)

2
+
𝜔�́�−𝓋

2
‖ + ∑ (𝓃+�́�

𝔦=�́�
𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 −

𝒯𝑖(𝜔𝔦)‖
2
) 

=
1

2
‖𝜔�́� − 𝒫ℳ𝑖

−1(0)(𝜔�́�)‖
2

+
1

2
‖𝜔�́� −𝓋‖

2
−
1

4
‖𝑃ℳ𝑖

−1(0)(𝜔ℎ́) − 𝓋‖
2

+ ∑ (𝓃+�́�
𝔦=�́�

𝔟𝔦𝜓𝔦 + ‖𝜔𝔦 −

𝒯𝑖(𝜔𝔦)‖
2
) 

 

‖𝜔𝓃+�́�+1 −
𝑃
ℳ𝑖
−1(0)

(𝜔ℎ́)+𝓋

2
‖
2

≤
1

2
(ℒ + 𝔟)2 +

1

2
(ℒ + 𝔟)2 −

1

4
𝔈2+ 

𝔈2

8
   =  (ℒ + 𝔟)2 +

𝔈2

8
 

As 𝓃 → ∞  we get  

ℒ2 ≤ lim
𝑛→∞

‖𝜔𝓃+�́�+1 −
𝑃
ℳ𝑖
−1(0)

(𝜔ℎ́)+𝓋

2
‖
2

≤ (ℒ + 𝔟)2 −  
𝔈2

8
< ℒ2    

Which is a contraction .So, 𝒫ℳ𝑖
−1(0)(𝜔𝑛) → 𝓋.That is 〈𝒫ℳ𝑖

−1(0)(𝜔𝓃)〉 converge  strongly to point 

in ℳ𝑖
−1(0)              

Theorem(2.3): Let  ℂ , ℳ𝔦 , 〈ℱ𝓃〉 , 〈𝒯
𝑖〉  and 〈𝜔𝓃〉  as in theorem 2.2 , 〈𝔟𝓃〉 be a sequence in 

(0,1] and  〈𝔞𝑛〉 be a sequence in  [𝔞, 𝔟]  such that 𝑜 < 𝔞 < 𝔟 < 1 𝑎𝑛𝑑 𝔞𝑛 + 𝔟𝑛 = 1 . If 𝔞𝑛 ≥ 𝔟𝑛  

and lim
𝓃→∞

𝔟𝑛 = 0  then the  technique f-point 〈𝜔𝓃〉  define  



IHJPAS. 37 (2) 2024 

 

360 
 

{

                                                                                     
   𝓌𝑛 = ∑ 𝒥𝓇𝓃,𝔦(𝜔𝑛)

𝒦
𝔦=0                                                                                 

     𝜔𝓃+1 = (𝔞𝑛 − 𝔟𝑛)ℱ(𝜔𝓃) + (1 − 𝔞𝑛)∑ 𝒯𝑖(𝜔𝓃) + 𝔟𝑛ℱ(𝓌𝑛)
𝒦
𝔦=0

   

With two conditions :  

(i) ‖𝓌𝓃‖
2 ≤ 𝜓𝓃 + ∑ ‖𝒥𝓇𝓃,𝔦(𝜔𝑛)‖

2𝑘
𝑖=1   and ∑ (𝔟𝓃𝜓𝓃 + ∑ ‖𝜔𝓃 − 𝒯

𝑖(𝜔𝓃)‖
2𝒦

𝔦=1 )∞
𝑖=0 < ∞,where 

〈𝜓𝓃〉 be a sequence in (0,∞). 

 (ii) ℳ𝑖
−1(0)⋂(⋂ 𝔣𝔦𝓍(𝒯𝑖))⋂(⋂ 𝔣𝔦𝓍(ℱ𝑛)) 

∞
𝑖=1 ≠ ∅ 𝑎𝑛𝑑   𝔞𝑛 ≥ 𝔟𝑛 

∞
𝑖=1 . 

 has converges weakly to a point 𝓋 ∈ ℳ𝑖
−1(0)  where  lim

𝑛→0
〈𝒫ℳ𝑖

−1(0)𝓍𝓃〉 = 𝓋   .  

Proof :Let 𝔯 ∈ ℳ𝑖
−1(0)⋂(⋂ 𝔣𝔦𝓍(𝒯𝑖))  ⋂(⋂ 𝔣𝔦𝓍(ℱ𝓃)

∞
𝑖=1 )  ∞

𝑖=1  

‖𝜔𝓃+1 − 𝔯‖
2 = ‖(𝔞𝑛 − 𝔟𝑛)ℱ(𝜔𝓃) + (1 − 𝔞𝓃)∑ 𝒯 𝔦(𝜔𝓃) + 𝔟𝑛ℱ(𝓌𝓃)

𝒦
𝑖=0 − 𝔯‖

2
  

≤ (𝔞𝑛 − 𝔟𝑛)‖ℱ(𝜔𝓃) − 𝔯‖
2 + (1 − 𝔞𝑛) ∑ ‖𝒯 𝔦(𝜔𝓃) − 𝔯‖

2𝒦
𝔦=0 + 𝔞𝓃‖ℱ(𝓌𝓃) − 𝔯‖

2  

‖𝜔𝓃+1 − 𝔯‖
2  ≤ (𝔞𝑛 − 𝔟𝑛)‖𝜔𝓃 − 𝔯‖

2 +(1 − 𝔞𝑛)∑ ‖𝜔𝓃 − 𝔯‖
2 + (1 − 𝔞𝑛)∑ 𝒦‖𝜔𝓃 −

𝒦
𝔦=0

𝑘
𝑖=0

𝒯𝑖(𝜔𝓃) − (𝔯 − 𝒯
𝔦(𝔯)‖

2
+ 𝔟𝒦‖𝓌𝓃 − 𝔯‖

2  , where 𝒦= 𝔰𝓊𝑝 { 𝒦𝔦 ∈ [0,1) , 𝔦 ∈ ℕ  

≤ (1 − 𝔟𝑛)∑ ‖𝜔𝓃 − 𝔯‖
2 +𝒦

𝑖=0 (1 − 𝔞𝑛)∑ 𝒦‖𝜔𝓃 − 𝒯
𝑖(𝜔𝓃)‖

2𝒦
𝔦=0 + 𝔟𝓃𝜓𝓃 +𝔟𝑛 ∑ ‖𝒥𝓇𝓃,𝔦(𝜔𝓃) −

𝒦
𝔦=0

𝔯‖
2
 

≤ (1 − 𝔟𝑛)∑ ‖𝜔𝓃 − 𝔯‖
2 +𝑘

𝑖=0 𝔟𝓃𝜓𝓃 + ∑ ‖𝜔𝓃 − 𝒯
𝑖(𝜔𝓃)‖

2𝒦
𝔦=0 + 𝔟𝓃𝜓𝓃 + 𝔟𝓃 ∑ ‖𝜔𝓃 − 𝔯‖

2𝒦
𝔦=0   

‖𝜔𝓃+1 − 𝔯‖
2 < ∑ ‖𝜔𝓃 − 𝔯‖

2 +𝒦
𝔦=0 𝔟𝓃𝜓𝓃 + ∑ ‖𝜔𝓃 − 𝒯

𝔦(𝜔)‖
2𝒦

𝔦=0    

Since   ∑ (∞
𝔦=0 𝔟𝓃𝜓𝓃 + ∑ ‖𝜔𝓃 − 𝒯

𝔦(𝜔𝓃)‖
2∞

𝔦=0 )  < ∞,we get , lim
𝓃→∞

‖𝜔𝓃 − 𝔯 ‖ exist . 

So that, the technique f-point 〈𝜔𝓃〉 is bounded ,so ∃ 〈𝜔𝓃𝒦〉 subsequence of 〈𝜔𝓃〉 , such that , 

𝜔𝓃𝒦 ⇀ 𝓋.  

Now, put 𝒬𝓃 = {‖𝜔𝓃 − 𝔯‖. ‖𝓌𝓃 − 𝔯‖,𝓃 ∈ ℕ}  and 𝒦 ∈ [0,1)  s.t  

(1 −𝒦)𝔞𝓃‖𝜔𝓃 −𝓌𝓃‖
2 ≤ (1 − 𝔞𝓃)‖𝜔𝓃 −𝓌𝓃‖

2 

≤ (1 − 𝔞𝑛)‖𝜔𝓃 − 𝔯‖
2 + (1 − 𝔞𝓃)‖𝓌𝑛 − 𝔯‖

2 + 2(1 − 𝔞𝓃)𝒬𝑛 

≤ 𝔟𝓃‖𝜔𝓃 − 𝔯‖
2 + 𝔟𝓃 [𝜓𝓃 + ∑ ‖𝒥𝓇𝓃,𝔦(𝜔𝓃) − 𝔯‖

2𝒦
𝔦=𝑜 ] + 2𝔟𝓃𝒬𝓃  

≤ 𝔟𝓃‖𝜔𝓃 − 𝔯‖
2 + 𝔟𝓃[𝜓𝓃 +∑ ‖𝜔𝓃 − 𝔯‖

2𝒦
𝔦=𝑜 ] + 2𝔟𝓃𝒬𝓃  

(1 −𝒦)𝔞𝓃‖𝜔𝓃 −𝓌𝓃‖
2 ≤

𝔟𝑛

𝔞𝑛
{∑ ‖𝜔𝓃 − 𝔯‖

2𝑘
𝑖=𝑜 + 𝜓𝓃 + 2𝔟𝓃𝒬𝓃}  

So, ‖𝜔𝓃 −𝓌𝑛‖ → 0 as 𝓃 → ∞ ,since 𝜔𝓃𝒦 ⇀ 𝓋 . Then 𝓌𝓃𝒦 ⇀ 𝓋 

Now, since  𝓌𝓃 = ∑ 𝒥𝓇𝓃,𝔦(𝜔𝓃)
𝒦
𝔦=0  , so 〈𝓏 − 𝒥𝓇𝓃𝒦,𝑖(𝜔𝓃𝒦), 𝓏 −́  ℕ𝓇𝓃𝒦,𝑖(𝜔𝓃𝒦)〉 ≥ 0 , �́� ∈ ℳ𝑖(𝓏)                                         

lim
𝓃→∞

‖ℕ𝓇𝔫,𝑖(𝜔𝓃)‖ = lim
𝓃→∞

‖
𝜔𝓃−𝒥𝓇𝓃,𝔦

(𝜔𝓃)

𝓇𝓃
‖ = 0   𝑎𝑠 𝓇𝓃 → ∞  

〈𝓏 − 𝓋, �́�〉 ≥ 0 ,    �́� ∈ ℳ𝑖(𝓏). as  𝒦 → ∞.Then, 𝒫ℳ𝑖
−1(0)𝜔𝓃 → �́� ∈ ℳ𝑖

−1 

Hence,〈𝜔𝓃𝒦 −𝒫ℳ𝑖
−1(0)(𝜔𝓃𝒦), 𝓉 − 𝒫ℳ𝑖

−1(0)(𝜔𝑛𝒦)〉 ≤ 0 , 𝓉 ∈ ℳ𝑖
−1(0) 

So, 〈𝓋 − �́�, 𝓉 − �́�〉 ≤ 0 , 𝓉 ∈ ℳ𝑖
−1(0).But  𝓋 ∈ ℳ𝑖

−1(0) . Then  

〈𝓋 − �́�, 𝓉 − �́�〉 ≤ 0 ⟹ ‖𝓋 − �́�‖2 ≤ 0  . That is 𝓋 = �́� 

Therefore, the technique f-point 〈𝜔𝓃〉  has convergence-w to the limit point of 𝒫ℳ𝑖
−1(0)(𝜔𝓃).     

 



IHJPAS. 37 (2) 2024 

 

361 
 

4. Conclusion 

1. A new technical methods of resolvent operators and metric projection of strictly pseudo 

contraction mapping are introduced  

2. The convergence weakly and convergence strongly of the proposed iterative method under 

some suitable conditions are studied. 

3. A common fixed points of different types of important mappings and  the set of zeros of 

maximal monotone operators are found. 

 

Acknowledgment 

Our researcher extends his Sincere thanks to the editor and members of the preparatory committee 

of the Ibn AL-Haitham Journal of Pure and Applied Sciences. 

  

Conflict of Interest  

There are no conflicts of interest. 

 

Funding  

There is no funding for the article. 

 
References  

1. Browder, F.E. Fixed-Point Theorems for Noncompact Mappings in Hilbert Space. Proc. Natl. Acad. Sci. 

1965, 53, 1272–1276. DOI: https://doi.org/10.1073/pnas.53.6.1272 

2. Browder, F.E. Convergence of Approximants to Fixed Points of Nonexpansive Nonlinear Mappings in 

Banach Spaces. Arch. Ration. Mech. Anal. 1967, 24, 82–90. 

3. Browder, F.E.; Petryshyn, W. V Construction of Fixed Points of Nonlinear Mappings in Hilbert Space. 

J. Math. Anal. Appl. 1967, 20, 197–228. DOI: https://doi.org/10.1016/0022-247X(67)90085-6.  

4. Mann, W.R. Mean Value Methods in Iteration. Proc. Am. Math. Soc. 1953, 4, 506–510. DOI: 

http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3 

5. Halpern, B. Fixed Points of Nonexpanding Maps. 1967. DOI: http://dx.doi.org/10.1090/S0002-9904-

1967-11864-0 

6. Goebel, K.; Kirk, W.A. Topics in Metric Fixed Point Theory; Cambridge university press, 1990; ISBN 

0521382890. 

7. Goebel, K.; Simeon, R. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings; 

Dekker, 1984; ISBN 0824772237. 

8. Bauschke, H.H. The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in 

Hilbert Space. J. Math. Anal. Appl. 1996, 202, 150–159. DOI : https://doi.org/10.1006/jmaa.1996.0308.  

9. Xu, H.-K. Iterative Algorithms for Nonlinear Operators. J. London Math. Soc. 2002, 66, 240–256.DOI: 

http://dx.doi.org/10.1112/S0024610702003332 

10. Xu, H.K. Remarks on an Iterative Method for Nonexpansive Mappings. Comm. Appl. Nonlinear Anal 

2003, 10, 67–75. DOI: https://doi.org/10.1016/j.jmaa.2005.11.057 

11. Byrne, C. A Unified Treatment of Some Iterative Algorithms in Signal Processing and Image 

Reconstruction. Inverse Probl. 2003, 20, 103. DOI: http://dx.doi.org/10.1088/0266-5611/20/1/006 

12. Kim, T.-H.; Xu, H.-K. Strong Convergence of Modified Mann Iterations. Nonlinear Anal. Theory, 

Methods Appl. 2005, 61, 51–60.DOI: https://doi.org/10.1016/j.na.2004.11.011 

13. Maibed, Z.H. Common Fixed Point Problem for Classes of Nonlinear Maps in Hilbert Space. In 

Proceedings of the IOP Conference Series: Materials Science and Engineering; IOP Publishing, 2020; 

Vol. 871, p. 12037. DOI: https://doi.org/10.1088/1757-899X/871/1/012037 

14. Kim,T.-H.;Xu, H.-K. Strong Convergence of Modified Mann Iterations for Asymptotically 

https://doi.org/10.1073/pnas.53.6.1272
http://dx.doi.org/10.1088/1757-899X/871/1/012037


IHJPAS. 37 (2) 2024 

 

362 
 

Nonexpansive Mappings and Semigroups. Nonlinear Anal. Theory, Methods Appl. 2006, 64, 1140–

1152. DOI: https://doi.org/10.1016/j.na.2005.05.059 . 

15. Xu, H.-K. Strong Convergence of an Iterative Method for Nonexpansive and Accretive Operators. J. 

Math. Anal. Appl. 2006, 314, 631–643. DOI: https://doi.org/10.1016/j.jmaa.2005.04.082 

16. Maibed, Z.H.; Thajil, A.Q. Zenali Iteration Method For Approximating Fixed Point of A δ ZA-Quasi 

Contractive Mappings. Ibn AL-Haitham J. Pure Appl. Sci. 2021, 34, 78–92.DOI: 

https://doi.org/10.30526/34.4.2705 

17. Hamed, M.M.; Jamil, Z.Z. Stability And Data Dependence Results For The Mann Iteration Schemes on 

N-Banach Space. Iraqi J. Sci. 2020, 1456–1460. DOI: https://doi.org/10.24996/ijs.2020.61.6.25 

18. Abed, S.S.; Ali, R.S.A. Fixed Point for Asymptotically Non-Expansive Mappings in 2-Banach Space. 

Ibn AL-Haitham J. Pure Appl. Sci. 2017, 27, 343–350. DOI https://doi.org/10.1006/jmaa.2001.7649 

19. Abed, S.S.; Maibed, Z.H. Proximal Schemes by Family of Szl–Widering Mappings. In Proceedings of 

the IOP Conference Series: Materials Science and Engineering; IOP Publishing, 2019; Vol. 571, p. 

12006. DOI: https://doi.org/10.1088/issn.1757-899X . 

20. Jamil, Z.Z.; Abed, M.B. On a Modified SP-Iterative Scheme for Approximating Fixed Point of a 

Contraction Mapping. Iraqi J. Sci. 2015, 56, 3230–3239. DOI: 

https://ijs.uobaghdad.edu.iq/index.php/eijs/article/view/9429 

21. Maibed, Z.H.; Thajil, A.Q. Equivalence of Some Iterations for Class of Quasi Contractive Mappings. In 

Proceedings of the Journal of Physics: Conference Series; IOP Publishing, 2021; Vol. 1879, p. 22115.  

DOI : http://dx.doi.org/10.1088/1742-6596/1879/2/022115 

22. Reich, S. Weak Convergence Theorems for Nonexpansive Mappings in Banach Spaces. J.Math. Anal. 

Appl 1979, 67, 274–276. DOI:https://doi.org/10.1016/0022-247X(79)90024-6 

23. A, Z.H.M. and A.M. On the Convergence of New Iteration Schems by Resolvent ZA-Jungck Mapping. 

J. Interdiscip. Math. to Appear 2022. DOI : https://doi.org/10.30526/36.1.2917 . 

24. T. Suzuki, W.T. Weak and Strong Convergence Theorems for Nonexpansive Mapping in Banach 

Spaces. Nonlinear Anal. 2001, 47, 2805–2815. http://dx.doi.org/10.1016/S0362-546X(01)00399-6 

25. Nakajo, K.; Takahashi, W. Strong Convergence Theorems for Nonexpansive Mappings and 

Nonexpansive Semigroups. J. Math. Anal. Appl. 2003, 279, 372–379. DOI: 

http://dx.doi.org/10.1016/S0022-247X(02)00458-4 .  

26. Takahashi, W.; Toyoda, M. Weak Convergence Theorems for Nonexpansive Mappings and Monotone 

Mappings. J. Optim. Theory Appl. 2003, 118, 417–428. DOI : 

http://dx.doi.org/10.1023/A:1025407607560 .  

27. Chen, R.; He, H. Viscosity Approximation of Common Fixed Points of Nonexpansive Semigroups in 

Banach Space. Appl. Math. Lett. 2007, 20, 751–757. DOI : DOI:10.1007/s10013-013-0033-3 . 

28. Marino, G.; Xu, H.-K. Weak and Strong Convergence Theorems for Strict Pseudo-Contractions in 

Hilbert Spaces. J. Math. Anal. Appl. 2007, 329, 336–346.DOI: 

https://doi.org/10.1016/j.jmaa.2006.06.055 . 

29. Rhoades, B.E. Fixed Point Iterations Using Infinite Matrices. Trans. Am. Math. Soc. 1974, 196, 161–

176.DOI: http://dx.doi.org/10.2307/1997020 . 

30. Acedo, G.L.; Xu, H.-K. Iterative Methods for Strict Pseudo-Contractions in Hilbert Spaces. Nonlinear 

Anal. Theory, Methods Appl. 2007, 67, 2258–2271.DOI : http://dx.doi.org/10.1016/j.na.2006.08.036 . 

31. Genel, A.; Lindenstrauss, J. An Example Concerning Fixed Points. Isr. J. Math. 1975, 22, 81–86. 

https://doi.org/10.1007/BF02757276 . 

32. Güler, O. On the Convergence of the Proximal Point Algorithm for Convex Minimization. SIAM J. 

Control Optim. 1991, 29, 403–419. DOI : http://dx.doi.org/10.1137/0329022 

33. Liu, L.-S. Ishikawa and Mann Iterative Process with Errors for Nonlinear Strongly Accretive Mappings 

in Banach Spaces. J. Math. Anal. Appl. 1995, 194, 114–125. https://doi.org/10.1006/jmaa.1998.5987 

http://dx.doi.org/10.1016/j.na.2005.05.059
http://dx.doi.org/10.1006/jmaa.2001.7649
http://dx.doi.org/10.1007/s10013-013-0033-3