IHJPAS. 36 (3) 2023 
 

283 

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

 

   

 

 

 
 

*Corresponding  Author: mrs_ zena.hussein@yahoo.com 

 

 

 

 

 

Abstract 

The focus of this paper reviewed generalized contraction mapping and nonexpansive maps and 

recall some theorems about the existence and uniqueness of common fixed points and coincidence 

fixed-point for such maps under some conditions. Moreover, some schemes of different types as 

one-step schemes, two-step schemes, and three-step schemes (Mann scheme algorithm, Ishukawa 

scheme algorithm, Noor scheme algorithm, 𝑆𝑃 −.scheme algorithm, 𝐶𝑅 − scheme algorithm 

Modified SP scheme algorithm Karahan scheme algorithm, and others. The convergence of these 

schemes has been studied. On the other hand, we also reviewed the convergence, valence, and 
stability theories of different types of near-plots in convex metric space. 

Keywords: Convergence, Fixed Point, Nonexpansive Map, Pseudocontractive Map and Iterative 

Methods. 

Introduction 

Fixed point theory is an important topic, and it has many applications in branches of mathematics 

various. In the year 1970, introduced Takahashi the idea of convexity in m-spaces and studied it 

as well as common f-point theorems for nonexpansive mappings. The convex m-space is a public, 

important, and, expansive space with a convex structure, where the Banach cone space is convex 

m-space. The principle of the Banach contraction states that they can approximate the contraction 

maps f-point by Picard proximal scheme. The seq ⟨𝓍𝑛 ⟩ of this scheme can be defined  as follows: 

Let ∅ ≠ ℳ be a closed-convex lies in ℋand  𝒥:  ℳ →  ℳ be a mapping: 

 𝒶0 ∈ ℳ,       𝒶𝑛+1 =  𝒥𝒶𝑛  ,  𝑛 ∈ 𝑁                                                                                                (1)   

Picard's proximal scheme for nonexpansive mappings does not converge to a f-point. Hence, to 

doi.org/10.30526/36.3.3006 

Article history: Received 10 September 2022, Accepted 1 November 2022, Published in July 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

A Review of the Some Fixed Point Theorems for Different Kinds of Maps 

 
  Zena Hussein Maibed*  

Department of Mathematics, College of Education 

for Pure Science Ibn Al-Haytham, University of 

Baghdad, Baghdad, Iraq. 

 

mrs_ zena.hussein@yahoo.com 

 

 

 

    Bayda Atiya Kalaf  

Department of Mathematics, College of Education 

for Pure Science Ibn Al-Haytham, University of 

Baghdad, Baghdad, Iraq. 

 

Hbama75@yahoo.com 

 

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IHJPAS. 36 (3) 2023 
 

284 
 
 

approximate the f-points of the non'expansion maps, a proximal scheme is introduced as: 

 𝒶0 𝜖 ℳ ,           𝒶𝑛+1 =  (1 −  𝛼𝑛  )𝒶𝑛 +  𝛼𝑛𝒥𝒶𝑛  ,  𝑛 ∈ 𝑁                                                         (2) 

Because the iterative Mann proximal scheme [1], fails to converge to the f-points of the spurious 

systolic maps, and for spurious systolic maps introduced Ishikawa proximal scheme to f-points. 

The sequence 〈𝓍𝑛〉 of the Ishikawa proximal scheme[2], defined as: 

𝒶0 ∈ ℳ    ,    𝒶𝑛+1 = ( 1 −  𝛼𝑛)𝒶𝑛 +  𝛼𝑛  𝒥𝒷𝑛 ,     𝒷𝑛 = ( 1 −  𝛽𝑛)𝒶𝑛 +  𝛽𝑛 ℐ𝒶𝑛  , 𝑛 ∈ 𝑁      (3) 

Noor,in 2000[3] introduced proximal scheme as: 

𝓌0 ∈ ℳ ,𝓌𝑛+1 = ( 1 −  𝛼𝑛)𝓌𝑛 +  𝛼𝑛  ℐ𝓊𝑛  , 𝓊𝑛 = ( 1 −  𝛽𝑛)𝓌𝑛 +  𝛽𝑛 𝒥𝓋𝑛 ,  

𝓋𝑛  = ( 1 −  𝛾𝑛)𝓌𝑛 +  𝛾𝑛 ℐ𝓌𝑛  , 𝑛 ∈ 𝑁                                                                                   (4) 

In[4],Agrawal introduced for nearly non'expansive maps,two steps as: 

𝓍0 ∈ ℳ,    𝓍𝑛+1 = (1 − 𝛼𝑛)𝒥𝓍𝑛 + 𝛼𝑛𝒥𝑡𝑛    ,  𝑡𝑛 = (1 − 𝛽𝑛)𝓍𝑛 + 𝛽𝑛𝒥𝓍𝑛 ,      𝑛 ∈ N           (5) 

𝑺𝑷 −iteration [5]: 

 𝓍0 ∈ ℳ ,      𝓍𝑛+1 =  𝛼𝑛 𝒥𝑦𝑛 +( 1 −  𝛼𝑛)𝑦𝑛 ,         𝑦𝑛 = 𝛽𝑛𝒥𝑧𝑛 + (1 − 𝛽𝑛)𝑧𝑛,     

𝑧𝑛 = 𝛾𝑛𝒥𝓍𝑛 + (1 − 𝛾𝑛 )𝓍𝑛              𝑛 ∈ 𝑁                                                                               (6) 

𝐂𝐑 −iteration [6]: 

𝓌0 ∈ ℳ ,     𝓌n+1 = ( 1 −  αn)𝓊n +  αn 𝒥𝓊n ,  𝓊n = ( 1 −  βn)𝒥𝓌n +  βn 𝒥𝓋n,                                       

 vn = ( 1 −  γn)𝓌n +  γn 𝒥𝓌n  ,     𝑛 ∈ N                                                                               (7) 

Modified 𝐒𝐏 iteration [7]: 

                                                                𝓍0 ∈ ℳ ,                                                                            (8) 

𝓍n+1 = 𝒥𝓎n , 

𝓎n = ( 1 −  αn)𝓏n +  αn 𝒥𝓏n , 

𝓏n = ( 1 −  βn)𝓍n +  βn 𝒥𝓍n 

  𝐊arahan iteration [8]: 

𝓌0 ∈ ℳ ,   𝓌n+1 = ( 1 −  αn)𝒥𝓌n +  αn 𝒥𝓊n ,   𝓊n = ( 1 −  βn)𝓌n +  βn 𝒥𝓋n , ,  

𝓋n = ( 1 −  γn)𝓌n +  γn 𝒥𝓌n ,          n ∈ N                                                               (9) 

Finally, [9] studied the existence of a f- point for type of contraction-maps and the convergence of 

a common f-point for Noor iteration in complete convex metric spaces(Com Con M-S). Then a 
lot of studies were carried out on this topic,see[10-18]. 

 



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Preliminaries 

         In this part,we introduce some concepts which is need in this work,see[8,11and12]. 

1.  A mapping𝒥 is called non'expansive if:  

‖𝒥𝒶 − 𝒹‖ ≤‖ 𝒶 − 𝒹 ‖ for all 𝒶,  𝒹 ∈  ℳ 

2. A mapping 𝒥 is called quasi'nonexpansive if: 

       𝐹(𝒥) ≠  ∅ and ‖𝒥𝒶 − 𝒥𝒷 ‖ ≤  ‖𝒶 −  𝒷‖  for all 𝒶, 𝒷 ∈  ℳ and 𝑦 ∈  𝐹(𝒥).  

3. It is easy to see that if 𝒥 is non'expansive with 𝐹(𝒥) ≠ ∅, then it is quasi'nonexpansive. 

4. A mapping 𝒥 is said to be e pseudocontractive if the inequality 

      ‖𝒶 − 𝒷‖ ≤ ‖𝒶 − 𝒷 + 𝑡[(𝐼 − 𝒥)𝒶 − (𝐼 − 𝒥)𝒷 ]‖  

      Hold for each 𝒶, 𝒷 ∈ ℳ and all 𝑡 > 0. 

 Some proximal scheme are used to approximate a f- point of Zamfirescu maps  are the most 

general contractive maps satisfying the condition: ∀𝒶,  𝒷 𝑙𝑖𝑒𝑠 𝑖𝑛 ℳ  at least one of the 

conditions is true: 

(𝑖) 𝒹 (𝒥𝒶,  𝒥𝒷) ≤  𝒫𝒹 (𝒶, 𝒷), 

(𝑖𝑖) 𝒹(𝒥𝒶,  𝒥𝒷) ≤  𝒬 [𝒹 (𝒶,  𝒥𝒶) +  𝒹(𝒷, 𝒥𝒷)], 

(𝑖𝑖𝑖) 𝒹 (𝒥𝒶,  𝒥𝒷) ≤  ℛ [𝒹 (𝒶,  𝒥𝒷) +  𝒹 (𝒷,  𝒥𝒶)].   

Where 0 ≤ 𝒫≤ 1, 0 ≤  𝒬 , and  ℛ ≤ 1/2 

Definition :Let ᶂ , ᶃ :  ℋ → ℋ be a two mappings. A point 𝒶 ∈  ℋ is called f- point of ᶂ if 

ᶂ (𝒶) = 𝒶, a common f-point of a pair (ᶂ ,  ᶃ) if ᶂ (𝒶) =  ᶃ(𝒶) = 𝒶 an a coincidence point of 

(ᶂ ,  ᶃ) 𝑖𝑓 ᶂ (𝒶) = ᶃ(𝒶). 

 Remarks :Amapping- Zamfirescu is  equivalent to the condition: 

𝑑 (𝑇𝒶,  𝑇𝒷) ≤ ℯ 𝑚𝑎𝑥 {𝑑 (𝒶,  𝒷) ,
{𝒹 (𝒶, T𝒶)+ 𝒹(𝒷,T𝒷)}

2
 ,

{𝒹 (𝒶, 𝑇𝒷) + 𝒹 (𝒷, 𝑇𝒶)}

2
 }                  

∀𝒶,  𝒷 ∈  ℋ,  0 <  ℯ <  1. 

Definition[12]: A mapping ℛ : ℋ × ℋ × [0,1] → ℋ is called convex structure on m-space,if for 

each (𝒶, 𝒷, 𝜆 ) ∈ ℋ × ℋ × [0,1] and 

 𝑢 ∈ ℋ,  𝒹(𝑢, ℛ(𝒶, 𝒷, 𝜆 )) ≤ 𝜆𝒹(𝑢, 𝒶) + (1 − 𝜆 )𝒹(𝑢, 𝒷).  

Definition [13]:  Letᶃ :  ℋ → ℋ be a mappings, {𝒦𝑛}𝑛=0
∞ ⊂ ℋ,and 𝜀𝑛 =

𝑑(𝒦𝑛+1, ᶂ (ᶃ, 𝒦𝑛 )),  𝑛 = 0,1,2, …  .Then  𝒦𝑛+1 = ᶂ (ᶃ, 𝒦𝑛 ) is said to be  𝑇-stable or stable with 

respect to ᶃ if and only if 



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286 
 
 

 lim
𝑛→∞

𝜀𝑛 = 0   implies  lim
𝑛→∞

𝒦𝑛 = 𝒫. 

Definition [14]: Let{ 𝒶n  }0
∞,  { 𝒷n  }0

∞ ∈ 𝑅     and converge to 𝒶 and 𝒷 a,respectively, and  

  𝑙𝑖𝑚
𝑛→∞

|𝒶𝑛−𝒶|

|𝒷𝑛−𝒷|
   = 𝓈, if 𝓈 =  0 , then { 𝒶n  }0

∞ ⇢ 𝒶 faster than{ 𝒷𝑛  }0
∞ ⇢ 𝒷and  if    0 ˂𝓈˂∞ , then it 

can be said that   𝒶𝑛  and { 𝒷𝑛   }0
∞  have the same rate of convergence.  

Lemma : [15]. If  0 ≤  𝒬 < 1 and {𝒩n}n
∞=0 is a positive R-sequence such that lim

n→∞
𝒩n =  0, then 

for any positive R-sequence {𝒽n}n
∞=0 satisfying 

𝒽𝑛+1 ≤  𝒬𝒽n + 𝒩n , n =  0,  1,  2,  .  .  .  ⟹ lim
n→∞

𝒽n  =  0. 

There are many studies on the iterations in other spaces see[18-21] 

Previous Results 

One of the most important previous results on this topic 

Theorem: In any metric space if 𝒥 satify the condition  

 𝒹(𝒶, 𝒥𝒷) +  𝒹(𝒷,  𝒥𝒷) ≤  𝑞𝒹(𝒶, 𝒷),                                                                                   (1) 

for all 𝒶, 𝒷 ∈  ℳ, where 2 ≤  q <  4. Then,𝒥has at least one fixed point. 

Theorem: Let 𝒥  be a mappingsatisfy  the condition  

𝒹(𝒥𝒶, 𝒥𝒷) +  𝒹(𝒶,  𝒥𝒷) +  𝒹(𝒷,  𝒥𝒷) ≤  𝑟𝒹(𝒶,  𝒷)     ∀ 𝒶,  𝒷 ∈  ℳ                                   (2) 

Then, 𝒥 has at least one f-point. 

Theorem: Consider a Com Con M-S. Suppose that  ᶂ , ᶃ are mappings of ℳ, and there exist 

ᶏ, ᶀ, ḉ, m as: 

2ᶀ –  |ḉ|  ≤  m <  2(ᶏ +  ᶀ +  ḉ) –  |ḉ|, 

ᶏ𝒹(ᶃ(x), ᶂ (x)) +  ᶀ𝒹(ᶃ(y), ᶂ(y)) +  ḉ𝒹(ᶂ (x), ᶂ(y) ) ≤  md(ᶃ(x), ᶃ(y)) 

then f has at least one f-point. 

In appreciably, a f-point iteration is  useful for applications if it satisfies the following 

requirements: 

(a) study data dependence results. 

(b) it converges to f- point. 

(c) it is 𝔍 -stable.  



IHJPAS. 36 (3) 2023 
 

287 
 
 

Theorem: Consider each of proximal processes Noor, Karhan and ModifiedSP.scheme  converge 

to 𝒷 ∈ 𝔍 where 𝔍 contraction map.Then the ModifiedSP.iteration  converges faster than Noor 

and Karhan  scheme. 

Theorem: Consider each of proximal processes Mann, Ishikawa  and Modified 𝑆𝑃-scheme 

converge to 𝒷 ∈ 𝔍 where 𝔍 contraction map.Then the ModifiedSP.iteration  converges faster 

than Mann, Ishikawa  scheme . 

Theorem:In a  Com Con M-S consider the  Mann proximal processes , converge to 𝒷 ∈ 𝔍 

where 𝔍 contraction map.Then the Mann scheme is T - stable  scheme . 

Theorem:In a  Com Con M-S consider the  Ishikawa proximal processes , converge to 𝒷 ∈ 𝔍 

where 𝔍 contraction map.Then the Ishikawa scheme is T - stable  scheme  

 

Theorem:In a H-S consider the  Ishikawa and Mann proximal processes such that converge it to 

𝒷 ∈ 𝔍 where 𝔍 quasi δ -contraction map.Then the Ishikawa scheme⇢ 𝒶 iff Mann scheme ⟶ 𝒶. 
 

Theorem:In a H-S consider the  Ishikawa and Modified𝒮𝒫.proximal processes such that converge 

it to 𝒷 ∈ 𝔍 where 𝔍 quasi δ -contraction map.Then Modified𝒮𝒫. iteration scheme⟶ 𝒶 iff Mann 
scheme ⇢ 𝒶. 
 

Theorem:In a H-S consider the  CR and Mann proximal processes such that converge it to 𝒷 ∈ 𝔍 
where 𝔍 quasi δ -contraction map.Then the CR a scheme⇢ 𝒶 iff Mann scheme ⇢ 𝒶. 
Theorem:In a H-S consider the  Noor and Mann proximal processes such that converge it to 𝒷 ∈ 𝔍 
where 𝔍 quasi δ -contraction map.Then the  Noor a scheme⟶ 𝒶 iff Mann scheme⟶ 𝒶. 
 

Conclusion 

A generalized review of contractionary mapping and non-expansion maps has been reviewed and 

some theories are recalled about the existence and uniqueness of the common fixed point and 

congruent fixed point of such maps under some conditions. Moreover, we also inferred the 

convergence and acceleration range of some schemes of different types such as one-step schemes, 

two-step schemes, and three-step schemes in convex metric space. 
 

 

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