78 

 

 

  

 

 

 Zenali  Iteration Method For Approximating Fixed Point of A 

𝜹𝓩𝓐 − Quasi Contractive mappings 

Zena Hussein Maibed                                                          Ali Qasem Thajil  

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

University of Baghdad, Iraq.  

      mrs_zena.hussein@yahoo.com                 Ali.Qasem1203a@ihcoedu.uobaghdad.edu.iq    

  

 

                                                                                     

 

Abstract. 

     This article will introduce a new iteration method called the zenali iteration method for the 

approximation of fixed points. We show that our iteration process is faster than the current leading 

iterations  like Mann, Ishikawa, Noor, D-  iterations, and   𝒦*-  iteration for new contraction 

mappings called 𝛿𝒵𝒜 − quasi contraction mappings. And we  proved that all these iterations 

(Mann, Ishikawa, Noor, D-  iterations and   𝒦*-  iteration) equivalent to approximate fixed points of 

𝛿𝒵𝒜 − quasi contraction. We support our analytic proof by a numerical example, data dependence 

result for contraction mappings type 𝛿𝒵𝒜 by employing zenali iteration also discussed.  

Keywords:  Mann and 𝒦*-  iteration, 𝛿𝒵𝒜 − quasi contraction mappings. 

1. Introduction  

      The fixed point theory is one of the most important theories that play an important and 

fundamental role to solve many problems in various fields of science and knowledge such as 

Geometry, game theory, chemistry, etc. numerical calculation of fixed points for nonlinear operators 

is also an active research problem at present for nonlinear analysis due to its applications in balance 

problems, variable inequality, image coding, computer simulation and more. For that, many authors 

have created a large number of algorithms to approximate the fixed point for different types of 

applications for example see [1-9]. The well-known Banach contraction theorem uses the Picard 

iteration mechanism for fixed point approximation. This paper consists of three sections section one 

converges the zenali iteration with all these iterations. In section two rate of converge, section three 

equivalent, section four numerical example with real datasets. Many of the other well-known 

iterative methods are those of Mann [10], Ishikawa [11], D- iteration [12], Picard S iteration [13] , 

𝒦*-iteration [14] , Noor iteration  [15]. 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.4.2705 

Article history: Received 15 April 2021, Accepted 30 May 20 12 , Published in October 2021. 

 

mailto:mrs_zena.hussein@yahoo.com
mailto:Ali.Qasem1203a@ihcoedu.uobaghdad.edu.iq


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

79 

 

Let ℳ be a uniformly convex Banach  space,  ∅ ≠ 𝒞 be a closed-convex subset of ℳ. We recall 

some definitions of  those iterations as: 

1- Let  < 𝓈𝑛 >, < 𝓉𝑛 > and  < 𝓊𝑛 >   are sequences lies in (0,1). The following iteration 〈dn〉 is 

called 𝐷- iteration  and defined as follows: 

d0 ∈ 𝒞, sn = ( 1 −  𝓊n)dn +  𝓊n 𝒯dn , 

tn  = ( 1 −  𝓉n)𝒯dn +  𝓉n 𝒯sn,   dn+1  = ( 1 −  𝓈n)𝒯sn +  𝓈n 𝒯tn. 

2-  Let  𝓏𝑛 ∈  𝒞. The following iteration 〈𝓏𝑛 〉 is called 𝑃icard iteration and defined as follows:                                                                                                            

𝓏𝑛+1  =  𝒯𝓏𝑛  , 𝑛 𝜖 𝑁. 

3- Let < 𝓈𝑛 >  be a sequence in (0,1). The following iteration is called  Mann iteration and defined 

as follows:                                                                                      

𝑟0 𝜖 𝒞, 𝑟𝑛+1  =  (1 −  𝓈𝑛   )𝑟𝑛  +  𝓈𝑛 𝒯𝑟𝑛   , 𝑛 𝜖𝑁. 

4- Let  < 𝓈𝑛 >, < 𝓉𝑛 >   and < 𝓊𝑛 >   be real sequences in (0,1). The following iteration is called 

Ishikawa  iteration and defined as follows   

𝑤0 ∈ 𝒞, 𝑤𝑛+1 = ( 1 −  𝓈𝑛 )𝑤𝑛 +  𝓈𝑛  𝒯𝑑𝑛  , 

𝑑𝑛  = ( 1 −  𝓉𝑛 )𝑤𝑛 +  𝓉𝑛  𝒯𝑤𝑛  , 𝑛 ∈. 

5-  Let  < 𝓈𝑛 >  and < 𝓉𝑛 >   are sequences lies in (0,1). The following iteration is called Picard 𝒮- 

iteration and defined as follows: 

ℎ𝑛 ∈ 𝒞,   ℎ𝑛+1 = 𝒯𝑙𝑛 , 

𝑙𝑛  = ( 1 −  𝓈𝑛 )𝒯ℎ𝑛 +  𝓈𝑛 𝒯𝑒𝑛 ,  �̆�𝑛  = ( 1 −  𝓉𝑛 )ℎ𝑛 +  𝓉𝑛  𝒯ℎ𝑛. 

6- Let  < 𝓈𝑛 >, < 𝓉𝑛 > and  < 𝓊𝑛 >   are sequences in (0,1). The following iteration 〈𝑞𝑛 〉 is called 

𝒦*-iteration, and defined as follows:        𝑞0 ∈ 𝒞, 

𝑞𝑛+1 = 𝒯𝑝𝑛  , 𝑝𝑛  = 𝒯(( 1 −  𝓈𝑛 )𝑜𝑛 +  𝓈𝑛 𝒯𝑜𝑛 )    and   𝑜𝑛  = ( 1 −  𝓉𝑛 )𝑞𝑛 +  𝓉𝑛 𝒯𝑞𝑛. 

7- Let  < 𝓈𝑛 >, < 𝓉𝑛 > and  < 𝓊𝑛 >   are sequences in (0,1). The following iteration 〈yn〉 is called 

Noor iteration  and defined as follows:           

𝑦0 ∈ 𝒞, 𝑦𝑛+1 = ( 1 −  𝓈𝑛)𝑦𝑛 +  𝓈𝑛  𝒯𝑞𝑛, 

𝑞𝑛  = ( 1 −  𝓈𝑛)𝑦𝑛 +  𝓈𝑛  𝒯𝑝𝑛, 

𝑝𝑛  = ( 1 −  𝓊𝑛 )𝑦𝑛 +  𝓊𝑛  𝒯𝑦𝑛 , n ∈ 𝑁. 

Definition1.1 [16]: Let < 𝒱𝑛 >, < 𝒰𝑛 >  are sequences lies in R converge to 𝒱 and  < 𝒰𝑛 > 

converge  to 𝒰, and Let < 𝒱𝑛 >   such that      𝒵 = lim
𝑛→∞

|𝒱𝑛−𝒱|

|𝒰𝑛−𝒰|
 

1. If 𝒵= 0. Then  The sequence < 𝒱𝑛 > is converge to 𝒱 faster then     < 𝒰𝑛 > converge to 𝒰. 

 2. If    0 ≺ 𝒵 ≺ ∞  →    < 𝓍𝑛 >  and  < 𝒰𝑛 >  have the same rate of convergence. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

80 

 

Lemma 1.2 [17]: Let ℳ be a uniformly convex Banach  space and 〈ℐ𝑛〉𝑛=0
∞   be any sequence such 

that 0 <  𝓅 ≤  ℐ𝑛  ≤  𝓇 <  1, for some 𝓅 , 𝓇 ∈  𝑅 and for all 𝑛 ≥  1, Let 〈𝒱𝑛〉𝑛=0
∞  and 〈𝒰𝑛 〉𝑛=0

∞ , be 

a nonnegative real sequences of ℳ such that lim
𝑛→∞

𝑠𝑢𝑝 ║ 𝒱𝑛║ ≤ 𝓇, lim
𝑛→∞

𝑠𝑢𝑝 ║ 𝒰𝑛║ ≤ 𝓇 and  

𝑙𝑖𝑚
𝑛→∞

║ ℐ𝑛  𝒱𝑛– (1 −  ℐ𝑛)  𝒰𝑛║ = 𝑟  for some 𝓇 ≥  0. Then,  𝑙𝑖𝑚
𝑛→∞

║ 𝒱𝑛 –  𝒰𝑛║ = 0. 

Lemma 1.3  [18]:  Let 〈𝒱𝑛 〉𝑛=0
∞  and 〈𝒰𝑛 〉𝑛=0

∞  be nonnegative real sequences satisfying the following 

condition: 

 𝒱𝑛+1 ≤ (1 −  𝑛 )𝒱𝑛 +  𝒰𝑛, where 𝑛 (0,1), for all  𝑛 ≥ 𝑛0,  ∑ 𝑛 =  ∞
∞
𝑛=1   

and  
𝒰𝑛  

𝑛  
    0 as n  . Then 𝑙𝑖𝑚

𝑛→∞
𝒱𝑛 = 0. 

2. Main Results  

 In this section, we introduced a new iteration process known as Zenali  Iteration  

and new contraction mappings called a  𝛿𝒵𝒜 − quasi contraction mappings. 

 

   Definition 2.1:  Let < 𝓈𝑛 >, < 𝓉𝑛 >  and < 𝓊𝑛 >   are sequences in (0,1) and 𝒯: 𝒞 → 𝒞. The 

following iteration is called  Zenali  iteration and defined as follows 

                                                 𝑥0 ∈ 𝒞,            𝑥𝑛+1 = 𝒯𝑦𝑛 

𝑦𝑛  = 𝒯(( 1 −  𝓈𝑛)𝑧𝑛 +  𝓈𝑛 𝒯𝑧𝑛), 

 𝑧𝑛  = 𝒯(( 1 −  𝓉𝑛 )𝑥𝑛 +  𝓉𝑛  𝒯𝑥𝑛 ). 

Definition 2.2:  Let 𝒯 be a self mapping on 𝒞, then 𝒯 called a 𝛿𝒵𝒜 − quasi contraction for all 

𝓍 , 𝓎 ∈ 𝒞, if    || 𝒯𝓍 –  𝒯𝓎 ||  ≤ 𝛿|| 𝓍–  𝓎 ||   +  𝒵𝒜 (𝓃𝓍, 𝓂𝓎)          where 

𝒜(𝓃𝓍, 𝓂𝓎) = 𝑚𝑖𝑛 {𝓃║𝓍 − 𝒯𝓍║, 𝓂║𝓎 − 𝒯𝓎║, 𝓃𝓂║𝓍 − 𝒯𝓎║, 𝓂𝓃║𝓎 − 𝒯𝓍║} 

for some 0 < 𝛿 ≤ 1,   𝒵 ≥ 0 and   𝓃, 𝓂 ≥ 0 . 

Lemma 2.3 :  Let 𝒞 be a nonempty convex and closed subset of a Banach space ℳ and let  𝒯: 𝒞 → 𝒞 

a 𝛿𝒵𝒜 − quasi contraction mapping. Suppose that 〈𝑥𝑛 〉 the Zenali iteration in 𝒞. 

If  ℱ(𝒯 )  ≠ ∅, then 1-  ║𝑧𝑛 –  𝓅 ║ ≤ ║𝑥𝑛 –  𝓅 ║ and ║𝑦𝑛 –  𝓅 ║ ≤ ║𝑥𝑛 –  𝓅 ║. 

2- 𝑙𝑖𝑚
𝑛→∞

║𝑥𝑛–  𝓅 ║ exists, for all  n ∈ N. 

Proof.  Let 𝓅 be a fixed point of 𝒯. Then the following inequalities hold 

║𝑧𝑛 –  𝓅 ║ = ║𝒯[(1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛]–  𝓅 ║   

            ≤ 𝛿║(1 −  𝓉𝑛  )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛–  𝓅 ║ + 𝒵𝒜 (𝓃[(1 −  𝓉𝑛  )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛], 𝓂 𝓅). 

Since ║𝒯𝓅–  𝓅 ║ → 0 as 𝑛 → ∞ then, 𝒵𝒜 (𝓃[(1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛], 𝓂 𝓅) = 0  

║𝑧𝑛 –  𝓅 ║ ≤ (1 −  𝓉𝑛 )║𝑥𝑛–  𝓅║ + 𝛿𝓉𝑛║𝑥𝑛 –  𝓅 ║ + 𝓉𝑛 𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝓅) 



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                  ≤ (1 − 𝓉𝑛 (1 − 𝛿))║𝑥𝑛 –  𝓅 ║ +  𝒵 𝑚𝑖𝑛 {𝓃║𝑥𝑛–  𝒯𝑥𝑛║ 

                   , 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑥𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑥𝑛║}        

║𝑧𝑛 –  𝓅 ║  ≤ ║𝑥𝑛 –  𝓅 ║                                                                                                     (2.1) 

And,  

║𝑦𝑛 − 𝑝 ║  = ║𝒯[(1 −  𝓈𝑛  ) 𝑧𝑛 +  𝓈𝑛  𝒯𝑧𝑛] − 𝑝 ║  

                ≤ 𝛿 ║(1 −  𝓈𝑛   ) 𝑧𝑛 +  𝓈𝑛 𝒯𝑧𝑛 − 𝑝 ║ + 𝒵𝒜 (𝓃[(1 − 𝓈𝑛  ) 𝑧𝑛 +  𝓈𝑛 𝒯𝑧𝑛], 𝓂𝓅) 

Since║𝒯𝓅–  𝓅 ║ → 0 as 𝑛 → ∞. Then,         𝒵𝒜(𝓃[(1 −  𝓈𝑛   ) 𝑧𝑛 +  𝓈𝑛  𝒯𝑧𝑛 ], 𝓂𝓅) = 0 

║𝑦𝑛 − 𝑝 ║ ≤ (1 −  𝓈𝑛  )║𝑧𝑛 –  𝓅 ║ + 𝛿𝓈𝑛║𝑧𝑛 –  𝓅 ║ + 𝓈𝑛𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅) 

                   ≤ (1 − 𝓈𝑛 (1 − 𝛿))║𝑧𝑛 –  𝓅 ║ +  𝒵𝑚𝑖𝑛 {𝓃║𝑧𝑛–  𝒯𝑧𝑛║ 

                     , 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑧𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑧𝑛║}            

║𝑦𝑛 − 𝑝 ║  ≤ ║𝑧𝑛 –  𝓅 ║.                                                                                                     (2.2) 

Using inequality (2.1) in (2.2), it follows that:  

║𝑦𝑛 –  𝓅 ║ ≤ ║𝑥𝑛 –  𝓅 ║                                                                                                       (2.3) 

And, ║𝑥𝑛+1 − 𝓅║ = ║𝒯𝑦𝑛 − 𝓅║ ≤  𝛿║𝑦𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑦𝑛 , 𝓂𝓅)      

                                   ≤  ║𝑦𝑛 − 𝓅║                                                                                         (2.4) 

Using inequality (2.3), inequality (2.4) becomes 

║𝑥𝑛+1 − 𝓅║ ≤  ║𝑥𝑛 − 𝓅║ for all   n ∈ N.                                                                         (2.5) 

So, {║𝑥𝑛 − 𝓅║} is decreasing, for each 𝓅 ∈ ℱ(𝒯 ), this implies that the sequence 

 𝑙𝑖𝑚
𝑛→∞

║𝑥𝑛 –  𝓅 ║ exists.                                                          ∎ 

Theorem 2.4 :   Let ℳ be a uniformly convex Banach space,  𝒞 be nonempty convex and closed 

subset of ℳ and let  𝒯: 𝒞 → 𝒞 a 𝛿𝒵𝒜 − quasi contraction mapping. Suppose that 〈𝑥𝑛 〉 the Zenali 

iteration in 𝒞. Then F(𝒯 ) ≠ ∅ if and only if  〈𝑥𝑛 〉 is bounded and 𝑙𝑖𝑚
𝑛→∞

║𝑥𝑛 –  𝒯𝑥𝑛  ║ = 0. 

Proof.  Since  𝓅 ∈ F(𝒯 ), from (2.5) we get  

║𝑥𝑛+1 − 𝓅║ ≤  ║𝑥𝑛 − 𝓅║ ≤ ⋯  ≤  ║𝑥0 − 𝓅║ for all n ∈ N.Thus, 〈𝑥𝑛 〉 is bounded set in 𝒞.  

Put,                            𝑟 = 𝑙𝑖𝑚
𝑛→∞

║𝑥𝑛 –  𝓅 ║                                                                             (2.6) 

And,  𝑙𝑖𝑚
𝑛→∞

║𝒯𝑥𝑛–  𝓅 ║ ≤ 𝑙𝑖𝑚
𝑛→∞

( 𝛿║𝑥𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝓅)) 

                                      ≤ 𝑙𝑖𝑚
𝑛→∞

( ║𝑥𝑛 − 𝓅║ +  𝒵𝑚𝑖𝑛 {𝓃║𝑥𝑛–  𝒯𝑥𝑛║ 

                                       , 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑥𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑥𝑛║}) 



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          𝑙𝑖𝑚
𝑛→∞

║𝒯𝑥𝑛 –  𝓅 ║ ≤ 𝑙𝑖𝑚
𝑛→∞

 ║𝑥𝑛 − 𝓅║                                                                           (2.7)  

Using inequality (2.6) in (2.7) becomes, 𝑙𝑖𝑚
𝑛→∞

║𝒯𝑥𝑛–  𝓅 ║ ≤ 𝑟                                         (2.8) 

From Equations(2.3) and (2.6), we have 

       𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓║y𝑛 –  𝓅 ║ ≤ r.                                                                                                  (2.9) 

Similarly by using (2.1) and (2.6),we have  

       𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝║z𝑛–  𝓅 ║ ≤ r                                                                                                (2.10) 

Now,                          𝑟 = 𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓║𝑥𝑛+1–  𝓅 ║= 𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓║𝒯y𝑛–  𝓅 ║ 

                                       ≤ 𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓( 𝛿║𝑦𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑦𝑛 , 𝓂𝓅)) 

                                       ≤ 𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓( ║𝑦𝑛 − 𝓅║ +  𝒵𝑚𝑖𝑛 {𝓃║𝑦𝑛 –  𝒯𝑦𝑛║ 

                                         , 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑦𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑦𝑛║}) 

Then, we get             𝑟  ≤  𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓║𝑦𝑛–  𝓅 ║.                                                                  (2.11) 

Having in mind (2.2), inequality (2.11) becomes,  𝑟  ≤  𝑙𝑖𝑚
𝑛→∞

𝑖𝑛𝑓║𝑧𝑛 –  𝓅 ║                  (2.12) 

From Eqs (2.10) and (2.12), we obtain  

                                   𝑟 =  𝑙𝑖𝑚
𝑛→∞

║𝑧𝑛 –  𝓅 ║ =  𝑙𝑖𝑚
𝑛→∞

(║𝒯((1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛 )–  𝓅 ║ ) 

                                       ≤ 𝛿 𝑙𝑖𝑚
𝑛→∞

║(1 −  𝓉𝑛   )(𝑥𝑛 −  𝓅  ) −  𝓉𝑛 ( 𝒯𝑥𝑛 –  𝓅) ║ 

                                        +𝒵𝒜 (𝓃[(1 −  𝓉𝑛  )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛] , 𝓂 𝓅). 

Since║𝒯𝓅–  𝓅 ║ → 0 as 𝑛 → ∞ then, 𝒵𝒜 (𝓃[(1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛] , 𝓂 𝓅)  = 0   

                                        ≤ 𝑙𝑖𝑚
𝑛→∞

 𝛿(1 − 𝓉𝑛 (1 − 𝛿))║𝑥𝑛 –  𝓅 ║ +  𝒵min {𝓃║𝑥𝑛–  𝒯𝑥𝑛║ 

                                           , 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑥𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑥𝑛║} 

                                        ≤ 𝑙𝑖𝑚
𝑛→∞

║𝑥𝑛–  𝓅 ║ = 𝑟  

So,                               𝑟 ≤ ║(1 −  𝓉𝑛   )(𝑥𝑛 −  𝓅  ) −  𝓉𝑛 ( 𝒯𝑥𝑛 –  𝓅) ║ ≤ 𝑟  

Then  ║(1 −  𝓉𝑛   )(𝑥𝑛 −  𝓅  ) −  𝓉𝑛 ( 𝒯𝑥𝑛–  𝓅)║ = 𝑟                                                       (2.13) 

Thus From Eqs (2.6), (2.8),  (2.13) and lemma(1.2 ) we obtain,  𝑙𝑖𝑚
𝑛→∞

║𝑥𝑛 – 𝒯𝑥𝑛  ║ = 0.    

Now, we prove that F(𝒯 ) ≠ ∅ 

Let 𝓅 ∈ A(𝒞 , 〈𝑥𝑛 〉 )  ⇒  r(𝒞 , 〈𝑥𝑛 〉 ) = r(𝓅 , 〈𝑥𝑛 〉 )         



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

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            r(𝒯𝓅 , 〈𝑥𝑛 〉 ) =   𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝 ║𝑥𝑛 –  𝒯𝓅║ = 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝[║𝑥𝑛 –  𝒯𝑥𝑛  ║ + ║𝒯𝑥𝑛 –  𝒯𝓅 ║]             

                       ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝[𝛿║𝑥𝑛 –  𝒯𝑥𝑛  ║ + 𝒵𝒜(𝓃𝑥𝑛 , 𝓂𝓅)  

                       ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝║𝑥𝑛–  𝓅 ║ =  r(𝓅 , 〈𝑥𝑛 〉 ) = r(𝒞 , 〈𝑥𝑛 〉 )                             

𝒯𝓅 ∈ A(𝒞 , 〈𝑥𝑛 〉 )   C a uniformly convex ⇒ A(C , 〈𝑥𝑛 〉 ) is a singleton 

           ⇒ 𝓅 =  𝒯𝓅 ⇒ 𝓅 ∈  F(𝒯 )  ⇒ F(𝒯 ) ≠ ∅.                                               ∎       

Lemma 2.5:    Let 𝒞 be a nonempty convex and closed subset of a Banach space ℳ and let  𝒯: 𝒞 →

𝒞 a 𝛿𝒵𝒜 − quasi contraction mapping. Suppose that 〈𝑟𝑛〉 the Mann iteration in 𝒞 If ℱ(𝒯 )  ≠ ∅, 

then 𝑙𝑖𝑚
𝑛→∞

║𝑟𝑛–  𝓅 ║ exists. 

 Proof:   Let 𝓅 be a c fixed point of. The following inequalities hold 

║𝑟𝑛+1 − 𝓅 ║ = ║(1 −  𝑎𝑛   )𝑟𝑛  +  𝑎𝑛  𝒯𝑟𝑛 − 𝓅║ 

                        ≤  (1 −  𝑎𝑛  )║𝑟𝑛 − 𝓅║ +  𝑎𝑛║𝒯𝑟𝑛 − 𝓅║ 

                        ≤  (1 −  𝑎𝑛  )║𝑟𝑛 − 𝓅║ +  𝛿𝑎𝑛║𝑟𝑛 − 𝓅║ + 𝑎𝑛 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝓅) 

                        ≤  (1 − 𝑎𝑛 (1 − 𝛿)║𝑟𝑛 − 𝓅║ 

║𝑟𝑛+1 − 𝓅 ║ ≤ ║𝑟𝑛 − 𝓅║                                                                                                 (2.14) 

So, {║𝑟𝑛 − 𝓅║} is decreasing, for each 𝓅 ∈ ℱ(𝒯 ), this implies that the sequence 𝑙𝑖𝑚
𝑛→∞

║𝑟𝑛 –  𝓅 ║ 

exists.                                                                ∎ 

Theorem 2.6: Let ℳ be a uniformly convex Banach space,  𝒞 be  nonempty convex and closed 

subset of  ℳ and let  𝒯: 𝒞 → 𝒞 a 𝛿𝒵𝒜 − quasi contraction mapping. Suppose that 〈𝑟𝑛〉 the Mann 

iteration in 𝒞. Then F(𝒯) ≠ ∅ if and only if 〈𝑟𝑛〉 is bounded and 𝑙𝑖𝑚
𝑛→∞

║𝑟𝑛–  𝒯𝑟𝑛║ = 0. 

Proof. Since 𝓅 ∈ F(𝒯 ), from(2.14) we get : 

║𝑟𝑛+1 − 𝓅║ ≤  ║𝑟𝑛 − 𝓅║ ≤ ⋯  ≤  ║𝑟0 − 𝓅║ for all n ∈ N. 

Thus, 〈𝑟𝑛 〉 is bounded set in 𝒞.   

Put,                        𝑟 = 𝑙𝑖𝑚
𝑛→∞

║𝑟𝑛 –  𝓅 ║                                                                               (2.15) 

𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝║𝒯𝑟𝑛 –  𝓅 ║ ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝(𝛿║𝑟𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝓅)) 

                                  ≤ 𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝║𝑟𝑛–  𝓅 ║                                                                        (2.16) 

Having in mind (2.15), inequality(2.16)becomes: 

𝑙𝑖𝑚
𝑛→∞

𝑠𝑢𝑝║𝒯𝑟𝑛 –  𝓅 ║ ≤ 𝑟                                                                                                     (2.17) 

 Now,    𝑟 = 𝑙𝑖𝑚
𝑛→∞

║𝑟𝑛+1–  𝓅 ║ = 𝑙𝑖𝑚
𝑛→∞

║(1 −  𝓈𝑛  )𝑟𝑛  +  𝓈𝑛 𝒯𝑟𝑛 − 𝓅║  



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                 = 𝑙𝑖𝑚
𝑛→∞

║(1 −  𝓈𝑛  )(𝑟𝑛 −  𝓅  ) +  𝓈𝑛( 𝒯𝑟𝑛 − 𝓅)║                                            (2.18) 

Thus From Eqs (2.15), (2.17),  (2.18) and lemma(1.2) we obtain,  𝑙𝑖𝑚
𝑛→∞

║𝑟𝑛–  𝒯𝑟𝑛║ = 0. 

Now, we prove that F(𝒯 ) ≠ ∅. By  the same proof  way of the previous theorem.              ∎ 

Now, we will study the  equivalent between many of iterations by using a  𝛿𝒵𝒜 − quasi contraction 

mappings.   

Theorem 2.7: Let  𝒞  closed  a nonempty convex and subset of a Banach space 𝑋,  𝒯 be a 𝛿𝒵𝒜 − 

quasi contraction mapping on 𝒞  and has a unique fixed point  𝓅. Consider  the Zenali iteration and 

Mann iteration with real sequences. Then the following a assertions are equivalent: 

The Mann iteration converges to  𝓅 . 

The Zenali iteration converges to  𝓅.  

Proof. We show that (i) → (ii) that is, if the Mann iteration converges, then the Zenali iteration does 

too. Since, the Mann iteration converges to  𝓅 

⟹ ║𝑟𝑛 − 𝓅║ → 0 𝑎𝑠 𝑛 ⟶ ∞. 

Now, consider Mann and the Zenali iterations, we have: 

 ║𝑟𝑛+1 − 𝑥𝑛+1 ║ = ║(1 −  𝓈𝑛   )𝑟𝑛  +  𝓈𝑛 𝒯𝑟𝑛 −   𝒯𝑦𝑛║ 

                ≤  (1 −  𝓈𝑛  )║𝑟𝑛 − 𝒯𝑦𝑛║ +  𝛿𝓈𝑛║𝑟𝑛 − 𝑦𝑛║ + 𝓈𝑛𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑦𝑛  ) 

                              ≤  (1 −  𝓈𝑛  )║𝑟𝑛 − 𝒯𝑟𝑛║ +  𝛿(1 −  𝓈𝑛  )║𝑟𝑛 − 𝑦𝑛║ 

                               +(1 −  𝓈𝑛  )𝐿𝓂 (𝑟𝑛 , 𝑦𝑛 ) +  𝛿𝓈𝑛║𝑟𝑛 − 𝑦𝑛║ + 𝓈𝑛𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑦𝑛  )             

                              =  (1 −  𝓈𝑛 )║𝑟𝑛 − 𝒯𝑟𝑛║ +  𝛿║𝑟𝑛 − 𝑦𝑛║ + 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑦𝑛  )           (2.19) 

║𝑟𝑛 − 𝑦𝑛  ║  = ║𝑟𝑛 − 𝒯[(1 −  𝓈𝑛  )𝑧𝑛 −  𝓈𝑛  𝒯𝑧𝑛]║ 

                     ≤ ║𝑟𝑛 − 𝒯𝑟𝑛║ + ║𝒯𝑟𝑛 − 𝒯[(1 −  𝓈𝑛   )𝑧𝑛 −  𝓈𝑛 𝒯𝑧𝑛]║ 

                     ≤ ║𝑟𝑛 − 𝒯𝑟𝑛║ + 𝛿║𝑟𝑛 − (1 −  𝓈𝑛  )𝑧𝑛 −  𝓈𝑛  𝒯𝑧𝑛║ 

                       +𝒵𝒜 (𝓃𝑟𝑛, 𝓂[(1 − 𝓈𝑛   ) 𝑧𝑛 +  𝓈𝑛 𝒯𝑧𝑛]) 

                     ≤  ║𝑟𝑛 − 𝒯𝑟𝑛║ +  𝛿(1 −  𝓈𝑛 )║𝑟𝑛 − 𝑧𝑛║ + 𝛿𝓈𝑛║𝑟𝑛 − 𝒯𝑟𝑛║ 

+𝛿 2𝓈𝑛║𝑟𝑛 − 𝑧𝑛║ + 𝛿𝓈𝑛𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝑧𝑛 ) + 𝒵𝒜 (𝓃𝑟𝑛 , 𝓂[(1 − 𝓈𝑛 ) 𝑧𝑛 + 𝓈𝑛 𝒯𝑧𝑛]) 

                     = (1 + 𝛿 𝓈𝑛  ) ║𝑟𝑛 − 𝒯𝑟𝑛║ + ( 𝛿(1 −  𝓈𝑛 ) + 𝛿
2𝓈𝑛)║𝑟𝑛 − 𝑧𝑛║ 

                       +𝛿𝓈𝑛 𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝑧𝑛  ) + 𝒵𝒜 (𝓃𝑟𝑛 , 𝓂[(1 − 𝓈𝑛  ) 𝑧𝑛 +  𝓈𝑛  𝒯𝑧𝑛])           (2.20) 

║𝑟𝑛 − 𝑧𝑛║ = ║𝑟𝑛 − 𝒯[(1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛 ]║ 

                   ≤ ║𝑟𝑛 − 𝒯𝑟𝑛║ + ║𝒯𝑟𝑛 − 𝒯[(1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛 ]║ 



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                   ≤ ║𝑟𝑛 − 𝒯𝑟𝑛║ + 𝛿║𝑟𝑛 − (1 −  𝓉𝑛   )𝑥𝑛 −  𝓉𝑛  𝒯𝑥𝑛║ + 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑥𝑛  ) 

                   ≤  ║𝑟𝑛 − 𝒯𝑟𝑛║ +  𝛿(1 −  𝓉𝑛  )║𝑟𝑛 − 𝑥𝑛║ + 𝛿𝓉𝑛║𝑟𝑛 − 𝒯𝑥𝑛║ + 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑥𝑛  )     

                   ≤  ║𝑟𝑛 − 𝒯𝑟𝑛║ +  𝛿(1 −  𝓉𝑛  )║𝑟𝑛 − 𝑥𝑛║ + 𝛿𝓉𝑛║𝑟𝑛 − 𝒯𝑟𝑛║ 

                    +𝛿 2𝓉𝑛║𝑟𝑛 − 𝑥𝑛║ + 𝛿𝓉𝑛 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑥𝑛  ) + 𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑥𝑛  ) 

                   = (1 + 𝛿 𝓉𝑛  ) ║𝑟𝑛 − 𝒯𝑟𝑛║ + ( 𝛿(1 −  𝓉𝑛  ) + 𝛿
2𝓉𝑛)║𝑟𝑛 − 𝑥𝑛║ 

                    +(1 + 𝛿𝓉𝑛  )𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝑥𝑛  )                                                                        (2.21) 

Substituting (2.21) in (2.20), we obtain  

║𝑟𝑛 − 𝑦𝑛  ║ ≤ (1 + 𝛿 𝓈𝑛  ) ║𝑟𝑛 − 𝒯𝑟𝑛║ + ( 𝛿(1 −  𝓈𝑛 ) + 𝛿
2𝓈𝑛) (1 + 𝛿 𝓉𝑛  ) ║𝑟𝑛 − 𝒯𝑟𝑛║ 

                       +( 𝛿(1 −  𝓈𝑛 ) + 𝛿
2𝑎𝑛)( 𝛿(1 −  𝓉𝑛 ) + 𝛿

2𝓉𝑛)║𝑟𝑛 − 𝑥𝑛║ 

                       +( 𝛿(1 −  𝓈𝑛 ) + 𝛿
2𝓈𝑛)(1 + 𝛿 𝓉𝑛 )𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑥𝑛  ) 

                       +(1 + 𝛿 𝓈𝑛 )𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑧𝑛  )                                                                    (2.22) 

Substituting (2.22) in (2.19), we obtain  

║𝑟𝑛+1 − 𝑥𝑛+1 ║ ≤ (1 −  𝓈𝑛  )║𝑟𝑛 − 𝒯𝑟𝑛║ + 𝛿(1 + 𝛿 𝓈𝑛 ) ║𝑟𝑛 − 𝒯𝑟𝑛║ 

                              +𝛿( 𝛿(1 − 𝓈𝑛  ) + 𝛿
2𝓈𝑛 ) (1 + 𝛿 𝓉𝑛  ) ║𝑟𝑛 − 𝒯𝑟𝑛║ 

                              +𝛿 2(1 − 𝓈𝑛 (1 − 𝛿))(1 − 𝓉𝑛(1 − 𝛿))║𝑟𝑛 − 𝑥𝑛║ 

                              +𝛿( 𝛿(1 −  𝓈𝑛  ) + 𝛿
2𝓈𝑛)(1 + 𝛿 𝓉𝑛  )𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝑥𝑛  ) 

                              +𝛿(1 + 𝛿 𝓈𝑛 )𝒵𝒜 (𝓃𝑟𝑛, 𝓂𝑧𝑛  ) + 𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝑦𝑛  ) 

                        ≤ [(1 − 𝓈𝑛 ) + 𝛿(1 + 𝛿 𝓈𝑛 ) + 𝛿( 𝛿(1 −  𝓈𝑛 ) + 𝛿
2𝓈𝑛)(1 + 𝛿𝓉𝑛 )] ║𝑟𝑛 − 𝒯𝑟𝑛║ 

                             +( (1 − 𝓈𝑛 (1 −  𝛿)║𝑟𝑛 − 𝑥𝑛║ + 𝛿( 𝛿(1 −  𝓈𝑛  ) + 𝛿
2𝓈𝑛)(1 + 𝛿𝓉𝑛  ) 

                            𝒵 𝑚𝑖𝑛 {𝓃║𝑟𝑛 − 𝒯𝑟𝑛║, 𝓂║𝑥𝑛 − 𝒯𝑥𝑛║, 𝓃𝓂║𝑟𝑛 − 𝒯𝑥𝑛║, 𝓂𝓃║𝑥𝑛 − 𝒯𝑟𝑛║} 

                              +𝛿(1 + 𝛿 𝓈𝑛 )𝒵 𝑚𝑖𝑛 {𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝑧𝑛 −  𝒯𝑧𝑛║, 𝓃𝓂║𝑟𝑛 − 𝒯𝑧𝑛║ 

                               , 𝓂𝓃║𝑧𝑛 −  𝒯𝑟𝑛║} + 𝒵 𝑚𝑖𝑛{𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝑦𝑛 −  𝒯𝑦𝑛║ 

                               , 𝓃𝓂║𝑟𝑛 − 𝒯𝑦𝑛║, 𝓂𝓃║𝑦𝑛 −  𝒯𝑟𝑛║} 

Let   𝜇𝑛 = 𝓈𝑛(1 − 𝛿) 𝜖 (0 , 1),             𝒱𝑛 = ║𝑟𝑛 − 𝑥𝑛║ 

𝒰𝑛 = [(1 − 𝓈𝑛 ) + 𝛿(1 + 𝛿𝓈𝑛) + 𝛿(𝛿(1 − 𝓈𝑛 ) + 𝛿
2𝓈𝑛)(1 + 𝛿𝓉𝑛  )]║𝑟𝑛 − 𝒯𝑟𝑛║ 

       +𝛿( 𝛿(1 −  𝓈𝑛  ) + 𝛿
2𝓈𝑛 )(1 + 𝛿 𝓉𝑛  )𝒵 𝑚𝑖𝑛 {𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝑥𝑛 −  𝒯𝑥𝑛║ 

        , 𝓃𝓂║𝑟𝑛 − 𝒯𝑥𝑛║, 𝓂𝓃║𝑥𝑛 −  𝒯𝑟𝑛║} + 𝛿(1 + 𝛿 𝓈𝑛  )𝒵 𝑚𝑖𝑛 {𝓃║𝑟𝑛 −  𝒯𝑟𝑛║ 

       , 𝓂║𝑧𝑛 −  𝒯𝑧𝑛║, 𝓃𝓂║𝑟𝑛 − 𝒯𝑧𝑛║ , 𝓂𝓃║𝑧𝑛 −  𝒯𝑟𝑛║} 



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86 

 

       +𝒵 𝑚𝑖𝑛{𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝑦𝑛 −  𝒯𝑦𝑛║  , 𝓃𝓂║𝑟𝑛 − 𝒯𝑦𝑛║, 𝓂𝓃║𝑦𝑛 −  𝒯𝑟𝑛║}                            

Furthermore,  using  𝒯𝓅 = 𝓅  and  ║𝑟𝑛 − 𝓅║ → 0, we have 

║𝑟𝑛 − 𝒯𝑟𝑛 ║    = ║𝑟𝑛 − 𝓅 +  𝒯𝓅 −  𝒯𝑟𝑛║ 

                          ≤  ║𝑟𝑛 − 𝓅║ +  𝛿║𝑟𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝓅 ) 

                          =  (1 + 𝛿)║𝑟𝑛 − 𝓅║ +  𝒵𝑚𝑖𝑛 {𝓃║𝑟𝑛 −  𝒯𝑟𝑛║ 

                            , 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑟𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑟𝑛║} 

Then, ║𝑟𝑛 − 𝒯𝑟𝑛  ║ → 0. 

Now,  because of these results, we get 𝒰𝑛 → 0  

By applying lemma( 1.3), we obtain 𝒱𝑛 = ║𝑟𝑛 − 𝑥𝑛║  → 0   as 𝑛  → 0. 

Consequently , ║𝑟𝑛+1 − 𝑥𝑛+1║  → 0   as 𝑛  → 0. Therefore,    

║𝑥𝑛 − 𝓅║ ≤ ║𝑟𝑛 − 𝑥𝑛  ║ + ║𝑟𝑛 − 𝓅║ → 0   as 𝑛  → ∞   

Now, we show that,  (ii) ⇒ (i). 

 Since, the Zenali iteration converges to  𝓅 ⟹ ║𝑥𝑛 − 𝓅║ → 0 𝑎𝑠 𝑛 ⟶ ∞. 

Now, consider the following 

║𝑥𝑛+1 − 𝑟𝑛+1 ║ ≤  ║𝑥𝑛+1 − 𝓅║ +  ║𝑟𝑛+1 − 𝓅║ 

                            = ║𝒯𝑦𝑛 − 𝓅║ + ║(1 −  𝓈𝑛  )𝑟𝑛  +  𝓈𝑛  𝒯𝑟𝑛 − 𝓅║  

                            ≤  𝛿║𝑦𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑦𝑛 , 𝓂𝓅 )  +  (1 −  𝓈𝑛  )║𝑟𝑛 − 𝓅║     

                               +𝛿𝓈𝑛║𝑟𝑛 − 𝓅║ + 𝓈𝑛𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝓅 ) 

                             =  𝛿║𝑦𝑛 − 𝓅║ + 𝒵𝒜 (𝓃𝑦𝑛 , 𝓂𝓅 ) + (1 − 𝓈𝑛 (1 − 𝛿))║𝑟𝑛 − 𝓅║ 

                               +𝓈𝑛𝐿𝓂 (𝑟𝑛 , 𝓅)                                                                                    (2.23) 

║𝑦𝑛 − 𝑝 ║  = ║𝒯[(1 −  𝓈𝑛  ) 𝑧𝑛 +  𝓈𝑛  𝒯𝑧𝑛] − 𝑝 ║  

                    ≤ 𝛿 ║(1 −  𝓈𝑛   ) 𝑧𝑛 +  𝓈𝑛  𝒯𝑧𝑛 − 𝑝 ║ + 𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅 ) 

≤ 𝛿 (1 − 𝓈𝑛 )║𝑧𝑛 − 𝓅║ + 𝛿
2𝓈𝑛║𝑧𝑛 − 𝓅║ + 𝛿𝓈𝑛 𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅 ) + 𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅 ) 

                    ≤ 𝛿 (1 − 𝓈𝑛(1 − 𝛿))║𝑧𝑛 − 𝓅║ + (𝛿𝓈𝑛 + 1)𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅 )                   (2.24) 

║𝑧𝑛 − 𝑝 ║  = ║𝒯[(1 −  𝓉𝑛   ) 𝑥𝑛 +  𝓉𝑛  𝒯𝑥𝑛] − 𝑝 ║  

                   ≤ 𝛿 ║(1 −  𝓉𝑛   ) 𝑥𝑛 +  𝓉𝑛  𝒯𝑥𝑛 − 𝑝 ║ + 𝒵𝒜 (𝓃((1 −  𝓉𝑛   ) 𝑥𝑛 +  𝓉𝑛  𝒯𝑥𝑛), 𝓂𝓅) 

                   ≤ 𝛿 (1 −  𝓉𝑛  )║𝑥𝑛 − 𝓅║ +  𝛿
2𝓉𝑛║𝑥𝑛 − 𝓅║ + 𝛿𝓉𝑛𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝓅) 

                    ≤ 𝛿 (1 − 𝓉𝑛 (1 − 𝛿))║𝑥𝑛 − 𝓅║ + (𝛿𝓉𝑛 + 1) + 𝛿𝓉𝑛 𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝓅)         (2.25) 

Substituting (2.25) in (2.24), we obtain 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

87 

 

║𝑦𝑛 − 𝑝 ║ ≤  𝛿
2 (1 − 𝓈𝑛 (1 − 𝛿)) (1 − 𝓉𝑛 (1 − 𝛿))║𝑥𝑛 − 𝓅║ + 𝛿 (1 − 𝓈𝑛 (1 − 𝛿)) 

                        (𝛿𝓉𝑛 + 1)𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝓅) + (𝛿𝓈𝑛 + 1)𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅)                         (2.26)  

Substituting (2.26) in (2.23), we obtain 

║𝑥𝑛+1 − 𝑟𝑛+1 ║ ≤ 𝛿
3 (1 − 𝓈𝑛 (1 − 𝛿)) (1 − 𝓉𝑛 (1 − 𝛿))║𝑥𝑛 − 𝓅║ +  (1 − 𝓈𝑛(1 − 𝛿)) 

                               ║𝑟𝑛 − 𝑥𝑛 + 𝑥𝑛 − 𝓅║ + 𝛿
2 (1 − 𝓈𝑛(1 − 𝛿))(𝛿𝓉𝑛 + 1)𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝓅) 

                               +𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅) + 𝛿(𝛿𝓈𝑛 + 1)𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝓅) + 𝓈𝑛𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝓅) 

                          ≤ [(1 − 𝓈𝑛(1 − 𝛿)) + (1 − 𝓈𝑛(1 − 𝛿))]║𝑥𝑛 − 𝓅║ 

                               + (1 − 𝓈𝑛 (1 − 𝛿))║𝑥𝑛 − 𝑟𝑛║ + 𝛿
2 (1 − 𝓈𝑛 (1 − 𝛿))(𝛿𝓉𝑛 + 1) 

                             𝒵𝑚𝑖𝑛{𝓃║𝑥𝑛 − 𝒯𝑥𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑦𝑛 − 𝒯𝓅║ , 𝓂𝓃║𝓅 −  𝒯𝑦𝑛║} 

                          +𝒵min {𝓃║𝑦𝑛 −  𝒯𝑦𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑥𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑥𝑛║} 

                          +𝛿(𝛿𝓈𝑛 + 1)𝒵min {𝓃║𝑧𝑛 −  𝒯𝑧𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑧𝑛 − 𝒯𝓅║ 

                            , 𝓂𝓃║𝓅 −  𝒯𝑧𝑛║} + 𝓈𝑛𝒵𝑚𝑖𝑛{𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝓅 −  𝒯𝓅║ 

                             , 𝓃𝓂║𝑟𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑟𝑛║} 

Let   𝜇𝑛 = 𝓈𝑛(1 − 𝛿) 𝜖 (0 , 1),             𝒱𝑛 = ║𝑥𝑛 − 𝑟𝑛║ 

𝒰𝑛 = [(1 − 𝓈𝑛(1 − 𝛿)) + (1 − 𝓈𝑛(1 − 𝛿))]║𝑥𝑛 − 𝓅║ + 𝛿
2 (1 − 𝓈𝑛 (1 − 𝛿))(𝛿𝓉𝑛 + 1) 

        𝒵𝑚𝑖𝑛{𝓃║𝑥𝑛 −  𝒯𝑥𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑥𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑥𝑛║} 

       +𝒵𝑚𝑖𝑛{𝓃║𝑦𝑛 −  𝒯𝑦𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑦𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑦𝑛║} 

       +𝛿(𝛿𝓈𝑛 + 1)𝒵𝑚𝑖𝑛{𝓃║𝑧𝑛 −  𝒯𝑧𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑧𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑧𝑛║} 

       +𝓈𝑛𝒵𝑚𝑖𝑛{𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝓅 −  𝒯𝓅║, 𝓃𝓂║𝑟𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑟𝑛║} 

     Since, ║𝑥𝑛 − 𝓅║ → 0   as 𝑛  → ∞. So, we get 𝒰𝑛 → 0, thus from lemma (1.3), we get 

 𝒱𝑛 = ║𝑥𝑛 − 𝑟𝑛║ → 0 as 𝑛  → 0. Consequently;║𝑥𝑛+1 − 𝑟𝑛+1║ → 0  as 𝑛  → 0 

Therefore, ║𝑟𝑛 − 𝓅║ ≤ ║𝑟𝑛 − 𝑥𝑛  ║ + ║𝑥𝑛 − 𝓅║ → 0   as 𝑛  → ∞.                 ∎ 

Now, we will prove that our new iteration is faster than many know iterations  By using    new 

contraction mappings.  

Theorem 2.8: Let  𝒯 be a 𝛿𝒵𝒜 − quasi contraction mapping on 𝒞. Suppose that the iterations 

Zenali iteration, Ishikawa iteration and Mann iteration converge to 𝓅 ∈ ℱ(𝒯 ) where 0 < 𝓋 ≤ 𝓊n, 

𝓈n , 𝓉n  < 1, ∀n ∈ N. Then the Zenali iteration converges faster than of Mann iteration and 

Ihikawa iteration. 

Proof. Consider Zenali iteration, we obtain  



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

88 

 

║�̃�𝑛+1 − 𝑝 ║ = ║𝒯𝑦𝑛 − 𝑝 ║ ≤ 𝛿 ║𝑦𝑛 − 𝑝 ║ +  𝒵𝒜 (𝓃𝑦𝑛 , 𝓂𝑝) 

                        =  𝛿 ║𝒯 ((1 −  𝓈n )𝑧𝑛 +  𝓈n𝒯𝑧𝑛) − 𝑝 ║ 

             ≤ 𝛿 2║ ((1 −  𝓈n )𝑧𝑛 +  𝓈n𝒯𝑧𝑛 ) − 𝑝 ║ + 𝒵𝒜 (𝓃(1 −  𝓈n )𝑧𝑛 +  𝓈n𝒯𝑧𝑛 ), 𝓂𝑝) 

Since ║𝒯𝓅–  𝓅 ║ → 0 as 𝑛 → ∞ then, 𝒵𝒜 (𝓃(1 −  𝓈n )𝑧𝑛 +  𝓈n𝒯𝑧𝑛), 𝓂 𝓅) = 0    

             =  𝛿 2║ ((1 −  𝓈n )(𝑧𝑛 − 𝑝) +  𝓈n(𝒯𝑧𝑛 − 𝑝) ║ 

             ≤ 𝛿 2 [ (1 −  𝓈n )║𝑧𝑛 − 𝑝║ + 𝛿 𝓈n║𝑧𝑛 − 𝑝║ +  𝒵𝒜 (𝓃𝑧𝑛 , 𝓂𝑝)] 

            = 𝛿 2 [ ((1 −  𝓈n ) + 𝛿 𝓈n] ║𝑧𝑛 − 𝑝║ 

             = 𝛿 2 [ (1 − 𝓈n(1 − 𝛿) ] ║𝒯 ((1 −  𝓉n )𝑥𝑛 +  𝓉n𝒯𝑥𝑛) − 𝑝║ 

                        ≤ 𝛿 2[ (1 − 𝓈n(1 − 𝛿) ] 𝛿║ ((1 −  𝓉n)(𝑥𝑛 − 𝑝) +  𝓉n(𝒯𝑥𝑛 − 𝑝)║ 

                        +𝒵𝒜 (𝓃(1 −  𝓉n )𝑥𝑛 +  𝓉n𝒯𝑥𝑛), 𝓂𝑝) 

≤ 𝛿 3 [ (1 − 𝛼𝑛 (1 − 𝛿)] [ ((1 − 𝓉n)║𝑥𝑛 − 𝑝║ + 𝛿𝓉n║𝑥𝑛 − 𝑝║ + 𝒵𝒜 (𝓃𝑥𝑛 , 𝓂𝑝)] 

          =  𝛿 3 [ (1 − 𝓈n(1 − 𝛿) ] [ ((1 −  𝓉n ) +  𝛿𝓉n]║𝑥𝑛 − 𝑝║ 

          ≤ 𝛿 3 [ (1 − 𝓋(1 − 𝛿) ] 2║𝑥𝑛 − 𝑝║ 

                ⁞ 

          ≤ (𝛿 3 [ 1 − 𝓋(1 − 𝛿) ] 2)𝑛║𝑥0 − 𝑝║ 

Suppose that  𝑍𝐴𝑛 = (𝛿
3  [1 − 𝓋(1 − 𝛿) ] 2)𝑛║𝑥0 − 𝑝║ 

Consider the Mann iteration, we have 

║𝑟𝑛+1 − 𝑝 ║ = ║(1 −  𝓈𝑛  )𝑟𝑛  +  𝓈𝑛 𝒯𝑟𝑛 − 𝓅║ 

             = ║ ((1 −  𝓈𝑛  )(𝑟𝑛 − 𝓅) + 𝓈𝑛(𝒯𝑟𝑛 − 𝓅) ║ 

             ≤  (1 −  𝓈𝑛  )║𝑟𝑛 − 𝓅║ +  𝓈𝑛║𝒯𝑟𝑛 − 𝓅║ 

             ≤  (1 −  𝓈𝑛  )║𝑟𝑛 − 𝓅║ +  𝛿𝓈𝑛║𝑟𝑛 − 𝓅║ +  𝒵𝒜 (𝓃𝑟𝑛 , 𝓂𝑝) 

                       ≤ [1 − 𝓋(1 − 𝛿)]║𝑟𝑛 − 𝓅║ + 𝒵min {𝓃║𝑟𝑛 −  𝒯𝑟𝑛║, 𝓂║𝓅 −  𝒯𝓅║ 

                        , 𝓃𝓂║𝑟𝑛 − 𝒯𝓅║, 𝓂𝓃║𝓅 −  𝒯𝑟𝑛║} 

                         =[  1 − 𝓋(1 − 𝛿)]║𝑟𝑛 − 𝓅║    

                    ⁞ 

            ≤  [  1 − 𝓋(1 − 𝛿)]𝑛║𝑟0 − 𝓅║ 

Suppose that  𝒮𝑛 = [  1 − 𝓋(1 − 𝛿)]
𝑛║𝑟0 − 𝓅║ 

Here, after simple compute, we have   



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

89 

 

𝑍𝐴𝑛

ℳ𝑛
 = 

(𝛿3  [1−𝓋(1−𝛿) ] 2)𝑛║𝑥0−𝑝║

[ 1−𝓋(1−𝛿))]𝑛║𝑤0−𝓅║
 → 0 as   𝑛 →  ∞.                 

Then, the Zenali iteration converges to 𝓅 faster than Ishikawa iteration and Mann iteration.  ∎ 

Theorem 2.9 : Let   𝒯 be a 𝛿𝒵𝒜 − quasi contraction self- mapping on 𝒞. Suppose that the  Zenali 

iteration and D - iteration converge to the same fixed point 𝓅 of 𝒯 where 0 < 𝓋 ≤ 𝓊𝑛, 𝓈𝑛  , 𝓉𝑛  < 1, 

∀𝑛 ∈ 𝑁.  Then, the Zenali iteration converges faster than D - iteration. 

Proof. Form D - iteration, we obtain  

║d𝑛+1 − 𝓅 ║ = ║(1 −  𝓈𝑛   )𝒯𝑠𝑛  +  𝓈𝑛 𝒯𝑡𝑛 − 𝓅 ║ 

                         ≤  𝛿(1 −  𝓈𝑛  )║𝑠𝑛 − 𝓅 ║ + (1 −  𝓈𝑛 )𝒵𝒜 (𝓃𝑠𝑛 , 𝓂𝑝) 

                           +𝛿𝓈𝑛║𝑡𝑛 − 𝓅 ║ + 𝓈𝑛𝒵𝒜 (𝓃𝑡𝑛 , 𝓂𝑝) 

                         =  𝛿[(1 −  𝓈𝑛  )║𝑠𝑛 − 𝓅 ║ + 𝓈𝑛║(1 −  𝓉𝑛  )𝒯𝑑𝑛 +  𝓉𝑛 𝒯𝑠𝑛 − 𝓅 ║] 

                         ≤ 𝛿[(1 −  𝓈𝑛  )║𝑠𝑛 − 𝓅 ║ + 𝛿𝓈𝑛 ((1 − 𝓉𝑛 )║𝑑𝑛 − 𝓅 ║ 

                           + 𝛿𝓈𝑛 𝓉𝑛║𝑠𝑛 − 𝓅 ║)] +  𝒵𝒜 (𝓃𝑠𝑛, 𝓂𝑝) +  𝒵𝒜 (𝓃𝑑𝑛 , 𝓂𝑝) 

                         ≤  𝛿[(1 −  𝓈𝑛  ) + 𝛿𝓈𝑛  𝓉𝑛)║𝑠𝑛 − 𝓅 ║ + 𝛿 𝓈𝑛 (1 − 𝓉𝑛 )║𝑑𝑛 − 𝓅 ║] 

                         = 𝛿[((1 − 𝓈𝑛 (1 − 𝛿𝓉𝑛 ))║(1 −  𝓊𝑛  )𝑑𝑛 +  𝓊𝑛 𝒯𝑑𝑛 − 𝓅 ║ 

                            +𝛿𝓈𝑛 (1 − 𝓉𝑛)║𝑑𝑛 − 𝓅 ║] 

                            ≤  𝛿[((1 −  𝓈𝑛 (1 − 𝛿𝓉𝑛  ))((1 −  𝓊𝑛  )║𝑑𝑛 − 𝓅 ║ +  𝛿𝓊𝑛║𝑑𝑛 − 𝓅 ║) 

                            +𝛿𝓈𝑛 (1 − 𝓉𝑛)║𝑑𝑛 − 𝓅 ║ +  𝒵𝒜 (𝓃𝑑𝑛 , 𝓂𝑝)] 

                         ≤ 𝛿[((1 −  𝓈𝑛 (1 − 𝛿𝓉𝑛  ))((1 −  𝓊𝑛  ) + 𝓊𝑛 ))║𝑑𝑛 − 𝓅 ║ 

                             + 𝛿𝓈𝑛(1 − 𝓉𝑛 )║𝑑𝑛 − 𝓅 ║] 

                         ≤ 𝛿[((1 −  𝓈𝑛(1 − 𝛿𝓉𝑛 ))║𝑑𝑛 − 𝓅 ║ + 𝛿 𝓈𝑛 (1 − 𝓉𝑛 )║𝑑𝑛 − 𝓅 ║] 

                        =  𝛿[((1 −  𝓈𝑛(1 − 𝛿))║𝑑𝑛 − 𝓅 ║] 

                         ≤  𝛿((1 −  𝓋(1 − 𝛿))║𝑑𝑛 − 𝓅 ║] 

                                    ⁞ 

                         ≤ [ 𝛿((1 −  𝓋(1 − 𝛿))]𝑛║𝑑0 − 𝓅 ║ 

Let 𝐷𝑛 =    [ 𝛿((1 −  𝓋(1 − 𝛿))]
𝑛║𝑑0 − 𝓅 ║ 

Form Zenali-Iteration, we have,   𝑍𝐴𝑛 = (𝛿
3  [1 − 𝓋(1 − 𝛿) ] 2)𝑛║𝑥0 − 𝑝║ 

𝑍𝐴𝑛

𝐷𝑛
 = 

(𝛿3  [1−𝓋(1−𝛿) ] 2)𝑛║𝑥0−𝑝║

 [ 𝛿((1− 𝓋(1−𝛿))]𝑛║𝑑0−𝓅 ║
 → 0 as   𝑛 →  ∞. Thus < 𝑥𝑛 > converges to 𝓅 faster than < 𝑑𝑛 >. 

So, the Zenali-Iteration converges faster than D - iteration.            ∎ 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

90 

 

We proof other iterations by the same proof way of the previous Theorem. 

Example 2.10: Let 𝒩 =  𝑅 and  𝒞 =  [0,100].  and 𝒯 be a mapping on 𝒞 defined by �̃� =

√�̃�2 − 9�̃� + 54 , for all 𝑥 𝜖 𝒞, such that  𝒯 is a 𝛿𝒵𝒜 − quasi contraction 𝓂apping and unique fixed 

point say 𝓅 = 6. 

Take < 𝓈𝑛 > =< 𝓉𝑛 > =  < 𝓊𝑛 > =  
3

4
 ,𝑣 =  

1

2
  with initial value 30. 

Table 1. Comparison speed of convergence among various iteration methods. 

n Zenali K*          D- Ishikawa Mann 

1 30 30 30 30 30 

2 13.9156 17.1404 21.1334 25.0120 27.1150 

3 6.1717 7.9203 13.2989 20.2548 24.2908 

4 6.0006 6.0388 7.8776 15.8509 21.5421 

5 6.0000 6.0004 6.1725 12.0133 18.8893 

6  6.0000 6.0087 9.0688 16.3607 

7   6.0007 7.2820 13.9954 

8   6.0000 6.4668 11.8476 

9    6.1601 9.8476 

10    6.0537 8.4901 

11    6.0179 7.4083 

12    6.0060 6.7247 

13    6.0020 6.3468 

14    6.0007 6.1587 

15    6.0002 6.0709 

16    6.0001 6.0313 

17    6.0000 6.0137 

18     6.0011 

19     6.0005 

20     6.0001 

21     6.0000 

 

3.Conclusion 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

91 

 

      In this section, a new  iteration method for approximation of fixed points and a new contraction 

mappings called δ𝒵𝒜 − quasi contraction mappings are introduced. Also, we proved that our 

iteration process is faster than the existing leading iterations  like Mann, Ishikawa, Noor, D- 

 iterations and   𝒦*-  iteration and proved that all these iterations are equivalent to approximate fixed 

points of δ𝒵𝒜 − quasi contraction.   

 

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