Zenali Iteration Method For Approximating Fixed Point of A δZA − Quasi Contractive mappings

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, Diterations, and K*iteration for new contraction mappings called δZA − quasi contraction mappings. And we proved that all these iterations (Mann, Ishikawa, Noor, Diterations 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.


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][2][3][4][5][6][7][8][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] , *-iteration [14] , Noor iteration [15].
Let ℳ be a uniformly convex Banach space, ∅ ≠ be a closed-convex subset of ℳ. We recall some definitions of those iterations as: 1-Let < >, < > and < > are sequences lies in (0,1). The following iteration 〈d n 〉 is called -iteration and defined as follows: d 0 ∈ , s n = ( 1 − n )d n + n d n , t n = ( 1 − n ) d n + n s n , d n+1 = ( 1 − n ) s n + n t n .

Main Results
In this section, we introduced a new iteration process known as Zenali Iteration and new contraction mappings called a − quasi contraction mappings.
Definition 2.1: Let < >, < > and < > are sequences in (0,1) and : → . The following iteration is called Zenali iteration and defined as follows Proof. Let be a fixed point of . Then the following inequalities hold ).
Since║ -║ → 0 as → ∞ then, Thus From Eqs (2.6), (2.8), (2.13) and lemma(1.2 ) we obtain, Proof: Let be a c fixed point of. The following inequalities hold So, {║ − ║} is decreasing, for each ∈ ℱ( ), this implies that the sequence Proof. Since ∈ F( ), from(2.14) we get : Thus, 〈 〉 is bounded set in . Now, we prove that F( ) ≠ ∅. By the same proof way of the previous theorem. ∎ Now, we will study the equivalent between many of iterations by using a − quasi contraction mappings.
Theorem 2.7: Let closed a nonempty convex and subset of a Banach space , be a − quasi contraction mapping on and has a unique fixed point . Consider the Zenali iteration and Mann iteration with real sequences. Then the following a assertions are equivalent: The Mann iteration converges to .
The Zenali iteration converges to .
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 Now, consider Mann and the Zenali iterations, we have: Now, because of these results, we get → 0 By applying lemma( 1.3), we obtain = ║ − ║ → 0 as → 0.

Now, consider the following
Now, we will prove that our new iteration is faster than many know iterations By using new contraction mappings. Theorem 2.8: Let be a − quasi contraction mapping on . Suppose that the iterations Zenali iteration, Ishikawa iteration and Mann iteration converge to ∈ ℱ( ) where 0 < ≤ n ,

Consider the Mann iteration, we have
Here, after simple compute, we have Then, the Zenali iteration converges to faster than Ishikawa iteration and Mann iteration. ∎ Theorem 2.9 : Let be a − quasi contraction self-mapping on . Suppose that the Zenali iteration and D -iteration converge to the same fixed point of where 0 < ≤ , , < 1, ∀ ∈ . Then, the Zenali iteration converges faster than D -iteration.
Proof. Form D -iteration, we obtain So, the Zenali-Iteration converges faster than D -iteration. ∎ We proof other iterations by the same proof way of the previous Theorem.
Take < > =< > = < > = 3 4 , = 1 2 with initial value 30. In this section, a new iteration method for approximation of fixed points and a new contraction mappings called δ − quasi contraction mappings are introduced. Also, we proved that our iteration process is faster than the existing leading iterations like Mann, Ishikawa, Noor, Diterations and *-iteration and proved that all these iterations are equivalent to approximate fixed points of δ − quasi contraction.