^{1}

^{2}

^{1}

^{2}

The purpose of this paper is to prove strong convergence and

In the recent years, numerous papers have been published on the strong convergence and

Mann [

A double Mann iteration, called Ishikawa iteration, was introduced by Ishikawa [

The three-step iteration, which is more general than Mann and Ishikawa iterations, was defined by Noor [

Rhoades and Soltuz [

Another two-step scheme, which is independent of (

The three-step iteration of (

In [

Several results have been proved for the strong convergence of the explicit iteration as well as the SP-iterative scheme of fixed points for different types of contractive-like operators in various spaces. See [

Kirk’s iterative procedure was defined by Kirk [

In order to reduce the cost of computations, Olatinwo [

The Kirk-Noor iteration was introduced by Chugh and Kumar [

In an attempt to generalize (

The Kirk-SP iteration was defined by Hussain et al. [

The Kirk-Multistep-SP, which generalized both Kirk-SP and Kirk-Thianwan schemes, was introduced by Akewe et al. [

Another modified form of explicit iteration is the implicit iteration. The implicit Mann iteration and implicit Ishikawa iteration were discussed by Ćirić et al. [

The implicit Noor iteration was defined by Chugh et al. [

The most generalized Banach operator used by several authors is the one proved by Zamfirescu [

Let

The equivalence form of (

Berinde [

In [

The equivalence form of (

Let

Let

if

if

if

Let

Let

Note that

We present our main results as follows.

Let

If we let

If

If

If

If

Throughout, the operator

Let

Let

Therefore, implicit Kirk-Multistep scheme (

Let

The strong convergence results for implicit Noor, implicit Ishikawa, and implicit Mann schemes are obvious from Theorem

Let

Let

Therefore, iterative scheme (

Let

The stability results for implicit Mann, implicit Ishikawa, and implicit Noor schemes with contractive condition (

We compare our iterative schemes with others by using the following example.

Let

For the implicit Kirk-Mann iteration (IKM), we have

The implicit multistep Kirk iteration (IMK) converges faster than the implicit Kirk-Noor iteration (IKN) for

Also,

The implicit Kirk-Noor iteration

Similarly, using Definition

For the Kirk-Mann iteration (KM), we have the following estimate:

Again, using Definition

The implicit Kirk-Mann iteration

For the comparison of implicit Kirk-Ishikawa iteration

The implicit Kirk-Ishikawa iteration

For the comparison of implicit Kirk-Noor iteration

The implicit Kirk-Noor iteration

For the comparison of implicit Kirk-Multistep iteration

The implicit Kirk-Multistep iteration

We have established and proved strong convergence and

Authors agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

The authors hereby declare that there are no competing interests.