The purpose of this paper is to introduce Kirktype new iterative schemes called KirkSP and KirkCR schemes and to study the convergence of these iterative schemes by employing certain quasicontractive operators. By taking an example, we will compare KirkSP, KirkCR, KirkMann, KirkIshikawa, and KirkNoor iterative schemes for aforementioned class of operators. Also, using computer programs in C++, we compare the abovementioned iterative schemes through examples of increasing, decreasing, sublinear, superlinear, and oscillatory functions.
There is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractivetype operator. Consequently, there is a theoretical and practical interest in approximating fixed points of various contractivetype operators. Let
The following iteration schemes are now well known:
In [
However, from [
Recently, Chugh and Kumar introduced the following iteration scheme [
(1) If
(2) If
(3) If
(4) If
In [
Let
The operators satisfying the condition (
Berinde in [
Let
However, in [
Also several authors [
Fixedpoint iterative schemes are designed to be applied in solving equations arising in physical formulation but there is no systematic study of numerical aspects of these iterative schemes. In computational mathematics, it is of vital interest to know which of the given iterative scheme converges faster to a desired solution, commonly known as rate of convergence. Rhoades in [
We will need the following lemmas and definition in the sequel.
If
Let
Suppose
Now, we define KirkSP and KirkCR iterative schemes as follows. Let
(5) Putting
(6) Putting
(7) Putting
(8) Putting
We now prove our main results.
Let
Using KirkSP iterative scheme (
Let
Using KirkCR iterative scheme (
Let
Using KirkNoor iterative scheme (
In [
Now, by providing Example
However, after interchanging the parameters the decreasing order of Kirktype iterative schemes is as follows: KirkCR, KirkSP, KirkNoor, KirkIshikawa, and KirkMann.
Let
It is clear that
Let
Therefore, by Definition
Similarly,
Again, similarly
Again, let
The following example shows comparison of simple iterative schemes with their corresponding Kirktype iterative schemes.
Let
KirkMann iterative scheme is faster than Mann iterative scheme,
KirkIshikawa iterative scheme is faster than Ishikawa iterative scheme,
KirkNoor iterative scheme is faster than Noor iterative scheme,
KirkSP iterative scheme is faster than SP iterative scheme,
KirkCR iterative scheme is faster than CR iterative scheme.
Let
It shows that KirkMann iterative scheme converges faster than Mann iterative scheme to the fixed point 0 of
Again, similarly,
Again,
Again,
In this section, with the help of computer programs in C++, we compare the rate of convergence of Kirktype iterative schemes, through examples. The outcome is listed in the form of Tables
Decreasing cum sublinear functions.
KirkCR iteration  KirkSP iteration  KirkNoor iteration  KirkMann iteration  KirkIshikawa iteration  












0  0.69857  0.710807  0.69857  0.723726  0.69857  0.710384  0.69857  0.710418  0.69857  0.698227 
1  0.800542  0.75984  0.78799  0.75565  0.800941  0.770859  0.800909  0.765142  0.812157  0.779333 
2  0.7492  0.754819  0.754002  0.754877  0.736164  0.752595  0.743004  0.75401  0.725717  0.750854 
3  0.754944  0.754878 


0.75745  0.754879  0.755859  0.754812  0.759396  0.754853 
4 




0.754877  0.754878  0.754952  0.754866  0.754905  0.754875 
5 






0.754891  0.754875  0.754881  0.754877 
6 






0.754881  0.754877 


7 






0.754879  0.754877 


8 










9 










10 










Increasing functions.
KirkCR iteration  KirkSP iteration  KirkNoor iteration  KirkMann iteration  KirkIshikawa iteration  












0  1.103589  1.116858  1.103589  1.17464  1.103589  1.451792  1.103589  1.241473  1.103589  1.365883 
1  1.142723  1.158453  1.165002  1.15863  1.300429  0.997618  1.193272  1.158697  1.253254  1.076407 
2  1.158561  1.158631 


1.102871  1.189602  1.158657  1.158641  1.128293  1.167741 
3  1.158631  1.15863 


1.171096  1.15929  1.158635  1.158633  1.162237  1.159321 
4 




1.15889  1.158731  1.158632  1.158631  1.158902  1.158763 
5 




1.15867  1.158655  1.158631  1.158631  1.158682  1.158667 
6 




1.15864  1.158638  1.158631  1.158631  1.158645  1.158643 
7 




1.158633  1.158633  1.158631  1.158631  1.158635  1.158635 
8 




1.158632  1.158632 


1.158632  1.158632 
9 




1.158631  1.158631 


1.158631  1.158631 
10 




1.158631  1.158631 


1.158631  1.158631 
11 




1.158631  1.158631 


1.158631  1.158631 
12 








1.158631  1.158631 
13 








1.15863  1.158631 
14 










15 










Functions with multiple zeros.
KirkCR iteration  KirkSP iteration  KirkNoor iteration  KirkMann iteration  KirkIshikawa iteration  












0  0.01  0.129763  0.01  0.181407  0.01  0.194268  0.01  0.0901  0.01  −0.04247 
1  0.757312  0.419604  0.670095  0.38479  0.649205  0.487941  0.827918  0.569045  1.086743  0.695256 
2  0.336859  0.382104  0.378484  0.381966  0.262204  0.363223  0.185722  0.402006  0.092869  0.341163 
3  0.381795  0.381966 


0.405485  0.381913  0.357596  0.385139  0.434066  0.381282 
4  0.381965  0.381966 


0.382031  0.38196  0.378054  0.382721  0.382812  0.381873 
5 




0.381974  0.381965  0.381033  0.382195  0.382081  0.381946 
6 




0.381968  0.381966  0.381683  0.382047  0.381991  0.38196 
7 






0.381866  0.381998  0.381973  0.381964 
8 






0.381926  0.38198  0.381968  0.381965 
9 






0.381949  0.381972  0.381967  0.381966 
10 






0.381958  0.381969 


11 






0.381962  0.381968 


12 






0.381964  0.381967 


13 






0.381965  0.381966 


14 






0.381965  0.381966 


15 










Superlinear functions.
KirkCR iteration  KirkSP iteration  KirkNoor iteration  KirkMann iteration  KirkIshikawa iteration  












0  0.988  0.994372  0.988  1.083583  0.988  1.092828  0.988  1.087853  0.988  1.093597 
1  0.999968  0.999996  0.994182  0.993672  0.992983  0.960331  0.993638  0.959083  0.992879  0.95975 
2 


0.999959  1.000023  0.998302  1.006115  0.998189  1.005282  0.998249  1.006211 
3 




0.999963  1  0.999972  0.999986  0.999962  1 
4 






1  0.999999 


5 










6 










7 










8 










9 










10 










Oscillatory functions.
KirkCR iteration  KirkSP iteration  KirkNoor iteration  KirkMann iteration  KirkIshikawa iteration  












0  0.25  0.25  0.25  0.25  0.25  0.25  0.25  0.25  0.25  0.25 
1  4  1.294298  4  0.966905  4  1.768962  4  2.90165  4  2.191915 
2  0.772619  0.995245  1.034228  1.000136  0.565303  0.889831  0.344631  1.425355  0.456222  0.833093 
3  1.004778  1  0.999864  1  1.123808  1  0.70158  1.063467  1.200346  1.000137 
4 






0.94032  1.008394  0.999863  1.000015 
5 






0.991676  1.001569  0.999985  1.000003 
6 






0.998434  1.000384  0.999997  1.000001 
7 






0.999616  1.000112  0.999999  1 
8 






0.999888  1.000037 


9 






0.999963  1.000014 


10 






0.999986  1.000005 


11 






0.999995  1.000002 


12 






0.999998  1.000001 


13 






0.999999  1 


14 










The function
Let
The function defined by
The function defined by
For detailed study, these programs are again executed after changing the parameters and some observations are made as given below.
The function defined by
(1) Taking initial guess
(2) Taking
(1) Taking initial guess
(2) Taking
(1) Taking initial guess
(2) Taking
(1) Taking initial guess
(2) Taking
(1) Taking initial guess
(2) Taking
The speed of iterative schemes depends on
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows:
KirkSP, KirkCR, KirkNoor, KirkIshikawa, and KirkMann.
(2) For initial guess away from the fixed point, KirkSP and KirkIshikawa iterative schemes show an increase while KirkCR, KirkNoor, and KirkMann iterative schemes show no change in the number of iterations to converge.
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows:
KirkSP, KirkCR, KirkMann, KirkNoor, and KirkIshikawa.
(2) For initial guess away from the fixed point, the number of iterations increases in case of KirkMann, KirkNoor, and KirkIshikawa iterative schemes. However, KirkSP and KirkCR schemes show no change in the number of iterations.
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows:
KirkSP, KirkCR, KirkNoor, KirkIshikwa, and KirkMann.
(2) For initial guess near the fixed point, KirkCR, KirkIshikawa, and KirkMann iterative schemes show a decrease while KirkNoor and KirkSP iterative schemes show no change in the number of iterations to converge.
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows:
KirkCR, KirkSP, KirkNoor, and KirkMann, while KirkNoor and KirkIshikawa iterative schemes show equivalence.
(2) For initial guess near the fixed point, KirkCR iterative scheme show an increase, while KirkSP, KirkIshikawa, KirkMann, and KirkNoor iterative schemes show no change in the number of iterations to converge.
(1) Decreasing order of rate of convergence of Kirk type iterative schemes is as follows:
KirkCR, KirkIshikawa, and KirkMann, while KirkCR, KirkSP, and KirkNoor iterative schemes show equivalence.
(2) For initial guess near the fixed point, KirkMann and KirkIshikawa iterative schemes show a decrease, while KirkCR, KirkSP, and KirkNoor iterative schemes show no change in the number of iterations to converge.
(9) It is observed from experiments that, on taking
(10) In Section
(11) Hence, KirkSP and KirkCR iterative schemes have a good potential for further applications.
The authors would like to thank the referees for valuable suggestions on the paper and N. Hussain gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.