Abstract
The purpose of this paper is to introduce Kirk-type new iterative schemes called Kirk-SP and Kirk-CR schemes and to study the convergence of these iterative schemes by employing certain quasi-contractive operators. By taking an example, we will compare Kirk-SP, Kirk-CR, Kirk-Mann, Kirk-Ishikawa, and Kirk-Noor iterative schemes for aforementioned class of operators. Also, using computer programs in C++, we compare the above-mentioned iterative schemes through examples of increasing, decreasing, sublinear, superlinear, and oscillatory functions.
Highlights
Introduction and PreliminariesThere is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractive-type operator
In 21, Qing and Rhoades by taking example showed that Ishikawa iteration is faster than Mann iteration for a certain quasi-contractive operator
Ciric et al 22, by providing an example, showed that Noor iterative scheme can be faster than Mann and Ishikawa iterative schemes for some quasi-contractive operator
Summary
There is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractive-type operator. 1.13 for all x, y ∈ X and δ ∈ 0, 1 He proved that this class is wider than the class of Zamfirescu operators and used the Ishikawa iteration scheme to approximate fixed points of this class of operators in an arbitrary Banach space given in the form of following theorem. Several authors 11–16 have studied the equivalence between different iterative schemes: Solutz 13, 14 proved that Picard, Mann, Ishikawa, and Noor iteration schemes are equilvalent for quasi-contractive operators. Chugh and Kumar 17 proved that, for quasi-contractive operators satisfying 1.13 , Picard, Mann, Ishikawa, Noor, and SP iterative schemes are equivalent. Motivated by the work of Olatinow 5 and Phuengrattana and Suantai 7 , in this paper, we introduce the Kirk-CR and Kirk-SP iterative schemes and study the strong convergence of these iterative schemes for quasi-contractive operators satisfying 1.14. By using C programming, comparison for rate of convergences between the above-mentioned Kirk type iterative schemes is shown for increasing, decreasing, sublinear, superlinear, and oscillatory functions, respectively
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