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Iterative methods for the class of quasi-contractive type operators and comparsion of their rate of convergence in convex metric spaces

Fukhar-ud-din Hafiz (King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia + The Islamia University of Bahawalpur, Department of Mathematics, Bahawalpur, Pakistan)
Berinde Vasile (North University of Baia Mare, Department of Mathematics and Computer Science, Victoria, Baia Mare, Romania + King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia)

We introduce modified Noor iterative method in a convex metric space and apply it to approximate fixed points of quasi-contractive operators introduced by Berinde [6]. Our results generalize and improve upon, among others, the corresponding results of Berinde [6], Bosede [9] and Phuengrattana and Suantai [20]. We also compare the rate of convergence of proposed iterative method to the iterative methods due to Noor [26], Ishikawa [14] and Mann [18]. It has been observed that the proposed method is faster than the other three methods. Incidently the results obtained herein provide analogues of the corresponding results of normed spaces and holds in CAT(0) spaces, simultaneously.

Keywords: Convex metric space, modified Noor iterative method, quasi-contractive operator, fixed point, convergence theorem, order of convergence.