On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval

Abstract

MSC: 26A18 47H10 54C05Keywords: Continuous functions Convergence theorem Fixed point Nondecreasing functions Rate of convergence a b s t r a c tIn this paper, we propose a new iteration, called the SP-iteration, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the SP-iteration of continuous functions on an arbitrary interval is given. We also compare the convergence speed of Mann, Ishikawa, Noor and SP-iterations. It is proved that the SP-iteration is equivalent to and converges faster than the others. Our results extend and improve the corresponding results of Borwein and Borwein [D. Borwein, J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991) 112-126], Qing and Qihou [Y. Qing, L. Qihou, The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. Math. Anal. Appl. 323 (2006) 1383-1386], Rhoades [B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741-750], and many others. Moreover, we also present numerical examples for the SP-iteration to compare with the Mann, Ishikawa and Noor iterations.

Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces

“…Let K be a nonempty closed convex subset of an arbitrary Banach space X and be a mapping satisfying (1.9). Let {s n } be defined through the Ishikawa iteration (1.4) and 0 Several authors [5,[8][9][10][11][12][13][14][15][16][17] have studied the equivalence between various iterative schemes. S. M. Solutz [15,16] proved that for quasi-contractive operators the itérative processes Picard, Mann, Ishikawa and Noor are équi-valent.…”

“…S. L. Singh [19] extended the work of Rhoades. Very recently, Phuengrattana and Suantai [5] proved that SP iterative scheme is equivalent to and faster than Mann, Ishikawa and Noor iterative schemes for increasing functions. Now, we introduce the following CR iterative process: Let X be a Banach space, a self map of X : T X X  and 0…”

On the convergence and data dependence results for multistep Picard-Mann iteration process in the class of contractive-like operators

“…(i) The Picard iteration [17] converges to p ∈ F (T ), (ii) The Mann iteration [13] converges to p ∈ F (T ), (iii) The SP iteration [16] converges to p ∈ F (T ), (iv) The multistep iteration (2.5) converges to p ∈ F (T ), (v) The multistep Picard-Mann iteration (2.8) converges to p ∈ F (T ).…”

“…If we take r = 2 and r = 3 in (2.7), respectively, we obtain the two-step iteration procedure given in [23] and SP iteration method in [16].…”

“…In 2007, Şoltuz [21] proved that the Mann [13], Ishikawa [9], Noor [14] and multistep (2.5) iterations are equivalent for quasi-contractive mappings in a normed space. In 2011, Chugh and Kumar [6] showed that the Picard [17], Mann [13], Ishikawa [9], new two step [23], Noor [14] and SP [16] iterations are equivalent for quasi-contractive mappings in a Banach space. In 2013, Karakaya et al [10] proved the data dependence results for the multistep (2.5) and CR [7] iteration processes for the class of contractive-like operators satisfying (2.4).…”