On the convergence of an iteration method for continuous mappings on an arbitrary interval
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On the convergence of an iteration method for continuous mappings on an arbitrary interval Nazli Kadioglu* and Isa Yildirim *
Correspondence: [email protected] Department of Mathematics, Ataturk University, Erzurum, 25240, Turkey
Abstract In this paper, we consider an iterative method for finding a fixed point of continuous mappings on an arbitrary interval. Then, we give the necessary and sufficient conditions for the convergence of the proposed iterative methods for continuous mappings on an arbitrary interval. We also compare the rate of convergence between iteration methods. Finally, we provide a numerical example which supports our theoretical results. MSC: 26A18; 47H10; 54C05 Keywords: continuous mapping; fixed point; convergence theorem
1 Introduction Let E be a closed interval on the real line and let f : E → E be a continuous mapping. A point p ∈ E is a fixed point of f if f (p) = p. We denote by F(f ) the set of fixed points of f . It is known that if E is also bounded, then F(f ) is nonempty. There are many iterative methods for finding a fixed point of f . For example, the Mann iteration (see []) is defined by u ∈ E and un+ = ( – αn )un + αn f (un )
(.)
for all n ≥ , where {αn }∞ n= is a sequence in [, ]. The Ishikawa iteration (see []) is defined by s ∈ E and ⎧ ⎨t = ( – b )s + b f (s ), n n n n n ⎩sn+ = ( – αn )sn + αn f (tn )
(.)
∞ for all n ≥ , where {αn }∞ n= , {bn }n= are sequences in [, ]. The Noor iteration (see []) is defined by w ∈ E and
⎧ ⎪ ⎪ ⎨rn = ( – an )wn + an f (wn ), ⎪ ⎪ ⎩
qn = ( – bn )wn + bn f (rn ),
(.)
wn+ = ( – αn )wn + αn f (qn )
© 2013 Kadioglu and Yildirim; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kadioglu and Yildirim Fixed Point Theory and Applications 2013, 2013:124 http://www.fixedpointtheoryandapplications.com/content/2013/1/124
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∞ ∞ for all n ≥ , where {αn }∞ n= , {bn }n= and {an }n= are sequences in [, ]. Clearly, Mann and Ishikawa iterations are special cases of the Noor iteration. In , Roades proved the convergence of the Mann iteration for the class of continuous and nondecreasing functions on a closed unit interval, and then he [] extended convergence results to Ishikawa iterations. He also proved that the Ishikawa iteration converges faster than the Mann iteration for the class of continuous and nondecreasing mappings. Later, in , Borwein and Borwein [] proved the convergence of the Mann iteration of continuous mappings on a bounded closed interval. Recently, Qing and Qihou [] extended their results to an arbitrary interval and to the Ishikawa iteration and a gave necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval. Recently, Phuengrattana and Suantai [] proved that the Mann, Ishikawa and
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