Open Access Research

The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Zhiqun Xue1*, Guiwen Lv1 and BE Rhoades2

Author Affiliations

1 Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P.R. China

2 Department of Mathematics, Indiana University, Bloomington, IN, 47405-7106, USA

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Fixed Point Theory and Applications 2012, 2012:206  doi:10.1186/1687-1812-2012-206

Published: 22 November 2012

Abstract

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1">View MathML</a> be a generalized Lipschitz Φ-hemi-contractive mapping with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6">View MathML</a> be four real sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a> and satisfy the conditions (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M8">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11">View MathML</a>. For some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14">View MathML</a> be any bounded sequences in D, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a> be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

MSC: 47H10.

Keywords:
generalized Lipschitz mapping; Φ-hemi-contractive mapping; Ishikawa iterative sequence with errors; uniformly smooth real Banach space