Strong convergence and stability of Picard iteration sequences for a general class of contractive-type mappings

• Published in: Fixed point theory and algorithms for sciences and engineering 2014-11, Vol.2014 (1), p.1-10, Article 233
• Main Author: Chidume, Charles E
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; Convergence; Differential Geometry; Mapping; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Nonlinearity; Operators; Picard iterations; Stability; Theorems; Topology
• ISSN: 1687-1812; 1687-1812; 2730-5422
• DOI: 10.1186/1687-1812-2014-233

Let ( E , ∥ ⋅ ∥ ) be a normed linear space, T : E → E be a mapping of E into itself satisfying the following contractive condition: ∥ T i x − T i y ∥ ⩽ a i ∥ x − y ∥ + φ i ( ∥ x − T x ∥ ) , for each x , y ∈ E , 0 ⩽ a i < 1 , where φ i : R + → R + is a sub-additive monotone increasing function with φ i ( 0 ) = 0 and φ i ( L u ) = L φ i ( u ) , L ⩾ 0 , u ∈ R + . It is shown that the Picard iteration process converges strongly to the unique fixed point of T . Furthermore, several classes of nonlinear operators studied by various authors are shown to belong to this class of mappings. Our theorem improves several recent important results. In particular, it improves a recent result of Akewe et al. (Fixed Point Theory Appl 2014:45, 2014), and a host of other results. MSC: accretive-type mappings, pseudocontractive mappings, Picard sequence, contractive-type mappings.