Abstract
We prove several strong convergence theorems for the Ishikawa iterative sequence with errors to a fixed point of strictly pseudocontractive mapping of Browder-Petryshyn type in Banach spaces and give sufficient and necessary conditions for the convergence of the scheme to a fixed point of the mapping. The results presented in this work give an affirmative answer to the open question raised by Zeng et al. 2006, and generalize the corresponding result of Zeng et al. 2006, Osilike and Udomene 2001, and others.
Highlights
Introduction and PreliminariesLet E be a real Banach space and E∗ its dual. ·, · denotes the generalized duality pairing between E and E∗
We prove several strong convergence theorems for the Ishikawa iterative sequence with errors to a fixed point of strictly pseudocontractive mapping of Browder-Petryshyn type in Banach spaces and give sufficient and necessary conditions for the convergence of the scheme to a fixed point of the mapping
F T is the fixed point set of T, that is, F T {x : T x x}
Summary
Let E be a real Banach space and E∗ its dual. ·, · denotes the generalized duality pairing between E and E∗. Let E be a real Banach space and E∗ its dual. ·, · denotes the generalized duality pairing between E and E∗. Let J : E → 2E∗ be the normalized duality mapping defined by the following:. It is well known that if E is smooth, J is single-valued. We denote a singlevalued selection of the normalized duality mapping by j. F T is the fixed point set of T , that is, F T {x : T x x}
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