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

摘要

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and T : D → D be a generalized Lipschitz Φ-hemi-contractive mapping with q ∈ F ( T ) ≠ ? . Let { a n } , { b n } , { c n } , { d n } be four real sequences in [ 0 , 1 ] and satisfy the conditions (i) a n , b n , d n → 0 as n → ∞ and c n = o ( a n ) ; (ii) ∑ n = 0 ∞ a n = ∞ . For some x 0 ∈ D , let { u n } , { v n } be any bounded sequences in D, and { x n } 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.