Abstract
Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and be a generalized Lipschitz Φ-hemi-contractive mapping with . Let , , , be four real sequences in and satisfy the conditions (i) as and ; (ii) . For some , let , be any bounded sequences in D, and 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.
Highlights
Introduction and preliminary LetE be a real Banach space and E* be its dual space
( ) T is called strongly pseudocontractive if there is a constant k ∈ (, ) such that for all x, y ∈ D, Tx – Ty, j(x – y) ≤ k x – y ; ( ) T is called φ-strongly pseudocontractive if for all x, y ∈ D, there exist j(x – y) ∈ J(x – y) and a strictly increasing continuous function φ : [, +∞) → [, +∞) with φ( ) = such that
It is obvious that -pseudocontractive mappings include φ-strongly pseudocontractive mappings, and strongly pseudocontractive mappings
Summary
Introduction and preliminary LetE be a real Banach space and E* be its dual space. The normalized duality mapping J : E → E* is defined byJ(x) = f ∈ E* : x, f = x = f , ∀x ∈ E, where ·, · denotes the generalized duality pairing. Banach space E, and T : D → D be a generalized Lipschitz -hemi-contractive mapping with q ∈ F(T) = ∅.
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