Abstract
Let be a closed convex subset of a real Banach space , is continuous pseudocontractive mapping, and is a fixed -Lipschitzian strongly pseudocontractive mapping. For any , let be the unique fixed point of . We prove that if has a fixed point and has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then converges to a fixed point of as approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).
Highlights
Introduction and preliminariesLet E be a real Banach space and let J denote the normalized duality mapping from E into 2E∗ given by J(x) = { f ∈ E∗, x, f = x f, x = f }, for all x ∈ E, where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing
We prove that if T has a fixed point and E has uniformly Gateaux differentiable norm, such that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, {xt} converges to a fixed point of T as t approaches to 0
Let E be a real Banach space and let T be a mapping with domain D(T) and range R(T) in E
Summary
YISHENG SONG AND RUDONG CHEN Received 20 March 2006; Revised 24 May 2006; Accepted 28 May 2006. Let K be a closed convex subset of a real Banach space E, T : K → K is continuous pseudocontractive mapping, and f : K → K is a fixed L-Lipschitzian strongly pseudocontractive mapping. For any t ∈ (0, 1), let xt be the unique fixed point of t f + (1 − t)T. We prove that if T has a fixed point and E has uniformly Gateaux differentiable norm, such that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, {xt} converges to a fixed point of T as t approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004)
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