Iterative Methods for Fixed Points of Asymptotically Weakly Contractive Maps

Abstract

Suppose K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E with P as the nonexpansive retraction. Suppose T : K → E is an asymptotically d-weakly contractive map with sequence {k_n }, k_n ≥ 1, lim k_n = 1 and with F(T) _n int (K) ≠ ø F(T):= {x ∈ K: Tx = x}. Suppose {x _n } is iteratively defined by x _n+1 = P((l − k_nα_n )x _n +k _n α _n T(PT) ^n−l x_n ), n = 1,2,...,x ₁ ∈ K, where α_n∈ (0,l) satisfies lim α_n = 0 and Σα_n = ∞. It is proved that {x _n } converges strongly to some x ^* ∈ F(T)∩ int K. Furthermore, if K is a closed convex subset of an arbitrary real Banach space and T is, in addition uniformly continuous, with F(T) ≠ ø, it is proved that {x_n } converges strongly to some x ^* ∈ F(T).

^†The author undertook this work when he was visiting the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, as a postdoctoral fellow.

^‡Present address: Department of Mathematics, University of Nigeria, Nsukka

Keywords:

• Asymptotically d-weakly contractive
• d-Weakly contractive
• Weakly contractive
• Nonexpansive retraction

2000 Mathematics Subject Classifications:

• 47H04
• 47H06
• 47H30
• 47J05
• 47J25

Notes

^†The author undertook this work when he was visiting the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, as a postdoctoral fellow.

^‡Present address: Department of Mathematics, University of Nigeria, Nsukka