Fixed point theorems for asymptotically pseudocontactive maps in certain Banach Spaces

Abstract

Let E be a real q-uniformly smooth Banach space (for example, L_p or lp spaces ,1‹p‹∞) and let T: E→E be asymptotically pseudocontractive mapping with nonempty fixed point set F(T) and a real sequence {k_n} _n=1^∞ . Let {x_n}_n≥1 be the sequence generated from an arbitrary x₁ in E by x_n+1=(1-α_n) x_n + α_nT ⁿx_n, n≥1. If the range of T is bounded and there exists a strictly increasing function Φ : [ o , ∞) →[ o , ∞) with Φ(0)=0 such that (Tⁿx_n-p, j(x_n-p)) ≤ k_n||x_n-p||² _ Φ(||x_n-p||), p∈F(T), then {x_n}^∞_n=1 converges strongly to p, provided that {k_n} and {α_n} satisfy certain properties. This result compliments the results of Chang, Park and Cho (2000), by dropping the Lipschitz condition on T.


2000 Mathematics Subject Classification : 47H09 47J05.





KEY WORDS: Fixed points, asymptotically pseudocontractive maps, modified Mann iterative methods, q-uniformly smooth spaces.


Global Journal of Mathematical Sciences Vol.3(2) 2004: 137-142