Fixed point theorems for asymptotically pseudocontactive maps in certain Banach Spaces

Abstract

Let E be a real q-uniformly smooth Banach space (for example, Lp 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 {kn} n=1∞ . Let {xn}n≥1 be the sequence generated from an arbitrary x1 in E by xn+1=(1-αn) xn + αnT nxn, 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 (Tnxn-p, j(xn-p)) ≤ kn||xn-p||2 _ Φ(||xn-p||), p∈F(T), then {xn}∞n=1 converges strongly to p, provided that {kn} and {αn} 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