Abstract :
Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, T :K → K
a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n 0 ⊂ [1,+∞),
limn→∞kn = 1 such that F(T ) = ∅. Let {αn}n 0 ⊂ [0, 1] be such that n 0 αn =∞, n 0 α2
n <∞ and n 0 αn(kn −1) <∞. Suppose {xn}n 0 is iteratively defined by xn+1 = (1 − αn)xn + αnT nxn,
n 0, and suppose there exists a strictly increasing continuous function φ : [0,+∞)→[0,+∞), φ(0) = 0
such that T nx − x∗, j (x − x∗) kn
x − x∗
2 − φ(
x − x∗
), ∀x ∈ K. It is proved that {xn}n 0 converges
strongly to x∗ ∈ F(T ). It is also proved that the sequence of iteration {xn} defined by xn+1 =
anxn+bnT nxn+cnun, n 0 (where {un}n 0 is a bounded sequence in K and {an}n 0, {bn}n 0, {cn}n 0
are sequences in [0, 1] satisfying appropriate conditions) converges strongly to a fixed point of T .
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