Strong convergence results for nonlinear mappings in real Banach spaces

Let X be a real Banach space, K be a nonempty closed convex subset of X, T : K → K be a nearly uniformly L-Lipschitzian mapping with sequence {an}. Let kn ⊂ [1, ∞) and En be sequences with limn→∞ kn = 1, limn→∞ En = 0 and F(T) = {ρ ∈ K : T ρ = ρ} 6= ∅. Let {αn}n≥0 be real sequence in [0, 1] satisfying the following conditions: (i)P n≥0 αn = ∞ (ii) limn→∞ αn = 0. For arbitrary x0 ∈ K, let {xn}n≥0 be iteratively defined by xn+1 = (1 − αn)xn + αnT nxn, n ≥ 0. If there exists a strictly increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that < T nx − T nρ, j(x − ρ) >≤ knkx − ρk 2 − Φ(kx − ρk) + En for all x ∈ K, then, {xn}n≥0 converges strongly to ρ ∈ F(T). It is also proved that the sequence of iteration {xn} defined by xn+1 = (1 − bn − dn)xn + bnT nxn + dnwn, n ≥ 0, where {wn}n≥0 is a bounded sequence in K and {bn}n≥0, {dn}n≥0 are sequences in [0,1] satisfying appropriate conditions, converges strongly to a fixed point of T.