An Iterative Algorithm for Two Asymptotically Pseudocontractive Mappings
Arif Rafiq, Ana Maria Acu, Florin Sofonea · 2009
Let K be a nonempty closed convex subset of a real Banach space E, Ti: K → K, i = 1, 2 be two uniformly L-Lipschitzian asymptotically pseudocontractive mappings with sequence {kn}n≥0 ⊂ [1,∞), n≥0 (kn − 1) < ∞ such that F (T1) ∩ F (T2) 6 = ϕ, where F (Ti) is the set of fixed points of Ti in K and p be a point in F (T1)∩F (T2). Let {αn}n≥0, {βn}n≥0 ⊂ [0, 1] be two sequences such that n≥0 αn = ∞ and lim n→∞αn = 0 = limn→∞βn. For arbitrary x0 ∈ K, let {xn}n≥0 be a sequence iteratively defined by xn+1 = (1 − αn)xn + αnTn1 yn, yn = (1 − βn)xn + βnTn2 xn, n ≥ 0. Suppose there exists a strictly increasing function φ: [0,∞) → [0,∞), φ(0) = 0 such that 〈Tni x − p, j(x − p) 〉 ≤ kn||x − p||2 − φ(||x − p||), ∀x ∈ K, i = 1, 2. Then {xn}n≥0 converges strongly to p ∈ F (T1) ∩ F (T2). The results proved in this paper significantly improve the results of Chang et al. [1].