ACTA UNIVERSITATIS APULENSIS No 20/2009 CONVERGENCE THEOREMS FOR UNIFORMLY L-LIPSCHITZIAN ASYMPTOTICALLY NONEXPANSIVE MAPPINGS
نویسنده
چکیده
LetK be a nonempty closed convex subset of a real Banach space E, T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂ [1,∞), lim n→∞ kn = 1 such that p ∈ F (T ) = {x ∈ K : Tx = x}. Let {an}n≥0, {bn}n≥0 , {cn}n≥0 be real sequences in [0, 1] satisfying the following conditions: (i) an + bn + cn = 1; (ii) ∑ n≥0 bn =∞; (iii) cn = o(bn); (iv) lim n→∞n = 0. For arbitrary x0 ∈ K let {xn}n≥0 be iteratively defined by xn+1 = anxn + bnTxn + cnun, n ≥ 0, where {un}n≥0 is a bounded sequence of error terms in K. Suppose there exists a strictly increasing function ψ : [0,∞)→ [0,∞), ψ(0) = 0 such that 〈Tx− p, j(x− p)〉 ≤ kn||x− p|| − ψ(||x− p||), ∀x ∈ K. Then {xn}n≥0 converges strongly to p ∈ F (T ). The results proved in this paper significantly improve the results of Ofoedu [11]. The remark 3 is important. 2000 Mathematics Subject Classification: Primary 47H10, 47H17: Secondary 54H25. 1.Introduction Let E be a real normed space and K be a nonempty convex subset of E. Let J denote the normalized duality mapping from E to 2E ∗ defined by J(x) = {f∗ ∈ E∗ : 〈x, f∗〉 = ||x|| and ||f∗|| = ||x||}, where E∗ denotes the dual space of E and 〈·, ·〉 denotes the generalized duality pairing. We shall denote the single-valued duality map by j. Let T : D(T ) ⊂ E → E be a mapping with domain D(T ) in E.
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