Strong convergence of new iterative algorithms for certain classes of asymptotically pseudocontractions
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RESEARCH
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Strong convergence of new iterative algorithms for certain classes of asymptotically pseudocontractions Micah Okwuchukwu Osilike* , Peter Uche Nwokoro and Eucharia Ezinwanne Chima * Correspondence: [email protected]; [email protected] Department of Mathematics, University of Nigeria, Nsukka, Nigeria
Abstract Let C be a nonempty closed convex subset of a real Hilbert space, and let T : C → C be an asymptotically k-strictly pseudocontractive mapping with ∞ ∞ F(T) = {x ∈ C : Tx = x} = ∅. Let {αn }∞ n=1 and {tn }n=1 be real sequences in (0, 1). Let {xn }n=1 be the sequence generated from an arbitrary x1 ∈ C by
νn = PC ((1 – tn )xn ), n ≥ 1, xn+1 = (1 – αn )νn + αn T n νn ,
n ≥ 1,
where PC : H → C is the metric projection. Under some appropriate mild conditions ∞ ∞ on {αn }∞ n=1 and {tn }n=1 , we prove that {xn }n=1 converges strongly to a fixed point of T. Furthermore, if T : C → C is uniformly L-Lipschitzian and asymptotically pseudocontractive with F(T) = ∅, we first prove that (I – T) is demiclosed at 0, and then ∞ prove that under some suitable conditions on the real sequences {αn }∞ n=1 , {βn }n=1 ∞ ∞ and {tn }n=1 in (0, 1), the sequence {xn }n=1 generated from an arbitrary x1 ∈ C by ⎧ ⎨νn = PC ((1 – tn )xn ), n ≥ 1, yn = (1 – βn )νn + βn T n νn , n ≥ 1, ⎩ xn+1 = (1 – αn )νn + αn T n yn , n ≥ 1, converges strongly to a fixed point of T. No compactness assumption is imposed on T or C and no further requirement is imposed on F(T). MSC: 47H09; 47J25; 65J15 Keywords: asymptotically pseudocontractive maps; fixed points; strong convergence; Hilbert spaces; iterative algorithm
1 Introduction Let H be a real Hilbert space with the inner product ·, · and the induced norm · . Let C be a nonempty closed convex subset of H. A mapping T : C → C is said to be L-Lipschitzian if there exists L ≥ such that
Tx – Ty ≤ L x – y ,
∀x, y ∈ C.
(.)
T is said to be a contraction if L ∈ [, ), and T is said to be nonexpansive if L = . T is said to be asymptotically nonexpansive (see, for example, []) if there exists a sequence ©2013 Osilike et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Osilike et al. Fixed Point Theory and Applications 2013, 2013:334 http://www.fixedpointtheoryandapplications.com/content/2013/1/334
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{kn }∞ n= ⊆ [, ∞) with limn→∞ kn = such that n T x – T n y ≤ kn x – y ,
∀x, y ∈ C.
(.)
It is well known (see, for example, []) that the class of nonexpansive mappings is a proper subclass of the class of asymptotically nonexpansive mappings. T is said to be asymptotically k-strictly pseudocontractive (see, for example, []) if there exist k ∈ [, ) and a sequence {kn }∞ n= ⊆ [, ∞), limn→∞ kn = such that n T x – T n y ≤ kn x – y + k x – T n x – y – T n y ,
∀x, y ∈ C.
(.
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