Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings

Applied Mathematics and Computation (Impact Factor: 1.55). 03/2009; 209(2):162-176. DOI: 10.1016/j.amc.2008.10.062
Source: DBLP


Let X be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. First purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping T and prove that this iterative process converges strongly to x∗∈F(T)≔{x∈X∣x=T(x)}, where x∗ is the unique solution in F(T) to the following variational inequality:〈f(x∗)-x∗,j(v-x∗)〉⩽0for allv∈F(T).Second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping T and prove that these iterative schemes strongly converge to the same point x∗∈F(T). Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature, see [L.C. Zeng, J.C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal. 64 (2006) 2507–2515; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291; Y.S. Song, R.D. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. (2006)].

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    • "for all í µí±¢, í µí±¢ † ∈ H. It is well known that, in a real Hilbert space H, the following equality holds: 󵄩 󵄩 󵄩 󵄩 󵄩 í µí¼‰í µí±¢ + (1 − í µí¼‰) í µí±¢ † 󵄩 󵄩 󵄩 󵄩 󵄩 2 = í µí¼‰‖í µí±¢‖ 2 + (1 − í µí¼‰) 󵄩 󵄩 󵄩 󵄩 󵄩 í µí±¢ † 󵄩 󵄩 󵄩 󵄩 󵄩 2 − í µí¼‰ (1 − í µí¼‰) 󵄩 󵄩 󵄩 󵄩 󵄩 í µí±¢ − í µí±¢ † 󵄩 󵄩 󵄩 󵄩 󵄩 2 (10) for all í µí±¢, í µí±¢ † ∈ H and í µí¼‰ ∈ [0] [1]. "
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