Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space

“…We know that J λ is nonexpansive and F (J λ ) = A −1 (0) for all λ > 0. An accretive operator A is said to be m-accretive if R(I + λA) = E for all λ > 0 (see also [1]). …”

Section: Viscosity Approximation Methods For Monotone Mappings Viscosmentioning

confidence: 99%

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

Plubtieng

2011

Mathematica Slovaca

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ABSTRACT. We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P C is a metric projection. We consider the iteration process {x n } of C defined by x 1 = x ∈ C is arbitrary andwhere f is a contraction on C, {S n } is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x n } converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.

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“…To deal with a family of mappings, the following conditions are introduced: Let C be a subset of a real Banach space E, and let {T n } ∞ n=1 be a family of mappings of C such that ∩ ∞ n=1 F(T n ) = ∅. Then, {T n } is said to satisfy the AKTT-condition [10] if for each bounded subset B of C,…”

Section: Lemma 22 [4]mentioning

confidence: 99%

Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings

Bunyawat

Suantai

2011

Fixed Point Theory Appl

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A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly k-strictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klin-eam and Suantai

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Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space

Cited by 307 publications

References 12 publications

Multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings

Multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings

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