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Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

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Qin X., Su Y., Shang M.

Open Mathematics

2007

tom 5

nr 2

345-357

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with {k n}, {l n} and {j n}. [1, ∞) such that Σn=1∞(k n − 1) < ∞, Σn=1∞(l n − 1) < ∞ and Σn=1∞(j n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T 1x = T 2x = T 3 x} = x} denotes the common fixed points set of T 1, T 2 and T 3. Let {α n}, {α′ n} and {α″ n} be real sequences in (0, 1) and ∈ ≤ {α n}, {α′ n}, {α″ n} ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x 1 ∈ K define the sequence {x n} by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_3 (PT_3 )^{n - 1} x_n + (1 - \alpha ''_n )x_n ), \hfill \\ y_n = P(\alpha '_n T_2 (PT_2 )^{n - 1} z_n + (1 - \alpha '_n )x_n ), \hfill \\ x_{n + 1} = P(\alpha _n T_1 (PT_1 )^{n - 1} y_n + (1 - \alpha _n )x_n ). \hfill \\ \end{gathered} \right.$$ (i) If the dual E* of E has the Kadec-Klee property then {x n} converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then {x n} converges strongly to a common fixed point p ∈ F.

Ograniczanie wyników

1 Open Mathematics

1 Qin X.

1 Shang M.

1 Su Y.

1 2007

Wyniki wyszukiwania

Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces