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.
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This paper is devoted to the numerical solution of nonlinear elliptic partial differential equations. Such problems describe various phenomena in science. An approach that exploits Hilbert space theory in the numerical study of elliptic PDEs is the idea of preconditioning operators. In this survey paper we briefly summarize the main lines of this theory with various applications.
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