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Iterative solution of nonlinear equations with strongly accretive operators in Banach spaces

Authors:

Zhou Haiyun

Institution:

Department of Basic Science, Ordnance Engineering College, Shijiazhuang 050003, P. R. China

Abstract:

Let X be a real Banach space with a uniformly convex dual X^*. Let T: X→X be a Lipschitzian and strongly accretive mapping with a Lipschitzian constant L≥1 and a strongly accretive constant k∈(0,1). Let {±_n} and {2_n} be two real sequences in 0,1] statisfying:

$\alpha _n \to 0 as n \to \infty ;$

(( i ))

$\beta _n< \frac{{k(1 - k)}}{{L(1 + L)}},for all n \geqslant 0;$

(( ii ))

$\prod\limits_{n = 0}^\infty {\alpha _n = \infty .}$

(( iii ))

Set Sx = f-Tx+x, ∀x∈X. Assume that {u_n} _n=0 ^∞ and {v_n} _n=0 ^∞ be two sequences in X satisfying ∥u _n∥ For arbitrary x₀∈X, the iteration sequence {x_n} is defined by

$(IS)_1 \left\{ \begin{gathered} x_{n + 1} = (1 - a_n )x_n + a_n Sy_n + u_n , \hfill \\ y_n = (1 - \beta _n )x_n + \beta _n Sx_n + v_n (n \geqslant 0), \hfill \\ \end{gathered} \right.$

then {x_n} converges strongly to the unique solution of the equation Tx=f. A related result deals with iterative approximation of fixed points of ϕ-hemicontractive mappings.

Keywords:

Ishikawa iteration with errors strongly accretive mapping hemicontractive mapping

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