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

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 {²n} 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 {un}n=0∞ and {vn}n=0∞ be two sequences in X satisfying ∥un∥ For arbitrary x0∈X, the iteration sequence {xn} 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 {xn} converges strongly to the unique solution of the equation Tx=f.