Iterative process to ϕ-hemicontractive operator and ϕ-strongly accretive operator equations

Abstract

Let E be an arbitrary real Banach space and K be a nonempty closed convex subsets of E. Let T:K→K be a uniformly continuous ϕ-hemicontractive operator with bounded range and {an}, {bn}, {cn}, {a′n}, {b′n}, {c′n} be sequences in [0, 1] satisfying: i) an+bn+cn=a′n+b′n+c′n=1. A n≥0; ‖)limbn=limb′n=limc′n=0; iii)\(\sum\limits_{n = 0}^\infty {b_n } = \infty \); IV) cn=0 (bn). For any given x0, u0, v0∈K, define the Ishikawa type iterative sequence {xn} as follows: $$\left\{ \begin{gathered} x_{n + 1} = a_n x_n + b_n Ty_n + c_n u_n , \hfill \\ y_n = a'_n x_n + b'_n Tx_n + c'_n v_n \left( {\forall n \geqslant 0} \right), \hfill \\ \end{gathered} \right.$$ where {un} and {vn} are bounded sequences in K. Then {xn} converges strongly to the unique fixed point of T. Related result deals with the convergence of Ishikawa type iterative sequence to the solution of ϕ-strongly accretive operator equations.