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

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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. Å 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.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan Normal University, 610066, Chengdu, P R China

    Ding Xie-ping Professor & Zhang Hong-lin

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  1. Ding Xie-ping Professor
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  2. Zhang Hong-lin
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Additional information

Paper from DING Xie-ping, Member of Editorial Committee, AMM

Foundation item: the National Natural Science Foundation of China (19871059)

Biography: DING Xie-ping (1938-)

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Xie-ping, D., Hong-lin, Z. Iterative process to ϕ-hemicontractive operator and ϕ-strongly accretive operator equations. Appl Math Mech 21, 1256–1263 (2000). https://doi.org/10.1007/BF02459246

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  • DOI: https://doi.org/10.1007/BF02459246

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