### Abstract

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L_{p} or l_{p} spaces, 1 < p < ∞) and K be a nonempty closed convex and bounded subset of E with φ ≠ int (K). Let T: K → K be a Lipschitzian pseudocontractive mapping such that for z ∈ int (K), ∥z - Tz∥ < ∥x - Tx∥, for all x ∈ ∂(K). Then for z_{0} ∈ K arbitrary, the iteration process (z_{n}) defined by z_{n+1}: = (1 - μ_{n+1})z+μ_{n+1}y_{n}; y_{n}: = (1 - α_{n})z_{n} + α_{n}Tz_{n} converges strongly to a fixed point of T, provided that (μ_{n}) and (α_{n}) satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T.

Original language | English |
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Pages (from-to) | 339-346 |

Number of pages | 8 |

Journal | Computers and Mathematics with Applications |

Volume | 44 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Jan 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

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*Computers and Mathematics with Applications*, vol. 44, no. 3-4, pp. 339-346. https://doi.org/10.1016/S0898-1221(02)00152-9

**Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps.** / Zegeye, H.; Prempeh, E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps

AU - Zegeye, H.

AU - Prempeh, E.

PY - 2002/1/1

Y1 - 2002/1/1

N2 - Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, Lp or lp spaces, 1 < p < ∞) and K be a nonempty closed convex and bounded subset of E with φ ≠ int (K). Let T: K → K be a Lipschitzian pseudocontractive mapping such that for z ∈ int (K), ∥z - Tz∥ < ∥x - Tx∥, for all x ∈ ∂(K). Then for z0 ∈ K arbitrary, the iteration process (zn) defined by zn+1: = (1 - μn+1)z+μn+1yn; yn: = (1 - αn)zn + αnTzn converges strongly to a fixed point of T, provided that (μn) and (αn) satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T.

AB - Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, Lp or lp spaces, 1 < p < ∞) and K be a nonempty closed convex and bounded subset of E with φ ≠ int (K). Let T: K → K be a Lipschitzian pseudocontractive mapping such that for z ∈ int (K), ∥z - Tz∥ < ∥x - Tx∥, for all x ∈ ∂(K). Then for z0 ∈ K arbitrary, the iteration process (zn) defined by zn+1: = (1 - μn+1)z+μn+1yn; yn: = (1 - αn)zn + αnTzn converges strongly to a fixed point of T, provided that (μn) and (αn) satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T.

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U2 - 10.1016/S0898-1221(02)00152-9

DO - 10.1016/S0898-1221(02)00152-9

M3 - Article

VL - 44

SP - 339

EP - 346

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 3-4

ER -