### Abstract

Let E be a real uniformly smooth Banach space. Let A: D(A) = E → 2^{E} be an accretive operator that satisfies the range condition and A^{-1}(0) ≠. Let {λ_{n}} and {θ_{n}} be two real sequences satisfying appropriate conditions, and for z ∈ E arbitrary, let the sequence {x_{n}} be generated from arbitrary x_{0} ∈ E by x_{n+1} = x_{n} - λ_{n} (u_{n} + θ_{n}(x_{n} - z)), u_{n} ∈ Ax_{n}, ≥ 0. Assume that {u_{n}} is bounded. It is proved that {x_{n}} converges strongly to Some x* ∈ A^{-1} (0). Furthermore, if K is a nonempty closed convex subset of E and T: K → K is a bounded continuous pseudocontractive map with F (T) := {Tx = x} ≠, it is proved that for arbitrary z ∈ K, the sequence {x_{n}} generated from x_{0} ∈ K by x_{n+1} = x_{n} - λ_{n} ((I - T)x_{n} + θ_{n} (x_{n} - z)), n ≥ 0,) where {λ_{n}} and {θ_{n}} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.

Original language | English |
---|---|

Pages (from-to) | 756-765 |

Number of pages | 10 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 282 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 15 2003 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

}

*Journal of Mathematical Analysis and Applications*, vol. 282, no. 2, pp. 756-765. https://doi.org/10.1016/S0022-247X(03)00252-X

**Iterative solution of nonlinear equations of accretive and pseudocontractive types.** / Chidume, C. E.; Zegeye, Hab.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Iterative solution of nonlinear equations of accretive and pseudocontractive types

AU - Chidume, C. E.

AU - Zegeye, Hab

PY - 2003/6/15

Y1 - 2003/6/15

N2 - Let E be a real uniformly smooth Banach space. Let A: D(A) = E → 2E be an accretive operator that satisfies the range condition and A-1(0) ≠. Let {λn} and {θn} be two real sequences satisfying appropriate conditions, and for z ∈ E arbitrary, let the sequence {xn} be generated from arbitrary x0 ∈ E by xn+1 = xn - λn (un + θn(xn - z)), un ∈ Axn, ≥ 0. Assume that {un} is bounded. It is proved that {xn} converges strongly to Some x* ∈ A-1 (0). Furthermore, if K is a nonempty closed convex subset of E and T: K → K is a bounded continuous pseudocontractive map with F (T) := {Tx = x} ≠, it is proved that for arbitrary z ∈ K, the sequence {xn} generated from x0 ∈ K by xn+1 = xn - λn ((I - T)xn + θn (xn - z)), n ≥ 0,) where {λn} and {θn} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.

AB - Let E be a real uniformly smooth Banach space. Let A: D(A) = E → 2E be an accretive operator that satisfies the range condition and A-1(0) ≠. Let {λn} and {θn} be two real sequences satisfying appropriate conditions, and for z ∈ E arbitrary, let the sequence {xn} be generated from arbitrary x0 ∈ E by xn+1 = xn - λn (un + θn(xn - z)), un ∈ Axn, ≥ 0. Assume that {un} is bounded. It is proved that {xn} converges strongly to Some x* ∈ A-1 (0). Furthermore, if K is a nonempty closed convex subset of E and T: K → K is a bounded continuous pseudocontractive map with F (T) := {Tx = x} ≠, it is proved that for arbitrary z ∈ K, the sequence {xn} generated from x0 ∈ K by xn+1 = xn - λn ((I - T)xn + θn (xn - z)), n ≥ 0,) where {λn} and {θn} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.

UR - http://www.scopus.com/inward/record.url?scp=0038793412&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038793412&partnerID=8YFLogxK

U2 - 10.1016/S0022-247X(03)00252-X

DO - 10.1016/S0022-247X(03)00252-X

M3 - Article

AN - SCOPUS:0038793412

VL - 282

SP - 756

EP - 765

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -