## Abstract

Let X and Y be real Banach spaces, K be a nonempty convex subset of X, and C : K → 2^{Y} be a multifunction such that for each u ∈ K, C (u) is a proper, closed and convex cone with int C (u) ≠ 0{combining long solidus overlay}, where int C (u) denotes the interior of C (u). Given the mappings T : K → 2^{L (X, Y)}, A : L (X, Y) → L (X, Y), f_{1} : L (X, Y) × K × K → Y, f_{2} : K × K → Y, and g : K → K, we introduce and consider the generalized implicit vector equilibrium problem: Find u^{*} ∈ K such that for any v ∈ K, there is s^{*} ∈ T u^{*} satisfying f_{1} (A s^{*}, v, g (u^{*})) + f_{2} (v, g (u^{*})) ∉ - int C (u^{*}). By using the KKM technique and the well-known Nadler's result, we prove some existence theorems of solutions for this class of generalized implicit vector equilibrium problems. Our theorems extend and improve the corresponding results of several authors.

Original language | English |
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Pages (from-to) | 1682-1691 |

Number of pages | 10 |

Journal | Computers and Mathematics with Applications |

Volume | 57 |

Issue number | 10 |

DOIs | |

State | Published - 05 2009 |

Externally published | Yes |

## Keywords

- Generalized implicit vector equilibrium problem
- Generalized implicit vector variational inequality
- Hausdorff metric
- KKM technique
- Nadler's theorem