J Glob Optim (2008) 40:463–473 DOI 10.1007/s10898-007-9177-6 Minty’s lemma and vector variational-like inequalities M. Chinaie · T. Jabarootian · M. Rezaie · J. Zafarani Received: 25 May 2007 / Accepted: 28 May 2007 / Published online: 7 July 2007 © Springer Science+Business Media, LLC 2007 Abstract In this paper, we consider two vector versions of Minty’s Lemma and obtain exis- tence theorems for three kinds of vector variational-like inequalities. The results presented in this paper are extension and improvement of the corresponding results of other authors. Keywords Fan’s KKM theorem · Minty’s Lemma · Vector variational-like inequality · Hausdorff metric Mathematics Subject Classification (2000) 47H04 · 47H05 · 49J40 · 49J53 1 Introduction Since Giannessi [9] introduced the vector variational inequality (VVI) in finite dimensional Euclidian space, many authors have intensively studied (VVI) and its various extensions. Several authors have investigated relationships between (VVI) and vector optimization prob- lems, vector complementarity problem. For details we refer to Chen [3], Chen and Yang [4], Daniilidis and Hadjisavvas [6], Giannessi [10,11], Giannessi and Maugeri [12], Giannessi and Maugeri [13], Huang and Fang [14], Konnov and Yao [18], Yang [22], Yang and Goh [23], and Zeng and Yao [24] and reference therein. The vector variational-like inequalities (VVLI), a generalization of (VVI) was studied by Ansari, Siddiqi and Yao [1], Chiang [5], Fang and Huang [8], Jabarootian and Zafarani [16], Lin [20], Yang [22]. Minty’s Lemma has J. Zafarani was partially supported by the Center of Excellence for Mathematics (University of Isfahan). M. Chinaie · M. Rezaie · J. Zafarani (B ) Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran e-mail: jzaf@math.ui.ac.ir T. Jabarootian Department of Mathematics, Islamic Azad University, Khomeiny Shahr Branch, Isfahan 84175-119, Iran 123