Math. Program., Ser. B (2009) 116:397–427 DOI 10.1007/s10107-007-0121-9 FULL LENGTH PAPER Error bounds for systems of lower semicontinuous functions in Asplund spaces Huynh van Ngai · Michel Théra Received: 6 June 2005 / Accepted: 25 October 2006 / Published online: 16 June 2007 © Springer-Verlag 2007 Abstract In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems. Keywords Fuzzy calculus · Metric regularity · Error bounds · Asplund space Mathematics Subject Classification (2000) 49J52 · 90C30 1 Introduction In all this contribution, we consider a Banach space X with norm ‖·‖ and we suppose given a finite family of extended-real-valued functions, f i (i = 1,..., n) defined on X . By an inequality/equality system we mean the problem of finding x ∈ X satisfying: f i (x ) ≤ 0 (i = 1,..., m) ; f j (x ) = 0 ( j = m + 1,..., n). (1.1) We use the symbol [ f i (x )] + to denote max( f i (x ), 0) Dedicated to Alfred Auslender. H. van Ngai Ecole Normale Supérieure de Quinhon, Quinhon, Vietnam e-mail: nghiakhiem@yahoo.com M. Théra (B ) Laboratoire XLIM, UMR-CNRS 6172, Université de Limoges, Limoges, France e-mail: michel.thera@unilim.fr 123