SIAM J. OPTIM. c  XXXX Society for Industrial and Applied Mathematics Vol. 0, No. 0, pp. 000–000 ERROR BOUNDS IN METRIC SPACES AND APPLICATION TO THE PERTURBATION STABILITY OF METRIC REGULARITY ∗ HUYNH VAN NGAI † AND MICHEL TH ´ ERA ‡ Abstract. This paper was motivated by the need to establish some new characterizations of the metric regularity of set-valued mappings. Through these new characterizations it was possible to investigate the global/local perturbation stability of the metric regularity and to extend a result by Ioffe [Set-Valued Anal., 9 (2001), pp. 101–109] on the perturbation stability of the global metric regularity when the image space is not necessarily complete. It was also possible to give a charac- terization of the local metric regularity and to derive a local version of the perturbation stability of the metric regularity. In this work we also describe an application of this perturbation stability and give a simple proof of a result on the error bound of 2-regular mappings established by Izmailov and Solodov [Math. Program., 89 (2001), pp. 413–435] and generalized by He and Sun [Math. Oper. Res., 30 (2005), pp. 701–717]. Key words. error bound, perturbation stability, metric regularity, generalized equations AMS subject classifications. 49J52, 49J53, 90C30 DOI. 10.1137/060675721 1. Introduction. Let X and Y be metric spaces endowed with metrics both denoted by d(·, ·). The open ball with center x and radius r> 0 is denoted by B(x, r). We recall that a set-valued (multivalued) mapping F : X ⇒ Y is a mapping which assigns to every x ∈ X a subset (possibly empty) F (x) of Y . As usual, we use the notation gph F := {(x, y) ∈ X × Y : y ∈ F (x)} for the graph of F , Dom F := {x ∈ X : F (x) = ∅} for the domain of F , and F -1 : Y ⇒ X for the inverse of F . This inverse (which always exists) is defined by F -1 (y) := {x ∈ X : y ∈ F (x)},y ∈ Y , and satisfies (x, y) ∈ gph F ⇐⇒ (y,x) ∈ gph F -1 . It is well known that a large amount of problems, for instance, inequalities and equal- ities systems, variational inequalities, or systems of optimality conditions, are covered by the solvability of a generalized equation (in the terminology of Robinson). For a given y ∈ Y , determine x ∈ X such that y ∈ F (x). In general F is of the form f + T , where f : X → Y and T : X ⇒ Y . An important subcase is furnished by variational inequalities, that is, the problem of finding a solution to the equation y ∈ f (x)+ N C (x), where T = N C is the normal- cone operator. For each x ∈ R n , the set N C (x) is the normal cone (in the sense of convex analysis) to a closed convex set C of R n at x. A central issue in variational analysis is to investigate the behavior of the set of solutions of a generalized equation associated to F , that is, the behavior of the ∗ Received by the editors November 22, 2006; accepted for publication (in revised form) June 25, 2007; published electronically DATE. http://www.siam.org/journals/siopt/x-x/67572.html † Department of Mathematics, University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam (uguiakhiem@yahoo.com). This author’s research was supported by XLIM (Department of Mathe- matics and Informatics), UMR 6172, Universit´ e de Limoges, and by PICS CNRS Formath Vietnam. ‡ Laboratoire XLIM, UMR-CNRS 6172, Universit´ e de Limoges, 87060 Limoges cedex, France (Michel.thera@unilim.fr). This author’s research was partially supported by “Fondation EADS” and by Agence Nationale de la Recherche under grant ANR NT05 - 1/43040. 1