SLOPES OF MULTIFUNCTIONS AND EXTENSIONS OF 1 METRIC REGULARITY 2 HUYNH VAN NGAI, ALEXANDER Y. KRUGER, AND MICHEL TH ´ ERA 3 Dedicated to Professor Phan Quoc Khanh on his 65th birthday Abstract. This article aims to demonstrate how the definitions of slopes can be extended to multi-valued mappings between metric spaces and ap- plied for characterizing metric regularity. Several kinds of local and nonlo- cal slopes are defined and several metric regularity properties for set-valued mappings between metric spaces are investigated. Keywords: variational analysis, error bounds, slope, multifunction, metric 4 regularity 5 Mathematics Subject Classification (2000): 49J52, 49J53, 58C06, 6 47H04, 54C60 7 1. Introduction 8 This article aims to demonstrate how the definitions of slopes which have 9 proved to be very useful tools for analyzing local properties of real-valued func- 10 tions [1–3, 6, 10–13, 15–17] can be extended to multi-valued mappings between 11 metric spaces and applied for characterizing metric regularity. 12 Several kinds of local and nonlocal slopes are defined in Section 2 following 13 the scheme developed in [10] for real-valued functions and extended in [4, 5] 14 to vector-valued functions. The idea is not quite new. Some elements of the 15 definitions introduced in the current article are present implicitly in many 16 publications [2, 3, 12, 13, 15, 16]. It seems the definitions can be useful and the 17 time has come to formulate them explicitly. 18 In this article we investigate several metric regularity properties for set- 19 valued mappings between metric spaces: 20 • conventional local metric regularity and uniform metric regularity for 21 mappings depending on a parameter (Section 3); 22 • metric regularity along a subspace (Section 4); 23 • metric multi-regularity for mappings into product spaces (Section 5) 24 and formulate the corresponding necessary and sufficient regularity criteria in 25 terms of slopes. For the definitions and characterizations of the mentioned 26 above extensions of metric regularity we refer the readers to [8,9]. 27 Our basic notation is standard, see [14, 18]. Depending on the context, X 28 and Y are either metric or normed spaces. Metrics in all spaces are denoted by 29 the same symbol d(·, ·). d(x, A) = inf a∈A ∥x − a∥ is the point-to-set distance 30 The research was partially supported by the Australian Research Council, grant DP110102011.