Abstract. Necessary conditions are developed for a general problem in the calculus of variations in which the Lagrangian function, although finite, need not be Lipschitz continuous or convex in the velocity argument. For the first time in such a broadly nonsmooth, nonconvex setting, a full subgradient version of Euler’s equation is derived for an arc that furnishes a local minimum in the classical weak sense, and the Weierstrass inequality is shown to accompany it when the arc gives a local minimum in the strong sense. The results are achieved through new techniques in nonsmooth analysis. 1

This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools.

. We discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis. Keywords: Smooth variational principle, fuzzy sum rules, mean value inequalities and partially smooth spaces. Short Title: Partially smooth variational principles. AMS (1991) subject classification: 49J50, 49J52. 1 Introduction Smooth variational analysis [7] has been highly successful in providing tools for the study of non-- smooth analysis and optimization problems: especially when married to viscosity concepts [10, 17]. Outside of smoothable Banach spaces (thus, notably in / L 1 spaces) general constructions such as those of Ioffe [25, 28, 29] require a largely non--constructive intersection over smooth or finite-- dimensional subspaces. Equally, outside of Asplund or Fr'echet spaces the most puissant results [41, 42] fail. Nonetheless, many problems inevitably lie in large (nonsmooth or non--Fr'echet) spaces, X. In such settings the ...

. We show, largely using convex examples, that most of the core results for limiting subdifferential calculus fail without additional restrictions in infinite dimensional Banach spaces. Key Words. Nonsmooth analysis, subdifferentials, coderivatives, extremal principle, open mapping theorem, metric regularity, multiplier rules, compactly Lipschitzian conditions. AMS (1991) subject classification: 26B05. y Research was supported by NSERC and by the Shrum Endowment at Simon Fraser University. z Research was supported by the National Science Fundation under grant DMS-9704203 and by funds from the Faculty Research and Creative Activities Support Fund, Western Michigan University. 1 Introduction Through the work of [3, 4, 26, 27] it has become clear that smooth subdifferentials characterize many important generalized derivative concepts such as Clarke's generalized gradient, Ioffe's geometric subdifferential and Mordukhovich 'es limiting subdifferential. This renewed interest in smooth s...

. We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson-Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson-Ursescu theorem we also establish a refined version of the multidirectional mean value inequality which is of independent interest. Key Words. Nonsmooth analysis, subdifferentials, coderivatives, implicit function theorem, solvability, stability, open mapping theorem, metric regularity, multidirectional mean value inequality. AMS (1991) subject classification: 26B05. 1 Research...

We prove a "typical" subdifferentiability principle and apply it to a variety of complete metric spaces of continuous functions on separable Banach spaces; so as to obtain existence of functions with maximal subdifferentials when ordered by inclusion. The relationship between continuous functions with maximal subdifferentials and nowhere monotone functions is discussed.

I will describe the basic ideas from Banach space theory that allow one to fruitfully apply notions from smooth analysis in spaces which do not admit smooth renorms (or "bumps") and so to perform Partially Smooth Variational Analysis. I shall also mention some open functional analytic questions relating to best approximation, viscosity and variational analysis.

Abstract. I will also discuss open problems and current challenges for the subject Boris Mordukhovich has played a key role in the development of modern Variational Analysis (VA) and its Applications. Modern non-smooth analysis is now roughly thirty-five years old. In this paper I shall attempt to analyse (briefly): where the subject stands today, where it should be going, and what it will take to get there? Summary: the first order theory is rather impressive, as are some applications. The second order theory is by comparison somewhat underdeveloped and wanting. “It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully,but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.”—Carl Friedrich Gauss (1777-1855) 1 1. Preliminaries I intend to discuss First-Order Theory, and then Higher-Order Theory mainly second-order and only mention passingly higher-order theory which really devolves to second-order theory. I’ll finish by touching on Applications of Variational Analysis both inside and outside Mathematics, mentioning both successes and limitations or failures. Each topic leads to open questions even in the convex case (CA). Some are