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1. CJM 2006 (vol 58 pp. 64)

Filippakis, Michael; Gasiński, Leszek; Papageorgiou, Nikolaos S.
 Multiplicity Results for Nonlinear Neumann Problems In this paper we study nonlinear elliptic problems of Neumann type driven by the $p$-Laplac\-ian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a $C^1$-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-$p$-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative $p$-Laplacian with Neumann boundary condition. Keywords:Nonsmooth critical point theory, locally Lipschitz function,, Clarke subdifferential, Neumann problem, strong resonance,, second deformation theorem, nonsmooth symmetric mountain pass theorem,, $p$-LaplacianCategories:35J20, 35J60, 35J85

2. CJM 2004 (vol 56 pp. 825)

Penot, Jean-Paul
 Differentiability Properties of Optimal Value Functions Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results. Keywords:differentiability, generic, marginal, performance function, subdifferentialCategories:26B05, 65K10, 54C60, 90C26, 90C48

3. CJM 1999 (vol 51 pp. 250)

Combari, C.; Poliquin, R.; Thibault, L.
 Convergence of Subdifferentials of Convexly Composite Functions In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlev\'e-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions. Keywords:epi-convergence, Mosco convergence, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functionsCategories:49A52, 58C06, 58C20, 90C30

4. CJM 1999 (vol 51 pp. 26)

Fabian, Marián; Mordukhovich, Boris S.
 Separable Reduction and Supporting Properties of FrÃ©chet-Like Normals in Banach Spaces We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals. Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chet-like normals and subdifferentials, supporting properties, Asplund spacesCategories:49J52, 58C20, 46B20