This paper focuses on second-order necessary optimality conditions for constrained optimization problems Banach spaces. For in the classical setting, where objective function is $$C^2$$ -smooth, we show that strengthened are valid if constraint set generalized polyhedral convex. a new just assumed to be $$C^1$$ -smooth and convex, establish sharp based Fréchet subdifferential of tangent set. Th...

We study the farthest point mapping in a p-normed space X in virtue of subdifferential of rx sup{{x − z p : z ∈ M}, where M is a weakly sequentially compact subset of X. We show that the set of all points in X which have farthest point in M contains a dense G δ subset of X.

We consider first order periodic differential inclusions in R . The presence of a subdifferential term incorporates in our framework differential variational inequalities in R . We establish the existence of extremal periodic solutions and we also obtain existence results for the “convex” and “nonconvex”problems.

Abstract In this paper, Sobolev-type conformable fractional stochastic evolution inclusions with Clarke subdifferential and nonlocal conditions are studied. By using calculus, analysis, properties of nonsmooth sufficient for controllability the considered problem established. Finally, an example is given to illustrate obtained results.

We revisit some basic concepts and ideas of the classical differential calculus convex analysis extending them to a broader frame. reformulate generalize notion Gateaux differentiability propose new notions generalized derivative subdifferential in an arbitrary topological vector space. Meaningful examples preserving key properties original are provided.

Since the early 1970's there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is \best" has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper it is shown that there are an innnite number of tangent cones with some of the nicest properties of Clarke's co...