Variational Principles and Mean Value Estimates

In this paper, for a continuous function defined on a closed subset of a Banach space and differentiable on the interior of the domain, the infimum of the norm of the first derivative is estimated through the variation of the function on the boundary of the domain. A similar result is obtained for locally Lipschitz functions. These results generalize the previously known estimates of the infimum of the norm of the derivative in terms of the variation of the function on the entire domain. Under the assumption that the considered function is twice differentiable on the interior of the domain, similar estimates for the infimum of the norm of the first-order derivative and the estimates of the values of the second-order derivative are obtained. The proof of these results is based on the application of the modifications of the Bishop–Phelps variational principle and the Borwein–Preiss smooth variational principle. As applications of these results, we obtain second-order necessary optimality conditions for minimizing sequences. This proposition is proved under an additional assumption on the norm of the Banach space valid for a wide class of function spaces. In addition, some of the described results are generalized for mappings, acting in Banach spaces ordered by pointed closed convex cones.