On optimality for Mayer type problem governed by a discrete inclusion system with Lipschitzian set-valued mappings

Set-valued optimization which is an extension of vector optimization to set-valued problems is a growing branch of applied mathematics. The application of vector optimization technics to set-valued problems and the investigation of optimality conditions has been of enormous interest in the research of optimization problems. In this paper we have considered a Mayer type problem governed by a discrete inclusion system with Lipschitzian set-valued mappings. A necessary condition for K-optimal solutions of the problem is given via local approximations which is considered the lower and upper tangent cones of a set and the lower derivative of the set-valued mappings.

Hausdorff dimension of sets of non-differentiability points of Cantor functions

Each number a in the segment (0, ½) produces a Cantor set, Ca, by putting b = 1 − 2a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervals which are subsequently generated. The distribution function of the uniform probability measure on Ca is a Cantor function, fa. When a = 1/3 = b, Ca is the standard Cantor set, C, and fa is the standard Cantor function, f. The upper derivative of f is infinite at each point of C and the lower derivative of f is infinite at most points of C in the following sense: the Hausdorff dimension of C is ln(2)/ln(3) and the Hausdorff dimension of S = {x ∈ C: the lower derivative of f is finite at x} is [ln(2)/ln(3)]2. The derivative of f is zero off C, the derivative of f is infinite on C — S, and S is the set of non-differentiability points of f. Similar results are established in this paper for all Ca: the Hausdorff dimension of Ca is ln (2)/ln (1/a) and the Hausdorff dimension of Sa is [ln (2)/ln (1/a)]2. Removing k segments of relative length b and leaving k + 1 intervals of relative length a produces a Cantor set of dimension ln(k + l)/ln(1/a); the dimension of the set of non-differentiability points of the corresponding Cantor function is [ln (k + l)/ln (1/a)]2.

On the Lower Derivate of a Set Function

In (5), the following theorem was proved in a very general setting:(1) An additive set function is non-negative whenever its lower derivative is non-negative.For a continuous additive function of intervals, theorem (1) can be improved as follows:(2) A continuous additive set function is non-negative whenever its lower derivative is non-negative except, perhaps, on a countable set.