Calculus of Variations with Classical and Fractional Derivatives

MSC 2010: 49K05, 26A33

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-Valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gˆateaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions.

Fractional Calculus of the Generalized Wright Function

Mathematics Subject Classification: 26A33, 33C20.

Fractional Calculus of P-transforms

MSC 2010: 44A20, 33C60, 44A10, 26A33, 33C20, 85A99

Principles of Extremum and Application to some Problems of Analysis

AMS subject classification: 41A17, 41A50, 49Kxx, 90C25.

Some Inequalities of the Uniform Ergodicity and Strong Stability of Homogeneous Markov Chains

2000 Mathematics Subject Classification: 60J45, 60K25

Operational Calculi for the Euler Operator

2000 Mathematics Subject Classification: 44A40, 44A35

Motzkin Decomposable Sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. We have provided several characterizations of the larger class of closed convex sets, Motzkin decomposable, in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed. Another result establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded. We characterize the class of the extended functions whose epigraphs are Motzkin decomposable sets showing, in particular, that these functions attain their global minima when they are bounded from below. Calculus of Motzkin decomposable sets and functions is provided.

Pseudodifferential Operators and Weighted Normed Symbol Spaces

2000 Mathematics Subject Classification: 35S05.