The purpose of this note is to prove an abstract operator version of certain existence theorems for differential and Volterra integral equations which deal with stability properties of solutions. We will do this in such a way that the results unify, as well as generalize, many of the basic theorems concerning stability of such equations. Thus, our main result(Theorem2)is an abstract version of ...

Interest in Sobolev type equations has recently increased signi cantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to ll up the theory but by the aspiration to comprehend nonclassical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if soluti...

We are to discuss how to view notions of computability for discontinuous functions. We con ne ourselves to real-valued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. I...

It has been known that the curvature of data spaces plays a role in data analysis. For example, the Frechet mean (intrinsic mean) always exists uniquely for a probability measure on a non-positively curved metric space. In this paper, we use the curvature of data spaces in a novel manner. A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. The p...

We prove a Lusternik-Schnirelmann type theorem for <em>C</em><sup>1</sup>- function φ : M → R, where is connected infinite dimensional Frechet manifold of class <em>C</em><sup>1</sup>. To this end, in context we the so-called Deformation Lemma and by using it derive result generalizing Minimax Principle.

Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that controls vanish on large parts underlying domain. Typical candidates for those integral Lebesgue spaces based $\ell_p$-metric $p\in[0,1)$ which nonconvex as well non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas Frechet, limiting, a...

The problems of modeling processes and systems the economy are considered, optimal control problem is posed, system nonlinear differential equations representing optimized target functional determined; a variational method for solving chosen, principle invariance possibilities provided by them dynamic, including economic, described. A critical analysis state in field economic given. features pr...