Vol 22, No 6 (2017)

For the Poisson problem -∆u+p x u- Ωu s r x, ds=ρf, u | Γ( Ω )=0 equivalence of positivity of the Green function and other classical properties is showed. Here Ω is an open set in R n , and Γ( Ω ) is the boundary of the Ω . For almost all x∈ Ω , r(x, ∙) is a measure satisfying certain symmetry condition. In particular this equation involves integral differential equation and the equation -∆u+p x u(x)- i=1 m p i x u h i x =ρf, where h i : Ω→Ω is a measurable mapping.

An inclusion with multi-valued mapping acting in spaces with vector-valued metrics is under discussion. It is shown that, if a multi-valued mapping F can be written as F(x)= Y(x, x), where the mapping Y is closed and metrically regular with some operator coefficient K with respect to one argument, Lipschitz with operator coefficient Q with respect to the other argument, and the spectral radius of the operator KQ is less than one, then the inclusion F(x) ∋ y is solvable. The estimations of the vector-valued distance from a solution x of the inclusion to a given element x 0 are derived. In the second part of the paper, these results are used to investigate an integral inclusion of the implicit type with respect to the unknown integrable function.

An inverse problem with mixed boundary value conditions for the Poisson equation for bodies of constant thickness is considered, aiming to reconstruct the sources density distribution. A stable solution of the problem is obtained.

In the article, a statement about estimation of the closeness of solutions of the perturbed inclusion to a given continuous function is formulated. An application of this statement to the study of perturbation of a linear boundary value problem for functional-differential equations is considered.

Linearly-quadratic mappings of finite-dimensional linear spaces over reals are considered. A criterion for a linearly-quadratic mapping to be a homeomorphism is obtained.

The paper is a study of the existence of equilibruim points in the dynamic Walrasian-Evans-Samuelson model. Sufficient conditions for the existence of the vector-function of equilibruim prices are derived from the existence theorems for coincidence points of Lipschitz continuous and covering mappings.

We propose a method of studying singular differential inclusions based on the representation of such an inclusion in the form of an operator inclusion in some space of measurable functions depending on the type of a given singularity. To the operator inclusion we apply the results on Lipschitz perturbations of multi-valued covering mappings. The article consists of three sections. In the first one we give the necessary definitions and formulate the theorem [A. Arutyunov, V.A. de Oliveira, F.L. Pereira, E. Zhukovskiy, S. Zhukovskiy // Applicable Analysis, 2015, 94, № 1] on the Lipschitz perturbations of multi-valued covering mappings. In the second section we introduce special metric spaces of integrable functions and obtain sufficient conditions of covering for the multi-valued Nemytskii operator in such spaces. Finally, using the mentioned results, we derive the existence conditions for the Cauchy problem for an implicit singular differential inclusion.

Generalized ( q1 , q2 ) -quasimetric spaces are considered. For contraction mappings in these spaces sufficient conditions for existence of fixed points are obtained.

In article the method of construction of integrated parameters for the systems having hierarchical structure is offered. Feature of researched systems will be their multidimensionality and heterogeneity of characteristics making them. The problem of construction of a complex estimation of a condition of such objects or processes is actual for various branches of knowledge (economy, ecology, medicine). From the mathematical point of view construction of integrated parameters concerns to problems multicriteria the analysis of hierarchies, therefore a first step at construction of an integrated parameter is decomposition of object on parts making it. Such decomposition is convenient for representing as the graph. We use a linear function - the enclosed linear convolution with weight factors of the importance of each criterion to make a scalar function from vector criterion. Weight factors are in turn determined by method of hierarchy’s analysis, allowing to translate qualitative gradation of a condition of system in quantitative. Further the integrated parameter is considered as the criterion function subject to improvement in some optimum image.

The article considers an example of aggregation (merging) of neighborhood systems in the problem of mathematical modeling of the ventilation system of the cement production workshop. The aim of simulation is to optimize the operation of the ventilation system according to the criteria of energy consumption and environmental friendliness.