Bukovinian Mathematical Journal最新文献_第3页

THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS 一类具有光滑符号的伪微分方程的非局部时间问题

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2021.01.09

R. Kolisnyk, V. Gorodetskyi, O. Martynyuk

{"title":"THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS","authors":"R. Kolisnyk, V. Gorodetskyi, O. Martynyuk","doi":"10.31861/bmj2021.01.09","DOIUrl":"https://doi.org/10.31861/bmj2021.01.09","url":null,"abstract":"In this paper we investigate the differential-operator equation\u0000$$\u0000partial u (t, x) / partial t + varphi (i partial / partial x) u (t, x) = 0, quad (t, x) in (0, + infty) times mathbb {R} equiv Omega,\u0000$$\u0000where the function $ varphi in C ^ {infty} (mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i partial / partial x $, in $ L_2 (mathbb {R}) $ it is established that the operator $ varphi (i partial / partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ partial u / partial t + sqrt {I- Delta} u = 0 $, $ Delta = D_x ^ 2 $, with the fractionation differentiation operator $ sqrt { I- Delta} = varphi (i partial / partial x) $, where $ varphi (sigma) = (1+ sigma ^ 2) ^ {1/2} $, $ sigma in mathbb {R} $ is attributed to the considered equation.\u0000\u0000Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, cdot) $ for $ t to + infty $ (solution stabilization) in spaces of type $ S '$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125226556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

STOKES SYSTEM WITH VARIABLE EXPONENTS OF NONLINEARITY 非线性变指数Stokes系统

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2022.02.03

O. Buhrii, M. Khoma

引用次数: 0

ON SOLUTIONS OF THE NONHOMOGENEOUS CAUCHY PROBLEM FOR PARABOLIC TYPE DIFFERENTIAL EQUATIONS IN A BANACH SPACE banach空间中抛物型微分方程非齐次柯西问题的解

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2022.02.02

V. Gorbachuk

引用次数: 0

OPTIMAL CONTROL IN THE MULTIPOINT BOUNDARY VALUE PROBLEM FOR 2B-PARABOLIC EQUATIONS 2b -抛物型方程多点边值问题的最优控制

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2022.01.10

I. Pukalskyi, I. Luste

{"title":"OPTIMAL CONTROL IN THE MULTIPOINT BOUNDARY VALUE PROBLEM FOR 2B-PARABOLIC EQUATIONS","authors":"I. Pukalskyi, I. Luste","doi":"10.31861/bmj2022.01.10","DOIUrl":"https://doi.org/10.31861/bmj2022.01.10","url":null,"abstract":"The potential theory method was used to study the existence of a solution of a multi-\u0000point boundary value problem for a 2b-parabolic equation. Using the Green’s function of a\u0000homogeneous boundary value problem for a 2b-parabolic equation, the integral Fredholm equation of the second kind is placed in accordance with the multipoint boundary value problem.\u0000Taking into account the constraints on the coeﬃcients of the nonlocal condition and using the\u0000sequential approximation method, an integrated image of the solution of the nonlocal problem\u0000at the initial moment of time and its estimation in the Holder spaces are found. Estimates of\u0000the solution of a nonlocal multipoint boundary value problem at ﬁxed moments of time given in\u0000a nonlocal condition are found by means of estimates of the components of the Green’s function of the general boundary value problem for the 2b-parabolic equation. Taking into account\u0000the obtained estimates and constraints on coeﬃcients in multipoint problem, estimates of the\u0000solution of the multipoint problem for the 2b-parabolic equations and its derivatives in Holder\u0000spaces are established. In addition, the uniqueness and integral image of the solution of the\u0000general multipoint problem for 2b-parabolic equations is justiﬁed. The obtained result is applied to the study of the optimal system control problem described by the general multipoint\u0000boundary value problem for 2b-parabolic equations. The case of simultaneous internal, initial\u0000and boundary value control of solutions to a multipoint parabolic boundary value problem is\u0000considered. The quality criterion is deﬁned by the sum of volume and surface integrals. The\u0000necessary and suﬃcient conditions for the existence of an optimal solution of the system described by the general multipoint boundary value problem for 2b-parabolic equations with limited\u0000internal, initial and boundary value control are established.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130760212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

CENTER PROBLEM FOR CUBIC DIFFERENTIAL SYSTEMS WITH THE LINE AT INFINITY AND AN AFFINE REAL INVARIANT STRAIGHT LINE OF TOTAL MULTIPLICITY FOUR 具有无穷远线和总倍数为4的仿射实不变直线的三次微分系统的中心问题

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2021.02.03

A. Suba, O. Vacaras

引用次数: 0

CENTER CONDITIONS FOR A CUBIC DIFFERENTIAL SYSTEM HAVING AN INTEGRATING FACTOR 具有积分因子的三次微分系统的中心条件

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2020.02.01

D. Cozma, A. Matei

引用次数: 1

Instability of unbounded solutions of evolution equations with operator coefficients commuting with rotation operators 算子系数与旋转算子可交换的演化方程无界解的不稳定性

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2019.01.099

V. Slyusarchuk

引用次数: 1

STABILITY OF CONTROLLED STOCHASTIC DYNAMIC SYSTEMS OF RANDOM STRUCTURE WITH MARKOV SWITCHES AND POISSON PERTURBATIONS 具有马尔可夫开关和泊松扰动的受控随机结构动态系统的稳定性

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2022.01.08

T. Lukashiv, I. Malyk

{"title":"STABILITY OF CONTROLLED STOCHASTIC DYNAMIC SYSTEMS OF RANDOM STRUCTURE WITH MARKOV SWITCHES AND POISSON PERTURBATIONS","authors":"T. Lukashiv, I. Malyk","doi":"10.31861/bmj2022.01.08","DOIUrl":"https://doi.org/10.31861/bmj2022.01.08","url":null,"abstract":"Lyapunov’s second method is used to study the problem of stability of controlled\u0000stochastic dynamical systems of random structure with Markov and Poisson perturbations. Markov switches reﬂect random eﬀects on the system at ﬁxed points in time.\u0000Poisson perturbations describe random eﬀects on the system at random times. In both\u0000cases there may be breaks in the phase trajectory of the ﬁrst kind.\u0000The conditions for the coeﬃcients of the system are written, which guarantee the\u0000existence and uniqueness of the solution of the stochastic system of a random structure, which is under the action of Markov switches and Poisson perturbations. The diﬀerences between these systems and systems that do not contain internal perturbations in the equation, which cause a change in the structure of the system, and external perturbations, which cause breaks in the phase trajectory at ﬁxed points in time, are discussed. The upper bound of the solution for the norm is obtained. The deﬁnition of the discrete Lyapunov operator based on the system and the Lyapunov function for the above-mentioned systems is given.\u0000Suﬃcient conditions of asymptotic stochastic stability in general, stability in l.i.m.\u0000and asymptotic stability in the l.i.m. for controlled stochastic dynamic systems of random structure with Markov switches and Poisson perturbations are obtained.\u0000A model example that reﬂects the features of the stability of the solution of a system\u0000with perturbations is considered: the conditions of asymptotic stability in the root mean\u0000square as a whole are established; the conditions of exponential stability and exponential instability are discussed. For linear systems, the necessary and suﬃcient stability conditions are determined in the example, based on the generalized Lyapunov exponent.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126711142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

ABOUT ONE CLASS OF FUNCTIONS WITH FRACTAL PROPERTIES 关于一类具有分形性质的函数

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/bmj2021.01.23

Y. Goncharenko, M. Pratsiovytyi, S. Dmytrenko, I. Lysenko, S. Ratushniak

引用次数: 4

APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS 用重复fejer和逼近泊松积分类

Bukovinian Mathematical Journal Pub Date : 1900-01-01 DOI: 10.31861/BMJ2020.02.10

JohnRichens, 郭利劭

{"title":"APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS","authors":"JohnRichens, 郭利劭","doi":"10.31861/BMJ2020.02.10","DOIUrl":"https://doi.org/10.31861/BMJ2020.02.10","url":null,"abstract":"The paper is devoted to the approximation by arithmetic means of Fourier sums of classes\u0000of periodic functions of high smoothness. The simplest example of a linear approximation\u0000of continuous periodic functions of a real variable is the approximation by partial sums of the\u0000Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class\u0000of continuous periodic functions. A signiﬁcant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to\u0000construct trigonometrical polynomials that would be uniformly convergent for each continuous\u0000function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely\u0000studied. One of the most important direction in this ﬁeld is the study of the asymptotic behavior\u0000of upper bounds of deviations of linear means of Fourier sums on diﬀerent classes of periodic\u0000functions. Methods of investigation of integral representations of deviations of trigonometric\u0000polynomials generated by linear methods of summation of Fourier series, were originated and\u0000developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and\u0000others.\u0000The aim of the work systematizes known results related to the approximation of classes\u0000of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for\u0000particular cases. In the paper is studied the approximative properties of repeated Fejer sums on\u0000the classes of periodic analytic functions of real variable. Under certain conditions, we obtained\u0000asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson\u0000integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky\u0000problem without any additional conditions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129637622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2