The problem of control of a parabolic system, which describes the heating of a given number of rods, is considered. The density functions of the internal heat sources of the rods are not exactly known, and only the segment of their change is given. Control are point heat sources that are located at the ends of the rods. The goal of the choice of control is to ensure that at a fixed time the modulus of the linear function determined using the average temperatures of the rods does not exceed the given value for any admissible functions of the density of internal heat sources. A technique has been developed for reducing this problem to a one-dimensional control problem under uncertainty. Necessary and sufficient termination conditions are found.
{"title":"On one problem of controlling the heating of a rod system under uncertainty","authors":"I. Izmest'ev, V. Ukhobotov","doi":"10.35634/vm220404","DOIUrl":"https://doi.org/10.35634/vm220404","url":null,"abstract":"The problem of control of a parabolic system, which describes the heating of a given number of rods, is considered. The density functions of the internal heat sources of the rods are not exactly known, and only the segment of their change is given. Control are point heat sources that are located at the ends of the rods. The goal of the choice of control is to ensure that at a fixed time the modulus of the linear function determined using the average temperatures of the rods does not exceed the given value for any admissible functions of the density of internal heat sources. A technique has been developed for reducing this problem to a one-dimensional control problem under uncertainty. Necessary and sufficient termination conditions are found.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84293376","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}
This paper addresses the problem of a roller-racer rolling on an oscillating plane. Equations of motion of the roller-racer in the form of a system of four nonautonomous differential equations are obtained. Two families of particular solutions are found which correspond to rectilinear motions of the roller-racer along and perpendicular to the plane's oscillations. Numerical estimates are given for the multipliers of solutions corresponding to the motion of the robot along the oscillations. Also, a special case is presented in which it is possible to obtain analytic expressions of the multipliers. In this case, it is shown that the motion along oscillations of a “folded” roller-racer is linearly orbitally stable as it moves with its joint ahead, and that all other motions are unstable. It is shown that, in a linear approximation, the family corresponding to the motion of the robot is perpendicular to the plane's oscillations, that is, it is unstable.
{"title":"Investigation of the orbital stability of rectilinear motions of roller-racer on a vibrating plane","authors":"E. Artemova, A. Kilin, Yu.V. Korobeinikova","doi":"10.35634/vm220408","DOIUrl":"https://doi.org/10.35634/vm220408","url":null,"abstract":"This paper addresses the problem of a roller-racer rolling on an oscillating plane. Equations of motion of the roller-racer in the form of a system of four nonautonomous differential equations are obtained. Two families of particular solutions are found which correspond to rectilinear motions of the roller-racer along and perpendicular to the plane's oscillations. Numerical estimates are given for the multipliers of solutions corresponding to the motion of the robot along the oscillations. Also, a special case is presented in which it is possible to obtain analytic expressions of the multipliers. In this case, it is shown that the motion along oscillations of a “folded” roller-racer is linearly orbitally stable as it moves with its joint ahead, and that all other motions are unstable. It is shown that, in a linear approximation, the family corresponding to the motion of the robot is perpendicular to the plane's oscillations, that is, it is unstable.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73958707","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}
A module $M$ is called pseudo semi-projective if, for all $alpha,betain mathrm{End}_R(M)$ with $mathrm{Im}(alpha)=mathrm{Im}(beta)$, there holds $alpha, mathrm{End}_R(M)=beta, mathrm{End}_R(M)$. In this paper, we study some properties of pseudo semi-projective modules and their endomorphism rings. It is shown that a ring $ R$ is a semilocal ring if and only if each semiprimitive finitely generated right $R$-module is pseudo semi-projective. Moreover, we show that if $M$ is a coretractable pseudo semi-projective module with finite hollow dimension, then $mathrm{End}_R(M)$ is a semilocal ring and every maximal right ideal of $mathrm{End}_R(M)$ has the form ${s in mathrm{End}_R(M) | mathrm{Im}(s) + mathrm{Ker}(h)ne M}$ for some endomorphism $h$ of $M$ with $h(M)$ hollow.
{"title":"Pseudo semi-projective modules and endomorphism rings","authors":"N. Ha","doi":"10.35634/vm220405","DOIUrl":"https://doi.org/10.35634/vm220405","url":null,"abstract":"A module $M$ is called pseudo semi-projective if, for all $alpha,betain mathrm{End}_R(M)$ with $mathrm{Im}(alpha)=mathrm{Im}(beta)$, there holds $alpha, mathrm{End}_R(M)=beta, mathrm{End}_R(M)$. In this paper, we study some properties of pseudo semi-projective modules and their endomorphism rings. It is shown that a ring $ R$ is a semilocal ring if and only if each semiprimitive finitely generated right $R$-module is pseudo semi-projective. Moreover, we show that if $M$ is a coretractable pseudo semi-projective module with finite hollow dimension, then $mathrm{End}_R(M)$ is a semilocal ring and every maximal right ideal of $mathrm{End}_R(M)$ has the form ${s in mathrm{End}_R(M) | mathrm{Im}(s) + mathrm{Ker}(h)ne M}$ for some endomorphism $h$ of $M$ with $h(M)$ hollow.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81518472","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}
{"title":"On totally global solvability of evolutionary Volterra equation of the second kind","authors":"A. Chernov","doi":"10.35634/vm220407","DOIUrl":"https://doi.org/10.35634/vm220407","url":null,"abstract":"","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78927960","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}
We consider a nonlinear control system in a finite-dimensional Euclidean space and on a finite time interval, which depends on a parameter. Reachable sets and integral funnels of a differential inclusion corresponding to a control system containing a parameter are studied. When studying numerous problems of control theory and differential games, constructing their solutions and estimating errors, various theoretical approaches and associated computational methods are used. The problems mentioned above include, for example, various types of approach problems, the resolving constructions of which can be described quite simply in terms of reachable sets and integral funnels. In this paper, we study the dependence of reachable sets and integral funnels on a parameter: the degree of this dependence on a parameter is estimated under certain conditions on the control system. The degree of dependence of the integral funnels is investigated for the change in their volume with a change in the parameter. To estimate this dependence, systems of sets in the phase space are introduced that approximate the reachable sets and integral funnels on a given time interval corresponding to a finite partition of this interval. In this case, the degree of dependence of the approximating system of sets on the parameter is first estimated, and then this estimate is used in estimating the dependence of the volume of the integral funnel of the differential inclusion on the parameter. This approach is natural and especially useful in the study of specific applied control problems, in solving which, in the end, one has to deal not with ideal reachable sets and integral funnels, but with their approximations corresponding to a discrete representation of the time interval.
{"title":"On the parametric dependence of the volume of integral funnels and their approximations","authors":"V. Ushakov, A. Ershov","doi":"10.35634/vm220307","DOIUrl":"https://doi.org/10.35634/vm220307","url":null,"abstract":"We consider a nonlinear control system in a finite-dimensional Euclidean space and on a finite time interval, which depends on a parameter. Reachable sets and integral funnels of a differential inclusion corresponding to a control system containing a parameter are studied. When studying numerous problems of control theory and differential games, constructing their solutions and estimating errors, various theoretical approaches and associated computational methods are used. The problems mentioned above include, for example, various types of approach problems, the resolving constructions of which can be described quite simply in terms of reachable sets and integral funnels.\u0000In this paper, we study the dependence of reachable sets and integral funnels on a parameter: the degree of this dependence on a parameter is estimated under certain conditions on the control system. The degree of dependence of the integral funnels is investigated for the change in their volume with a change in the parameter. To estimate this dependence, systems of sets in the phase space are introduced that approximate the reachable sets and integral funnels on a given time interval corresponding to a finite partition of this interval. In this case, the degree of dependence of the approximating system of sets on the parameter is first estimated, and then this estimate is used in estimating the dependence of the volume of the integral funnel of the differential inclusion on the parameter. This approach is natural and especially useful in the study of specific applied control problems, in solving which, in the end, one has to deal not with ideal reachable sets and integral funnels, but with their approximations corresponding to a discrete representation of the time interval.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85285937","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}
Let $G=(V,E)$ be a graph of order $p$ and size $q$ having no isolated vertices. A bijection $fcolon Ehm{rightarrow}left{1,2,3,ldots,q right}$ is called a local antimagic labeling if for all $uvin E$, we have $w(u)neq w(v)$, the weight $w(u)=sum_{ein E(u)}f(e)$, where $E(u)$ is the set of edges incident to $u$. A graph $G$ is local antimagic, if $G$ has a local antimagic labeling. The local antimagic chromatic number $chi_{la}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local antimagic labelings of $G$. In this paper, we completely determine the local antimagic chromatic number for the corona product of wheel and null graphs.
{"title":"Local antimagic chromatic number for the corona product of wheel and null graphs","authors":"Rathinavel Shankar, M. Nalliah","doi":"10.35634/vm220308","DOIUrl":"https://doi.org/10.35634/vm220308","url":null,"abstract":"Let $G=(V,E)$ be a graph of order $p$ and size $q$ having no isolated vertices. A bijection $fcolon Ehm{rightarrow}left{1,2,3,ldots,q right}$ is called a local antimagic labeling if for all $uvin E$, we have $w(u)neq w(v)$, the weight $w(u)=sum_{ein E(u)}f(e)$, where $E(u)$ is the set of edges incident to $u$. A graph $G$ is local antimagic, if $G$ has a local antimagic labeling. The local antimagic chromatic number $chi_{la}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local antimagic labelings of $G$. In this paper, we completely determine the local antimagic chromatic number for the corona product of wheel and null graphs.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80802148","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}
In the first part of the paper, the nonlinear metric space $langleoverline{rm G}^infty[a,b],drangle$ is defined and studied. It consists of functions defined on the interval $[a,b]$ and taking the values in the extended numeric axis $overline{mathbb R}$. For any $xinoverline{rm G}^infty[a,b]$ and $tin(a,b)$ there are limit numbers $x(t-0),x(t+0) inoverline{mathbb R}$ (and numbers $x(a+0),x(b-0)inoverline{mathbb R}$). The completeness of the space is proved. It is the closure of the space of step functions in the metric $d$. In the second part of the work, the nonlinear space ${rm RL}[a,b]$ is defined and studied. Every piecewise smooth function defined on $[a,b]$ is contained in ${rm RL}[a,b]$. Every function $xin{rm RL}[a,b]$ has bounded variation. All one-sided derivatives (with values in the metric space $langleoverline{mathbb R},varrhorangle$) are defined for it. The function of left-hand derivatives is continuous on the left, and the function of right-hand derivatives is continuous on the right. Both functions extended to the entire interval $[a,b]$ belong to the space $overline{rm G}^infty[a,b]$. In the final part of the paper, two subspaces of the space ${rm RL}[a,b]$ are defined and studied. In subspaces, promising formulations for the simplest variational problems are stated and discussed.
{"title":"On nonlinear metric spaces of functions of bounded variation","authors":"V. N. Baranov, V. Rodionov","doi":"10.35634/vm220301","DOIUrl":"https://doi.org/10.35634/vm220301","url":null,"abstract":"In the first part of the paper, the nonlinear metric space $langleoverline{rm G}^infty[a,b],drangle$ is defined and studied. It consists of functions defined on the interval $[a,b]$ and taking the values in the extended numeric axis $overline{mathbb R}$. For any $xinoverline{rm G}^infty[a,b]$ and $tin(a,b)$ there are limit numbers $x(t-0),x(t+0) inoverline{mathbb R}$ (and numbers $x(a+0),x(b-0)inoverline{mathbb R}$). The completeness of the space is proved. It is the closure of the space of step functions in the metric $d$. In the second part of the work, the nonlinear space ${rm RL}[a,b]$ is defined and studied. Every piecewise smooth function defined on $[a,b]$ is contained in ${rm RL}[a,b]$. Every function $xin{rm RL}[a,b]$ has bounded variation. All one-sided derivatives (with values in the metric space $langleoverline{mathbb R},varrhorangle$) are defined for it. The function of left-hand derivatives is continuous on the left, and the function of right-hand derivatives is continuous on the right. Both functions extended to the entire interval $[a,b]$ belong to the space $overline{rm G}^infty[a,b]$. In the final part of the paper, two subspaces of the space ${rm RL}[a,b]$ are defined and studied. In subspaces, promising formulations for the simplest variational problems are stated and discussed.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77476025","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}
In this paper directional derivative sets and differentials of a given set valued map are studied. Different type relations between directional derivative sets and differentials of a set valued map are specified. It is established that every compact subset of lower derivative set can be used for lower approximation of given set valued map. Upper and lower contingent cones of some plane sets are calculated and compared.
{"title":"Upper and lower directional derivative sets and differentials of the set valued maps","authors":"Nihal Ege, A. Huseyin, N. Huseyin","doi":"10.35634/vm220304","DOIUrl":"https://doi.org/10.35634/vm220304","url":null,"abstract":"In this paper directional derivative sets and differentials of a given set valued map are studied. Different type relations between directional derivative sets and differentials of a set valued map are specified. It is established that every compact subset of lower derivative set can be used for lower approximation of given set valued map. Upper and lower contingent cones of some plane sets are calculated and compared.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88070541","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}
In this paper, we consider a boundary value problem for differential equations of Langevin type with the Caputo fractional derivative in a Banach space. It is assumed that the nonlinear part of the equation is a Caratheodory type map. Equations of this type generalize equations of motion in various kinds of media, for example, viscoelastic media or in media where a drag force is expressed using a fractional derivative. We will use the theory of fractional mathematical analysis, the properties of the Mittag-Leffler function, as well as the theory of measures of non-compactness and condensing operators to solve the problem. The initial problem is reduced to the problem of the existence of fixed points of the corresponding resolving integral operator in the space of continuous functions. We will use Sadovskii type fixed point theorem to prove the existence of fixed points of the resolving operator. We will show that the resolving integral operator is condensing with respect to the vector measure of non-compactness in the space of continuous functions and transforms a closed ball in this space into itself.
{"title":"On a boundary value problem for a class of fractional Langevin type differential equations in a Banach space","authors":"G. Petrosyan","doi":"10.35634/vm220305","DOIUrl":"https://doi.org/10.35634/vm220305","url":null,"abstract":"In this paper, we consider a boundary value problem for differential equations of Langevin type with the Caputo fractional derivative in a Banach space. It is assumed that the nonlinear part of the equation is a Caratheodory type map. Equations of this type generalize equations of motion in various kinds of media, for example, viscoelastic media or in media where a drag force is expressed using a fractional derivative. We will use the theory of fractional mathematical analysis, the properties of the Mittag-Leffler function, as well as the theory of measures of non-compactness and condensing operators to solve the problem. The initial problem is reduced to the problem of the existence of fixed points of the corresponding resolving integral operator in the space of continuous functions. We will use Sadovskii type fixed point theorem to prove the existence of fixed points of the resolving operator. We will show that the resolving integral operator is condensing with respect to the vector measure of non-compactness in the space of continuous functions and transforms a closed ball in this space into itself.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79683225","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}
Set-valued mappings acting from a partially ordered space $X=(X,leq)$ to a set $Y$ on which a reflexive binary relation $vartheta$ is given (this relation is not supposed to be antisymmetric or transitive, i.e., $vartheta$ is not an order in $Y$), are considered. For such mappings, analogues of the concepts of covering and monotonicity are introduced. These concepts are used to study the inclusion $F(x)ni tilde{y},$ where $Fcolon X rightrightarrows Y,$ $tilde{y}in Y.$ It is assumed that for some given $x_0 in X,$ there exists $y_{0} in F(x_{0})$ such that $(tilde{y},y_{0}) in vartheta.$ Conditions for the existence of a solution $xin X$ satisfying the inequality $xleq x_0$ are obtained, as well as those for the existence of minimal and least solutions. The property of stability of solutions of the considered inclusion to changes of the set-valued mapping $F$ and of the element $widetilde{y}$ is also defined and investigated. Namely, the sequence of “perturbed” inclusions $F_i(x)ni tilde{y}_i,$ $iin mathbb{N},$ is assumed, and the conditions of existence of solutions $x_i in X$ such that for any increasing sequence of integers ${i_n}$ there holds $sup_{n in mathbb{N}}{x_{i_{n}}}= x,$ where $x in X$ is a solution of the initial inclusion, are derived.
考虑从部分有序空间$X=(X,leq)$到给定自反二元关系$vartheta$的集合$Y$的集值映射(该关系不应该是反对称的或传递的，即$vartheta$不是$Y$中的一个顺序)。对于这样的映射，引入了覆盖和单调性概念的类似概念。这些概念用于研究包含$F(x)ni tilde{y},$，其中$Fcolon X rightrightarrows Y,$$tilde{y}in Y.$假设对于某些给定的$x_0 in X,$存在$y_{0} in F(x_{0})$，使得$(tilde{y},y_{0}) in vartheta.$解$xin X$满足不等式$xleq x_0$的存在条件，以及最小解和最小解的存在条件。定义并研究了所考虑的集值映射$F$和元素$widetilde{y}$的变化包含解的稳定性。也就是说，假设“扰动”包含序列$F_i(x)ni tilde{y}_i,$$iin mathbb{N},$，并推导出解$x_i in X$存在的条件，使得对于任何递增的整数序列${i_n}$存在$sup_{n in mathbb{N}}{x_{i_{n}}}= x,$，其中$x in X$是初始包含的解。
{"title":"On one inclusion with a mapping acting from a partially ordered set to a set with a reflexive binary relation","authors":"S. Benarab, E. Panasenko","doi":"10.35634/vm220302","DOIUrl":"https://doi.org/10.35634/vm220302","url":null,"abstract":"Set-valued mappings acting from a partially ordered space $X=(X,leq)$ to a set $Y$ on which a reflexive binary relation $vartheta$ is given (this relation is not supposed to be antisymmetric or transitive, i.e., $vartheta$ is not an order in $Y$), are considered. For such mappings, analogues of the concepts of covering and monotonicity are introduced. These concepts are used to study the inclusion $F(x)ni tilde{y},$ where $Fcolon X rightrightarrows Y,$ $tilde{y}in Y.$ It is assumed that for some given $x_0 in X,$ there exists $y_{0} in F(x_{0})$ such that $(tilde{y},y_{0}) in vartheta.$ Conditions for the existence of a solution $xin X$ satisfying the inequality $xleq x_0$ are obtained, as well as those for the existence of minimal and least solutions. The property of stability of solutions of the considered inclusion to changes of the set-valued mapping $F$ and of the element $widetilde{y}$ is also defined and investigated. Namely, the sequence of “perturbed” inclusions $F_i(x)ni tilde{y}_i,$ $iin mathbb{N},$ is assumed, and the conditions of existence of solutions $x_i in X$ such that for any increasing sequence of integers ${i_n}$ there holds $sup_{n in mathbb{N}}{x_{i_{n}}}= x,$ where $x in X$ is a solution of the initial inclusion, are derived.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79168475","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}
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