This paper presents new existence and stability results for fixed point inclusions associated with multi valued operators satisfying a Feng-Liu type contraction on a set endowed with two b-metrics. The obtained results extend and improve several recent contributions in this area. Examples are also given to support our findings.
The behavior of a defocusing semilinear integro-differential equation of hyperbolic type with zero initial conditions is investigated. By introducing a new variable and new functions, the problem is reduced to a form convenient for analysis. Using integration over characteristics, the necessary conditions for the problem data are obtained. Next, using d'Alembert's formula, the problem is reduced to solving a Volterra-type integral equation. The existence and uniqueness of a solution to this problem are proven using the contraction mapping method.
We introduce and study the so-called weakly $\sqrt{J}U$ rings (hereafter abbreviated as $W\sqrt{J}U$ rings for short), in which every unit is of the form $j+1$ or $j-1$ for some $j$ in $\sqrt{J(R)} : = \{x \in R\colon x^n \in J(R) \text{ for some } n\ge 1\}$. This class of rings non-trivially generalizes the classes of $\sqrt{J}U$, $UU$, $JU$, $WUU$ and $WJU$ rings, respectively. We investigate their basic properties showing that they are Dedekind-finite, that $M_n(R)$ is never $W\sqrt{J}U$ for $n\ge 2$, and that when $\operatorname{char}(R)>0$ it must be equal to $2^\alpha 3^\beta$ for some $\alpha, \beta \in \mathbb{N} \cup\{0\}$. Moreover, for group rings $RG$, we prove that if $RG$ is $W\sqrt{J}U$, then $R$ is $W\sqrt{J}U$ and $G$ is a torsion group. In addition, when $R$ has positive characteristic and $G$ is a locally finite $p$-group, we give a complete characterization like this: $RG$ is a $W\sqrt{J}U$ ring if, and only if, either $R$ is a $\sqrt{J}U$ ring and $G$ is a $2$-group, or $R$ is a $W\sqrt{J}U$ ring with $3\in J(R)$ and $G$ is a $3$-group, or $R\cong R_1\times R_2$ with $R_1$ a $\sqrt{J}U$ ring, $R_2$ a $W\sqrt{J}U$ ring and $G$ a trivial group. Our results substantially improve on recent achievements due to Saini and Udar in Czech. Math. J. (2025).
The paper is devoted to the comparison of two approaches to the numerical implementation of the option price determined in the Black-Scholes model. On the example of the European call option, in the paper, calculations of the option price are done using the Monte-Carlo method and the finite difference method. The application of the Monte-Carlo method is based on the presentation of the option price as a probabilistic characteristic of the underlying asset and the numerical solution of the stochastic differential equation (SDE) describing the change in the price of this underlying asset. In our work, the numerical solution of the SDE is constructed by the Euler-Maruyama method, and then the option price is found by the Monte-Carlo method. The use of the finite difference method is based on the relationship between stochastic differential equations and deterministic equations for the probabilistic characteristics of SDE solutions. Due to this relationship, the option price satisfies the partial differential equation. The application of these two approaches is demonstrated on the example of a specific equation for the price of the underlying asset. The numerical experiments made it possible to compare and evaluate the advantages and disadvantages of each approach.
In a finite-dimensional Euclidean space, we consider a nonstationary simple pursuit problem involving a group of pursuers and two evaders, with all participants having equal capabilities, under the condition that the evaders use the same control. The set of admissible controls for each player is a ball of radius one centered at the origin. The target sets are the origin. The goal of the group of pursuers is to capture at least one evader by a given number of pursuers. The goal of the evaders is the opposite. Sufficient conditions for capture and sufficient conditions for evasion are obtained.
Let X-0 subset of R-n be a nonempty open set and X-0 subset of X subset of X & strns;(0). In the author's previous paper, the Banach space G(X) of continuous bounded functions f: X -> R was defined, which was endowed with a special norm ||& centerdot;||. The definition of the norm involves an n-dimensional vector (Delta x)(-1 )Delta f, which is an analogue of the relation (triangle f)/(triangle x) , which generates the concept of the derivative of a function of one variable. The vector (Delta x)(-1 )Delta f can be associated with the vector grad f(& centerdot;) (the matrix Delta x is invertible). In this paper, on an alternative basis, we define a space H(X) such that G(X) = H(X). The paper presents two criteria for continuous differentiability and smoothness of functions of several variables (in terms of the spaces G(X) and H(X), respectively, without elements of traditional differential calculus). In the final section, for n 2 and delta is an element of (0, 1], instead of the expression (Delta x)(-delta)Delta f, which makes sense only for n = 1 or for delta = 1, a new construction is defined that generates a space Hp(X) such that p = delta(-1), H-1(X) = H(X) and F-delta ' (X) subset of H-p(X). By F-delta '(X) we denote the space of functions f: X -> R with Lipschitz-Holder condition such that sup |f (x)-f(y)|/||x-y||(delta) < infinity (where x, y is an element of X, x not equal y).
The problem of sequential bypassing megacities (non-empty finite sets) under decomposition conditions is investigated: a set of tasks is specified by a system of clusters, the order of service of which is set. Each cluster defines a partial routing task with precedence conditions and cost functions allowing task list dependence. The statement is oriented towards engineering applications connected with sheet cutting on CNC-machines; this refers to the task of cutting in zones that are determined by technological requirements. The procedure of compositional solutions optimization is constructed, which is based on the separate application of a broadly understood dynamic programming and a specific rule linking the conditions of partial problems by assigning terminal components of additive criteria based on the extremum functions of these partial problems. The constructed algorithm implemented on a multi-core PC; computational experiment in solving problems of appreciable dimension (hundreds of megacities) showed the possibility of solving the problem at time acceptable for engineering practice.
We consider controlled initial-boundary value problems for semilinear partial differential equations with first- and second-order time derivatives, represented as a Cauchy problem for a semilinear evolution equation in a Hilbert space with an unbounded skew-adjoint operator and an incoming linear control function. Based on the author's previously obtained results for this Cauchy problem, sufficient conditions are established for the exact controllability of the equations under consideration to a given final state (as well as to given intermediate states at intermediate times) over an arbitrarily fixed (without additional conditions) time interval. Furthermore, a theorem on the stability of the control process is proved for an abstract evolution equation.
The paper is devoted to the problem of numerically calculating the non-convexity measure of reachable sets of control systems represented in pixel form. The non-convexity measure considered in the paper in terms of alpha-sets is unstable in the Hausdorff metric. Therefore, a modified non-convexity measure is introduced, which coincides with the original one for weakly Vial convex sets and for polyhedra with a finite number of vertices. It is proved that the introduced modified non-convexity measure coincides with the original non-convexity measure alpha, calculated for a neighborhood of the set under consideration. The minimum thickness of this neighborhood is, in fact, a parameter of this modified measure. It is shown that pi-sets are unstable with respect to the modified non-convexity measure. However, it is proved that convex sets (0-sets) have a stable modified non-convexity measure with respect to small perturbations in the Hausdorff metric. The concept of a pseudo-Chebyshev layer is introduced. Using this concept, an asymptotic estimate is obtained for the change in the nonconvexity measure alpha arising from a small displacement of the polygon's vertices. A conjecture is put forward regarding the stability of the modified nonconvexity measure for alpha-sets with alpha < pi.
A group of mobile agents on a straight line is considered. First- and second-order integrators are used as agent models. Decentralized control protocols are proposed that provide both uniform and specified nonlinear uniform (uniform with respect to some function) deployment of agents on a straight-line segment. The construction of these protocols is based on information received by agents about the position of their neighbors, and this information comes not directly, but through auxiliary agents. In addition, there is a constant delay in signals from neighbors and communications between agents can be destroyed and restored at arbitrary instants of time. Using methods of positive systems theory and the Lyapunov–Krasovsky functional method, it is proved that the proposed control algorithms are robust with respect to communication delay and network topology switching.
The problem of eigenvalue spectrum assignment is considered in a generalized formulation. The system coefficients are block matrices. For block matrix bilinear control systems, we obtain sufficient conditions for resolving the problem of arbitrary matrix coefficient assignment for the characteristic matrix polynomial when the coefficients of the system have a special form, namely, the state matrix is a lower block Hessenberg matrix, and the matrix coefficients at the controller contain some zero blocks. The main result generalizes the corresponding theorem for block matrix bilinear control systems with a lower block Frobenius matrix and for block matrix linear control system closed-loop by linear static output feedback. An example is presented to illustrate the result.
The article represents a purely algebraic (i.e., derivative-free) approach to the description of physical and chemical processes in the continuum approximation as a generalization of Godunov's scheme. This approach is based on dividing the domain into finite volumes with the continuous medium inside. The mathematical model of the studied physical and chemical processes in each volume consists of the conservation laws and phenomenological laws. According to the molecular-kinetic theory, all macroparameters (density, temperature, pressure, etc.) are constant in open finite volumes and discontinuous on their faces. The advantage of the algebraic approach (so-called computational macromechanics) is simpler and more accurate mathematical description of the modeled phenomena, which is important for black-box software. One of the problems is the formulation of phenomenological laws of the gradient type (Fourier, Fick, Newton, etc.), since all macroparameters are defined in finite volumes in the computational macromechanics, not in points. First, the article represents an expression for the specific heat flux in the ideal gas. Then, an equation for the thermal conductivity coefficient is obtained in the discontinuous approximation under the assumption that the continuous temperature changes linearly in each finite volume, and the thermal conductivity coefficient depends linearly on the temperature. The variants for constructing difference schemes of computational macromechanics are considered. In conclusion, the results of 1D computational experiment illustrating theoretical analysis are presented.
A nonlinear diffusion equation with two spatial variables and several time delay variables is considered. The problem is discretized. Constructions of the alternating directions method with piecewise linear interpolation and extrapolation by continuation are presented. This method has the second order of smallness with respect to the time discretization step $\Delta$ and the space discretization step $h$. As a result, the method is reduced to solving two tridiagonal systems of linear algebraic equations at each time step, which have diagonal dominance. These systems are efficiently solved using the sweep method. The order of the residual without interpolation of the method is studied. Under certain assumptions, the convergence of the method with the order $O(\Delta^2+h^2)$ is justified. The results of numerical modeling for a diffusion equation with two delay variables are presented. The computable orders of convergence for each discretization step in the examples turned out to be close to the theoretically obtained orders of convergence for the corresponding discretization steps.
The article considers a nonlinear equation with the Laplace–Monge–Ampère operator in cases of power-law or exponential nonlinearity. To construct exact solutions, it is proposed to apply the method of reduction to ordinary differential equations using a quadratic auxiliary function of spatial variables. Multidimensional anisotropic exact solutions are obtained, which are expressed explicitly in terms of elementary and special functions and/or solutions of ordinary differential equations. A number of examples are provided to illustrate the obtained results.
The paper considers non-local control and optimal control problems for a linear time-varying dynamic system described by an ordinary differential equation with undivided multipoint conditions and with an integral condition imposed on the components of the state vector in some part of the time interval of the system operation. For the problem of optimal control, it is assumed that the quality criterion is specified over the whole time interval of system operation and has the meaning of the norm of some normed space. Conditions for complete controllability of a dynamic system with imposed non-local conditions are formulated. Conditions are obtained under which a solution to the control problem with such nonlocal conditions exists, and this solution is constructed. A constructive approach to solving control problems and a method for constructing an optimal control function are proposed. Continuity and non-uniqueness of the control function are shown. A solution to a specific control problem is given as an illustration.
This paper is devoted to the study of the asymptotic behavior of a class of control systems with multidimensional periodic nonlinearities and a countable set of equilibria. Such systems are known as synchronization or pendulum-like systems. Their stability is defined as the convergence of any solution to one of the equilibria. The dynamics of synchronization systems cannot be studied using classical methods designed for systems with a single equilibrium state. Therefore, within the framework of the classical methods of A.M. Lyapunov and V.M. Popov, special methods have been developed to generate conditions for the convergence of solutions and conditions for the absence of oscillations of a certain frequency. For systems with multidimensional nonlinearities, these conditions take the form of matrix frequency inequalities with variable parameters. This paper substantiates the optimal choice of variable parameters, which allows obtaining improved estimates of stability regions and regions of absence of high-frequency oscillations in the parameter space of specific systems.
The problem of sequentially traversing megacities with precedence conditions and cost functions that allow dependence on the task list is considered. It is assumed that the entire set of tasks is decomposed into a sum of non-empty subsets (groups); it is necessary to sequentially solve partial problems of visiting megacities in each of the groups. The order of visiting the groups themselves is specified a priori. It is assumed that the precedence conditions of the overall problem are localized in the aforementioned groups. The formulation is oriented towards the engineering problem of tool control during shape sheet metal cutting of parts by zones on CNC machines. The main method used is dynamic programming under decomposition conditions, where optimal procedures are implemented for each of the partial problems separately, after which a special gluing of the obtained solutions is performed. This resolves the issue of decomposing the original “large” problem into a system of partial problems of moderate dimension. Based on theoretical constructs, a workable algorithm is constructed and implemented on a PC. Solutions to model examples are presented.
The lattice non-Hermitian Bogolyubov–de Gennes model of superconductivity is considered, based on the periodic Kitaev model, with local potentials of two types. One of them generates non-reciprocal electron–electron and hole–hole transitions, while the other generates a non-reciprocal electron–hole transition. Conditions for the existence and analytical formulas for eigenfunctions with zero eigenvalue describing Majorana states are found.
The oscillatory properties of solutions of an arbitrary nonlinear differential system (with a zero solution) are considered. For such a system, the lower, upper, lowest, and upperest indices of oscillation, wandering, and rotation are determined. The connections of the numerical values of those indices with both the corresponding complete and completest oscillatory properties of that system and with the properties opposite to them: non-oscillation, non-wandering, and non-rotation are studied. The logical connections of all the listed properties with each other are also studied, and the absence of individual such connections is confirmed by concrete counterexamples. In addition, a similar connection is studied between the presence of any oscillatory property in a system and the unit or zero value of the measure of the corresponding property for the system itself (the concepts of such measures of a probabilistic nature for a differential system have only recently been introduced into consideration).
A first-order ordinary differential equation, solved with respect to the derivative, is considered. It is assumed that its right-hand side is continuous on a set consisting of a connected open subset of a two-dimensional Euclidean space and some part of its boundary. A theory is presented devoted to solving the questions of existence, continuability and uniqueness of solutions of the BIVP that is the initial value problem posed at a boundary point. This theory will allow to supplement the existing theory of first-order ODEs, in which the Cauchy problem is posed at an interior point (IIVP). The main results on existence or non-existence of a BIVP solution were obtained in 2020, therefore in this paper they are only systematized and supplemented. The results related to continuability, as well as the uniqueness or non-uniqueness of solutions to the BIVP are new. Theorems on the formal and local uniqueness of solutions to the Cauchy problem are proved. The differences between BIVP and IIVP are shown. For example, the non-equivalence of the concepts of formal and local uniqueness for BIVP and IIVP is demonstrated. This non-equivalence leads to the appearance of so-called hidden points of non-uniqueness along with points of non-uniqueness and uniqueness.
