Ukrains’kyi Matematychnyi Zhurnal

Vladislav Fedorovych Babenko (to his 75th birthday)

2024-11-05T10:11:04+02:00

On some spectral properties of nonlocal boundary-value problems for nonlinear differential inclusion

2024-11-06T10:12:44+02:00

UDC 517.9

We study the solutions to the Sturm–Liouville boundary-value problem for a nonlinear differential inclusion with nonlocal conditions. The maximal and minimal solutions are demonstrated. The analysis of eigenvalues and eigenfunctions is performed. It is discussed whether multiple solutions may exist for the inhomogeneous Sturm–Liouville boundary-value problem for differential equation with nonlocal conditions.

Involute-evolute curves with modified orthogonal frame in Galilean space $G_{3}$

2024-11-06T10:12:48+02:00

UDC 514

We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space $G_3$. The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

Sharp starlike and convex radius for the differential operator of analytic functions

2024-11-06T10:12:47+02:00

UDC 517.5

Under given coefficient conditions for analytic functions $f$ in the unit disk $\mathbf{D}$, we first obtain the starlike and convex radius for the linear combination of the differential operator $zf'$ of analytic functions $f$ in $\mathbf{D}.$ Then we obtain the starlike and convex radius for the differential operator $zf'$. Furthermore, we present the starlike and convex radius for the linear combination of the differential operator $zf'$ and analytic functions $f$ in $\mathbf{D}$. Our results imply the related result obtained by Gavrilov [Mat. Zametki, 7, 295–298 (1970)].

Weakly nonlinear Fredholm integrodifferential equations with degenerate kernel in Banach spaces

2024-11-06T10:12:52+02:00

UDC 517.968.2

We consider weakly nonlinear Fredholm integrodifferential equations with degenerate kernel in Banach spaces. Necessary conditions are established for the existence of a solution, which turns into the generating solution for $\varepsilon = 0$. We also obtain sufficient conditions for the existence of at least one solution and construct a convergent iterative algorithm for its determination.

Generalization of some integral inequalities in multiplicative calculus with their computational analysis

2024-11-06T10:12:43+02:00

UDC 517.9

We focus on generalizing some multiplicative integral inequalities for twice differentiable functions. First, we derive a multiplicative integral identity for multiplicatively twice differentiable functions. Then, with the help of the integral identity, we prove a family of integral inequalities, such as Simpson, Hermite–Hadamard, midpoint, trapezoid, and Bullen types inequalities for multiplicatively convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities to prove the validity of the results for multiplicatively convex functions. The generalized forms obtained in our research offer valuable tools for researchers in various fields.

On the application of the averaging method to one problem of optimal control with unfixed time

2024-11-06T10:12:54+02:00

UDC 517.9

We consider the problem of optimal control for a system of differential equations with rapidly oscillating coefficients and a coercive target functional. The final time is not fixed. It is defined as the first time of hitting of a given closed bounded subset of the phase space by the phase point. The solvability of this problem is proved, and the convergence of the optimal solutions of the original problem to the optimal process of the problem with averaged parameters is justified.

New sequence spaces derived from the Catalan–Motzkin matrix and related matrix transformations

2024-11-06T10:12:38+02:00

UDC 512.6

We consider the domain of a conservative matrix involving Catalan and Motzkin numbers in the sequence spaces $c$ and $c_{0}.$ Moreover, the $\alpha$-, $\beta$-, and $\gamma$-duals are given and some matrix mappings are presented on the resulting spaces.

Stabilization of a class of $\psi$-Caputo fractional homogeneous polynomial systems

2024-11-06T10:12:51+02:00

UDC 517.9

In a constructive way, we study the problem of stabilization of $\psi$-Caputo fractional homogeneous polynomial systems. By using the Lyapunov functions, we construct stabilizing feedback laws for the analyzed fractional system. A numerical example is given to illustrate the efficiency of the obtained result.

Inequalities for the geometric-mean distance metric

2024-11-05T10:12:35+02:00

UDC 514

We study a hyperbolic-type metric $h_{G,c}$ introduced by Dovgoshey, Hariri, and Vuorinen and determine the best constant $c>0$ for which this function $h_{G,c}$ is a metric in specifically chosen $G$. We also present several sharp inequalities between $h_{G,c}$ and other hyperbolic-type metrics and offer several results obtained for the ball inclusion.

Estimation of the fundamental solution of a new class for non-Archimedean pseudodifferential equations

2024-11-05T10:12:36+02:00

UDC 517.9

We investigate the equation with the Vladimirov–Taibleson pseudodifferential operator for functions with $p$-adic time and space variables, which generalizes the $p$-adic wave equation in the cases where the orders of the time and space derivatives do not coincide. We prove the existence and uniqueness of the solution to the corresponding Cauchy problem. Some properties of this solution are established, including, in particular, the finite domain of dependence, which resembles the behavior of classical hyperbolic equations. We also deduce an $L^1$-estimate for the solution. On the other hand, we prove an estimate for the fundamental solution of the problem, which is an analog of the corresponding estimates for parabolic-type equations with real time and $p$-adic space variables.

Investigation of the approximate solution of one class of curvilinear integral equations by the projection method

2024-11-05T10:12:34+02:00

UDC 517.9

We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined.