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Functional Differential Equations: Theory and Applications–Dedicated to the Memory of Nikolay V. Azbelev on the Occasion of His 100th Birthday Anniversary

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 9016

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Prof. Dr. Vladimir P. Maksimov

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Guest Editor

Department of Information Systems and Mathematical Methods in Economics, Perm State University, 614990 Perm, Russia
Interests: functional differential equations with ordinary derivatives; continuous-discrete systems with aftereffect; boundary value problems, control problems; attainability sets; reliable computing experiment in the study of linear problems; economic dynamics models

Prof. Dr. Alexander Domoshnitsky

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Guest Editor

Department of Mathematics, Ariel University, Ariel 40700, Israel
Interests: functional differential equations; general theory; boundary value problems; positivity of solutions; nonoscillation; distances between adjacent zeros of solutions; distribution of zeros; Sturm’s theorem; stability; feedback control; delay differential equations; integro-differential equations; impulsive equations; applications of equations with memory in technology and medicine
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Special Issue Information

Dear Colleagues,

This Issue is dedicated to the memory of Professor Nikolay Azbelev on the occasion of his 100^th birthday anniversary.

Professor N. Azbelev, a well-known Russian mathematician, has been a leading figure in the integral and functional differential equations profession for about five decades. He is one of creators of the contemporary Theory of Functional Differential Equations. Today, this theory is being developed by the efforts of researchers from many countries in order to find more relevant applications.

The key idea of the Issue is to present new high-quality results in all major sections of the theory and in its applications. The topic of the Issue covers the following sections of the theory, but is not limited to them: general theory, boundary value problems, control problems, stability, asymptotic properties of solutions, oscillation/non-oscillation, numerical methods, and reliable computing.

Prof. Dr. Vladimir P. Maksimov
Prof. Dr. Alexander Domoshnitsky
Guest Editors

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Keywords

functional differential equations
general theory
boundary value problems
control problems
stability
asymptotic properties of solutions
oscillation/non-oscillation
numerical methods and reliable computing
applied problems

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Published Papers (6 papers)

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Research

17 pages, 353 KB

Open AccessArticle

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

by Evgeny Zhukovskiy and Elena Panasenko

Mathematics 2024, 12(1), 64; https://doi.org/10.3390/math12010064 - 24 Dec 2023

Cited by 2 | Viewed by 1120

Abstract

The equation

G (x, x) = \tilde{y},

where

G : X \times X \to Y,

and

X, Y

are vector metric spaces (meaning that the values of a distance between the points in these spaces belong [...] Read more.

The equation

G (x, x) = \tilde{y},

where

G : X \times X \to Y,

and

X, Y

are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones

E_{+}, M_{+}

of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a “model” equation, namely,

g (t, t) = 0,

where a continuous map

g : E_{+} \times E_{+} \to M_{+}

is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation. Full article

(This article belongs to the Special Issue Functional Differential Equations: Theory and Applications–Dedicated to the Memory of Nikolay V. Azbelev on the Occasion of His 100th Birthday Anniversary)

18 pages, 438 KB

Open AccessArticle

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

by Eugene Bravyi

Mathematics 2023, 11(24), 4980; https://doi.org/10.3390/math11244980 - 17 Dec 2023

Viewed by 1108

Abstract

Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions. Full article

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42 pages, 800 KB

Open AccessArticle

Stability Conditions for Linear Semi-Autonomous Delay Differential Equations

by Vera Malygina and Kirill Chudinov

Mathematics 2023, 11(22), 4654; https://doi.org/10.3390/math11224654 - 15 Nov 2023

Cited by 1 | Viewed by 1377

Abstract

We present a new method for obtaining stability conditions for certain classes of delay differential equations. The method is based on the transition from an individual equation to a family of equations, and next the selection of a representative of this family, the test equation, asymptotic properties of which determine those of all equations in the family. This approach allows us to obtain the conditions that are the criteria for the stability of all equations of a given family. These conditions are formulated in terms of the parameters of the class of equations being studied, and are effectively verifiable. The main difference of the proposed method from the known general methods (using Lyapunov–Krasovsky functionals, Razumikhin functions, and Azbelev W-substitution) is the emphasis on the exactness of the result; the difference from the known exact methods is a significant expansion of the range of applicability. The method provides an algorithm for checking stability conditions, which is carried out in a finite number of operations and allows the use of numerical methods. Full article

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14 pages, 319 KB

Open AccessArticle

Numerical Method for Solving the Nonlinear Superdiffusion Equation with Functional Delay

by Vladimir Pimenov and Andrei Lekomtsev

Mathematics 2023, 11(18), 3941; https://doi.org/10.3390/math11183941 - 16 Sep 2023

Cited by 1 | Viewed by 1291

Abstract

For a space-fractional diffusion equation with a nonlinear superdiffusion coefficient and with the presence of a delay effect, the grid numerical method is constructed. Interpolation and extrapolation procedures are used to account for the functional delay. At each time step, the algorithm reduces to solving a linear system with a main matrix that has diagonal dominance. The convergence of the method in the maximum norm is proved. The results of numerical experiments with constant and variable delays are presented. Full article

13 pages, 577 KB

Open AccessArticle

On the Optimal Control Problem for Vibrations of the Rod/String Consisting of Two Non-Homogeneous Sections with the Condition at an Intermediate Time

by Vanya Barseghyan and Svetlana Solodusha

Mathematics 2022, 10(23), 4444; https://doi.org/10.3390/math10234444 - 24 Nov 2022

Cited by 4 | Viewed by 1506

Abstract

We consider an optimal boundary control problem for a one-dimensional wave equation consisting of two non-homogenous segments with piecewise constant characteristics. The wave equation describes the longitudinal vibrations of a non-homogeneous rod or the transverse vibrations of a non-homogeneous string with given initial, intermediate, and final conditions. We assume that wave travel time for each of the sections is the same. The control is carried out by shifting one end with the other end fixed. The quality criterion is set on the entire time interval. A constructive approach to building an optimal boundary control is proposed. The results obtained are illustrated with an analytical example. Full article

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12 pages, 457 KB

Open AccessFeature PaperArticle

Negativity of Green’s Functions to Focal and Two-Point Boundary Value Problems for Equations of Second Order with Delay and Impulses in Their Derivatives

by Alexander Domoshnitsky, Sergey Malev and Vladimir Raichik

Mathematics 2022, 10(19), 3683; https://doi.org/10.3390/math10193683 - 8 Oct 2022

Viewed by 1482

Abstract

We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green’s functions are obtained. Full article

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Functional Differential Equations: Theory and Applications–Dedicated to the Memory of Nikolay V. Azbelev on the Occasion of His 100th Birthday Anniversary

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