Vol 69, No 1 (2023): Differential and Functional Differential Equations


Smoothness of solutions to the damping problem for nonstationary control system with delay of neutral type on the whole interval

Adkhamova A.S.


We consider the damping problem for a nonstationary control system described by a system of differential-difference equations of neutral type with smooth matrix coe cients and several delays. This problem is equivalent to the boundary-value problem for a system of second-order differentialdifference equations, which has a unique generalized solution. It is proved that the smoothness of this solution can be violated on the considered interval and is preserved only on some subintervals. Su cient conditions for the initial function are obtained to ensure the smoothness of the generalized solution over the entire interval.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):1-17
pages 1-17 views

Quasilinear elliptic and parabolic systems with nondiagonal principal matrices and strong nonlinearities in the gradient. Solvability and regularity problems

Arkhipova A.A.


We consider nondiagonal elliptic and parabolic systems of equations with strongly nonlinear terms in the gradient. We review and comment known solvability and regularity results and describe the last author’s results in this field.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):18-31
pages 18-31 views

The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems

Ashyralyev A., Ashyralyyev C.


This is a discussion on the second-order accuracy difference schemes for approximate solution of the integral-type time-nonlocal parabolic problems. The theorems on the stability of r-modified Crank-Nicolson difference schemes and second-order accuracy implicit difference scheme for approximate solution of the integral-type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second-order accuracy in t difference schemes for the one and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):32-49
pages 32-49 views

Shadowing property for nonautonomous dynamical systems

Blank M.L.


A new approach based on the analysis of the influence of a single perturbation is proposed as a test for the shadowing property for a broad class of dynamical systems (in particular, non-autonomous) under a variety of perturbations. Applications for several interesting cases are considered in detail.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):50-61
pages 50-61 views

Absence of positive solutions of some nonlinear inequalities with transformations of the argument in a half-space

Galakhov E.I., Salieva O.A.


We prove the absence of positive solutions for some semilinear elliptic partial differential inequalities with transformations of the argument in a half-space. The proofs are based on the test functions method.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):62-72
pages 62-72 views

Spectral properties of operators in the problem on normal oscillations of a mixture of viscous compressible fluids

Zakora D.A.


In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):73-97
pages 73-97 views

Entropy and renormalized solutions for a nonlinear elliptic problem in Musielak-Orlicz spaces

Kozhevnikova L.M.


In this paper, we establish the equivalence of entropy and renormalized solutions of second-order elliptic equations with nonlinearities defined by the Musielak-Orlicz functions and the right-hand side from the space L1(Ω). In nonreflexive Musielak-Orlicz-Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):98-115
pages 98-115 views

On asymptotic properties of solutions for differential equations of neutral type

Malygina V.V., Chudinov K.M.


The stability of systems of linear autonomous functional differential equations of neutral type is studied. The study is based on the well-known representation of the solution in the form of an integral operator, the kernel of which is the Cauchy function of the equation under study. The definitions of Lyapunov, asymptotic, and exponential stability are formulated in terms of the corresponding properties of the Cauchy function, which allows us to clarify a number of traditional concepts without loss of generality. Along with the concept of asymptotic stability, a new concept of strong asymptotic stability is introduced. The main results are related to the stability with respect to the initial function from the spaces of summable functions. In particular, it is established that strong asymptotic stability with initial data from the space \(L_1\) is equivalent to the exponential estimate of the Cauchy function and, moreover, exponential stability with respect to initial data from the spaces \(L_p\) for any \(p\ge1.\)

Contemporary Mathematics. Fundamental Directions. 2023;69(1):116-133
pages 116-133 views

L2-estimates of error in homogenization of parabolic equations with correctors taken into account

Pastukhova S.E.


We consider second-order parabolic equations with bounded measurable \(\varepsilon\)-periodic coefficients. To solve the Cauchy problem in the layer \( R^d \times(0,T) \) with the nonhomogeneous equation, we obtain approximations in the norm \(\|\cdot\|_{L^2(R^d\times(0,T))}\) with remainder of order \(\varepsilon^2\) as \(\varepsilon \to 0.\)

Contemporary Mathematics. Fundamental Directions. 2023;69(1):134-151
pages 134-151 views

Smoothness of generalized solutions to the Dirichlet problem for strongly elliptic functional differential equations with orthotropic contractions on the boundary of adjacent subdomains

Tasevich A.L.


The paper is devoted to the study of the smoothness of generalized solutions of the first boundaryvalue problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a circle, the coe cients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group on a circle are found in explicit form for any right-hand side from the Lebesgue space.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):152-165
pages 152-165 views

Integro-differential equations in Banach spaces and analytic resolving families of operators

Fedorov V.E., Godova A.D.


We study a class of equations in Banach spaces with a Riemann–Liouville-type integro-differential operator with an operator-valued convolution kernel. The properties of \(k\)-resolving operators of such equations are studied and the class \(\mathcal A_{m,K,\chi}\) of linear closed operators is defined such that the belonging to this class is necessary and, in the case of commutation of the operator with the convolution kernel, is sufficient for the existence of analytic in the sector \(k\)-resolving families of operators of the equation under study. Under certain additional conditions on the convolution kernel, we prove theorems on the unique solvability of the nonhomogeneous linear equation of the class under consideration if the nonhomogeneity is continuous in the norm of the graph of the operator from the equation or Hölder continuous. We obtain the theorem on sufficient conditions on an additive perturbation of an operator of the class \(\mathcal A_{m,K,\chi}\) in order that the perturbed operator also belong to such a class. Abstract results are used in the study of initial-boundary value problems for a system of partial differential equations with several fractional Riemann–Liouville derivatives of different orders with respect to time and for an equation with a fractional Prabhakar derivative with respect to time.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):166-184
pages 166-184 views

Method of search functionals and its applications in fixed point and coincidence theory

Fomenko T.N.


The paper contains a survey of several results from the author’s papers and joint papers by the author and Yu. N. Zakharyan, both on the zero existence and approximation for single-valued and multivalued (α, β)-search functionals, and also on the zero existence preservation for parametric family of such functionals, under the parameter changing. Some corollaries of these results in the fixed point and coincidence theory of single-valued and multi-valued mappings of metric spaces are also given. The comparison is provided with some known results by other authors. In the concluding part of the paper, we investigate the problem on the existence of a parameter-continuous single-valued branch of zeros for a parametric family of search functionals. A theorem on the existence of solution of this problem is proved.

Contemporary Mathematics. Fundamental Directions. 2023;69(1):185-200
pages 185-200 views

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