Mathematical Notes最新文献

Abstract

We study a problem of damping a control system described by functional-differential equations of natural order (n) and neutral type with nonsmooth complex coefficients on an arbitrary tree with global delay. The latter means that the delay propagates through internal vertices of the tree. Minimization of the energy functional of the system leads to a variational problem. We establish its equivalence to a certain self-adjoint boundary value problem on the tree for equations of order (2n) with nonlocal quasi-derivatives and multidirectional shifts of the argument as well as Kirchhoff-type conditions emerging at the internal vertices. The unique solvability of both problems is proved.

Abstract

For semilinear partial differential equations, we consider the solution in the form of a plane wave traveling with a constant velocity. This solution is determined from an ordinary differential equation. A wave that stabilizes at infinity to equilibria corresponds to a phase trajectory connecting fixed points. The fundamental problem of the possibility of using such solutions in applications is their stability in the linear approximation. The stability problem is solved for a wave that corresponds to a trajectory from a saddle to a node. It is known that the velocity is determined ambiguously in this case. In this paper, a method is indicated for finding the limit of the velocity of stable waves for parabolic and hyperbolic equations, which can easily be implemented numerically.

Abstract

We obtain necessary and sufficient conditions for the weight under which the Haar system is a basis in a weighted Lebesgue space with variable exponent.

Abstract

In this paper, we study the convergence of generalized pseudo-spectrum associated with bounded operators in a Hilbert space. We prove that the approximate generalized pseudo-spectrum converges to the exact set under norm convergence. To prove this result, we use the Hausdorff distance and the assumption that the generalized resolvent operator is not constant on any open subset of the generalized resolvent set.

Abstract

Sufficient conditions are given for the unique solvability of nonlocal problems for abstract singular equations that are formulated in terms of the zeros of the modified Bessel function and the resolvent of the operator coefficient of the equations under consideration. Examples are presented.

Abstract

A ring of coupled oscillators with delayed feedback with various types of coupling between the oscillators is considered. For each type of coupling, the asymptotic behavior of the model solutions with respect to a large parameter is constructed for a wide variety of initial conditions. It is shown that the studying the behavior of solutions to the original infinite-dimensional models can be reduced to studying the dynamics of the constructed finite-dimensional mappings. High quality conclusions about the dynamics of the original systems are made. It is shown that the behavior of solutions significantly varies with variations in the type of coupling. Conditions on the system parameters are found under which the synchronization, two-cluster synchronization, and more complex modes are possible.

Abstract

The study of the zero existence problem for a nonnegative set-valued functional on a metric space is continued. The zero existence problem for a functional related by a certain (theta)-continuity condition to a parametric family of ((alpha,beta))-search functionals on an open subset of a metric space is examined. A theorem containing several sufficient conditions for this functional to have zeros is proved.

As corollaries of this result, theorems on the existence of coincidence and fixed points are also proved for set-valued mappings related by the (theta)-continuity condition to families of set-valued mappings with the property that the existence of coincidence and fixed points in an open subset of a metric space is preserved under parameter variation. For uniformly convex metric spaces, analogs of M. Edelstein’s 1972 asymptotic center theorem and M. Frigon’s 1996 fixed point theorem for nonexpansive mappings of Banach spaces are obtained and compared with the main results of the paper.

Abstract

We study the solvability of a Cauchy type problem for linear and quasilinear equations with Hilfer fractional derivatives solved for the higher-order derivative. The linear operator acting on the unknown function in the equation is assumed to be bounded. The unique solvability of the Cauchy type problem for a linear inhomogeneous equation is proved. Using the resulting solution formula, we reduce the Cauchy type problem for the quasilinear differential equation to an integro-differential equation of the form (y=G(y)). Under the local Lipschitz condition for the nonlinear operator in the equation, the contraction property of the operator (G) in a suitably chosen metric function space on a sufficiently small interval is proved. Thus, we prove a theorem on the existence of a unique local solution of the Cauchy type problem for the quasilinear equation. The result on the unique global solvability of this problem is obtained by proving the contraction property of a sufficiently high power of the operator (G) in a special function space on the original interval provided that the Lipschitz condition is satisfied for the nonlinear operator in the equation. We use the general results to study Cauchy type problems for a quasilinear system of ordinary differential equations and for a quasilinear system of integro-differential equations.

Abstract

In this paper, a modification of a renewal-reward process (X(t)) with dependent components is mathematically constructed and the stationary characteristics of this process are studied. Stochastic processes with dependent components have rarely been studied in the literature owing to their complex mathematical structure. We partially fill the gap by studying the effect of the dependence assumption on the stationary properties of the process (X(t)). To this end, first, we obtain explicit formulas for the ergodic distribution and the stationary moments of the process. Then we analyze the asymptotic behavior of the stationary moments of the process by using the basic results of the renewal theory and the Laplace transform method. Based on the analysis, we obtain two-term asymptotic expansions of the stationary moments. Moreover, we present two-term asymptotic expansions for the expectation, variance, and standard deviation of the process (Xleft(tright)). Finally, the asymptotic results obtained are examined in special cases.

Abstract

We consider an (n)-dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain period for the representative point to be returned to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching instants and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching instant for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results.