Asymptotics and Quotient Spaces of Solutions of Operator-Difference Equations and Differential Equations with Small Delay

Formerly, in order to conduct the in-depth study of differential equations with delay, the author proposed the method of splitting the solution space reducing such equations to the systems of operator-difference equations. Using this method, the author assumed new conditions, i.e. the absolute domains for coefficients sufficient for the existence of special (slowly changing) solutions, and proved the presence of approximating and asymptotically approximating properties in them, as well as the asymptotic one-dimensional space of solutions of the initial problems for linear scalar differential equations with insignificantly retarded argument and the corresponding operator-difference equation systems (special solutions correspond, to the solutions with a slowly changing first component and a relatively small second component). For the purposes of the single-point representation of the obtained results and other data related to the theory of dynamic systems (the distance between the solution values tends to zero alongside the unlimited increase in argument), throughout this research paper the author uses the concept of the asymptotic equivalence of solutions for dynamic systems, as it was introduced by the author in their previous research. In order to shape the new mathematical objects, the concept of asymptotic Hausdorff equivalence of solutions for dynamic systems is introduced (the distance between solution values tends to zero with unlimited increase in argument of one solution and monotonic transformation of argument of another solution).

Analyzing stability of equilibrium points in impulsive neural network models involving generalized piecewise alternately advanced and retarded argument

In this paper, we investigate the models of the impulsive cellular neural network with piecewise alternately advanced and retarded argument of generalized argument (in short IDEPCAG). To ensure the existence, uniqueness and global exponential stability of the equilibrium state, several new sufficient conditions are obtained, which extend the results of the previous literature. The method is based on utilizing Banach’s fixed point theorem and a new IDEPCAG’s Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of proposed results. We end the article with a brief conclusion.

Transmission problem for the Sturm-Liouville equation involving a retarded argument

In this work, spectral properties of a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary conditions and in the transmission conditions at the point of discontinuity are investigated. To this aim, asymptotic formulas for the eigenvalues and eigenfunctions are obtained.

Modeling of discrete control of antiviral immune response in conditions of uncertainty

In this paper is considered numerically solving a problem of discrete control with uncertainty of the immune response in infectious disease. The problem of discrete control is described by a nonlinear system of ordinary differential equations with a retarded argument. Control under conditions of uncertainty implies that the parameter values are unknown and parameter estimation is adjusted as new experimental values become available. In order to solve this problem, the algorithm that allows us to construct a control function and at the same time to identify parameters within mathematical model of the infectious disease is proposed. By means of the proposed algorithm, the treatment programs based on an immunotherapy are constructed and parameters of the mathematical model of antiviral immune response are identified. It is shown that immunotherapy allows us to effectively cure the acute form of a disease.

Bounds for distance between eigenvalues of boundary value problems with retarded argument

In this study we are concerned with spectrum of boundary value problems with retarded argument with discontinuous weight function, two supplementary transmission conditions at the point of discontinuity, spectral and physical parameters in the boundary condition and we obtain bounds for the distance between eigenvalues. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. In the special case that ω (x) ≡ 1, the transmission coefficients γ1 = δ1, γ2 = δ2 and retarded argument Δ ≡ 0 in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.