This paper presents a survey on the recent contributions to linear time- invariant delay-differential systems in the behavioral approach. In this survey both systems with commensurate and with noncommensurate delays will be considered. The emphasis lies on the investigation of the relationship between various systems descriptions. While this can be understood in a completely algebraic setting for systems with commensurate delays, this is not the case for systems with noncommensurate delays. In the...

The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the $n-$th order neutral differential equation $${(x\left(t\right)-px(t-\tau ))}^{\left(n\right)}-q\left(t\right)x\left(\sigma \left(t\right)\right)=0,$$ where $\sigma \left(t\right)$ is a delayed or advanced argument.

The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear...

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input $\to$ state” map in $L_2$-norms is established. A structure of the reachable sets for arbitrary $T\>0$ is studied. In general case, only the first component $u(·,T)$ of the complete state $\{u(·,T),u_t(·,T)\}$ may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation....

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input → state" map in L2-norms is established. A structure of the reachable sets for arbitrary T>0 is studied. In general case, only the first component $u(·,T)$ of the complete state $\{u(·,T),u_t(·,T)\}$ may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation....

We consider a system of delay differential equations modelling the tumor-immune system competition with negative immune response and three positive stationary points. The dynamics of the first two positive solutions are studied in terms of the local stability. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence and stability of a limit cycle bifurcating from the second positive stationary point, when the delay (taken as a parameter) crosses some...

This paper is concerned with periodic solutions of first-order nonlinear functional differential equations with deviating arguments. Some new sufficient conditions for the existence of periodic solutions are obtained. The paper extends and improves some well-known results.

We consider a boundary value problem for a differential equation with deviating arguments and p-Laplacian: $-{\left({\varphi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+d/dtgradF\left(x\right)+g(t,x\left(t\right),x\left(\delta \left(t\right)\right)$, x’(t), x’(τ(t))) = 0, t ∈ [0,1]; $x\left(t\right)=\underline{\phi }\left(t\right),$ t ≤ 0; $x\left(t\right)=\overline{\phi }\left(t\right)$, t ≥ 1. An existence result is obtained with the help of the Leray-Schauder degree theory, with no restriction on the damping forces d/dt grad F(x).

We study the local and global existence of mild solutions to a class of semilinear fractional Cauchy problems in the α-norm assuming that the operator in the linear part is the generator of a compact analytic C₀-semigroup. A suitable notion of mild solution for this class of problems is also introduced. The results obtained are a generalization and continuation of some recent results on this issue.