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Abstract: In this paper, some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions with minimal period 4 of the delay differential equation by using relative Morse index and some considerate analysis techniques. [Copyright &y& Elsevier]

Abstract: We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays. [Copyright &y& Elsevier]

This article is concerned with the stability of θ-methods for linear delay-integro-differential systems. The linear θ-methods for the modified linear delay-integro-differential systems are established. It is shown that the linear θ-methods are GP-stable if θ∈[1/2, 1]. Finally, some numerical experiments are given to demonstrate the conclusions. [ABSTRACT FROM AUTHOR]

Abstract: We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with -boundary conditions is called -stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR2s) kind, and the Top Order Methods (TOMs), are considered in this paper. By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be -stable. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the numerical solution of delay integro-differential equations. The adaptation of linear multistep methods is considered. The emphasis is on the linear stability of numerical methods. It is shown that every A-stable, strongly 0-stable linear multistep method of Pouzet type can preserve the delay-independent stability of the underlying linear systems. In addition, some delay-dependent stability conditions for the stability of numerical methods are also given. [Copyright &y& Elsevier]

The article presents the study of a semigroup method to semilinear partial functional differential equations with infinite delay. The study aims to understand the solution semigroup generated by a particular partial functional differential equation with infinite delay. The paper presents some basic results on existence, uniqueness and properties of integral solutions and then establishes properties of the solution operator in nonlinear case. It utilizes the Crandall and Liggett theorem in order to compute the nonlinear semigroup by an exponential formula.

Abstract: The Mackey-Glass equation, , is considered, with variable coefficients and a nonconstant delay. Under rather natural assumptions all solutions are positive and bounded. Persistence and extinction conditions are presented for this equation. In the case when there exists a constant positive equilibrium, local asymptotic stability of the constant solution and oscillation about this equilibrium are analyzed. The results are illustrated by numerical examples. In particular, it is demonstrated that with delay in both terms, a solution with positive initial conditions may become negative. [Copyright &y& Elsevier]

This article solves the absolute stabilization problem for delay systems in the class of finite-dimensional controllers. The absolute stabilization problem for the class of control objects described by delay differential equations was solved in a previous study. For various applications, it is important to consider a version of this problem with the additional condition that the controller should be governed by ordinary differential equations, i.e., have a finite-dimensional state space as a dynamical system. Such controllers are usually referred to as finite-dimensional dynamical controllers.