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A new definition of a stability nonlinear delay differential equations is introduced and studied. This type of stability generalizes the known in the literature practical stability and it is a generalization in two directions - the behavior of two nonzero solutions are compared; - the solutions are with different initial data - different initial time intervals as well as different initial functions. The introduced concept of stability is an extension of the studied up today types of stability and it gives a new tool for more adequate investigating of real world problems in science, engineering, technologies. Several sufficient conditions are obtained. Lyapunov functions as well as comparison results for scalar ordinary differential equations have been employed. [ABSTRACT FROM AUTHOR]

A method to study asymptotic properties of solutions to systems of differential equations with distributed time-delays and Boolean-type nonlinearities (step functions) is offered. Such systems arise in many applications, but this paper deals with specific examples of such systems coming from genetic regulatory networks. A challenge is to analyze stable stationary points which belong to the discontinuity set of the system (thresholds). The paper describes an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. The basic technical tool consists in replacing step functions by the so-called “logoid functions” and investigating the smooth systems thus obtained. [ABSTRACT FROM AUTHOR]