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We consider a differential equation with a state-dependent delay motivated by a queueing process. The time delay is determined by an algebraic equation involving the length of the queue for which a discontinuous differential equation holds. The new type of state-dependent delay raises some problems that are studied in this paper. We formulate an appropriate framework to handle the system, and show that the solutions define a Lipschitz continuous semiflow in the phase space. The second main result guarantees the existence of slowly oscillating periodic solutions. [ABSTRACT FROM AUTHOR]

By applying M-matrix theory and some new analysis techniques, we have succeeded in establishing the asymptotic stability of the zero solution of the "pure-delay type" neutral system with variable delays. The criteria obtained in this paper extend and improve the existing ones. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of two-dimensional Riemann problems for Hamilton-Jacobi equations. The solutions to such problems are fundamental building blocks for constructing solutions to more general problems, in particular, for numerical construction using methods such as front tracking. Here we prove the existence of a particular class of Riemann problem for which the viscosity solutions contain closed characteristic orbits, enclosing furthermore a periodic sonic structure, which n turn encloses a parabolic structure. The existence of such examples elucidates the difficulties encountered in designing construction methods for viscosity solutions to Riemann problems n dimension ≥ 2. This investigation was prompted by the discovery of numerical evidence of examples displaying an even richer internal structure. [ABSTRACT FROM AUTHOR]